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# 4. The model file¶

## 4.1. Conventions¶

A model file contains a list of commands and of blocks. Each command and each element of a block is terminated by a semicolon (;). Blocks are terminated by end;.

If Dynare encounters an unknown expression at the beginning of a line or after a semicolon, it will parse the rest of that line as native MATLAB code, even if there are more statements separated by semicolons present. To prevent cryptic error messages, it is strongly recommended to always only put one statement/command into each line and start a new line after each semicolon. 1

Lines of codes can be commented out line by line or as a block. Single-line comments begin with // and stop at the end of the line. Multiline comments are introduced by /* and terminated by */.

Examples

// This is a single line comment

var x; // This is a comment about x

/* This is another inline comment about alpha */  alpha = 0.3;

/*
This comment is spanning
two lines.
*/

Note that these comment marks should not be used in native MATLAB code regions where the % should be preferred instead to introduce a comment. In a verbatim block, see Verbatim inclusion, this would result in a crash since // is not a valid MATLAB statement).

Most Dynare commands have arguments and several accept options, indicated in parentheses after the command keyword. Several options are separated by commas.

In the description of Dynare commands, the following conventions are observed:

• Optional arguments or options are indicated between square brackets: ‘[]’;

• Repeated arguments are indicated by ellipses: “…”;

• Mutually exclusive arguments are separated by vertical bars: ‘|’;

• INTEGER indicates an integer number;

• INTEGER_VECTOR indicates a vector of integer numbers separated by spaces, enclosed by square brackets;

• DOUBLE indicates a double precision number. The following syntaxes are valid: 1.1e3, 1.1E3, 1.1d3, 1.1D3. In some places, infinite Values Inf and -Inf are also allowed;

• NUMERICAL_VECTOR indicates a vector of numbers separated by spaces, enclosed by square brackets;

• EXPRESSION indicates a mathematical expression valid outside the model description (see Expressions);

• MODEL_EXPRESSION (sometimes MODEL_EXP) indicates a mathematical expression valid in the model description (see Expressions and Model declaration);

• MACRO_EXPRESSION designates an expression of the macro processor (see Macro expressions);

• VARIABLE_NAME (sometimes VAR_NAME) indicates a variable name starting with an alphabetical character and can’t contain: ‘()+-*/^=!;:@#.’ or accentuated characters;

• PARAMETER_NAME (sometimes PARAM_NAME) indicates a parameter name starting with an alphabetical character and can’t contain: ‘()+-*/^=!;:@#.’ or accentuated characters;

• LATEX_NAME (sometimes TEX_NAME) indicates a valid LaTeX expression in math mode (not including the dollar signs);

• FUNCTION_NAME indicates a valid MATLAB function name;

• FILENAME indicates a filename valid in the underlying operating system; it is necessary to put it between quotes when specifying the extension or if the filename contains a non-alphanumeric character;

• QUOTED_STRING indicates an arbitrary string enclosed between (single) quotes.

## 4.2. Variable declarations¶

While Dynare allows the user to choose their own variable names, there are some restrictions to be kept in mind. First, variables and parameters must not have the same name as Dynare commands or built-in functions. In this respect, Dynare is not case-sensitive. For example, do not use Ln or Sigma_e to name your variable. Not conforming to this rule might yield hard-to-debug error messages or crashes. Second, when employing user-defined steady state files it is recommended to avoid using the name of MATLAB functions as this may cause conflicts. In particular, when working with user-defined steady state files, do not use correctly-spelled greek names like alpha, because there are MATLAB functions of the same name. Rather go for alppha or alph. Lastly, please do not name a variable or parameter i. This may interfere with the imaginary number i and the index in many loops. Rather, name investment invest. Using inv is also not recommended as it already denotes the inverse operator. Commands for declaring variables and parameters are described below.

Command: var VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)]...;
Command: var(deflator=MODEL_EXPR) VAR_NAME (... same options apply)
Command: var(log_deflator=MODEL_EXPR) VAR_NAME (... same options apply)

This required command declares the endogenous variables in the model. See Conventions for the syntax of VAR_NAME and MODEL_EXPR. Optionally it is possible to give a LaTeX name to the variable or, if it is nonstationary, provide information regarding its deflator. The variables in the list can be separated by spaces or by commas. var commands can appear several times in the file and Dynare will concatenate them. Dynare stores the list of declared parameters, in the order of declaration, in a column cell array M_.endo_names.

Options

If the model is nonstationary and is to be written as such in the model block, Dynare will need the trend deflator for the appropriate endogenous variables in order to stationarize the model. The trend deflator must be provided alongside the variables that follow this trend.

deflator = MODEL_EXPR

The expression used to detrend an endogenous variable. All trend variables, endogenous variables and parameters referenced in MODEL_EXPR must already have been declared by the trend_var, log_trend_var, var and parameters commands. The deflator is assumed to be multiplicative; for an additive deflator, use log_deflator.

log_deflator = MODEL_EXPR

Same as deflator, except that the deflator is assumed to be additive instead of multiplicative (or, to put it otherwise, the declared variable is equal to the log of a variable with a multiplicative trend).

long_name = QUOTED_STRING

This is the long version of the variable name. Its value is stored in M_.endo_names_long (a column cell array, in the same order as M_.endo_names). In case multiple long_name options are provided, the last one will be used. Default: VAR_NAME.

NAME = QUOTED_STRING

This is used to create a partitioning of variables. It results in the direct output in the .m file analogous to: M_.endo_partitions.NAME = QUOTED_STRING;.

Example (variable partitioning)

var c gnp cva (country=US', state=VA')
cca (country=US', state=CA', long_name=Consumption CA');
var(deflator=A) i b;
var c $C$ (long_name=Consumption');

Command: varexo VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)...];

This optional command declares the exogenous variables in the model. See Conventions for the syntax of VAR_NAME. Optionally it is possible to give a LaTeX name to the variable. Exogenous variables are required if the user wants to be able to apply shocks to her model. The variables in the list can be separated by spaces or by commas. varexo commands can appear several times in the file and Dynare will concatenate them.

Options

long_name = QUOTED_STRING

Like long_name but value stored in M_.exo_names_long.

NAME = QUOTED_STRING

Like partitioning but QUOTED_STRING stored in M_.exo_partitions.NAME.

Example

varexo m gov;

Remarks

An exogenous variable is an innovation, in the sense that this variable cannot be predicted from the knowledge of the current state of the economy. For instance, if logged TFP is a first order autoregressive process:

$a_t = \rho a_{t-1} + \varepsilon_t$

then logged TFP $$a_t$$ is an endogenous variable to be declared with var, its best prediction is $$\rho a_{t-1}$$, while the innovation $$\varepsilon_t$$ is to be declared with varexo.

Command: varexo_det VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)...];

This optional command declares exogenous deterministic variables in a stochastic model. See Conventions for the syntax of VARIABLE_NAME. Optionally it is possible to give a LaTeX name to the variable. The variables in the list can be separated by spaces or by commas. varexo_det commands can appear several times in the file and Dynare will concatenate them.

It is possible to mix deterministic and stochastic shocks to build models where agents know from the start of the simulation about future exogenous changes. In that case stoch_simul will compute the rational expectation solution adding future information to the state space (nothing is shown in the output of stoch_simul) and forecast will compute a simulation conditional on initial conditions and future information.

Note that exogenous deterministic variables cannot appear with a lead or a lag in the model.

Options

long_name = QUOTED_STRING

Like long_name but value stored in M_.exo_det_names_long.

NAME = QUOTED_STRING

Like partitioning but QUOTED_STRING stored in M_.exo_det_partitions.NAME.

Example

varexo m gov;
varexo_det tau;

Command: parameters PARAM_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)...];

This command declares parameters used in the model, in variable initialization or in shocks declarations. See Conventions for the syntax of PARAM_NAME. Optionally it is possible to give a LaTeX name to the parameter.

The parameters must subsequently be assigned values (see Parameter initialization).

The parameters in the list can be separated by spaces or by commas. parameters commands can appear several times in the file and Dynare will concatenate them.

Options

long_name = QUOTED_STRING

Like long_name but value stored in M_.param_names_long.

NAME = QUOTED_STRING

Like partitioning but QUOTED_STRING stored in M_.param_partitions.NAME.

Example

parameters alpha, bet;

Command: change_type(var|varexo|varexo_det|parameters) VAR_NAME | PARAM_NAME...;

Changes the types of the specified variables/parameters to another type: endogenous, exogenous, exogenous deterministic or parameter. It is important to understand that this command has a global effect on the .mod file: the type change is effective after, but also before, the change_type command. This command is typically used when flipping some variables for steady state calibration: typically a separate model file is used for calibration, which includes the list of variable declarations with the macro processor, and flips some variable.

Example

var y, w;
parameters alpha, beta;
...
change_type(var) alpha, beta;
change_type(parameters) y, w;

Here, in the whole model file, alpha and beta will be endogenous and y and w will be parameters.

Command: predetermined_variables VAR_NAME...;

In Dynare, the default convention is that the timing of a variable reflects when this variable is decided. The typical example is for capital stock: since the capital stock used at current period is actually decided at the previous period, then the capital stock entering the production function is k(-1), and the law of motion of capital must be written:

k = i + (1-delta)*k(-1)

Put another way, for stock variables, the default in Dynare is to use a “stock at the end of the period” concept, instead of a “stock at the beginning of the period” convention.

The predetermined_variables is used to change that convention. The endogenous variables declared as predetermined variables are supposed to be decided one period ahead of all other endogenous variables. For stock variables, they are supposed to follow a “stock at the beginning of the period” convention.

Note that Dynare internally always uses the “stock at the end of the period” concept, even when the model has been entered using the predetermined_variables command. Thus, when plotting, computing or simulating variables, Dynare will follow the convention to use variables that are decided in the current period. For example, when generating impulse response functions for capital, Dynare will plot k, which is the capital stock decided upon by investment today (and which will be used in tomorrow’s production function). This is the reason that capital is shown to be moving on impact, because it is k and not the predetermined k(-1) that is displayed. It is important to remember that this also affects simulated time series and output from smoother routines for predetermined variables. Compared to non-predetermined variables they might otherwise appear to be falsely shifted to the future by one period.

Example

The following two program snippets are strictly equivalent.

Using default Dynare timing convention:

var y, k, i;
...
model;
y = k(-1)^alpha;
k = i + (1-delta)*k(-1);
...
end;

Using the alternative timing convention:

var y, k, i;
predetermined_variables k;
...
model;
y = k^alpha;
k(+1) = i + (1-delta)*k;
...
end;

Command: trend_var(growth_factor = MODEL_EXPR) VAR_NAME [$LATEX_NAME$]...;

This optional command declares the trend variables in the model. See Conventions for the syntax of MODEL_EXPR and VAR_NAME. Optionally it is possible to give a LaTeX name to the variable.

The variable is assumed to have a multiplicative growth trend. For an additive growth trend, use log_trend_var instead.

Trend variables are required if the user wants to be able to write a nonstationary model in the model block. The trend_var command must appear before the var command that references the trend variable.

trend_var commands can appear several times in the file and Dynare will concatenate them.

If the model is nonstationary and is to be written as such in the model block, Dynare will need the growth factor of every trend variable in order to stationarize the model. The growth factor must be provided within the declaration of the trend variable, using the growth_factor keyword. All endogenous variables and parameters referenced in MODEL_EXPR must already have been declared by the var and parameters commands.

Example

trend_var (growth_factor=gA) A;

Command: log_trend_var(log_growth_factor = MODEL_EXPR) VAR_NAME [$LATEX_NAME$]...;

Same as trend_var, except that the variable is supposed to have an additive trend (or, to put it otherwise, to be equal to the log of a variable with a multiplicative trend).

Command: model_local_variable VARIABLE_NAME [LATEX_NAME]... ;

This optional command declares a model local variable. See Conventions for the syntax of VARIABLE_NAME. As you can create model local variables on the fly in the model block (see Model declaration), the interest of this command is primarily to assign a LATEX_NAME to the model local variable.

Example

model_local_variable GDP_US $GDPUS$;

### 4.2.1. On-the-fly Model Variable Declaration¶

Endogenous variables, exogenous variables, and parameters can also be declared inside the model block. You can do this in two different ways: either via the equation tag or directly in an equation.

To declare a variable on-the-fly in an equation tag, simply state the type of variable to be declared (endogenous, exogenous, or parameter followed by an equal sign and the variable name in single quotes. Hence, to declare a variable c as endogenous in an equation tag, you can type [endogenous='c'].

To perform on-the-fly variable declaration in an equation, simply follow the symbol name with a vertical line (|, pipe character) and either an e, an x, or a p. For example, to declare a parameter named alphaa in the model block, you could write alphaa|p directly in an equation where it appears. Similarly, to declare an endogenous variable c in the model block you could write c|e. Note that in-equation on-the-fly variable declarations must be made on contemporaneous variables.

On-the-fly variable declarations do not have to appear in the first place where this variable is encountered.

Example

The following two snippets are equivalent:

model;
[endogenous='k',name='law of motion of capital']
k(+1) = i|e + (1-delta|p)*k;
y|e = k^alpha|p;
...
end;
delta = 0.025;
alpha = 0.36;

var k, i, y;
parameters delta, alpha;
delta = 0.025;
alpha = 0.36;
...
model;
[name='law of motion of capital']
k(1) = i|e + (1-delta|p)*k;
y|e = k|e^alpha|p;
...
end;

## 4.3. Expressions¶

Dynare distinguishes between two types of mathematical expressions: those that are used to describe the model, and those that are used outside the model block (e.g. for initializing parameters or variables, or as command options). In this manual, those two types of expressions are respectively denoted by MODEL_EXPRESSION and EXPRESSION.

Unlike MATLAB or Octave expressions, Dynare expressions are necessarily scalar ones: they cannot contain matrices or evaluate to matrices. 2

Expressions can be constructed using integers (INTEGER), floating point numbers (DOUBLE), parameter names (PARAMETER_NAME), variable names (VARIABLE_NAME), operators and functions.

The following special constants are also accepted in some contexts:

Constant: inf

Represents infinity.

Constant: nan

“Not a number”: represents an undefined or unrepresentable value.

### 4.3.1. Parameters and variables¶

Parameters and variables can be introduced in expressions by simply typing their names. The semantics of parameters and variables is quite different whether they are used inside or outside the model block.

#### 4.3.1.1. Inside the model¶

Parameters used inside the model refer to the value given through parameter initialization (see Parameter initialization) or homotopy_setup when doing a simulation, or are the estimated variables when doing an estimation.

Variables used in a MODEL_EXPRESSION denote current period values when neither a lead or a lag is given. A lead or a lag can be given by enclosing an integer between parenthesis just after the variable name: a positive integer means a lead, a negative one means a lag. Leads or lags of more than one period are allowed. For example, if c is an endogenous variable, then c(+1) is the variable one period ahead, and c(-2) is the variable two periods before.

When specifying the leads and lags of endogenous variables, it is important to respect the following convention: in Dynare, the timing of a variable reflects when that variable is decided. A control variable — which by definition is decided in the current period — must have no lead. A predetermined variable — which by definition has been decided in a previous period — must have a lag. A consequence of this is that all stock variables must use the “stock at the end of the period” convention.

Leads and lags are primarily used for endogenous variables, but can be used for exogenous variables. They have no effect on parameters and are forbidden for local model variables (see Model declaration).

#### 4.3.1.2. Outside the model¶

When used in an expression outside the model block, a parameter or a variable simply refers to the last value given to that variable. More precisely, for a parameter it refers to the value given in the corresponding parameter initialization (see Parameter initialization); for an endogenous or exogenous variable, it refers to the value given in the most recent initval or endval block.

### 4.3.2. Operators¶

The following operators are allowed in both MODEL_EXPRESSION and EXPRESSION:

• Binary arithmetic operators: +, -, *, /, ^

• Unary arithmetic operators: +, -

• Binary comparison operators (which evaluate to either 0 or 1): <, >, <=, >=, ==, !=

Note the binary comparison operators are differentiable everywhere except on a line of the 2-dimensional real plane. However for facilitating convergence of Newton-type methods, Dynare assumes that, at the points of non-differentiability, the partial derivatives of these operators with respect to both arguments is equal to 0 (since this is the value of the partial derivatives everywhere else).

The following special operators are accepted in MODEL_EXPRESSION (but not in EXPRESSION):

Operator: STEADY_STATE (MODEL_EXPRESSION)

This operator is used to take the value of the enclosed expression at the steady state. A typical usage is in the Taylor rule, where you may want to use the value of GDP at steady state to compute the output gap.

Exogenous and exogenous deterministic variables may not appear in MODEL_EXPRESSION.

Warning

The concept of a steady state is ambiguous in a perfect foresight context with permament and potentially anticipated shocks occuring. Dynare will use the contents of oo_.steady_state as its reference for calls to the STEADY_STATE()-operator. In the presence of endval, this implies that the terminal state provided by the user is used. This may be a steady state computed by Dynare (if endval is followed by steady) or simply the terminal state provided by the user (if endval is not followed by steady). Put differently, Dynare will not automatically compute the steady state conditional on the specificed value of the exogenous variables in the respective periods.

Operator: EXPECTATION (INTEGER) (MODEL_EXPRESSION)

This operator is used to take the expectation of some expression using a different information set than the information available at current period. For example, EXPECTATION(-1)(x(+1)) is equal to the expected value of variable x at next period, using the information set available at the previous period. See Auxiliary variables for an explanation of how this operator is handled internally and how this affects the output.

### 4.3.3. Functions¶

#### 4.3.3.1. Built-in functions¶

The following standard functions are supported internally for both MODEL_EXPRESSION and EXPRESSION:

Function: exp(x)

Natural exponential.

Function: log(x)
Function: ln(x)

Natural logarithm.

Function: log10(x)

Base 10 logarithm.

Function: sqrt(x)

Square root.

Function: cbrt(x)

Cube root.

Function: sign(x)

Signum function, defined as:

$\begin{split}\textrm{sign}(x) = \begin{cases} -1 &\quad\text{if }x<0\\ 0 &\quad\text{if }x=0\\ 1 &\quad\text{if }x>0 \end{cases}\end{split}$

Note that this function is not continuous, hence not differentiable, at $$x=0$$. However, for facilitating convergence of Newton-type methods, Dynare assumes that the derivative at $$x=0$$ is equal to $$0$$. This assumption comes from the observation that both the right- and left-derivatives at this point exist and are equal to $$0$$, so we can remove the singularity by postulating that the derivative at $$x=0$$ is $$0$$.

Function: abs(x)

Absolute value.

Note that this continuous function is not differentiable at $$x=0$$. However, for facilitating convergence of Newton-type methods, Dynare assumes that the derivative at $$x=0$$ is equal to $$0$$ (even if the derivative does not exist). The rational for this mathematically unfounded definition, rely on the observation that the derivative of $$\mathrm{abs}(x)$$ is equal to $$\mathrm{sign}(x)$$ for any $$x\neq 0$$ in $$\mathbb R$$ and from the convention for the value of $$\mathrm{sign}(x)$$ at $$x=0$$).

Function: sin(x)
Function: cos(x)
Function: tan(x)
Function: asin(x)
Function: acos(x)
Function: atan(x)

Trigonometric functions.

Function: max(a, b)
Function: min(a, b)

Maximum and minimum of two reals.

Note that these functions are differentiable everywhere except on a line of the 2-dimensional real plane defined by $$a=b$$. However for facilitating convergence of Newton-type methods, Dynare assumes that, at the points of non-differentiability, the partial derivative of these functions with respect to the first (resp. the second) argument is equal to $$1$$ (resp. to $$0$$) (i.e. the derivatives at the kink are equal to the derivatives observed on the half-plane where the function is equal to its first argument).

Function: normcdf(x)
Function: normcdf(x, mu, sigma)

Gaussian cumulative density function, with mean mu and standard deviation sigma. Note that normcdf(x) is equivalent to normcdf(x,0,1).

Function: normpdf(x)
Function: normpdf(x, mu, sigma)

Gaussian probability density function, with mean mu and standard deviation sigma. Note that normpdf(x) is equivalent to normpdf(x,0,1).

Function: erf(x)

Gauss error function.

#### 4.3.3.2. External functions¶

Any other user-defined (or built-in) MATLAB or Octave function may be used in both a MODEL_EXPRESSION and an EXPRESSION, provided that this function has a scalar argument as a return value.

To use an external function in a MODEL_EXPRESSION, one must declare the function using the external_function statement. This is not required for external functions used in an EXPRESSION outside of a model block or steady_state_model block.

Command: external_function(OPTIONS...);

This command declares the external functions used in the model block. It is required for every unique function used in the model block.

external_function commands can appear several times in the file and must come before the model block.

Options

name = NAME

The name of the function, which must also be the name of the M-/MEX file implementing it. This option is mandatory.

nargs = INTEGER

The number of arguments of the function. If this option is not provided, Dynare assumes nargs = 1.

first_deriv_provided [= NAME]

If NAME is provided, this tells Dynare that the Jacobian is provided as the only output of the M-/MEX file given as the option argument. If NAME is not provided, this tells Dynare that the M-/MEX file specified by the argument passed to NAME returns the Jacobian as its second output argument. When this option is not provided, Dynare will use finite difference approximations for computing the derivatives of the function, whenever needed.

second_deriv_provided [= NAME]

If NAME is provided, this tells Dynare that the Hessian is provided as the only output of the M-/MEX file given as the option argument. If NAME is not provided, this tells Dynare that the M-/MEX file specified by the argument passed to NAME returns the Hessian as its third output argument. NB: This option can only be used if the first_deriv_provided option is used in the same external_function command. When this option is not provided, Dynare will use finite difference approximations for computing the Hessian derivatives of the function, whenever needed.

Example

external_function(name = funcname);
external_function(name = otherfuncname, nargs = 2, first_deriv_provided, second_deriv_provided);
external_function(name = yetotherfuncname, nargs = 3, first_deriv_provided = funcname_deriv);

### 4.3.4. A few words of warning in stochastic context¶

The use of the following functions and operators is strongly discouraged in a stochastic context: max, min, abs, sign, <, >, <=, >=, ==, !=.

The reason is that the local approximation used by stoch_simul or estimation will by nature ignore the non-linearities introduced by these functions if the steady state is away from the kink. And, if the steady state is exactly at the kink, then the approximation will be bogus because the derivative of these functions at the kink is bogus (as explained in the respective documentations of these functions and operators).

Note that extended_path is not affected by this problem, because it does not rely on a local approximation of the mode.

## 4.4. Parameter initialization¶

When using Dynare for computing simulations, it is necessary to calibrate the parameters of the model. This is done through parameter initialization.

The syntax is the following:

PARAMETER_NAME = EXPRESSION;

Here is an example of calibration:

parameters alpha, beta;

beta = 0.99;
alpha = 0.36;
A = 1-alpha*beta;

Internally, the parameter values are stored in M_.params:

MATLAB/Octave variable: M_.params

Contains the values of model parameters. The parameters are in the order that was used in the parameters command, hence ordered as in M_.param_names.

The parameter names are stored in M_.param_names:

MATLAB/Octave variable: M_.param_names

Cell array containing the names of the model parameters.

MATLAB/Octave command: get_param_by_name('PARAMETER_NAME');

Given the name of a parameter, returns its calibrated value as it is stored in M_.params.

MATLAB/Octave command: set_param_value('PARAMETER_NAME', MATLAB_EXPRESSION);

Sets the calibrated value of a parameter to the provided expression. This does essentially the same as the parameter initialization syntax described above, except that it accepts arbitrary MATLAB/Octave expressions, and that it works from MATLAB/Octave scripts.

## 4.5. Model declaration¶

The model is declared inside a model block:

Block: model ;
Block: model(OPTIONS...);

The equations of the model are written in a block delimited by model and end keywords.

There must be as many equations as there are endogenous variables in the model, except when computing the unconstrained optimal policy with ramsey_model, ramsey_policy or discretionary_policy.

The syntax of equations must follow the conventions for MODEL_EXPRESSION as described in Expressions. Each equation must be terminated by a semicolon (‘;’). A normal equation looks like:

MODEL_EXPRESSION = MODEL_EXPRESSION;

When the equations are written in homogenous form, it is possible to omit the ‘=0’ part and write only the left hand side of the equation. A homogenous equation looks like:

MODEL_EXPRESSION;

Inside the model block, Dynare allows the creation of model-local variables, which constitute a simple way to share a common expression between several equations. The syntax consists of a pound sign (#) followed by the name of the new model local variable (which must not be declared as in Variable declarations, but may have been declared by model_local_variable), an equal sign, and the expression for which this new variable will stand. Later on, every time this variable appears in the model, Dynare will substitute it by the expression assigned to the variable. Note that the scope of this variable is restricted to the model block; it cannot be used outside. To assign a LaTeX name to the model local variable, use the declaration syntax outlined by model_local_variable. A model local variable declaration looks like:

#VARIABLE_NAME = MODEL_EXPRESSION;

It is possible to tag equations written in the model block. A tag can serve different purposes by allowing the user to attach arbitrary informations to each equation and to recover them at runtime. For instance, it is possible to name the equations with a name-tag, using a syntax like:

model;

[name = 'Budget constraint'];
c + k = k^theta*A;

end;

Here, name is the keyword indicating that the tag names the equation. If an equation of the model is tagged with a name, the resid command will display the name of the equations (which may be more informative than the equation numbers) in addition to the equation number. Several tags for one equation can be separated using a comma:

model;

[name='Taylor rule',mcp = 'r > -1.94478']
r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e;

end;

Options

linear

Declares the model as being linear. It spares oneself from having to declare initial values for computing the steady state of a stationary linear model. This option can’t be used with non-linear models, it will NOT trigger linearization of the model.

use_dll

Instructs the preprocessor to create dynamic loadable libraries (DLL) containing the model equations and derivatives, instead of writing those in M-files. You need a working compilation environment, i.e. a working mex command (see Compiler installation for more details). Using this option can result in faster simulations or estimations, at the expense of some initial compilation time. Alternatively, this option can be given to the dynare command (see Dynare invocation). 3

block

Perform the block decomposition of the model, and exploit it in computations (steady-state, deterministic simulation, stochastic simulation with first order approximation and estimation). See https://archives.dynare.org/DynareWiki/FastDeterministicSimulationAndSteadyStateComputation for details on the algorithms used in deterministic simulation and steady-state computation.

bytecode

Instead of M-files, use a bytecode representation of the model, i.e. a binary file containing a compact representation of all the equations.

cutoff = DOUBLE

Threshold under which a jacobian element is considered as null during the model normalization. Only available with option block. Default: 1e-15

mfs = INTEGER

Controls the handling of minimum feedback set of endogenous variables. Only available with option block. Possible values:

0

All the endogenous variables are considered as feedback variables (Default).

1

The endogenous variables assigned to equation naturally normalized (i.e. of the form $$x=f(Y)$$ where $$x$$ does not appear in $$Y$$) are potentially recursive variables. All the other variables are forced to belong to the set of feedback variables.

2

In addition of variables with mfs = 1 the endogenous variables related to linear equations which could be normalized are potential recursive variables. All the other variables are forced to belong to the set of feedback variables.

3

In addition of variables with mfs = 2 the endogenous variables related to non-linear equations which could be normalized are potential recursive variables. All the other variables are forced to belong to the set of feedback variables.

no_static

Don’t create the static model file. This can be useful for models which don’t have a steady state.

differentiate_forward_vars
differentiate_forward_vars = ( VARIABLE_NAME [VARIABLE_NAME ...] )

Tells Dynare to create a new auxiliary variable for each endogenous variable that appears with a lead, such that the new variable is the time differentiate of the original one. More precisely, if the model contains x(+1), then a variable AUX_DIFF_VAR will be created such that AUX_DIFF_VAR=x-x(-1), and x(+1) will be replaced with x+AUX_DIFF_VAR(+1).

The transformation is applied to all endogenous variables with a lead if the option is given without a list of variables. If there is a list, the transformation is restricted to endogenous with a lead that also appear in the list.

This option can useful for some deterministic simulations where convergence is hard to obtain. Bad values for terminal conditions in the case of very persistent dynamics or permanent shocks can hinder correct solutions or any convergence. The new differentiated variables have obvious zero terminal conditions (if the terminal condition is a steady state) and this in many cases helps convergence of simulations.

parallel_local_files = ( FILENAME [, FILENAME]... )

Declares a list of extra files that should be transferred to slave nodes when doing a parallel computation (see Parallel Configuration).

balanced_growth_test_tol = DOUBLE

Tolerance used for determining whether cross-derivatives are zero in the test for balanced growth path (the latter is documented on https://archives.dynare.org/DynareWiki/RemovingTrends). Default: 1e-6

Example (Elementary RBC model)

var c k;
varexo x;
parameters aa alph bet delt gam;

model;
c =  - k + aa*x*k(-1)^alph + (1-delt)*k(-1);
c^(-gam) = (aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam)/(1+bet);
end;

Example (Use of model local variables)

The following program:

model;
# gamma = 1 - 1/sigma;
u1 = c1^gamma/gamma;
u2 = c2^gamma/gamma;
end;

…is formally equivalent to:

model;
u1 = c1^(1-1/sigma)/(1-1/sigma);
u2 = c2^(1-1/sigma)/(1-1/sigma);
end;

Example (A linear model)

model(linear);
x = a*x(-1)+b*y(+1)+e_x;
y = d*y(-1)+e_y;
end;

Dynare has the ability to output the original list of model equations to a LaTeX file, using the write_latex_original_model command, the list of transformed model equations using the write_latex_dynamic_model command, and the list of static model equations using the write_latex_static_model command.

Command: write_latex_original_model(OPTIONS);

This command creates two LaTeX files: one containing the model as defined in the model block and one containing the LaTeX document header information.

If your .mod file is FILENAME.mod, then Dynare will create a file called FILENAME/latex/original.tex, which includes a file called FILENAME/latex/original_content.tex (also created by Dynare) containing the list of all the original model equations.

If LaTeX names were given for variables and parameters (see Variable declarations), then those will be used; otherwise, the plain text names will be used.

Time subscripts (t, t+1, t-1, …) will be appended to the variable names, as LaTeX subscripts.

Compiling the TeX file requires the following LaTeX packages: geometry, fullpage, breqn.

Options

write_equation_tags

Write the equation tags in the LaTeX output. The equation tags will be interpreted with LaTeX markups.

Command: write_latex_dynamic_model ;
Command: write_latex_dynamic_model(OPTIONS);

This command creates two LaTeX files: one containing the dynamic model and one containing the LaTeX document header information.

If your .mod file is FILENAME.mod, then Dynare will create a file called FILENAME/latex/dynamic.tex, which includes a file called FILENAME/latex/dynamic_content.tex (also created by Dynare) containing the list of all the dynamic model equations.

If LaTeX names were given for variables and parameters (see Variable declarations), then those will be used; otherwise, the plain text names will be used.

Time subscripts (t, t+1, t-1, …) will be appended to the variable names, as LaTeX subscripts.

Note that the model written in the TeX file will differ from the model declared by the user in the following dimensions:

• The timing convention of predetermined variables (see predetermined_variables) will have been changed to the default Dynare timing convention; in other words, variables declared as predetermined will be lagged on period back,

• The EXPECTATION operators will have been removed, replaced by auxiliary variables and new equations (as explained in the documentation of EXPECTATION),

• Endogenous variables with leads or lags greater or equal than two will have been removed, replaced by new auxiliary variables and equations,

• Exogenous variables with leads or lags will also have been replaced by new auxiliary variables and equations.

For the required LaTeX packages, see write_latex_original_model.

Options

write_equation_tags
Command: write_latex_static_model(OPTIONS);

This command creates two LaTeX files: one containing the static model and one containing the LaTeX document header information.

If your .mod file is FILENAME.mod, then Dynare will create a file called FILENAME/latex/static.tex, which includes a file called FILENAME/latex/static_content.tex (also created by Dynare) containing the list of all the steady state model equations.

If LaTeX names were given for variables and parameters (see Variable declarations), then those will be used; otherwise, the plain text names will be used.

Note that the model written in the TeX file will differ from the model declared by the user in the some dimensions (see write_latex_dynamic_model for details).

Also note that this command will not output the contents of the optional steady_state_model block (see steady_state_model); it will rather output a static version (i.e. without leads and lags) of the dynamic model declared in the model block. To write the LaTeX contents of the steady_state_model see write_latex_steady_state_model.

For the required LaTeX packages, see write_latex_original_model.

Options

write_equation_tags
Command: write_latex_steady_state_model()

This command creates two LaTeX files: one containing the steady state model and one containing the LaTeX document header information.

If your .mod file is FILENAME.mod, then Dynare will create a file called FILENAME/latex/steady_state.tex, which includes a file called FILENAME/latex/steady_state_content.tex (also created by Dynare) containing the list of all the steady state model equations.

If LaTeX names were given for variables and parameters (see Variable declarations), then those will be used; otherwise, the plain text names will be used.

Note that the model written in the .tex file will differ from the model declared by the user in some dimensions (see write_latex_dynamic_model for details).

For the required LaTeX packages, see write_latex_original_model.

## 4.6. Auxiliary variables¶

The model which is solved internally by Dynare is not exactly the model declared by the user. In some cases, Dynare will introduce auxiliary endogenous variables—along with corresponding auxiliary equations—which will appear in the final output.

The main transformation concerns leads and lags. Dynare will perform a transformation of the model so that there is only one lead and one lag on endogenous variables and no leads/lags on exogenous variables.

This transformation is achieved by the creation of auxiliary variables and corresponding equations. For example, if x(+2) exists in the model, Dynare will create one auxiliary variable AUX_ENDO_LEAD = x(+1), and replace x(+2) by AUX_ENDO_LEAD(+1).

A similar transformation is done for lags greater than 2 on endogenous (auxiliary variables will have a name beginning with AUX_ENDO_LAG), and for exogenous with leads and lags (auxiliary variables will have a name beginning with AUX_EXO_LEAD or AUX_EXO_LAG respectively).

Another transformation is done for the EXPECTATION operator. For each occurrence of this operator, Dynare creates an auxiliary variable defined by a new equation, and replaces the expectation operator by a reference to the new auxiliary variable. For example, the expression EXPECTATION(-1)(x(+1)) is replaced by AUX_EXPECT_LAG_1(-1), and the new auxiliary variable is declared as AUX_EXPECT_LAG_1 = x(+2).

Auxiliary variables are also introduced by the preprocessor for the ramsey_model and ramsey_policy commands. In this case, they are used to represent the Lagrange multipliers when first order conditions of the Ramsey problem are computed. The new variables take the form MULT_i, where i represents the constraint with which the multiplier is associated (counted from the order of declaration in the model block).

Auxiliary variables are also introduced by the differentiate_forward_vars option of the model block. The new variables take the form AUX_DIFF_FWRD_i, and are equal to x-x(-1) for some endogenous variable x.

Finally, auxiliary variables will arise in the context of employing the diff-operator.

Once created, all auxiliary variables are included in the set of endogenous variables. The output of decision rules (see below) is such that auxiliary variable names are replaced by the original variables they refer to.

The number of endogenous variables before the creation of auxiliary variables is stored in M_.orig_endo_nbr, and the number of endogenous variables after the creation of auxiliary variables is stored in M_.endo_nbr.

See https://git.dynare.org/Dynare/dynare/-/wikis/Auxiliary-variables for more technical details on auxiliary variables.

## 4.7. Initial and terminal conditions¶

For most simulation exercises, it is necessary to provide initial (and possibly terminal) conditions. It is also necessary to provide initial guess values for non-linear solvers. This section describes the statements used for those purposes.

In many contexts (deterministic or stochastic), it is necessary to compute the steady state of a non-linear model: initval then specifies numerical initial values for the non-linear solver. The command resid can be used to compute the equation residuals for the given initial values.

Used in perfect foresight mode, the types of forward-looking models for which Dynare was designed require both initial and terminal conditions. Most often these initial and terminal conditions are static equilibria, but not necessarily.

One typical application is to consider an economy at the equilibrium at time 0, trigger a shock in first period, and study the trajectory of return to the initial equilibrium. To do that, one needs initval and shocks (see Shocks on exogenous variables).

Another one is to study how an economy, starting from arbitrary initial conditions at time 0 converges towards equilibrium. In this case models, the command histval permits to specify different historical initial values for variables with lags for the periods before the beginning of the simulation. Due to the design of Dynare, in this case initval is used to specify the terminal conditions.

Block: initval ;
Block: initval(OPTIONS...);

The initval block has two main purposes: providing guess values for non-linear solvers in the context of perfect foresight simulations and providing guess values for steady state computations in both perfect foresight and stochastic simulations. Depending on the presence of histval and endval blocks it is also used for declaring the initial and terminal conditions in a perfect foresight simulation exercise. Because of this interaction of the meaning of an initval block with the presence of histval and endval blocks in perfect foresight simulations, it is strongly recommended to check that the constructed oo_.endo_simul and oo_.exo_simul variables contain the desired values after running perfect_foresight_setup and before running perfect_foresight_solver. In the presence of leads and lags, these subfields of the results structure will store the historical values for the lags in the first column/row and the terminal values for the leads in the last column/row.

The initval block is terminated by end; and contains lines of the form:

VARIABLE_NAME = EXPRESSION;

In a deterministic (i.e. perfect foresight) model

First, both the oo_.endo_simul and oo_.exo_simul variables storing the endogenous and exogenous variables will be filled with the values provided by this block. If there are no other blocks present, it will therefore provide the initial and terminal conditions for all the endogenous and exogenous variables, because it will also fill the last column/row of these matrices. For the intermediate simulation periods it thereby provides the starting values for the solver. In the presence of a histval block (and therefore absence of an endval block), this histval block will provide/overwrite the historical values for the state variables (lags) by setting the first column/row of oo_.endo_simul and oo_.exo_simul. This implies that the initval block in the presence of histval only sets the terminal values for the variables with leads and provides initial values for the perfect foresight solver.

Because of these various functions of initval it is often necessary to provide values for all the endogenous variables in an initval block. Initial and terminal conditions are strictly necessary for lagged/leaded variables, while feasible starting values are required for the solver. It is important to be aware that if some variables, endogenous or exogenous, are not mentioned in the initval block, a zero value is assumed. It is particularly important to keep this in mind when specifying exogenous variables using varexo that are not allowed to take on the value of zero, like e.g. TFP.

Note that if the initval block is immediately followed by a steady command, its semantics are slightly changed. The steady command will compute the steady state of the model for all the endogenous variables, assuming that exogenous variables are kept constant at the value declared in the initval block. These steady state values conditional on the declared exogenous variables are then written into oo_.endo_simul and take up the potential roles as historical and terminal conditions as well as starting values for the solver. An initval block followed by steady is therefore formally equivalent to an initval block with the specified values for the exogenous variables, and the endogenous variables set to the associated steady state values conditional on the exogenous variables.

In a stochastic model

The main purpose of initval is to provide initial guess values for the non-linear solver in the steady state computation. Note that if the initval block is not followed by steady, the steady state computation will still be triggered by subsequent commands (stoch_simul, estimation…).

As such, initval allows specifying the initial instrument value for steady state finding when providing an analytical conditional steady state file for ramsey_model-computations.

It is not necessary to declare 0 as initial value for exogenous stochastic variables, since it is the only possible value.

The subsequently computed steady state (not the initial values, use histval for this) will be used as the initial condition at all the periods preceeding the first simulation period for the three possible types of simulations in stochastic mode:

To start simulations at a particular set of starting values that are not a computed steady state, use histval.

Options

all_values_required

Issues an error and stops processing the .mod file if there is at least one endogenous or exogenous variable that has not been set in the initval block.

Example
initval;
c = 1.2;
k = 12;
x = 1;
end;

Block: endval ;
Block: endval(OPTIONS...);

This block is terminated by end; and contains lines of the form:

VARIABLE_NAME = EXPRESSION;

The endval block makes only sense in a deterministic model and cannot be used together with histval. Similar to the initval command, it will fill both the oo_.endo_simul and oo_.exo_simul variables storing the endogenous and exogenous variables with the values provided by this block. If no initval block is present, it will fill the whole matrices, therefore providing the initial and terminal conditions for all the endogenous and exogenous variables, because it will also fill the first and last column/row of these matrices. Due to also filling the intermediate simulation periods it will provide the starting values for the solver as well.

If an initval block is present, initval will provide the historical values for the variables (if there are states/lags), while endval will fill the remainder of the matrices, thereby still providing i) the terminal conditions for variables entering the model with a lead and ii) the initial guess values for all endogenous variables at all the simulation dates for the perfect foresight solver.

Note that if some variables, endogenous or exogenous, are NOT mentioned in the endval block, the value assumed is that of the last initval block or steady command (if present). Therefore, in contrast to initval, omitted variables are not automatically assumed to be 0 in this case. Again, it is strongly recommended to check the constructed oo_.endo_simul and oo_.exo_simul variables after running perfect_foresight_setup and before running perfect_foresight_solver to see whether the desired outcome has been achieved.

Like initval, if the endval block is immediately followed by a steady command, its semantics are slightly changed. The steady command will compute the steady state of the model for all the endogenous variables, assuming that exogenous variables are kept constant to the value declared in the endval block. These steady state values conditional on the declared exogenous variables are then written into oo_.endo_simul and therefore take up the potential roles as historical and terminal conditions as well as starting values for the solver. An endval block followed by steady is therefore formally equivalent to an endval block with the specified values for the exogenous variables, and the endogenous variables set to the associated steady state values.

Options

all_values_required

Example

var c k;
varexo x;

model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
end;

initval;
c = 1.2;
k = 12;
x = 1;
end;

endval;
c = 2;
k = 20;
x = 2;
end;

perfect_foresight_setup(periods=200);
perfect_foresight_solver;

In this example, the problem is finding the optimal path for consumption and capital for the periods $$t=1$$ to $$T=200$$, given the path of the exogenous technology level x. c is a forward-looking variable and the exogenous variable x appears with a lead in the expected return of physical capital, while k is a purely backward-looking (state) variable.

The initial equilibrium is computed by steady conditional on x=1, and the terminal one conditional on x=2. The initval block sets the initial condition for k (since it is the only backward-looking variable), while the endval block sets the terminal condition for c (since it is the only forward-looking endogenous variable). The starting values for the perfect foresight solver are given by the endval block. See below for more details.

Example

var c k;
varexo x;

model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
end;

initval;
k = 12;
end;

endval;
c = 2;
x = 1.1;
end;

perfect_foresight_setup(periods=200);
perfect_foresight_solver;

In this example, there is no steady command, hence the conditions are exactly those specified in the initval and endval blocks. We need terminal conditions for c and x, since both appear with a lead, and an initial condition for k, since it appears with a lag.

Setting x=1.1 in the endval block without a shocks block implies that technology is at $$1.1$$ in $$t=1$$ and stays there forever, because endval is filling all entries of oo_.endo_simul and oo_.exo_simul except for the very first one, which stores the initial conditions and was set to $$0$$ by the initval block when not explicitly specifying a value for it.

Because the law of motion for capital is backward-looking, we need an initial condition for k at time $$0$$. Due to the presence of endval, this cannot be done via a histval block, but rather must be specified in the initval block. Similarly, because the Euler equation is forward-looking, we need a terminal condition for c at $$t=201$$, which is specified in the endval block.

As can be seen, it is not necessary to specify c and x in the initval block and k in the endval block, because they have no impact on the results. Due to the optimization problem in the first period being to choose c,k at $$t=1$$ given the predetermined capital stock k inherited from $$t=0$$ as well as the current and future values for technology x, the values for c and x at time $$t=0$$ play no role. The same applies to the choice of c,k at time $$t=200$$, which does not depend on k at $$t=201$$. As the Euler equation shows, that choice only depends on current capital as well as future consumption c and technology x, but not on future capital k. The intuitive reason is that those variables are the consequence of optimization problems taking place in at periods $$t=0$$ and $$t=201$$, respectively, which are not modeled here.

Example

initval;
c = 1.2;
k = 12;
x = 1;
end;

endval;
c = 2;
k = 20;
x = 1.1;
end;

In this example, initial conditions for the forward-looking variables x and c are provided, together with a terminal condition for the backward-looking variable k. As shown in the previous example, these values will not affect the simulation results. Dynare simply takes them as given and basically assumes that there were realizations of exogenous variables and states that make those choices equilibrium values (basically initial/terminal conditions at the unspecified time periods $$t<0$$ and $$t>201$$).

The above example suggests another way of looking at the use of steady after initval and endval. Instead of saying that the implicit unspecified conditions before and after the simulation range have to fit the initial/terminal conditions of the endogenous variables in those blocks, steady specifies that those conditions at $$t<0$$ and $$t>201$$ are equal to being at the steady state given the exogenous variables in the initval and endval blocks. The endogenous variables at $$t=0$$ and $$t=201$$ are then set to the corresponding steady state equilibrium values.

The fact that c at $$t=0$$ and k at $$t=201$$ specified in initval and endval are taken as given has an important implication for plotting the simulated vector for the endogenous variables, i.e. the rows of oo_.endo_simul: this vector will also contain the initial and terminal conditions and thus is 202 periods long in the example. When you specify arbitrary values for the initial and terminal conditions for forward- and backward-looking variables, respectively, these values can be very far away from the endogenously determined values at $$t=1$$ and $$t=200$$. While the values at $$t=0$$ and $$t=201$$ are unrelated to the dynamics for $$0<t<201$$, they may result in strange-looking large jumps. In the example above, consumption will display a large jump from $$t=0$$ to $$t=1$$ and capital will jump from $$t=200$$ to $$t=201$$ when using rplot or manually plotting oo_.endo_val.

Block: histval ;
Block: histval(OPTIONS...);

In a deterministic perfect foresight context

In models with lags on more than one period, the histval block permits to specify different historical initial values for different periods of the state variables. In this case, the initval block takes over the role of specifying terminal conditions and starting values for the solver. Note that the histval block does not take non-state variables.

This block is terminated by end; and contains lines of the form:

VARIABLE_NAME(INTEGER) = EXPRESSION;

EXPRESSION is any valid expression returning a numerical value and can contain already initialized variable names.

By convention in Dynare, period 1 is the first period of the simulation. Going backward in time, the first period before the start of the simulation is period 0, then period -1, and so on.

State variables not initialized in the histval block are assumed to have a value of zero at period 0 and before. Note that histval cannot be followed by steady.

Example

model;
x=1.5*x(-1)-0.6*x(-2)+epsilon;
log(c)=0.5*x+0.5*log(c(+1));
end;

histval;
x(0)=-1;
x(-1)=0.2;
end;

initval;
c=1;
x=1;
end;

In this example, histval is used to set the historical conditions for the two lags of the endogenous variable x, stored in the first column of oo_.endo_simul. The initval block is used to set the terminal condition for the forward looking variable c, stored in the last column of oo_.endo_simul. Moreover, the initval block defines the starting values for the perfect foresight solver for both endogenous variables c and x.

In a stochastic simulation context

In the context of stochastic simulations, histval allows setting the starting point of those simulations in the state space. As for the case of perfect foresight simulations, all not explicitly specified variables are set to 0. Moreover, as only states enter the recursive policy functions, all values specified for control variables will be ignored. This can be used

• In stoch_simul, if the periods option is specified. Note that this only affects the starting point for the simulation, but not for the impulse response functions. When using the loglinear option, the histval block nevertheless takes the unlogged starting values.

• In forecast as the initial point at which the forecasts are computed. When using the loglinear option, the histval block nevertheless takes the unlogged starting values.

• In conditional_forecast for a calibrated model as the initial point at which the conditional forecasts are computed. When using the loglinear option, the histval-block nevertheless takes the unlogged starting values.

• In Ramsey policy, where it also specifies the values of the endogenous states (including lagged exogenous) at which the objective function of the planner is computed. Note that the initial values of the Lagrange multipliers associated with the planner’s problem cannot be set (see evaluate_planner_objective).

Options

all_values_required

Example

var x y;
varexo e;

model;
x = y(-1)^alpha*y(-2)^(1-alpha)+e;

end;

initval;
x = 1;
y = 1;
e = 0.5;
end;

histval;
y(0) = 1.1;
y(-1) = 0.9;
end;

stoch_simul(periods=100);

Command: resid ;

This command will display the residuals of the static equations of the model, using the values given for the endogenous in the last initval or endval block (or the steady state file if you provided one, see Steady state).

Command: initval_file(OPTIONS...);

In a deterministic setup, this command is used to specify a path for all endogenous and exogenous variables. The length of these paths must be equal to the number of simulation periods, plus the number of leads and the number of lags of the model (for example, with 50 simulation periods, in a model with 2 lags and 1 lead, the paths must have a length of 53). Note that these paths cover two different things:

• The constraints of the problem, which are given by the path for exogenous and the initial and terminal values for endogenous

• The initial guess for the non-linear solver, which is given by the path for endogenous variables for the simulation periods (excluding initial and terminal conditions)

In perfect foresight and stochastic contexts, steady uses the first observation loaded by initval_file as guess value to solve for the steady state of the model. This first observation is determined by the first_obs option when it is used.

Don’t mix initval_file with initval statements. However, after initval_file, you can modify the historical initial values with histval or histval_file statement.

There can be several initval_file statements in a model file. Each statement resets oo_.initval_series.

Options

datafile = FILENAME
filename = FILENAME (deprecated)

The name of the file containing the data. It must be included in quotes if the filename contains a path or an extension. The command accepts the following file formats:

• M-file (extension .m): for each endogenous and exogenous variable, the file must contain a row or column vector of the same name.

• MAT-file (extension .mat): same as for M-files.

• Excel file (extension .xls or .xlsx): for each endogenous and exogenous variable, the file must contain a column of the same name. NB: Octave only supports the .xlsx file extension and must have the io package installed (easily done via octave by typing ‘pkg install -forge io’). The first column may contain the date of each observation.

• CSV files (extension .csv): for each endogenous and exogenous variable, the file must contain a column of the same name. The first column may contain the date of each observation.

first_obs = {INTEGER | DATE}

The observation number or the date (see The dates class) of the first observation to be used in the file

first_simulation_period = {INTEGER | DATE}

The observation number in the file or the date (see dates) at which the simulation (or the forecast) is starting. This option avoids to have to compute the maximum number of lags in the model. The observation corresponding to the first period of simulation doesn’t need to exist in the file as the only dates necessary for initialization are before that date.

last_obs = {INTEGER | DATE}

The observaton number or the date (see The dates class) of the last observation to be used in the file.

nobs = INTEGER

The number of observations to be used in the file (starting with first of first_obs observation).

series = DSERIES NAME

The name of a DSERIES containing the data (see The dseries class)

Example 1

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

initval_file(datafile=mydata.csv);

perfect_foresight_setup(periods=200);
perfect_foresight_solver;

The initial and terminal values are taken from file mydata.csv (nothing guarantees that these vales are the steady state of the model). The guess value for the trajectories are also taken from the file. The file must contain at least 203 observations of variables c, x and e. If there are more than 203 observations available in the file, the first 203 are used by perfect_foresight_setup(periods=200). Note that the values for the auxiliary variable corresponding to x(-2) are automatically computed by initval_file.

Example 2

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

initval_file(datafile=mydata.csv,
first_obs=10);

perfect_foresight_setup(periods=200);
perfect_foresight_solver;

The initial and terminal values are taken from file mydata.csv starting with the 10th observation in the file. There must be at least 212 observations in the file.

Example 3

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

ds = dseries(mydata.csv);
lds = log(ds);

initval_file(series=lds,
first_obs=2010Q1);

perfect_foresight_setup(periods=200);
perfect_foresight_solver;

The initial and terminal values are taken from dseries lds. All observations are loaded starting with the 1st quarter of 2010 until the end of the file. There must be data available at least until 2050Q3.

Example 4

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

initval_file(datafile=mydata.csv,
first_simulation_period=2010Q1);

perfect_foresight_setup(periods=200);
perfect_foresight_solver;

The initial and terminal values are taken from file mydata.csv. The observations in the file must have dates. All observations are loaded from the 3rd quarter of 2009 until the end of the file. There must be data available in the file at least until 2050Q1.

Example 5

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

initval_file(datafile=mydata.csv,
last_obs = 212);

perfect_foresight_setup(periods=200);
perfect_foresight_solver;

The initial and terminal values are taken from file mydata.csv. The first 212 observations are loaded and the first 203 observations will be used by perfect_foresight_setup(periods=200).

Example 6

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

initval_file(datafile=mydata.csv,
first_obs = 10,
nobs = 203);

perfect_foresight_setup(periods=200);
perfect_foresight_solver;

The initial and terminal values are taken from file mydata.csv. Observations 10 to 212 are loaded.

Example 7

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

initval_file(datafile=mydata.csv,
first_obs = 10);

The values of the 10th observation of mydata.csv are used as guess value to compute the steady state. The exogenous variables are set to values found in the file or zero if these variables aren’t present.

Command: histval_file(OPTIONS...);

This command is equivalent to histval, except that it reads its input from a file, and is typically used in conjunction with smoother2histval.

Options

datafile = FILENAME
filename = FILENAME (deprecated)

The name of the file containing the data. The command accepts the following file formats:

• M-file (extension .m): for each endogenous and exogenous variable, the file must contain a row or column vector of the same name.

• MAT-file (extension .mat): same as for M-files.

• Excel file (extension .xls or .xlsx): for each endogenous and exogenous variable, the file must contain a column of the same name. NB: Octave only supports the .xlsx file extension and must have the io package installed (easily done via octave by typing ‘pkg install -forge io’). The first column may contain the date of each observation.

• CSV files (extension .csv): for each endogenous and exogenous variable, the file must contain a column of the same name. The first column may contain the date of each observation.

first_obs = {INTEGER | DATE}

The observation number or the date (see The dates class) of the first observation to be used in the file

first_simulation_period = {INTEGER | DATE}

The observation number in the file or the date (see The dates class) at which the simulation (or the forecast) is starting. This option avoids to have to compute the maximum number of lags in the model. The observation corresponding to the first period of simulation doesn’t need to exist in the file as the only dates necessary for initialization are before that date.

last_obs = {INTEGER | DATE}

The observation number or the date (see The dates class) of the last observation to be used in the file.

nobs = INTEGER

The number of observations to be used in the file (starting with first of first_obs observation).

series = DSERIES NAME

The name of a DSERIES containing the data (see The dseries class)

Example 1

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

x = 0;
c = exp(c*x/(1 - d));
end;

histval_file(datafile=mydata.csv);

stoch_simul(order=1,periods=100);

The initial values for the stochastic simulation are taken from the two first rows of file mydata.csv.

Example 2

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

histval_file(datafile=mydata.csv,
first_obs=10);

stoch_simul(order=1,periods=100);

The initial values for the stochastic simulation are taken from rows 10 and 11 of file mydata.csv.

Example 3

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

histval_file(datafile=mydata.csv,
first_obs=2010Q1);

stoch_simul(order=1,periods=100);

The initial values for the stochastic simulation are taken from observations 2010Q1 and 2010Q2 of file mydata.csv.

Example 4

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

histval_file(datafile=mydata.csv,
first_simulation_period=2010Q1)

stoch_simul(order=1,periods=100);

The initial values for the stochastic simulation are taken from observations 2009Q3 and 2009Q4 of file mydata.csv.

Example 5

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

histval_file(datafile=mydata.csv,
last_obs = 4);

stoch_simul(order=1,periods=100);

The initial values for the stochastic simulation are taken from the two first rows of file mydata.csv.

Example 6

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

initval_file(datafile=mydata.csv,
first_obs = 10,
nobs = 4);

stoch_simul(order=1,periods=100);

The initial values for the stochastic simulation are taken from rows 10 and 11 of file mydata.csv.

Example 7

var c x;
varexo e;
parameters a b c d;

a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;

model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;

initval_file(datafile=mydata.csv,
first_obs=10);

histval_file(datafile=myotherdata.csv);

perfect_foresight_setup(periods=200);
perfect_foresight_solver;

Historical initial values for the simulation are taken from the two first rows of file myotherdata.csv.

Terminal values and guess values for the simulation are taken from file mydata.csv starting with the 12th observation in the file. There must be at least 212 observations in the file.

## 4.8. Shocks on exogenous variables¶

In a deterministic context, when one wants to study the transition of one equilibrium position to another, it is equivalent to analyze the consequences of a permanent shock and this in done in Dynare through the proper use of initval and endval.

Another typical experiment is to study the effects of a temporary shock after which the system goes back to the original equilibrium (if the model is stable…). A temporary shock is a temporary change of value of one or several exogenous variables in the model. Temporary shocks are specified with the command shocks.

In a stochastic framework, the exogenous variables take random values in each period. In Dynare, these random values follow a normal distribution with zero mean, but it belongs to the user to specify the variability of these shocks. The non-zero elements of the matrix of variance-covariance of the shocks can be entered with the shocks command. Or, the entire matrix can be directly entered with Sigma_e (this use is however deprecated).

If the variance of an exogenous variable is set to zero, this variable will appear in the report on policy and transition functions, but isn’t used in the computation of moments and of Impulse Response Functions. Setting a variance to zero is an easy way of removing an exogenous shock.

Note that, by default, if there are several shocks or mshocks blocks in the same .mod file, then they are cumulative: all the shocks declared in all the blocks are considered; however, if a shocks or mshocks block is declared with the overwrite option, then it replaces all the previous shocks and mshocks blocks.

Block: shocks ;
Block: shocks(overwrite);

See above for the meaning of the overwrite option.

In deterministic context

For deterministic simulations, the shocks block specifies temporary changes in the value of exogenous variables. For permanent shocks, use an endval block.

The block should contain one or more occurrences of the following group of three lines:

var VARIABLE_NAME;
periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...;
values DOUBLE | (EXPRESSION)  [[,] DOUBLE | (EXPRESSION) ]...;

It is possible to specify shocks which last several periods and which can vary over time. The periods keyword accepts a list of several dates or date ranges, which must be matched by as many shock values in the values keyword. Note that a range in the periods keyword can be matched by only one value in the values keyword. If values represents a scalar, the same value applies to the whole range. If values represents a vector, it must have as many elements as there are periods in the range.

Note that shock values are not restricted to numerical constants: arbitrary expressions are also allowed, but you have to enclose them inside parentheses.

The feasible range of periods is from 0 to the number of periods specified in perfect_foresight_setup.

Warning

Note that the first endogenous simulation period is period 1. Thus, a shock value specified for the initial period 0 may conflict with (i.e. may overwrite or be overwritten by) values for the initial period specified with initval or endval (depending on the exact context). Users should always verify the correct setting of oo_.exo_simul after perfect_foresight_setup.

Example (with scalar values)

shocks;

var e;
periods 1;
values 0.5;
var u;
periods 4:5;
values 0;
var v;
periods 4:5 6 7:9;
values 1 1.1 0.9;
var w;
periods 1 2;
values (1+p) (exp(z));

end;

Example (with vector values)

xx = [1.2; 1.3; 1];

shocks;
var e;
periods 1:3;
values (xx);
end;

In stochastic context

For stochastic simulations, the shocks block specifies the non zero elements of the covariance matrix of the shocks of exogenous variables.

You can use the following types of entries in the block:

• Specification of the standard error of an exogenous variable.

var VARIABLE_NAME; stderr EXPRESSION;

• Specification of the variance of an exogenous variable.

var VARIABLE_NAME = EXPRESSION;

• Specification the covariance of two exogenous variables.

var VARIABLE_NAME, VARIABLE_NAME = EXPRESSION;

• Specification of the correlation of two exogenous variables.

corr VARIABLE_NAME, VARIABLE_NAME = EXPRESSION;

In an estimation context, it is also possible to specify variances and covariances on endogenous variables: in that case, these values are interpreted as the calibration of the measurement errors on these variables. This requires the varobs command to be specified before the shocks block.

Example

shocks;
var e = 0.000081;
var u; stderr 0.009;
corr e, u = 0.8;
var v, w = 2;
end;

In stochastic optimal policy context

When computing conditional welfare in a ramsey_model or discretionary_policy context, welfare is conditional on the state values inherited by planner when making choices in the first period. The information set of the first period includes the respective exogenous shock realizations. Thus, their known value can be specified using the perfect foresight syntax. Note that i) all other values specified for periods than period 1 will be ignored and ii) the value of lagged shocks (e.g. in the case of news shocks) is specified with histval.

Example

shocks;
var u; stderr 0.008;
var u;
periods 1;
values 1;
end;

Mixing deterministic and stochastic shocks

It is possible to mix deterministic and stochastic shocks to build models where agents know from the start of the simulation about future exogenous changes. In that case stoch_simul will compute the rational expectation solution adding future information to the state space (nothing is shown in the output of stoch_simul) and forecast will compute a simulation conditional on initial conditions and future information.

Example

varexo_det tau;
varexo e;
...
shocks;
var e; stderr 0.01;
var tau;
periods 1:9;
values -0.15;
end;

stoch_simul(irf=0);

forecast;

Block: mshocks ;
Block: mshocks(overwrite);

The purpose of this block is similar to that of the shocks block for deterministic shocks, except that the numeric values given will be interpreted in a multiplicative way. For example, if a value of 1.05 is given as shock value for some exogenous at some date, it means 5% above its steady state value (as given by the last initval or endval block).

The syntax is the same as shocks in a deterministic context.

This command is only meaningful in two situations:

• on exogenous variables with a non-zero steady state, in a deterministic setup,

• on deterministic exogenous variables with a non-zero steady state, in a stochastic setup.

See above for the meaning of the overwrite option.

Block: heteroskedastic_shocks ;
Block: heteroskedastic_shocks(overwrite);

In estimation context, it implements heteroskedastic filters, where the standard error of shocks may unexpectedly change in every period. The standard deviation of shocks may be either provided directly or set/modified in each observed period by a scale factor. If std0 is the usual standard error for shock1, then:

• using a scale factor in period t implies: std(shock1|t)=std0(shock1)*scale(t)

• using a provided value in period t implies: std(shock1|t)=value(t).

The block has a similar syntax as the shocks block in a perfect foresight context. It should contain one or more occurrences of the following group of three lines (for setting values):

var VARIABLE_NAME;
periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...;
values DOUBLE | (EXPRESSION)  [[,] DOUBLE | (EXPRESSION) ]...;

OR (for setting scale factors):

var VARIABLE_NAME;
periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...;
scales DOUBLE | (EXPRESSION)  [[,] DOUBLE | (EXPRESSION) ]...;

NOTE: scales and values cannot be simultaneously set for the same shock in the same period, but it is possible to set values for some periods and scales for other periods for the same shock. There can be only one scales and values directive each for a given shock, so all affected periods must be set in one statement.

Example

heteroskedastic_shocks;

var e1;
periods 86:87, 89:97;
scales 0.5, 0;

var e1;
periods 88;
values 0.1;

var e2;
periods 86:87 88:97;
values 0.04 0.01;

end;

Special variable: Sigma_e

This special variable specifies directly the covariance matrix of the stochastic shocks, as an upper (or lower) triangular matrix. Dynare builds the corresponding symmetric matrix. Each row of the triangular matrix, except the last one, must be terminated by a semi-colon ;. For a given element, an arbitrary EXPRESSION is allowed (instead of a simple constant), but in that case you need to enclose the expression in parentheses. The order of the covariances in the matrix is the same as the one used in the varexo declaration.

Example

varexo u, e;

Sigma_e = [ 0.81 (phi*0.9*0.009);
0.000081];

This sets the variance of u to 0.81, the variance of e to 0.000081, and the correlation between e and u to phi.

Warning

The use of this special variable is deprecated and is strongly discouraged. You should use a shocks block instead.

MATLAB/Octave command: get_shock_stderr_by_name('EXOGENOUS_NAME');

Given the name of an exogenous variable, returns its standard deviation, as set by a previous shocks block.

MATLAB/Octave command: set_shock_stderr_value('EXOGENOUS_NAME', MATLAB_EXPRESSION);

Sets the standard deviation of an exgonous variable. This does essentially the same as setting the standard error via a shocks block, except that it accepts arbitrary MATLAB/Octave expressions, and that it works from MATLAB/Octave scripts.

## 4.9. Other general declarations¶

Command: dsample INTEGER [INTEGER];

Reduces the number of periods considered in subsequent output commands.

Command: periods INTEGER

This command is now deprecated (but will still work for older model files). It is not necessary when no simulation is performed and is replaced by an option periods in perfect_foresight_setup, simul and stoch_simul.

This command sets the number of periods in the simulation. The periods are numbered from 1 to INTEGER. In perfect foresight simulations, it is assumed that all future events are perfectly known at the beginning of period 1.

Example

periods 100;

There are two ways of computing the steady state (i.e. the static equilibrium) of a model. The first way is to let Dynare compute the steady state using a nonlinear Newton-type solver; this should work for most models, and is relatively simple to use. The second way is to give more guidance to Dynare, using your knowledge of the model, by providing it with a method to compute the steady state, either using a steady_state_model block or writing matlab routine.

### 4.10.1. Finding the steady state with Dynare nonlinear solver¶

Command: steady ;
Command: steady(OPTIONS...);

This command computes the steady state of a model using a nonlinear Newton-type solver and displays it. When a steady state file is used steady displays the steady state and checks that it is a solution of the static model.

More precisely, it computes the equilibrium value of the endogenous variables for the value of the exogenous variables specified in the previous initval or endval block.

steady uses an iterative procedure and takes as initial guess the value of the endogenous variables set in the previous initval or endval block.

For complicated models, finding good numerical initial values for the endogenous variables is the trickiest part of finding the equilibrium of that model. Often, it is better to start with a smaller model and add new variables one by one.

Options

maxit = INTEGER

Determines the maximum number of iterations used in the non-linear solver. The default value of maxit is 50.

tolf = DOUBLE

Convergence criterion for termination based on the function value. Iteration will cease when the residuals are smaller than tolf. Default: eps^(1/3)

solve_algo = INTEGER

Determines the non-linear solver to use. Possible values for the option are:

0

Use fsolve (under MATLAB, only available if you have the Optimization Toolbox; always available under Octave).

1

Use a Newton-like algorithm with line-search.

2

Splits the model into recursive blocks and solves each block in turn using the same solver as value 1.

3

Use Chris Sims’ solver.

4

Splits the model into recursive blocks and solves each block in turn using a trust-region solver with autoscaling.

5

Newton algorithm with a sparse Gaussian elimination (SPE) (requires bytecode option, see Model declaration).

6

Newton algorithm with a sparse LU solver at each iteration (requires bytecode and/or block option, see Model declaration).

7

Newton algorithm with a Generalized Minimal Residual (GMRES) solver at each iteration (requires bytecode and/or block option, see Model declaration).

8

Newton algorithm with a Stabilized Bi-Conjugate Gradient (BICGSTAB) solver at each iteration (requires bytecode and/or block option, see Model declaration).

9

Trust-region algorithm with autoscaling (same as value 4, but applied to the entire model, without splitting).

10

Levenberg-Marquardt mixed complementarity problem (LMMCP) solver (Kanzow and Petra (2004)). The complementarity conditions are specified with an mcp equation tag, see lmmcp.

11

PATH mixed complementarity problem solver of Ferris and Munson (1999). The complementarity conditions are specified with an mcp equation tag, see lmmcp. Dynare only provides the interface for using the solver. Due to licence restrictions, you have to download the solver’s most current version yourself from http://pages.cs.wisc.edu/~ferris/path.html and place it in MATLAB’s search path.

12

Specialized version of 2 for models where all the equations have one endogenous variable on the left hand side and where each equation determines a different endogenous variable. Only expressions allowed on the left hand side are the natural logarithm of an endogenous variable, the first difference of an endogenous variable (with the diff operator), or the first difference of the logarithm of an endogenous variable. Univariate blocks are solved by evaluating the expression on the right hand side.

14

Specialized version of 4 for models where all the equations have one endogenous variable on the left hand side and where each equation determines a different endogenous variable. Only expressions allowed on the left hand side are the natural logarithm of an endogenous variable, the first difference of an endogenous variable (with the diff operator), or the first difference of the logarithm of an endogenous variable.. Univariate blocks are solved by evaluating the expression on the right hand side.

Default value is 4.

homotopy_mode = INTEGER

Use a homotopy (or divide-and-conquer) technique to solve for the steady state. If you use this option, you must specify a homotopy_setup block. This option can take three possible values:

1

In this mode, all the parameters are changed simultaneously, and the distance between the boundaries for each parameter is divided in as many intervals as there are steps (as defined by the homotopy_steps option); the problem is solved as many times as there are steps.

2

Same as mode 1, except that only one parameter is changed at a time; the problem is solved as many times as steps times number of parameters.

3

Dynare tries first the most extreme values. If it fails to compute the steady state, the interval between initial and desired values is divided by two for all parameters. Every time that it is impossible to find a steady state, the previous interval is divided by two. When it succeeds to find a steady state, the previous interval is multiplied by two. In that last case homotopy_steps contains the maximum number of computations attempted before giving up.

homotopy_steps = INTEGER

Defines the number of steps when performing a homotopy. See homotopy_mode option for more details.

homotopy_force_continue = INTEGER

This option controls what happens when homotopy fails.

0

steady fails with an error message

1

steady keeps the values of the last homotopy step that was successful and continues. BE CAREFUL: parameters and/or exogenous variables are NOT at the value expected by the user

Default is 0.

nocheck

Don’t check the steady state values when they are provided explicitly either by a steady state file or a steady_state_model block. This is useful for models with unit roots as, in this case, the steady state is not unique or doesn’t exist.

markowitz = DOUBLE

Value of the Markowitz criterion (:math:(0,infty)) used to select the pivot with sparse Gaussian elimination (solve_algo = 5). This criterion governs the tradeoff between selecting the pivot resulting in the most accurate solution (low markowitz values) and the one that preserves maximum sparsity (high markowitz values). Default: 0.5.

Example

After computation, the steady state is available in the following variable:

MATLAB/Octave variable: oo_.steady_state

Contains the computed steady state. Endogenous variables are ordered in the order of declaration used in the var command (which is also the order used in M_.endo_names).

MATLAB/Octave command: get_mean('ENDOGENOUS_NAME' [, 'ENDOGENOUS_NAME']... );

Returns the steady of state of the given endogenous variable(s), as it is stored in oo_.steady_state. Note that, if the steady state has not yet been computed with steady, it will first try to compute it.

Block: homotopy_setup ;

This block is used to declare initial and final values when using a homotopy method. It is used in conjunction with the option homotopy_mode of the steady command.

The idea of homotopy (also called divide-and-conquer by some authors) is to subdivide the problem of finding the steady state into smaller problems. It assumes that you know how to compute the steady state for a given set of parameters, and it helps you finding the steady state for another set of parameters, by incrementally moving from one to another set of parameters.

The purpose of the homotopy_setup block is to declare the final (and possibly also the initial) values for the parameters or exogenous that will be changed during the homotopy. It should contain lines of the form:

VARIABLE_NAME, EXPRESSION, EXPRESSION;

This syntax specifies the initial and final values of a given parameter/exogenous.

There is an alternative syntax:

VARIABLE_NAME, EXPRESSION;

Here only the final value is specified for a given parameter/exogenous; the initial value is taken from the preceeding initval block.

A necessary condition for a successful homotopy is that Dynare must be able to solve the steady state for the initial parameters/exogenous without additional help (using the guess values given in the initval block).

If the homotopy fails, a possible solution is to increase the number of steps (given in homotopy_steps option of steady).

Example

In the following example, Dynare will first compute the steady state for the initial values (gam=0.5 and x=1), and then subdivide the problem into 50 smaller problems to find the steady state for the final values (gam=2 and x=2):

var c k;
varexo x;

parameters alph gam delt bet aa;
alph=0.5;
delt=0.02;
aa=0.5;
bet=0.05;

model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
end;

initval;
x = 1;
k = ((delt+bet)/(aa*x*alph))^(1/(alph-1));
c = aa*x*k^alph-delt*k;
end;

homotopy_setup;
gam, 0.5, 2;
x, 2;
end;

steady(homotopy_mode = 1, homotopy_steps = 50);

### 4.10.2. Providing the steady state to Dynare¶

If you know how to compute the steady state for your model, you can provide a MATLAB/Octave function doing the computation instead of using steady. Again, there are two options for doing that:

• The easiest way is to write a steady_state_model block, which is described below in more details. See also fs2000.mod in the examples directory for an example. The steady state file generated by Dynare will be called +FILENAME/steadystate.m.

• You can write the corresponding MATLAB function by hand. If your MOD-file is called FILENAME.mod, the steady state file must be called FILENAME_steadystate.m. See NK_baseline_steadystate.m in the examples directory for an example. This option gives a bit more flexibility (loops and conditional structures can be used), at the expense of a heavier programming burden and a lesser efficiency.

Note that both files allow to update parameters in each call of the function. This allows for example to calibrate a model to a labor supply of 0.2 in steady state by setting the labor disutility parameter to a corresponding value (see NK_baseline_steadystate.m in the examples directory). They can also be used in estimation where some parameter may be a function of an estimated parameter and needs to be updated for every parameter draw. For example, one might want to set the capital utilization cost parameter as a function of the discount rate to ensure that capacity utilization is 1 in steady state. Treating both parameters as independent or not updating one as a function of the other would lead to wrong results. But this also means that care is required. Do not accidentally overwrite your parameters with new values as it will lead to wrong results.

Block: steady_state_model ;

When the analytical solution of the model is known, this command can be used to help Dynare find the steady state in a more efficient and reliable way, especially during estimation where the steady state has to be recomputed for every point in the parameter space.

Each line of this block consists of a variable (either an endogenous, a temporary variable or a parameter) which is assigned an expression (which can contain parameters, exogenous at the steady state, or any endogenous or temporary variable already declared above). Each line therefore looks like:

VARIABLE_NAME = EXPRESSION;

Note that it is also possible to assign several variables at the same time, if the main function in the right hand side is a MATLAB/Octave function returning several arguments:

[ VARIABLE_NAME, VARIABLE_NAME... ] = EXPRESSION;

Dynare will automatically generate a steady state file (of the form +FILENAME/steadystate.m) using the information provided in this block.

Steady state file for deterministic models

The steady_state_model block also works with deterministic models. An initval block and, when necessary, an endval block, is used to set the value of the exogenous variables. Each initval or endval block must be followed by steady to execute the function created by steady_state_model and set the initial, respectively terminal, steady state.

Example

var m P c e W R k d n l gy_obs gp_obs y dA;
varexo e_a e_m;

parameters alp bet gam mst rho psi del;

...
// parameter calibration, (dynamic) model declaration, shock calibration...
...

dA = exp(gam);
gst = 1/dA; // A temporary variable
m = mst;

// Three other temporary variables
khst = ( (1-gst*bet*(1-del)) / (alp*gst^alp*bet) )^(1/(alp-1));
xist = ( ((khst*gst)^alp - (1-gst*(1-del))*khst)/mst )^(-1);
nust = psi*mst^2/( (1-alp)*(1-psi)*bet*gst^alp*khst^alp );

n  = xist/(nust+xist);
P  = xist + nust;
k  = khst*n;

l  = psi*mst*n/( (1-psi)*(1-n) );
c  = mst/P;
d  = l - mst + 1;
y  = k^alp*n^(1-alp)*gst^alp;
R  = mst/bet;

// You can use MATLAB functions which return several arguments
[W, e] = my_function(l, n);

gp_obs = m/dA;
gy_obs = dA;
end;

### 4.10.3. Replace some equations during steady state computations¶

When there is no steady state file, Dynare computes the steady state by solving the static model, i.e. the model from the .mod file from which leads and lags have been removed.

In some specific cases, one may want to have more control over the way this static model is created. Dynare therefore offers the possibility to explicitly give the form of equations that should be in the static model.

More precisely, if an equation is prepended by a [static] tag, then it will appear in the static model used for steady state computation, but that equation will not be used for other computations. For every equation tagged in this way, you must tag another equation with [dynamic]: that equation will not be used for steady state computation, but will be used for other computations.

This functionality can be useful on models with a unit root, where there is an infinity of steady states. An equation (tagged [dynamic]) would give the law of motion of the nonstationary variable (like a random walk). To pin down one specific steady state, an equation tagged [static] would affect a constant value to the nonstationary variable. Another situation where the [static] tag can be useful is when one has only a partial closed form solution for the steady state.

Example

This is a trivial example with two endogenous variables. The second equation takes a different form in the static model:

var c k;
varexo x;
...
model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
[dynamic] c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
[static] k = ((delt+bet)/(x*aa*alph))^(1/(alph-1));
end;

## 4.11. Getting information about the model¶

Command: check ;
Command: check(OPTIONS...);

Computes the eigenvalues of the model linearized around the values specified by the last initval, endval or steady statement. Generally, the eigenvalues are only meaningful if the linearization is done around a steady state of the model. It is a device for local analysis in the neighborhood of this steady state.

A necessary condition for the uniqueness of a stable equilibrium in the neighborhood of the steady state is that there are as many eigenvalues larger than one in modulus as there are forward looking variables in the system. An additional rank condition requires that the square submatrix of the right Schur vectors corresponding to the forward looking variables (jumpers) and to the explosive eigenvalues must have full rank.

Note that the outcome may be different from what would be suggested by sum(abs(oo_.dr.eigval)) when eigenvalues are very close to qz_criterium.

Options

solve_algo = INTEGER

See solve_algo, for the possible values and their meaning.

qz_zero_threshold = DOUBLE

Value used to test if a generalized eigenvalue is $$0/0$$ in the generalized Schur decomposition (in which case the model does not admit a unique solution). Default: 1e-6.

Output

check returns the eigenvalues in the global variable oo_.dr.eigval.

MATLAB/Octave variable: oo_.dr.eigval

Contains the eigenvalues of the model, as computed by the check command.

Command: model_diagnostics ;

This command performs various sanity checks on the model, and prints a message if a problem is detected (missing variables at current period, invalid steady state, singular Jacobian of static model).

Command: model_info ;
Command: model_info(OPTIONS...);

This command provides information about the model.

When used outside the context of the block option of the model block, it will provide a list of predetermined state variables, forward-looking variables, and purely static variables.

When used in conjunction with the block option of the model block, it displays:

• The normalization of the model: an endogenous variable is attributed to each equation of the model;

• The block structure of the model: for each block model_info indicates its type, the equations number and endogenous variables belonging to this block.

There are five different types of blocks depending on the simulation method used:

• EVALUATE FORWARD

In this case the block contains only equations where the endogenous variable attributed to the equation appears at current period on the left hand side and where no forward looking endogenous variables appear. The block has the form: $$y_{j,t} = f_j(y_t, y_{t-1}, \ldots, y_{t-k})$$.

• EVALUATE BACKWARD

The block contains only equations where the endogenous variable attributed to the equation appears at current period on the left hand side and where no backward looking endogenous variables appear. The block has the form: $$y_{j,t} = f_j(y_t, y_{t+1}, \ldots, y_{t+k})$$.

• SOLVE BACKWARD x

The block contains only equations where the endogenous variable attributed to the equation does not appear at current period on the left hand side and where no forward looking endogenous variables appear. The block has the form: $$g_j(y_{j,t}, y_t, y_{t-1}, \ldots, y_{t-k})=0$$. x is equal to SIMPLE if the block has only one equation. If several equations appear in the block, x is equal to COMPLETE.

• SOLVE FORWARD x

The block contains only equations where the endogenous variable attributed to the equation does not appear at current period on the left hand side and where no backward looking endogenous variables appear. The block has the form: $$g_j(y_{j,t}, y_t, y_{t+1}, \ldots, y_{t+k})=0$$. x is equal to SIMPLE if the block has only one equation. If several equations appear in the block, x is equal to COMPLETE.

• SOLVE TWO BOUNDARIES x

The block contains equations depending on both forward and backward variables. The block looks like: $$g_j(y_{j,t}, y_t, y_{t-1}, \ldots, y_{t-k} ,y_t, y_{t+1}, \ldots, y_{t+k})=0$$. x is equal to SIMPLE if the block has only one equation. If several equations appear in the block, x is equal to COMPLETE.

Options

static

Prints out the block decomposition of the static model. Without the static option, model_info displays the block decomposition of the dynamic model.

incidence

Displays the gross incidence matrix and the reordered incidence matrix of the block decomposed model.

Prints the equations and the Jacobian matrix of the dynamic model stored in the bytecode binary format file. Can only be used in conjunction with the bytecode option of the model block.

Prints the equations and the Jacobian matrix of the static model stored in the bytecode binary format file. Can only be used in conjunction with the bytecode option of the model block.

## 4.12. Deterministic simulation¶

When the framework is deterministic, Dynare can be used for models with the assumption of perfect foresight. Typically, the system is supposed to be in a state of equilibrium before a period ‘1’ when the news of a contemporaneous or of a future shock is learned by the agents in the model. The purpose of the simulation is to describe the reaction in anticipation of, then in reaction to the shock, until the system returns to the old or to a new state of equilibrium. In most models, this return to equilibrium is only an asymptotic phenomenon, which one must approximate by an horizon of simulation far enough in the future. Another exercise for which Dynare is well suited is to study the transition path to a new equilibrium following a permanent shock. For deterministic simulations, the numerical problem consists of solving a nonlinar system of simultaneous equations in n endogenous variables in T periods. Dynare offers several algorithms for solving this problem, which can be chosen via the stack_solve_algo option. By default (stack_solve_algo=0), Dynare uses a Newton-type method to solve the simultaneous equation system. Because the resulting Jacobian is in the order of n by T and hence will be very large for long simulations with many variables, Dynare makes use of the sparse matrix capacities of MATLAB/Octave. A slower but potentially less memory consuming alternative (stack_solve_algo=6) is based on a Newton-type algorithm first proposed by Laffargue (1990) and Boucekkine (1995), which uses relaxation techniques. Thereby, the algorithm avoids ever storing the full Jacobian. The details of the algorithm can be found in Juillard (1996). The third type of algorithms makes use of block decomposition techniques (divide-and-conquer methods) that exploit the structure of the model. The principle is to identify recursive and simultaneous blocks in the model structure and use this information to aid the solution process. These solution algorithms can provide a significant speed-up on large models.

Warning

Be careful when employing auxiliary variables in the context of perfect foresight computations. The same model may work for stochastic simulations, but fail for perfect foresight simulations. The issue arises when an equation suddenly only contains variables dated t+1 (or t-1 for that matter). In this case, the derivative in the last (first) period with respect to all variables will be 0, rendering the stacked Jacobian singular.

Example

Consider the following specification of an Euler equation with log utility:

Lambda = beta*C(-1)/C;
Lambda(+1)*R(+1)= 1;

Clearly, the derivative of the second equation with respect to all endogenous variables at time t is zero, causing perfect_foresight_solver to generally fail. This is due to the use of the Lagrange multiplier Lambda as an auxiliary variable. Instead, employing the identical

beta*C/C(+1)*R(+1)= 1;

will work.

Command: perfect_foresight_setup ;
Command: perfect_foresight_setup(OPTIONS...);

Prepares a perfect foresight simulation, by extracting the information in the initval, endval and shocks blocks and converting them into simulation paths for exogenous and endogenous variables.

This command must always be called before running the simulation with perfect_foresight_solver.

Options

periods = INTEGER

Number of periods of the simulation.

datafile = FILENAME

Used to specify path for all endogenous and exogenous variables. Strictly equivalent to initval_file.

Output

The paths for the exogenous variables are stored into oo_.exo_simul.

The initial and terminal conditions for the endogenous variables and the initial guess for the path of endogenous variables are stored into oo_.endo_simul.

Command: perfect_foresight_solver ;
Command: perfect_foresight_solver(OPTIONS...);

Computes the perfect foresight (or deterministic) simulation of the model.

Note that perfect_foresight_setup must be called before this command, in order to setup the environment for the simulation.

Options

maxit = INTEGER

Determines the maximum number of iterations used in the non-linear solver. The default value of maxit is 50.

tolf = DOUBLE

Convergence criterion for termination based on the function value. Iteration will cease when it proves impossible to improve the function value by more than tolf. Default: 1e-5

tolx = DOUBLE

Convergence criterion for termination based on the change in the function argument. Iteration will cease when the solver attempts to take a step that is smaller than tolx. Default: 1e-5

noprint

Don’t print anything. Useful for loops.

print

Print results (opposite of noprint).

stack_solve_algo = INTEGER

Algorithm used for computing the solution. Possible values are:

0

Newton method to solve simultaneously all the equations for every period, using sparse matrices (Default).

1

Use a Newton algorithm with a sparse LU solver at each iteration (requires bytecode and/or block option, see Model declaration).

2

Use a Newton algorithm with a Generalized Minimal Residual (GMRES) solver at each iteration (requires bytecode and/or block option, see Model declaration)

3

Use a Newton algorithm with a Stabilized Bi-Conjugate Gradient (BICGSTAB) solver at each iteration (requires bytecode and/or block option, see Model declaration).

4

Use a Newton algorithm with a optimal path length at each iteration (requires bytecode and/or block option, see Model declaration).

5

Use a Newton algorithm with a sparse Gaussian elimination (SPE) solver at each iteration (requires bytecode option, see Model declaration).

6

Use the historical algorithm proposed in Juillard (1996): it is slower than stack_solve_algo=0, but may be less memory consuming on big models (not available with bytecode and/or block options).

7

Allows the user to solve the perfect foresight model with the solvers available through option solve_algo (See solve_algo for a list of possible values, note that values 5, 6, 7 and 8, which require bytecode and/or block options, are not allowed). For instance, the following commands:

perfect_foresight_setup(periods=400);
perfect_foresight_solver(stack_solve_algo=7, solve_algo=9)

trigger the computation of the solution with a trust region algorithm.

robust_lin_solve

Triggers the use of a robust linear solver for the default stack_solve_algo=0.

solve_algo

See solve_algo. Allows selecting the solver used with stack_solve_algo=7.

no_homotopy

By default, the perfect foresight solver uses a homotopy technique if it cannot solve the problem. Concretely, it divides the problem into smaller steps by diminishing the size of shocks and increasing them progressively until the problem converges. This option tells Dynare to disable that behavior. Note that the homotopy is not implemented for purely forward or backward models.

markowitz = DOUBLE

Value of the Markowitz criterion, used to select the pivot. Only used when stack_solve_algo = 5. Default: 0.5.

minimal_solving_periods = INTEGER

Specify the minimal number of periods where the model has to be solved, before using a constant set of operations for the remaining periods. Only used when stack_solve_algo = 5. Default: 1.

lmmcp

Solves the perfect foresight model with a Levenberg-Marquardt mixed complementarity problem (LMMCP) solver (Kanzow and Petra (2004)), which allows to consider inequality constraints on the endogenous variables (such as a ZLB on the nominal interest rate or a model with irreversible investment). This option is equivalent to stack_solve_algo=7 and solve_algo=10. Using the LMMCP solver avoids the need for min/max operators and explicit complementary slackness conditions in the model as they will typically introduce a singularity into the Jacobian. This is done by setting the problem up as a mixed complementarity problem (MCP) of the form:

\begin{align}\begin{aligned}\begin{split}LB = X &\Rightarrow F(X)>0\\\end{split}\\\begin{split}LB\leq X \leq UB &\Rightarrow F(X)=0\\\end{split}\\X =UB &\Rightarrow F(X)<0.\end{aligned}\end{align}

where $$X$$ denotes the vector of endogenous variables, $$F(X)$$ the equations of the model, $$LB$$ denotes a lower bound, and $$UB$$ an upper bound. Such a setup is implemented by attaching an equation tag (see Model declaration) with the mcp keyword to the affected equations. This tag states that the equation to which the tag is attached has to hold unless the inequality constraint within the tag is binding.

For instance, a ZLB on the nominal interest rate would be specified as follows in the model block:

model;
...
[mcp = 'r > -1.94478']
r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e;
...
end;

where 1.94478 is the steady state level of the nominal interest rate and r is the nominal interest rate in deviation from the steady state. This construct implies that the Taylor rule is operative, unless the implied interest rate r<=-1.94478, in which case the r is fixed at -1.94478 (thereby being equivalent to a complementary slackness condition). By restricting the value of r coming out of this equation, the mcp tag also avoids using max(r,-1.94478) for other occurrences of r in the rest of the model. Two things are important to keep in mind. First, because the mcp tag effectively replaces a complementary slackness condition, it cannot be simply attached to any equation. Rather, it must be attached to the correct affected equation as otherwise the solver will solve a different problem than originally intended. Second, the sign of the residual of the dynamic equation must conform to the MCP setup outlined above. In case of the ZLB, we are dealing with a lower bound. Consequently, the dynamic equation needs to return a positive residual. Dynare by default computes the residual of an equation LHS=RHS as residual=LHS-RHS, while an implicit equation LHS is interpreted as LHS=0. For the above equation this implies

residual= r - (rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e);

which is correct, since it will be positive if the implied interest rate rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e is below r=-1.94478. In contrast, specifying the equation as

rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e = r;

would be wrong.

Note that in the current implementation, the content of the mcp equation tag is not parsed by the preprocessor. The inequalities must therefore be as simple as possible: an endogenous variable, followed by a relational operator, followed by a number (not a variable, parameter or expression).

endogenous_terminal_period

The number of periods is not constant across Newton iterations when solving the perfect foresight model. The size of the nonlinear system of equations is reduced by removing the portion of the paths (and associated equations) for which the solution has already been identified (up to the tolerance parameter). This strategy can be interpreted as a mix of the shooting and relaxation approaches. Note that round off errors are more important with this mixed strategy (user should check the reported value of the maximum absolute error). Only available with option stack_solve_algo==0.

linear_approximation

Solves the linearized version of the perfect foresight model. The model must be stationary. Only available with option stack_solve_algo==0 or stack_solve_algo==7.

Output

The simulated endogenous variables are available in global matrix oo_.endo_simul.

Command: simul ;
Command: simul(OPTIONS...);

Short-form command for triggering the computation of a deterministic simulation of the model. It is strictly equivalent to a call to perfect_foresight_setup followed by a call to perfect_foresight_solver.

Options

Accepts all the options of perfect_foresight_setup and perfect_foresight_solver.

MATLAB/Octave variable: oo_.endo_simul

This variable stores the result of a deterministic simulation (computed by perfect_foresight_solver or simul) or of a stochastic simulation (computed by stoch_simul with the periods option or by extended_path). The variables are arranged row by row, in order of declaration (as in M_.endo_names). Note that this variable also contains initial and terminal conditions, so it has more columns than the value of periods option.

MATLAB/Octave variable: oo_.exo_simul

This variable stores the path of exogenous variables during a simulation (computed by perfect_foresight_solver, simul, stoch_simul or extended_path). The variables are arranged in columns, in order of declaration (as in M_.exo_names). Periods are in rows. Note that this convention regarding columns and rows is the opposite of the convention for oo_.endo_simul!

## 4.13. Stochastic solution and simulation¶

In a stochastic context, Dynare computes one or several simulations corresponding to a random draw of the shocks.

The main algorithm for solving stochastic models relies on a Taylor approximation, up to third order, of the expectation functions (see Judd (1996), Collard and Juillard (2001a, 2001b), and Schmitt-Grohé and Uríbe (2004)). The details of the Dynare implementation of the first order solution are given in Villemot (2011). Such a solution is computed using the stoch_simul command.

As an alternative, it is possible to compute a simulation to a stochastic model using the extended path method presented by Fair and Taylor (1983). This method is especially useful when there are strong nonlinearities or binding constraints. Such a solution is computed using the extended_path command.

### 4.13.1. Computing the stochastic solution¶

Command: stoch_simul [VARIABLE_NAME...];
Command: stoch_simul(OPTIONS...) [VARIABLE_NAME...];

Solves a stochastic (i.e. rational expectations) model, using perturbation techniques.

More precisely, stoch_simul computes a Taylor approximation of the model around the deterministic steady state and solves of the the decision and transition functions for the approximated model. Using this, it computes impulse response functions and various descriptive statistics (moments, variance decomposition, correlation and autocorrelation coefficients). For correlated shocks, the variance decomposition is computed as in the VAR literature through a Cholesky decomposition of the covariance matrix of the exogenous variables. When the shocks are correlated, the variance decomposition depends upon the order of the variables in the varexo command.

The Taylor approximation is computed around the steady state (see Steady state).

The IRFs are computed as the difference between the trajectory of a variable following a shock at the beginning of period 1 and its steady state value. More details on the computation of IRFs can be found at https://archives.dynare.org/DynareWiki/IrFs.

Variance decomposition, correlation, autocorrelation are only displayed for variables with strictly positive variance. Impulse response functions are only plotted for variables with response larger than $$10^{-10}$$.

Variance decomposition is computed relative to the sum of the contribution of each shock. Normally, this is of course equal to aggregate variance, but if a model generates very large variances, it may happen that, due to numerical error, the two differ by a significant amount. Dynare issues a warning if the maximum relative difference between the sum of the contribution of each shock and aggregate variance is larger than 0.01%.

The covariance matrix of the shocks is specified with the shocks command (see Shocks on exogenous variables).

When a list of VARIABLE_NAME is specified, results are displayed only for these variables.

The stoch_simul command with a first order approximation can benefit from the block decomposition of the model (see block).

Options

ar = INTEGER

Order of autocorrelation coefficients to compute and to print. Default: 5.

drop = INTEGER

Number of points (burnin) dropped at the beginning of simulation before computing the summary statistics. Note that this option does not affect the simulated series stored in oo_.endo_simul and the workspace. Here, no periods are dropped. Default: 100.

hp_filter = DOUBLE

Uses HP filter with $$\lambda =$$ DOUBLE before computing moments. If theoretical moments are requested, the spectrum of the model solution is filtered following the approach outlined in Uhlig (2001). Default: no filter.

one_sided_hp_filter = DOUBLE

Uses the one-sided HP filter with $$\lambda =$$ DOUBLE described in Stock and Watson (1999) before computing moments. This option is only available with simulated moments. Default: no filter.

bandpass_filter

Uses a bandpass filter with the default passband before computing moments. If theoretical moments are requested, the spectrum of the model solution is filtered using an ideal bandpass filter. If empirical moments are requested, the Baxter and King (1999) filter is used. Default: no filter.

bandpass_filter = [HIGHEST_PERIODICITY LOWEST_PERIODICITY]

Uses a bandpass filter before computing moments. The passband is set to a periodicity of to LOWEST_PERIODICITY, e.g. $$6$$ to $$32$$ quarters if the model frequency is quarterly. Default: [6,32].

filtered_theoretical_moments_grid = INTEGER

When computing filtered theoretical moments (with either option hp_filter or option bandpass_filter), this option governs the number of points in the grid for the discrete Inverse Fast Fourier Transform. It may be necessary to increase it for highly autocorrelated processes. Default: 512.

irf = INTEGER

Number of periods on which to compute the IRFs. Setting irf=0 suppresses the plotting of IRFs. Default: 40.

irf_shocks = ( VARIABLE_NAME [[,] VARIABLE_NAME ...] )

The exogenous variables for which to compute IRFs. Default: all.

relative_irf

Requests the computation of normalized IRFs. At first order, the normal shock vector of size one standard deviation is divided by the standard deviation of the current shock and multiplied by 100. The impulse responses are hence the responses to a unit shock of size 1 (as opposed to the regular shock size of one standard deviation), multiplied by 100. Thus, for a loglinearized model where the variables are measured in percent, the IRFs have the interpretation of the percent responses to a 100 percent shock. For example, a response of 400 of output to a TFP shock shows that output increases by 400 percent after a 100 percent TFP shock (you will see that TFP increases by 100 on impact). Given linearity at order=1, it is straightforward to rescale the IRFs stored in oo_.irfs to any desired size. At higher order, the interpretation is different. The relative_irf option then triggers the generation of IRFs as the response to a 0.01 unit shock (corresponding to 1 percent for shocks measured in percent) and no multiplication with 100 is performed. That is, the normal shock vector of size one standard deviation is divided by the standard deviation of the current shock and divided by 100. For example, a response of 0.04 of log output (thus measured in percent of the steady state output level) to a TFP shock also measured in percent then shows that output increases by 4 percent after a 1 percent TFP shock (you will see that TFP increases by 0.01 on impact).

irf_plot_threshold = DOUBLE

Threshold size for plotting IRFs. All IRFs for a particular variable with a maximum absolute deviation from the steady state smaller than this value are not displayed. Default: 1e-10.

nocorr

Don’t print the correlation matrix (printing them is the default).

nodecomposition

Don’t compute (and don’t print) unconditional variance decomposition.

nofunctions

Don’t print the coefficients of the approximated solution (printing them is the default).

nomoments

Don’t print moments of the endogenous variables (printing them is the default).

nograph

Do not create graphs (which implies that they are not saved to the disk nor displayed). If this option is not used, graphs will be saved to disk (to the format specified by graph_format option, except if graph_format=none) and displayed to screen (unless nodisplay option is used).

graph

Re-enables the generation of graphs previously shut off with nograph.

nodisplay

Do not display the graphs, but still save them to disk (unless nograph is used).

graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )

Specify the file format(s) for graphs saved to disk. Possible values are eps (the default), pdf, fig and none (under Octave, fig is unavailable). If the file format is set equal to none, the graphs are displayed but not saved to the disk.

noprint
print

See print.

order = INTEGER

Order of Taylor approximation. Note that for third order and above, the k_order_solver option is implied and only empirical moments are available (you must provide a value for periods option). Default: 2 (except after an estimation command, in which case the default is the value used for the estimation).

k_order_solver

Use a k-order solver (implemented in C++) instead of the default Dynare solver. This option is not yet compatible with the bytecode option (see Model declaration). Default: disabled for order 1 and 2, enabled for order 3 and above.

periods = INTEGER

If different from zero, empirical moments will be computed instead of theoretical moments. The value of the option specifies the number of periods to use in the simulations. Values of the initval block, possibly recomputed by steady, will be used as starting point for the simulation. The simulated endogenous variables are made available to the user in a vector for each variable and in the global matrix oo_.endo_simul (see oo_.endo_simul). The simulated exogenous variables are made available in oo_.exo_simul (see oo_.exo_simul). Default: 0.

qz_criterium = DOUBLE

Value used to split stable from unstable eigenvalues in reordering the Generalized Schur decomposition used for solving first order problems. Default: 1.000001 (except when estimating with lik_init option equal to 1: the default is 0.999999 in that case; see Estimation based on likelihood).

qz_zero_threshold = DOUBLE
replic = INTEGER

Number of simulated series used to compute the IRFs. Default: 1 if order=1, and 50 otherwise.

simul_replic = INTEGER

Number of series to simulate when empirical moments are requested (i.e. periods $$>$$ 0). Note that if this option is greater than 1, the additional series will not be used for computing the empirical moments but will simply be saved in binary form to the file FILENAME_simul in the FILENAME/Output-folder. Default: 1.

solve_algo = INTEGER

See solve_algo, for the possible values and their meaning.

aim_solver

Use the Anderson-Moore Algorithm (AIM) to compute the decision rules, instead of using Dynare’s default method based on a generalized Schur decomposition. This option is only valid for first order approximation. See AIM website for more details on the algorithm.

conditional_variance_decomposition = INTEGER
conditional_variance_decomposition = [INTEGER1:INTEGER2]
conditional_variance_decomposition = [INTEGER1 INTEGER2 ...]

Computes a conditional variance decomposition for the specified period(s). The periods must be strictly positive. Conditional variances are given by $$var(y_{t+k}\vert t)$$. For period 1, the conditional variance decomposition provides the decomposition of the effects of shocks upon impact.

The results are stored in oo_.conditional_variance_decomposition (see oo_.conditional_variance_decomposition). In the presence of measurement error, the oo_.conditional_variance_decomposition field will contain the variance contribution after measurement error has been taken out, i.e. the decomposition will be conducted of the actual as opposed to the measured variables. The variance decomposition of the measured variables will be stored in oo_.conditional_variance_decomposition_ME (see oo_.conditional_variance_decomposition_ME). The variance decomposition is only conducted, if theoretical moments are requested, i.e. using the periods=0-option. Only available at order<3 and without pruning''. In case of order=2, Dynare provides a second-order accurate approximation to the true second moments based on the linear terms of the second-order solution (see Kim, Kim, Schaumburg and Sims (2008)). Note that the unconditional variance decomposition i.e. at horizon infinity) is automatically conducted if theoretical moments are requested and if nodecomposition is not set (see oo_.variance_decomposition).

pruning

Discard higher order terms when iteratively computing simulations of the solution. At second order, Dynare uses the algorithm of Kim, Kim, Schaumburg and Sims (2008), while at third order its generalization by Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018) is used. Not available above third order. When specified, theoretical moments are based on the pruned state space, i.e. the computation of second moments uses all terms as in Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018), page 10 as opposed to simply providing a second-order accurate result based on the linear solution as in Kim, Kim, Schaumburg and Sims (2008).

partial_information

Computes the solution of the model under partial information, along the lines of Pearlman, Currie and Levine (1986). Agents are supposed to observe only some variables of the economy. The set of observed variables is declared using the varobs command. Note that if varobs is not present or contains all endogenous variables, then this is the full information case and this option has no effect. More references can be found here .

sylvester = OPTION

Determines the algorithm used to solve the Sylvester equation for block decomposed model. Possible values for OPTION are:

default

Uses the default solver for Sylvester equations (gensylv) based on Ondra Kamenik’s algorithm (see here for more information).

fixed_point

Uses a fixed point algorithm to solve the Sylvester equation (gensylv_fp). This method is faster than the default one for large scale models.

Default value is default.

sylvester_fixed_point_tol = DOUBLE

The convergence criterion used in the fixed point Sylvester solver. Its default value is 1e-12.

dr = OPTION

Determines the method used to compute the decision rule. Possible values for OPTION are:

default

Uses the default method to compute the decision rule based on the generalized Schur decomposition (see Villemot (2011) for more information).

cycle_reduction

Uses the cycle reduction algorithm to solve the polynomial equation for retrieving the coefficients associated to the endogenous variables in the decision rule. This method is faster than the default one for large scale models.

logarithmic_reduction

Uses the logarithmic reduction algorithm to solve the polynomial equation for retrieving the coefficients associated to the endogenous variables in the decision rule. This method is in general slower than the cycle_reduction.

Default value is default.

dr_cycle_reduction_tol = DOUBLE

The convergence criterion used in the cycle reduction algorithm. Its default value is 1e-7.

dr_logarithmic_reduction_tol = DOUBLE

The convergence criterion used in the logarithmic reduction algorithm. Its default value is 1e-12.

dr_logarithmic_reduction_maxiter = INTEGER

The maximum number of iterations used in the logarithmic reduction algorithm. Its default value is 100.

loglinear

See loglinear. Note that ALL variables are log-transformed by using the Jacobian transformation, not only selected ones. Thus, you have to make sure that your variables have strictly positive steady states. stoch_simul will display the moments, decision rules, and impulse responses for the log-linearized variables. The decision rules saved in oo_.dr and the simulated variables will also be the ones for the log-linear variables.

tex

Requests the printing of results and graphs in TeX tables and graphics that can be later directly included in LaTeX files.

dr_display_tol = DOUBLE

Tolerance for the suppression of small terms in the display of decision rules. Rows where all terms are smaller than dr_display_tol are not displayed. Default value: 1e-6.

contemporaneous_correlation

Saves the contemporaneous correlation between the endogenous variables in oo_.contemporaneous_correlation. Requires the nocorr option not to be set.

spectral_density

Triggers the computation and display of the theoretical spectral density of the (filtered) model variables. Results are stored in oo_.SpectralDensity, defined below. Default: do not request spectral density estimates.

hp_ngrid = INTEGER

Deprecated option. It has the same effect as filtered_theoretical_moments_grid.

Output

This command sets oo_.dr, oo_.mean, oo_.var, oo_.var_list, and oo_.autocorr, which are described below.

If the periods option is present, sets oo_.skewness, oo_.kurtosis, and oo_.endo_simul (see oo_.endo_simul), and also saves the simulated variables in MATLAB/Octave vectors of the global workspace with the same name as the endogenous variables.

If option irf is different from zero, sets oo_.irfs (see below) and also saves the IRFs in MATLAB/Octave vectors of the global workspace (this latter way of accessing the IRFs is deprecated and will disappear in a future version).

If the option contemporaneous_correlation is different from 0, sets oo_.contemporaneous_correlation, which is described below.

Example

shocks;
var e;
stderr 0.0348;
end;

stoch_simul;

Performs the simulation of the 2nd-order approximation of a model with a single stochastic shock e, with a standard error of 0.0348.

Example

stoch_simul(irf=60) y k;

Performs the simulation of a model and displays impulse response functions on 60 periods for variables y and k.

MATLAB/Octave variable: oo_.mean

After a run of stoch_simul, contains the mean of the endogenous variables. Contains theoretical mean if the periods option is not present, and simulated mean otherwise. The variables are arranged in declaration order.

MATLAB/Octave variable: oo_.var

After a run of stoch_simul, contains the variance-covariance of the endogenous variables. Contains theoretical variance if the periods option is not present and simulated variance otherwise. Only available for order<4. At order=2 it will be be a second-order accurate approximation (i.e. ignoring terms of order 3 and 4 that would arise when using the full second-order policy function). At order=3, theoretical moments are only available with pruning. The variables are arranged in declaration order.

MATLAB/Octave variable: oo_.var_list

The list of variables for which results are displayed.

MATLAB/Octave variable: oo_.skewness

After a run of stoch_simul contains the skewness (standardized third moment) of the simulated variables if the periods option is present. The variables are arranged in declaration order.

MATLAB/Octave variable: oo_.kurtosis

After a run of stoch_simul contains the excess kurtosis (standardized fourth moment) of the simulated variables if the periods option is present. The variables are arranged in declaration order.

MATLAB/Octave variable: oo_.autocorr

After a run of stoch_simul, contains a cell array of the autocorrelation matrices of the endogenous variables. The element number of the matrix in the cell array corresponds to the order of autocorrelation. The option ar specifies the number of autocorrelation matrices available. Contains theoretical autocorrelations if the periods option is not present and simulated autocorrelations otherwise. Only available for order<4. At order=2 it will be be a second-order accurate approximation. At order=3, theoretical moments are only available with pruning. The field is only created if stationary variables are present.

The element oo_.autocorr{i}(k,l) is equal to the correlation between $$y^k_t$$ and $$y^l_{t-i}$$, where $$y^k$$ (resp. $$y^l$$) is the $$k$$-th (resp. $$l$$-th) endogenous variable in the declaration order.

Note that if theoretical moments have been requested, oo_.autocorr{i} is the same than oo_.gamma_y{i+1}.

MATLAB/Octave variable: oo_.gamma_y

After a run of stoch_simul, if theoretical moments have been requested (i.e. if the periods option is not present), this variable contains a cell array with the following values (where ar is the value of the option of the same name):

oo_.gamma{1}

Variance/covariance matrix.

oo_.gamma{i+1} (for i=1:ar)

Autocorrelation function. See oo_.autocorr for more details. Beware, this is the autocorrelation function, not the autocovariance function.

oo_.gamma{ar+2}

Unconditional variance decomposition, see oo_.variance_decomposition.

oo_.gamma{ar+3}

If a second order approximation has been requested, contains the vector of the mean correction terms.

Only available at order<4. In case order=2, the theoretical second moments are a second order accurate approximation of the true second moments. See conditional_variance_decomposition. At order=3, theoretical moments are only available with pruning.

MATLAB/Octave variable: oo_.variance_decomposition

After a run of stoch_simul when requesting theoretical moments (periods=0), contains a matrix with the result of the unconditional variance decomposition (i.e. at horizon infinity). The first dimension corresponds to the endogenous variables (in the order of declaration after the command or in M_.endo_names) and the second dimension corresponds to exogenous variables (in the order of declaration). Numbers are in percent and sum up to 100 across columns. In the presence of measurement error, the field will contain the variance contribution after measurement error has been taken out, i.e. the decomposition will be conducted of the actual as opposed to the measured variables.

MATLAB/Octave variable: oo_.variance_decomposition_ME

Field set after a run of stoch_simul when requesting theoretical moments (periods=0) if measurement error is present. It is similar to oo_.variance_decomposition, but the decomposition will be conducted of the measured variables. The field contains a matrix with the result of the unconditional variance decomposition (i.e. at horizon infinity). The first dimension corresponds to the observed endoogenous variables (in the order of declaration after the command) and the second dimension corresponds to exogenous variables (in the order of declaration), with the last column corresponding to the contribution of measurement error. Numbers are in percent and sum up to 100 across columns.

MATLAB/Octave variable: oo_.conditional_variance_decomposition

After a run of stoch_simul with the conditional_variance_decomposition option, contains a three-dimensional array with the result of the decomposition. The first dimension corresponds to the endogenous variables (in the order of declaration after the command or in M_.endo_names if not specified), the second dimension corresponds to the forecast horizons (as declared with the option), and the third dimension corresponds to the exogenous variables (in the order of declaration). In the presence of measurement error, the field will contain the variance contribution after measurement error has been taken out, i.e. the decomposition will be conductedof the actual as opposed to the measured variables.

MATLAB/Octave variable: oo_.conditional_variance_decomposition_ME

Field set after a run of stoch_simul with the conditional_variance_decomposition option if measurement error is present. It is similar to oo_.conditional_variance_decomposition, but the decomposition will be conducted of the measured variables. It contains a three-dimensional array with the result of the decomposition. The first dimension corresponds to the endogenous variables (in the order of declaration after the command or in M_.endo_names if not specified), the second dimension corresponds to the forecast horizons (as declared with the option), and the third dimension corresponds to the exogenous variables (in the order of declaration), with the last column corresponding to the contribution of the measurement error.

MATLAB/Octave variable: oo_.contemporaneous_correlation

After a run of stoch_simul with the contemporaneous_correlation option, contains theoretical contemporaneous correlations if the periods option is not present, and simulated contemporaneous correlations otherwise. Only available for order<4. At order=2 it will be be a second-order accurate approximation. At order=3, theoretical moments are only available with pruning. The variables are arranged in declaration order.

MATLAB/Octave variable: oo_.SpectralDensity

After a run of stoch_simul with option spectral_density, contains the spectral density of the model variables. There will be a nvars by nfrequencies subfield freqs storing the respective frequency grid points ranging from $$0$$ to $$2\pi$$ and a same sized subfield density storing the corresponding density.

MATLAB/Octave variable: oo_.irfs

After a run of stoch_simul with option irf different from zero, contains the impulse responses, with the following naming convention: VARIABLE_NAME_SHOCK_NAME.

For example, oo_.irfs.gnp_ea contains the effect on gnp of a one-standard deviation shock on ea.

MATLAB/Octave command: get_irf('EXOGENOUS_NAME' [, 'ENDOGENOUS_NAME']... );

Given the name of an exogenous variables, returns the IRFs for the requested endogenous variable(s), as they are stored in oo_.irfs.

The approximated solution of a model takes the form of a set of decision rules or transition equations expressing the current value of the endogenous variables of the model as function of the previous state of the model and shocks observed at the beginning of the period. The decision rules are stored in the structure oo_.dr which is described below.

MATLAB/Octave variable: oo_.dr

Structure storing the decision rules. The subfields for different orders of approximation are explained below.

Command: extended_path ;
Command: extended_path(OPTIONS...);

Simulates a stochastic (i.e. rational expectations) model, using the extended path method presented by Fair and Taylor (1983). Time series for the endogenous variables are generated by assuming that the agents believe that there will no more shocks in the following periods.

This function first computes a random path for the exogenous variables (stored in oo_.exo_simul, see oo_.exo_simul) and then computes the corresponding path for endogenous variables, taking the steady state as starting point. The result of the simulation is stored in oo_.endo_simul (see oo_.endo_simul). Note that this simulation approach does not solve for the policy and transition equations but for paths for the endogenous variables.

Options

periods = INTEGER

The number of periods for which the simulation is to be computed. No default value, mandatory option.

solver_periods = INTEGER

The number of periods used to compute the solution of the perfect foresight at every iteration of the algorithm. Default: 200.

order = INTEGER

If order is greater than 0 Dynare uses a gaussian quadrature to take into account the effects of future uncertainty. If order $$=S$$ then the time series for the endogenous variables are generated by assuming that the agents believe that there will no more shocks after period $$t+S$$. This is an experimental feature and can be quite slow. A non-zero value is not compatible with either the bytecode or the block option of the model block. Default: 0.

hybrid

Use the constant of the second order perturbation reduced form to correct the paths generated by the (stochastic) extended path algorithm.

lmmcp

Solves the perfect foresight model with a Levenberg-Marquardt mixed complementarity problem (LMMCP) solver (Kanzow and Petra (2004)), which allows to consider inequality constraints on the endogenous variables (such as a ZLB on the nominal interest rate or a model with irreversible investment). For specifying the necessary mcp-tag, see lmmcp.

### 4.13.2. Typology and ordering of variables¶

Dynare distinguishes four types of endogenous variables:

Purely backward (or purely predetermined) variables

Those that appear only at current and past period in the model, but not at future period (i.e. at $$t$$ and $$t-1$$ but not $$t+1$$). The number of such variables is equal to M_.npred.

Purely forward variables

Those that appear only at current and future period in the model, but not at past period (i.e. at $$t$$ and $$t+1$$ but not $$t-1$$). The number of such variables is stored in M_.nfwrd.

Mixed variables

Those that appear at current, past and future period in the model (i.e. at $$t$$, $$t+1$$ and $$t-1$$). The number of such variables is stored in M_.nboth.

Static variables

Those that appear only at current, not past and future period in the model (i.e. only at $$t$$, not at $$t+1$$ or $$t-1$$). The number of such variables is stored in M_.nstatic.

Note that all endogenous variables fall into one of these four categories, since after the creation of auxiliary variables (see Auxiliary variables), all endogenous have at most one lead and one lag. We therefore have the following identity:

M_.npred + M_.both + M_.nfwrd + M_.nstatic = M_.endo_nbr

MATLAB/Octave variable: M_.state_var

Vector of numerical indices identifying the state variables in the vector of declared variables. M_.endo_names(M_.state_var) therefore yields the name of all variables that are states in the model declaration, i.e. that show up with a lag.

Internally, Dynare uses two orderings of the endogenous variables: the order of declaration (which is reflected in M_.endo_names), and an order based on the four types described above, which we will call the DR-order (“DR” stands for decision rules). Most of the time, the declaration order is used, but for elements of the decision rules, the DR-order is used.

The DR-order is the following: static variables appear first, then purely backward variables, then mixed variables, and finally purely forward variables. Inside each category, variables are arranged according to the declaration order.

MATLAB/Octave variable: oo_.dr.order_var

This variables maps DR-order to declaration order.

MATLAB/Octave variable: oo_.dr.inv_order_var

This variable contains the inverse map.

In other words, the k-th variable in the DR-order corresponds to the endogenous variable numbered oo_.dr.order_var(k) in declaration order. Conversely, k-th declared variable is numbered oo_.dr.inv_order_var(k) in DR-order.

Finally, the state variables of the model are the purely backward variables and the mixed variables. They are ordered in DR-order when they appear in decision rules elements. There are M_.nspred = M_.npred + M_.nboth such variables. Similarly, one has M_.nsfwrd = M_.nfwrd + M_.nboth, and M_.ndynamic = M_.nfwrd + M_.nboth + M_.npred.

### 4.13.3. First-order approximation¶

The approximation has the stylized form:

$y_t = y^s + A y^h_{t-1} + B u_t$

where $$y^s$$ is the steady state value of $$y$$ and $$y^h_t=y_t-y^s$$.

MATLAB/Octave variable: oo.dr.state_var

Vector of numerical indices identifying the state variables in the vector of declared variables, given the current parameter values for which the decision rules have been computed. It may differ from M_.state_var in case a state variable drops from the model given the current parameterization, because it only gets 0 coefficients in the decision rules. See M_.state_var.

The coefficients of the decision rules are stored as follows:

• $$y^s$$ is stored in oo_.dr.ys. The vector rows correspond to all endogenous in the declaration order.

• $$A$$ is stored in oo_.dr.ghx. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to state variables in DR-order, as given by oo_.dr.state_var. (N.B.: if the block option to the model block has been specified, then rows are in declaration order, and columns are ordered according to oo_.dr.state_var which may differ from DR-order.)

• $$B$$ is stored oo_.dr.ghu. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to exogenous variables in declaration order. (N.B.: if the block option to the model block has been specified, then rows are in declaration order.)

Of course, the shown form of the approximation is only stylized, because it neglects the required different ordering in $$y^s$$ and $$y^h_t$$. The precise form of the approximation that shows the way Dynare deals with differences between declaration and DR-order, is

$y_t(\mathrm{oo\_.dr.order\_var}) = y^s(\mathrm{oo\_.dr.order\_var}) + A \cdot y_{t-1}(\mathrm{oo\_.dr.order\_var(k2)}) - y^s(\mathrm{oo\_.dr.order\_var(k2)}) + B\cdot u_t$

where $$\mathrm{k2}$$ selects the state variables, $$y_t$$ and $$y^s$$ are in declaration order and the coefficient matrices are in DR-order. Effectively, all variables on the right hand side are brought into DR order for computations and then assigned to $$y_t$$ in declaration order.

### 4.13.4. Second-order approximation¶

The approximation has the form:

$y_t = y^s + 0.5 \Delta^2 + A y^h_{t-1} + B u_t + 0.5 C (y^h_{t-1}\otimes y^h_{t-1}) + 0.5 D (u_t \otimes u_t) + E (y^h_{t-1} \otimes u_t)$

where $$y^s$$ is the steady state value of $$y$$, $$y^h_t=y_t-y^s$$, and $$\Delta^2$$ is the shift effect of the variance of future shocks. For the reordering required due to differences in declaration and DR order, see the first order approximation.

The coefficients of the decision rules are stored in the variables described for first order approximation, plus the following variables:

• $$\Delta^2$$ is stored in oo_.dr.ghs2. The vector rows correspond to all endogenous in DR-order.

• $$C$$ is stored in oo_.dr.ghxx. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to the Kronecker product of the vector of state variables in DR-order.

• $$D$$ is stored in oo_.dr.ghuu. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to the Kronecker product of exogenous variables in declaration order.

• $$E$$ is stored in oo_.dr.ghxu. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to the Kronecker product of the vector of state variables (in DR-order) by the vector of exogenous variables (in declaration order).

### 4.13.5. Third-order approximation¶

The approximation has the form:

$y_t = y^s + G_0 + G_1 z_t + G_2 (z_t \otimes z_t) + G_3 (z_t \otimes z_t \otimes z_t)$

where $$y^s$$ is the steady state value of $$y$$, and $$z_t$$ is a vector consisting of the deviation from the steady state of the state variables (in DR-order) at date $$t-1$$ followed by the exogenous variables at date $$t$$ (in declaration order). The vector $$z_t$$ is therefore of size $$n_z$$ = M_.nspred + M_.exo_nbr.

The coefficients of the decision rules are stored as follows:

• $$y^s$$ is stored in oo_.dr.ys. The vector rows correspond to all endogenous in the declaration order.

• $$G_0$$ is stored in oo_.dr.g_0. The vector rows correspond to all endogenous in DR-order.

• $$G_1$$ is stored in oo_.dr.g_1. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to state variables in DR-order, followed by exogenous in declaration order.

• $$G_2$$ is stored in oo_.dr.g_2. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to the Kronecker product of state variables (in DR-order), followed by exogenous (in declaration order). Note that the Kronecker product is stored in a folded way, i.e. symmetric elements are stored only once, which implies that the matrix has $$n_z(n_z+1)/2$$ columns. More precisely, each column of this matrix corresponds to a pair $$(i_1, i_2)$$ where each index represents an element of $$z_t$$ and is therefore between $$1$$ and $$n_z$$. Only non-decreasing pairs are stored, i.e. those for which $$i_1 \leq i_2$$. The columns are arranged in the lexicographical order of non-decreasing pairs. Also note that for those pairs where $$i_1 \neq i_2$$, since the element is stored only once but appears two times in the unfolded $$G_2$$ matrix, it must be multiplied by 2 when computing the decision rules.

• $$G_3$$ is stored in oo_.dr.g_3. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to the third Kronecker power of state variables (in DR-order), followed by exogenous (in declaration order). Note that the third Kronecker power is stored in a folded way, i.e. symmetric elements are stored only once, which implies that the matrix has $$n_z(n_z+1)(n_z+2)/6$$ columns. More precisely, each column of this matrix corresponds to a tuple $$(i_1, i_2, i_3)$$ where each index represents an element of $$z_t$$ and is therefore between $$1$$ and $$n_z$$. Only non-decreasing tuples are stored, i.e. those for which $$i_1 \leq i_2 \leq i_3$$. The columns are arranged in the lexicographical order of non-decreasing tuples. Also note that for tuples that have three distinct indices (i.e. $$i_1 \neq i_2$$ and $$i_1 \neq i_3$$ and $$i_2 \neq i_3$$), since these elements are stored only once but appears six times in the unfolded $$G_3$$ matrix, they must be multiplied by 6 when computing the decision rules. Similarly, for those tuples that have two equal indices (i.e. of the form $$(a,a,b)$$ or $$(a,b,a)$$ or $$(b,a,a)$$), since these elements are stored only once but appears three times in the unfolded $$G_3$$ matrix, they must be multiplied by 3 when computing the decision rules.

### 4.13.6. Higher-order approximation¶

Higher-order approximations are simply a generalization of what is done at order 3.

The steady state is stored in oo_.dr.ys and the constant correction is stored in oo_.dr.g_0. The coefficient for orders 1, 2, 3, 4… are respectively stored in oo_.dr.g_0, oo_.dr.g_1, oo_.dr.g_2, oo_.dr.g_3, oo_.dr.g_4… The columns of those matrices correspond to multidimensional indices of state variables, in such a way that symmetric elements are never repeated (for more details, see the description of oo_.dr.g_3 in the third-order case).

## 4.14. Occasionally binding constraints (OCCBIN)¶

Dynare allows simulating models with up to two occasionally-binding constraints by relying on a piecewise linear solution as in Guerrieri and Iacoviello (2015). It also allows estimating such models employing either the inversion filter of Cuba-Borda, Guerrieri, Iacoviello, and Zhong (2019) or the piecewise Kalman filter of Giovannini, Pfeiffer, and Ratto (2021). To trigger computations involving occasionally-binding constraints requires

1. defining and naming the occasionally-binding constraints using an occbin_constraints-block

2. specifying the model equations for the respective regimes in the model-block using appropriate equation tags.

3. potentially specifying a sequence of surprise shocks using a shocks(surprise)-block

4. setting up Occbin simulations or estimation with occbin_setup

5. triggering a simulation with occbin_solver or running estimation or calib_smoother.

All of these elements are discussed in the following.

Block: occbin_constraints ;

The occbin_constraints-block specifies the occasionally-binding constraints. It contains one or two of the following lines:

name ‘STRING’; bind EXPRESSION; [relax EXPRESSION;] [error_bind EXPRESSION;] [error_relax EXPRESSION;]

STRING is the name of constraint that is used to reference the constraint in relax/bind equation-tags to identify the respective regime (see below). The bind-expresssion is mandatory and defines a logical condition that is evaluated in the baseline/steady state regime to check whether the specified constraint becomes binding. In contrast, the relax-expression is optional and specifies a logical condition that is evaluated in the binding regime to check whether the regime returns to the baseline/steady state regime. If not specified, Dynare will simply check in the binding regime whether the bind-expression evaluates to false. However, there are cases where the bind-expression cannot be evaluated in the binding regime(s), because the variables involved are constant by definition so that e.g. the value of the Lagrange multiplier on the complementary slackness condition needs to be checked. In these cases, it is necessary to provide an explicit condition that can be evaluated in the binding regime that allows to check whether it should be left.

Note that the baseline regime denotes the steady state of the model where the economy will settle in the long-run without shocks. For that matter, it may be one where e.g. a borrowing constraint is binding. In that type of setup, the bind-condition is used to specify the condition when this borrowing constraint becomes non-binding so that the alternative regime is entered.

Three things are important to keep in mind when specifying the expressions. First, feasible expressions may only contain contemporaneous endogenous variables. If you want to include leads/lags or exogenous variables, you need to define an auxiliary variable. Second, Dynare will at the current stage not linearly approximate the entered expressions. Because Occbin will work with a linearized model, consistency will often require the user to enter a linearized constraint. Otherwise, the condition employed for checking constraint violations may differ from the one employed within model simulations based on the piecewise-linear model solution. Third, in contrast to the original Occbin replication codes, the variables used in expressions are not automatically demeaned, i.e. they refer to the levels, not deviations from the steady state. To access the steady state level of a variable, the STEADY_STATE()-operator can be used.

Finally, it’s worth keeping in mind that for each simulation period, Occbin will check the respective conditions for whether the current regime should be left. Small numerical differences from the cutoff point for a regime can sometimes lead to oscillations between regimes and cause a spurious periodic solution. Such cases may be prevented by introducing a small buffer between the two regimes, e.g.

occbin_constraints;
name 'ELB'; bind inom <= iss-1e8; relax inom > iss+1e-8;
end;

The error_bind and error_relax-options are optional and allow specifying numerical criteria for the size of the respective constraint violations employed in numerical routines. By default, Dynare will simply use the absolute value of the bind and relax inequalities. But occasionnally, user-specified expressions perform better.

Example

occbin_constraints;
name 'IRR'; bind log_Invest-log(steady_state(Invest))<log(phi); relax Lambda<0;
end;

IRR is a constraint for irreversible investment that becomes binding if investment drops below its steady state by more than 0.025 percent in the non-binding regime. The constraint will be relaxed whenever the associated Lagrange multiplier Lambda in the binding regime becomes negative. Note that the constraint here takes on a linear form to be consistent with a piecewise linear model solution

The specification of the model equations belonging to the respective regimes is done in the model-block, with equation tags indicating to which regime a particular equation belongs. All equations that differ across regimes must have a name-tag attached to them that allows uniquely identifying different versions of the same equation. The name of the constraints specified is then used in conjunction with a bind or relax tag to indicate to which regime a particular equation belongs. In case of more than one occasionally-binding constraint, if an equation belongs to several regimes (e.g. both constraints binding), the constraint name tags must be separated by a comma. If only one name tag is present, the respective equation is assumed to hold for both states of the other constraint.

Example

[name='investment',bind='IRR,INEG']
[name='investment',relax='IRR']
Lambda=0;
[name='investment',bind='IRR',relax='INEG']

The three entered equations for the investment condition define the model equation for all four possible combinations of the two constraints. The first equation defines the model equation in the regime where both the IRR and INEG constraint are binding. The second equation defines the model equation for the regimes where the IRR constraint is non-binding, regardless of whether the INEG constraint is binding or not. Finally, the last equation defines the model equation for the final regime where the IRR constraint is binding, but the INEG one is not.

Block: shocks(surprise) ;
Block: shocks(surprise,overwrite);

The shocks(surprise)-block allows specifying a sequence of temporary changes in the value of exogenous variables that in each period come as a surprise to agents, i.e. are not anticipated. Note that to actually use the specified shocks in subsequent commands like occbin_solver, the block needs to be followed by a call to occbin_setup.

The block mirrors the perfect foresight syntax in that it should contain one or more occurrences of the following group of three lines:

var VARIABLE_NAME;
periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...;
values DOUBLE | (EXPRESSION)  [[,] DOUBLE | (EXPRESSION) ]...;

Example (with vector values and overwrite option)

shockssequence = randn(100,1)*0.02;

shocks(surprise,overwrite);
var epsilon;
periods 1:100;
values (shockssequence);
end;

Command: occbin_setup ;
Command: occbin_setup(OPTIONS...);

Prepares a simulation with occasionally binding constraints. This command will also translate the contents of a shocks(surprise)-block for use in subsequent commands.

In order to conduct estimation with occasionally binding constraints, it needs to be prefaced by a call to occbin_setup to trigger the use of either the inversion filter or the piecewise Kalman filter (default). An issue that can arise in the context of estimation is a structural shock dropping out of the model in a particular regime. For example, at the zero lower bound on interest rates, the monetary policy shock in the Taylor rule will not appear anymore. This may create a problem of stochastic singularity if there are then more observables than shocks. To avoid this issue, the data points for the zero interest rate should be set to NaN and the standard deviation of the associated shock set to 0 for the corresponding periods using the heteroskedastic_shocks-block.

Note that models with unit roots will require the user to specify the diffuse_filter-option as otherwise Blanchard-Kahn errors will be triggered. For the piecewise Kalman filter, the initialization steps in the diffuse filter will always rely on the model solved for the baseline regime, without checking whether this is the actual regime in the first period(s).

Example

occbin_setup(likelihood_inversion_filter,smoother_inversion_filter);
estimation(smoother,heteroskedastic_filter,...);

The above piece of code sets up an estimation employing the inversion filter for both the likelihood evaluation and the smoother, while also accounting for heteroskedastic_shocks using the heteroskedastic_filter-option.

Be aware that Occbin has largely command-specific options, i.e. there are separate options to control the behavior of Occbin when called by the smoother or when computing the likelihood. These latter commands will not inherit the options potentially previously set for simulations.

Options

simul_periods = INTEGER

Number of periods of the simulation. Default: 100.

simul_maxit = INTEGER

Maximum number of iterations when trying to find the regimes of the piecewise solution. Default: 30.

simul_check_ahead_periods = INTEGER

Number of periods for which to check ahead for return to the baseline regime. This number should be chosen large enough, because Occbin requires the simulation to return to the baseline regime at the end of time. Default: 200.

simul_curb_retrench

Instead of basing the initial regime guess for the current iteration on the last iteration, update the guess only one period at a time. This will slow down the iterations, but may lead to more robust convergence behavior. Default: not enabled.

simul_periodic_solution

Accept a periodic solution where the solution alternates between two sets of results across iterations, i.e. is not found to be unique. This is sometimes caused by spurious numerical errors that lead to oscillations between regiems and may be prevented by allowing for a small buffer in regime transitions. Default: not enabled.

simul_debug

Provide additional debugging information during solving. Default: not enabled.

smoother_periods = INTEGER

Number of periods employed during the simulation when called by the smoother (equivalent of simul_periods). Default: 100.

smoother_maxit = INTEGER

Maximum number of iterations employed during the simulation when called by the smoother (equivalent of simul_maxit). Default: 30.

smoother_check_ahead_periods = INTEGER

Number of periods for which to check ahead for return to the baseline regime during the simulation when called by the smoother (equivalent of simul_check_ahead_periods). Default: 200.

smoother_curb_retrench

Have the smoother invoke the simul_curb_retrench-option during simulations. Default: not enabled.

smoother_periodic_solution

Accept periodic solution where solution alternates between two sets of results (equivalent of simul_periodic_solution). Default: not enabled.

likelihood_periods = INTEGER

Number of periods employed during the simulation when computing the likelihood (equivalent of simul_periods). Default: 100.

likelihood_maxit = INTEGER

Maximum number of iterations employed during the simulation when computing the likelihood (equivalent of simul_maxit). Default: 30.

likelihood_check_ahead_periods = INTEGER

Number of periods for which to check ahead for return to the baseline regime during the simulation when computing the likelihood (equivalent of simul_check_ahead_periods). Default: 200.

likelihood_curb_retrench

Have the likelihood computation invoke the simul_curb_retrench-option during simulations. Default: not enabled.

likelihood_periodic_solution

Accept periodic solution where solution alternates between two sets of results (equivalent of simul_periodic_solution). Default: not enabled.

likelihood_inversion_filter

Employ the inversion filter of Cuba-Borda, Guerrieri, Iacoviello, and Zhong (2019) when estimating the model. Default: not enabled.

likelihood_piecewise_kalman_filter

Employ the piecewise Kalman filter of Giovannini, Pfeiffer, and Ratto (2021) when estimating the model. Note that this filter is incompatible with univariate Kalman filters, i.e. kalman_algo=2,4. Default: enabled.

likelihood_max_kalman_iterations

Maximum number of iterations of the outer loop for the piecewise Kalman filter. Default: 10.

smoother_inversion_filter

Employ the inversion filter of Cuba-Borda, Guerrieri, Iacoviello, and Zhong (2019) when running the smoother. The underlying assumption is that the system starts at the steady state. In this case, the inversion filter will provide the required smoother output. Default: not enabled.

smoother_piecewise_kalman_filter

Employ the piecewise Kalman filter of Giovannini, Pfeiffer, and Ratto (2021) when running the smoother. Default: enabled.

filter_use_relaxation

Triggers relaxation within the guess and verify algorithm used in the update step of the piecewise Kalman filter. When old and new guess regime differ to much, use a new guess closer to the previous guess. In case of multiple solutions, tends to provide an occasionally binding regime with a shorter duration (typically preferable). Specifying this option may slow down convergence. Default: not enabled.

Output

The paths for the exogenous variables are stored into options_.occbin.simul.SHOCKS.

Command: occbin_solver ;
Command: occbin_solver(OPTIONS...);

Computes a simulation with occasionally-binding constraints based on a piecewise-linear solution.

Note that occbin_setup must be called before this command in order for the simulation to take into account previous shocks(surprise)-commands.

Options

simul_periods = INTEGER
simul_maxit = INTEGER
simul_check_ahead_periods = INTEGER
simul_curb_retrench
simul_debug

Output

The command outputs various objects into oo_.occbin.

MATLAB/Octave variable: oo_.occbin.simul.piecewise

Matrix storing the simulations based on the piecewise-linear solution. The variables are arranged by column, in order of declaration (as in M_.endo_names), while the the rows correspond to the simul_periods.

MATLAB/Octave variable: oo_.occbin.simul.linear

Matrix storing the simulations based on the linear solution, i.e. ignoring the occasionally binding constraint(s). The variables are arranged column by column, in order of declaration (as in M_.endo_names), while the the rows correspond to the simul_periods.

MATLAB/Octave variable: oo_.occbin.simul.shocks_sequence

Matrix storing the shock sequence employed during the simulation. The shocks are arranged column by column, with their order in M_.exo_names stored in oo_.occbin.exo_pos. The the rows correspond to the number of shock periods specified in a surprise(shocks)-block, which may be smaller than simul_periods.

MATLAB/Octave variable: oo_.occbin.simul.regime_history

Structure storing information on the regime history, conditional on the shock that happened in the respective period (stored along the rows). type is equal to either smoother or simul, depending on whether the output comes from a run of simulations or the smoother. The subfield regime contains a vector storing the regime state, while the the subfield regimestart indicates the expected start of the respective regime state. For example, if row 40 contains [1,0] for regime2 and [1,6] for regimestart2, it indicates that - after the shock in period 40 has occurred - the second constraint became binding (1) and is expected to revert to non-binding (0) after six periods including the current one, i.e. period 45.

MATLAB/Octave variable: oo_.occbin.simul.ys

Command: occbin_graph [VARIABLE_NAME...];
Command: occbin_graph(OPTIONS...) [VARIABLE_NAME...];

Plots a graph comparing the simulation results of the piecewise-linear solution with the occasionally binding contraints to the linear solution ignoring the constraint.

Options

noconstant

Omit the steady state in the graphs.

Command: occbin_write_regimes ;
Command: occbin_write_regimes(OPTIONS...);

Write the information on the regime history stored in oo_.occbin.simul.regime_history or oo_.occbin.smoother.regime_history into an Excel file stored in the FILENAME/Output-folder.

Options

periods = INTEGER

Number of periods for which to write the expected regime durations. Default: write all available periods.

filename = FILENAME

Name of the Excel-file to write. Default: FILENAME_occbin_regimes.

simul

Selects the regime history from the last run of simulations. Default: enabled.

smoother

Selects the regime history from the last run of the smoother. Default: use simul.

## 4.15. Estimation based on likelihood¶

Provided that you have observations on some endogenous variables, it is possible to use Dynare to estimate some or all parameters. Both maximum likelihood (as in Ireland (2004)) and Bayesian techniques (as in Fernández-Villaverde and Rubio-Ramírez (2004), Rabanal and Rubio-Ramirez (2003), Schorfheide (2000) or Smets and Wouters (2003)) are available. Using Bayesian methods, it is possible to estimate DSGE models, VAR models, or a combination of the two techniques called DSGE-VAR.

Note that in order to avoid stochastic singularity, you must have at least as many shocks or measurement errors in your model as you have observed variables.

The estimation using a first order approximation can benefit from the block decomposition of the model (see block).

Command: varobs VARIABLE_NAME...;

This command lists the name of observed endogenous variables for the estimation procedure. These variables must be available in the data file (see estimation_cmd).

Alternatively, this command is also used in conjunction with the partial_information option of stoch_simul, for declaring the set of observed variables when solving the model under partial information.

Only one instance of varobs is allowed in a model file. If one needs to declare observed variables in a loop, the macro processor can be used as shown in the second example below.

Example

varobs C y rr;

Declares endogenous variables C, y and rr as observed variables.

Example (with a macro processor loop)

varobs
@#for co in countries
GDP_@{co}
@#endfor
;

This block specifies linear trends for observed variables as functions of model parameters. In case the loglinear option is used, this corresponds to a linear trend in the logged observables, i.e. an exponential trend in the level of the observables.

Each line inside of the block should be of the form:

VARIABLE_NAME(EXPRESSION);

In most cases, variables shouldn’t be centered when observation_trends is used.

Example

observation_trends;
Y (eta);
P (mu/eta);
end;

Block: estimated_params ;

This block lists all parameters to be estimated and specifies bounds and priors as necessary.

Each line corresponds to an estimated parameter.

In a maximum likelihood or a method of moments estimation, each line follows this syntax:

stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME
, INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND ];

In a Bayesian MCMC or a penalized method of moments estimation, each line follows this syntax:

stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME | DSGE_PRIOR_WEIGHT
[, INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND]], PRIOR_SHAPE,
PRIOR_MEAN, PRIOR_STANDARD_ERROR [, PRIOR_3RD_PARAMETER [,
PRIOR_4TH_PARAMETER [, SCALE_PARAMETER ] ] ];

The first part of the line consists of one of the four following alternatives:

• stderr VARIABLE_NAME

Indicates that the standard error of the exogenous variable VARIABLE_NAME, or of the observation error/measurement errors associated with endogenous observed variable VARIABLE_NAME, is to be estimated.

• corr VARIABLE_NAME1, VARIABLE_NAME2

Indicates that the correlation between the exogenous variables VARIABLE_NAME1 and VARIABLE_NAME2, or the correlation of the observation errors/measurement errors associated with endogenous observed variables VARIABLE_NAME1 and VARIABLE_NAME2, is to be estimated. Note that correlations set by previous shocks-blocks or estimation-commands are kept at their value set prior to estimation if they are not estimated again subsequently. Thus, the treatment is the same as in the case of deep parameters set during model calibration and not estimated.

• PARAMETER_NAME

The name of a model parameter to be estimated

• DSGE_PRIOR_WEIGHT

Special name for the weigh of the DSGE model in DSGE-VAR model.

The rest of the line consists of the following fields, some of them being optional:

INITIAL_VALUE

Specifies a starting value for the posterior mode optimizer or the maximum likelihood estimation. If unset, defaults to the prior mean.

LOWER_BOUND

Specifies a lower bound for the parameter value in maximum likelihood estimation. In a Bayesian estimation context, sets a lower bound only effective while maximizing the posterior kernel. This lower bound does not modify the shape of the prior density, and is only aimed at helping the optimizer in identifying the posterior mode (no consequences for the MCMC). For some prior densities (namely inverse gamma, gamma, uniform, beta or Weibull) it is possible to shift the support of the prior distributions to the left or the right using prior_3rd_parameter. In this case the prior density is effectively modified (note that the truncated Gaussian density is not implemented in Dynare). If unset, defaults to minus infinity (ML) or the natural lower bound of the prior (Bayesian estimation).

UPPER_BOUND

Same as lower_bound, but specifying an upper bound instead.

PRIOR_SHAPE

A keyword specifying the shape of the prior density. The possible values are: beta_pdf, gamma_pdf, normal_pdf, uniform_pdf, inv_gamma_pdf, inv_gamma1_pdf, inv_gamma2_pdf and weibull_pdf. Note that inv_gamma_pdf is equivalent to inv_gamma1_pdf.

PRIOR_MEAN

The mean of the prior distribution.

PRIOR_STANDARD_ERROR

The standard error of the prior distribution.

PRIOR_3RD_PARAMETER

A third parameter of the prior used for generalized beta distribution, generalized gamma, generalized Weibull and for the uniform distribution. Default: 0.

PRIOR_4TH_PARAMETER

A fourth parameter of the prior used for generalized beta distribution and for the uniform distribution. Default: 1.

SCALE_PARAMETER

A parameter specific scale parameter for the jumping distribution’s covariance matrix of the Metropolis-Hasting algorithm.

Note that INITIAL_VALUE, LOWER_BOUND, UPPER_BOUND, PRIOR_MEAN, PRIOR_STANDARD_ERROR, PRIOR_3RD_PARAMETER, PRIOR_4TH_PARAMETER and SCALE_PARAMETER can be any valid EXPRESSION. Some of them can be empty, in which Dynare will select a default value depending on the context and the prior shape.

In case of the uniform distribution, it can be specified either by providing an upper and a lower bound using PRIOR_3RD_PARAMETER and PRIOR_4TH_PARAMETER or via mean and standard deviation using PRIOR_MEAN, PRIOR_STANDARD_ERROR. The other two will automatically be filled out. Note that providing both sets of hyperparameters will yield an error message.

As one uses options more towards the end of the list, all previous options must be filled: for example, if you want to specify SCALE_PARAMETER, you must specify PRIOR_3RD_PARAMETER and PRIOR_4TH_PARAMETER. Use empty values, if these parameters don’t apply.

Example

corr eps_1, eps_2, 0.5,  ,  , beta_pdf, 0, 0.3, -1, 1;

Sets a generalized beta prior for the correlation between eps_1 and eps_2 with mean 0 and variance 0.3. By setting PRIOR_3RD_PARAMETER to -1 and PRIOR_4TH_PARAMETER to 1 the standard beta distribution with support [0,1] is changed to a generalized beta with support [-1,1]. Note that LOWER_BOUND and UPPER_BOUND are left empty and thus default to -1 and 1, respectively. The initial value is set to 0.5.

Example

corr eps_1, eps_2, 0.5,  -0.5,  1, beta_pdf, 0, 0.3, -1, 1;

Sets the same generalized beta distribution as before, but now truncates this distribution to [-0.5,1] through the use of LOWER_BOUND and UPPER_BOUND.

Parameter transformation

Sometimes, it is desirable to estimate a transformation of a parameter appearing in the model, rather than the parameter itself. It is of course possible to replace the original parameter by a function of the estimated parameter everywhere is the model, but it is often unpractical.

In such a case, it is possible to declare the parameter to be estimated in the parameters statement and to define the transformation, using a pound sign (#) expression (see Model declaration).

Example

parameters bet;

model;
# sig = 1/bet;
c = sig*c(+1)*mpk;
end;

estimated_params;
bet, normal_pdf, 1, 0.05;
end;

Block: estimated_params_init ;
Block: estimated_params_init(OPTIONS...);

This block declares numerical initial values for the optimizer when these ones are different from the prior mean. It should be specified after the estimated_params block as otherwise the specified starting values are overwritten by the latter.

Each line has the following syntax:

stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME, INITIAL_VALUE;

Options

use_calibration

For not specifically initialized parameters, use the deep parameters and the elements of the covariance matrix specified in the shocks block from calibration as starting values for estimation. For components of the shocks block that were not explicitly specified during calibration or which violate the prior, the prior mean is used.

See estimated_params, for the meaning and syntax of the various components.

Block: estimated_params_bounds ;

This block declares lower and upper bounds for parameters in maximum likelihood estimation.

Each line has the following syntax:

stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME, LOWER_BOUND, UPPER_BOUND;

See estimated_params, for the meaning and syntax of the various components.

Command: estimation [VARIABLE_NAME...];
Command: estimation(OPTIONS...) [VARIABLE_NAME...];

This command runs Bayesian or maximum likelihood estimation.

The following information will be displayed by the command:

• Results from posterior optimization (also for maximum likelihood)

• Marginal log data density

• Posterior mean and highest posterior density interval (shortest credible set) from posterior simulation

• Convergence diagnostic table when only one MCM chain is used or Metropolis-Hastings convergence graphs documented in Pfeifer (2014) in case of multiple MCM chains

• Table with numerical inefficiency factors of the MCMC

• Graphs with prior, posterior, and mode

• Graphs of smoothed shocks, smoothed observation errors, smoothed and historical variables

Note that the posterior moments, smoothed variables, k-step ahead filtered variables and forecasts (when requested) will only be computed on the variables listed after the estimation command. Alternatively, one can choose to compute these quantities on all endogenous or on all observed variables (see consider_all_endogenous, consider_all_endogenous_and_auxiliary, and consider_only_observed options below). If no variable is listed after the estimation command, then Dynare will interactively ask which variable set to use.

Also, during the MCMC (Bayesian estimation with mh_replic $$>0$$) a (graphical or text) waiting bar is displayed showing the progress of the Monte-Carlo and the current value of the acceptance ratio. Note that if the load_mh_file option is used (see below) the reported acceptance ratio does not take into account the draws from the previous MCMC. In the literature there is a general agreement for saying that the acceptance ratio should be close to one third or one quarter. If this not the case, you can stop the MCMC (Ctrl-C) and change the value of option mh_jscale (see below).

Note that by default Dynare generates random numbers using the algorithm mt199937ar (i.e. Mersenne Twister method) with a seed set equal to 0. Consequently the MCMCs in Dynare are deterministic: one will get exactly the same results across different Dynare runs (ceteris paribus). For instance, the posterior moments or posterior densities will be exactly the same. This behaviour allows to easily identify the consequences of a change on the model, the priors or the estimation options. But one may also want to check that across multiple runs, with different sequences of proposals, the returned results are almost identical. This should be true if the number of iterations (i.e. the value of mh_replic) is important enough to ensure the convergence of the MCMC to its ergodic distribution. In this case the default behaviour of the random number generators in not wanted, and the user should set the seed according to the system clock before the estimation command using the following command:

set_dynare_seed('clock');

so that the sequence of proposals will be different across different runs.

Finally, Dynare does not always properly distinguish between maximum likelihood and Bayesian estimation in its field names. While there is an important conceptual distinction between frequentist confidence intervals and Bayesian highest posterior density intervals (HPDI) as well as between posterior density and likelilhood, Dynare sometimes uses the Bayesian terms as a stand-in in its display of maximum likelihood results. An example is the storage of the output of the forecast-option of estimation with ML, which will use HPDinf/HPDsup to denote the confidence interval.

Algorithms

The Monte Carlo Markov Chain (MCMC) diagnostics are generated by the estimation command if mh_replic is larger than 2000 and if option nodiagnostic is not used. If mh_nblocks is equal to one, the convergence diagnostics of Geweke (1992,1999) is computed. It uses a chi-square test to compare the means of the first and last draws specified by geweke_interval after discarding the burn-in of mh_drop. The test is computed using variance estimates under the assumption of no serial correlation as well as using tapering windows specified in taper_steps. If mh_nblocks is larger than 1, the convergence diagnostics of Brooks and Gelman (1998) are used instead. As described in section 3 of Brooks and Gelman (1998) the univariate convergence diagnostics are based on comparing pooled and within MCMC moments (Dynare displays the second and third order moments, and the length of the Highest Probability Density interval covering 80% of the posterior distribution). Due to computational reasons, the multivariate convergence diagnostic does not follow Brooks and Gelman (1998) strictly, but rather applies their idea for univariate convergence diagnostics to the range of the posterior likelihood function instead of the individual parameters. The posterior kernel is used to aggregate the parameters into a scalar statistic whose convergence is then checked using the Brooks and Gelman (1998) univariate convergence diagnostic.

The inefficiency factors are computed as in Giordano et al.(2011) based on Parzen windows as in e.g. Andrews (1991).

Options

datafile = FILENAME

The datafile: a .m file, a .mat file, a .csv file, or a .xls/.xlsx file (under Octave, the io package from Octave-Forge is required for the .csv and .xlsx formats and the .xls file extension is not supported). Note that the base name (i.e. without extension) of the datafile has to be different from the base name of the model file. If there are several files named FILENAME, but with different file endings, the file name must be included in quoted strings and provide the file ending like:

estimation(datafile='../fsdat_simul.mat',...);

dirname = FILENAME

Directory in which to store estimation output. To pass a subdirectory of a directory, you must quote the argument. Default: <mod_file>.

xls_sheet = QUOTED_STRING

The name of the sheet with the data in an Excel file.

xls_range = RANGE

The range with the data in an Excel file. For example, xls_range=B2:D200.

nobs = INTEGER

The number of observations following first_obs to be used. Default: all observations in the file after first_obs.

nobs = [INTEGER1:INTEGER2]

Runs a recursive estimation and forecast for samples of size ranging of INTEGER1 to INTEGER2. Option forecast must also be specified. The forecasts are stored in the RecursiveForecast field of the results structure (see RecursiveForecast). The respective results structures oo_ are saved in oo_recursive_ (see oo_recursive_) and are indexed with the respective sample length.

first_obs = INTEGER

The number of the first observation to be used. In case of estimating a DSGE-VAR, first_obs needs to be larger than the number of lags. Default: 1.

first_obs = [INTEGER1:INTEGER2]

Runs a rolling window estimation and forecast for samples of fixed size nobs starting with the first observation ranging from INTEGER1 to INTEGER2. Option forecast must also be specified. This option is incompatible with requesting recursive forecasts using an expanding window (see nobs). The respective results structures oo_ are saved in oo_recursive_ (see oo_recursive_) and are indexed with the respective first observation of the rolling window.

prefilter = INTEGER

A value of 1 means that the estimation procedure will demean each data series by its empirical mean. If the loglinear option without the logdata option is requested, the data will first be logged and then demeaned. Default: 0, i.e. no prefiltering.

presample = INTEGER

The number of observations after first_obs to be skipped before evaluating the likelihood. These presample observations do not enter the likelihood, but are used as a training sample for starting the Kalman filter iterations. This option is incompatible with estimating a DSGE-VAR. Default: 0.

loglinear

Computes a log-linear approximation of the model instead of a linear approximation. As always in the context of estimation, the data must correspond to the definition of the variables used in the model (see Pfeifer (2013) for more details on how to correctly specify observation equations linking model variables and the data). If you specify the loglinear option, Dynare will take the logarithm of both your model variables and of your data as it assumes the data to correspond to the original non-logged model variables. The displayed posterior results like impulse responses, smoothed variables, and moments will be for the logged variables, not the original un-logged ones. Default: computes a linear approximation.

logdata

Dynare applies the $$log$$ transformation to the provided data if a log-linearization of the model is requested (loglinear) unless logdata option is used. This option is necessary if the user provides data already in logs, otherwise the $$log$$ transformation will be applied twice (this may result in complex data).

plot_priors = INTEGER

Control the plotting of priors.

0

No prior plot.

1

Prior density for each estimated parameter is plotted. It is important to check that the actual shape of prior densities matches what you have in mind. Ill-chosen values for the prior standard density can result in absurd prior densities.

Default value is 1.

nograph
posterior_nograph

Suppresses the generation of graphs associated with Bayesian IRFs (bayesian_irf), posterior smoothed objects (smoother), and posterior forecasts (forecast).

posterior_graph

Re-enables the generation of graphs previously shut off with posterior_nograph.

nodisplay
graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
no_init_estimation_check_first_obs

Do not check for stochastic singularity in first period. If used, ESTIMATION CHECKS does not return an error if the check fails only in first observation. This should only be used when observing stock variables (e.g. capital) in first period, on top of their associated flow (e.g. investment). Using this option may lead to a crash or provide undesired/wrong results for badly specified problems (e.g. the additional variable observed in first period is not predetermined).

lik_init = INTEGER

Type of initialization of Kalman filter:

1

For stationary models, the initial matrix of variance of the error of forecast is set equal to the unconditional variance of the state variables.

2

For nonstationary models: a wide prior is used with an initial matrix of variance of the error of forecast diagonal with 10 on the diagonal (follows the suggestion of Harvey and Phillips(1979)).

3

For nonstationary models: use a diffuse filter (use rather the diffuse_filter option).

4

The filter is initialized with the fixed point of the Riccati equation.

5

Use i) option 2 for the non-stationary elements by setting their initial variance in the forecast error matrix to 10 on the diagonal and all covariances to 0 and ii) option 1 for the stationary elements.

Default value is 1. For advanced use only.

lik_algo = INTEGER

For internal use and testing only.

conf_sig = DOUBLE

Level of significance of the confidence interval used for classical forecasting after estimation. Default: 0.9.

mh_conf_sig = DOUBLE

Confidence/HPD interval used for the computation of prior and posterior statistics like: parameter distributions, prior/posterior moments, conditional variance decomposition, impulse response functions, Bayesian forecasting. Default: 0.9.

mh_replic = INTEGER

Number of replications for each chain of the Metropolis-Hastings algorithm. The number of draws should be sufficient to achieve convergence of the MCMC and to meaningfully compute posterior objects. Default: 20000.

sub_draws = INTEGER

Number of draws from the MCMC that are used to compute posterior distribution of various objects (smoothed variable, smoothed shocks, forecast, moments, IRF). The draws used to compute these posterior moments are sampled uniformly in the estimated empirical posterior distribution (i.e. draws of the MCMC). sub_draws should be smaller than the total number of MCMC draws available. Default: min(posterior_max_subsample_draws, (Total number of draws)*(number of chains) ).

posterior_max_subsample_draws = INTEGER

Maximum number of draws from the MCMC used to compute posterior distribution of various objects (smoothed variable, smoothed shocks, forecast, moments, IRF), if not overriden by option sub_draws. Default: 1200.

mh_nblocks = INTEGER

Number of parallel chains for Metropolis-Hastings algorithm. Default: 2.

mh_drop = DOUBLE

The fraction of initially generated parameter vectors to be dropped as a burn-in before using posterior simulations. Default: 0.5.

mh_jscale = DOUBLE

The scale parameter of the jumping distribution’s covariance matrix (Metropolis-Hastings or TaRB-algorithm). The default value is rarely satisfactory. This option must be tuned to obtain, ideally, an acceptance ratio of 25%-33%. Basically, the idea is to increase the variance of the jumping distribution if the acceptance ratio is too high, and decrease the same variance if the acceptance ratio is too low. In some situations it may help to consider parameter-specific values for this scale parameter. This can be done in the estimated_params block.

Note that mode_compute=6 will tune the scale parameter to achieve an acceptance rate of AcceptanceRateTarget. The resulting scale parameter will be saved into a file named MODEL_FILENAME_mh_scale.mat in the FILENAME/Output-folder. This file can be loaded in subsequent runs via the posterior_sampler_options option scale_file. Both mode_compute=6 and scale_file will overwrite any value specified in estimated_params with the tuned value. Default: 0.2.

Note also that for the Random Walk Metropolis Hastings algorithm, it is possible to use option mh_tune_jscale, to automatically tune the value of mh_jscale. In this case, the mh_jscale option must not be used.

mh_init_scale = DOUBLE

The scale to be used for drawing the initial value of the Metropolis-Hastings chain. Generally, the starting points should be overdispersed for the Brooks and Gelman (1998) convergence diagnostics to be meaningful. Default: 2*mh_jscale.

It is important to keep in mind that mh_init_scale is set at the beginning of Dynare execution, i.e. the default will not take into account potential changes in mh_jscale introduced by either mode_compute=6 or the posterior_sampler_options option scale_file. If mh_init_scale is too wide during initalization of the posterior sampler so that 100 tested draws are inadmissible (e.g. Blanchard-Kahn conditions are always violated), Dynare will request user input of a new mh_init_scale value with which the next 100 draws will be drawn and tested. If the nointeractive option has been invoked, the program will instead automatically decrease mh_init_scale by 10 percent after 100 futile draws and try another 100 draws. This iterative procedure will take place at most 10 times, at which point Dynare will abort with an error message.

mh_tune_jscale [= DOUBLE]

Automatically tunes the scale parameter of the jumping distribution’s covariance matrix (Metropolis-Hastings), so that the overall acceptance ratio is close to the desired level. Default value is 0.33. It is not possible to match exactly the desired acceptance ratio because of the stochastic nature of the algorithm (the proposals and the initial conditions of the markov chains if mh_nblocks>1). This option is only available for the Random Walk Metropolis Hastings algorithm. Must not be used in conjunction with mh_jscale = DOUBLE.

mh_tune_guess = DOUBLE

Specifies the initial value for the mh_tune_jscale option. Default: 0.2. Must not be set if mh_tune_jscale is not used.

mh_recover

Attempts to recover a Metropolis-Hastings simulation that crashed prematurely, starting with the last available saved mh-file. Shouldn’t be used together with load_mh_file or a different mh_replic than in the crashed run. Since Dynare 4.5 the proposal density from the previous run will automatically be loaded. In older versions, to assure a neat continuation of the chain with the same proposal density, you should provide the mode_file used in the previous run or the same user-defined mcmc_jumping_covariance when using this option. Note that under Octave, a neat continuation of the crashed chain with the respective last random number generator state is currently not supported.

mh_posterior_mode_estimation

Skip optimizer-based mode-finding and instead compute the mode based on a run of a MCMC. The MCMC will start at the prior mode and use the prior variances to compute the inverse Hessian.

mode_file = FILENAME

Name of the file containing previous value for the mode. When computing the mode, Dynare stores the mode (xparam1) and the hessian (hh, only if cova_compute=1) in a file called MODEL_FILENAME_mode.mat in the FILENAME/Output-folder. After a successful run of the estimation command, the mode_file will be disabled to prevent other function calls from implicitly using an updated mode-file. Thus, if the .mod file contains subsequent estimation commands, the mode_file option, if desired, needs to be specified again.

mode_compute = INTEGER | FUNCTION_NAME

Specifies the optimizer for the mode computation:

0

The mode isn’t computed. When the mode_file option is specified, the mode is simply read from that file.

When mode_file option is not specified, Dynare reports the value of the log posterior (log likelihood) evaluated at the initial value of the parameters.

When mode_file is not specified and there is no estimated_params block, but the smoother option is used, it is a roundabout way to compute the smoothed value of the variables of a model with calibrated parameters.

1

Uses fmincon optimization routine (available under MATLAB if the Optimization Toolbox is installed; available under Octave if the optim package from Octave-Forge, version 1.6 or above, is installed).

2

Uses the continuous simulated annealing global optimization algorithm described in Corana et al.(1987) and Goffe et al.(1994).

3

Uses fminunc optimization routine (available under MATLAB if the Optimization Toolbox is installed; available under Octave if the optim package from Octave-Forge is installed).

4

Uses Chris Sims’s csminwel.

5

Uses Marco Ratto’s newrat. This value is not compatible with non linear filters or DSGE-VAR models. This is a slice optimizer: most iterations are a sequence of univariate optimization step, one for each estimated parameter or shock. Uses csminwel for line search in each step.

6

Uses a Monte-Carlo based optimization routine (see https://archives.dynare.org/DynareWiki/MonteCarloOptimization for more details).

7

Uses fminsearch, a simplex-based optimization routine (available under MATLAB if the Optimization Toolbox is installed; available under Octave if the optim package from Octave-Forge is installed).

8

Uses Dynare implementation of the Nelder-Mead simplex-based optimization routine (generally more efficient than the MATLAB or Octave implementation available with mode_compute=7).

9

Uses the CMA-ES (Covariance Matrix Adaptation Evolution Strategy) algorithm of Hansen and Kern (2004), an evolutionary algorithm for difficult non-linear non-convex optimization.

10

Uses the simpsa algorithm, based on the combination of the non-linear simplex and simulated annealing algorithms as proposed by Cardoso, Salcedo and Feyo de Azevedo (1996).

11

This is not strictly speaking an optimization algorithm. The (estimated) parameters are treated as state variables and estimated jointly with the original state variables of the model using a nonlinear filter. The algorithm implemented in Dynare is described in Liu and West (2001), and works with k order local approximations of the model.

12

Uses the particleswarm optimization routine (available under MATLAB if the Global Optimization Toolbox is installed; not available under Octave).

13

Uses the lsqnonlin non-linear least squares optimization routine (available under MATLAB if the Optimization Toolbox is installed; available under Octave if the optim package from Octave-Forge is installed). Only supported for method_of_moments.

101

Uses the SolveOpt algorithm for local nonlinear optimization problems proposed by Kuntsevich and Kappel (1997).

102

Uses simulannealbnd optimization routine (available under MATLAB if the Global Optimization Toolbox is installed; not available under Octave)

FUNCTION_NAME

It is also possible to give a FUNCTION_NAME to this option, instead of an INTEGER. In that case, Dynare takes the return value of that function as the posterior mode.

Default value is 4.

silent_optimizer

Instructs Dynare to run mode computing/optimization silently without displaying results or saving files in between. Useful when running loops.

mcmc_jumping_covariance = OPTION

Tells Dynare which covariance to use for the proposal density of the MCMC sampler. OPTION can be one of the following:

hessian

Uses the Hessian matrix computed at the mode.

prior_variance

Uses the prior variances. No infinite prior variances are allowed in this case.

identity_matrix

Uses an identity matrix.

FILENAME

Loads an arbitrary user-specified covariance matrix from FILENAME.mat. The covariance matrix must be saved in a variable named jumping_covariance, must be square, positive definite, and have the same dimension as the number of estimated parameters.

Note that the covariance matrices are still scaled with mh_jscale. Default value is hessian.

mode_check

Tells Dynare to plot the posterior density for values around the computed mode for each estimated parameter in turn. This is helpful to diagnose problems with the optimizer. Note that for order>1 the likelihood function resulting from the particle filter is not differentiable anymore due to the resampling step. For this reason, the mode_check plot may look wiggly.

mode_check_neighbourhood_size = DOUBLE

Used in conjunction with option mode_check, gives the width of the window around the posterior mode to be displayed on the diagnostic plots. This width is expressed in percentage deviation. The Inf value is allowed, and will trigger a plot over the entire domain (see also mode_check_symmetric_plots). Default:0.5.

mode_check_symmetric_plots = INTEGER

Used in conjunction with option mode_check, if set to 1, tells Dynare to ensure that the check plots are symmetric around the posterior mode. A value of 0 allows to have asymmetric plots, which can be useful if the posterior mode is close to a domain boundary, or in conjunction with mode_check_neighbourhood_size = Inf when the domain in not the entire real line. Default: 1.

mode_check_number_of_points = INTEGER

Number of points around the posterior mode where the posterior kernel is evaluated (for each parameter). Default is 20.

prior_trunc = DOUBLE

Probability of extreme values of the prior density that is ignored when computing bounds for the parameters. Default: 1e-32.

huge_number = DOUBLE

Value for replacing infinite values in the definition of (prior) bounds when finite values are required for computational reasons. Default: 1e7.

load_mh_file

Tells Dynare to add to previous Metropolis-Hastings simulations instead of starting from scratch. Since Dynare 4.5 the proposal density from the previous run will automatically be loaded. In older versions, to assure a neat continuation of the chain with the same proposal density, you should provide the mode_file used in the previous run or the same user-defined mcmc_jumping_covariance when using this option. Shouldn’t be used together with mh_recover. Note that under Octave, a neat continuation of the chain with the last random number generator state of the already present draws is currently not supported.

load_results_after_load_mh

This option is available when loading a previous MCMC run without adding additional draws, i.e. when load_mh_file is specified with mh_replic=0. It tells Dynare to load the previously computed convergence diagnostics, marginal data density, and posterior statistics from an existing _results file instead of recomputing them.

mh_initialize_from_previous_mcmc

This option allows to pick initial values for new MCMC from a previous one, where the model specification, the number of estimated parameters, (some) prior might have changed (so a situation where load_mh_file would not work). If an additional parameter is estimated, it is automatically initialized from prior_draw. Note that, if this option is used to skip the optimization step, you should use a sampling method which does not require a proposal density, like slice. Otherwise, optimization should always be done beforehand or a mode file with an appropriate posterior covariance matrix should be used.

mh_initialize_from_previous_mcmc_directory = FILENAME

If mh_initialize_from_previous_mcmc is set, users must provide here the path to the standard FNAME folder from where to load prior definitions and last MCMC values to be used to initialize the new MCMC.

Example: if previous project directory is /my_previous_dir and FNAME is mymodel, users should set the option as

mh_initialize_from_previous_mcmc_directory = '/my_previous_dir/mymodel'

Dynare will then look for the last record file into

/my_previous_dir/mymodel/metropolis/mymodel_mh_history_<LAST>.mat

and for the prior definition file into

/my_previous_dir/mymodel/prior/definition.mat

mh_initialize_from_previous_mcmc_record = FILENAME

If mh_initialize_from_previous_mcmc is set, and whenever the standard file or directory tree is not applicable to load initial values, users may directly provide here the path to the record file from which to load values to be used to initialize the new MCMC.

mh_initialize_from_previous_mcmc_prior = FILENAME

If mh_initialize_from_previous_mcmc is set, and whenever the standard file or directory tree is not applicable to load initial values, users may directly provide here the path to the prior definition file, to get info in the priors used in previous MCMC.

optim = (NAME, VALUE, ...)

A list of NAME and VALUE pairs. Can be used to set options for the optimization routines. The set of available options depends on the selected optimization routine (i.e. on the value of option mode_compute):

1, 3, 7, 12, 13

Available options are given in the documentation of the MATLAB Optimization Toolbox or in Octave’s documentation.

2

Available options are:

'initial_step_length'

Initial step length. Default: 1.

'initial_temperature'

Initial temperature. Default: 15.

'MaxIter'

Maximum number of function evaluations. Default: 100000.

'neps'

Number of final function values used to decide upon termination. Default: 10.

'ns'

Number of cycles. Default: 10.

'nt'

Number of iterations before temperature reduction. Default: 10.

'step_length_c'

Step length adjustment. Default: 0.1.

'TolFun'

Stopping criteria. Default: 1e-8.

'rt'

Temperature reduction factor. Default: 0.1.

'verbosity'

Controls verbosity of display during optimization, ranging from 0 (silent) to 3 (each function evaluation). Default: 1

4

Available options are:

'InitialInverseHessian'

Initial approximation for the inverse of the Hessian matrix of the posterior kernel (or likelihood). Obviously this approximation has to be a square, positive definite and symmetric matrix. Default: '1e-4*eye(nx)', where nx is the number of parameters to be estimated.

'MaxIter'

Maximum number of iterations. Default: 1000.

'NumgradAlgorithm'

Possible values are 2, 3 and 5, respectively, corresponding to the two, three and five points formula used to compute the gradient of the objective function (see Abramowitz and Stegun (1964)). Values 13 and 15 are more experimental. If perturbations on the right and the left increase the value of the objective function (we minimize this function) then we force the corresponding element of the gradient to be zero. The idea is to temporarily reduce the size of the optimization problem. Default: 2.

'NumgradEpsilon'

Size of the perturbation used to compute numerically the gradient of the objective function. Default: 1e-6.

'TolFun'

Stopping criteria. Default: 1e-7.

'verbosity'

Controls verbosity of display during optimization. Set to 0 to set to silent. Default: 1.

'SaveFiles'

Controls saving of intermediate results during optimization. Set to 0 to shut off saving. Default: 1.

5

Available options are:

'Hessian'

Triggers three types of Hessian computations. 0: outer product gradient; 1: default Dynare Hessian routine; 2: ’mixed’ outer product gradient, where diagonal elements are obtained using second order derivation formula and outer product is used for correlation structure. Both {0} and {2} options require univariate filters, to ensure using maximum number of individual densities and a positive definite Hessian. Both {0} and {2} are quicker than default Dynare numeric Hessian, but provide decent starting values for Metropolis for large models (option {2} being more accurate than {0}). Default: 1.

'MaxIter'

Maximum number of iterations. Default: 1000.

'TolFun'

Stopping criteria. Default: 1e-5 for numerical derivatives, 1e-7 for analytic derivatives.

'verbosity'

Controls verbosity of display during optimization. Set to 0 to set to silent. Default: 1.

'SaveFiles'

Controls saving of intermediate results during optimization. Set to 0 to shut off saving. Default: 1.

6

Available options are:

'AcceptanceRateTarget'

A real number between zero and one. The scale parameter of the jumping distribution is adjusted so that the effective acceptance rate matches the value of option 'AcceptanceRateTarget'. Default: 1.0/3.0.

'InitialCovarianceMatrix'

Initial covariance matrix of the jumping distribution. Default is 'previous' if option mode_file is used, 'prior' otherwise.

'nclimb-mh'

Number of iterations in the last MCMC (climbing mode). Default: 200000.

'ncov-mh'

Number of iterations used for updating the covariance matrix of the jumping distribution. Default: 20000.

'nscale-mh'

Maximum number of iterations used for adjusting the scale parameter of the jumping distribution. Default: 200000.

'NumberOfMh'

Number of MCMC run sequentially. Default: 3.

8

Available options are:

'InitialSimplexSize'

Initial size of the simplex, expressed as percentage deviation from the provided initial guess in each direction. Default: .05.

'MaxIter'

Maximum number of iterations. Default: 5000.

'MaxFunEvals'

Maximum number of objective function evaluations. No default.

'MaxFunvEvalFactor'

Set MaxFunvEvals equal to MaxFunvEvalFactor times the number of estimated parameters. Default: 500.

'TolFun'

Tolerance parameter (w.r.t the objective function). Default: 1e-4.

'TolX'

Tolerance parameter (w.r.t the instruments). Default: 1e-4.

'verbosity'

Controls verbosity of display during optimization. Set to 0 to set to silent. Default: 1.

9

Available options are:

'CMAESResume'

Resume previous run. Requires the variablescmaes.mat from the last run. Set to 1 to enable. Default: 0.

'MaxIter'

Maximum number of iterations.

'MaxFunEvals'

Maximum number of objective function evaluations. Default: Inf.

'TolFun'

Tolerance parameter (w.r.t the objective function). Default: 1e-7.

'TolX'

Tolerance parameter (w.r.t the instruments). Default: 1e-7.

'verbosity'

Controls verbosity of display during optimization. Set to 0 to set to silent. Default: 1.

'SaveFiles'

Controls saving of intermediate results during optimization. Set to 0 to shut off saving. Default: 1.

10

Available options are:

'EndTemperature'

Terminal condition w.r.t the temperature. When the temperature reaches EndTemperature, the temperature is set to zero and the algorithm falls back into a standard simplex algorithm. Default: 0.1.

'MaxIter'

Maximum number of iterations. Default: 5000.

'MaxFunvEvals'

Maximum number of objective function evaluations. No default.

'TolFun'

Tolerance parameter (w.r.t the objective function). Default: 1e-4.

'TolX'

Tolerance parameter (w.r.t the instruments). Default: 1e-4.

'verbosity'

Controls verbosity of display during optimization. Set to 0 to set to silent. Default: 1.

101

Available options are:

'LBGradientStep'

Lower bound for the stepsize used for the difference approximation of gradients. Default: 1e-11.

'MaxIter'

Maximum number of iterations. Default: 15000

'SpaceDilation'

Coefficient of space dilation. Default: 2.5.

'TolFun'

Tolerance parameter (w.r.t the objective function). Default: 1e-6.

'TolX'

Tolerance parameter (w.r.t the instruments). Default: 1e-6.

'verbosity'

Controls verbosity of display during optimization. Set to 0 to set to silent. Default: 1.

102

Available options are given in the documentation of the MATLAB Global Optimization Toolbox.

Example

To change the defaults of csminwel (mode_compute=4):

estimation(..., mode_compute=4,optim=('NumgradAlgorithm',3,'TolFun',1e-5),...);

nodiagnostic

Does not compute the convergence diagnostics for Metropolis-Hastings. Default: diagnostics are computed and displayed.

bayesian_irf

Triggers the computation of the posterior distribution of IRFs. The length of the IRFs are controlled by the irf option. Results are stored in oo_.PosteriorIRF.dsge (see below for a description of this variable).

relative_irf
dsge_var = DOUBLE

Triggers the estimation of a DSGE-VAR model, where the weight of the DSGE prior of the VAR model is calibrated to the value passed (see Del Negro and Schorfheide (2004)). It represents the ratio of dummy over actual observations. To assure that the prior is proper, the value must be bigger than $$(k+n)/T$$, where $$k$$ is the number of estimated parameters, $$n$$ is the number of observables, and $$T$$ is the number of observations.

NB: The previous method of declaring dsge_prior_weight as a parameter and then calibrating it is now deprecated and will be removed in a future release of Dynare. Some of objects arising during estimation are stored with their values at the mode in oo_.dsge_var.posterior_mode.

dsge_var

Triggers the estimation of a DSGE-VAR model, where the weight of the DSGE prior of the VAR model will be estimated (as in Adjemian et al.(2008)). The prior on the weight of the DSGE prior, dsge_prior_weight, must be defined in the estimated_params section.

NB: The previous method of declaring dsge_prior_weight as a parameter and then placing it in estimated_params is now deprecated and will be removed in a future release of Dynare.

dsge_varlag = INTEGER

The number of lags used to estimate a DSGE-VAR model. Default: 4.

posterior_sampling_method = NAME

Selects the sampler used to sample from the posterior distribution during Bayesian estimation. Default:’random_walk_metropolis_hastings’.

'random_walk_metropolis_hastings'

Instructs Dynare to use the Random-Walk Metropolis-Hastings. In this algorithm, the proposal density is recentered to the previous draw in every step.

'tailored_random_block_metropolis_hastings'

Instructs Dynare to use the Tailored randomized block (TaRB) Metropolis-Hastings algorithm proposed by Chib and Ramamurthy (2010) instead of the standard Random-Walk Metropolis-Hastings. In this algorithm, at each iteration the estimated parameters are randomly assigned to different blocks. For each of these blocks a mode-finding step is conducted. The inverse Hessian at this mode is then used as the covariance of the proposal density for a Random-Walk Metropolis-Hastings step. If the numerical Hessian is not positive definite, the generalized Cholesky decomposition of Schnabel and Eskow (1990) is used, but without pivoting. The TaRB-MH algorithm massively reduces the autocorrelation in the MH draws and thus reduces the number of draws required to representatively sample from the posterior. However, this comes at a computational cost as the algorithm takes more time to run.

'independent_metropolis_hastings'

Use the Independent Metropolis-Hastings algorithm where the proposal distribution - in contrast to the Random Walk Metropolis-Hastings algorithm - does not depend on the state of the chain.

'slice'

Instructs Dynare to use the Slice sampler of Planas, Ratto, and Rossi (2015). Note that 'slice' is incompatible with prior_trunc=0.

posterior_sampler_options = (NAME, VALUE, ...)

A list of NAME and VALUE pairs. Can be used to set options for the posterior sampling methods. The set of available options depends on the selected posterior sampling routine (i.e. on the value of option posterior_sampling_method):

'random_walk_metropolis_hastings'

Available options are:

'proposal_distribution'

Specifies the statistical distribution used for the proposal density.

'rand_multivariate_normal'

Use a multivariate normal distribution. This is the default.

'rand_multivariate_student'

Use a multivariate student distribution.

'student_degrees_of_freedom'

Specifies the degrees of freedom to be used with the multivariate student distribution. Default: 3.

'use_mh_covariance_matrix'

Indicates to use the covariance matrix of the draws from a previous MCMC run to define the covariance of the proposal distribution. Requires the load_mh_file option to be specified. Default: 0.

'scale_file'

Provides the name of a _mh_scale.mat file storing the tuned scale factor from a previous run of mode_compute=6.

'save_tmp_file'

Save the MCMC draws into a _mh_tmp_blck file at the refresh rate of the status bar instead of just saving the draws when the current _mh*_blck file is full. Default: 0

'independent_metropolis_hastings'

Takes the same options as in the case of random_walk_metropolis_hastings.

'slice'

'rotated'

Triggers rotated slice iterations using a covariance matrix from initial burn-in iterations. Requires either use_mh_covariance_matrix or slice_initialize_with_mode. Default: 0.

'mode_files'

For multimodal posteriors, provide the name of a file containing a nparam by nmodes variable called xparams storing the different modes. This array must have one column vector per mode and the estimated parameters along the row dimension. With this info, the code will automatically trigger the rotated and mode options. Default: [].

'slice_initialize_with_mode'

The default for slice is to set mode_compute=0 and start the chain(s) from a random location in the prior space. This option first runs the mode-finder and then starts the chain from the mode. Together with rotated, it will use the inverse Hessian from the mode to perform rotated slice iterations. Default: 0.

'initial_step_size'

Sets the initial size of the interval in the stepping-out procedure as fraction of the prior support, i.e. the size will be initial_step_size * (UB-LB). initial_step_size must be a real number in the interval [0,1]. Default: 0.8.

'use_mh_covariance_matrix'

See use_mh_covariance_matrix. Must be used with 'rotated'. Default: 0.

'save_tmp_file'

See save_tmp_file. Default: 1.

'tailored_random_block_metropolis_hastings'

'proposal_distribution'

Specifies the statistical distribution used for the proposal density. See proposal_distribution.

new_block_probability = DOUBLE

Specifies the probability of the next parameter belonging to a new block when the random blocking in the TaRB Metropolis-Hastings algorithm is conducted. The higher this number, the smaller is the average block size and the more random blocks are formed during each parameter sweep. Default: 0.25.

mode_compute = INTEGER

Specifies the mode-finder run in every iteration for every block of the TaRB Metropolis-Hastings algorithm. See mode_compute. Default: 4.

optim = (NAME, VALUE,...)

Specifies the options for the mode-finder used in the TaRB Metropolis-Hastings algorithm. See optim.

'scale_file'

See scale_file..

'save_tmp_file'

See save_tmp_file. Default: 1.

moments_varendo

Triggers the computation of the posterior distribution of the theoretical moments of the endogenous variables. Results are stored in oo_.PosteriorTheoreticalMoments (see oo_.PosteriorTheoreticalMoments). The number of lags in the autocorrelation function is controlled by the ar option.

contemporaneous_correlation

See contemporaneous_correlation. Results are stored in oo_.PosteriorTheoreticalMoments. Note that the nocorr option has no effect.

no_posterior_kernel_density

Shuts off the computation of the kernel density estimator for the posterior objects (see density field).

conditional_variance_decomposition = INTEGER
conditional_variance_decomposition = [INTEGER1:INTEGER2]
conditional_variance_decomposition = [INTEGER1 INTEGER2 ...]

Computes the posterior distribution of the conditional variance decomposition for the specified period(s). The periods must be strictly positive. Conditional variances are given by $$var(y_{t+k}\vert t)$$. For period 1, the conditional variance decomposition provides the decomposition of the effects of shocks upon impact. The results are stored in oo_.PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecomposition.. Note that this option requires the option moments_varendo to be specified. In the presence of measurement error, the field will contain the variance contribution after measurement error has been taken out, i.e. the decomposition will be conducted of the actual as opposed to the measured variables. The variance decomposition of the measured variables will be stored in oo_.PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecompositionME.

filtered_vars

Triggers the computation of the posterior distribution of filtered endogenous variables/one-step ahead forecasts, i.e. $$E_{t}{y_{t+1}}$$. Results are stored in oo_.FilteredVariables (see below for a description of this variable)

smoother

Triggers the computation of the posterior distribution of smoothed endogenous variables and shocks, i.e. the expected value of variables and shocks given the information available in all observations up to the final date ($$E_{T}{y_t}$$). Results are stored in oo_.SmoothedVariables, oo_.SmoothedShocks and oo_.SmoothedMeasurementErrors. Also triggers the computation of oo_.UpdatedVariables, which contains the estimation of the expected value of variables given the information available at the current date ($$E_{t}{y_t}$$). See below for a description of all these variables.

smoother_redux

Triggers a faster computation of the smoothed endogenous variables and shocks for large models. It runs the smoother only for the state variables (i.e. with the same representation used for likelihood computations) and computes the remaining variables ex-post. Static unobserved objects (filtered, smoothed, updated, k-step ahead) are recovered, but there are exceptions to a full recovery, depending on how static unobserved variables depend on the restricted state space adopted. For example, lagged shocks which are ONLY used to recover NON-observed static variables will not be recovered). For such exceptions, only the following output is provided:

FilteredVariablesKStepAhead: will be fully recovered

SmoothedVariables, FilteredVariables, UpdatedVariables: recovered for all periods beyond period d+1,

where d denotes the number of diffuse filtering steps.

FilteredVariablesKStepAheadVariances, Variance, and State_uncertainty cannot be recovered, and ZERO is provided as output.

If you need variances for those variables, either do not set the option, or declare the variable as observed, using NaNs as data points.

forecast = INTEGER

Computes the posterior distribution of a forecast on INTEGER periods after the end of the sample used in estimation. If no Metropolis-Hastings is computed, the result is stored in variable oo_.forecast and corresponds to the forecast at the posterior mode. If a Metropolis-Hastings is computed, the distribution of forecasts is stored in variables oo_.PointForecast and oo_.MeanForecast. See Forecasting, for a description of these variables.

tex

See tex.

kalman_algo = INTEGER

0

Automatically use the Multivariate Kalman Filter for stationary models and the Multivariate Diffuse Kalman Filter for non-stationary models.

1

Use the Multivariate Kalman Filter.

2

Use the Univariate Kalman Filter.

3

Use the Multivariate Diffuse Kalman Filter.

4

Use the Univariate Diffuse Kalman Filter.

Default value is 0. In case of missing observations of single or all series, Dynare treats those missing values as unobserved states and uses the Kalman filter to infer their value (see e.g. Durbin and Koopman (2012), Ch. 4.10) This procedure has the advantage of being capable of dealing with observations where the forecast error variance matrix becomes singular for some variable(s). If this happens, the respective observation enters with a weight of zero in the log-likelihood, i.e. this observation for the respective variable(s) is dropped from the likelihood computations (for details see Durbin and Koopman (2012), Ch. 6.4 and 7.2.5 and Koopman and Durbin (2000)). If the use of a multivariate Kalman filter is specified and a singularity is encountered, Dynare by default automatically switches to the univariate Kalman filter for this parameter draw. This behavior can be changed via the use_univariate_filters_if_singularity_is_detected option.

fast_kalman_filter

Select the fast Kalman filter using Chandrasekhar recursions as described by Herbst (2015). This setting is only used with kalman_algo=1 or kalman_algo=3. In case of using the diffuse Kalman filter (kalman_algo=3/lik_init=3), the observables must be stationary. This option is not yet compatible with analytic_derivation.

kalman_tol = DOUBLE

Numerical tolerance for determining the singularity of the covariance matrix of the prediction errors during the Kalman filter (minimum allowed reciprocal of the matrix condition number). Default value is 1e-10.

diffuse_kalman_tol = DOUBLE

Numerical tolerance for determining the singularity of the covariance matrix of the prediction errors ($$F_{\infty}$$) and the rank of the covariance matrix of the non-stationary state variables ($$P_{\infty}$$) during the Diffuse Kalman filter. Default value is 1e-6.

filter_covariance

Saves the series of one step ahead error of forecast covariance matrices. With Metropolis, they are saved in oo_.FilterCovariance, otherwise in oo_.Smoother.Variance. Saves also k-step ahead error of forecast covariance matrices if filter_step_ahead is set.

filter_step_ahead = [INTEGER1:INTEGER2]
filter_step_ahead = [INTEGER1 INTEGER2 ...]

Triggers the computation k-step ahead filtered values, i.e. $$E_{t}{y_{t+k}}$$. Stores results in oo_.FilteredVariablesKStepAhead. Also stores 1-step ahead values in oo_.FilteredVariables. oo_.FilteredVariablesKStepAheadVariances is stored if filter_covariance.

filter_decomposition

Triggers the computation of the shock decomposition of the above k-step ahead filtered values. Stores results in oo_.FilteredVariablesShockDecomposition.

smoothed_state_uncertainty

Triggers the computation of the variance of smoothed estimates, i.e. $$var_T(y_t)$$. Stores results in oo_.Smoother.State_uncertainty.

diffuse_filter

Uses the diffuse Kalman filter (as described in Durbin and Koopman (2012) and Koopman and Durbin (2003) for the multivariate and Koopman and Durbin (2000) for the univariate filter) to estimate models with non-stationary observed variables. This option will also reset the qz_criterium to count unit root variables towards the stable variables. Trying to estimate a model with unit roots will otherwise result in a Blanchard-Kahn error.

When diffuse_filter is used the lik_init option of estimation has no effect.

When there are nonstationary exogenous variables in a model, there is no unique deterministic steady state. For instance, if productivity is a pure random walk:

$a_t = a_{t-1} + e_t$

any value of $$\bar a$$ of $$a$$ is a deterministic steady state for productivity. Consequently, the model admits an infinity of steady states. In this situation, the user must help Dynare in selecting one steady state, except if zero is a trivial model’s steady state, which happens when the linear option is used in the model declaration. The user can either provide the steady state to Dynare using a steady_state_model block (or writing a steady state file) if a closed form solution is available, see steady_state_model, or specify some constraints on the steady state, see equation_tag_for_conditional_steady_state, so that Dynare computes the steady state conditionally on some predefined levels for the non stationary variables. In both cases, the idea is to use dummy values for the steady state level of the exogenous non stationary variables.

Note that the nonstationary variables in the model must be integrated processes (their first difference or k-difference must be stationary).

heteroskedastic_filter

Runs filter, likelihood, and smoother using heteroskedastic definitions provided in a heteroskedastic_shocks block.

selected_variables_only

Only run the classical smoother on the variables listed just after the estimation command. This option is incompatible with requesting classical frequentist forecasts and will be overridden in this case. When using Bayesian estimation, the smoother is by default only run on the declared endogenous variables. Default: run the smoother on all the declared endogenous variables.

cova_compute = INTEGER

When 0, the covariance matrix of estimated parameters is not computed after the computation of posterior mode (or maximum likelihood). This increases speed of computation in large models during development, when this information is not always necessary. Of course, it will break all successive computations that would require this covariance matrix. Otherwise, if this option is equal to 1, the covariance matrix is computed and stored in variable hh of MODEL_FILENAME_mode.mat. Default is 1.

solve_algo = INTEGER

See solve_algo.

order = INTEGER

Order of approximation around the deterministic steady state. When greater than 1, the likelihood is evaluated with a particle or nonlinear filter (see Fernández-Villaverde and Rubio-Ramírez (2005)). Default is 1, i.e. the likelihood of the linearized model is evaluated using a standard Kalman filter.

irf = INTEGER

See irf. Only used if bayesian_irf is passed.

irf_shocks = ( VARIABLE_NAME [[,] VARIABLE_NAME ...] )

See irf_shocks. Only used if bayesian_irf is passed.

irf_plot_threshold = DOUBLE

See irf_plot_threshold. Only used if bayesian_irf is passed.

aim_solver
sylvester = OPTION
sylvester_fixed_point_tol = DOUBLE

See sylvester_fixed_point_tol .

lyapunov = OPTION

Determines the algorithm used to solve the Lyapunov equation to initialized the variance-covariance matrix of the Kalman filter using the steady-state value of state variables. Possible values for OPTION are:

default

Uses the default solver for Lyapunov equations based on Bartels-Stewart algorithm.

fixed_point

Uses a fixed point algorithm to solve the Lyapunov equation. This method is faster than the default one for large scale models, but it could require a large amount of iterations.

doubling

Uses a doubling algorithm to solve the Lyapunov equation (disclyap_fast). This method is faster than the two previous one for large scale models.

square_root_solver

Uses a square-root solver for Lyapunov equations (dlyapchol). This method is fast for large scale models (available under MATLAB if the Control System Toolbox is installed; available under Octave if the control package from Octave-Forge is installed)

Default value is default.

lyapunov_fixed_point_tol = DOUBLE

This is the convergence criterion used in the fixed point Lyapunov solver. Its default value is 1e-10.

lyapunov_doubling_tol = DOUBLE

This is the convergence criterion used in the doubling algorithm to solve the Lyapunov equation. Its default value is 1e-16.

use_penalized_objective_for_hessian

Use the penalized objective instead of the objective function to compute numerically the hessian matrix at the mode. The penalties decrease the value of the posterior density (or likelihood) when, for some perturbations, Dynare is not able to solve the model (issues with steady state existence, Blanchard and Kahn conditions, …). In pratice, the penalized and original objectives will only differ if the posterior mode is found to be near a region where the model is ill-behaved. By default the original objective function is used.

analytic_derivation

Triggers estimation with analytic gradient at order=1. The final hessian at the mode is also computed analytically. Only works for stationary models without missing observations, i.e. for kalman_algo<3. Optimizers that rely on analytic gradients are mode_compute=1,3,4,5,101.

ar = INTEGER

See ar. Only useful in conjunction with option moments_varendo.

endogenous_prior

Use endogenous priors as in Christiano, Trabandt and Walentin (2011). The procedure is motivated by sequential Bayesian learning. Starting from independent initial priors on the parameters, specified in the estimated_params block, the standard deviations observed in a “pre-sample”, taken to be the actual sample, are used to update the initial priors. Thus, the product of the initial priors and the pre-sample likelihood of the standard deviations of the observables is used as the new prior (for more information, see the technical appendix of Christiano, Trabandt and Walentin (2011)). This procedure helps in cases where the regular posterior estimates, which minimize in-sample forecast errors, result in a large overprediction of model variable variances (a statistic that is not explicitly targeted, but often of particular interest to researchers).

use_univariate_filters_if_singularity_is_detected = INTEGER

Decide whether Dynare should automatically switch to univariate filter if a singularity is encountered in the likelihood computation (this is the behaviour if the option is equal to 1). Alternatively, if the option is equal to 0, Dynare will not automatically change the filter, but rather use a penalty value for the likelihood when such a singularity is encountered. Default: 1.

keep_kalman_algo_if_singularity_is_detected

With the default use_univariate_filters_if_singularity_is_detected=1, Dynare will switch to the univariate Kalman filter when it encounters a singular forecast error variance matrix during Kalman filtering. Upon encountering such a singularity for the first time, all subsequent parameter draws and computations will automatically rely on univariate filter, i.e. Dynare will never try the multivariate filter again. Use the keep_kalman_algo_if_singularity_is_detected option to have the use_univariate_filters_if_singularity_is_detected only affect the behavior for the current draw/computation.

rescale_prediction_error_covariance

Rescales the prediction error covariance in the Kalman filter to avoid badly scaled matrix and reduce the probability of a switch to univariate Kalman filters (which are slower). By default no rescaling is done.

qz_zero_threshold = DOUBLE
taper_steps = [INTEGER1 INTEGER2 ...]

Percent tapering used for the spectral window in the Geweke (1992,1999) convergence diagnostics (requires mh_nblocks=1). The tapering is used to take the serial correlation of the posterior draws into account. Default: [4 8 15].

geweke_interval = [DOUBLE DOUBLE]

Percentage of MCMC draws at the beginning and end of the MCMC chain taken to compute the Geweke (1992,1999) convergence diagnostics (requires mh_nblocks=1) after discarding the first mh_drop = DOUBLE percent of draws as a burnin. Default: [0.2 0.5].

raftery_lewis_diagnostics

Triggers the computation of the Raftery and Lewis (1992) convergence diagnostics. The goal is deliver the number of draws required to estimate a particular quantile of the CDF q with precision r with a probability s. Typically, one wants to estimate the q=0.025 percentile (corresponding to a 95 percent HPDI) with a precision of 0.5 percent (r=0.005) with 95 percent certainty (s=0.95). The defaults can be changed via raftery_lewis_qrs. Based on the theory of first order Markov Chains, the diagnostics will provide a required burn-in (M), the number of draws after the burnin (N) as well as a thinning factor that would deliver a first order chain (k). The last line of the table will also deliver the maximum over all parameters for the respective values.

raftery_lewis_qrs = [DOUBLE DOUBLE DOUBLE]

Sets the quantile of the CDF q that is estimated with precision r with a probability s in the Raftery and Lewis (1992) convergence diagnostics. Default: [0.025 0.005 0.95].

consider_all_endogenous

Compute the posterior moments, smoothed variables, k-step ahead filtered variables and forecasts (when requested) on all the endogenous variables. This is equivalent to manually listing all the endogenous variables after the estimation command.

consider_all_endogenous_and_auxiliary

Compute the posterior moments, smoothed variables, k-step ahead filtered variables and forecasts (when requested) on all the endogenous variables and the auxiliary variables introduced by the preprocessor. This option is useful when e.g. running smoother2histval on the results of the Kalman smoother.

consider_only_observed

Compute the posterior moments, smoothed variables, k-step ahead filtered variables and forecasts (when requested) on all the observed variables. This is equivalent to manually listing all the observed variables after the estimation command.

number_of_particles = INTEGER

Number of particles used when evaluating the likelihood of a non linear state space model. Default: 1000.

resampling = OPTION

Determines if resampling of the particles is done. Possible values for OPTION are:

none

No resampling.

systematic

Resampling at each iteration, this is the default value.

generic

Resampling if and only if the effective sample size is below a certain level defined by resampling_threshold * number_of_particles.

resampling_threshold = DOUBLE

A real number between zero and one. The resampling step is triggered as soon as the effective number of particles is less than this number times the total number of particles (as set by number_of_particles). This option is effective if and only if option resampling has value generic.

resampling_method = OPTION

Sets the resampling method. Possible values for OPTION are: kitagawa, stratified and smooth.

filter_algorithm = OPTION

Sets the particle filter algorithm. Possible values for OPTION are:

sis

Sequential importance sampling algorithm, this is the default value.

apf

Auxiliary particle filter.

gf

Gaussian filter.

gmf

Gaussian mixture filter.

cpf

Conditional particle filter.

nlkf

Use a standard (linear) Kalman filter algorithm with the nonlinear measurement and state equations.

proposal_approximation = OPTION

Sets the method for approximating the proposal distribution. Possible values for OPTION are: cubature, montecarlo and unscented. Default value is unscented.

distribution_approximation = OPTION

Sets the method for approximating the particle distribution. Possible values for OPTION are: cubature, montecarlo and unscented. Default value is unscented.

cpf_weights = OPTION

Controls the method used to update the weights in conditional particle filter, possible values are amisanotristani (Amisano et al. (2010)) or murrayjonesparslow (Murray et al. (2013)). Default value is amisanotristani.

nonlinear_filter_initialization = INTEGER

Sets the initial condition of the nonlinear filters. By default the nonlinear filters are initialized with the unconditional covariance matrix of the state variables, computed with the reduced form solution of the first order approximation of the model. If nonlinear_filter_initialization=2, the nonlinear filter is instead initialized with a covariance matrix estimated with a stochastic simulation of the reduced form solution of the second order approximation of the model. Both these initializations assume that the model is stationary, and cannot be used if the model has unit roots (which can be seen with the check command prior to estimation). If the model has stochastic trends, user must use nonlinear_filter_initialization=3, the filters are then initialized with an identity matrix for the covariance matrix of the state variables. Default value is nonlinear_filter_initialization=1 (initialization based on the first order approximation of the model).

particle_filter_options = (NAME, VALUE, ...)

A list of NAME and VALUE pairs. Can be used to set some fine-grained options for the particle filter routines. The set of available options depends on the selected filter routine.

Available options are:

'pruning'

Enable pruning for particle filter-related simulations. Default: false.

'liu_west_delta'

Set the value for delta for the Liu/West online filter. Default: 0.99.

'unscented_alpha'

Set the value for alpha for unscented transforms. Default: 1.

'unscented_beta'

Set the value for beta for unscented transforms. Default: 2.

'unscented_kappa'

Set the value for kappa for unscented transforms. Default: 1.

'initial_state_prior_std'

Value of the diagonal elements for the initial covariance of the state variables when employing nonlinear_filter_initialization=3. Default: 1.

'mixture_state_variables'

Number of mixture components in the Gaussian-mixture filter (gmf) for the state variables. Default: 5.

'mixture_structural_shocks'

Number of mixture components in the Gaussian-mixture filter (gmf) for the structural shocks. Default: 1.

'mixture_measurement_shocks'

Number of mixture components in the Gaussian-mixture filter (gmf) for the measurement errors. Default: 1.

Note

If no mh_jscale parameter is used for a parameter in estimated_params, the procedure uses mh_jscale for all parameters. If mh_jscale option isn’t set, the procedure uses 0.2 for all parameters. Note that if mode_compute=6 is used or the posterior_sampler_option called scale_file is specified, the values set in estimated_params will be overwritten.

“Endogenous” prior restrictions

It is also possible to impose implicit “endogenous” priors about IRFs and moments on the model during estimation. For example, one can specify that all valid parameter draws for the model must generate fiscal multipliers that are bigger than 1 by specifying how the IRF to a government spending shock must look like. The prior restrictions can be imposed via irf_calibration and moment_calibration blocks (see IRF/Moment calibration). The way it works internally is that any parameter draw that is inconsistent with the “calibration” provided in these blocks is discarded, i.e. assigned a prior density of 0. When specifying these blocks, it is important to keep in mind that one won’t be able to easily do model_comparison in this case, because the prior density will not integrate to 1.

Output

After running estimation, the parameters M_.params and the variance matrix M_.Sigma_e of the shocks are set to the mode for maximum likelihood estimation or posterior mode computation without Metropolis iterations. After estimation with Metropolis iterations (option mh_replic > 0 or option load_mh_file set) the parameters M_.params and the variance matrix M_.Sigma_e of the shocks are set to the posterior mean.

Depending on the options, estimation stores results in various fields of the oo_ structure, described below. In the following variables, we will adopt the following shortcuts for specific field names:

MOMENT_NAME

This field can take the following values:

HPDinf

Lower bound of a 90% HPD interval. 4

HPDsup

Upper bound of a 90% HPD interval.

HPDinf_ME

Lower bound of a 90% HPD interval 5 for observables when taking measurement error into account (see e.g. Christoffel et al. (2010), p.17).

HPDsup_ME

Upper bound of a 90% HPD interval for observables when taking measurement error into account.

Mean

Mean of the posterior distribution.

Median

Median of the posterior distribution.

Std

Standard deviation of the posterior distribution.

Variance

Variance of the posterior distribution.

deciles

Deciles of the distribution.

density

Non parametric estimate of the posterior density following the approach outlined in Skoeld and Roberts (2003). First and second columns are respectively abscissa and ordinate coordinates.

ESTIMATED_OBJECT

This field can take the following values:

measurement_errors_corr

Correlation between two measurement errors.

measurement_errors_std

Standard deviation of measurement errors.

parameters

Parameters.

shocks_corr

Correlation between two structural shocks.

shocks_std

Standard deviation of structural shocks.

MATLAB/Octave variable: oo_.MarginalDensity.LaplaceApproximation

Variable set by the estimation command. Stores the marginal data density based on the Laplace Approximation.

MATLAB/Octave variable: oo_.MarginalDensity.ModifiedHarmonicMean

Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Stores the marginal data density based on Geweke (1999) Modified Harmonic Mean estimator.

MATLAB/Octave variable: oo_.posterior.optimization

Variable set by the estimation command if mode-finding is used. Stores the results at the mode. Fields are of the form:

oo_.posterior.optimization.OBJECT

where OBJECT is one of the following:

mode

Parameter vector at the mode.

Variance

Inverse Hessian matrix at the mode or MCMC jumping covariance matrix when used with the MCMC_jumping_covariance option.

log_density

Log likelihood (ML)/log posterior density (Bayesian) at the mode when used with mode_compute>0.

MATLAB/Octave variable: oo_.posterior.metropolis

Variable set by the estimation command if mh_replic>0 is used. Fields are of the form:

oo_.posterior.metropolis.OBJECT

where OBJECT is one of the following:

mean

Mean parameter vector from the MCMC.

Variance

Covariance matrix of the parameter draws in the MCMC.

MATLAB/Octave variable: oo_.FilteredVariables

Variable set by the estimation command, if it is used with the filtered_vars option.

After an estimation without Metropolis, fields are of the form:

oo_.FilteredVariables.VARIABLE_NAME

After an estimation with Metropolis, fields are of the form:

oo_.FilteredVariables.MOMENT_NAME.VARIABLE_NAME

MATLAB/Octave variable: oo_.FilteredVariablesKStepAhead

Variable set by the estimation command, if it is used with the filter_step_ahead option. The k-steps are stored along the rows while the columns indicate the respective variables. The third dimension of the array provides the observation for which the forecast has been made. For example, if filter_step_ahead=[1 2 4] and nobs=200, the element (3,5,204) stores the four period ahead filtered value of variable 5 computed at time t=200 for time t=204. The periods at the beginning and end of the sample for which no forecasts can be made, e.g. entries (1,5,1) and (1,5,204) in the example, are set to zero. Note that in case of Bayesian estimation the variables will be ordered in the order of declaration after the estimation command (or in general declaration order if no variables are specified here). In case of running the classical smoother, the variables will always be ordered in general declaration order. If the selected_variables_only option is specified with the classical smoother, non-requested variables will be simply left out in this order.

MATLAB/Octave variable: oo_.FilteredVariablesKStepAheadVariances

Variable set by the estimation command, if it is used with the filter_step_ahead option. It is a 4 dimensional array where the k-steps are stored along the first dimension, while the fourth dimension of the array provides the observation for which the forecast has been made. The second and third dimension provide the respective variables. For example, if filter_step_ahead=[1 2 4] and nobs=200, the element (3,4,5,204) stores the four period ahead forecast error covariance between variable 4 and variable 5, computed at time t=200 for time t=204. Padding with zeros and variable ordering is analogous to oo_.FilteredVariablesKStepAhead.

MATLAB/Octave variable: oo_.Filtered_Variables_X_step_ahead

Variable set by the estimation command, if it is used with the filter_step_ahead option in the context of Bayesian estimation. Fields are of the form:

oo_.Filtered_Variables_X_step_ahead.VARIABLE_NAME

The n-th entry stores the k-step ahead filtered variable computed at time n for time n+k.

MATLAB/Octave variable: oo_.FilteredVariablesShockDecomposition

Variable set by the estimation command, if it is used with the filter_step_ahead option. The k-steps are stored along the rows while the columns indicate the respective variables. The third dimension corresponds to the shocks in declaration order. The fourth dimension of the array provides the observation for which the forecast has been made. For example, if filter_step_ahead=[1 2 4] and nobs=200, the element (3,5,2,204) stores the contribution of the second shock to the four period ahead filtered value of variable 5 (in deviations from the mean) computed at time t=200 for time t=204. The periods at the beginning and end of the sample for which no forecasts can be made, e.g. entries (1,5,1) and (1,5,204) in the example, are set to zero. Padding with zeros and variable ordering is analogous to oo_.FilteredVariablesKStepAhead.

MATLAB/Octave variable: oo_.PosteriorIRF.dsge

Variable set by the estimation command, if it is used with the bayesian_irf option. Fields are of the form:

oo_.PosteriorIRF.dsge.MOMENT_NAME.VARIABLE_NAME_SHOCK_NAME

MATLAB/Octave variable: oo_.SmoothedMeasurementErrors

Variable set by the estimation command, if it is used with the smoother option. Fields are of the form:

oo_.SmoothedMeasurementErrors.VARIABLE_NAME

MATLAB/Octave variable: oo_.SmoothedShocks

Variable set by the estimation command (if used with the smoother option), or by the calib_smoother command.

After an estimation without Metropolis, or if computed by calib_smoother, fields are of the form:

oo_.SmoothedShocks.VARIABLE_NAME

After an estimation with Metropolis, fields are of the form:

oo_.SmoothedShocks.MOMENT_NAME.VARIABLE_NAME

MATLAB/Octave variable: oo_.SmoothedVariables

Variable set by the estimation command (if used with the smoother option), or by the calib_smoother command.

After an estimation without Metropolis, or if computed by calib_smoother, fields are of the form:

oo_.SmoothedVariables.VARIABLE_NAME

After an estimation with Metropolis, fields are of the form:

oo_.SmoothedVariables.MOMENT_NAME.VARIABLE_NAME

MATLAB/Octave command: get_smooth('VARIABLE_NAME' [, 'VARIABLE_NAME']...);

Returns the smoothed values of the given endogenous or exogenous variable(s), as they are stored in the oo_.SmoothedVariables and oo_.SmoothedShocks variables.

MATLAB/Octave variable: oo_.UpdatedVariables

Variable set by the estimation command (if used with the smoother option), or by the calib_smoother command. Contains the estimation of the expected value of variables given the information available at the current date.

After an estimation without Metropolis, or if computed by calib_smoother, fields are of the form:

oo_.UpdatedVariables.VARIABLE_NAME

After an estimation with Metropolis, fields are of the form:

oo_.UpdatedVariables.MOMENT_NAME.VARIABLE_NAME

MATLAB/Octave command: get_update('VARIABLE_NAME' [, 'VARIABLE_NAME']...);

Returns the updated values of the given variable(s), as they are stored in the oo_.UpdatedVariables variable.

MATLAB/Octave variable: oo_.FilterCovariance

Three-dimensional array set by the estimation command if used with the smoother and Metropolis, if the filter_covariance option has been requested. Contains the series of one-step ahead forecast error covariance matrices from the Kalman smoother. The M_.endo_nbr times M_.endo_nbr times T+1 array contains the variables in declaration order along the first two dimensions. The third dimension of the array provides the observation for which the forecast has been made. Fields are of the form:

oo_.FilterCovariance.MOMENT_NAME

Note that density estimation is not supported.

MATLAB/Octave variable: oo_.Smoother.Variance

Three-dimensional array set by the estimation command (if used with the smoother) without Metropolis, or by the calib_smoother command, if the filter_covariance option has been requested. Contains the series of one-step ahead forecast error covariance matrices from the Kalman smoother. The M_.endo_nbr times M_.endo_nbr times T+1 array contains the variables in declaration order along the first two dimensions. The third dimension of the array provides the observation for which the forecast has been made.

MATLAB/Octave variable: oo_.Smoother.State_uncertainty

Three-dimensional array set by the estimation command (if used with the smoother option) without Metropolis, or by the calib_smoother command, if the smoothed_state_uncertainty option has been requested. Contains the series of covariance matrices for the state estimate given the full data from the Kalman smoother. The M_.endo_nbr times M_.endo_nbr times T array contains the variables in declaration order along the first two dimensions. The third dimension of the array provides the observation for which the smoothed estimate has been made.

MATLAB/Octave variable: oo_.Smoother.SteadyState

Variable set by the estimation command (if used with the smoother) without Metropolis, or by the calib_smoother command. Contains the steady state component of the endogenous variables used in the smoother in order of variable declaration.

MATLAB/Octave variable: oo_.Smoother.TrendCoeffs

Variable set by the estimation command (if used with the smoother) without Metropolis, or by the calib_smoother command. Contains the trend coefficients of the observed variables used in the smoother in order of declaration of the observed variables.

MATLAB/Octave variable: oo_.Smoother.Trend

Variable set by the estimation command (if used with the smoother option), or by the calib_smoother command. Contains the trend component of the variables used in the smoother.

Fields are of the form:

oo_.Smoother.Trend.VARIABLE_NAME

MATLAB/Octave variable: oo_.Smoother.Constant

Variable set by the estimation command (if used with the smoother option), or by the calib_smoother command. Contains the constant part of the endogenous variables used in the smoother, accounting e.g. for the data mean when using the prefilter option.

Fields are of the form:

oo_.Smoother.Constant.VARIABLE_NAME

MATLAB/Octave variable: oo_.Smoother.loglinear

Indicator keeping track of whether the smoother was run with the loglinear option and thus whether stored smoothed objects are in logs.

MATLAB/Octave variable: oo_.PosteriorTheoreticalMoments

Variable set by the estimation command, if it is used with the moments_varendo option. Fields are of the form:

oo_.PosteriorTheoreticalMoments.dsge.THEORETICAL_MOMENT.ESTIMATED_OBJECT.MOMENT_NAME.VARIABLE_NAME

where THEORETICAL_MOMENT is one of the following:

covariance

Variance-covariance of endogenous variables.

contemporaneous_correlation

Contemporaneous correlation of endogenous variables when the contemporaneous_correlation option is specified.

correlation

Auto- and cross-correlation of endogenous variables. Fields are vectors with correlations from 1 up to order options_.ar.

VarianceDecomposition

Decomposition of variance (unconditional variance, i.e. at horizon infinity). 6

VarianceDecompositionME

Same as VarianceDecomposition, but contains the decomposition of the measured as opposed to the actual variable. The joint contribution of the measurement error will be saved in a field named ME.

ConditionalVarianceDecomposition

Only if the conditional_variance_decomposition option has been specified. In the presence of measurement error, the field will contain the variance contribution after measurement error has been taken out, i.e. the decomposition will be conducted of the actual as opposed to the measured variables.

ConditionalVarianceDecompositionME

Only if the conditional_variance_decomposition option has been specified. Same as ConditionalVarianceDecomposition, but contains the decomposition of the measured as opposed to the actual variable. The joint contribution of the measurement error will be saved in a field names ME.

MATLAB/Octave variable: oo_.posterior_density

Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form:

oo_.posterior_density.PARAMETER_NAME

MATLAB/Octave variable: oo_.posterior_hpdinf

Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form:

oo_.posterior_hpdinf.ESTIMATED_OBJECT.VARIABLE_NAME

MATLAB/Octave variable: oo_.posterior_hpdsup

Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form:

oo_.posterior_hpdsup.ESTIMATED_OBJECT.VARIABLE_NAME

MATLAB/Octave variable: oo_.posterior_mean

Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form:

oo_.posterior_mean.ESTIMATED_OBJECT.VARIABLE_NAME

MATLAB/Octave variable: oo_.posterior_mode

Variable set by the estimation command during mode-finding. Fields are of the form:

oo_.posterior_mode.ESTIMATED_OBJECT.VARIABLE_NAME

MATLAB/Octave variable: oo_.posterior_std_at_mode

Variable set by the estimation command during mode-finding. It is based on the inverse Hessian at oo_.posterior_mode. Fields are of the form:

oo_.posterior_std_at_mode.ESTIMATED_OBJECT.VARIABLE_NAME

MATLAB/Octave variable: oo_.posterior_std

Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form:

oo_.posterior_std.ESTIMATED_OBJECT.VARIABLE_NAME

MATLAB/Octave variable: oo_.posterior_var

Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form:

oo_.posterior_var.ESTIMATED_OBJECT.VARIABLE_NAME

MATLAB/Octave variable: oo_.posterior_median

Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form:

oo_.posterior_median.ESTIMATED_OBJECT.VARIABLE_NAME

Example

Here are some examples of generated variables:

oo_.posterior_mode.parameters.alp
oo_.posterior_mean.shocks_std.ex
oo_.posterior_hpdsup.measurement_errors_corr.gdp_conso

MATLAB/Octave variable: oo_.dsge_var.posterior_mode

Structure set by the dsge_var option of the estimation command after mode_compute.

The following fields are saved:

PHI_tilde

Stacked posterior DSGE-BVAR autoregressive matrices at the mode (equation (28) of Del Negro and Schorfheide (2004)).

SIGMA_u_tilde

Posterior covariance matrix of the DSGE-BVAR at the mode (equation (29) of Del Negro and Schorfheide (2004)).

iXX

Posterior population moments in the DSGE-BVAR at the mode ( $$inv(\lambda T \Gamma_{XX}^*+ X'X)$$).

prior

Structure storing the DSGE-BVAR prior.

PHI_star

Stacked prior DSGE-BVAR autoregressive matrices at the mode (equation (22) of Del Negro and Schorfheide (2004)).

SIGMA_star

Prior covariance matrix of the DSGE-BVAR at the mode (equation (23) of Del Negro and Schorfheide (2004)).

ArtificialSampleSize

Size of the artifical prior sample ( $$inv(\lambda T)$$).

DF

Prior degrees of freedom ( $$inv(\lambda T-k-n)$$).

iGXX_star

Inverse of the theoretical prior “covariance” between X and X ($$\Gamma_{xx}^*$$ in Del Negro and Schorfheide (2004)).

MATLAB/Octave variable: oo_.RecursiveForecast

Variable set by the forecast option of the estimation command when used with the nobs = [INTEGER1:INTEGER2] option (see nobs).

Fields are of the form:

oo_.RecursiveForecast.FORECAST_OBJECT.VARIABLE_NAME

where FORECAST_OBJECT is one of the following 7 :

Mean

Mean of the posterior forecast distribution.

HPDinf/HPDsup

Upper/lower bound of the 90% HPD interval taking into account only parameter uncertainty (corresponding to oo_.MeanForecast).

HPDTotalinf/HPDTotalsup.

Upper/lower bound of the 90% HPD interval taking into account both parameter and future shock uncertainty (corresponding to oo_.PointForecast)

VARIABLE_NAME contains a matrix of the following size: number of time periods for which forecasts are requested using the nobs = [INTEGER1:INTEGER2] option times the number of forecast horizons requested by the forecast option. i.e., the row indicates the period at which the forecast is performed and the column the respective k-step ahead forecast. The starting periods are sorted in ascending order, not in declaration order.

MATLAB/Octave variable: oo_.convergence.geweke

Variable set by the convergence diagnostics of the estimation command when used with mh_nblocks=1 option (see mh_nblocks).

Fields are of the form:

oo_.convergence.geweke.VARIABLE_NAME.DIAGNOSTIC_OBJECT

where DIAGNOSTIC_OBJECT is one of the following:

posteriormean

Mean of the posterior parameter distribution.

posteriorstd

Standard deviation of the posterior parameter distribution.

nse_iid

Numerical standard error (NSE) under the assumption of iid draws.

rne_iid

Relative numerical efficiency (RNE) under the assumption of iid draws.

nse_x

Numerical standard error (NSE) when using an x% taper.

rne_x

Relative numerical efficiency (RNE) when using an x% taper.

pooled_mean

Mean of the parameter when pooling the beginning and end parts of the chain specified in geweke_interval and weighting them with their relative precision. It is a vector containing the results under the iid assumption followed by the ones using the taper_steps option (see taper_steps).

pooled_nse

NSE of the parameter when pooling the beginning and end parts of the chain and weighting them with their relative precision. See pooled_mean.

prob_chi2_test

p-value of a chi-squared test for equality of means in the beginning and the end of the MCMC chain. See pooled_mean. A value above 0.05 indicates that the null hypothesis of equal means and thus convergence cannot be rejected at the 5 percent level. Differing values along the taper_steps signal the presence of significant autocorrelation in draws. In this case, the estimates using a higher tapering are usually more reliable.

Command: unit_root_vars VARIABLE_NAME...;

This command is deprecated. Use estimation option diffuse_filter instead for estimating a model with non-stationary observed variables or steady option nocheck to prevent steady to check the steady state returned by your steady state file.

Dynare also has the ability to estimate Bayesian VARs:

Command: bvar_density ;

Computes the marginal density of an estimated BVAR model, using Minnesota priors.

See bvar-a-la-sims.pdf, which comes with Dynare distribution, for more information on this command.

## 4.16. Estimation based on moments¶

Provided that you have observations on some endogenous variables, it is possible to use Dynare to estimate some or all parameters using a method of moments approach. Both the Simulated Method of Moments (SMM) and the Generalized Method of Moments (GMM) are available. The general idea is to minimize the distance between unconditional model moments and corresponding data moments (so called orthogonality or moment conditions). For SMM, Dynare computes model moments via stochastic simulations based on the perturbation approximation up to any order, whereas for GMM model moments are computed in closed-form based on the pruned state-space representation of the perturbation solution up to third order. The implementation of SMM is inspired by Born and Pfeifer (2014) and Ruge-Murcia (2012), whereas the one for GMM is adapted from Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018) and Mutschler (2018). Successful estimation heavily relies on the accuracy and efficiency of the perturbation approximation, so it is advised to tune this as much as possible (see Computing the stochastic solution). The method of moments estimator is consistent and asymptotically normally distributed given certain regularity conditions (see Duffie and Singleton (1993) for SMM and Hansen (1982) for GMM). For instance, it is required to have at least as many moment conditions as estimated parameters (over-identified or just identified). Moreover, the Jacobian of the moments with respect to the estimated parameters needs to have full rank. Performing identification analysis helps to check this regularity condition.

In the over-identified case of declaring more moment conditions than estimated parameters, the choice of weighting_matrix matters for the efficiency of the estimation, because the estimated orthogonality conditions are random variables with unequal variances and usually non-zero cross-moment covariances. A weighting matrix allows to re-weight moments to put more emphasis on moment conditions that are more informative or better measured (in the sense of having a smaller variance). To achieve asymptotic efficiency, the weighting matrix needs to be chosen such that, after appropriate scaling, it has a probability limit proportional to the inverse of the covariance matrix of the limiting distribution of the vector of orthogonality conditions. Dynare uses a Newey-West-type estimator with a Bartlett kernel to compute an estimate of this so-called optimal weighting matrix. Note that in this over-identified case, it is advised to perform the estimation in at least two stages by setting e.g. weighting_matrix=['DIAGONAL','DIAGONAL'] so that the computation of the optimal weighting matrix benefits from the consistent estimation of the previous stages. The optimal weighting matrix is used to compute standard errors and the J-test of overidentifying restrictions, which tests whether the model and selection of moment conditions fits the data sufficiently well. If the null hypothesis of a “valid” model is rejected, then something is (most likely) wrong with either your model or selection of orthogonality conditions.

In case the (presumed) global minimum of the moment distance function is located in a region of the parameter space that is typically considered unlikely (dilemma of absurd parameters), you may opt to choose the penalized_estimator option. Similar to adding priors to the likelihood, this option incorporates prior knowledge (i.e. the prior mean) as additional moment restrictions and weights them by their prior precision to guide the minimization algorithm to more plausible regions of the parameter space. Ideally, these regions are characterized by only slightly worse values of the objective function. Note that adding prior information comes at the cost of a loss in efficiency of the estimator.

Command: varobs VARIABLE_NAME...;

Required. All variables used in the matched_moments block need to be observable. See varobs for more details.

Block: matched_moments ;

This block specifies the product moments which are used in estimation. Currently, only linear product moments (e.g. $$E[y_t], E[y_t^2], E[x_t y_t], E[y_t y_{t-1}], E[y_t^3 x^2_{t-4}]$$) are supported. For other functions like $$E[\log(y_t)e^{x_t}]$$ you need to declare auxiliary endogenous variables.

Each line inside of the block should be of the form:

VARIABLE_NAME(LEAD/LAG)^POWER*VARIABLE_NAME(LEAD/LAG)^POWER*...*VARIABLE_NAME(LEAD/LAG)^POWER;

where VARIABLE_NAME is the name of a declared observable variable, LEAD/LAG is either a negative integer for lags or a positive one for leads, and POWER is a positive integer indicating the exponent on the variable. You can omit LEAD/LAG equal to 0 or POWER equal to 1.

Example

For $$E[c_t], E[y_t], E[c_t^2], E[c_t y_t], E[y_t^2], E[c_t c_{t+3}], E[y_{t+1}^2 c^3_{t-4}], E[c^3_{t-5} y_{t}^2]$$ use the following block:

matched_moments;
c;
y;
c*c;
c*y;
y^2;
c*c(3);
y(1)^2*c(-4)^3;
c(-5)^3*y(0)^2;
end;

Limitations

1. For GMM, Dynare can only compute the theoretical mean, covariance, and autocovariances (i.e. first and second moments). Higher-order moments are only supported for SMM.

2. By default, the product moments are not demeaned, unless the prefilter option is set to 1. That is, by default, c*c corresponds to $$E[c_t^2]$$ and not to $$Var[c_t]=E[c_t^2]-E[c_t]^2$$.

Output

Dynare translates the matched_moments block into a cell array M_.matched_moments where:

• the first column contains a vector of indices for the chosen variables in declaration order

• the second column contains the corresponding vector of leads and lags

• the third column contains the corresponding vector of powers

During the estimation phase, Dynare will eliminate all redundant or duplicate orthogonality conditions in M_.matched_moments and display which conditions were removed. In the example above, this would be the case for the last row, which is the same as the second-to-last one. The original block is saved in M_.matched_moments_orig.

Block: estimated_params ;

Required. See estimated_params for the meaning and syntax.

Block: estimated_params_init ;

See estimated_params_init for the meaning and syntax.

Block: estimated_params_bounds ;

See estimated_params_bounds for the meaning and syntax.

Command: method_of_moments(OPTIONS...);

This command runs the method of moments estimation. The following information will be displayed in the command window:

• Overview of options chosen by the user

• Estimation results for each stage and iteration

• Value of minimized moment distance objective function

• Result of the J-test

• Table of data moments and estimated model moments

Necessary options

mom_method = SMM|GMM

“Simulated Method of Moments” is triggered by SMM and “Generalized Method of Moments” by GMM.

datafile = FILENAME

The name of the file containing the data. See datafile for the meaning and syntax.

Options common for SMM and GMM

order = INTEGER

Order of perturbation approximation. For GMM only orders 1|2|3 are supported. For SMM, you can choose an arbitrary order. Note that the order set in other functions will not overwrite the default. Default: 1.

pruning

Discard higher order terms when iteratively computing simulations of the solution. See pruning for more details. Default: not set for SMM, always set for GMM.

penalized_estimator

This option includes deviations of the estimated parameters from the prior mean as additional moment restrictions and weights them by their prior precision. Default: not set.

weighting_matrix = ['WM1','WM2',...,'WMn']

Determines the weighting matrix used at each estimation stage. The number of elements will define the number of stages, i.e. weighting_matrix = ['DIAGONAL','DIAGONAL','OPTIMAL'] performs a three-stage estimation. Possible values for WM are:

IDENTITY_MATRIX

Sets the weighting matrix equal to the identity matrix.

OPTIMAL

Uses the optimal weighting matrix computed by a Newey-West-type estimate with a Bartlett kernel. At the first stage, the data-moments are used as initial estimate of the model moments, whereas at subsequent stages the previous estimate of model moments will be used when computing the optimal weighting matrix.

DIAGONAL

Uses the diagonal of the OPTIMAL weighting matrix. This choice puts weights on the specified moments instead of on their linear combinations.

FILENAME

The name of the mat-file (extension .mat) containing a user-specified weighting matrix. The file must include a positive definite square matrix called weighting_matrix with both dimensions equal to the number of orthogonality conditions.

Default value is ['DIAGONAL','OPTIMAL'].

weighting_matrix_scaling_factor = DOUBLE

Scaling of weighting matrix in objective function. This value should be chosen to obtain values of the objective function in a reasonable numerical range to prevent over- and underflows. Default: 1.

bartlett_kernel_lag = INTEGER

Bandwidth of kernel for computing the optimal weighting matrix. Default: 20.

se_tolx = DOUBLE

Step size for numerical differentiation when computing standard errors with a two-sided finite difference method. Default: 1e-5.

verbose

Display and store intermediate estimation results in oo_.mom. Default: not set.

SMM-specific options

burnin = INTEGER

Number of periods dropped at the beginning of simulation. Default: 500.

bounded_shock_support

Trim shocks in simulations to $$\pm 2$$ standard deviations. Default: not set.

seed = INTEGER

Common seed used in simulations. Default: 24051986.

simulation_multiple = INTEGER

Multiple of data length used for simulation. Default: 7.

GMM-specific options

analytic_standard_errors

Compute standard errors using analytical derivatives of moments with respect to estimated parameters. Default: not set, i.e. standard errors are computed using a two-sided finite difference method, see se_tolx.

General options

dirname = FILENAME

Directory in which to store estimation output. See dirname for more details. Default: <mod_file>.

graph_format = FORMAT

Specify the file format(s) for graphs saved to disk. See graph_format for more details. Default: eps.

nodisplay

See nodisplay. Default: not set.

nograph

See nograph. Default: not set.

noprint

See noprint. Default: not set.

plot_priors = INTEGER

Control the plotting of priors. See plot_priors for more details. Default: 1, i.e. plot priors.

prior_trunc = DOUBLE

See prior_trunc for more details. Default: 1e-10.

tex

See tex. Default: not set.

Data options

first_obs = INTEGER

See first_obs. Default: 1.

nobs = INTEGER

See nobs. Default: all observations are considered.

prefilter = INTEGER

A value of 1 means that the estimation procedure will demean each data series by its empirical mean and each model moment by its theoretical mean. See prefilter for more details. Default: 0, i.e. no prefiltering.

logdata

See logdata. Default: not set.

xls_sheet = QUOTED_STRING
xls_range = RANGE

Optimization options

huge_number = DOUBLE

See huge_number. Default: 1e7.

mode_compute = INTEGER | FUNCTION_NAME

See mode_compute. Default: 13, i.e. lsqnonlin if the Matlab Optimization Toolbox or the Octave optim-package are present, 4, i.e. csminwel otherwise.

additional_optimizer_steps = [INTEGER|FUNCTION_NAME,INTEGER|FUNCTION_NAME,...]

Vector of additional minimization algorithms run after mode_compute. If verbose option is set, then the additional estimation results are saved into the oo_.mom structure prefixed with verbose_. Default: no additional optimization iterations.

optim = (NAME, VALUE, ...)

See optim.

silent_optimizer

See silent_optimizer. Default: not set.

Numerical algorithms options

aim_solver

See aim_solver. Default: not set.

k_order_solver

See k_order_solver. Default: disabled for order 1 and 2, enabled for order 3 and above.

dr = OPTION

See dr. Default: default, i.e. generalized Schur decomposition.

dr_cycle_reduction_tol = DOUBLE

See dr_cycle_reduction_tol. Default: 1e-7.

dr_logarithmic_reduction_tol = DOUBLE

See dr_logarithmic_reduction_tol. Default: 1e-12.

dr_logarithmic_reduction_maxiter = INTEGER

See dr_logarithmic_reduction_maxiter. Default: 100.

lyapunov = OPTION

See lyapunov. Default: default, i.e. based on Bartlets-Stewart algorithm.

lyapunov_complex_threshold = DOUBLE

See lyapunov_complex_threshold. Default: 1e-15.

lyapunov_fixed_point_tol = DOUBLE

See lyapunov_fixed_point_tol. Default: 1e-10.

lyapunov_doubling_tol = DOUBLE

See lyapunov_doubling_tol. Default: 1e-16.

sylvester = OPTION

See sylvester. Default: default, i.e. uses gensylv.

sylvester_fixed_point_tol = DOUBLE

See sylvester_fixed_point_tol. Default: 1e-12.

qz_criterium = DOUBLE

See qz_criterium. Default: 0.999999 as it is assumed that the observables are weakly stationary.

qz_zero_threshold = DOUBLE

See qz_zero_threshold. Default: 1e-6.

schur_vec_tol = DOUBLE

Tolerance level used to find nonstationary variables in Schur decomposition of the transition matrix. Default: 1e-11.

mode_check

Plots univariate slices through the moments distance objective function around the computed minimum for each estimated parameter. This is helpful to diagnose problems with the optimizer. Default: not set.

mode_check_neighbourhood_size = DOUBLE

See mode_check_neighbourhood_size. Default: 0.5.

mode_check_symmetric_plots = INTEGER

See mode_check_symmetric_plots. Default: 1.

mode_check_number_of_points = INTEGER

See mode_check_number_of_points. Default: 20.

Output

method_of_moments stores user options in a structure called options_mom_ in the global workspace. After running the estimation, the parameters M_.params and the covariance matrices of the shocks M_.Sigma_e and of the measurement errors M_.H are set to the parameters that minimize the quadratic moments distance objective function. The estimation results are stored in the oo_.mom structure with the following fields:

MATLAB/Octave variable: oo_.mom.data_moments

Variable set by the method_of_moments command. Stores the mean of the selected empirical moments of data. NaN values due to leads/lags or missing data are omitted when computing the mean. Vector of dimension equal to the number of orthogonality conditions.

MATLAB/Octave variable: oo_.mom.m_data

Variable set by the method_of_moments command. Stores the selected empirical moments at each point in time. NaN values due to leads/lags or missing data are replaced by the corresponding mean of the moment. Matrix of dimension time periods times number of orthogonality conditions.

MATLAB/Octave variable: oo_.mom.Sw

Variable set by the method_of_moments command. Stores the Cholesky decomposition of the currently used weighting matrix. Square matrix of dimensions equal to the number of orthogonality conditions.

MATLAB/Octave variable: oo_.mom.model_moments

Variable set by the method_of_moments command. Stores the implied selected model moments given the current parameter guess. Model moments are computed in closed-form from the pruned state-space system for GMM, whereas for SMM these are based on averages of simulated data. Vector of dimension equal to the number of orthogonality conditions.

MATLAB/Octave variable: oo_.mom.Q

Variable set by the method_of_moments command. Stores the scalar value of the quadratic moment’s distance objective function.

MATLAB/Octave variable: oo_.mom.model_moments_params_derivs

Variable set by the method_of_moments command. Stores the analytically computed Jacobian matrix of the derivatives of the model moments with respect to the estimated parameters. Only for GMM with analytic_standard_errors. Matrix with dimension equal to the number of orthogonality conditions times number of estimated parameters.

MATLAB/Octave variable: oo_.mom.gmm_stage_*_mode
MATLAB/Octave variable: oo_.mom.smm_stage_*_mode
MATLAB/Octave variable: oo_.mom.verbose_gmm_stage_*_mode
MATLAB/Octave variable: oo_.mom.verbose_smm_stage_*_mode

Variables set by the method_of_moments command when estimating with GMM or SMM. Stores the estimated values at stages 1, 2,…. The structures contain the following fields:

• measurement_errors_corr: estimated correlation between two measurement errors

• measurement_errors_std: estimated standard deviation of measurement errors

• parameters: estimated model parameters

• shocks_corr: estimated correlation between two structural shocks.

• shocks_std: estimated standard deviation of structural shocks.

If the verbose option is set, additional fields prefixed with verbose_ are saved for all additional_optimizer_steps.

MATLAB/Octave variable: oo_.mom.gmm_stage_*_std_at_mode
MATLAB/Octave variable: oo_.mom.smm_stage_*_std_at_mode
MATLAB/Octave variable: oo_.mom.verbose_gmm_stage_*_std_at_mode
MATLAB/Octave variable: oo_.mom.verbose_smm_stage_*_std_at_mode

Variables set by the method_of_moments command when estimating with GMM or SMM. Stores the estimated standard errors at stages 1, 2,…. The structures contain the following fields:

• measurement_errors_corr: standard error of estimated correlation between two measurement errors

• measurement_errors_std: standard error of estimated standard deviation of measurement errors

• parameters: standard error of estimated model parameters

• shocks_corr: standard error of estimated correlation between two structural shocks.

• shocks_std: standard error of estimated standard deviation of structural shocks.

If the verbose option is set, additional fields prefixed with verbose_ are saved for all additional_optimizer_steps.

MATLAB/Octave variable: oo_.mom.J_test

Variable set by the method_of_moments command. Structure where the value of the test statistic is saved into a field called j_stat, the degress of freedom into a field called degrees_freedom and the p-value of the test statistic into a field called p_val.

## 4.17. Model Comparison¶

Command: model_comparison FILENAME[(DOUBLE)]...;
Command: model_comparison(marginal_density = ESTIMATOR) FILENAME[(DOUBLE)]...;

This command computes odds ratios and estimate a posterior density over a collection of models (see e.g. Koop (2003), Ch. 1). The priors over models can be specified as the DOUBLE values, otherwise a uniform prior over all models is assumed. In contrast to frequentist econometrics, the models to be compared do not need to be nested. However, as the computation of posterior odds ratios is a Bayesian technique, the comparison of models estimated with maximum likelihood is not supported.

It is important to keep in mind that model comparison of this type is only valid with proper priors. If the prior does not integrate to one for all compared models, the comparison is not valid. This may be the case if part of the prior mass is implicitly truncated because Blanchard and Kahn conditions (instability or indeterminacy of the model) are not fulfilled, or because for some regions of the parameters space the deterministic steady state is undefined (or Dynare is unable to find it). The compared marginal densities should be renormalized by the effective prior mass, but this not done by Dynare: it is the user’s responsibility to make sure that model comparison is based on proper priors. Note that, for obvious reasons, this is not an issue if the compared marginal densities are based on Laplace approximations.

Options

marginal_density = ESTIMATOR

Specifies the estimator for computing the marginal data density. ESTIMATOR can take one of the following two values: laplace for the Laplace estimator or modifiedharmonicmean for the Geweke (1999) Modified Harmonic Mean estimator. Default value: laplace

Output

The results are stored in oo_.Model_Comparison, which is described below.

Example

model_comparison my_model(0.7) alt_model(0.3);

This example attributes a 70% prior over my_model and 30% prior over alt_model.

MATLAB/Octave variable: oo_.Model_Comparison

Variable set by the model_comparison command. Fields are of the form:

oo_.Model_Comparison.FILENAME.VARIABLE_NAME

where FILENAME is the file name of the model and VARIABLE_NAME is one of the following:

Prior

(Normalized) prior density over the model.

Log_Marginal_Density

Logarithm of the marginal data density.

Bayes_Ratio

Ratio of the marginal data density of the model relative to the one of the first declared model

Posterior_Model_Probability

Posterior probability of the respective model.

## 4.18. Shock Decomposition¶

Command: shock_decomposition [VARIABLE_NAME]...;
Command: shock_decomposition(OPTIONS...) [VARIABLE_NAME]...;

This command computes the historical shock decomposition for a given sample based on the Kalman smoother, i.e. it decomposes the historical deviations of the endogenous variables from their respective steady state values into the contribution coming from the various shocks. The variable_names provided govern for which variables the decomposition is plotted.

Note that this command must come after either estimation (in case of an estimated model) or stoch_simul (in case of a calibrated model).

Options

parameter_set = OPTION

Specify the parameter set to use for running the smoother. Possible values for OPTION are:

• calibration

• prior_mode

• prior_mean

• posterior_mode

• posterior_mean

• posterior_median

• mle_mode

Note that the parameter set used in subsequent commands like stoch_simul will be set to the specified parameter_set. Default value: posterior_mean if Metropolis has been run, mle_mode if MLE has been run.

datafile = FILENAME

See datafile. Useful when computing the shock decomposition on a calibrated model.

first_obs = INTEGER
nobs = INTEGER

See nobs.

prefilter = INTEGER
loglinear
diffuse_kalman_tol = DOUBLE
diffuse_filter
xls_sheet = QUOTED_STRING
xls_range = RANGE
use_shock_groups [= NAME]

Uses shock grouping defined by the string instead of individual shocks in the decomposition. The groups of shocks are defined in the shock_groups block. If no group name is given, default is assumed.

colormap = VARIABLE_NAME

Controls the colormap used for the shocks decomposition graphs. VARIABLE_NAME must be the name of a MATLAB/Octave variable that has been declared beforehand and whose value will be passed to the MATLAB/Octave colormap function (see the MATLAB/Octave manual for the list of acceptable values).

nograph

See nograph. Suppresses the display and creation only within the shock_decomposition command, but does not affect other commands. See plot_shock_decomposition for plotting graphs.

init_state = BOOLEAN

If equal to 0, the shock decomposition is computed conditional on the smoothed state variables in period 0, i.e. the smoothed shocks starting in period 1 are used. If equal to 1, the shock decomposition is computed conditional on the smoothed state variables in period 1. Default: 0.

with_epilogue

If set, then also compute the decomposition for variables declared in the epilogue block (see Epilogue Variables).

Output

MATLAB/Octave variable: oo_.shock_decomposition

The results are stored in the field oo_.shock_decomposition, which is a three dimensional array. The first dimension contains the M_.endo_nbr endogenous variables. The second dimension stores in the first M_.exo_nbr columns the contribution of the respective shocks. Column M_.exo_nbr+1 stores the contribution of the initial conditions, while column M_.exo_nbr+2 stores the smoothed value of the respective endogenous variable in deviations from their steady state, i.e. the mean and trends are subtracted. The third dimension stores the time periods. Both the variables and shocks are stored in the order of declaration, i.e. M_.endo_names and M_.exo_names, respectively.

Block: shock_groups ;
Block: shock_groups(OPTIONS...);

Shocks can be regrouped for the purpose of shock decomposition. The composition of the shock groups is written in a block delimited by shock_groups and end.

Each line defines a group of shocks as a list of exogenous variables:

SHOCK_GROUP_NAME   = VARIABLE_1 [[,] VARIABLE_2 [,]...];
'SHOCK GROUP NAME' = VARIABLE_1 [[,] VARIABLE_2 [,]...];

Options

name = NAME

Specifies a name for the following definition of shock groups. It is possible to use several shock_groups blocks in a model file, each grouping being identified by a different name. This name must in turn be used in the shock_decomposition command. If no name is given, default is used.

Example

varexo e_a, e_b, e_c, e_d;
...

shock_groups(name=group1);
supply = e_a, e_b;
'aggregate demand' = e_c, e_d;
end;

shock_decomposition(use_shock_groups=group1);

This example defines a shock grouping with the name group1, containing a set of supply and demand shocks and conducts the shock decomposition for these two groups.

Command: realtime_shock_decomposition [VARIABLE_NAME]...;
Command: realtime_shock_decomposition(OPTIONS...) [VARIABLE_NAME]...;

This command computes the realtime historical shock decomposition for a given sample based on the Kalman smoother. For each period $$T=[\texttt{presample},\ldots,\texttt{nobs}]$$, it recursively computes three objects:

• Real-time historical shock decomposition $$Y(t\vert T)$$ for $$t=[1,\ldots,T]$$, i.e. without observing data in $$[T+1,\ldots,\texttt{nobs}]$$. This results in a standard shock decomposition being computed for each additional datapoint becoming available after presample.

• Forecast shock decomposition $$Y(T+k\vert T)$$ for $$k=[1,\ldots,forecast]$$, i.e. the $$k$$-step ahead forecast made for every $$T$$ is decomposed in its shock contributions.

• Real-time conditional shock decomposition of the difference between the real-time historical shock decomposition and the forecast shock decomposition. If vintage is equal to 0, it computes the effect of shocks realizing in period $$T$$, i.e. decomposes $$Y(T\vert T)-Y(T\vert T-1)$$. Put differently, it conducts a $$1$$-period ahead shock decomposition from $$T-1$$ to $$T$$, by decomposing the update step of the Kalman filter. If vintage>0 and smaller than nobs, the decomposition is conducted of the forecast revision $$Y(T+k\vert T+k)-Y(T+k\vert T)$$.

Like shock_decomposition it decomposes the historical deviations of the endogenous variables from their respective steady state values into the contribution coming from the various shocks. The variable_names provided govern for which variables the decomposition is plotted.

Note that this command must come after either estimation (in case of an estimated model) or stoch_simul (in case of a calibrated model).

Options

parameter_set = OPTION

See parameter_set for possible values.

datafile = FILENAME

See datafile.

first_obs = INTEGER
nobs = INTEGER

See nobs.

use_shock_groups [= NAME]
colormap = VARIABLE_NAME
nograph

See nograph. Only shock decompositions are computed and stored in oo_.realtime_shock_decomposition, oo_.conditional_shock_decomposition and oo_.realtime_forecast_shock_decomposition but no plot is made (See plot_shock_decomposition).

presample = INTEGER

Data point above which recursive realtime shock decompositions are computed, i.e. for $$T=[\texttt{presample+1} \ldots \texttt{nobs}]$$.

forecast = INTEGER

Compute shock decompositions up to $$T+k$$ periods, i.e. get shock contributions to k-step ahead forecasts.

save_realtime = INTEGER_VECTOR

Choose for which vintages to save the full realtime shock decomposition. Default: 0.

fast_realtime = INTEGER
fast_realtime = [INTEGER1:INTEGER2]
fast_realtime = [INTEGER1 INTEGER2 ...]

Runs the smoother only for the data vintages provided by the specified integer (vector).

with_epilogue

Output

MATLAB/Octave variable: oo_.realtime_shock_decomposition

Structure storing the results of realtime historical decompositions. Fields are three-dimensional arrays with the first two dimension equal to the ones of oo_.shock_decomposition. The third dimension stores the time periods and is therefore of size T+forecast. Fields are of the form:

oo_.realtime_shock_decomposition.OBJECT

where OBJECT is one of the following:

pool

Stores the pooled decomposition, i.e. for every real-time shock decomposition terminal period $$T=[\texttt{presample},\ldots,\texttt{nobs}]$$ it collects the last period’s decomposition $$Y(T\vert T)$$ (see also plot_shock_decomposition). The third dimension of the array will have size nobs+forecast.

time_*

Stores the vintages of realtime historical shock decompositions if save_realtime is used. For example, if save_realtime=[5] and forecast=8, the third dimension will be of size 13.

MATLAB/Octave variable: oo_.realtime_conditional_shock_decomposition

Structure storing the results of real-time conditional decompositions. Fields are of the form:

oo_.realtime_conditional_shock_decomposition.OBJECT

where OBJECT is one of the following:

pool

Stores the pooled real-time conditional shock decomposition, i.e. collects the decompositions of $$Y(T\vert T)-Y(T\vert T-1)$$ for the terminal periods $$T=[\texttt{presample},\ldots,\texttt{nobs}]$$. The third dimension is of size nobs.

time_*

Store the vintages of $$k$$-step conditional forecast shock decompositions $$Y(t\vert T+k)$$, for $$t=[T \ldots T+k]$$. See vintage. The third dimension is of size 1+forecast.

MATLAB/Octave variable: oo_.realtime_forecast_shock_decomposition

Structure storing the results of realtime forecast decompositions. Fields are of the form:

oo_.realtime_forecast_shock_decomposition.OBJECT

where OBJECT is one of the following:

pool

Stores the pooled real-time forecast decomposition of the $$1$$-step ahead effect of shocks on the $$1$$-step ahead prediction, i.e. $$Y(T\vert T-1)$$.

time_*

Stores the vintages of $$k$$-step out-of-sample forecast shock decompositions, i.e. $$Y(t\vert T)$$, for $$t=[T \ldots T+k]$$. See vintage.

Command: plot_shock_decomposition [VARIABLE_NAME]...;
Command: plot_shock_decomposition(OPTIONS...) [VARIABLE_NAME]...;

This command plots the historical shock decomposition already computed by shock_decomposition or realtime_shock_decomposition. For that reason, it must come after one of these commands. The variable_names provided govern which variables the decomposition is plotted for.

Further note that, unlike the majority of Dynare commands, the options specified below are overwritten with their defaults before every call to plot_shock_decomposition. Hence, if you want to reuse an option in a subsequent call to plot_shock_decomposition, you must pass it to the command again.

Options

use_shock_groups [= NAME]
colormap = VARIABLE_NAME
nodisplay
nograph
graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
detail_plot

Plots shock contributions using subplots, one per shock (or group of shocks). Default: not activated

interactive

screen_shocks

For large models (i.e. for models with more than 16 shocks), plots only the shocks that have the largest historical contribution for chosen selected variable_names. Historical contribution is ranked by the mean absolute value of all historical contributions.

steadystate

If passed, the the $$y$$-axis value of the zero line in the shock decomposition plot is translated to the steady state level. Default: not activated

type = qoq | yoy | aoa

For quarterly data, valid arguments are: qoq for quarter-on-quarter plots, yoy for year-on-year plots of growth rates, aoa for annualized variables, i.e. the value in the last quarter for each year is plotted. Default value: empty, i.e. standard period-on-period plots (qoq for quarterly data).

fig_name = STRING

Specifies a user-defined keyword to be appended to the default figure name set by plot_shock_decomposition. This can avoid to overwrite plots in case of sequential calls to plot_shock_decomposition.

write_xls

Saves shock decompositions to Excel-file in the main directory, named FILENAME_shock_decomposition_TYPE_FIG_NAME.xls. This option requires your system to be configured to be able to write Excel files. 8

realtime = INTEGER

Which kind of shock decomposition to plot. INTEGER can take the following values:

If no vintage is requested, i.e. vintage=0 then the pooled objects from realtime_shock_decomposition will be plotted and the respective vintage otherwise. Default: 0.

vintage = INTEGER

Selects a particular data vintage in $$[presample,\ldots,nobs]$$ for which to plot the results from realtime_shock_decomposition selected via the realtime option. If the standard historical shock decomposition is selected (realtime=0), vintage will have no effect. If vintage=0 the pooled objects from realtime_shock_decomposition will be plotted. If vintage>0, it plots the shock decompositions for vintage $$T=\texttt{vintage}$$ under the following scenarios:

• realtime=1: the full vintage shock decomposition $$Y(t\vert T)$$ for $$t=[1,\ldots,T]$$

• realtime=2: the conditional forecast shock decomposition from $$T$$, i.e. plots $$Y(T+j\vert T+j)$$ and the shock contributions needed to get to the data $$Y(T+j)$$ conditional on $$T=$$ vintage, with $$j=[0,\ldots,\texttt{forecast}]$$.

• realtime=3: plots unconditional forecast shock decomposition from $$T$$, i.e. $$Y(T+j\vert T)$$, where $$T=\texttt{vintage}$$ and $$j=[0,\ldots,\texttt{forecast}]$$.

Default: 0.

plot_init_date = DATE

If passed, plots decomposition using plot_init_date as initial period. Default: first observation in estimation

plot_end_date = DATE

If passed, plots decomposition using plot_end_date as last period. Default: last observation in estimation

diff

If passed, plot the decomposition of the first difference of the list of variables. If used in combination with flip, the diff operator is first applied. Default: not activated

flip

If passed, plot the decomposition of the opposite of the list of variables. If used in combination with diff, the diff operator is first applied. Default: not activated

max_nrows

Maximum number of rows in the subplot layout of detailed shock decomposition graphs. Note that columns are always 3. Default: 6

with_epilogue
init2shocks
init2shocks = NAME

Use the information contained in an init2shocks block, in order to attribute initial conditions to shocks. The name of the block can be explicitly given, otherwise it defaults to the default block.

Block: init2shocks ;
Block: init2shocks(OPTIONS...);

This blocks gives the possibility of attributing the initial condition of endogenous variables to the contribution of exogenous variables in the shock decomposition.

For example, in an AR(1) process, the contribution of the initial condition on the process variable can naturally be assigned to the innovation of the process.

Each line of the block should have the syntax:

VARIABLE_1 [,] VARIABLE_2;

Where VARIABLE_1 is an endogenous variable whose initial condition will be attributed to the exogenous VARIABLE_2.

The information contained in this block is used by the plot_shock_decomposition command when given the init2shocks option.

Options

name = NAME

Specifies a name for the block, that can be referenced from plot_shock_decomposition, so that several such blocks can coexist in a single model file. If the name is unspecified, it defaults to default.

Example

var y y_s R pie dq pie_s de A y_obs pie_obs R_obs;
varexo e_R e_q e_ys e_pies e_A;
...

model;
dq = rho_q*dq(-1)+e_q;
A = rho_A*A(-1)+e_A;
...
end;

...

init2shocks;
dq e_q;
A e_A;
end;

shock_decomposition(nograph);

plot_shock_decomposition(init2shocks) y_obs R_obs pie_obs dq de;

In this example, the initial conditions of dq and A will be respectively attributed to e_q and e_A.

Command: initial_condition_decomposition [VARIABLE_NAME]...;
Command: initial_condition_decomposition(OPTIONS...) [VARIABLE_NAME]...;

This command computes and plots the decomposition of the effect of smoothed initial conditions of state variables. The variable_names provided govern which variables the decomposition is plotted for.

Further note that, unlike the majority of Dynare commands, the options specified below are overwritten with their defaults before every call to initial_condition_decomposition. Hence, if you want to reuse an option in a subsequent call to initial_condition_decomposition, you must pass it to the command again.

Options

colormap = VARIABLE_NAME
nodisplay
graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
detail_plot

Plots shock contributions using subplots, one per shock (or group of shocks). Default: not activated

steadystate

If passed, the the $$y$$-axis value of the zero line in the shock decomposition plot is translated to the steady state level. Default: not activated

type = qoq | yoy | aoa

For quarterly data, valid arguments are: qoq for quarter-on-quarter plots, yoy for year-on-year plots of growth rates, aoa for annualized variables, i.e. the value in the last quarter for each year is plotted. Default value: empty, i.e. standard period-on-period plots (qoq for quarterly data).

fig_name = STRING

Specifies a user-defined keyword to be appended to the default figure name set by plot_shock_decomposition. This can avoid to overwrite plots in case of sequential calls to plot_shock_decomposition.

write_xls

Saves shock decompositions to Excel-file in the main directory, named FILENAME_shock_decomposition_TYPE_FIG_NAME_initval.xls. This option requires your system to be configured to be able to write Excel files. 8

plot_init_date = DATE

If passed, plots decomposition using plot_init_date as initial period. Default: first observation in estimation

plot_end_date = DATE

If passed, plots decomposition using plot_end_date as last period. Default: last observation in estimation

diff

If passed, plot the decomposition of the first difference of the list of variables. If used in combination with flip, the diff operator is first applied. Default: not activated

flip

If passed, plot the decomposition of the opposite of the list of variables. If used in combination with diff, the diff operator is first applied. Default: not activated

Command: squeeze_shock_decomposition [VARIABLE_NAME]...;

For large models, the size of the information stored by shock decompositions (especially various settings of realtime decompositions) may become huge. This command allows to squeeze this information in two possible ways:

• Automatic (default): only the variables for which plotting has been explicitly required with plot_shock_decomposition will have their decomposition left in oo_ after this command is run;

• If a list of variables is passed to the command, then only those variables will have their decomposition left in oo_ after this command is run.

## 4.19. Calibrated Smoother¶

Dynare can also run the smoother on a calibrated model:

Command: calib_smoother [VARIABLE_NAME]...;
Command: calib_smoother(OPTIONS...) [VARIABLE_NAME]...;

This command computes the smoothed variables (and possible the filtered variables) on a calibrated model.

A datafile must be provided, and the observable variables declared with varobs. The smoother is based on a first-order approximation of the model.

By default, the command computes the smoothed variables and shocks and stores the results in oo_.SmoothedVariables and oo_.SmoothedShocks. It also fills oo_.UpdatedVariables.

Options

datafile = FILENAME

See datafile.

filtered_vars

Triggers the computation of filtered variables. See filtered_vars, for more details.

filter_step_ahead = [INTEGER1:INTEGER2]
prefilter = INTEGER
parameter_set = OPTION

See parameter_set for possible values. Default: calibration.

loglinear

See loglinear.

first_obs = INTEGER
filter_decomposition
filter_covariance
smoother_redux
kalman_algo = INTEGER
diffuse_filter = INTEGER
diffuse_kalman_tol = DOUBLE
xls_sheet = QUOTED_STRING
xls_range = RANGE

## 4.20. Forecasting¶

On a calibrated model, forecasting is done using the forecast command. On an estimated model, use the forecast option of estimation command.

It is also possible to compute forecasts on a calibrated or estimated model for a given constrained path of the future endogenous variables. This is done, from the reduced form representation of the DSGE model, by finding the structural shocks that are needed to match the restricted paths. Use conditional_forecast, conditional_forecast_paths and plot_conditional_forecast for that purpose.

Finally, it is possible to do forecasting with a Bayesian VAR using the bvar_forecast command.

Command: forecast [VARIABLE_NAME...];
Command: forecast(OPTIONS...) [VARIABLE_NAME...];

This command computes a simulation of a stochastic model from an arbitrary initial point.

When the model also contains deterministic exogenous shocks, the simulation is computed conditionally to the agents knowing the future values of the deterministic exogenous variables.

forecast must be called after stoch_simul.

forecast plots the trajectory of endogenous variables. When a list of variable names follows the command, only those variables are plotted. A 90% confidence interval is plotted around the mean trajectory. Use option conf_sig to change the level of the confidence interval.

Options

periods = INTEGER

Number of periods of the forecast. Default: 5.

conf_sig = DOUBLE

Level of significance for confidence interval. Default: 0.90.

nograph
nodisplay
graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )

Initial Values

forecast computes the forecast taking as initial values the values specified in histval (see histval). When no histval block is present, the initial values are the one stated in initval. When initval is followed by command steady, the initial values are the steady state (see steady).

Output

The results are stored in oo_.forecast, which is described below.

Example

varexo_det tau;

varexo e;
...
shocks;
var e; stderr 0.01;
var tau;
periods 1:9;
values -0.15;
end;

stoch_simul(irf=0);

forecast;

MATLAB/Octave variable: oo_.forecast

Variable set by the forecast command, or by the estimation command if used with the forecast option and ML or if no Metropolis-Hastings has been computed (in that case, the forecast is computed for the posterior mode). Fields are of the form:

oo_.forecast.FORECAST_MOMENT.VARIABLE_NAME

where FORECAST_MOMENT is one of the following:

HPDinf

Lower bound of a 90% HPD interval 9 of forecast due to parameter uncertainty, but ignoring the effect of measurement error on observed variables. In case of ML, it stores the lower bound of the confidence interval.

HPDsup

Upper bound of a 90% HPD forecast interval due to parameter uncertainty, but ignoring the effect of measurement error on observed variables. In case of ML, it stores the upper bound of the confidence interval.

HPDinf_ME

Lower bound of a 90% HPD interval 10 of forecast for observed variables due to parameter uncertainty and measurement error. In case of ML, it stores the lower bound of the confidence interval.

HPDsup_ME

Upper bound of a 90% HPD interval of forecast for observed variables due to parameter uncertainty and measurement error. In case of ML, it stores the upper bound of the confidence interval.

Mean

Mean of the posterior distribution of forecasts.

MATLAB/Octave variable: oo_.PointForecast

Set by the estimation command, if it is used with the forecast option and if either mh_replic > 0 or the load_mh_file option are used.

Contains the distribution of forecasts taking into account the uncertainty about both parameters and shocks.

Fields are of the form:

oo_.PointForecast.MOMENT_NAME.VARIABLE_NAME

MATLAB/Octave variable: oo_.MeanForecast

Set by the estimation command, if it is used with the forecast option and if either mh_replic > 0 or load_mh_file option are used.

Contains the distribution of forecasts where the uncertainty about shocks is averaged out. The distribution of forecasts therefore only represents the uncertainty about parameters.

Fields are of the form:

oo_.MeanForecast.MOMENT_NAME.VARIABLE_NAME

Command: conditional_forecast(OPTIONS...);

This command computes forecasts on an estimated or calibrated model for a given constrained path of some future endogenous variables. This is done using the reduced form first order state-space representation of the DSGE model by finding the structural shocks that are needed to match the restricted paths. Consider the augmented state space representation that stacks both predetermined and non-predetermined variables into a vector $$y_{t}$$:

$y_t=Ty_{t-1}+R\varepsilon_t$

Both $$y_t$$ and $$\varepsilon_t$$ are split up into controlled and uncontrolled ones, and we assume without loss of generality that the constrained endogenous variables and the controlled shocks come first :

$\begin{split}\begin{pmatrix} y_{c,t}\\ y_{u,t} \end{pmatrix} = \begin{pmatrix} T_{c,c} & T_{c,u}\\ T_{u,c} & T_{u,u} \end{pmatrix} \begin{pmatrix} y_{c,t-1}\\ y_{u,t-1} \end{pmatrix} + \begin{pmatrix} R_{c,c} & R_{c,u}\\ R_{u,c} & R_{u,u} \end{pmatrix} \begin{pmatrix} \varepsilon_{c,t}\\ \varepsilon_{u,t} \end{pmatrix}\end{split}$

where matrices $$T$$ and $$R$$ are partitioned consistently with the vectors of endogenous variables and innovations. Provided that matrix $$R_{c,c}$$ is square and full rank (a necessary condition is that the number of free endogenous variables matches the number of free innovations), given $$y_{c,t}$$, $$\varepsilon_{u,t}$$ and $$y_{t-1}$$ the first block of equations can be solved for $$\varepsilon_{c,t}$$:

$\varepsilon_{c,t} = R_{c,c}^{-1}\bigl( y_{c,t} - T_{c,c}y_{c,t} - T_{c,u}y_{u,t} - R_{c,u}\varepsilon_{u,t}\bigr)$

and $$y_{u,t}$$ can be updated by evaluating the second block of equations:

$y_{u,t} = T_{u,c}y_{c,t-1} + T_{u,u}y_{u,t-1} + R_{u,c}\varepsilon_{c,t} + R_{u,u}\varepsilon_{u,t}$

By iterating over these two blocks of equations, we can build a forecast for all the endogenous variables in the system conditional on paths for a subset of the endogenous variables. If the distribution of the free innovations $$\varepsilon_{u,t}$$ is provided (i.e. some of them have positive variances) this exercise is replicated (the number of replication is controlled by the option replic described below) by drawing different sequences of free innovations. The result is a predictive distribution for the uncontrolled endogenous variables, $$y_{u,t}$$, that Dynare will use to report confidence bands around the point conditional forecast.

A few things need to be noted. First, the controlled exogenous variables are set to zero for the uncontrolled periods. This implies that there is no forecast uncertainty arising from these exogenous variables in uncontrolled periods. Second, by making use of the first order state space solution, even if a higher-order approximation was performed, the conditional forecasts will be based on a first order approximation. Since the controlled exogenous variables are identified on the basis of the reduced form model (i.e. after solving for the expectations), they are unforeseen shocks from the perspective of the agents in the model. That is, agents expect the endogenous variables to return to their respective steady state levels but are surprised in each period by the realisation of shocks keeping the endogenous variables along a predefined (unexpected) path. Fourth, if the structural innovations are correlated, because the calibrated or estimated covariance matrix has non zero off diagonal elements, the results of the conditional forecasts will depend on the ordering of the innovations (as declared after varexo). As in VAR models, a Cholesky decomposition is used to factorise the covariance matrix and identify orthogonal impulses. It is preferable to declare the correlations in the model block (explicitly imposing the identification restrictions), unless you are satisfied with the implicit identification restrictions implied by the Cholesky decomposition.

This command has to be called after estimation or stoch_simul.

Use conditional_forecast_paths block to give the list of constrained endogenous, and their constrained future path. Option controlled_varexo is used to specify the structural shocks which will be matched to generate the constrained path.

Use plot_conditional_forecast to graph the results.

Options

parameter_set = OPTION

See parameter_set for possible values. No default value, mandatory option.

controlled_varexo = (VARIABLE_NAME...)

Specify the exogenous variables to use as control variables. No default value, mandatory option.

periods = INTEGER

Number of periods of the forecast. Default: 40. periods cannot be smaller than the number of constrained periods.

replic = INTEGER

Number of simulations used to compute the conditional forecast uncertainty. Default: 5000.

conf_sig = DOUBLE

Level of significance for confidence interval. Default: 0.80.

Output

The results are stored in oo_.conditional_forecast, which is described below.

Example

var y a;
varexo e u;
...
estimation(...);

conditional_forecast_paths;
var y;
periods 1:3, 4:5;
values 2, 5;
var a;
periods 1:5;
values 3;
end;

conditional_forecast(parameter_set = calibration, controlled_varexo = (e, u), replic = 3000);

plot_conditional_forecast(periods = 10) a y;

MATLAB/Octave variable: oo_.conditional_forecast.cond

Variable set by the conditional_forecast command. It stores the conditional forecasts. Fields are periods+1 by 1 vectors storing the steady state (time 0) and the subsequent periods forecasts periods. Fields are of the form:

oo_.conditional_forecast.cond.FORECAST_MOMENT.VARIABLE_NAME

where FORECAST_MOMENT is one of the following:

Mean

Mean of the conditional forecast distribution.

ci

Confidence interval of the conditional forecast distribution. The size corresponds to conf_sig.

MATLAB/Octave variable: oo_.conditional_forecast.uncond

Variable set by the conditional_forecast command. It stores the unconditional forecasts. Fields are of the form:

oo_.conditional_forecast.uncond.FORECAST_MOMENT.VARIABLE_NAME

MATLAB/Octave variable: forecasts.instruments

Variable set by the conditional_forecast command. Stores the names of the exogenous instruments.

MATLAB/Octave variable: oo_.conditional_forecast.controlled_variables

Variable set by the conditional_forecast command. Stores the position of the constrained endogenous variables in declaration order.

MATLAB/Octave variable: oo_.conditional_forecast.controlled_exo_variables

Variable set by the conditional_forecast command. Stores the values of the controlled exogenous variables underlying the conditional forecasts to achieve the constrained endogenous variables. Fields are [number of constrained periods] by 1 vectors and are of the form:

oo_.conditional_forecast.controlled_exo_variables.FORECAST_MOMENT.SHOCK_NAME

MATLAB/Octave variable: oo_.conditional_forecast.graphs

Variable set by the conditional_forecast command. Stores the information for generating the conditional forecast plots.

Block: conditional_forecast_paths ;

Describes the path of constrained endogenous, before calling conditional_forecast. The syntax is similar to deterministic shocks in shocks, see conditional_forecast for an example.

The syntax of the block is the same as for the deterministic shocks in the shocks blocks (see Shocks on exogenous variables). Note that you need to specify the full path for all constrained endogenous variables between the first and last specified period. If an intermediate period is not specified, a value of 0 is assumed. That is, if you specify only values for periods 1 and 3, the values for period 2 will be 0. Currently, it is not possible to have uncontrolled intermediate periods.

It is however possible to have different number of controlled periods for different variables. In that case, the order of declaration of endogenous controlled variables and of controlled_varexo matters: if the second endogenous variable is controlled for less periods than the first one, the second controlled_varexo isn’t set for the last periods.

In case of the presence of observation_trends, the specified controlled path for these variables needs to include the trend component. When using the loglinear option, it is necessary to specify the logarithm of the controlled variables.

Block: filter_initial_state ;

This block specifies the initial values of the endogenous states at the beginning of the Kalman filter recursions. That is, if the Kalman filter recursion starts with time t=1 being the first observation, this block provides the state estimate at time 0 given information at time 0, $$E_0(x_0)$$. If nothing is specified, the initial condition is assumed to be at the steady state (which is the unconditional mean for a stationary model).

This block is terminated by end;.

Each line inside of the block should be of the form:

VARIABLE_NAME(INTEGER)=EXPRESSION;

EXPRESSION is any valid expression returning a numerical value and can contain parameter values. This allows specifying relationships that will be honored during estimation. INTEGER refers to the lag with which a variable appears. By convention in Dynare, period 1 is the first period. Going backwards in time, the first period before the start of the simulation is period 0, then period -1, and so on. Note that the filter_initial_state block does not take non-state variables.

Example

filter_initial_state;
k(0)= ((1/bet-(1-del))/alp)^(1/(alp-1))*l_ss;
P(0)=2.5258;
m(0)= mst;
end;

Command: plot_conditional_forecast [VARIABLE_NAME...];
Command: plot_conditional_forecast(periods = INTEGER) [VARIABLE_NAME...];

Plots the conditional (plain lines) and unconditional (dashed lines) forecasts.

To be used after conditional_forecast.

Options

periods = INTEGER

Number of periods to be plotted. Default: equal to periods in conditional_forecast. The number of periods declared in plot_conditional_forecast cannot be greater than the one declared in conditional_forecast.

Command: bvar_forecast ;

This command computes (out-of-sample) forecasts for an estimated BVAR model, using Minnesota priors.

See bvar-a-la-sims.pdf, which comes with Dynare distribution, for more information on this command.

If the model contains strong non-linearities or if some perfectly expected shocks are considered, the forecasts and the conditional forecasts can be computed using an extended path method. The forecast scenario describing the shocks and/or the constrained paths on some endogenous variables should be build. The first step is the forecast scenario initialization using the function init_plan:

MATLAB/Octave command: HANDLE = init_plan(DATES);

Creates a new forecast scenario for a forecast period (indicated as a dates class, see dates class members). This function return a handle on the new forecast scenario.

The forecast scenario can contain some simple shocks on the exogenous variables. This shocks are described using the function basic_plan:

MATLAB/Octave command: HANDLE = basic_plan(HANDLE, VAR_NAME', SHOCK_TYPE', DATES, MATLAB VECTOR OF DOUBLE | [DOUBLE | EXPR [DOUBLE | EXPR] ] );

Adds to the forecast scenario a shock on the exogenous variable indicated between quotes in the second argument. The shock type has to be specified in the third argument between quotes: ’surprise’ in case of an unexpected shock or ’perfect_foresight’ for a perfectly anticipated shock. The fourth argument indicates the period of the shock using a dates class (see dates class members). The last argument is the shock path indicated as a MATLAB vector of double. This function return the handle of the updated forecast scenario.

The forecast scenario can also contain a constrained path on an endogenous variable. The values of the related exogenous variable compatible with the constrained path are in this case computed. In other words, a conditional forecast is performed. This kind of shock is described with the function flip_plan:

MATLAB/Octave command: HANDLE = flip_plan(HANDLE, VAR_NAME', VAR_NAME', SHOCK_TYPE', DATES, MATLAB VECTOR OF DOUBLE | [DOUBLE | EXPR [DOUBLE | EXPR] ] );

Adds to the forecast scenario a constrained path on the endogenous variable specified between quotes in the second argument. The associated exogenous variable provided in the third argument between quotes, is considered as an endogenous variable and its values compatible with the constrained path on the endogenous variable will be computed. The nature of the expectation on the constrained path has to be specified in the fourth argument between quotes: ’surprise’ in case of an unexpected path or ’perfect_foresight’ for a perfectly anticipated path. The fifth argument indicates the period where the path of the endogenous variable is constrained using a dates class (see dates class members). The last argument contains the constrained path as a MATLAB vector of double. This function return the handle of the updated forecast scenario.

Once the forecast scenario if fully described, the forecast is computed with the command det_cond_forecast:

MATLAB/Octave command: DSERIES = det_cond_forecast(HANDLE[, DSERIES [, DATES]]);

Computes the forecast or the conditional forecast using an extended path method for the given forecast scenario (first argument). The past values of the endogenous and exogenous variables provided with a dseries class (see dseries class members) can be indicated in the second argument. By default, the past values of the variables are equal to their steady-state values. The initial date of the forecast can be provided in the third argument. By default, the forecast will start at the first date indicated in the init_plan command. This function returns a dset containing the historical and forecast values for the endogenous and exogenous variables.

Example

% conditional forecast using extended path method
% with perfect foresight on r path

var y r;
varexo e u;
...
smoothed = dseries('smoothed_variables.csv');

fplan = init_plan(2013Q4:2029Q4);
fplan = flip_plan(fplan, 'y', 'u', 'surprise', 2013Q4:2014Q4,  [1 1.1 1.2 1.1 ]);
fplan = flip_plan(fplan, 'r', 'e', 'perfect_foresight', 2013Q4:2014Q4,  [2 1.9 1.9 1.9 ]);

dset_forecast = det_cond_forecast(fplan, smoothed);

plot(dset_forecast.{'y','u'});
plot(dset_forecast.{'r','e'});

Command: smoother2histval ;
Command: smoother2histval(OPTIONS...);

The purpose of this command is to construct initial conditions (for a subsequent simulation) that are the smoothed values of a previous estimation.

More precisely, after an estimation run with the smoother option, smoother2histval will extract the smoothed values (from oo_.SmoothedVariables, and possibly from oo_.SmoothedShocks if there are lagged exogenous), and will use these values to construct initial conditions (as if they had been manually entered through histval).

Options

period = INTEGER

Period number to use as the starting point for the subsequent simulation. It should be between 1 and the number of observations that were used to produce the smoothed values. Default: the last observation.

infile = FILENAME

Load the smoothed values from a _results.mat file created by a previous Dynare run. Default: use the smoothed values currently in the global workspace.

invars = ( VARIABLE_NAME [VARIABLE_NAME ...] )

A list of variables to read from the smoothed values. It can contain state endogenous variables, and also exogenous variables having a lag. Default: all the state endogenous variables, and all the exogenous variables with a lag.

outfile = FILENAME

Write the initial conditions to a file. Default: write the initial conditions in the current workspace, so that a simulation can be performed.

outvars = ( VARIABLE_NAME [VARIABLE_NAME ...] )

A list of variables which will be given the initial conditions. This list must have the same length than the list given to invars, and there will be a one-to-one mapping between the two list. Default: same value as option invars.

Use cases

There are three possible ways of using this command:

• Everything in a single file: run an estimation with a smoother, then run smoother2histval (without the infile and outfile options), then run a stochastic simulation.

• In two files: in the first file, run the smoother and then run smoother2histval with the outfile option; in the second file, run histval_file to load the initial conditions, and run a (deterministic or stochastic) simulation.

• In two files: in the first file, run the smoother; in the second file, run smoother2histval with the infile option equal to the _results.mat file created by the first file, and then run a (deterministic or stochastic) simulation.

## 4.21. Optimal policy¶

Dynare has tools to compute optimal policies for various types of objectives. You can either solve for optimal policy under commitment with ramsey_model, for optimal policy under discretion with discretionary_policy or for optimal simple rules with osr (also implying commitment).

Command: planner_objective MODEL_EXPRESSION ;

This command declares the policy maker objective, for use with ramsey_model or discretionary_policy.

You need to give the one-period objective, not the discounted lifetime objective. The discount factor is given by the planner_discount option of ramsey_model and discretionary_policy. The objective function can only contain current endogenous variables and no exogenous ones. This limitation is easily circumvented by defining an appropriate auxiliary variable in the model.

With ramsey_model, you are not limited to quadratic objectives: you can give any arbitrary nonlinear expression.

With discretionary_policy, the objective function must be quadratic.

Command: evaluate_planner_objective ;

This command computes, displays, and stores the value of the planner objective function under Ramsey policy or discretion in oo_.planner_objective_value. It will provide both unconditional welfare and welfare conditional on the initial (i.e. period 0) values of the endogenous and exogenous state variables inherited by the planner. In a deterministic context, the respective initial values are set using initval or histval (depending on the exact context).

In a stochastic context, if no initial state values have been specified with histval, their values are taken to be the steady state values. Because conditional welfare is computed conditional on optimal policy by the planner in the first endogenous period (period 1), it is conditional on the information set in the period 1. This information set includes both the predetermined states inherited from period 0 (specified via histval for both endogenous and lagged exogenous states) as well as the period 1 values of the exogenous shocks. The latter are specified using the perfect foresight syntax of the shocks-block.

At the current stage, the stochastic context does not support the pruning option. At order>3, only the computation of conditional welfare with steady state Lagrange multipliers is supported. Note that at order=2, the output is based on the second-order accurate approximation of the variance stored in oo_.var.

Example (stochastic context)

var a ...;
varexo u;

model;
a = rho*a(-1)+u+u(-1);
...
end;

histval;
u(0)=1;
a(0)=-1;
end;

shocks;
var u; stderr 0.008;
var u;
periods 1;
values 1;
end;

evaluate_planner_objective;

MATLAB/Octave variable: oo_.planner_objective_value.unconditional

Scalar storing the value of unconditional welfare. In a perfect foresight context, it corresponds to welfare in the long-run, approximated as welfare in the terminal simulation period.

MATLAB/Octave variable: oo_.planner_objective_value.conditional

In a perfect foresight context, this field will be a scalar storing the value of welfare conditional on the specified initial condition and zero initial Lagrange multipliers.

In a stochastic context, it will have two subfields:

MATLAB/Octave variable: oo_.planner_objective_value.conditional.steady_initial_multiplier

Stores the value of the planner objective when the initial Lagrange multipliers associated with the planner’s problem are set to their steady state values (see ramsey_policy).

MATLAB/Octave variable: oo_.planner_objective_value.conditional.zero_initial_multiplier

Stores the value of the planner objective when the initial Lagrange multipliers associated with the planner’s problem are set to 0, i.e. it is assumed that the planner exploits its ability to surprise private agents in the first period of implementing Ramsey policy. This value corresponds to the planner implementing optimal policy for the first time and committing not to re-optimize in the future.

### 4.21.1. Optimal policy under commitment (Ramsey)¶

Dynare allows to automatically compute optimal policy choices of a Ramsey planner who takes the specified private sector equilibrium conditions into account and commits to future policy choices. Doing so requires specifying the private sector equilibrium conditions in the model-block and a planner_objective as well as potentially some instruments to facilitate computations.

Warning

Be careful when employing forward-looking auxiliary variables in the context of timeless perspective Ramsey computations. They may alter the problem the Ramsey planner will solve for the first period, although they seemingly leave the private sector equilibrium unaffected. The reason is the planner optimizes with respect to variables dated t and takes the value of time 0 variables as given, because they are predetermined. This set of initially predetermined variables will change with forward-looking definitions. Thus, users are strongly advised to use model-local variables instead.

Example

Consider a perfect foresight example where the Euler equation for the return to capital is given by

1/C=beta*1/C(+1)*(R(+1)+(1-delta))

The job of the Ramsey planner in period 1 is to choose $$C_1$$ and $$R_1$$, taking as given $$C_0$$. The above equation may seemingly equivalently be written as

1/C=beta*1/C(+1)*(R_cap);
R_cap=R(+1)+(1-delta);

due to perfect foresight. However, this changes the problem of the Ramsey planner in the first period to choosing $$C_1$$ and $$R_1$$, taking as given both $$C_0$$ and $$R^{cap}_0$$. Thus, the relevant return to capital in the Euler equation of the first period is not a choice of the planner anymore due to the forward-looking nature of the definition in the second line!

A correct specification would be to instead define R_cap as a model-local variable:

1/C=beta*1/C(+1)*(R_cap);
#R_cap=R(+1)+(1-delta);

Command: ramsey_model(OPTIONS...);

This command computes the First Order Conditions for maximizing the policy maker objective function subject to the constraints provided by the equilibrium path of the private economy.

The planner objective must be declared with the planner_objective command.

This command only creates the expanded model, it doesn’t perform any computations. It needs to be followed by other instructions to actually perform desired computations. Examples are calls to steady to compute the steady state of the Ramsey economy, to stoch_simul with various approximation orders to conduct stochastic simulations based on perturbation solutions, to estimation in order to estimate models under optimal policy with commitment, and to perfect foresight simulation routines.

See Auxiliary variables, for an explanation of how Lagrange multipliers are automatically created.

Options

This command accepts the following options:

planner_discount = EXPRESSION

Declares or reassigns the discount factor of the central planner optimal_policy_discount_factor. Default: 1.0.

planner_discount_latex_name = LATEX_NAME

Sets the LaTeX name of the optimal_policy_discount_factor parameter.

instruments = (VARIABLE_NAME,...)

Declares instrument variables for the computation of the steady state under optimal policy. Requires a steady_state_model block or a _steadystate.m file. See below.

Dynare takes advantage of the fact that the Lagrange multipliers appear linearly in the equations of the steady state of the model under optimal policy. Nevertheless, it is in general very difficult to compute the steady state with simply a numerical guess in initval for the endogenous variables.

It greatly facilitates the computation, if the user provides an analytical solution for the steady state (in steady_state_model block or in a _steadystate.m file). In this case, it is necessary to provide a steady state solution CONDITIONAL on the value of the instruments in the optimal policy problem and declared with the option instruments. The initial value of the instrument for steady state finding in this case is set with initval. Note that computing and displaying steady state values using the steady-command or calls to resid must come after the ramsey_model statement and the initval-block.

Note that choosing the instruments is partly a matter of interpretation and you can choose instruments that are handy from a mathematical point of view but different from the instruments you would refer to in the analysis of the paper. A typical example is choosing inflation or nominal interest rate as an instrument.

Block: ramsey_constraints ;

This block lets you define constraints on the variables in the Ramsey problem. The constraints take the form of a variable, an inequality operator (> or <) and a constant.

Example

ramsey_constraints;
i > 0;
end;

Command: ramsey_policy [VARIABLE_NAME...];
Command: ramsey_policy(OPTIONS...) [VARIABLE_NAME...];

This command is deprecated and formally equivalent to the calling sequence

ramsey_model;
stoch_simul;
evaluate_planner_objective;

It computes an approximation of the policy that maximizes the policy maker’s objective function subject to the constraints provided by the equilibrium path of the private economy and under commitment to this optimal policy. The Ramsey policy is computed by approximating the equilibrium system around the perturbation point where the Lagrange multipliers are at their steady state, i.e. where the Ramsey planner acts as if the initial multipliers had been set to 0 in the distant past, giving them time to converge to their steady state value. Consequently, the optimal decision rules are computed around this steady state of the endogenous variables and the Lagrange multipliers.

Note that the variables in the list after the ramsey_policy or stoch_simul-command can also contain multiplier names, but in a case-sensititve way (e.g. MULT_1). In that case, Dynare will for example display the IRFs of the respective multipliers when irf>0.

The planner objective must be declared with the planner_objective command.

Options

This command accepts all options of stoch_simul, plus:

planner_discount = EXPRESSION
instruments = (VARIABLE_NAME,...)

Declares instrument variables for the computation of the steady state under optimal policy. Requires a steady_state_model block or a _steadystate.m file. See below.

Output

This command generates all the output variables of stoch_simul. For specifying the initial values for the endogenous state variables (except for the Lagrange multipliers), see above.

### 4.21.2. Optimal policy under discretion¶

Command: discretionary_policy [VARIABLE_NAME...];
Command: discretionary_policy(OPTIONS...) [VARIABLE_NAME...];

This command computes an approximation of the optimal policy under discretion. The algorithm implemented is essentially an LQ solver, and is described by Dennis (2007).

You must ensure that your objective is quadratic. Regarding the model, it must either be linear or solved at first order with an analytical steady state provided. In the first case, you should set the linear option of the model block.

It is possible to use the estimation command after the discretionary_policy command, in order to estimate the model with optimal policy under discretion and evaluate_planner_objective to compute welfare.

Options

This command accepts the same options as ramsey_policy, plus:

discretionary_tol = NON-NEGATIVE DOUBLE

Sets the tolerance level used to assess convergence of the solution algorithm. Default: 1e-7.

maxit = INTEGER

Maximum number of iterations. Default: 3000.

### 4.21.3. Optimal Simple Rules (OSR)¶

Command: osr [VARIABLE_NAME...];
Command: osr(OPTIONS...) [VARIABLE_NAME...];

This command computes optimal simple policy rules for linear-quadratic problems of the form:

$\min_\gamma E(y'_tWy_t)$

such that:

$A_1 E_ty_{t+1}+A_2 y_t+ A_3 y_{t-1}+C e_t=0$

where:

• $$E$$ denotes the unconditional expectations operator;

• $$\gamma$$ are parameters to be optimized. They must be elements of the matrices $$A_1$$, $$A_2$$, $$A_3$$, i.e. be specified as parameters in the params command and be entered in the model block;

• $$y$$ are the endogenous variables, specified in the var command, whose (co)-variance enters the loss function;

• $$e$$ are the exogenous stochastic shocks, specified in the varexo- ommand;

• $$W$$ is the weighting matrix;

The linear quadratic problem consists of choosing a subset of model parameters to minimize the weighted (co)-variance of a specified subset of endogenous variables, subject to a linear law of motion implied by the first order conditions of the model. A few things are worth mentioning. First, $$y$$ denotes the selected endogenous variables’ deviations from their steady state, i.e. in case they are not already mean 0 the variables entering the loss function are automatically demeaned so that the centered second moments are minimized. Second, osr only solves linear quadratic problems of the type resulting from combining the specified quadratic loss function with a first order approximation to the model’s equilibrium conditions. The reason is that the first order state-space representation is used to compute the unconditional (co)-variances. Hence, osr will automatically select order=1. Third, because the objective involves minimizing a weighted sum of unconditional second moments, those second moments must be finite. In particular, unit roots in $$y$$ are not allowed.

The subset of the model parameters over which the optimal simple rule is to be optimized, $$\gamma$$, must be listed with osr_params.

The weighting matrix $$W$$ used for the quadratic objective function is specified in the optim_weights block. By attaching weights to endogenous variables, the subset of endogenous variables entering the objective function, $$y$$, is implicitly specified.

The linear quadratic problem is solved using the numerical optimizer specified with opt_algo.

Options

The osr command will subsequently run stoch_simul and accepts the same options, including restricting the endogenous variables by listing them after the command, as stoch_simul (see Stochastic solution and simulation) plus

opt_algo = INTEGER

Specifies the optimizer for minimizing the objective function. The same solvers as for mode_compute (see mode_compute) are available, except for 5, 6, and 10.

optim = (NAME, VALUE, ...)

A list of NAME and VALUE pairs. Can be used to set options for the optimization routines. The set of available options depends on the selected optimization routine (i.e. on the value of option opt_algo). See optim.

maxit = INTEGER

Determines the maximum number of iterations used in opt_algo=4. This option is now deprecated and will be removed in a future release of Dynare. Use optim instead to set optimizer-specific values. Default: 1000.

tolf = DOUBLE

Convergence criterion for termination based on the function value used in opt_algo=4. Iteration will cease when it proves impossible to improve the function value by more than tolf. This option is now deprecated and will be removed in a future release of Dynare. Use optim instead to set optimizer-specific values. Default: e-7.

silent_optimizer
huge_number = DOUBLE

Value for replacing the infinite bounds on parameters by finite numbers. Used by some optimizers for numerical reasons (see huge_number). Users need to make sure that the optimal parameters are not larger than this value. Default: 1e7.

The value of the objective is stored in the variable oo_.osr.objective_function and the value of parameters at the optimum is stored in oo_.osr.optim_params. See below for more details.

After running osr the parameters entering the simple rule will be set to their optimal value so that subsequent runs of stoch_simul will be conducted at these values.

Command: osr_params PARAMETER_NAME...;

This command declares parameters to be optimized by osr.

Block: optim_weights ;

This block specifies quadratic objectives for optimal policy problems.

More precisely, this block specifies the nonzero elements of the weight matrix $$W$$ used in the quadratic form of the objective function in osr.

An element of the diagonal of the weight matrix is given by a line of the form:

VARIABLE_NAME EXPRESSION;

An off-the-diagonal element of the weight matrix is given by a line of the form:

VARIABLE_NAME,  VARIABLE_NAME EXPRESSION;

Example

var y inflation r;
varexo y_ inf_;

parameters delta sigma alpha kappa gammarr gammax0 gammac0 gamma_y_ gamma_inf_;

delta =  0.44;
kappa =  0.18;
alpha =  0.48;
sigma = -0.06;

gammarr = 0;
gammax0 = 0.2;
gammac0 = 1.5;
gamma_y_ = 8;
gamma_inf_ = 3;

model(linear);
y  = delta * y(-1)  + (1-delta)*y(+1)+sigma *(r - inflation(+1)) + y_;
inflation  =   alpha * inflation(-1) + (1-alpha) * inflation(+1) + kappa*y + inf_;
r = gammax0*y(-1)+gammac0*inflation(-1)+gamma_y_*y_+gamma_inf_*inf_;
end;

shocks;
var y_; stderr 0.63;
var inf_; stderr 0.4;
end;

optim_weights;
inflation 1;
y 1;
y, inflation 0.5;
end;

osr_params gammax0 gammac0 gamma_y_ gamma_inf_;
osr y;

Block: osr_params_bounds ;

This block declares lower and upper bounds for parameters in the optimal simple rule. If not specified the optimization is unconstrained.

Each line has the following syntax:

PARAMETER_NAME, LOWER_BOUND, UPPER_BOUND;

Note that the use of this block requires the use of a constrained optimizer, i.e. setting opt_algo to 1, 2, 5 or 9.

Example

osr_params_bounds;
gamma_inf_, 0, 2.5;
end;

osr(opt_algo=9) y;

MATLAB/Octave variable: oo_.osr.objective_function

After an execution of the osr command, this variable contains the value of the objective under optimal policy.

MATLAB/Octave variable: oo_.osr.optim_params

After an execution of the osr command, this variable contains the value of parameters at the optimum, stored in fields of the form oo_.osr.optim_params.PARAMETER_NAME.

MATLAB/Octave variable: M_.osr.param_names

After an execution of the osr command, this cell contains the names of the parameters.

MATLAB/Octave variable: M_.osr.param_indices

After an execution of the osr command, this vector contains the indices of the OSR parameters in M_.params.

MATLAB/Octave variable: M_.osr.param_bounds

After an execution of the osr command, this two by number of OSR parameters matrix contains the lower and upper bounds of the parameters in the first and second column, respectively.

MATLAB/Octave variable: M_.osr.variable_weights

After an execution of the osr command, this sparse matrix contains the weighting matrix associated with the variables in the objective function.

MATLAB/Octave variable: M_.osr.variable_indices

After an execution of the osr command, this vector contains the indices of the variables entering the objective function in M_.endo_names.

## 4.22. Sensitivity and identification analysis¶

Dynare provides an interface to the global sensitivity analysis (GSA) toolbox (developed by the Joint Research Center (JRC) of the European Commission), which is now part of the official Dynare distribution. The GSA toolbox can be used to answer the following questions:

1. What is the domain of structural coefficients assuring the stability and determinacy of a DSGE model?

2. Which parameters mostly drive the fit of, e.g., GDP and which the fit of inflation? Is there any conflict between the optimal fit of one observed series versus another?

3. How to represent in a direct, albeit approximated, form the relationship between structural parameters and the reduced form of a rational expectations model?

The discussion of the methodologies and their application is described in Ratto (2008).

With respect to the previous version of the toolbox, in order to work properly, the GSA toolbox no longer requires that the Dynare estimation environment is set up.

### 4.22.1. Performing sensitivity analysis¶

Command: dynare_sensitivity ;
Command: dynare_sensitivity(OPTIONS...);

This command triggers sensitivity analysis on a DSGE model.

Sampling Options

Nsam = INTEGER

Size of the Monte-Carlo sample. Default: 2048.

ilptau = INTEGER

If equal to 1, use $$LP_\tau$$ quasi-Monte-Carlo. If equal to 0, use LHS Monte-Carlo. Default: 1.

pprior = INTEGER

If equqal to 1, sample from the prior distributions. If equal to 0, sample from the multivariate normal $$N(\bar{\theta},\Sigma)$$, where $$\bar{\theta}$$ is the posterior mode and $$\Sigma=H^{-1}$$, $$H$$ is the Hessian at the mode. Default: 1.

prior_range = INTEGER

If equal to 1, sample uniformly from prior ranges. If equal to 0, sample from prior distributions. Default: 1.

morris = INTEGER

If equal to 0, ANOVA mapping (Type I error) If equal to 1, Screening analysis (Type II error). If equal to 2, Analytic derivatives (similar to Type II error, only valid when identification=1). Default: 1 when identification=1, 0 otherwise.

morris_nliv = INTEGER

Number of levels in Morris design. Default: 6.

morris_ntra = INTEGER

Number trajectories in Morris design. Default: 20.

ppost = INTEGER

If equal to 1, use Metropolis posterior sample. If equal to 0, do not use Metropolis posterior sample. Default: 0.

NB: This overrides any other sampling option.

neighborhood_width = DOUBLE

When pprior=0 and ppost=0, allows for the sampling of parameters around the value specified in the mode_file, in the range $$\texttt{xparam1} \pm \left \vert \texttt{xparam1} \times \texttt{neighborhood\_width} \right \vert$$. Default: 0.

Stability Mapping Options

stab = INTEGER

If equal to 1, perform stability mapping. If equal to 0, do not perform stability mapping. Default: 1.

load_stab = INTEGER

If equal to 1, load a previously created sample. If equal to 0, generate a new sample. Default: 0.

alpha2_stab = DOUBLE

Critical value for correlations $$\rho$$ in filtered samples: plot couples of parmaters with $$\left\vert\rho\right\vert>$$ alpha2_stab. Default: 0.

pvalue_ks = DOUBLE

The threshold $$pvalue$$ for significant Kolmogorov-Smirnov test (i.e. plot parameters with $$pvalue<$$ pvalue_ks). Default: 0.001.

pvalue_corr = DOUBLE

The threshold $$pvalue$$ for significant correlation in filtered samples (i.e. plot bivariate samples when $$pvalue<$$ pvalue_corr). Default: 1e-5.

Reduced Form Mapping Options

redform = INTEGER

If equal to 1, prepare Monte-Carlo sample of reduced form matrices. If equal to 0, do not prepare Monte-Carlo sample of reduced form matrices. Default: 0.

load_redform = INTEGER

If equal to 1, load previously estimated mapping. If equal to 0, estimate the mapping of the reduced form model. Default: 0.

logtrans_redform = INTEGER

If equal to 1, use log-transformed entries. If equal to 0, use raw entries. Default: 0.

threshold_redform = [DOUBLE DOUBLE]

The range over which the filtered Monte-Carlo entries of the reduced form coefficients should be analyzed. The first number is the lower bound and the second is the upper bound. An empty vector indicates that these entries will not be filtered. Default: empty.

ksstat_redform = DOUBLE

Critical value for Smirnov statistics $$d$$ when reduced form entries are filtered. Default: 0.001.

alpha2_redform = DOUBLE

Critical value for correlations $$\rho$$ when reduced form entries are filtered. Default: 1e-5.

namendo = (VARIABLE_NAME...)

List of endogenous variables. ‘:’ indicates all endogenous variables. Default: empty.

namlagendo = (VARIABLE_NAME...)

List of lagged endogenous variables. ‘:’ indicates all lagged endogenous variables. Analyze entries [namendo $$\times$$ namlagendo] Default: empty.

namexo = (VARIABLE_NAME...)

List of exogenous variables. ‘:’ indicates all exogenous variables. Analyze entries [namendo $$\times$$ namexo]. Default: empty.

RMSE Options

rmse = INTEGER

If equal to 1, perform RMSE analysis. If equal to 0, do not perform RMSE analysis. Default: 0.

load_rmse = INTEGER

If equal to 1, load previous RMSE analysis. If equal to 0, make a new RMSE analysis. Default: 0.

lik_only = INTEGER

If equal to 1, compute only likelihood and posterior. If equal to 0, compute RMSE’s for all observed series. Default: 0.

var_rmse = (VARIABLE_NAME...)

List of observed series to be considered. ‘:’ indicates all observed variables. Default: varobs.

pfilt_rmse = DOUBLE

Filtering threshold for RMSE’s. Default: 0.1.

istart_rmse = INTEGER

Value at which to start computing RMSE’s (use 2 to avoid big intitial error). Default: presample+1.

alpha_rmse = DOUBLE

Critical value for Smirnov statistics $$d$$: plot parameters with $$d>$$ alpha_rmse. Default: 0.001.

alpha2_rmse = DOUBLE

Critical value for correlation $$\rho$$: plot couples of parmaters with $$\left\vert\rho\right\vert=$$ alpha2_rmse. Default: 1e-5.

datafile = FILENAME

See datafile.

nobs = INTEGER
nobs = [INTEGER1:INTEGER2]

See nobs.

first_obs = INTEGER
prefilter = INTEGER
presample = INTEGER
nograph
nodisplay
graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
conf_sig = DOUBLE

See conf_sig.

loglinear

See loglinear.

mode_file = FILENAME
kalman_algo = INTEGER

Identification Analysis Options

identification = INTEGER

If equal to 1, performs identification analysis (forcing redform=0 and morris=1) If equal to 0, no identification analysis. Default: 0.

morris = INTEGER

See morris.

morris_nliv = INTEGER
morris_ntra = INTEGER
load_ident_files = INTEGER

Loads previously performed identification analysis. Default: 0.

useautocorr = INTEGER

Use autocorrelation matrices in place of autocovariance matrices in moments for identification analysis. Default: 0.

ar = INTEGER

Maximum number of lags for moments in identification analysis. Default: 1.

diffuse_filter = INTEGER

### 4.22.2. IRF/Moment calibration¶

The irf_calibration and moment_calibration blocks allow imposing implicit “endogenous” priors about IRFs and moments on the model. The way it works internally is that any parameter draw that is inconsistent with the “calibration” provided in these blocks is discarded, i.e. assigned a prior density of 0. In the context of dynare_sensitivity, these restrictions allow tracing out which parameters are driving the model to satisfy or violate the given restrictions.

IRF and moment calibration can be defined in irf_calibration and moment_calibration blocks:

Block: irf_calibration ;
Block: irf_calibration(OPTIONS...);

This block allows defining IRF calibration criteria and is terminated by end;. To set IRF sign restrictions, the following syntax is used:

VARIABLE_NAME(INTEGER), EXOGENOUS_NAME, -;
VARIABLE_NAME(INTEGER:INTEGER), EXOGENOUS_NAME, +;

To set IRF restrictions with specific intervals, the following syntax is used:

VARIABLE_NAME(INTEGER), EXOGENOUS_NAME, [EXPRESSION, EXPRESSION];
VARIABLE_NAME(INTEGER:INTEGER), EXOGENOUS_NAME, [EXPRESSION, EXPRESSION];

When (INTEGER:INTEGER) is used, the restriction is considered to be fulfilled by a logical OR. A list of restrictions must always be fulfilled with logical AND.

Options

relative_irf

Example

irf_calibration;
y(1:4), e_ys, [-50, 50]; //[first year response with logical OR]
@#for ilag in 21:40
R_obs(@{ilag}), e_ys, [0, 6]; //[response from 5th to 10th years with logical AND]
@#endfor
end;

Block: moment_calibration ;
Block: moment_calibration(OPTIONS...);

This block allows defining moment calibration criteria. This block is terminated by end;, and contains lines of the form:

VARIABLE_NAME1, VARIABLE_NAME2(+/-INTEGER), [EXPRESSION, EXPRESSION];
VARIABLE_NAME1, VARIABLE_NAME2(+/-INTEGER), +/-;
VARIABLE_NAME1, VARIABLE_NAME2(+/-(INTEGER:INTEGER)), [EXPRESSION, EXPRESSION];
VARIABLE_NAME1, VARIABLE_NAME2((-INTEGER:+INTEGER)), [EXPRESSION, EXPRESSION];

When (INTEGER:INTEGER) is used, the restriction is considered to be fulfilled by a logical OR. A list of restrictions must always be fulfilled with logical AND. The moment restrictions generally apply to auto- and cross-correlations between variables. The only exception is a restriction on the unconditional variance of an endogenous variable, specified as shown in the example below.

Example

moment_calibration;
y_obs,y_obs, [0.5, 1.5]; //[unconditional variance]
y_obs,y_obs(-(1:4)), +; //[sign restriction for first year autocorrelation with logical OR]
@#for ilag in -2:2
y_obs,R_obs(@{ilag}), -; //[-2:2 cross correlation with logical AND]
@#endfor
@#for ilag in -4:4
y_obs,pie_obs(@{ilag}), -; //[-4_4 cross correlation with logical AND]
@#endfor
end;

### 4.22.3. Performing identification analysis¶

Command: identification ;
Command: identification(OPTIONS...);

This command triggers:

1. Theoretical identification analysis based on

• moments as in Iskrev (2010)

• spectral density as in Qu and Tkachenko (2012)

• minimal system as in Komunjer and Ng (2011)

• reduced-form solution and linear rational expectation model as in Ratto and Iskrev (2011)

Note that for orders 2 and 3, all identification checks are based on the pruned state space system as in Mutschler (2015). That is, theoretical moments and spectrum are computed from the pruned ABCD-system, whereas the minimal system criteria is based on the first-order system, but augmented by the theoretical (pruned) mean at order 2 or 3.

2. Identification strength analysis based on (theoretical or simulated) curvature of moment information matrix as in Ratto and Iskrev (2011)

3. Parameter checks based on nullspace and multicorrelation coefficients to determine which (combinations of) parameters are involved

General Options

order = 1|2|3

Order of approximation. At orders 2 and 3 identification is based on the pruned state space system. Note that the order set in other functions does not overwrite the default. Default: 1.

parameter_set = OPTION

See parameter_set for possible values. Default: prior_mean.

prior_mc = INTEGER

Size of Monte-Carlo sample. Default: 1.

prior_range = INTEGER

Triggers uniform sample within the range implied by the prior specifications (when prior_mc>1). Default: 0.

advanced = INTEGER

If set to 1, shows a more detailed analysis, comprised of an analysis for the linearized rational expectation model as well as the associated reduced form solution. Further performs a bruteforce search of the groups of parameters best reproducing the behavior of each single parameter. The maximum dimension of the group searched is triggered by max_dim_cova_group. Default: 0.

max_dim_cova_group = INTEGER

In the brute force search (performed when advanced=1) this option sets the maximum dimension of groups of parameters that best reproduce the behavior of each single model parameter. Default: 2.

gsa_sample_file = INTEGER|FILENAME

If equal to 0, do not use sample file. If equal to 1, triggers gsa prior sample. If equal to 2, triggers gsa Monte-Carlo sample (i.e. loads a sample corresponding to pprior=0 and ppost=0 in the dynare_sensitivity options). If equal to FILENAME uses the provided path to a specific user defined sample file. Default: 0.

diffuse_filter

Deals with non-stationary cases. See diffuse_filter.

Numerical Options

analytic_derivation_mode = INTEGER

Different ways to compute derivatives either analytically or numerically. Possible values are:

• 0: efficient sylvester equation method to compute analytical derivatives

• 1: kronecker products method to compute analytical derivatives (only at order=1)

• -1: numerical two-sided finite difference method to compute all identification Jacobians (numerical tolerance level is equal to options_.dynatol.x)

• -2: numerical two-sided finite difference method to compute derivatives of steady state and dynamic model numerically, the identification Jacobians are then computed analytically (numerical tolerance level is equal to options_.dynatol.x)

Default: 0.

normalize_jacobians = INTEGER

If set to 1: Normalize Jacobian matrices by rescaling each row by its largest element in absolute value. Normalize Gram (or Hessian-type) matrices by transforming into correlation-type matrices. Default: 1

tol_rank = DOUBLE

Tolerance level used for rank computations. Default: 1.e-10.

tol_deriv = DOUBLE

Tolerance level for selecting non-zero columns in Jacobians. Default: 1.e-8.

tol_sv = DOUBLE

Tolerance level for selecting non-zero singular values. Default: 1.e-3.

schur_vec_tol = DOUBLE

Identification Strength Options

no_identification_strength

Disables computations of identification strength analysis based on sample information matrix.

periods = INTEGER

When the analytic Hessian is not available (i.e. with missing values or diffuse Kalman filter or univariate Kalman filter), this triggers the length of stochastic simulation to compute Simulated Moments Uncertainty. Default: 300.

replic = INTEGER

When the analytic Hessian is not available, this triggers the number of replicas to compute Simulated Moments Uncertainty. Default: 100.

Moments Options

no_identification_moments

Disables computations of identification check based on Iskrev (2010)’s J, i.e. derivative of first two moments.

ar = INTEGER

Number of lags of computed autocovariances/autocorrelations (theoretical moments) in Iskrev (2010)’s J criteria. Default: 1.

useautocorr = INTEGER

If equal to 1, compute derivatives of autocorrelation. If equal to 0, compute derivatives of autocovariances. Default: 0.

Spectrum Options

no_identification_spectrum

Disables computations of identification check based on Qu and Tkachenko (2012)’s G, i.e. Gram matrix of derivatives of first moment plus outer product of derivatives of spectral density.

grid_nbr = INTEGER

Number of grid points in [-pi;pi] to approximate the integral to compute Qu and Tkachenko (2012)’s G criteria. Default: 5000.

Minimal State Space System Options

no_identification_minimal

Disables computations of identification check based on Komunjer and Ng (2011)’s D, i.e. minimal state space system and observational equivalent spectral density transformations.

Misc Options

nograph
nodisplay
graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
tex

See tex.

Debug Options

load_ident_files = INTEGER

If equal to 1, allow Dynare to load previously computed analyzes. Default: 0.

lik_init = INTEGER
kalman_algo = INTEGER
no_identification_reducedform

Disables computations of identification check based on steady state and reduced-form solution.

checks_via_subsets = INTEGER

If equal to 1: finds problematic parameters in a bruteforce fashion: It computes the rank of the Jacobians for all possible parameter combinations. If the rank condition is not fullfilled, these parameter sets are flagged as non-identifiable. The maximum dimension of the group searched is triggered by max_dim_subsets_groups. Default: 0.

max_dim_subsets_groups = INTEGER

Sets the maximum dimension of groups of parameters for which the above bruteforce search is performed. Default: 4.

### 4.22.4. Types of analysis and output files¶

The sensitivity analysis toolbox includes several types of analyses. Sensitivity analysis results are saved locally in <mod_file>/gsa, where <mod_file>.mod is the name of the Dynare model file.

#### 4.22.4.1. Sampling¶

The following binary files are produced:

• <mod_file>_prior.mat: this file stores information about the analyses performed sampling from the prior, i.e. pprior=1 and ppost=0;

• <mod_file>_mc.mat: this file stores information about the analyses performed sampling from multivariate normal, i.e. pprior=0 and ppost=0;

• <mod_file>_post.mat: this file stores information about analyses performed using the Metropolis posterior sample, i.e. ppost=1.

#### 4.22.4.2. Stability Mapping¶

Figure files produced are of the form <mod_file>_prior_*.fig and store results for stability mapping from prior Monte-Carlo samples:

• <mod_file>_prior_stable.fig: plots of the Smirnov test and the correlation analyses confronting the cdf of the sample fulfilling Blanchard-Kahn conditions (blue color) with the cdf of the rest of the sample (red color), i.e. either instability or indeterminacy or the solution could not be found (e.g. the steady state solution could not be found by the solver);

• <mod_file>_prior_indeterm.fig: plots of the Smirnov test and the correlation analyses confronting the cdf of the sample producing indeterminacy (red color) with the cdf of the rest of the sample (blue color);

• <mod_file>_prior_unstable.fig: plots of the Smirnov test and the correlation analyses confronting the cdf of the sample producing explosive roots (red color) with the cdf of the rest of the sample (blue color);

• <mod_file>_prior_wrong.fig: plots of the Smirnov test and the correlation analyses confronting the cdf of the sample where the solution could not be found (e.g. the steady state solution could not be found by the solver - red color) with the cdf of the rest of the sample (blue color);

• <mod_file>_prior_calib.fig: plots of the Smirnov test and the correlation analyses splitting the sample fulfilling Blanchard-Kahn conditions, by confronting the cdf of the sample where IRF/moment restrictions are matched (blue color) with the cdf where IRF/moment restrictions are NOT matched (red color);

Similar conventions apply for <mod_file>_mc_*.fig files, obtained when samples from multivariate normal are used.

#### 4.22.4.3. IRF/Moment restrictions¶

The following binary files are produced:

• <mod_file>_prior_restrictions.mat: this file stores information about the IRF/moment restriction analysis performed sampling from the prior ranges, i.e. pprior=1 and ppost=0;

• <mod_file>_mc_restrictions.mat: this file stores information about the IRF/moment restriction analysis performed sampling from multivariate normal, i.e. pprior=0 and ppost=0;

• <mod_file>_post_restrictions.mat: this file stores information about IRF/moment restriction analysis performed using the Metropolis posterior sample, i.e. ppost=1.

Figure files produced are of the form <mod_file>_prior_irf_calib_*.fig and <mod_file>_prior_moment_calib_*.fig and store results for mapping restrictions from prior Monte-Carlo samples:

• <mod_file>_prior_irf_calib_<ENDO_NAME>_vs_<EXO_NAME>_<PERIOD>.fig: plots of the Smirnov test and the correlation analyses splitting the sample fulfilling Blanchard-Kahn conditions, by confronting the cdf of the sample where the individual IRF restriction <ENDO_NAME> vs. <EXO_NAME> at period(s) <PERIOD> is matched (blue color) with the cdf where the IRF restriction is NOT matched (red color)

• <mod_file>_prior_irf_calib_<ENDO_NAME>_vs_<EXO_NAME>_ALL.fig: plots of the Smirnov test and the correlation analyses splitting the sample fulfilling Blanchard-Kahn conditions, by confronting the cdf of the sample where ALL the individual IRF restrictions for the same couple <ENDO_NAME> vs. <EXO_NAME> are matched (blue color) with the cdf where the IRF restriction is NOT matched (red color)

• <mod_file>_prior_irf_restrictions.fig: plots visual information on the IRF restrictions compared to the actual Monte Carlo realization from prior sample.

• <mod_file>_prior_moment_calib_<ENDO_NAME1>_vs_<ENDO_NAME2>_<LAG>.fig: plots of the Smirnov test and the correlation analyses splitting the sample fulfilling Blanchard-Kahn conditions, by confronting the cdf of the sample where the individual acf/ccf moment restriction <ENDO_NAME1> vs. <ENDO_NAME2> at lag(s) <LAG> is matched (blue color) with the cdf where the IRF restriction is NOT matched (red color)

• <mod_file>_prior_moment_calib_<ENDO_NAME>_vs_<EXO_NAME>_ALL.fig: plots of the Smirnov test and the correlation analyses splitting the sample fulfilling Blanchard-Kahn conditions, by confronting the cdf of the sample where ALL the individual acf/ccf moment restrictions for the same couple <ENDO_NAME1> vs. <ENDO_NAME2> are matched (blue color) with the cdf where the IRF restriction is NOT matched (red color)

• <mod_file>_prior_moment_restrictions.fig: plots visual information on the moment restrictions compared to the actual Monte Carlo realization from prior sample.

Similar conventions apply for <mod_file>_mc_*.fig and <mod_file>_post_*.fig files, obtained when samples from multivariate normal or from posterior are used.

#### 4.22.4.4. Reduced Form Mapping¶

When the option threshold_redform is not set, or it is empty (the default), this analysis estimates a multivariate smoothing spline ANOVA model (the ’mapping’) for the selected entries in the transition matrix of the shock matrix of the reduce form first order solution of the model. This mapping is done either with prior samples or with MC samples with neighborhood_width. Unless neighborhood_width is set with MC samples, the mapping of the reduced form solution forces the use of samples from prior ranges or prior distributions, i.e.: pprior=1 and ppost=0. It uses 250 samples to optimize smoothing parameters and 1000 samples to compute the fit. The rest of the sample is used for out-of-sample validation. One can also load a previously estimated mapping with a new Monte-Carlo sample, to look at the forecast for the new Monte-Carlo sample.

The following synthetic figures are produced:

• <mod_file>_redform_<endo name>_vs_lags_*.fig: shows bar charts of the sensitivity indices for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus lagged endogenous variables (namlagendo); suffix log indicates the results for log-transformed entries;

• <mod_file>_redform_<endo name>_vs_shocks_*.fig: shows bar charts of the sensitivity indices for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus exogenous variables (namexo); suffix log indicates the results for log-transformed entries;

• <mod_file>_redform_gsa(_log).fig: shows bar chart of all sensitivity indices for each parameter: this allows one to notice parameters that have a minor effect for any of the reduced form coefficients.

Detailed results of the analyses are shown in the subfolder <mod_file>/gsa/redform_prior for prior samples and in <mod_file>/gsa/redform_mc for MC samples with option neighborhood_width, where the detailed results of the estimation of the single functional relationships between parameters $$\theta$$ and reduced form coefficient (denoted as $$y$$ hereafter) are stored in separate directories named as:

• <namendo>_vs_<namlagendo>, for the entries of the transition matrix;

• <namendo>_vs_<namexo>, for entries of the matrix of the shocks.

The following files are stored in each directory (we stick with prior sample but similar conventions are used for MC samples):

• <mod_file>_prior_<namendo>_vs_<namexo>.fig: histogram and CDF plot of the MC sample of the individual entry of the shock matrix, in sample and out of sample fit of the ANOVA model;

• <mod_file>_prior_<namendo>_vs_<namexo>_map_SE.fig: for entries of the shock matrix it shows graphs of the estimated first order ANOVA terms $$y = f(\theta_i)$$ for each deep parameter $$\theta_i$$;

• <mod_file>_prior_<namendo>_vs_<namlagendo>.fig: histogram and CDF plot of the MC sample of the individual entry of the transition matrix, in sample and out of sample fit of the ANOVA model;

• <mod_file>_prior_<namendo>_vs_<namlagendo>_map_SE.fig: for entries of the transition matrix it shows graphs of the estimated first order ANOVA terms $$y = f(\theta_i)$$ for each deep parameter $$\theta_i$$;

• <mod_file>_prior_<namendo>_vs_<namexo>_map.mat, <mod_file>_<namendo>_vs_<namlagendo>_map.mat: these files store info in the estimation;

When option logtrans_redform is set, the ANOVA estimation is performed using a log-transformation of each y. The ANOVA mapping is then transformed back onto the original scale, to allow comparability with the baseline estimation. Graphs for this log-transformed case, are stored in the same folder in files denoted with the _log suffix.

When the option threshold_redform is set, the analysis is performed via Monte Carlo filtering, by displaying parameters that drive the individual entry y inside the range specified in threshold_redform. If no entry is found (or all entries are in the range), the MCF algorithm ignores the range specified in threshold_redform and performs the analysis splitting the MC sample of y into deciles. Setting threshold_redform=[-inf inf] triggers this approach for all y’s.

Results are stored in subdirectories of <mod_file>/gsa/redform_prior named

• <mod_file>_prior_<namendo>_vs_<namlagendo>_threshold, for the entries of the transition matrix;

• <mod_file>_prior_<namendo>_vs_<namexo>_threshold, for entries of the matrix of the shocks.

The files saved are named:

• <mod_file>_prior_<namendo>_vs_<namexo>_threshold.fig, <mod_file>_<namendo>_vs_<namlagendo>_threshold.fig: graphical outputs;

• <mod_file>_prior_<namendo>_vs_<namexo>_threshold.mat, <mod_file>_<namendo>_vs_<namlagendo>_threshold.mat: info on the analysis;

#### 4.22.4.5. RMSE¶

The RMSE analysis can be performed with different types of sampling options:

1. When pprior=1 and ppost=0, the toolbox analyzes the RMSEs for the Monte-Carlo sample obtained by sampling parameters from their prior distributions (or prior ranges): this analysis provides some hints about what parameter drives the fit of which observed series, prior to the full estimation;

2. When pprior=0 and ppost=0, the toolbox analyzes the RMSEs for a multivariate normal Monte-Carlo sample, with covariance matrix based on the inverse Hessian at the optimum: this analysis is useful when maximum likelihood estimation is done (i.e. no Bayesian estimation);

3. When ppost=1 the toolbox analyzes the RMSEs for the posterior sample obtained by Dynare’s Metropolis procedure.

The use of cases 2 and 3 requires an estimation step beforehand. To facilitate the sensitivity analysis after estimation, the dynare_sensitivity command also allows you to indicate some options of the estimation command. These are:

• datafile

• nobs

• first_obs

• prefilter

• presample

• nograph

• nodisplay

• graph_format

• conf_sig

• loglinear

• mode_file

Binary files produced my RMSE analysis are:

• <mod_file>_prior_*.mat: these files store the filtered and smoothed variables for the prior Monte-Carlo sample, generated when doing RMSE analysis (pprior=1 and ppost=0);

• <mode_file>_mc_*.mat: these files store the filtered and smoothed variables for the multivariate normal Monte-Carlo sample, generated when doing RMSE analysis (pprior=0 and ppost=0).

Figure files <mod_file>_rmse_*.fig store results for the RMSE analysis.

• <mod_file>_rmse_prior*.fig: save results for the analysis using prior Monte-Carlo samples;

• <mod_file>_rmse_mc*.fig: save results for the analysis using multivariate normal Monte-Carlo samples;

• <mod_file>_rmse_post*.fig: save results for the analysis using Metropolis posterior samples.

The following types of figures are saved (we show prior sample to fix ideas, but the same conventions are used for multivariate normal and posterior):

• <mod_file>_rmse_prior_params_*.fig: for each parameter, plots the cdfs corresponding to the best 10% RMSEs of each observed series (only those cdfs below the significance threshold alpha_rmse);

• <mod_file>_rmse_prior_<var_obs>_*.fig: if a parameter significantly affects the fit of var_obs, all possible trade-off’s with other observables for same parameter are plotted;

• <mod_file>_rmse_prior_<var_obs>_map.fig: plots the MCF analysis of parameters significantly driving the fit the observed series var_obs;

• <mod_file>_rmse_prior_lnlik*.fig: for each observed series, plots in BLUE the cdf of the log-likelihood corresponding to the best 10% RMSEs, in RED the cdf of the rest of the sample and in BLACK the cdf of the full sample; this allows one to see the presence of some idiosyncratic behavior;

• <mod_file>_rmse_prior_lnpost*.fig: for each observed series, plots in BLUE the cdf of the log-posterior corresponding to the best 10% RMSEs, in RED the cdf of the rest of the sample and in BLACK the cdf of the full sample; this allows one to see idiosyncratic behavior;

• <mod_file>_rmse_prior_lnprior*.fig: for each observed series, plots in BLUE the cdf of the log-prior corresponding to the best 10% RMSEs, in RED the cdf of the rest of the sample and in BLACK the cdf of the full sample; this allows one to see idiosyncratic behavior;

• <mod_file>_rmse_prior_lik.fig: when lik_only=1, this shows the MCF tests for the filtering of the best 10% log-likelihood values;

• <mod_file>_rmse_prior_post.fig: when lik_only=1, this shows the MCF tests for the filtering of the best 10% log-posterior values.

#### 4.22.4.6. Screening Analysis¶

Screening analysis does not require any additional options with respect to those listed in Sampling Options. The toolbox performs all the analyses required and displays results.

The results of the screening analysis with Morris sampling design are stored in the subfolder <mod_file>/gsa/screen. The data file <mod_file>_prior stores all the information of the analysis (Morris sample, reduced form coefficients, etc.).

Screening analysis merely concerns reduced form coefficients. Similar synthetic bar charts as for the reduced form analysis with Monte-Carlo samples are saved:

• <mod_file>_redform_<endo name>_vs_lags_*.fig: shows bar charts of the elementary effect tests for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus lagged endogenous variables (namlagendo);

• <mod_file>_redform_<endo name>_vs_shocks_*.fig: shows bar charts of the elementary effect tests for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus exogenous variables (namexo);

• <mod_file>_redform_screen.fig: shows bar chart of all elementary effect tests for each parameter: this allows one to identify parameters that have a minor effect for any of the reduced form coefficients.

#### 4.22.4.7. Identification Analysis¶

Setting the option identification=1, an identification analysis based on theoretical moments is performed. Sensitivity plots are provided that allow to infer which parameters are most likely to be less identifiable.

Prerequisite for properly running all the identification routines, is the keyword identification; in the Dynare model file. This keyword triggers the computation of analytic derivatives of the model with respect to estimated parameters and shocks. This is required for option morris=2, which implements Iskrev (2010) identification analysis.

For example, the placing:

identification;
dynare_sensitivity(identification=1, morris=2);

in the Dynare model file triggers identification analysis using analytic derivatives as in Iskrev (2010), jointly with the mapping of the acceptable region.

The identification analysis with derivatives can also be triggered by the single command:

identification;

This does not do the mapping of acceptable regions for the model and uses the standard random sampler of Dynare. Additionally, using only identification; adds two additional identification checks: namely, of Qu and Tkachenko (2012) based on the spectral density and of Komunjer and Ng (2011) based on the minimal state space system. It completely offsets any use of the sensitivity analysis toolbox.

## 4.23. Markov-switching SBVAR¶

Given a list of variables, observed variables and a data file, Dynare can be used to solve a Markov-switching SBVAR model according to Sims, Waggoner and Zha (2008). 11 Having done this, you can create forecasts and compute the marginal data density, regime probabilities, IRFs, and variance decomposition of the model.

The commands have been modularized, allowing for multiple calls to the same command within a <mod_file>.mod file. The default is to use <mod_file> to tag the input (output) files used (produced) by the program. Thus, to call any command more than once within a <mod_file>.mod file, you must use the *_tag options described below.

Command: markov_switching(OPTIONS...);

Declares the Markov state variable information of a Markov-switching SBVAR model.

Options

chain = INTEGER

The Markov chain considered. Default: none.

number_of_regimes = INTEGER

Specifies the total number of regimes in the Markov Chain. This is a required option.

duration = DOUBLE | [ROW VECTOR OF DOUBLES]

The duration of the regimes or regimes. This is a required option. When passed a scalar real number, it specifies the average duration for all regimes in this chain. When passed a vector of size equal number_of_regimes, it specifies the average duration of the associated regimes (1:number_of_regimes) in this chain. An absorbing state can be specified through the restrictions option.

restrictions = [[ROW VECTOR OF 3 DOUBLES],[ROW VECTOR OF 3 DOUBLES],...]

Provides restrictions on this chain’s regime transition matrix. Its vector argument takes three inputs of the form: [current_period_regime, next_period_regime, transition_probability].

The first two entries are positive integers, and the third is a non-negative real in the set [0,1]. If restrictions are specified for every transition for a regime, the sum of the probabilities must be 1. Otherwise, if restrictions are not provided for every transition for a given regime the sum of the provided transition probabilities msut be <1. Regardless of the number of lags, the restrictions are specified for parameters at time t since the transition probability for a parameter at t is equal to that of the parameter at t-1.

In case of estimating a MS-DSGE model, 12 in addition the following options are allowed:

parameters = [LIST OF PARAMETERS]

This option specifies which parameters are controlled by this Markov Chain.

number_of_lags = DOUBLE

Provides the number of lags that each parameter can take within each regime in this chain.

Example

markov_switching(chain=1, duration=2.5, restrictions=[[1,3,0],[3,1,0]]);

Specifies a Markov-switching BVAR with a first chain with 3 regimes that all have a duration of 2.5 periods. The probability of directly going from regime 1 to regime 3 and vice versa is 0.

Example

markov_switching(chain=2, number_of_regimes=3, duration=[0.5, 2.5, 2.5],
parameter=[alpha, rho], number_of_lags=2, restrictions=[[1,3,0],[3,3,1]]);

Specifies a Markov-switching DSGE model with a second chain with 3 regimes that have durations of 0.5, 2.5, and 2.5 periods, respectively. The switching parameters are alpha and rho. The probability of directly going from regime 1 to regime 3 is 0, while regime 3 is an absorbing state.

Command: svar(OPTIONS...);

Each Markov chain can control the switching of a set of parameters. We allow the parameters to be divided equation by equation and by variance or slope and intercept.

Options

coefficients

Specifies that only the slope and intercept in the given equations are controlled by the given chain. One, but not both, of coefficients or variances must appear. Default: none.

variances

Specifies that only variances in the given equations are controlled by the given chain. One, but not both, of coefficients or variances must appear. Default: none.

equations

Defines the equation controlled by the given chain. If not specified, then all equations are controlled by chain. Default: none.

chain = INTEGER

Specifies a Markov chain defined by markov_switching. Default: none.

Command: sbvar(OPTIONS...);

To be documented. For now, see the wiki: https://archives.dynare.org/DynareWiki/SbvarOptions

Options

datafile, freq, initial_year, initial_subperiod, final_year, final_subperiod, data, vlist, vlistlog, vlistper, restriction_fname, nlags, cross_restrictions, contemp_reduced_form, real_pseudo_forecast, no_bayesian_prior, dummy_obs, nstates, indxscalesstates, alpha, beta, gsig2_lmdm, q_diag, flat_prior, ncsk, nstd, ninv, indxparr, indxovr, aband, indxap, apband, indximf, indxfore, foreband, indxgforhat, indxgimfhat, indxestima, indxgdls, eq_ms, cms, ncms, eq_cms, tlindx, tlnumber, cnum, forecast, coefficients_prior_hyperparameters

Block: svar_identification ;

This block is terminated by end; and contains lines of the form:

UPPER_CHOLESKY;
LOWER_CHOLESKY;
EXCLUSION CONSTANTS;
EXCLUSION LAG INTEGER; EQUATION INTEGER, VARIABLE_NAME [,VARIABLE_NAME...];
RESTRICTION EQUATION INTEGER, EXPRESSION = EXPRESSION;

To be documented. For now, see the wiki: https://archives.dynare.org/DynareWiki/MarkovSwitchingInterface

Command: ms_estimation(OPTIONS...);

Triggers the creation of an initialization file for, and the estimation of, a Markov-switching SBVAR model. At the end of the run, the $$A^0$$, $$A^+$$, $$Q$$ and $$\zeta$$ matrices are contained in the oo_.ms structure.

General Options

file_tag = FILENAME

The portion of the filename associated with this run. This will create the model initialization file, init_<file_tag>.dat. Default: <mod_file>.

output_file_tag = FILENAME

The portion of the output filename that will be assigned to this run. This will create, among other files, est_final_<output_file_tag>.out, est_intermediate_<output_file_tag>.out. Default: <file_tag>.

no_create_init

Do not create an initialization file for the model. Passing this option will cause the Initialization Options to be ignored. Further, the model will be generated from the output files associated with the previous estimation run (i.e. est_final_<file_tag>.out, est_intermediate_<file_tag>.out or init_<file_tag>.dat, searched for in sequential order). This functionality can be useful for continuing a previous estimation run to ensure convergence was reached or for reusing an initialization file. NB: If this option is not passed, the files from the previous estimation run will be overwritten. Default: off (i.e. create initialization file)

Initialization Options

coefficients_prior_hyperparameters = [DOUBLE1 DOUBLE2 ... DOUBLE6]

Sets the hyper parameters for the model. The six elements of the argument vector have the following interpretations:

1

Overall tightness for $$A^0$$ and $$A^+$$.

2

Relative tightness for $$A^+$$.

3

Relative tightness for the constant term.

4

Tightness on lag decay (range: 1.2 - 1.5); a faster decay produces better inflation process.

5

Weight on nvar sums of coeffs dummy observations (unit roots).

6

Weight on single dummy initial observation including constant.

Default: [1.0 1.0 0.1 1.2 1.0 1.0]

freq = INTEGER | monthly | quarterly | yearly

Frequency of the data (e.g. monthly, 12). Default: 4.

initial_year = INTEGER

The first year of data. Default: none.

initial_subperiod = INTEGER

The first period of data (i.e. for quarterly data, an integer in [1,4]). Default: 1.

final_year = INTEGER

The last year of data. Default: Set to encompass entire dataset.

final_subperiod = INTEGER

The final period of data (i.e. for monthly data, an integer in [1,12]. Default: When final_year is also missing, set to encompass entire dataset; when final_year is indicated, set to the maximum number of subperiods given the frequency (i.e. 4 for quarterly data, 12 for monthly,…).

datafile = FILENAME

See datafile.

xls_sheet = QUOTED_STRING
xls_range = RANGE
nlags = INTEGER

The number of lags in the model. Default: 1.

cross_restrictions

Use cross $$A^0$$ and $$A^+$$ restrictions. Default: off.

contemp_reduced_form

Use contemporaneous recursive reduced form. Default: off.

no_bayesian_prior

Do not use Bayesian prior. Default: off (i.e. use Bayesian prior).

alpha = INTEGER

Alpha value for squared time-varying structural shock lambda. Default: 1.

beta = INTEGER

Beta value for squared time-varying structural shock lambda. Default: 1.

gsig2_lmdm = INTEGER

The variance for each independent $$\lambda$$ parameter under SimsZha restrictions. Default: 50^2.

specification = sims_zha | none

This controls how restrictions are imposed to reduce the number of parameters. Default: Random Walk.

Estimation Options

convergence_starting_value = DOUBLE

This is the tolerance criterion for convergence and refers to changes in the objective function value. It should be rather loose since it will gradually be tightened during estimation. Default: 1e-3.

convergence_ending_value = DOUBLE

The convergence criterion ending value. Values much smaller than square root machine epsilon are probably overkill. Default: 1e-6.

convergence_increment_value = DOUBLE

Determines how quickly the convergence criterion moves from the starting value to the ending value. Default: 0.1.

max_iterations_starting_value = INTEGER

This is the maximum number of iterations allowed in the hill-climbing optimization routine and should be rather small since it will gradually be increased during estimation. Default: 50.

max_iterations_increment_value = DOUBLE

Determines how quickly the maximum number of iterations is increased. Default: 2.

max_block_iterations = INTEGER

The parameters are divided into blocks and optimization proceeds over each block. After a set of blockwise optimizations are performed, the convergence criterion is checked and the blockwise optimizations are repeated if the criterion is violated. This controls the maximum number of times the blockwise optimization can be performed. Note that after the blockwise optimizations have converged, a single optimization over all the parameters is performed before updating the convergence value and maximum number of iterations. Default: 100.

max_repeated_optimization_runs = INTEGER

The entire process described by max_block_iterations is repeated until improvement has stopped. This is the maximum number of times the process is allowed to repeat. Set this to 0 to not allow repetitions. Default: 10.

function_convergence_criterion = DOUBLE

The convergence criterion for the objective function when max_repeated_optimizations_runs is positive. Default: 0.1.

parameter_convergence_criterion = DOUBLE

The convergence criterion for parameter values when max_repeated_optimizations_runs is positive. Default: 0.1.

number_of_large_perturbations = INTEGER

The entire process described by max_block_iterations is repeated with random starting values drawn from the posterior. This specifies the number of random starting values used. Set this to 0 to not use random starting values. A larger number should be specified to ensure that the entire parameter space has been covered. Default: 5.

number_of_small_perturbations = INTEGER

The number of small perturbations to make after the large perturbations have stopped improving. Setting this number much above 10 is probably overkill. Default: 5.

number_of_posterior_draws_after_perturbation = INTEGER

The number of consecutive posterior draws to make when producing a small perturbation. Because the posterior draws are serially correlated, a small number will result in a small perturbation. Default: 1.

max_number_of_stages = INTEGER

The small and large perturbation are repeated until improvement has stopped. This specifies the maximum number of stages allowed. Default: 20.

random_function_convergence_criterion = DOUBLE

The convergence criterion for the objective function when number_of_large_perturbations is positive. Default: 0.1.

random_parameter_convergence_criterion = DOUBLE

The convergence criterion for parameter values when number_of_large_perturbations is positive. Default: 0.1.

Example

ms_estimation(datafile=data, initial_year=1959, final_year=2005,
nlags=4, max_repeated_optimization_runs=1, max_number_of_stages=0);

ms_estimation(file_tag=second_run, datafile=data, initial_year=1959,
final_year=2005, nlags=4, max_repeated_optimization_runs=1,
max_number_of_stages=0);

ms_estimation(file_tag=second_run, output_file_tag=third_run,
no_create_init, max_repeated_optimization_runs=5,
number_of_large_perturbations=10);

Command: ms_simulation ;
Command: ms_simulation(OPTIONS...);

Simulates a Markov-switching SBVAR model.

Options

file_tag = FILENAME

The portion of the filename associated with the ms_estimation run. Default: <mod_file>.

output_file_tag = FILENAME

The portion of the output filename that will be assigned to this run. Default: <file_tag>.

mh_replic = INTEGER

The number of draws to save. Default: 10,000.

drop = INTEGER

The number of burn-in draws. Default: 0.1*mh_replic*thinning_factor.

thinning_factor = INTEGER

The total number of draws is equal to thinning_factor*mh_replic+drop. Default: 1.

adaptive_mh_draws = INTEGER

Tuning period for Metropolis-Hastings draws. Default: 30,000.

save_draws

Save all elements of $$A^0$$, $$A^+$$, $$Q$$, and $$\zeta$$, to a file named draws_<<file_tag>>.out with each draw on a separate line. A file that describes how these matrices are laid out is contained in draws_header_<<file_tag>>.out. A file called load_flat_file.m is provided to simplify loading the saved files into the corresponding variables A0, Aplus, Q, and Zeta in your MATLAB/Octave workspace. Default: off.

Example

ms_simulation(file_tag=second_run);
ms_simulation(file_tag=third_run, mh_replic=5000, thinning_factor=3);

Command: ms_compute_mdd ;
Command: ms_compute_mdd(OPTIONS...);

Computes the marginal data density of a Markov-switching SBVAR model from the posterior draws. At the end of the run, the Muller and Bridged log marginal densities are contained in the oo_.ms structure.

Options

file_tag = FILENAME
output_file_tag = FILENAME
simulation_file_tag = FILENAME`

The portion of the filename associ