text:
Chapter 12 Fluid Dynamics and Its Biological and Medical Applications

# 12.2 Bernoulli’s Equation

### Summary

• Explain the terms in Bernoulli’s equation.
• Explain how Bernoulli’s equation is related to conservation of energy.
• Explain how to derive Bernoulli’s principle from Bernoulli’s equation.
• Calculate with Bernoulli’s principle.
• List some applications of Bernoulli’s principle.

When a fluid flows into a narrower channel, its speed increases. That means its kinetic energy also increases. Where does that change in kinetic energy come from? The increased kinetic energy comes from the net work done on the fluid to push it into the channel and the work done on the fluid by the gravitational force, if the fluid changes vertical position. Recall the work-energy theorem,

${W_{\text{net}}\:=}$ ${\frac{1}{2}}$ ${mv^2\:-}$ ${\frac{1}{2}}$ ${mv_0^2}.$

There is a pressure difference when the channel narrows. This pressure difference results in a net force on the fluid: recall that pressure times area equals force. The net work done increases the fluid’s kinetic energy. As a result, the pressure will drop in a rapidly-moving fluid, whether or not the fluid is confined to a tube.

There are a number of common examples of pressure dropping in rapidly-moving fluids. Shower curtains have a disagreeable habit of bulging into the shower stall when the shower is on. The high-velocity stream of water and air creates a region of lower pressure inside the shower, and standard atmospheric pressure on the other side. The pressure difference results in a net force inward pushing the curtain in. You may also have noticed that when passing a truck on the highway, your car tends to veer toward it. The reason is the same—the high velocity of the air between the car and the truck creates a region of lower pressure, and the vehicles are pushed together by greater pressure on the outside. (See Figure 1.) This effect was observed as far back as the mid-1800s, when it was found that trains passing in opposite directions tipped precariously toward one another.

### MAKING CONNECTIONS: TAKE-HOME INVESTIGATION WITH A SHEET OF PAPER

Hold the short edge of a sheet of paper parallel to your mouth with one hand on each side of your mouth. The page should slant downward over your hands. Blow over the top of the page. Describe what happens and explain the reason for this behavior.

# Bernoulli’s Equation

The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli’s equation, named after its discoverer, the Swiss scientist Daniel Bernoulli (1700–1782). Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant:

${P\:+}$ ${\frac{1}{2}}$ ${\rho{v}^2+\rho{g}h=\textbf{ constant,}}$

where ${P}$ is the absolute pressure, ${\rho}$ is the fluid density, ${v}$ is the velocity of the fluid, ${h}$ is the height above some reference point, and ${g}$ is the acceleration due to gravity. If we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Let the subscripts 1 and 2 refer to any two points along the path that the bit of fluid follows; Bernoulli’s equation becomes

${P_1\:+}$ ${\frac{1}{2}}$ ${\rho{v}_1^2+\rho{g}h_1=P_2\:+}$ ${\frac{1}{2}}$ ${\rho{v}_2^2+\rho{g}h_2.}$

Bernoulli’s equation is a form of the conservation of energy principle. Note that the second and third terms are the kinetic and potential energy with ${m}$ replaced by ${\rho}.$ In fact, each term in the equation has units of energy per unit volume. We can prove this for the second term by substituting ${\rho=m/V}$ into it and gathering terms:

${\frac{1}{2}}$ ${\rho{v}^2\:=}$ ${\frac{\frac{1}{2}mv^2}{V}}$ ${=}$ ${\frac{\text{KE}}{V}}.$

So ${\frac{1}{2}\rho{v}^2}$ is the kinetic energy per unit volume. Making the same substitution into the third term in the equation, we find

${\rho{gh}\:=}$ ${\frac{mgh}{V}}$ ${=}$ ${\frac{\text{PE}_{\textbf{g}}}{V}},$

so ${\rho{gh}}$ is the gravitational potential energy per unit volume. Note that pressure ${P}$ has units of energy per unit volume, too. Since ${P=F/A},$ its units are ${\text{N/m}^2}.$ If we multiply these by m/m, we obtain ${\textbf{N}\cdotp\text{m/m}^3=\text{J/m}^3},$ or energy per unit volume. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.

### MAKING CONNECTIONS: CONSERVATION OF ENERGY

Conservation of energy applied to fluid flow produces Bernoulli’s equation. The net work done by the fluid’s pressure results in changes in the fluid’s $\text{KE}$ and ${\text{PE}_{\textbf{g}}}$ per unit volume. If other forms of energy are involved in fluid flow, Bernoulli’s equation can be modified to take these forms into account. Such forms of energy include thermal energy dissipated because of fluid viscosity.

The general form of Bernoulli’s equation has three terms in it, and it is broadly applicable. To understand it better, we will look at a number of specific situations that simplify and illustrate its use and meaning.

# Bernoulli’s Equation for Static Fluids

Let us first consider the very simple situation where the fluid is static—that is, ${v_1=v_2=0}.$ Bernoulli’s equation in that case is

${P_1+\rho{gh}_1=P_2+\rho{gh}_2}.$

We can further simplify the equation by taking ${h_2=0}$ (we can always choose some height to be zero, just as we often have done for other situations involving the gravitational force, and take all other heights to be relative to this). In that case, we get

${P_2=P_1+\rho{gh}_1}.$

This equation tells us that, in static fluids, pressure increases with depth. As we go from point 1 to point 2 in the fluid, the depth increases by ${h_1},$ and consequently, ${P_2}$ is greater than ${P_1}$ by an amount ${\rho{gh}_1}.$ In the very simplest case, ${P_1}$ is zero at the top of the fluid, and we get the familiar relationship ${P=\rho{gh}}.$ (Recall that ${P=\rho{gh}}$ and ${\rho\text{PE}_{\textbf{g}}=mgh.}$ ) Bernoulli’s equation includes the fact that the pressure due to the weight of a fluid is ${\rho{gh}}.$ Although we introduce Bernoulli’s equation for fluid flow, it includes much of what we studied for static fluids in the preceding chapter.

# Bernoulli’s Principle—Bernoulli’s Equation at Constant Depth

Another important situation is one in which the fluid moves but its depth is constant—that is, ${h_1=h_2}.$ Under that condition, Bernoulli’s equation becomes

${P_1\:+}$ ${\frac{1}{2}}$ ${\rho{v}_1^2=P_2\:+}$ ${\frac{1}{2}}$ ${\rho{v}_2^2}.$

Situations in which fluid flows at a constant depth are so important that this equation is often called Bernoulli’s principle. It is Bernoulli’s equation for fluids at constant depth. (Note again that this applies to a small volume of fluid as we follow it along its path.) As we have just discussed, pressure drops as speed increases in a moving fluid. We can see this from Bernoulli’s principle. For example, if ${v_2}$ is greater than ${v_1}$ in the equation, then ${P_2}$ must be less than ${P_1}$ for the equality to hold.

### Example 1: Calculating Pressure: Pressure Drops as a Fluid Speeds Up

In Chapter 12.1 Example 2, we found that the speed of water in a hose increased from 1.96 m/s to 25.5 m/s going from the hose to the nozzle. Calculate the pressure in the hose, given that the absolute pressure in the nozzle is ${1.01\times10^5\text{ N/m}^2}$ (atmospheric, as it must be) and assuming level, frictionless flow.

Strategy

Level flow means constant depth, so Bernoulli’s principle applies. We use the subscript 1 for values in the hose and 2 for those in the nozzle. We are thus asked to find ${P_1}.$

Solution

Solving Bernoulli’s principle for ${P_1}$ yields

${P_1=P_2\:+}$ ${\frac{1}{2}}$ ${\rho{v}_2^2\:-}$ ${\frac{1}{2}}$ ${\rho{v}_1^2=P_2\:+}$ ${\frac{1}{2}}$ ${\rho(v_2^2-v_1^2)}.$

Substituting known values,

$\begin{array}{lcl} {P_1} & {=} & {1.01\times10^5\text{ N/m}^2} \\ {} & {} & {+\frac{1}{2}(10^3\text{ kg/m}^3)[(25.5\text{ m/s})^2-(1.96\text{ m/s})^2]} \\ {} & {=} & {4.24\times10^5\text{ N/m}^2.} \end{array}$

Discussion

This absolute pressure in the hose is greater than in the nozzle, as expected since vv is greater in the nozzle. The pressure ${P_2}$ in the nozzle must be atmospheric since it emerges into the atmosphere without other changes in conditions.

# Applications of Bernoulli’s Principle

There are a number of devices and situations in which fluid flows at a constant height and, thus, can be analyzed with Bernoulli’s principle.

## Entrainment

People have long put the Bernoulli principle to work by using reduced pressure in high-velocity fluids to move things about. With a higher pressure on the outside, the high-velocity fluid forces other fluids into the stream. This process is called entrainment. Entrainment devices have been in use since ancient times, particularly as pumps to raise water small heights, as in draining swamps, fields, or other low-lying areas. Some other devices that use the concept of entrainment are shown in Figure 2.

## Wings and Sails

The airplane wing is a beautiful example of Bernoulli’s principle in action. Figure 3(a) shows the characteristic shape of a wing. The wing is tilted upward at a small angle and the upper surface is longer, causing air to flow faster over it. The pressure on top of the wing is therefore reduced, creating a net upward force or lift. (Wings can also gain lift by pushing air downward, utilizing the conservation of momentum principle. The deflected air molecules result in an upward force on the wing — Newton’s third law.) Sails also have the characteristic shape of a wing. (See Figure 3(b).) The pressure on the front side of the sail, ${P_{\text{front}}},$ is lower than the pressure on the back of the sail, ${P_{\text{back}}}.$ This results in a forward force and even allows you to sail into the wind.

### Making Connections: Take-Home Investigation with Two Strips of Paper

For a good illustration of Bernoulli’s principle, make two strips of paper, each about 15 cm long and 4 cm wide. Hold the small end of one strip up to your lips and let it drape over your finger. Blow across the paper. What happens? Now hold two strips of paper up to your lips, separated by your fingers. Blow between the strips. What happens?

## Velocity measurement

Figure 4 shows two devices that measure fluid velocity based on Bernoulli’s principle. The manometer in Figure 4(a) is connected to two tubes that are small enough not to appreciably disturb the flow. The tube facing the oncoming fluid creates a dead spot having zero velocity ( ${v_1=0}$ ) in front of it, while fluid passing the other tube has velocity ${v_2}.$ This means that Bernoulli’s principle as stated in ${P_1+\frac{1}{2}\rho{v}_1^2=P_2+\frac{1}{2}\rho{v}_2^2}$ becomes

${P_1=P_2\:+}$ ${\frac{1}{2}}$ ${\rho{v}_2^2}.$

Thus pressure ${P_2}$ over the second opening is reduced by ${\frac{1}{2}\rho{v}_2^2},$ and so the fluid in the manometer rises by ${h}$ on the side connected to the second opening, where

${h\:\propto}$ ${\frac{1}{2}}$ ${\rho{v}_2^2}.$

(Recall that the symbol ${\propto}$ means “proportional to.”) Solving for ${v_2},$ we see that

${v_\propto\sqrt{h}}.$

Figure 4(b) shows a version of this device that is in common use for measuring various fluid velocities; such devices are frequently used as air speed indicators in aircraft.

# Summary

• Bernoulli’s equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid:
${P_1\:+}$ ${\frac{1}{2}}$ ${\rho{v}_1^2+\rho{gh}_1=P_2\:+}$ ${\frac{1}{2}}$ ${\rho{v}_2^2+\rho{gh}_2}.$
• Bernoulli’s principle is Bernoulli’s equation applied to situations in which depth is constant. The terms involving depth (or height h ) subtract out, yielding
${P_1\:+}$ ${\frac{1}{2}}$ ${\rho{v}_1^2=P_2\:+}$ ${\frac{1}{2}}$ ${\rho{v}_2^2}.$
• Bernoulli’s principle has many applications, including entrainment, wings and sails, and velocity measurement.

### Conceptual Questions

1: You can squirt water a considerably greater distance by placing your thumb over the end of a garden hose and then releasing, than by leaving it completely uncovered. Explain how this works.

2: Water is shot nearly vertically upward in a decorative fountain and the stream is observed to broaden as it rises. Conversely, a stream of water falling straight down from a faucet narrows. Explain why, and discuss whether surface tension enhances or reduces the effect in each case.

3: Look back to Figure 1. Answer the following two questions. Why is ${P_{\textbf{o}}}$ less than atmospheric? Why is ${P_{\textbf{o}}}$ greater than ${P_{\textbf{i}}}?$

4: Give an example of entrainment not mentioned in the text.

5: Many entrainment devices have a constriction, called a Venturi, such as shown in Figure 5. How does this bolster entrainment?

6: Some chimney pipes have a T-shape, with a crosspiece on top that helps draw up gases whenever there is even a slight breeze. Explain how this works in terms of Bernoulli’s principle.

7: Is there a limit to the height to which an entrainment device can raise a fluid? Explain your answer.

8: Why is it preferable for airplanes to take off into the wind rather than with the wind?

9: Roofs are sometimes pushed off vertically during a tropical cyclone, and buildings sometimes explode outward when hit by a tornado. Use Bernoulli’s principle to explain these phenomena.

10: Why does a sailboat need a keel?

11: It is dangerous to stand close to railroad tracks when a rapidly moving commuter train passes. Explain why atmospheric pressure would push you toward the moving train.

12: Water pressure inside a hose nozzle can be less than atmospheric pressure due to the Bernoulli effect. Explain in terms of energy how the water can emerge from the nozzle against the opposing atmospheric pressure.

13: A perfume bottle or atomizer sprays a fluid that is in the bottle. (Figure 6.) How does the fluid rise up in the vertical tube in the bottle?

14: If you lower the window on a car while moving, an empty plastic bag can sometimes fly out the window. Why does this happen?

### Problems & Exercises

1: Verify that pressure has units of energy per unit volume.

2: Suppose you have a wind speed gauge like the pitot tube shown in [link](b). By what factor must wind speed increase to double the value of ${h}$ in the manometer? Is this independent of the moving fluid and the fluid in the manometer?

3: If the pressure reading of your pitot tube is 15.0 mm Hg at a speed of 200 km/h, what will it be at 700 km/h at the same altitude?

4: Calculate the maximum height to which water could be squirted with the hose in [link] example if it: (a) Emerges from the nozzle. (b) Emerges with the nozzle removed, assuming the same flow rate.

5: Every few years, winds in Boulder, Colorado, attain sustained speeds of 45.0 m/s (about 100 mi/h) when the jet stream descends during early spring. Approximately what is the force due to the Bernoulli effect on a roof having an area of ${220\text{ m}^2}?$ Typical air density in Boulder is ${1.14\text{ kg/m}^3},$ and the corresponding atmospheric pressure is ${8.89\times10^4\text{ N/m}^2}.$ (Bernoulli’s principle as stated in the text assumes laminar flow. Using the principle here produces only an approximate result, because there is significant turbulence.)

6: (a) Calculate the approximate force on a square meter of sail, given the horizontal velocity of the wind is 6.00 m/s parallel to its front surface and 3.50 m/s along its back surface. Take the density of air to be ${1.29\text{ kg/m}^3}.$ (The calculation, based on Bernoulli’s principle, is approximate due to the effects of turbulence.) (b) Discuss whether this force is great enough to be effective for propelling a sailboat.

7: (a) What is the pressure drop due to the Bernoulli effect as water goes into a 3.00-cm-diameter nozzle from a 9.00-cm-diameter fire hose while carrying a flow of 40.0 L/s? (b) To what maximum height above the nozzle can this water rise? (The actual height will be significantly smaller due to air resistance.)

8: (a) Using Bernoulli’s equation, show that the measured fluid speed vv for a pitot tube, like the one in Figure 4(b), is given by

${v\:=}$ ${\left(\frac{2\rho^{\prime}gh}{\rho}\right)}$ ${^{^{^{1/2}}}},$

where ${h}$ is the height of the manometer fluid, ${\rho^{\prime}}$ is the density of the manometer fluid, ${\rho}$ is the density of the moving fluid, and ${g}$ is the acceleration due to gravity. (Note that ${v}$ is indeed proportional to the square root of ${h},$ as stated in the text.) (b) Calculate ${v}$ for moving air if a mercury manometer’s ${h}$ is 0.200 m.

## Glossary

Bernoulli’s equation
the equation resulting from applying conservation of energy to an incompressible frictionless fluid: P + 1/2pv2 + pgh = constant , through the fluid
Bernoulli’s principle
Bernoulli’s equation applied at constant depth: P1 + 1/2pv12 = P2 + 1/2pv22

### Solutions

Problems & Exercises

1:

$\begin{array}{lcl} {P} & {=} & {\frac{\text{Force}}{\text{Area}},} \\ {(P)_{\text{units}}} & {=} & {\text{N/m}^2=\textbf{N}\cdotp\text{m/m}^3=\text{J/m}^3} \\ {} & {=} & {\text{energy/volume}} \end{array}$

3:

184 mm Hg

5:

${2.54\times10^5\text{ N}}$

7:

(a) ${1.58\times10^6\text{ N/m}^2}$

(b) 163 m


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# Chapter 5 Speculative asset price bubbles

## 5.1 Introduction

This chapter is devoted to such an important economic phenomenon as the speculative price bubbles. In section 5.2, we will define the speculative bubbles and discuss their importance for the society. In section 5.3, a typology of the early warning systems is presented. In section 5.4, the techniques, which allow dating the periods of expansions (bubbles) and recessions (no bubbles) are presented. Sections 5.5 and 5.6 show the ways how to timely recognize signs of the building-up speculative bubbles using the early warning systems with exogenous and endogenous bubble chronologies, correspondingly.

## 5.2 Speculative bubbles

The trajectory of the market economy is not smooth. From time to time, the economy experiences booms and crises. The examples of large crises include the Great Depression 1929–1932 that took place worldwide and to some extent was responsible for the outbreak of World War II; the Lost Decade (1991–2000 or even 1991–2010) in Japan; as well as the Great Recession, or Financial Crisis, 2008–2009 and the COVID-19 sanitary and economic crisis of 2020 that shook virtually entire world.

In many cases, the economic crises are triggered by the speculations. The speculation can be defined as purchasing an asset not for own use but with an expectation that its price will increase further and that the asset can be sold at a later stage in order to realize capital gains. Speculations arise because financial capital is looking for ways of earning profit. It cannot lie idle, so the capital holders are always screening various markets in order to find those having the highest profit margin. So, speculations can emerge in virtually any market. From the early economic history, we know examples of speculating with tulip bulbs —the so-called "tulip mania’’— in The Netherlands in the 1630s. Speculations with stocks are also known at least from the early 18th century, such as the Mississippi Bubble 1718–1720 and the South Sea bubble of 1719–1720. See, for example, Garber (1990). In 2001, a large crisis occured at the stock market, which was related to the speculations with stocks of the nascent Internet companies and, therefore, was nicknamed the DotCom crisis. The housing also often becomes an object for speculations. For example, the Great Recession 2008–2009 was mainly triggered by the speculations with subprime mortgage loans in the USA.

The speculative asset price bubble is defined as "large, sustained mispricings of financial or real assets’’ (Brunnermeier and Oehmke 2013). Alternatively, the speculative bubbles refer to the periods in which the price of an asset exceeds its fundamental value due to investors’ beliefs that they can sell the asset at a higher price in the future. The fundamental value of an asset is its value determined by its dividend stream. By contrast, the speculative component of the asset’s value is that based on the expectations of reselling it at a higher price.

How do the economists explain the emergence of speculative price bubbles? There are several well-established theories that explain seemingly unsustainable developments by rational behavior of agents (Brunnermeier 2008; Brunnermeier and Oehmke 2013; Glaeser and Nathanson 2015). Brunnermeier and Oehmke (2013) provide a classification of the existing approaches to explaining the buildup of asset price bubbles. They distinguish between five types of theories.

The first explanation is the rational bubbles theory. It is argued that investors hold bubble assets since they expect the price of the asset to grow under the assumption that any investor in the future also expects asset price increases, even if the bubble may burst with a certain probability. Price increases then become self-fulfilling prophecies, which implies explosive growth of asset valuations (Blanchard and Watson 1982).

Second, overlapping generations models allow the existence of bubbles, because they can restore efficiency of an economy, which is inherently not Pareto optimal (Tirole 1985). The idea is that even fiat money —which has no intrinsic value— can have a positive price because it is a store of value. Storing value, and thus increasing the price of money, might be a rational option in situations of over-accumulation of private capital, i.e., when capital accumulation exceeds the "golden rule.’’ Then, transferring capital to future generations and giving fiat money a positive price (which is then a bubble asset) can increase economic efficiency.

Third, the persistence of bubbles can be explained in the framework of the informational frictions theory. In this type of model, two sources of risk are introduced that encourage traders to invest in bubble assets. On the one hand, the fundamental risk arises from uncertainty of an unexpected jump of the fundamental value. In this case, investors who bet against a bubble lose money. On the other hand, risk-neutral traders, who are aware of the bubble, can still trade in order to benefit from speculative price increases, as long as they weigh the risk of a bursting bubble less than the chances of price increases (Abreu and Brunnermeier 2003). In both cases, investors delay the bursting of the bubble and allow it to grow even larger.

Fourth, according to the delegated investment theory, most households delegate the management of their assets to the fund managers, who often are concerned only about short-run price movements (Shleifer and Vishny 1997) and may have incentives to buy overpriced assets (Allen and Gorton 1993).

Fifth, the theory of the heterogeneous-beliefs bubbles implies that investors, possibly due to psychological biases, have different beliefs about the future. These beliefs when combined-with short-sale constraints-can lead to overpricing when optimists push up prices, while pessimists are unable to counteract this (Miller 1977). Regardless of the driving forces behind a bubble, house price bubbles can be defined as periods in which housing prices run well above their fundamental values, i.e., the development of rents. In addition, as pointed out by Brunnermeier and Oehmke (2013), asset values grow explosively during speculative bubble periods. While the notion "well above’’ is rather open to interpretation, an explosive house price growth can be formally tested using statistical methods.

An important role in the formation of speculative asset price bubbles is played by the financialization. The financialization can be defined as a way of capital accumulation, when profit-making takes place increasingly through financial channels rather than through trade and production (Aalbers 2016). Due to the slowdown of economic growth, the capital accumulation becomes more and more focused on the growth of finance, benefiting the financial markets agents (investors) rather than the real economy. The financialization embraces increasingly the non-financial firms (Internet, car industry, real estate sector, etc.) and households forcing them to accept risks and act as capitalists. It became a means of the capital switching in the sense of Harvey (1985), which channels the capital from the primary circuit (production, manufacturing, industrial sector), secondary (built environment for production, e.g., infrastructure, and for consumption, e.g., housing), or tertiary circuit (social infrastructure, i.e., investment in technology, science, conditions of employees, health and education) to the quaternary circuit of capital (Aalbers 2016). This capital switching inevitably leads to over-accumulation and to crises.

The crises are costly — they destroy jobs and wealth. The bursting bubbles and resulting crises lead to huge GDP losses, increases in unemployment, bankruptcies of firms and private households, and other negative consequences for the whole society or its important parts. Therefore, it is important to be able to foresee them. If we will be able to identify timely the arrival of crises, then we can hope to mitigate their negative consequences.

## 5.3 Early warning system

An early warning system is a tool for forecasting the booms and crises. It has two main elements: 1) an expansion/recession (boom/bust) dates, or bubble chronology; and a forecasting model. In the first step, the periods of bubbles and no bubbles, or expansions and recessions are established. In the second, step a forecasting model is estimated, which allows predicting the bubbles. Figure 5.1 depicts a typology of different early warning systems.

It distinguishes between the early warning systems with exogenous and endogenous chronologies. In the EWS with exogenous chronology, the determination of the bubble chronology step is done separately and independently on the forecasting step, while in the EWS with endogenous endogenous both these steps are carried out simultaneously. Figure 5.1 displays three families of methods used in order to establish the bubble chronology (ad hoc approach, expert judgment, and statistical testing), two families of forecasting techniques requiring exogenous chronologies (signalling approach and discrete-choice models), and three forecasting methods with endogenous chronology (TAR, STAR, and Markov switching). Below we will discuss each of these methods in more detail.

## 5.4 Dating the speculative bubbles

Nobody knows when exactly do the speculative bubbles (or expansions and recessions in case of business cycles) start and end. In fact, what we observe are periodic declines and increases of asset prices or economic performance. However, without special techniques it is impossible to tell apart small short-lived fluctuations from the cyclical movements. Therefore, some algorithm is needed in order to establish the chronology of bubbles or business cycles. As shown in Figure , there are three families of methods establishing bubble chronologies: 1) expert judgment; 2) ad hoc dating; and 3) formal statistical tests.

### 5.4.1 Expert judgment

A group of experts decides on the dates of beginning and ending of booms and busts. The most known such a group is the NBER’s Business Cycle Dating Committee.12 It maintains a chronology of the U.S. business cycle. The chronology comprises alternating dates of peaks and troughs in economic activity. The way the experts make their dating decisions is perfectly described by the NBER itself:

The Committee applies its judgment… and has no fixed rule to determine whether a contraction is only a short interruption of an expansion, or an expansion is only a short interruption of a contraction.

The NBER does not have a fixed definition of economic activity. It examines and compares the behavior of various measures of broad activity: real GDP, economy-wide employment, and real income. The NBER also can consider indicators that do not cover the entire economy, such as real sales and the Fed’s index of industrial production. A well-defined peak or trough in real sales or industrial production helps to determine overall peak or trough dates, particularly if the economy-wide indicators are in conflict or do not have well-defined peaks or troughs.

Figure 5.2 plots the US real GDP together with the NBER business cycle chronology depicted as shaded areas.

Sources: FRED, NBER, and own representation.

It can be seen that the NBER chronology accurately picks up the declines in the real GDP. An especially deep and protracted decline is observed in 2008–2009, the episode known as the Great Recession.

The NBER is an established authority and its chronology is a widely recognized one. Other countries have no such expert committee at the national level. Fortunately, there is an international organization — Economic Cycle Research Institute (ECRI), which produces business cycle chronologies for various countries.13

Each technique has its advantages and disadvantages. What are the strengths and weaknesses of the expert judgment approach? It has the following advantages: Experts make use of different sources of information. The wise judgment can be superior to a thoughtless use of sophisticated techniques. The disadvantages of the expert judgment approach to dating the bubbles are as follows: The expert’s procedure is not transparent. It is impossible to replicate. It is a very time consuming process: it takes many months or even years to identify the cyclical phases.

### 5.4.2 Ad hoc dating methods

The simplest method of dating business cycles and speculative asset price bubble is the rule of thumb using the two-quarter definition. According to this rule, recession is defined as two consecutive quarters of decline in real GDP.

A more sophisticated technique is the procedure developed by Bry and Boschan (1971). It tries to emulate the decision making process of the NBER using the monthly series of industrial production. The Bry-Boschan technique does not use any formal statistical framework to do the dating. Instead it translates the NBER method into a set of simple decision rules. It basically comprises two stages: 1) selecting the candidates for turning points and 2) applying a censoring rule to eliminate the turning points, which do not satisfy some criteria (e.g., minimum duration).

A quarterly version of Bry-Boschan technique was developed by Harding and Pagan (2001). Each point in the sample is checked for being a local maximum or minimum:

$$$y_t=\left\{ \begin{array}{l} \mbox{peak, if } y_t=\max\big(y_{t-K}, \ldots, y_{t+L}\big)\\ \mbox{trough, if } y_t=\min\big(y_{t-K}, \ldots, y_{t+L}\big)\\ \mbox{neither peak nor trough, otherwise} \end{array} \right.$$$ Normally, one takes $$K=L=2$$ for quarterly data (6 for monthly data). Eliminate candidate peaks and troughs, which do not satisfy two restrictions: 1) minimum phase duration should be 6 months (2 quarters) and 2) complete cycle must last at least 15 months (5 quarters).

The most important appeal of the Bry-Boschan technique is its simplicity and transparency. It is also very robust in the sense that changing the sample will not affect the dates. Although it is sensitive to the choice of criteria and censoring rules. Arbitrariness is the major problem related to the ad hoc techniques.

Crises occurring at the currency markets have their specific features and, therefore, require somewhat different tools in order to identify them. A currency crisis is defined as speculative pressures in foreign exchange markets. In order to identify the periods of a currency crisis, constructed crisis indices need to reflect both successful and unsuccessful speculative attacks on domestic currency. Following Eichengreen, Rose, and Wyplosz (1994) recent studies use as crisis indicator the speculative exchange market pressure (EMP), which is a weighed average of depreciation rate of nominal exchange rates, percentage change in international reserves, and change in interest rate. Different weighting schemes are possible. For example, it is reasonable to use for this purpose the inverse of the standard deviations of the component variables ($$\frac{1}{\sigma_i}$$, where $$i=\{\text{exchange rate}, \text{reserves}, \text{interest rate}\}$$) in order to equalize their volatilities.

Currency crisis is identified when the EMP index exceeds a particular threshold, such as 2 or 3 standard deviations ($$\sigma$$) above its mean ($$\mu$$). Figure 5.3 illustrates this approach using the monthly Russian financial data over the period 1997–2018.

Sources: Central Bank of Russia, Federal Reserve Bank of St. Louis (FRED), and own calculations.

Three panels show the dynamics of the exchange rate of ruble, international official reserves of Russia, and short-term interest rate (immediate Rates: less than 24 hours). The lower right panel displays the EMP index calculated using these three series. The dashed (dotted) line reflects the mean plus two (three) standard deviations. The periods, when the EMP index intersects these lines, are treated as currency crises. In this case, it happens twice: in 1998, when a large economic crisis occured in Russia, and 2014, when oil prices went down, Crimean crisis started, and anti-Russian sanctions were introduced by the Western countries. However, in the second case, the EMP crossed only the $$\mu + 2\sigma$$ line, but not the $$\mu + 3\sigma$$ line. Thus, no currency crisis would be detected, if we used the stricter criterion.

### 5.4.3 Formal statistical tests

Formal statistical tests are free from arbitrariness of both the expert judgment and ad hoc methods. Recently, they have been increasingly used to detect the asset price speculative bubbles for stock prices, house prices, and commodity prices.

Here, we will discuss the so-called explosive root test developed by Phillips, Shi, and Yu (2015) in order to identify multiple speculative bubbles. The test is based on the idea that house prices are fundamentally determined by future rents in the period of ownership and, moreover, by future asset price increases. Under the assumption of perfectly informed and rational agents, price increases are again solely determined by dividends earned in the period after the next sale.

In the real estate context, this implies that house prices are —in the long run— tied to the development of rents. Under the standard no arbitrage condition, the house price can thus be expressed as

$$$P_t = \frac{E_t [P_{t+1} + R_{t+1}]}{1 + i} \tag{5.1}$$$ where $$P_t$$ is the real estate price, $$R_t$$ denotes the rental income, $$t$$ is the current time period, $$i$$ is the risk-free interest rate, and $$E[\cdot]$$ is the rational expectation, conditional on the information available (Homm and Breitung 2012). The fundamental price, $$P^F$$, can be determined by forward iteration of equation (5.1):

$$$P^F_t = \sum_{n=1}^{\infty} \frac{1}{(1+i)^n}E_t[R_{t+n}] \tag{5.2}$$$

However, there is a unique solution for equation (5.2) under the transversality condition, according to which the present value of a payment occurring infinitely far in the future must be zero,

$$$\lim_{k\rightarrow\infty} E_t \Big[\frac{1}{(1 + i)^k} P_{t+k}\Big]= 0 \tag{5.3}$$$

If the actual price process contains additional elements, like a bubble component $$B$$, the pricing equation becomes $$$P_t = \lim_{k\rightarrow\infty} E_t \Big[\frac{1}{(1 + i)^k}\Big(\hat{P}_{t+k} + (1 + i)^k B_t)\Big)\Big]+ P^F_t \tag{5.4}$$$

In this case, there are infinitely many solutions. Today’s house price can be decomposed in two elements —one covering the fundamental value, determined by future rental income and another that is related to potentially speculative motivations. In case of a speculative bubble, any rational investors should expect the house price to increase at rate $$i$$. Because all rational investors expect other investors to pay a price $$P^F_{t+1}+ B_{t+1}$$, they are willing to pay $$P^F_t+ B_t$$ in period $$t$$.

The fundamental component in equation (5.4) cannot be observed. Hence, assumptions have to be made in order to characterize the time series properties of the fundamental price, $$P^F_t$$. One can impose a plausible assumption that housing rent, $$R_t$$, follows a random walk with drift: $$$R_t = \mu + R_{t-1} + u_t$$$ where $$\mu$$ is the drift and $$u_t$$ is a white noise process. Under this assumption the fundamental price can be expressed as

$$$P^F_t = \frac{1 + i}{i^2}\mu + \frac{1}{i} R_t$$$ where $$i$$ is the risk-free rate (Homm and Breitung 2012). As a result, if $$R_t$$ follows a random walk with drift, so does the fundamental component of the housing price, $$P^F_t$$. This permits distinguishing the fundamental price from the speculative bubble that can be described as an explosive autoregressive process. Thus, a test procedure boils down to testing the null hypothesis of a random walk against the alternative of an explosive process.

The basic idea of the test is to analyze the roots of an autoregressive process. The explosive root is tested against the alternative of a unit root (random walk). The latter reflects the rational expectations hypothesis.

Figure 5.4 compares two artificially generated time series: a random walk and an explosive process.

It can be seen that at some point the explosive process rapidly at an ever increasing rate diverges into infinity.

The test is based on the rolling regression model of the following form:

$$$\Delta y_t = \alpha_{r_1,r_2} + \beta_{r_1,r_2} y_{t-1} + \sum_{k=1}^K\phi_{r_1,r_2} + \varepsilon_t$$$ where $$y_t$$ is the variable to be tested for explosive roots; $$k$$ is the lag order; $$\alpha,\beta,\phi$$ are parameters to be estimated; and $$\varepsilon_t$$ is the disturbance term. Under the null hypothesis (random walk), $$\beta_{r_1,r_2}=0$$, while under the alternative hypothesis (explosive process), $$\beta_{r_1,r_2}>0$$.

The sample of this rolling-window regression starts from the $$r_1$$-th fraction and ends at the $$r_2$$-th fraction of the total sample ($$T$$): $$r_2 = r_1 + r_w$$ and $$r_w > 0$$ is the (fractional) window size of the regression.

Based on this regression, an augmented Dickey-Fuller (ADF) test is conducted on a forward expanding sample sequence. Phillips, Shi, and Yu (2015) test is the supremum value of the ADF statistic sequence:

$$$SADF(r_0) = \sup_{r_2 \in[r_0,1]}ADF_0^{r_2}$$$ where $$ADF_0^{r_2}$$ is the ADF statistic for a sample running from 0 to $$r_2$$.

Figure 5.5 shows the house price-to-rent ratios in various countries and the corresponding speculative bubbles detected using this time series and the explosive root test of Phillips, Shi, and Yu (2015).

Source: OECD and own calculations.

The bubble episodes are quite rare events. However, they can last for 10 years or more, as the example of many countries in the early 2000s and of Scandinavian countries in the 2010s shows.

## 5.5 Early warning systems with exogenous chronology

As pointed out in section 5.3, the two main elements of an early warning system are: a cycle or bubble chronology and a forecasting model. The chronology can be exogenous, when it is not a part of the forecasting model; or endogenous, when it is obtained when estimating the forecasting model. Here, we will consider the early warning systems with exogenous chronology. While the procedure used in order to detect speculative bubbles or business cycle turning points were discussed in the previous section, in this section we will concentrate on the prediction of bubbles and turning points. The EWS with exogenous chronology can use the following forecasting approaches: signalling approach and discrete-choice models — see Figure 5.1.

### 5.5.1 Signalling approach

The signalling approach was suggested by Kaminsky, Lizondo, and Reinhart (1998) and Reinhart and Kaminsky (1999). The idea behind this approach is to find indicators that allow the best prediction of crises. The main criterion used to choose the indicators is their behavior: a good indicator tends to deviate from its "normal’’ path prior to a crisis. Every time an indicator exceeds certain threshold value (e.g., mean plus several standard deviations), it is interpreted as a warning signal that a crisis can take place within several periods.

The threshold values are calculated in such a way as to strike the balance between the risk of producing too many false alarms and the risk of missing the crisis altogether. A reasonable lead between the signal and the crisis must be defined: any signal given within some period (window) before the beginning of the crisis is labeled a good signal, while any other signal sent outside that window is labeled a false alarm or noise. Thus, a designer of a bubble prediction model based on the signalling approach needs leading indicators that are highly correlated with the speculative bubble chronology. The examples of economic indicators: business confidence indicators, stock exchange indices, interest spreads (difference between the long-term and short-term interest rates), etc.

The forecasting performance of each indicator can be measured and compared in order to identify the best indicators, that is, indicators, which allow the most precise predictions of speculative price bubbles or business cycle turning points.

Table 5.1 describes the four possible forecasting situations.

Table 5.1: Crisis detection: signals vs. crises
Crisis No crisis
Signal sent correct signal, $$A$$ false alarm, $$B$$
Signal not sent missing signal, $$C$$ correct signal, $$D$$

The correct signal takes place, if the signal is sent and the crisis occurs or if the signal is not sent and there is no crisis. Otherwise, when the signal is sent, but no crisis occurs, we talk about a false alarm; whereas when no signal is sent and the crisis occurs, we call it missing signal. The forecast accuracy tests are based on this classification of cases.

For example, Kuipers score uses this classification in the following way:

$$$KS = \frac{N_A}{N_A+N_C} - \frac{N_B}{N_B+N_D}$$$ where $$N_A$$ and $$N_D$$ are the number of correct identifications; $$N_B$$ is the number of false alarms; and $$N_C$$ is the number of missing signals. The Kuipers score, $$KS$$, varies between -1 (all false) and 1 (all correct).

The individual indicators can be combined to a composite indicator using their $$KS$$ measures as weights. The signalling approach is typically used to predict the currency crises. However, there are also studies that use it in order to forecast stock and house price bubbles (Gerdesmeier, Reimers, and Roffia 2010; Dreger and Kholodilin 2013).

### 5.5.2 Discrete-choice models

An alternative to signals approach uses the crisis chronology as dependent variable. The chronology is defined as binary, or discrete-choice, variable: $$$C_t=\left\{ \begin{array}{ll} 1 & \mbox{if crisis in period t} \\ 0 & \mbox{otherwise} \end{array} \right.$$$ In such cases, the discrete-choice models are used such as linear probability model as well as logit and probit models.

### 5.5.3 Linear probability model

The probability of being in the period of recession or of speculative bubble is defined as:

\begin{align} \Pr(C_t=1|x_t) & = F(x_t,\beta) \nonumber \\ \Pr(C_t=0|x_t)& = 1-F(x_t,\beta) \tag{5.5} \end{align} where $$x_t$$ is a vector of indicators at time $$t$$ ($$t=1,\ldots,T$$); and $$\beta$$ is $$k\times 1$$ parameter vector.

The problem at this point is to devise a suitable model for the r.h.s. of equation. One possibility is to retain the familiar linear regression:

$$$F(x_t,\beta) = x_t'\beta$$$

In that case, we have a **linear probability model*, which is the multiple linear regression model when the dependent variable, $$y$$, is binary rather than continuous: $$$C_t = \beta_0 + \beta_1 x_{1t} + \ldots + \beta_K x_{Kt} + \varepsilon_t$$$ where $$C_t$$ is a binary crisis variable.

Because dependent variable is binary, $$$E(C_t|x_{1t},\ldots,x_{Kt})=\Pr(C|x_{1t},\ldots,x_{Kt})$$$ So, for the linear probability model: $$$\Pr(C_t|x_{1t},\ldots,x_{Kt}) = \beta_0 + \beta_1 x_{1t} + \ldots + \beta_K x_{Kt}$$$ Regression coefficient $$\beta_k$$ on a regressor $$x_{kt}$$ is the change in the probability that $$C_t=1$$ associated with a unit change in $$x_{kt}$$.

Regression coefficients of linear probability model can be estimated by OLS. The usual (heteroskedasticity-robust) OLS standard errors can be used for confidence intervals and hypothesis tests. The usual goodness-of-fit measure, $$R^2$$, used for the linear models makes no sense for a linear probability model. When dependent variable is continuous, it is possible to imagine a situation, in which $$R^2=1$$: All the data lie exactly on the regression line. However, this is impossible, when the dependent variable is binary, unless the regressors are also binary.

Figure 5.6 shows an example of a LPM estimated for the artificially generated data.

The LPM is a simple and yet powerful model. Nevertheless, it also has shortcomings. First, the linearity that makes the linear probability model easy to use is also its major flaw. Because probabilities cannot be less than 0 or exceed 1, the effect on the probability that $$C_t=1$$ of a given change in $$x_t$$ must be nonlinear.

Second, in the linear probability model, the effect of a given change in income is constant, which leads to predicted probabilities that drop below 0 or exceed 1. But this is a nonsense!

### 5.5.4 Logit and probit models

The logit and probit regressions are nonlinear regression models specifically designed for binary dependent variables. Because a regression with a binary dependent variable $$C_t$$ models the probability that $$C_t=1$$, it makes sense to adopt a nonlinear formulation that forces the predicted values to lie between 0 and 1. Because cumulative probability distribution (c.d.f.) functions produce probabilities between 0 and 1, they are used in logit and probit regressions. The probit regression uses the standard normal c.d.f., while the logit regression uses the logistic c.d.f.

The probit model uses the standard normal c.d.f.: $$$\Pr(C_t=1|x_t) = \frac{1}{\sqrt{2\pi}}\int^{x_t'\beta}_{-\infty} \exp^{-\frac{z^2}{2}}dz = \Phi(x_t'\beta) \tag{5.6}$$$ where $$\Phi$$ is the standard normal distribution function.

The logit regression uses the logistic c.d.f.:

$$$\Pr(C_t=1|x_t) = \frac{\exp(x_t'\beta)}{1+\exp(x_t'\beta)} = \Lambda(x_t'\beta) \tag{5.7}$$$ where $$\Lambda$$ is the logistic distribution function.

Figure 5.7 compares transition functions of logit and probit models.

The logit transition function has heavier tails than the probit one.

The question, which distribution (logistic or exponential) to use, is a natural one. The logistic distribution is similar to the normal, except in tails, which are considerably heavier. It closely resembles a Student distribution with 7 degrees of freedom. Therefore, for intermediate values of $$x_t'\beta$$ (say, between -1.2 and +1.2), the two distributions tend to give similar probabilities. The logistic distribution tends to give larger probabilities to $$C_t=1$$ when $$x_t'\beta$$ is extremely small (and smaller probabilities to $$C_t=1$$ when $$x_t'\beta$$ is very large) than the normal distribution.

We should expect different predictions from the two models, if the sample contains 1) very few responses ($$C_t$$’s equal to 1) or very few nonresponses ($$C_t$$’s equal to 0); or 2) very wide variation in an important independent variable, particularly if (1) is also true.

It is virtually impossible to justify the choice of one distribution or another on theoretical grounds.

All three models — linear probability, logit, and probit — are just approximations to the unknown population regression function $$F(C_t|x_t)=\Pr(C_t=1|x_t)$$. The linear probability model is the easiest to use and to interpret, but it cannot capture the nonlinear nature of the true population regression function. Logit and probit regressions model this nonlinearity in probabilities, but their regression coefficients are more difficult to interpret. So which should be used in practice? There is no one right answer, and different researchers use different methods.

These models are estimated for the US data, using the NBER business cycle chronology as the dependent variable.

Figure 5.8 compares conditional probabilities of recession obtained using logit and probit models to the NBER business cycle chronology.

Both models produce quite accurate forecasts of the NBER recession dates. The logit model appears to be slightly more accurate than the probit model: it has higher recession probabilities during the recession phases of the NBER and lower probabilities during the NBER expansion phases.

## 5.6 Early warning systems with endogenous chronology

This class of models allows moving away from a priori dating of crises. Here, three forecasting model classes will be considered: TAR, STAR, and Markov switching; see Figure 5.1. These are non-linear models, which permit identifying crises in an endogenous way, as switches between the alternating regimes of crises and non-crises are conditioned upon some variable. While in TAR and STAR models, the transition variable is observable and must be somewhat arbitrarily chosen by the forecaster, in the Markov-switching models, it is unobservable.

The models considered in this section have their advantages and disadvantages. Their advantage is that they reduce arbitrariness in defining crisis periods. They do not need any separately established bubble chronology, since it is estimated simultaneously with other parameters of these models. The models with endogenous chronology have two disadvantages: 1) they are more computation-intensive and unstable compared to the EWS with exogenous chronology and 2) their data requirements are higher (for example, more observations are needed).

### 5.6.1 Threshold autoregression

We start from the simplest non-linear model, which allows simultaneously detecting and predicting speculative bubbles. It is known as threshold autoregressive model (TAR) and was suggested by Tong (1978) and Tong (1990). In this model, the regime shifts (from expansion to recession or from bubble to no-bubble) in the dependent variable, $$y_t$$, are triggered by an observable transition variable, $$x_t$$, crossing a threshold $$c$$:

$$$y_t=\bigg(\alpha_0+\sum_{p=1}^P \alpha_{p}y_{t-p}\bigg)I(x_t;c) + \bigg(\beta_0+\sum_{p=1}^P\beta_{p}y_{t-p}\bigg) \bigg(1-I(x_t;c)\bigg)+\varepsilon_t$$$

where $$y_t$$ is the dependent variable of interest (e.g., real GDP growth rates or price-to-rent ratio); $$\alpha$$’s and $$\beta$$’s are the regime-dependent parameters; $$\varepsilon_t$$ is the error term, $$\varepsilon_t \sim IID(0,\sigma^2)$$; and $$I(\cdot)$$ is the indicator function.

The indicator function, $$I(x_t;c)$$, is defined as: $$$I(x;c)=\left\{ \begin{array}{rcl} 1, & \mbox{if} & g(x_t)>c \\ 0, & \mbox{if} & g(x_t)\leq c \end{array} \right. \tag{5.8}$$$ If the indicator function of transition variable exceeds certain threshold, then economy is in one regime (e.g., crisis), otherwise economy is in another regime (e.g., no crisis).

Figure 5.9 illustrates the indicator (or transition) function of the TAR model:

The switch between two regimes is discontinuous: an infinitely small change in the transition variable is enough to jump into another regime. One may argue that the economy is seldom confronted with such a situation. There can be many different types of transition variable, $$x_t$$. For $$x_t=t$$ a model with a structural break at time $$t=c$$ occurs. For the purposes of analysis of currency and financial crises other transition variables may be more useful.

In fact, the TAR model separately estimates two linear models for two different subsamples, depending on the threshold value of the transition variable. The recession subsample is two times shorter than the expansion subsample. In both regimes, the intercept is negative. However, in the recession regime, is almost three times larger in absolute value than that in the expansion regime. The parameter estimates of different explanatory variables are different across regimes.

Figure 5.10 shows the conditional probabilities of recession resulting from the TAR model presented above.

The upper panel depicts the growth rates of the real industrial production and the NBER business cycle chronology, where recessions are denoted by the shaded areas. The lower panel compares the conditional recession probabilities to the NBER chronology. The TAR recession probabilities in most cases coincide with the NBER dates, but predict much longer recessions than really observed. This means that the TAR model in this particular case produces many false alarms.

If the transition variable is a lagged endogenous variable, $$y_{t-d}$$, with delay $$d>0$$, self-exciting threshold autoregressive (SETAR) model results: $$$y_t=\bigg(\alpha_0+\sum_{i=1}^p \alpha_p y_{t-i}\bigg)I(y_{t-d};c) + \bigg(\beta_0+\sum_{i=1}^p \beta_p y_{t-i}\bigg)\bigg(1-I(y_{t-d};c)\bigg) +\varepsilon_t$$$ where $$\varepsilon_t \sim IID(0,\sigma^2)$$.

In SETAR model, regime-generating process is not assumed to be exogenous but directly linked to the lagged endogenous variable $$y_{t-d}$$.

### 5.6.2 Smooth transition autoregression

Smooth transition autoregressive model (STAR), was first suggested in Teräsvirta and Anderson (1992) and van Dijk, Teräsvirta, and Franses (2002). It has the following form: $$$y_t=\bigg(\alpha_0+\sum_{i=1}^p \alpha_p y_{t-i}\bigg)F(x_t;\gamma,c) + \bigg(\beta_0+\sum_{i=1}^p \beta_p y_{t-i}\bigg)\bigg(1-F(x_t;\gamma,c)\bigg) +\varepsilon_t \tag{5.9}$$$ where $$y_t$$ is a dependent variable (e.g., real GDP growth rates or price-to-rent ratio); $$x_t$$ is the transition variable; and $$F(x_t;\gamma,c)$$ is a transition function, which, unlike the abrupt step-wise indicator function in TAR, is smooth. It is a continuous function bounded between 0 and 1.

As transition variable one can use either a lagged dependent variable $$y_{t-d}$$ taken with some delay $$d>0$$, or an exogenous observed variable, $$x_t$$; or (possibly nonlinear) function of lagged exogenous variable $$g(x_t)$$;

There are two main types of STAR transition function: logistic and exponential functions.

Logistic function is defined as: $$$\begin{array}{ll} F(x_t;\gamma,c)=\frac{1}{1+\exp[-\gamma(x_t-c)]}, & \gamma>0 \end{array} \tag{5.10}$$$

Exponential function is defined as: $$$\begin{array}{ll} F(x_t;\gamma,c)=1-\exp[-\gamma(x_t-c)^2], & \gamma>0 \end{array} \tag{5.11}$$$ where $$c$$ is the threshold between the regimes; and $$\gamma$$ is the smoothness parameter: the smaller $$\gamma$$ the smoother transition between regimes.

Note that when $$\gamma\longrightarrow \infty$$, STAR transition function $$F(x_t;\gamma,c)$$ becomes indicator function of TAR, $$I(x_t;c)$$. When $$\gamma\longrightarrow 0$$, STAR model converges to a linear AR(p).

STAR model with logistic transition function is called logistic STAR (LSTAR). The model with exponential transition function is known as exponential STAR (ESTAR). There is no economic theory allowing to distinguish between LSTAR and ESTAR. The choice of transition function as well as of delay parameter $$d$$ and lag structure of the model should be based on the data.

The STAR model can be interpreted as a regime-switching model that allows for two regimes associated with the extreme values of the transition function, $$F(x_t;\gamma,c)=0$$ and $$F(x_t;\gamma,c)=1$$, where the transition from one regime to the other is smooth. The regime that occurs at $$t$$ can be determined by observed variable $$x_t$$ and corresponding value of $$F(x_t;\gamma,c)$$. Figure 5.11 displays the LSTAR and ESTAR transition functions.

These transition functions are very different. While LSTAR transition function is low at low values of the transition variable, $$x_t$$, relative to the threshold value $$c$$, it gets higher at the higher values of the transition variable. Such a trajectory corresponds very well to the business cycles and asset price bubbles: during expansions (bubbles), the transition variable is higher, while during recessions (no-bubbles), it is lower. In the ESTAR model, regimes are associated with small and large absolute deviations of transition variable from threshold $$c$$: when $$x_t$$ is either too small or too high ($$|x_t-c|$$ is large), regime 1 occurs, while when $$x_t$$ is close to the threshold ($$|x_t-c|$$ is small), regime 2 takes place. Thus, the ESTAR transition variable is high at the extreme values of the transition variable and low at its middle values. Such a trajectory is more appropriate for the currency crises: when the exchange rate is extremely under- or overvalued, that is, it violates upper/lower limit of a specified band of fluctuations, the economy is in the crisis regime, while the exchange rate takes intermediate values, the economy is in no-crisis regime.

Figure 5.12 shows the effect the smoothness parameter, $$\gamma$$, exerts on the smoothness of the transition function.

Smaller values of $$\gamma$$ produce smoother transition. Recall that at very high values of $$\gamma$$ the transition function converges to the TAR stepwise transition function.

Figure 5.13 shows the effects exerted by the change in the threshold parameter, $$c$$, on the LSTAR and ESTAR transition functions.

Larger $$c$$ values shift the transition function rightward without changing the form of the function.

One useful output of the STAR models are the conditional probabilities of recessions or bubbles, depending on how we define the regime of interest. These are computed using equations (5.10) and (5.11).

Conditional probabilities for the LSTAR model: $$$\hat{F}(x_t)=\frac{1}{1+\exp[-\hat{\gamma}(x_t-\hat{c})]}$$$ where $$\hat{\gamma}$$ and $$\hat{c}$$ denote the estimated parameter values.

Conditional probabilities for the ESTAR model: $$$\hat{F}(x_t)=1-\exp[-\hat{\gamma}(x_t-\hat{c})^2]$$$

Figure 5.14 compares the conditional probabilities of recessions obtained using the LSTAR model to the NBER business cycle chronology.

The LSTAR-based recession probabilities quite accurately depict the NBER recession phases (shaded areas): the probabilities approaching 1 almost always precede or coincide with the NBER recession periods.

The STAR approach is very flexible. It allows for some useful and interesting extensions. For example, in the previous examples we considered the models with only two regimes: expansions vs. recessions or bubbles vs. no-bubbles. However, other STAR models can be constructed, which have more than two regimes. For instance, the model can contain three regimes: expansions, mild recessions, and deep recessions (depressions). This is especially useful, when some recessions are very deep (like the Great Recession or COVID-19 crisis) compared to other economic downturns. The STAR models with multiple regimes were elaborated by van Dijk and Franses (1999).

Another extension is the STAR model with an asymmetric transition function. In such a model, the switch between regimes can be different depending on the starting point. For example, during recession the economy declines swiftly, while during expansion the recovery can be slow. The STAR model with asymmetric transition was introduced by Siliverstovs (2005).

### 5.6.3 Markov switching

Markov-switching models (MS), like TAR and STAR, allow to diminish the degree of arbitrariness by estimating the crisis chronology and not relying on the ad hoc definitions. Unlike in TAR and STAR, however, the Markov switching implies the existence of an unobserved state variable, which governs switches between regimes. The regimes can be expansions vs. recessions or bubbles vs. no-bubbles.

One of the first MS models was suggested by Hamilton (1989). It fostered a great interest in Markov switching models as a tool to characterize macroeconomic fluctuations. There is a large number of extensions and refinements of the original model (Krolzig 1997). The basic Markov-switching model can be formulated as follows: $$$y_t=\mu(s_t)+\sum^P_{p=1} \alpha_p y_{t-p}+\varepsilon_t \tag{5.12}$$$ where $$y_t$$ is some dependent variable (e.g., real GDP growth or price-to-rent ratio); $$\varepsilon_t \sim NID(0,\sigma^2)$$; and $$\mu(s_t)$$ is the regime-specific intercept that switches between two states:

$$$\mu(s_t)=\left\{ \begin{array}{rcll} \mu_1<0, & \mbox{if} & s_t=1 & \mbox{expansion} \\ \mu_2>0, & \mbox{if} & s_t=2 & \mbox{recession} \end{array} \right.$$$

In this simple model, the variance of the disturbance term, $$\sigma^2_{\varepsilon}$$, is assumed to be identical in both regimes. In more sophisticated models, all parameters can be regime-dependent: intercept (or mean), $$\mu(s_t)$$; autoregressive coefficients, $$\alpha_p(s_t)$$; and residual variance, $$\sigma^2_{\varepsilon}(s_t)$$. Thus, during the bubble periods, the growth rate of the price-to-rent ratio, $$\frac{\mu(s_t=1)}{1-\sum^P_{p=1} \alpha_p(s_t=1)}$$, can be higher than during the no-bubble periods, $$\frac{\mu(s_t=0)}{1-\sum^P_{p=1} \alpha_p(s_t=0)}$$. In addition, bubble regimes can have higher volatility than the no-bubble ones: $$\sigma^2_{\varepsilon}(s_t=1)>\sigma^2_{\varepsilon}(s_t=0)$$.

When the financial and currency crises are considered, the two states may be defined differently. Stock market: for a stock exchange index the low values of the intercept $$\mu(s_t)$$ can represent “crashes”, while high values — the “booms’’. In the foreign exchange market, the”high volatiliy" regime is associated with crises, while the “low volatility” regime is associated with normal times. Consecutively, when modelling currency crises, a Markov-switching model with state-dependent residual variance $$\sigma^2_{\varepsilon}(s_t)$$ is needed.

## 5.7 Forecast performance measures

In this chapter, we presented a wide range of alternative indicators and models that can be used in order to detect and forecast the episodes like crises or speculative bubbles. How to determine which of the models has the best forecast accuracy? There are several alternative measures of forecast accuracy. They measure how well an indicator or model allow forecasting speculative bubbles or turning points of business cycles.

The fraction correctly predicted, or Ben-Akiva and Lerman’ measure of fit, is computed as the average proportion of correct predictions made by the forecasting model:

$$$FCP = \frac{\sum_{t=1}^T \left(C_t \hat{F}_t + (1-C_t)(1-\hat{F}_t)\right)}{T} \tag{5.13}$$$ where $$C_t$$ is the binary dependent variable (e.g., recessions or bubbles); $$\hat{F}_t$$ is the predicted probability of crisis. The difficulty in this computation is that in unbalanced samples, the less frequent outcome will usually be predicted very badly by the standard procedure, and this measure does not pick up that point.

Cramer (1999) proposed an alternative measure that directly measures this failure:

\begin{align} \lambda = E(\hat{F}_t|C_t=1) - E(\hat{F}_t| C_t=0) \nonumber \\ = \frac{\sum_{t=1}^T C_t\hat{F}_t }{T_1} - \frac{\sum_{t=1}^T (1- C_t)\hat{F}_t }{T_0} \tag{5.14} \end{align} where $$T_1$$ is number of periods, for which $$C_t=1$$, $$T_0$$ is number of observations, for which $$C_t=0$$. Thus, the first r.h.s. term is the average $$\hat{F}_t$$ for $$C_t=1$$, while the second r.h.s. term is the average $$\hat{F}_t$$ for $$C_t=0$$. The Cramer’s measure heavily penalizes the incorrect predictions and, since each proportion is taken within the subsample, it is not unduly influenced by large size of the group of more frequent outcomes.

The next two measures of forecast accuracy are widely used by the practitioners.

Log probability score is defined as: \begin{align} LPS = -\frac{1}{T}\sum_{t=1}^T \Big[(1-\hat{F}_t)\ln(1-C_t) + \hat{F}_t\ln(C_t)\Big] \tag{5.15} \end{align}

Quadratic probability score is computed as: \begin{align} QPS = \frac{1}{T}\sum_{t=1}^T (\hat{F}_t-C_t)^2 \tag{5.16} \end{align}

The QPS varies between 0 (perfect performance) and 1 (bad performance).

Finally, for the sake of completeness, we should mention the Kuipers score, which was already introduced in section 5.5.1: $$$KS = \frac{N_A}{N_A+N_C} - \frac{N_B}{N_B+N_D} \tag{5.17}$$$ The Kuipers score can take values between -1 (all false) and 1 (all correct).

## Exercises

1. Which of the approaches to determining the turning points of housing cycles is the best one: ad hoc, expert judgement, or statistical tests? Explain your reasoning.
2. Assume that the actual housing price is 50,000 rubles per m$$^2$$ and that, according to your estimations, the fundamental price is 40,000 rubles per m$$^2$$. What can you conclude?

## Key terms

 speculative asset price bubbles volatility logit model booms and busts formal statistical tests probit model early warning system multiple bubbles test threshold autoregression expert judgment random walk smooth transition autoregression NBER explosive root Markov switching ECRI fundamental price structural break ad hoc dating exogenous chronology forecast accuracy measures recession chronology endogenous chronology fraction correctly predicted peaks and troughs signalling approach Kuipers score currency crisis discrete-choice model log probability score exchange market pressure linear probability model quadratic probability score

### References

Aalbers, Manuel B. 2016. “The Financialization of Home and the Mortgage Market Crisis.” In The Financialization of Housing, 40–63. Routledge.

Abreu, Dilip, and Markus K Brunnermeier. 2003. “Bubbles and Crashes.” Econometrica 71 (1): 173–204.

Allen, Franklin, and Gary Gorton. 1993. “Churning Bubbles.” The Review of Economic Studies 60 (4): 813–36.

Blanchard, Olivier J, and Mark W Watson. 1982. “Bubbles, Rational Expectations and Financial Markets.” In Crises in the Economic and Financial Structure: Bubbles, Bursts, and Shocks, edited by P Wachtel. D.C.Heath; Company, Lexington.

Brunnermeier, Markus K. 2008. “Bubbles.” In The New Palgrave Dictionary of Economics: Volume 1–8, edited by S N Durlauf and L E Blume, 2:578–83. Springer.

Brunnermeier, Markus K, and Martin Oehmke. 2013. “Bubbles, Financial Crises, and Systemic Risk.” In Handbook of the Economics of Finance, 2:1221–88. Elsevier.

Bry, Gerhard, and Charlotte Boschan. 1971. Cyclical Analysis of Time Series: Selected Procedures and Computer Programs. NBER Books. National Bureau of Economic Research, Inc.

Cramer, Jan Salomon. 1999. “Predictive Performance of the Binary Logit Model in Unbalanced Samples.” Journal of the Royal Statistical Society: Series D (the Statistician) 48 (1): 85–94.

Dreger, Christian, and Konstantin A. Kholodilin. 2013. “An Early Warning System to Predict Speculative House Price Bubbles.” Economics — the Open-Access, Open-Assessment E-Journal 7: 1–26.

Eichengreen, Barry, Andrew K. Rose, and Charles Wyplosz. 1994. “Speculative Attacks on Pegged Exchange Rates: An Empirical Exploration with Special Reference to the European Monetary System.” NBER Working Papers 4898. National Bureau of Economic Research, Inc.

Garber, Peter M. 1990. “Famous First Bubbles.” Journal of Economic Perspectives 4 (2): 35–54.

Gerdesmeier, Dieter, Hans-Eggert Reimers, and Barbara Roffia. 2010. “Asset Price Misalignments and the Role of Money and Credit.” International Finance 13 (3): 377–407.

Glaeser, Edward L, and Charles G Nathanson. 2015. “Housing Bubbles.” In Handbook of Regional and Urban Economics, 5:701–51. Elsevier.

Hamilton, James D. 1989. “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle.” Econometrica 57 (2): 357–84.

Harding, Don, and Adrian Pagan. 2001. “Extracting, Using and Analysing Cyclical Information.” MPRA Paper 15. University Library of Munich, Germany.

Harvey, David. 1985. The Urbanization of Capital. The Johns Hopkins University Press.

Homm, Ulrich, and Jörg Breitung. 2012. “Testing for Speculative Bubbles in Stock Markets: A Comparison of Alternative Methods.” Journal of Financial Econometrics 10 (1): 198–231.

Kaminsky, Graciela, Saul Lizondo, and Carmen M. Reinhart. 1998. “Leading Indicators of Currency Crises.” IMF Staff Papers 45 (1): 1–48.

Krolzig, Hans-Martin. 1997. Markov-Switching Vector Autoregressions. Modelling, Statistical Inference and Application to Business Cycle Analysis. Springer Verlag.

Miller, Edward M. 1977. “Risk, Uncertainty, and Divergence of Opinion.” The Journal of Finance 32 (4): 1151–68.

Phillips, Peter CB, Shuping Shi, and Jun Yu. 2015. “Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500.” International Economic Review 56 (4): 1043–78.

Reinhart, Carmen M., and Graciela L. Kaminsky. 1999. “The Twin Crises: The Causes of Banking and Balance-of-Payments Problems.” American Economic Review 89 (3): 473–500.

Shleifer, Andrei, and Robert W Vishny. 1997. “The Limits of Arbitrage.” The Journal of Finance 52 (1): 35–55.

Siliverstovs, Boriss. 2005. “The Bi-Parameter Smooth Transition Autoregressive Model.” Economics Bulletin 3 (23): 1–11.

Teräsvirta, T, and H M Anderson. 1992. “Characterizing Nonlinearities in Business Cycles Using Smooth Transition Autoregressive Models.” Journal of Applied Econometrics 7 (S): S119–36.

Tirole, Jean. 1985. “Asset Bubbles and Overlapping Generations.” Econometrica: Journal of the Econometric Society 53 (6): 1499–1528.

Tong, H. 1978. “On a Threshold Model.” In Pattern Recognition and Signal Processing, edited by C. H. Chen. Sijhoff & Noordhoff.

Tong, H. 1990. Non-Linear Time Series: A Dynamical System Approach. Oxford University Press.

van Dijk, Dick, and Philip Hans Franses. 1999. “Modeling Multiple Regimes in the Business Cycle.” Macroeconomic Dynamics 3 (03): 311–40.

van Dijk, Dick, Timo Teräsvirta, and Philip Hans Franses. 2002. “Smooth Transition Autoregressive Models — a Survey of Recent Developments.” Econometric Reviews 21 (1): 1–47.

1. The dates of the peaks and troughs of the US business cycle can be found here: http://www.nber.org/cycles/recessions.html.↩

2. This is the website of the ECRI: https://www.businesscycle.com/.↩

format: plain


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## Ampère's Law and Symmetry Argument (20 minutes)

This lecture is the introduction to the Ampère's Law Activity.

• Total current as the flux of current density through a gate (see Acting Out Current Density Activity).
• Ampère's law relates the circulation of the magnetic field around a closed loop and the total current enclosed by the loop.

$$\int{\vec{B}\cdot{\vec{dr}}}=\mu_{0}I_{encl}$$

• Finding the magnetic field using Ampère's Law is an inverse problem and can only be done when there is a sufficient amount of symmetry in the current distribution so that the field be pulled out of the flux integral.
• In order to pull the field out of the integral, one must use the symmetry to make assumptions about the components and the dependence of the field (e.g. that there is only an azimuthal component and it is only dependent on the radius). To justify these assumptions, we build on the idea of proof by contradiction, where one assumes the opposite of what we want to show and demonstrates a contradiction.
• The idea is to make arguments about the functional dependence and direction of the magnetic field by comparing the assumptions made about the field and what an observer placed near the current density in various locations and facing various directions would expect to see based on how she perceives the current distribution to change.

Example: Infinite sheet of current that extends infinitely in the $x$ and $y$ directions and has finite thickness in the $z$ direction.

• What Amperian loop do we want? (can be done as a SWBQ where students are asked to Draw the Amperian loop)
• One that is parallel or perpendicular to field at every point and where the magnitude is constant ($\vec{B}=\pm B(z)\, \hat{y}$ from symmetry arguments below).
• constant $z$, same infinitesimal distance ($dz$) above and below sheet
• Symmetry arguments for directional components
• Magnetic field must be perpendicular to the current at all points (cross product in Biot-Savart Law): $\hat{J}=\hat{x} \Rightarrow \hat{B}\perp\hat{x}$
• Magnetic field does not have component in the $z$ direction.
• If an observer standing on a point above the surface facing in the direction of the current were to close her eyes, reverse the direction of the current ($-\hat{x}$), rotate 180$^\circ$ and open her eyes, there would be nothing to indicate that she has turned. Therefore, every measurement that she takes must be the same in both orientations.
• If she were to measure the magnetic field in the $+\hat{z}$ direction, close her eyes, reverse the direction of the current, rotate, open her eyes, and measure again, the current would have reversed direction and thus, the magnetic field would measure in the $-z$ direction.
• This is a contradiction, thus the magnetic field cannot have a component in the $z$ direction: $\hat{B}\neq\hat{z}$
• Therefore, the magnetic field is purely in the $y$ direction.
• Symmetry arguments for coordinate dependence
• Magnetic field does not have a dependence on $x$ or $y$.
• If an observer were to close her eyes, move in the $x$ or $y$ direction, and open her eyes, the current distribution would look the same.
• Suppose she measures the magnitude of the magnetic field to be $B=3$, closes her eyes, moves in the $x$ or $y$ direction, opens her eyes, and measures again. If the magnetic field depended on either $x$ or $y$, she would measure a different magnitude.
• This is a contradiction, thus the magnitude of the magnetic field cannot depend on $x$ or $y$: $B\neq B(x)\neq(y)$
• If she were to move up or down ($z$ direction), the world may or may not look the same. Therefore, the magnitude of the magnetic field only depends on $z$.

This problem can drag on if followed through all the way to the end, so it is better to stop after the symmetry arguments and the Amperian loop is chosen. This takes them from an inverse problem to a point where you can pull B out of the integral. If students are really struggling with the integral or with finding the current, you can stop the class and continue working on the example.

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SECTION-5

MACHINE DESIGN MCQ

##### Q31. In a transverse double fillet welded lap joint, the allowable tensile stress for the well metal is equal to

(a) 1.414 F/h × l
(b) 0.707 F/h× l
(c) 4.414 F/h× l
(d) 3F/h ×l
where h = Thickness of the plate, l = Length of weld.

Ans:(b) 0.707 F/h× l

##### Q32. In a transverse single fillet welded lap joint, the allowable tensile stress for the weld metal is equal to

(a) 1.414 F/h × l
(b) 0.707 F/h× l
(c) 4.414 F/h× l
(d) 3F/h ×l

Ans:(a) 1.414 F/h × l

##### Q33. When fillet welds are used on one side of the lap joint, a bending moment is introduced because of the eccentricity of the forces. The maximum normal stress due to bending moment is equal to

(a) 1.414 F/h × l
(b) 0.707 F/h× l
(c) 4.414 F/h× l
(d) 3F/h ×l

Ans:(d) 3F/h ×l

##### Q34. When fillet welds are used on one side of the lap joint, a bending moment is introduced because of the eccentricity of the forces, the resultant stress at the root of the weld is equal to

(a) 1.414 F/h × l
(b) 0.707 F/h× l
(c) 4.414 F/h× l
(d) 3F/h ×l

Ans:(c) 4.414 F/h× l

##### Q35. The diameter ‘d’ of the rivet for the given thickness ‘t’ of the plate and to take up a given load F is given by

(a) d = 1.91 $\sqrt{t}$  when d and t are in mm

(b) d = 6 $\sqrt{t}$ when d and t are in cm

(c) both (a) and (b)
(d) none of the above.

Ans:(d) none of the above.

##### Q36. Unwin’s empirical formula is used to find

(a) pitch of a riveted joint
(b) diameter of the rivet hole, when thickness of the plates is less than 8 mm
(c) diameter of the rivet hole, when thickness of the plate is more than 8 mm
(d) none of the above.

Ans:(c) diameter of the rivet hole, when thickness of the plate is more than 8 mm

##### Q37. In Lozenge joint

(a) diamond riveting is used
(b) number of rivets in each row is the same
(c) number of rivets in each row goes no decreasing as we proceed from the outermost row to innermost row
(d) none of the above.

Ans:(a) diamond riveting is used

##### Q38. Two plates are joined by a lap joint. The pull required to crush the rivet per pitch length is equal to

(a) n × d × t × σc
(b) π(d – t) × t × σc
$\text { (c) } n \times\left(\frac{\pi}{4} d^{2}-d t\right) \sigma_{c}$

$\text { (d) } n \times \frac{\pi}{4} d^{2} \times \sigma_{c} \text {. }$

where σc = Permissible crushing stress, n = Number of rivets per pitch length under crushing,
and d = Diameter of rivet.

Ans:(a) n × d × t × σc

##### Q39. Which of the following screw thread is used for power transmission in either direction?

(d) All of the above.

##### Q40. Efficiency of a square threaded screw, neglecting collar friction is given by

(a) tan α/tan φ
(b) tan α/tan (α+φ)
(c) tan φ/tan α
(d) tan φ/tan (α+φ)
where α = Helix angle, and φ = Angle of friction.

Ans:(b) tan α/tan (α+φ)

1      2      3      4      5      6      7      8

DEAR READERS IF YOU FIND ANY ERROR/TYPING MISTAKE PLEASE LET ME KNOW IN THE COMMENT SECTIONS OR E-MAIL: [email protected]

##### Read More Sections of Machine Design

Each section contains maximum 80 Questions. To practice more questions visit other sections.

Machine Design MCQ – Section-1

Machine Design MCQ – Section-2

Machine Design MCQ – Section-3

Machine Design MCQ – Section-4

Machine Design MCQ – Section-5

Machine Design MCQ – Section-6

Machine Design MCQ – Section-7


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How do you find the slope of a line given the points (2,2) and (5,8) on the line?

May 24, 2017

$m = \frac{6}{3} = 2$

Explanation:

Slope means the steepness or gradient of a line.

It can be described as the " "("change in the y-values")/("change in the x-values")

which is sometimes explained as $\left(\text{rise")/("run}\right)$

The formula is $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

The points are (2,2)" as $\left({x}_{1} , {y}_{1}\right) \text{ and } \left(5 , 8\right)$ as $\left({x}_{2} , {y}_{2}\right)$

$m = \frac{8 - 2}{5 - 2} = \frac{6}{3} = 2$

May 24, 2017

$m = 2$

Explanation:

The slope formula determines a slope from two points.

It says that given two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$, you find the slope from:

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

If we let $\left(2 , 2\right) = \left({x}_{1} , {y}_{1}\right)$ and $\left(5 , 8\right) = \left({x}_{2} , {y}_{2}\right)$, then we can plug these values into the slope formula.

Note that it does not matter which of the two points you decide to be the "first" or the "second".

$m = \frac{8 - 2}{5 - 2} = \frac{6}{3} = 2$


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# Does Conway's PRIMEGAME generate all prime powers of 2?

Most sites I have visited reading on this interesting topic state something along the lines

"the only powers of two (other than 2 itself) that occur in this sequence are those with prime exponent" (MathWorld)

or

"After 2, this sequence contains the following powers of 2: [...] which are the prime powers of 2." (Wikipedia)

These careful formulations would imply that the set of powers of 2 generated in the sequence is a subset of prime powers of 2.

However, the OEIS seems absolutely certain that the two sets are equal: http://oeis.org/A034785

This result is also cited on other sites I do not consider very reliable for exact wording, like http://esolangs.org/wiki/Fractran.

Honestly, I have not understood the internal mechanics of PRIMEGAME enough to answer my own question yet. However, I think it makes a significant difference in the interestingness of PRIMEGAME. Why would sites like MathWorld not state the full fact?

-
The MathWorld article, directly after the passage you quote, says: "$2^2$, $2^3$, $2^5$, $2^7$, ..." Unless MathWorld is known to be fast-and-loose with ellipses, that would strongly suggest to me that the sequence eventually includes every prime power of two. –  Chris Pressey Dec 14 '12 at 12:41
Yes, PRIMEGAME outputs 2^k if and only if k is prime. Here's one explanation by Conway himself: mathematik.uni-bielefeld.de/~sillke/NEWS/fractran See also oeis.org/wiki/Conway's_PRIMEGAME The original paper is well worth reading if you can track it down. –  JɛﬀE Dec 14 '12 at 14:59
@JɛﬀE comment->answer ? –  Suresh Venkat Dec 14 '12 at 16:21
note [complexity-theory angle] its very inefficient. in analysis article decomposing it into subroutine primitives, "Using these, the author shows that the Conway program is equivalent to a well-known (although highly inefficient) procedure for inspecting the next number for primality. His running analysis shows that inspecting the thousandth prime (8831) would require 468 056 052 atomic steps." Richard K. Guy, Math. Mag. 56 (1983), no. 1, 26--33. –  vzn Dec 14 '12 at 20:09

Yes, PRIMEGAME outputs $2^k$ if and only if $k$ is prime.

Conway's original paper is well worth reading if you can track it down. You can also find a very clear exposition in Richard Guy's paper Conway's prime producing machine (Mathematics Magazine 56(1):26–33, 1983), including the wonderful cartoon below. (Yes, that's Conway with the Alexander horns, referring to a famous drawing by Simon Fraser.) Conway himself posted a concise proof on the math-fun mailing list. There's also a brief explanation at the OEIS blog.

-
Great picture!!! –  Danny May 29 at 11:01


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Physics Help Forum elastic collision

 Kinematics and Dynamics Kinematics and Dynamics Physics Help Forum

 Apr 15th 2009, 12:42 PM #1 Junior Member   Join Date: Feb 2009 Posts: 8 elastic collision I missed a day of class and can't figure out how to do this problem. Any explanations would help me out greatly. A proton (atomic mass 1 u) with a speed of 500 m/s collides elastically with another proton at rest. The original proton is scattered at 60 degrees from its original direction. What are the speeds of the two particles after the collision?
 Apr 15th 2009, 11:47 PM #2 Senior Member   Join Date: Apr 2008 Location: HK Posts: 886 This is a 2-D problem for conservation of momentum Take the direction the proton travels as positive x-direction and perpendicular to x as y direction One important fact you have to notice is that when two objects of same masses collide with each other, their angle between their new travelling directions is 90 degrees Then by $\displaystyle m_1u_1 + m_2u_2=m_1v_1+m_2v_2$ In x-direction $\displaystyle 1(500)+1(0)=1cos60^ov_1 + 1cos30^ov_2$ In y-direction $\displaystyle 1(0)+1(0)=1sin60^ov_1 - 1sin30^ov_2$

 Tags collision, elastic

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 Similar Physics Forum Discussions Thread Thread Starter Forum Replies Last Post teddybear9 Energy and Work 1 Feb 1st 2012 08:24 AM Aladdin Kinematics and Dynamics 2 Nov 5th 2009 10:43 AM chrisfnet Kinematics and Dynamics 3 Oct 27th 2009 03:28 PM rjsci13 Kinematics and Dynamics 1 Oct 13th 2009 03:50 AM SengNee Advanced Mechanics 2 Jul 25th 2008 07:40 PM


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nLab Tate's acyclicity theorem

Contents

Context

Analytic geometry

analytic geometry (complex, rigid, global)

Basic concepts

analytic function

analytification

GAGA

Contents

Statement

Let $X = Spec_{an}(A)$ be an affinoid space with affinoid algebra $A$.

Then for every finite cover of $X$ by affinoid domains, the corresponding Cech cohomology with coefficients in the structure sheaf $A$, or with coefficients in any finite Banach module over $A$, is concentrated in degree 0.

References

• S. Bosch, U. Güntzer, Reinhold Remmert, section 8.2 of Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)

• Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

• pdf

• Doosung Park, Affinoid domains, lecture notes (pdf)

Last revised on July 18, 2014 at 00:46:35. See the history of this page for a list of all contributions to it.


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# Limit theorems for stationary Markov processes with L2-spectral gap

Limit theorems for stationary Markov processes with L2-spectral gap - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

1 IRMAR - Institut de Recherche Mathématique de Rennes

Abstract : Let $X t, Y t {t\in \mathbb{T}}$ be a discrete or continuous-time Markov process with state space $\mathbb{X} \times \mathbb{R}^d$ where $\mathbb{X}$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $X t, Y t {t\in \mathbb{T}}$ is assumed to be a Markov additive process. In particular, this implies that the first component $X t {t\in \mathbb{T}}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process $Y t {t\in \mathbb{T}}$ is shown to satisfy the following classical limit theorems: a the central limit theorem, b the local limit theorem, c the one-dimensional Berry-Esseen theorem, d the one-dimensional first-order Edgeworth expansion, provided that we have $\sup\{ t\in0,1\cap \mathbb{T} : \mathbb{E}{\pi,0}|Y t| ^{\alpha} < 1\}$ with the expected order with respect to the independent case up to some $\varepsilon > 0$ for c and d. For the statements b and d, a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process $X t {t\in \mathbb{T}}$ has an invariant probability distribution $\pi$, is stationary and has the $\mathbb{L}^2\pi$-spectral gap property that is, $X t {t\in \mathbb{N}}$ is $ho$-mixing in the discrete-time case. The case where $X t {t\in \mathbb{T}}$ is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M-estimators associated with $ho$-mixing Markov chains.

Keywords : Markov additive process central limit theorems Berry-Esseen bound Edgeworth expansion spectral method $ho$-mixing M-estimator

Autor: Déborah Ferré - Loïc Hervé - James Ledoux -

Fuente: https://hal.archives-ouvertes.fr/

DESCARGAR PDF


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# An Inside Job

Algebra Level pending

$$(x+3)(x^2+ax-2) = x^3 + bx^2 - 6$$ for some real numbers $$a$$ and $$b .$$ What is $$a ?$$

×


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# Write Files from a Database¶

Last Updated: July 30, 2014

Reading GSSHA files is only half the story. GsshaPy is also able read a GsshaPy database and write the data back to the proper file formats. It is necessary to write the data back to the original file format to be able to execute the GSSHA simulation after modifying some input file in the database.

## Retrieve Object from Database¶

Like reading to the database, we need and instance of a gsshapy.orm.ProjectFile class to call the write() method on. When writing, the ProjectFile instance is created by querying the database for the project file we wish to write. Issue the following command in the Python prompt:

>>> newProjectFile = session.query(ProjectFile).first()

The query instantiates the new project file object with the data from the database query.

## Write to File¶

Now we call the write() method on the new instance of gsshapy.orm.ProjectFile, newProjectFile. This method requires three arguments: an SQLAlchemy session object, a directory to write to, and the name you wish the file to be saved as. Define these attributes and call the write method as follows:

>>> writeDirectory = '/path_to/tutorial-data/write'
>>> name = 'mymodel'
>>> newProjectFile.write(session=session, directory=writeDirectory, name=name)

If all has gone well, you will find a copy of the project file in the write directory. If you compare the file with the original you will notice some differences. Notice that most of the path prefixes have been changed to match the name of the project file. This is a GSSHA convention that is preserved by GsshaPy. If you change only the project file using GsshaPy, be sure it is written out with the same name as the original. Paths are stored as relative in the GsshaPy database. Consequently, all the paths will be written out again as relative paths.

Tip

If you need to prepend a directory to the paths in the project file, use the appendDirectory() method of a gsshapy.orm.ProjectFile object.


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# What quantity of heat is necessary to convert 50 g of ice at 0 degrees C into steam at 100 degrees C?

## The heat of fusion of ice is 80 cal/g., the heat of vaporization of water is 540 cal/g., and the specific heat of water is 1 cal/g.*degrees C.

Aug 5, 2016

The amount of heat required is 36 kcal.

There are three separate heats involved in this problem:

1. ${q}_{1}$ = heat required to melt the ice to water at 0 °C
2. ${q}_{2}$ = heat required to warm the water from 0 °C to 100 °C
3. ${q}_{3}$ = heat required to convert the water to steam at 100 °C

q = q_1 + q_2 + q_3 = mΔ_text(fus)H + mcΔT + mΔ_text(vap)H

q_1 = mΔ_"fus"H = 50 color(red)(cancel(color(black)("g"))) × 80color(white)(l) "cal"·color(red)(cancel(color(black)("g"^"-1"))) = "4 000 cal"

q_2 = mcΔT = 50 color(red)(cancel(color(black)("g"))) × 1 color(white)(l)"cal"·color(red)(cancel(color(black)("g"^"-1""°C"^"-1"))) × 100 color(red)(cancel(color(black)("°C"))) = "5 000 cal"

q_3 = mΔ_"vap"H = 50 color(red)(cancel(color(black)("g"))) × 540color(white)(l) "cal"·color(red)(cancel(color(black)("g"^"-1"))) = "27 000 cal"

$q = {q}_{1} + {q}_{2} + {q}_{3} = \text{4 000 cal + 5 000 cal + 27 000 cal" = "36 000 cal" = "36 kcal}$


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### - Art Gallery -

Carreau fluid is a type of generalized Newtonian fluid where viscosity, $${\displaystyle \mu _{\operatorname {eff} }}$$, depends upon the shear rate, $${\dot {\gamma }}$$, by the following equation:

$${\displaystyle \mu _{\operatorname {eff} }({\dot {\gamma }})=\mu _{\operatorname {\inf } }+(\mu _{0}-\mu _{\operatorname {\inf } })\left(1+\left(\lambda {\dot {\gamma }}\right)^{2}\right)^{\frac {n-1}{2}}}$$

Where: $$\mu _{0}$$ , $${\displaystyle \mu _{\operatorname {\inf } }}$$ , $$\lambda$$ and n are material coefficients.

$$\mu _{0}$$ = viscosity at zero shear rate (Pa.s)

$${\displaystyle \mu _{\operatorname {\inf } }}$$ = viscosity at infinite shear rate (Pa.s)

$$\lambda$$ = relaxation time (s)

n = power index

At low shear rate ($${\displaystyle {\dot {\gamma }}\ll 1/\lambda }$$ ) a Carreau fluid behaves as a Newtonian fluid with viscosity $$\mu_0$$ . At intermediate shear rates ( $${\displaystyle {\dot {\gamma }}\gtrsim 1/\lambda }$$), a Carreau fluid behaves as a Power-law fluid. At high shear rate, which depends on the power index n and the infinite shear-rate viscosity $${\displaystyle \mu _{\operatorname {\inf } }}$$, a Carreau fluid behaves as a Newtonian fluid again with viscosity $${\displaystyle \mu _{\operatorname {\inf } }}$$.

The model was first proposed by Pierre Carreau.

Navier-Stokes equations
Fluid
Cross fluid
Power-law fluid
Generalized Newtonian fluid

References

Kennedy, P. K., Flow Analysis of Injection Molds. New York. Hanser. ISBN 1-56990-181-3

Physics Encyclopedia

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Index