PDF Chapter 11: Hypothesis Testing and the Wald Test

Chapter 11: Hypothesis Testing and the Wald Test

Chapter 11 Outline

? No Money Illusion Theory: Taking Stock ? No Money Illusion Theory: Calculating Prob[Results IF H0 True]

o Clever Algebraic Manipulation

o Wald Test Restricted Regression Reflects H0 Unrestricted Regression Reflects H1 Comparing the Restricted Sum of Squared Residuals and

the Unrestricted Sum of Squared Residuals: The F-Statistic

o Let Statistical Software Do the Work

? Testing the Significance of the "Entire" Model ? Equivalence of Two-Tailed t-Tests and Wald Tests (F-Tests)

o Two-Tailed t-Test

o Wald Test

? Three Important Distributions: Normal, Student-t, and F

Chapter 11 Prep Questions

1. Consider the log form of the constant elasticity demand model:

log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) + et

Show that if P + I + CP = 0, then

log(Qt) = log(Const) + P[log(Pt) - log(ChickPt)] + I[log(It) -

log(ChickPt)] + et

Hint: If the coefficients sum to 0, solve for CP in terms P and I; then,

substitute this expression for CP into the log form of the constant elasticity

demand model.

2. Review how the ordinary least squares (OLS) estimation procedure determines

the value of the parameter estimates. What criterion does this procedure use to

determine the value of the parameter estimates?

3. Recall that the presence of a random variable brings forth both bad news and

good news.

a. What is the bad news?

b. What is the good news?

4. Focus on our beef consumption data:

Beef Consumption Data: Monthly time series data of beef consumption, beef

prices, income, and chicken prices from 1985 and 1986.

Qt

Quantity of beef demanded in month t (millions of pounds)

Pt

Price of beef in month t (cents per pound)

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It

Disposable income in month t (billions of chained 1985 dollars)

ChickPt Price of chicken in month t (cents per pound)

Consider the log form of the constant elasticity demand model:

Model: log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) +

et

a. Use the ordinary least squares (OLS) estimation procedure to estimate

the parameters of the constant elasticity demand model.

[Link to MIT-BeefDemand-1985-1986.wf1 goes here.]

1) What does the sum of squared residuals equal? 2) What is the ordinary least squares (OLS) estimate for

i) P? ii) I? iii) CP? iv) log(Const)? What criterion do these estimates satisfy? b. Now, consider a different restriction. Restrict the value of CP to 0. 1) Incorporate this restriction into the constant elasticity demand model. What is the equation describing the restriction model? 2) We wish to use the ordinary least squares (OLS) estimation procedure to estimate the remaining parameters. i) What dependent variable should we use? ii) What explanatory variables should we use? iii) Run the regression. iv) What does the sum of squared residuals equal? 3) Compared to unrestricted regression, part a, has the sum of squared residuals risen or fallen? Explain why. c. Now, restrict the value of CP to 1. 1) Incorporate this restriction into the constant elasticity demand model. What is the equation describing the restriction model? 2) We wish to use the ordinary least squares (OLS) estimation procedure to estimate the remaining parameters. i) What dependent variable should we use? ii) What explanatory variables should we use? iii) Run the regression. iv) What does the sum of squared residuals equal?

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3) Compared to unrestricted regression, part a, has the sum of squared residuals risen or fallen? Explain why.

No Money Illusion Theory: Taking Stock The money illusion theory contends that whenever all prices and income change by the same proportion the quantity demanded is unaffected. In terms of elasticities, this means that a good's elasticities (own price, income, and cross price) sum to 0. Let us briefly review the steps that we undertook in the last chapter to assess this theory.

Project: Assess the no money illusion theory. Since the linear demand model is intrinsically inconsistent with the no

money illusion theory, we cannot use it to assess the theory. The constant elasticity demand model can be used, however:

Constant Elasticity Demand Model: Q = Const PP I I ChickPCP P = (Own) Price Elasticity of Demand I = Income Elasticity of Demand CP = Cross Price Elasticity of Demand

When the elasticities sum to 0, no money illusion exists: No Money Illusion Theory: The elasticities sum to 0: P + I + CP = 0. Next, we converted the constant elasticity demand model into a linear

relationship by taking natural logarithms: log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) + et

We then used the ordinary least squares (OLS) estimation procedure to estimate the elasticities:

[Link to MIT-BeefDemand-1985-1986.wf1 goes here.]

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Ordinary Least Squares (OLS)

Dependent Variable: LogQ

Explanatory Variable(s): Estimate

SE t-Statistic

LogP

-0.411812 0.093532 -4.402905

LogI

0.508061 0.266583 1.905829

LogChickP

0.124724 0.071415 1.746465

Const

9.499258 2.348619 4.044615

Prob 0.0003 0.0711 0.0961 0.0006

Number of Observations 24

Degrees of Freedom 20

Estimated Equation: EstLogQ = 9.50 - .41LogP + .51LogI + .12LogChick Interpretation of Estimates:

bP = -.41: (Own) Price Elasticity of Demand = -.41 bI = .51: Income Elasticity of Demand = .51 bChickP = .12: Cross Price Elasticity of Demand = .12 Critical Result: Sum of the elasticity estimates (bP + bI + bCP = -.41 + .51 + .12

= .22) does not equal 0; the estimate is .22 from 0. This evidence suggests that money illusion is present and the no money illusion theory is incorrect. Table 11.1: Beef Demand Regression Results ? Constant Elasticity Model

If all prices and income increase by 1 percent, the quantity of beef demanded would increase by .22 percent. The sum of the elasticity estimates does not equal 0; more specifically, the sum lies .22 from 0. The nonzero sum suggests that money illusion exists.

However, as a consequence of random influences, we could never expect the sum of the elasticity estimates to equal exactly precisely 0, even if the sum of the actual elasticities did equal 0. Consequently, we followed the hypothesis testing procedure. We played the cynic in order to construct the null and alternative hypotheses. Finally, we needed to calculate the probability that the results would be like those we obtained (or even stronger), if the cynic is correct and null hypothesis is actually true; that is, we needed to calculate Prob[Results IF H0 True].

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No Money Illusion Theory: Calculating Prob[Results IF H0 True]

Clever Algebraic Manipulation In the last chapter, we explored one way to calculate this probability, the clever algebraic manipulation approach. First, we cleverly defined a new coefficient that equals 0 if and only if the null hypothesis is true:

Clever = P + I + CP We then reformulated the null and alternative hypotheses in terms of the new coefficient, Clever:

H0: P + I + CP = 0 Clever = 0 Money illusion not present H1: P + I + CP 0 Clever 0 Money illusion present After incorporating the new coefficient into the model, we used the ordinary least squares (OLS) estimation procedure to estimate the value of the new coefficient. Since the null hypothesis is now expressed as the new, clever coefficient equaling 0, the new coefficient's tails probability reported in the regression printout is the probability that we need: Prob[Results IF H0 True] = .4325 We shall now explore two other ways to calculate this probability: ? Wald (F-distribution) test ? Letting statistical software do the work

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