1. Lecture 11 Preview: Hypothesis Testing and the Wald Test Wald Test Let Statistical Software Do the Work Testing the Significance of the “Entire” Model No Money Illusion Theory: Calculating Prob[Results IF H 0 True] Clever Algebraic Manipulation Equivalence of Two-Tailed Student-t Tests and Wald Tests (F-Tests) Three Important Distributions: Normal, Student-t, and F Restricted Regression Reflects H 0 Unrestricted Regression Reflects H 1 Comparing the Restricted Sum of Squared Residuals and the Unrestricted Sum of Squared Residuals: The F-statistic Two-Tailed t-Test Wald Test

2. Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogP  0.411812 0.093532-4.4029050.0003 LogI0.5080610.2665831.9058290.0711 LogChickP0.1247240.0714151.7464650.0961 Const9.4992582.3486194.0446150.0006 Number of Observations24Degrees of Freedom20 Step 1: Collect data, run the regression, and interpret the estimates Model: No Money Illusion Theory:  P +  I +  CP = 0 Taking logs: log(Q t ) = log(  Const ) +  P log(P t ) +  I log(I t ) +  CP log(ChickP t ) + e t Interpreting the coefficient estimates: b I = Income Elasticity of Demand =.51 b P = (Own) Price Elasticity of Demand = .41 b CP = Cross Price Elasticity of Demand =.12 What does the sum of the elasticity estimates equal? b P + b I + b CP = .41 +.51 +.12 Critical Regression Result: The sum of the elasticity estimates is.22; the sum is.22 from 0. A 1 percent increase in all prices and income increases the quantity demanded by.22 percent Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses: H 0 :  P +  I +  CP = 0 Cynic’s view is correct: No money illusion H 1 :  P +  I +  CP  0 Cynic’s view is incorrect: Money illusion present Cynic’s view: Despite the results, money illusion is not present. Step 0: Construct a model reflecting the theory to be tested. No Money Illusion Theory: Increasing all prices and income by the same proportion does not affect the quantity of goods demanded. =.22  EViews The evidence suggests that the no money illusion theory is incorrect.

3. Step 3: Formulate the question to assess the cynic’s view. Specific Question: Critical regression result: The sum of elasticity estimates equals.22, the sum is.22 from 0. What is the probability that the sum of elasticity estimates in one regression would be at least.22 from 0, if H 0 were true (if the sum of the actual elasticities equaled 0)? Question: How can we calculate Prob[Results IF H 0 True]? H 0 :  P +  I +  CP = 0 Cynic is correct: No money illusion H 1 :  P +  I +  CP ≠ 0 Cynic is incorrect: Money illusion present Generic Question: What is the probability that the results would be like those we actually obtained (or even stronger), if the cynic were correct and H 0 actually true? Answer: There are three ways: Clever Algebraic Manipulation: Wald Test Let Statistical Software Do the Work Step 4: Use the properties of the estimation procedure to calculate Prob[Results IF H 0 True]. Prob[Results IF H 0 True] =.43

4. Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogPLessLogChickP  0.467358 0.062229-7.5102840.0000 LogILessLogChickP0.3019060.0689034.3816060.0003 Const11.368760.26048243.645160.0000 Number of Observations24Degrees of Freedom21 Sum of Squared Residuals.004825 Calculating Prob[Results IF H 0 True] – Method 2: Wald Test log(Q t ) = log(  Const ) +  P log(P t ) +  I log(I t ) +  CP log(ChickP t ) + e t Model and Hypotheses: H 0 :  P +  I +  CP = 0 No money illusion. H 1 :  P +  I +  CP  0 Money illusion present First, consider the restricted model, the model that is consistent with the null hypothesis: H 0 :  P +  I +  CP = 0log(Q t ) = log(  Const ) +  P log(P t ) +  I log(I t ) +  CP log(ChickP t ) + e t  CP =  (  P +  I ) = log(  Const ) +  P log(P t ) +  I log(I t )  (  P +  I ) log(ChickP t ) + e t = log(  Const ) +  P log(P t ) +  I log(I t )   P log(ChickP t )   I log(ChickP t ) + e t = log(  Const ) +  P log(P t )   P log(ChickP t ) +  I log(I t )   I log(ChickP t ) + e t = log(  Const ) +  P [log(P t )  log(ChickP t )] +  I [log(I t )  log(ChickP t )] + e t SSR R =.004825 DF R = 24  3 = 21 Restricted Regression:  EViews log(Q t )= log(  Const ) +  P LogPLessLogChickP t +  I LogILessLogChickP t + e t Two Regressions: A Restricted Regression and an Unrestricted Regression

5. Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogP  0.411812 0.093532-4.4029050.0003 LogI0.5080610.2665831.9058290.0711 LogChickP0.1247240.0714151.7464650.0961 Const9.4992582.3486194.0446150.0006 Number of Observations24Degrees of Freedom20 Sum of Squared Residuals.004675 log(Q t ) = log(  Const ) +  P log(P t ) +  I log(I t ) +  CP log(ChickP t ) + e t Model and Hypotheses: H 0 :  P +  I +  CP = 0 H 1 :  P +  I +  CP  0 Next, consider the unrestricted model, the model that is consistent with the alternative hypothesis. Here we allow the estimates to take on any values; their values are unrestricted: Unrestricted Regression: SSR U =.004675 DF U = 24  4 = 20 Compare the sum of squared residuals from the restricted and unrestricted regressions: SSR R =.004825 In general, SSR R  SSR U Is this a coincidence?No. The OLS estimation procedure chooses the estimates of the constant and coefficients so as to minimize the sum of squared residuals. The unrestricted regression places no restrictions on the estimates. Imposing a restriction limits the ability to make the sum of squared residuals as small as possible. Note that SSR R > SSR U A restriction can never decrease the sum of squared residuals; a restriction can only increase the sum. No money illusion. Money illusion present  EViews  Lab 11.1

6. SSR R will be larger than SSR U.  Restriction not actually true  Restriction should cost much in terms of SSR  Expect SSR R to be much larger than SSR U  Expect F-statistic to be large Depends on whether or not H 0 is actually true. If H 0 is not actually true  Restriction actually true  Restriction should cost little in terms of SSR  Expect SSR R to be only a little larger than SSR U  Expect F-statistic to be small If H 0 is actually true But, by how much? Restricted regression Unrestricted regression How do we decide if the SSR R is much or just a little larger than SSR U ? Question: Can the F-statistic be negative? No. Because SSR R  SSR U H 0 :  P +  I +  CP = 0 H 1 :  P +  I +  CP  0 No money illusion. Money illusion present

7. Calculating the F-Statistic: SSR R =.004875DF R = 21 SSR U =.004625DF U = 20 SSR R  SSR U =.000150DF R  DF U = 1 H 0 :  P +  I +  CP = 0 No money illusion H 1 :  P +  I +  CP  0 Money illusion present SSR R =.004875 DF R = 21 SSR U =.004625 DF U = 20 Restricted regressionUnrestricted regression =.000150/1.004675/20 =.000150.000234 =.64 Cynic’s View Sure, SSR R is larger than SSR U and the F-statistic is not 0, but this is just the “luck of the draw.” In fact, H 0 is true, money illusion is not present, the restriction is true. H 0 :  P +  I +  CP = 0 No money illusion Restriction actually true H 1 :  P +  I +  CP  0 Money illusion present Restriction actually not true Question Assess the Cynic’s View Generic Question: What is the probability that the results would be like those we actually obtained (or even stronger), if the cynic were correct and H 0 actually true? Specific Question: Regression results: F-statistic equals.64. What is the probability that the F-statistic from one pair of regressions would be.64 or more, if H 0 were true (if no money illusion were present or equivalently if the restriction were true)? Answer: Prob[Results IF Cynic Correct] or equivalently Prob[Results IF H 0 True]

8. H 0 :  P +  I +  CP = 0 No money illusion Restriction actually true H 1 :  P +  I +  CP  0 Money illusion present Restriction actually not true =.64.000150.004675 = / 1 / 20 Prob[Results IF H 0 True]  Lab11.3 .43 Calculating Prob[Results IF H 0 True] Using a Simulation Prob[Results IF H 0 True] small  Unlikely that H 0 is true Prob[Results IF H 0 True] large  Likely that H 0 is true  Do not reject H 0  Reject H 0 Specific Question: Regression results: F-statistic equals.64. What is the probability that the F- statistic from one pair of regressions would be.64 or more, if H 0 were true (if money illusion is not present or equivalently if the restriction is true)? Answer: Prob[Results IF H 0 True] SSR Dem =.004675 SSR Num =.000150 DF Dem = 20 DF Num = 1 Calculating Prob[Results IF H 0 True] Using the F-Distribution Prob[Results IF H 0 True] DF Num = 1 DF Dem = 20.64.43 The F-distribution is described by the degrees of freedom of the numerator and the degrees of freedom of the denominator: Question: Using the “traditional” significance levels, would you reject H 0 ? Answer: No.  Lab 11.4 But, what if we could not use a simulation? =.43

9. Wald Test Degrees of Freedom ValueNumDemProb F-statistic0.6416441200.4325 Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogP  0.411812 0.093532-4.4029050.0003 LogI0.5080610.2665831.9058290.0711 LogChickP0.1247240.0714151.7464650.0961 Const9.4992582.3486194.0446150.0006 Number of Observations24Degrees of Freedom20 Method 1 – Clever Algebra Approach: Prob[Results IF H 0 True] What is the probability that the coefficient estimate from one regression would be at least.22 from 0, if H 0 were true (if there were no money illusion, if  Clever actually equaled 0)? =.43 Method 2 – Wald Approach: Prob[Results IF H 0 True] What is the probability that the F-statistic from one pair of regressions would be.64 or more, if H 0 were true (if there were no money illusion, if actual elasticity sum to 0)? =.43 Conclusion: While the approaches differ, the two methods produce identical results. In fact, it can be shown rigorously that methods 1 and 2 are equivalent.Calculating Prob[Results IF H 0 True] – Method 3: Let statistical software do the work Coefficient restriction: C(1) + C(2) + C(3) = 0 The F-statistic equals.64. If the restriction we specified were actually true, the probability that the F-statistic from one repetition of the experiment would be.64 or more equals.43. The probabilities in all three methods are identical. Run the unrestricted regression: Note: The F-statistics in the second and third methods are identical. C(1) C(2) C(3) C(4) EViews Numbering Convention:  EViews

10. Comparing methods 1, 2, and 3: Method 1 – Clever Algebra Approach: Prob[Results IF H 0 True] What is the probability that the coefficient estimate from one regression would be at least.22 from 0, if H 0 were true (if there were no money illusion, if  Clever actually equaled 0)? =.43 Method 2 – Wald Approach: Prob[Results IF H 0 True] What is the probability that the F-statistic from one pair of regressions would be.64 or more, if H 0 were true (if there were no money illusion, if actual elasticity sum to 0)? =.43 Method 3 – Statistical Software Approach: Prob[Results IF H 0 True]=.43 Conclusion: While the approaches differ, the methods produce identical results.

11. Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogP  0.411812 0.093532-4.4029050.0003 LogI0.5080610.2665831.9058290.0711 LogChickP0.1247240.0714151.7464650.0961 Const9.4992582.3486194.0446150.0006 Number of Observations24Degrees of Freedom20 Sum of Squared Residuals.004675 Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb Const 12.305760.0057522139.5390.0000 Number of Observations24Degrees of Freedom23 Sum of Squared Residuals.018261 Testing the “Entire” Model Model: log(Q t ) = log(  Const ) +  P log(P t ) +  I log(I t ) +  CP log(ChickP t ) + e H 0 :  P = 0 and  I = 0 and  CP = 0 The explanatory variables have no effect on the dependent variable. H 1 :  P  0 and/or  I  0 and/or  CP  0 At least one explanatory variable has an effect on the dependent variable. Restricted Regression. H 0 :  P = 0 and  I = 0 and  CP = 0 Unrestricted Regression. H 1 :  P  0 and/or  I  0 and/or  CP  0 NB: If the null hypothesis is actually true, we have a good or bad model? log(Q t ) = log(  Const ) SSR U =.004675 DF U = 24  4 = 20 DF R = 24  1 = 23 SSR R =.018261 log(Q t ) = log(  Const ) +  P log(P t ) +  I log(I t ) +  CP log(ChickP t ) + e t  EViews Very bad!

12. Prob[Results IF H 0 True]: The F-statistic equals 19.4. What is the probability that the F-statistic from one pair of regressions would be 19.4 or more, if H 0 were true (if prices and income actually had no effect on the quantity demanded, if  P,  I, and  CP actually equaled 0)? Prob[Results IF H 0 True] smallProb[Results IF H 0 True] large H 0 :  P = 0 and  I = 0 and  CP = 0 The explanatory variables have no effect on the dependent variable. H 1 :  P  0 and/or  I  0 and/or  CP  0 At least one explanatory variable has an effect on the dependent variable. SSR R =. 018261 DF R = 24  1 = 23 SSR U =.004675DF U = 24 – 4 = 20 SSR R  SSR U =. 013586DF R  DF U = 3 =. 013586/3.004675/20 =.004529.000234 =19.4 Calculate the F-statistic:  Unlikely H 0 is true  Reject H 0  Likely H 0 is true  Do not reject H 0 SSR Dem =.000234DF Dem SSR Num =.013586DF Num = 20 = 3 Calculating Prob[Results IF H 0 True]  Lab 11.5 Prob[Results IF H 0 True] <.0001

13. Wald Test Degrees of Freedom ValueNumDemProb F-statistic19.372233200.0000 Ordinary Least Squares (OLS) Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogP  0.411812 0.093532-4.4029050.0003 LogI0.5080610.2665831.9058290.0711 LogChickP0.1247240.0714151.7464650.0961 Const9.4992582.3486194.0446150.0006 Number of Observations24Degrees of Freedom20 Sum of Squared Residuals.004675 F-Statistic19.3722Prob[F-Statistic].000004 Prob[Results IF H 0 True]: What is the probability that the F-statistic from one pair of regressions would be 19.4 or more, if H 0 were true (if prices and income actually have no effect on the quantity demanded, if  P,  I, and  CP actually equaled 0)? In fact, statistical software calculates this F-statistic and probability automatically: Prob[Results IF H 0 True] <.0001  Prob[Results IF H 0 True] small H 0 :  P = 0 and  I = 0 and  CP = 0 The explanatory variables have no effect on the dependent variable. H 1 :  P  0 and/or  I  0 and/or  CP  0 At least one explanatory variable has an effect on the dependent variable.  Unlikely H 0 is true  Reject H 0 Coefficient restriction: C(1) = C(2) = C(3) = 0 Using the “traditional” significance levels would you reject H 0 ?  EViews Could we let statistical software do the work? C(1) C(2) C(3) C(4) EViews Numbering Convention: Aside: Can a probability equal precisely 0? No.  EViews

14. Dependent Variable: LogQ Explanatory Variable(s): EstimateSEt-StatisticProb LogP  0.411812 0.093532-4.4029050.0003 LogI0.5080610.2665831.9058290.0711 LogChickP0.1247240.0714151.7464650.0961 Const9.4992582.3486194.0446150.0006 Number of Observations24Degrees of Freedom20 Equivalence of Two-Tailed t-Tests and Wald Tests Claim: A two-tailed t-test is a special case of a Wald Test log(Q t ) = log(  Const ) +  P log(P t ) +  I log(I t ) +  CP log(ChickP t ) + e t We shall illustrate this by considering the constant elasticity demand model. H 0 :  CP = 0 H 1 :  CP  0 Price of chicken does not affect the quantity of beef demanded Price of chicken does affect the quantity of beef demanded To justify the claim, focus attention on the coefficient of the price of chicken,  CP, and consider the following “two-tail” hypotheses: Prob[Results IF H 0 True]: What is the probability that the coefficient estimate in one regression would be at least.12 from 0, if H 0 were true (if the actual coefficient,  CP, equaled 0; that is, if the price of chicken had no effect on the quantity demanded)? First, we shall calculate the probability using a Student t-test and then a Wald test. Critical Regression Result: The LogChickP coefficient estimate equals.12. The estimate does not equal 0; the estimate is.12 from 0. This evidence suggests that the chicken price does affect the quantity of beef demanded.  EViews

15. Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogP  0.411812 0.093532-4.4029050.0003 LogI0.5080610.2665831.9058290.0711 LogChickP0.1247240.0714151.7464650.0961 Const9.4992582.3486194.0446150.0006 Number of Observations24Degrees of Freedom20 Use the estimation procedure’s general properties to calculate Prob[Results IF H 0 True]. H 0 :  CP = 0 Price of chicken has no effect on the quantity of beef demanded H 1 :  CP  0 Price of chicken has an effect on the quantity of beef demanded b CP t-distribution OLS estimation procedure unbiased Mean[b CP ] =  CP SE[b CP ] If H 0 were true Number of observations Number of parameters Standard Error DF= 24  4 = 20= 0=.0714 Prob[Results IF H 0 True] Tails Probability: Probability that a coefficient estimate, b CP, resulting from one regression would will lie at least.12 from 0, if the coefficient,  CP, actually equaled 0. Estimate was.12: What is the probability that the coefficient estimate in one regression would be at least.12 from 0, if H 0 were true (if the coefficient,  CP actually equaled 0)?.0961/2 Mean = 0 SE =.0714 DF = 20.120 Tails Probability =.0961.0961/2 =.0961.12

16. Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogP  0.305725 0.074513-4.1029630.0005 LogI 0.8697060.1758954.9444660.0001 Const 6.4073021.6168413.9628520.0007 Number of Observations24Degrees of Freedom21 Sum of Squared Residuals.005388 Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogP  0.411812 0.093532-4.4029050.0003 LogI0.5080610.2665831.9058290.0711 LogChickP0.1247240.0714151.7464650.0961 Const9.4992582.3486194.0446150.0006 Number of Observations24Degrees of Freedom20 Sum of Squared Residuals.004675 Wald Test Model: log(Q t ) = log(  Const ) +  P log(P t ) +  I log(I t ) +  CP log(ChickP t ) + e t “Two-tail” hypothesis: H 0 :  CP = 0 Price of chicken does not affect the quantity of beef demanded H 1 :  CP  0 Price of chicken does affect the quantity of beef demanded Restricted Regression – the null hypothesis:  CP = 0 Unrestricted Regression – the alternative hypothesis:  CP  0 SSR R =.005388 SSR U =.004675 DF R = 24  3 = 21 DF U = 24  4 = 20 log(Q t ) = log(  Const ) +  P log(P t ) +  I log(I t ) + e t  EViews log(Q t ) = log(  Const ) +  P log(P t ) +  I log(I t ) +  CP log(ChickP t ) + e t

17. Wald Test Degrees of Freedom ValueNumDemProb F-statistic3.050139120.0961 SSR R =.005388 DF R = 24  3 = 21 SSR U =.004675DF U = 24 – 4 = 20 SSR R  SSR U =.000713DF R  DF U = 1 =. 000713/1.004675/20 =3.05 H 0 :  CP = 0 Price of chicken has no effect on the quantity demanded. Restricted regression H 1 :  CP  0 Price of chicken has an effect on the quantity demanded. Unrestricted regression Calculate the F-statistic: DF Num = 1 DF Dem = 20 3.05.0961 Wald Test: Prob[Results IF H 0 True] =.0961 The two probabilities are identical. It can be shown rigorously that the two-tailed Student-t test is a special case of the Wald test in which the restriction requires the coefficient to equal 0. Let software calculate Prob[Results IF H 0 True]: Prob[Results IF H 0 True]: The F-statistic equals 3.05. What is the probability that the F-statistic from one pair of regressions would be 3.05 or more, if the H 0 were true (if the actual coefficient,  CP, equaled 0; that is, if the price of chicken actually has no effect on the quantity demanded)? Two-tailed Student-t Test: Prob[Results IF H 0 True] =.0961 Summary of Prob[Results IF H 0 True] Calculations Prob[Results IF H 0 True] =.0961  EViews

18. Three Important Distributions: Normal, Student-t, and F Theories Involving a Single Variable  Student-t distribution Theories Involving Two or More Variables  F distribution  which is described by the  Mean SE DF DF of the Numerator DF of the Denominator If SD is known  Normal distribution  which is described by the If SD is not known  which is described by the  Mean SD