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Amherst College Department of Economics Economics 360 Fall 2012 Wednesday, October 10 Handout: One-Tailed Tests, Two-Tailed Tests, and Logarithms Preview • A One-Tailed Hypothesis Test: The Downward Sloping Demand Curve • One-Tailed versus Two-Tailed Tests • A Two-Tailed Hypothesis Test: The Budget Theory of Demand • Summary: One-Tailed and Two-Tailed Tests • Logarithms: A Useful Econometric Tool o Linear Model o Log Dependent Variable Model o Log Explanatory Variable Model o Log-Log (Constant Elasticity) Model One Tail Hypothesis Test: Downward Sloping Linear Demand Curve Theory: A higher price decreases the quantity demanded; demand curve is downward sloping. Step 0: Construct a model reflecting the theory to be tested: GasConst = βConst + βPPriceDollarst + et where GasCons = Quantity of Gasoline Demanded PriceDollars = Price (Chained 1990 Dollars) βP reflects the change in the quantity demanded resulting from a ________________ in the price. The theory suggests that βP should be ______________. A higher price __________ the quantity demanded; the demand curve is ____________ sloping. Step 1: Collect data, run the regression, and interpret the estimates We have collected data for the price of gasoline and gasoline consumption in the U.S. during the 1990’s: GasConst PriceDollarst Year 1990 1991 1992 1993 1994 U. S. gasoline consumption in year t (millions of gallons per day) Real price of gasoline in year t (dollars per gallon – chained 2000 dollars) Real Price ($ per gallon) 1.43 1.35 1.31 1.25 1.23 Gasoline Consumption (Millions of gals) 303.9 301.9 305.3 314.0 319.2 Year 1995 1996 1997 1998 1999 Real Price ($ per gallon) 1.25 1.31 1.29 1.10 1.19 Gasoline Consumption (Millions of gals) 327.1 331.4 336.7 346.7 354.1 2 Dependent Variable: GASCONS Included observations: 10 Coefficient PRICEDOLLARS -151.6556 C 516.7801 Std. Error 47.57295 60.60223 t-Statistic -3.187853 8.527410 Prob. 0.0128 0.0000 Estimated Equation: EstGasCons = _______ − _______PriceDollars. Interpretation: We estimate that a $1 increase in the real price of gasoline ______________ the quantity of gasoline demanded by _______ million gallons. Critical Result: The coefficient estimate equals _______. The _____________ sign of the coefficient estimate suggests that a higher price ____________ the quantity demanded. This evidence ____________ the downward sloping demand theory. Step 2: Play the cynic, challenge the evidence, and construct the null and alternative hypotheses. Cynic’s view: The price actually has no effect on the quantity of gasoline demanded; the negative coefficient estimate obtained from the data was just “the luck of the draw.” In fact, the actual coefficient, βP, equals 0. Now, we construct the null and alternative hypotheses: Cynic’s view is correct: Price has no effect on quantity demanded. H0: βP = 0 H1: βP < 0 Cynic’s view is incorrect: A higher price decreases quantity demanded. Step 3: Formulate the question to assess the cynic’s view. Question for the Cynic: • Generic Question: What is the probability that the results would be like those we actually obtained (or even stronger), if the null hypothesis were actually true (if the cynic is correct and the price actually has no impact on the quantity demanded)? • Specific Question: The regression’s coefficient estimate was −151.7. What is the probability that the coefficient estimate in one regression would be −151.7 or less, if H0 were actually true (if the actual coefficient, βP, equals 0)? Answer: Prob[Results IF Cynic Correct] or Prob[Results IF H0 True] The magnitude of this probability determines whether we accept or reject the null hypothesis: Prob[Results IF H0 True] small Prob[Results IF H0 True] large ↓ ___________ that H0 is true ↓ ___________ that H0 is true ↓ ___________ H0 ↓ ___________ H0 3 Step 4: Use the general properties of the estimation procedure, the probability distribution of the estimates, to calculate Prob[Results IF H0 True]. • • • • Since ordinary least squares estimation procedure for the coefficient value is unbiased, the mean of the probability distribution for the estimate equals _______________________. If the null hypothesis were true, the actual price coefficient would equal _______. The standard error equals _______. The degrees of freedom equal __________________. OLS estimation procedure unbiased Assume H0 is true Mean[bP] = ___ = ___ Dependent Variable: GASCONS Included observations: 10 Coefficient PRICEDOLLARS -151.6556 C 516.7801 Student t-distribution Mean = ______ SE = ______ DF = ______ ______ ______ bP ______ ______ Standard error ______ ______ Number of Observations Number of Parameters DF = ___ − ___ = ___ SE[bP] = ____ Std. Error 47.57295 60.60223 t-Statistic -3.187853 8.527410 Prob. 0.0128 0.0000 Recall that the Prob. column reports the “tails probability:” Tails Probability: The probability that the coefficient estimate, bP, resulting from one regression would lie at least _________ from ____, if the actual coefficient, βP, equals ____. Prob[Results IF H0 True] = Step 5: Decide on the standard of proof, a significance level The significance level is the dividing line between the probability being small and the probability being large. Prob[Results IF H0 True] Prob[Results IF H0 True] Less Than Significance Level Greater Than Significance Level ↓ ↓ Prob[Results IF H0 True] large Prob[Results IF H0 True] small ↓ Unlikely that H0 is true ↓ Likely that H0 is true ↓ Reject H0 ↓ Do not reject H0 Do we reject the null hypothesis at a 10 percent (.10) significance level? ______ Do we reject the null hypothesis at a 5 percent (.05) significance level? ______ Do we reject the null hypothesis at a 1 percent (.01) significance level? ______ Do the results lend support to the downward sloping demand curve theory? ______ 4 Two Tailed Hypothesis Test: Budget Theory of Demand Budget Theory of Demand: Total expenditures for gasoline are constant. That is, when the gasoline price changes, demanders adjust the quantity demanded so as to keep their total gasoline expenditures constant: P×Q = Constant Question: What economic concept is relevant here? Claim: The price elasticity of demand is the relevant concept. To understand why we begin with the verbal definition of price elasticity: Verbal Definition: The price elasticity of demand equals the percent change in the quantity demanded resulting from a one percent change in price. How can we convert the verbal definition of the price elasticity of demand into a rigorous mathematical definition? Price Elasticity = _____________ Change in Quantity resulting from a 1 ___________ Change in the Price = Percent Change in the Quantity Percent Change in the Price Calculating percent changes. If X increases from 200 to 220, there is a 10 percent increase: 200 → 220 20 Percent Change in X = 200 × 100 = .1× 100 = 10 percent We can now generalize this: X → X + ΔX ΔX Percent Change in X = X × 100 ΔQ Q × 100 = ΔP P × 100 = ΔQ P ΔP Q dQ P = dP Q Substituting for the percent changes Simplifying Taking limits as ΔP approaches 0 5 Step 0: Construct a model reflecting the theory to be tested: Constant Price Elasticity Model: Q = βConstP dQ P Price Elasticity of Demand = dP Q βP βP−1 = βConst βP P βP−1 = βConst βPP = βP P βP−1 P P Q Taking the derivative of Q with respect to P P βConstPβP P βP Substituting βConstP βP Substituting βConstP βP = βP for Q. for Q. Simplifying. The price elasticity of demand just equals the value of βP. Question: What does the budget theory of demand postulate about βP? P×Q = Constant Solving for Q Q = _____________ β Compare this equation to the constant price elasticity demand model: Q = βConstP P. Clearly, βConst = ___________ βP = ___ Answer: The budget theory of demand postulates that the price elasticity of demand equals −1.0. Budget Theory of Demand: βP = ___ Logarithms allow us to converting a constant price elasticity model into a linear model: βConst PβP Q = log(Q) = log(βConst) + βPlog(P) Step 1: Collect data, run the regression, and interpret the estimates We have already collected the data, but now we must generate two new variables: the logarithm of quantity and the logarithm of price: • LogQ = log(GasCons) • LogP = log(PriceDollars) Dependent Variable: LOGQ Included observations: 10 LOGP C Coefficient -0.585623 5.918487 Std. Error 0.183409 0.045315 t-Statistic -3.192988 130.6065 Prob. 0.0127 0.0000 Interpretation: We estimate that a 1 percent increase in the price _________________ the quantity demand by _______ percent. That is, the estimate for the price elasticity of demand equals ________. 6 Critical Result: The coefficient estimate equals ______. The coefficient estimate does not equal _____; the estimate lies __________ from _____. Theory ______ Evidence ______ Price Elasticity ______ 0 The critical result is that the estimate lies ________ from where the theory claims it should be. This evidence suggests that the budget theory of demand is _______________. Step 2: Play the cynic, challenge the evidence, and construct the null and alternative hypotheses. The cynic always challenges the evidence: Cynic’s view: Sure the coefficient estimate from regression suggests that the price elasticity of demand does not equal −1.0, but this is just “the luck of the draw.” In fact, the actual price elasticity of demand equals −1.0. We shall now construct the null and alternative hypotheses to address this question: H0: βP = −1.0 Cynic’s view is correct: Actual price elasticity of demand equals −1.0 H1: βP ≠ −1.0 Cynic’s view is incorrect: Actual price elasticity of demand does not equal −1.0 Following the cynic’s lead, the null hypothesis always challenges the evidence. On the other hand, the alternative hypothesis is consistent with the evidence. Step 3: Formulate the question to assess the cynic’s view. Question for the Cynic: • Generic Question: What is the probability that the results would be like those we actually obtained (or even stronger), if the cynic is correct and the actual price elasticity of demand equals −1.0? • Specific Question: The regression’s coefficient estimate was −.586: What is the probability that the coefficient estimate, bP, in one regression would be at least .414 from −1.0, if H0 were actually true (if the actual coefficient, βP, equals −1.0)? Answer: Prob[Results IF Cynic Correct] or Prob[Results IF H0 True] 7 Step 4: Use the general properties of the estimation procedure, the probability distribution of the estimates, to calculate Prob[Results IF H0 True]. • • • • Since ordinary least squares estimation procedure for the coefficient value is unbiased, the mean of the probability distribution for the estimate equals _______________________. If the null hypothesis were true, the actual price coefficient would equal _____. The standard error equals _______. The degrees of freedom equal __________________. OLS estimation procedure unbiased Student t-distribution Mean = ______ SE = ______ DF = ______ ______ bCbP ______ Assume H0 is true Mean[bP] = ___ = ___ ______ Standard error ______ ______ Number of Observations ______ Number of Parameters DF = ___ − ___ = ___ SE[bP] = ____ Question: Can we use can use the “tails probability” as reported in the regression printout to compute this probability? Answer: ________ Tails Probability: The probability that the coefficient estimate, bP, resulting from one regression would lie at least _________ from ____, if the actual coefficient, βP, equals ____. NB: The tails probability is calculated on the premise that the actual value of the coefficient equals _____. Econometrics Lab: Prob[Results IF H0 True] ≈ Left Tail ↓ _____ Right Tail ↓ + _____ = _____ 8 Hypothesis Testing: Using Regression Printouts with Clever Algebraic Manipulations Clever Definition: βClever = βP + 1.0 βP = −1.0 if and only if βClever = 0 Now, recall our equation for the constant price elasticity model: where LogQ = log(GasCons) LogQ = c + βPLogP LogP = log(Price) Next, a little algebra: βClever = βP + 1.0, βP = βClever − 1.0. Substituting βClever − 1.0 for βP LogQ = c + (βClever − 1.0) LogP LogQ LogQ + LogP = c + βCleverLogP − LogP = c + βCleverLogP LogQPlusLogP = c + βCleverLogP where LogQPlusLogP = LogQ + LogP We can now express the hypotheses in terms of βC. Recall that βP = −1.0 if and only if βClever = 0: H0: βP = −1.0 ⇒ H0: βClever = 0 Actual price elasticity of demand equals −1.0 H1: βP ≠ −1.0 ⇒ H1: βClever ≠ 0 Actual price elasticity of demand does not equal −1.0 Dependent Variable: LOGQPLUSLOGP Included observations: 10 Coefficient LOGP 0.414377 C 5.918487 Std. Error 0.183409 0.045315 t-Statistic 2.259308 130.6065 Prob. 0.0538 0.0000 Critical Result: The coefficient estimate equals ________. The coefficient estimate ___________________ equal _____; the estimate is _______ from _____. Question for the Cynic: • Specific Question: The regression’s coefficient estimate was .414: What is the probability that the coefficient estimate, bClever, in one regression would be at least .414 from 0, if H0 were actually true (if the actual coefficient, βClever, equals 0)? Answer: Prob[Results IF H0 True] 9 Next, calculate Prob[Results IF H0 True] focusing on βClever: • Since ordinary least squares estimation procedure for the coefficient value is unbiased, the mean of the probability distribution for the estimate ______ equals _______________________. • If the null hypothesis were true, the actual price coefficient would equal _________. • The standard error equals ______. ______ • The degrees of freedom equal __________________. OLS estimation procedure unbiased Assume H0 is true Mean[bClever] = ___ = ___ Standard error Student t-distribution Mean = ______ SE = ______ DF = ______ ______ bCbClever ______ ______ Number of Observations SE[bClever] = ____ ______ Number of Parameters DF = ___ − ___ = ___ Tails Probability: The probability that the coefficient estimate, bP, resulting from one regression would lie at least _________ from ____, if the actual coefficient, βClever, equals ____. Answer: Prob[Results IF H0 True] = ________ Question: Is this the same answer as we calculated with the Econometrics Lab? _____ Step 5: Decide upon the standard of proof, what constitutes proof beyond a reasonable doubt. Decide on the significance level, the dividing line between small and large probability: Prob[Results IF H0 True] Less Than Significance Level ↓ Prob[Results IF H0 True] small Prob[Results IF H0 True] Greater Than Significance Level ↓ Prob[Results IF H0 True] large ↓ Unlikely that H0 is true ↓ Likely that H0 is true ↓ Reject H0 ↓ Do not reject H0 Do we reject the null hypothesis at a 10 percent (.10) significance level? ______ Do we reject the null hypothesis at a 5 percent (.05) significance level? ______ Do we reject the null hypothesis at a 1 percent (.01) significance level? ______ What is your assessment of the budget theory of demand? 10 Summary: One Tailed Versus Two Tailed Tests – Which Is Appropriate? Theory: Coefficient is less than or greater than a specific value (often 0) ↓ One tailed test appropriate Theory: Coefficient equals a specific value ↓ Two tailed test appropriate Theory: β > c or β < c Theory: β = c Probability Distribution Probability Distribution H0: β = c H1: β ≠ c H0: β = c H1: β > c b c Probability Distribution b c Prob[Results IF H0 True] = Probability of obtaining results like those we actually got (or even stronger), if H0 is true H0: β = c H1: β < c Prob[Results IF H0 True] b c Logarithms: A Useful Econometric Tool Small Large Reject H0 Do not reject H0 1 Logarithms provide a very convenient way to fine tune our theories by expressing them in terms of percentages rather than “natural” units. Linear Model: yt = βConst + βxxt + et Coefficient estimate: Estimates the (natural) unit change in y resulting from a one (natural) unit change in x Log Dependent Variable Model: log(yt) = βConst + βxxt + et Coefficient estimate multiplied by 100: Estimates the percent change in y resulting from a one (natural) unit change in x Log Explanatory Variable Model: yt = βConst + βxlog(xt) + et Coefficient estimate divided by 100: Estimates the (natural) unit change in y resulting from a one percent change in x Log-Log (Constant Elasticity) Model: log(yt) = βConst + βxlog(xt) + et Coefficient estimate: Estimates the percent change in y resulting from a one percent change in x 1 The log notation refers to the natural logarithm (logarithm base e), not the logarithm base 10. 11 Using Logarithms – An Illustration: Wages and High School Education Basic Theory: Additional years of high school education increase the wage. Wage and Education Data: Cross section data of wages and education for 212 workers included in the March 2007 Current Population Survey residing in the Northeast region of the United States who have completed the ninth, tenth, eleventh, or twelfth grades, but have not continued on to college or junior college. Waget HSEduct Wage rate earned by worker t (dollars per hour) Highest high school grade completed by worker t (9, 10, 11, or 12 years) Linear model: Waget = βConst + βEHSEduct + et This model includes no logarithms. Wage is expressed in dollars (natural units) and education in years (natural units). Dependent Variable: Wage Explanatory Variable: HSEduc Dependent Variable: WAGE Included observations: 212 HSEDUC C Coefficient 1.645899 -3.828617 Std. Error 0.555890 6.511902 t-Statistic 2.960834 -0.587941 Prob. 0.0034 0.5572 Estimated Equation: Wage = −3.83 + 1.65HSEduc. Coefficient Interpretation: One ___________________ increase in HSEduc ↓ Increases Wage by ________ ___________________ Log dependent variable model: LogWaget = β Const + βEHSEduct + et The dependent variable (LogWage) is expressed in terms of the logarithm of dollars; the explanatory variable (HSEduc) is expressed in years (natural units). Dependent Variable: LogWage Explanatory Variable: HSEduc Dependent Variable: LOGWAGE Included observations: 212 Coefficient HSEDUC 0.113824 C 1.329791 Std. Error 0.033231 0.389280 t-Statistic 3.425227 3.416030 Prob. 0.0007 0.0008 Estimated Equation: LogWage = 1.33 + .114HSEduc. Coefficient Interpretation: One ___________________ increase in HSEduc ↓ Increases Wage by ________ ___________________ 12 Log explanatory variable model: Waget = βConst + βELogHSEduct + et The dependent variable (Wage) is expressed in terms of dollars (natural units); the explanatory variable (LogHSEduc) is expressed in terms of the log of years. Dependent Variable: Wage Explanatory Variable: LogHSEduc Dependent Variable: WAGE Included observations: 212 LOGHSEDUC C Coefficient 17.30943 -27.10445 Std. Error 5.923282 14.55474 t-Statistic 2.922270 -1.862242 Prob. 0.0039 0.0640 Estimated Equation: Wage = −27.1 + 17.31LogHSEduc. Coefficient Interpretation: One ___________________ increase in HSEduc ↓ Increases Wage by ________ ___________________ Log-log (constant elasticity) model: LogWaget = β Const + β ELogHSEduct + et Both the dependent and explanatory variables are expressed in terms of logs. This is just the constant elasticity model that we discussed earlier. Dependent Variable: LogWage Explanatory Variable: LogHSEduc Dependent Variable: LOGWAGE Included observations: 212 Coefficient LOGHSEDUC 1.195654 C -0.276444 Std. Error 0.354177 0.870286 t-Statistic 3.375868 -0.317647 Prob. 0.0009 0.7511 Estimated Equation: LogWage = −.28 + 1.20LogHSEduc. Coefficient Interpretation: One ___________________ increase in HSEduc ↓ Increases Wage by ________ ___________________