# Lecture 9 Preview: One-Tailed Tests, Two-Tailed Tests, and Logarithms A One-Tailed Hypothesis Test: The Downward Sloping Demand Curve A Two-Tailed Hypothesis.

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Lecture 9 Preview: One-Tailed Tests, Two-Tailed Tests, and Logarithms A One-Tailed Hypothesis Test: The Downward Sloping Demand Curve A Two-Tailed Hypothesis Test: The Budget Theory of Demand Logarithms: A Useful Econometric Tool to Fine Tuning Hypotheses One-Tailed versus Two-Tailed Tests Summary: One-Tailed and Two-Tailed Tests Linear Model Log Dependent Variable Model Log Explanatory Variable Model Log-Log (Constant Elasticity) Model Hypothesis Testing Using Clever Algebraic Manipulations

Dependent Variable: GasCons Explanatory Variable(s):EstimateSEt-StatisticProb PriceDollars  151.6556 47.57295-3.1878530.0128 Const 516.780160.602238.5274100.0000 Number of Observations10 One Tailed Hypothesis Test: Downward Sloping Market Demand Curve Step 1: Collect data, run the regression, and interpret the estimates Theory: A higher price decreases the quantity demanded; demand curve is downward sloping. GasCon t =  Const +  P PriceDollars t + e t  P reflects the change in quantity demanded resulting from a \$1 increase in the price GasCons t = Quantity of Gasoline Demanded PriceDollars t = Price of Gasoline Step 0: Construct a model reflecting the theory to be tested The theory suggests that  P should be negative. Theory:  P < 0. b P = Estimated coefficient  P =  151.7 Estimated Equation: Interpretation: We estimate that a \$1 increase in the real price of gasoline decreases the quantity of gasoline demanded by 151.7 million gallons. Critical Result: The coefficient estimate equals  151.7. The negative sign of the coefficient estimate suggests that a higher price reduces the quantity demanded.  EViews EstGasCons = 516.8  151.7PriceDollars P D Q Gasoline Prices and Consumption in the 1990’s GasCons t U. S. gasoline consumption (millions of gallons per day) PriceDollars t Price of gasoline (2000 dollars per gallon)Gasoline Real PriceConsumptionReal PriceConsumption Year(\$ per gallon)(Millions of gals)Year(\$ per gallon)(Millions of gals) 19901.43303.919951.25327.1 19911.35301.919961.31331.4 19921.31305.319971.29336.7 19931.25314.019981.10346.7 19941.23319.219991.19354.1 This evidence supports the downward sloping demand theory. A higher price decreases the quantity demanded; the demand curve is downward sloping.

Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses: H 0 :  P = 0 Cynic is correct: Price has no impact on the quantity demanded H 1 :  P < 0 Cynic is incorrect: A higher price decreases the quantity demanded Step 3: Formulate the question to assess the cynic’s view. 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 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,  P, equals 0. Answer: Prob[Results IF Cynic Correct] or equivalently Prob[Results IF H 0 True] 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, b P, in one regression would be  151.7 or less, if H 0 were true (if the actual coefficient,  P, equaled 0)?  Do not reject H 0  Reject H 0

Dependent Variable: GasCons Explanatory Variable(s):EstimateSEt-StatisticProb PriceDollars  151.6556 47.57295-3.1878530.0128 Const 516.780160.602238.5274100.0000 Number of Observations10 Step 4: Use the estimation procedure’s general properties to calculate Prob[Results IF H 0 True]. H 0 :  P = 0 Cynic is correct: Price has no impact on the quantity demanded H 1 :  P < 0 Cynic is incorrect: A higher price decreases the quantity demanded bPbP t-distribution OLS estimation procedure unbiased Mean[b P ] =  P SE[b P ] If H 0 were true Number of observations Number of parameters Standard Error DF= 10  2 = 8= 0= 47.6 Prob[Results IF H 0 True] = Tails Probability: Probability that a coefficient estimate, b P, resulting from one regression would will lie at least 151.7 from 0, if the actual coefficient,  P, equaled 0. Estimate was  151.7 : What is the probability that the coefficient estimate in one regression would be  151.7 or less, if H 0 were true (if the actual coefficient,  P, equaled 0)?.0128/2 Mean = 0 SE = 47.6 DF = 8  151.7 0 Tails Probability =.0128 =.0064.0128 2 151.7

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 H 0 True] small  Unlikely that H 0 is true  Prob[Results IF H 0 True] large  Likely that H 0 is true Prob[Results IF H 0 True] Less Than Significance Level Prob[Results IF H 0 True] Greater Than Significance Level H 0 :  P = 0 Cynic is correct: Price has no impact on the quantity demanded H 1 :  P < 0 Cynic is incorrect: A higher price decreases the quantity demanded At the “traditional” significance levels we could reject the null hypothesis; that is, we could reject that notion that an increase in the price has no effect on the quantity demanded.  Do not reject H 0  Reject H 0 Prob[Results IF H 0 True] =.0064 Would we reject H 0 at a 10 percent (.10) significance level?Yes. Would we reject H 0 at a 1 percent (.01) significance level?Yes. Would we reject H 0 at a 5 percent (.05) significance level?Yes. Does this lend support to the downward sloping demand curve theory? Yes.

bxbx t-distribution Mean = 0 SE =.5196 DF = 1 1.20 Review: One Tailed Tests Thus far, we have considered only one tailed tests because the two theories we considered were only concerned with the sign of the coefficient. We have focused on only one side, one tail, of the probability distribution because the theory postulated that the actual coefficient was either positive or negative. Quiz Score Theory: These one tailed tests are appropriate for most economic theories because most economic theories postulate that the explanatory variable has a positive influence or a negative influence on the dependent variable, suggesting that the coefficient is positive or negative. bPbP t-distribution.0064 Mean = 0 SE = 47.6 DF = 8  151.7 0 Demand Curve Theory: As we shall see, however, some economic theories suggest that the coefficient of the explanatory variable equals a specific value (rather than be positive or negative). In this cases, we are concerned with both sides (or both tails) of the probability distribution..13 To assess the quiz score theory we estimated the probability that the coefficient estimate, b x, in one regression would be 1.2 or more, if H 0 were true (if the actual coefficient,  x, equaled 0). To assess the demand curve theory we estimated the probability that the coefficient estimate, b P, in one regression would be  151.7 or less, if H 0 were true (if the actual coefficient,  P, equaled 0).  x > 0 b x = 1.2  P < 0b P =  151.7

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: Question: What economic concept is relevant here?Claim: The Price Elasticity of Demand. Verbal Definition of the Price Elasticity: The percent change in the quantity resulting from a one percent change in price. Converting the Verbal Definition into a Mathematical Definition. Price Elasticity = Percent Change in the Quantity resulting from a 1 Percent Change in Price. Calculating percent changes: If X increases from 200 to 220, it increases by 10 percent. Percent change in X In general, percent change in X P  Q = Constant

Constant price elasticity model: Claim:  P equals the price elasticity of demand. log(Q) = log(  Const ) +  P log(P) A linear expression for the constant price elasticity model: Consider the P’s in the numerator: Now, simplify. Price elasticity of demand Step 0: Construct a model reflecting the theory to be tested =  P NB: Whenever the dependent variable and the explanatory variable are logarithms, the coefficient of the explanatory variable,  P, is the elasticity. = Budget Theory of Demand: P  Q = Constant Q = Constant  P  1  Const = Constant  P =  1 Q =  Const P PP Budget Theory of Demand:  P =  1.0 LogQ = c +  P LogP Q =  Const P PP Answer: The budget theory of demand postulates that the price elasticity of demand equals  1.0. Question: What does the budget theory of demand postulate about the price elasticity of demand,  P.

Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogP  0.585623 0.183409-3.1929880.0127 Const 5.9184870.045315130.60650.0000 Number of Observations10 Budget Theory of Demand:  P =  1.0. Step 1: Collect data, run the regression, and interpret the results Interpretation: We estimate that a 1 percent increase in the price decreases the quantity demand by.586 percent. That is, the estimate for the price elasticity of demand equals .586. Critical Result: The coefficient estimate equals .586.  EViews Gasoline Prices and Consumption in the 1990’s GasCons U. S. gasoline consumption (millions of gallons per day) PriceDollars Price of gasoline (2000 dollars per gallon) Generate new variablesLogQ = log(GasCons)LogP = log(PriceDollars) The coefficient estimate does not equal  1.0. Critical Result: The estimate lies.414 from where the theory claims it should be. The evidence suggests that the budget theory of demand is incorrect.

Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses: H 0 :  P =  1.0 Cynic’s view is correct: Actual price elasticity of demand equals  1.0 H 1 :  P ≠  1.0 Cynic’s view is incorrect: Actual price elasticity of demand does not equal  1.0 Step 3: Formulate the question to assess the cynic’s view. 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 Cynic’s view: Sure the coefficient estimate, .586, suggests that the price elasticity of demand does not equal  1.0, but this is just the “luck of the draw.” The actual elasticity of demand equals  1.0. Answer: Prob[Results IF Cynic Correct] or equivalently Prob[Results IF H 0 True] 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, b P, in one regression would be at least.414 from  1.0, if H 0 were actually true (if the actual coefficient,  P, equals  1.0)?  Do not reject H 0  Reject H 0

Dependent Variable: LogQ Explanatory Variable(s):EstimateSEt-StatisticProb LogP  0.585623 0.183409-3.1929880.0127 Const 5.9184870.045315130.60650.0000 Number of Observations10 Step 4: Use the estimation procedure’s general properties to calculate Prob[Results IF H 0 True]. H 0 :  P =  1.0 Cynic’s view is correct: Actual price elasticity of demand equals  1.0 H 1 :  P ≠  1.0 Cynic’s view is incorrect: Actual price elasticity of demand does not equal  1.0 bPbP t-distribution OLS estimation procedure unbiased Mean[b P ] =  P SE[b P ] If H 0 were true Number of observations Number of parameters Standard error DF= 10  2 = 8 =  1.0 =.183 Prob[Results IF H 0 True] Tails Probability: Probability that a coefficient estimate, b P, resulting from one regression would will lie at least.586 from 0, if the actual coefficient,  P, equals 0. Estimate was .586: What is the probability that the coefficient estimate in one regression would be at least.414 from  1.0, if H 0 were true (if the actual coefficient,  P, equals  1.0)?.027 Mean =  1.00 SE =.183 DF = 8 .586  1.0 But this does not help, since H 0 :  P =  1.0, not 0..027 .027 +.027 =.054.414 Econometrics Lab:  Lab 9.1a  Lab 9.1b  1.414 The tails probability is based on the premise that the coefficient’s actual value equals 0.

Dependent Variable: LogQPlusLogP Explanatory Variable(s):EstimateSEt-StatisticProb LogP 0.4143770.1834092.2593080.0538 Const 5.9184870.045315130.60650.0000 Number of Observations10 An Aside. Hypothesis Testing Using Regression Printouts: Clever Algebraic Manipulations Question: Can we exploit tails probability to calculate the Prob[Results IF H 0 True]?  Clever =  P + 1.0  Clever = 0 if and only if  P =  1.0 LogQ = c +  P LogPSince  Clever =  P + 1.0,  P = LogQ = c + (  Clever  1.0)LogP LogQ = c +  Clever LogP  LogP LogQ + LogP = c +  Clever LogP LogQPlusLogP = c +  Clever LogP where LogQPlusLogP = LogQ + LogP We can now express the null and alternative hypotheses in terms of  Clever : H 0 :  P =  1.0 H 1 :  P   1.0 Actual price elasticity of demand equals  1.0 Actual price elasticity of demand does not equal  1.0  H 0 :  Clever = 0  H 1 :  Clever  0 We now generate the new variable, LogQPlusLogP. Question: Why do we define  Clever this way? Critical Result: The coefficient estimate equals.414. The coefficient estimate does not equal 0; the estimate is.414 from 0.  EViews Answer: Yes, cleverly define a new coefficient so that H 0 can be expressed as the coefficient equaling 0:  Clever  1.0

Dependent Variable: LogQPlusLogP Explanatory Variable(s):EstimateSEt-StatisticProb LogP 0.4143770.1834092.2593080.0538 Const 5.9184870.045315130.60650.0000 Number of Observations10 b Clever t-distribution OLS estimation procedure unbiased Mean[b Clever ] =  Clever SE[b Clever ] If H 0 were true Number of observations Number of parameters Standard Error DF= 10  2 = 8= 0=.183 Prob[Results IF H 0 True] Tails Probability: Probability that a coefficient estimate, b Clever, resulting from one regression would will lie at least.414 from 0, if the actual coefficient,  Clever, equals 0. Specific Question: The regression’s coefficient estimate was.414: What is the probability that the coefficient estimate, b Clever, in one regression would be at least.414 from 0, if H 0 were actually true (if the actual coefficient,  Clever, equals 0)?.0538/2 Mean = 0 SE =.183 DF = 8.4140 Tails Probability =.0538.0538/2 .054.414 H 0 :  P =  1.0 or  Clever = 0 Actual price elasticity of demand equals  1.0 H 1 :  P ≠  1.0 or  Clever ≠ 0 Actual price elasticity of demand does not equal  1.0 NB: Same answer as before. Answer: Prob[Results IF H 0 True].

H 0 :  P =  1.0 or  Clever = 0 Actual price elasticity of demand equals  1.0 H 1 :  P ≠  1.0 or  Clever ≠ 0 Actual price elasticity of demand does not equal  1.0 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 H 0 True] small  Unlikely that H 0 is true  Prob[Results IF H 0 True] large  Likely that H 0 is true Prob[Results IF H 0 True] Less Than Significance Level Prob[Results IF H 0 True] Greater Than Significance Level Prob[Results IF H 0 True] .054  Do not reject H 0  Reject H 0 Prob[Results IF H 0 True] .054 Would we reject H 0 at a 10 percent (.10) significance level?Yes. Would we reject H 0 at a 1 percent (.01) significance level?No. Would we reject H 0 at a 5 percent (.05) significance level?No.

Summary: One Tail Versus Two Tail 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 Prob[Results IF H 0 True] Equals the probability of obtaining results like those we actually got (or even stronger), if H 0 were true. Small  Reject H 0 Large  Do not reject H 0 Prob[Results IF H 0 True]

Linear Model: y t = b Const + b x x t + e t Coefficient estimate, b x : Estimates the (natural) unit change in y resulting from a one (natural) unit change in x Log Dependent Variable Model: log(y t ) = b Const + b x x t + e t Coefficient estimate multiplied by 100: Estimates the percent change in y resulting from a one (natural) unit change in x Coefficient estimate divided by 100: Estimates the (natural) unit change in y resulting from a one percent change in x Coefficient estimate: Estimates the percent change in y resulting from a one percent change in x Log Explanatory Variable Model: y t = b Const + b x log(x t ) + e t Log-Log (Constant Elasticity) Model: log(y t ) = b Const + b x log(x t ) + e t Logarithms: A Useful Econometric Tool Logarithms provide a very convenient way to fine tune our theories by expressing them in terms of percentages rather than “natural” units. Preview

Dependent Variable: Wage Explanatory Variable(s):EstimateSEt-StatisticProb HSEduc 1.6458990.5558902.9608340.0034 Const  3.828617 6.511902-0.5879410.5572 Number of Observations212 Dependent Variable: LogWage Explanatory Variable(s):EstimateSEt-StatisticProb HSEduc 0.1138240.0332313.4252270.0007 Const 1.3297910.3892803.4160300.0008 Number of Observations212 Illustration: Wages and Education of Non-College Educated Workers Basic Theory: Additional years of education increases wage rate. Interpretation of Coefficient Estimate Log Dependent Variable Model: Linear Model: Wage t =  Const +  E HSEduc t + e t LogWage t =  Const +  E HSEduc t + e t  EViews 1 year (natural unit) increase in High School Education  Increases Wage by about \$1.65 (natural units) Interpretation of Coefficient Estimate 1 year (natural unit) increase in High School Education  Increases Wage by about 11.4 percent Wage t =Wage rate (Dollars per hour). HSEduc t =Highest high school grade completed (9, 10, 11, or 12) Data:

Dependent Variable: Wage Explanatory Variable(s):EstimateSEt-StatisticProb LogHSEduc 17.309435.9232822.9222700.0039 Const  27.10445 14.55474-1.8622420.0640 Number of Observations212 Dependent Variable: LogWage Explanatory Variable(s):EstimateSEt-StatisticProb LogHSEduc 1.1956540.3541773.3758680.0009 Const  0.276444 0.870286-0.3176470.7511 Number of Observations212 Log Explanatory Variable Model: Log-Log (Elasticity) Model: Wage t =  Const +  E LogHSEduc t + e t LogWage t =  Const +  E LogHSEduc t + e t Interpretation of Coefficient Estimate 1 percent increase in High School Education  Increases Wage by about \$.17 (natural units) Interpretation of Coefficient Estimate 1 percent increase in High School Education  Increases Salary by about 1.19 percent  EViews

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