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Lecture 9 Preview: One-Tailed Tests, Two-Tailed Tests, and Logarithms

Lecture 9 Preview: One-Tailed Tests, Two-Tailed Tests, and Logarithms. 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. Hypothesis Testing Using Clever Algebraic Manipulations.

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Lecture 9 Preview: One-Tailed Tests, Two-Tailed Tests, and Logarithms

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  1. Lecture 9 Preview: One-Tailed Tests, Two-Tailed Tests, and Logarithms 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 Hypothesis Testing Using Clever Algebraic Manipulations Summary: One-Tailed and Two-Tailed Tests Logarithms: A Useful Econometric Tool to Fine Tuning Hypotheses Linear Model Log Dependent Variable Model Log Explanatory Variable Model Log-Log (Constant Elasticity) Model

  2. One Tailed Hypothesis Test: Downward Sloping Market 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= Quantity of Gasoline Demanded GasCont = Const + PPriceDollarst + et PriceDollarst = Price of Gasoline P reflects the change in quantity demanded resulting from a $1 increase in the price The theory suggests that P should be negative. Theory: P < 0. A higher price decreases the quantity demanded; the demand curve is downward sloping. Step 1: Collect data, run the regression, and interpret the estimates EViews P Gasoline Prices and Consumption in the 1990’s GasConst U. S. gasoline consumption (millions of gallons per day) PriceDollarst Price of gasoline (2000 dollars per gallon) Gasoline Gasoline Real Price Consumption Real Price Consumption Year ($ per gallon) (Millions of gals) Year ($ per gallon) (Millions of gals) 1990 1.43 303.9 1995 1.25 327.1 1991 1.35 301.9 1996 1.31 331.4 1992 1.31 305.3 1997 1.29 336.7 1993 1.25 314.0 1998 1.10 346.7 1994 1.23 319.2 1999 1.19 354.1 Estimated Equation: EstGasCons = 516.8  151.7PriceDollars bP = Estimated coefficient P = 151.7 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. D This evidence supports the downward sloping demand theory. Q

  3. Step 2: Play the cynic and challenge the results; 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, P, equals 0. H0: P = 0 Cynic is correct: Price has no impact on the quantity demanded H1: P < 0 Cynic is incorrect: A higher price decreases the quantity demanded Step 3: Formulate the question to 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 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, bP, in one regression would be 151.7 or less, if H0 were true (if the actual coefficient, P, equaled 0)? Answer: Prob[Results IF Cynic Correct] or equivalently Prob[Results IF H0 True] Prob[Results IF H0 True] small Prob[Results IF H0 True] large Unlikely that H0 is true Likely that H0 is true Reject H0  Do not reject H0

  4. H0: P = 0 Cynic is correct: Price has no impact on the quantity demanded H1: P < 0 Cynic is incorrect: A higher price decreases the quantity demanded Step 4: Use the estimation procedure’s general properties to calculate Prob[Results IF H0 True]. Estimate was 151.7 : What is the probability that the coefficient estimate in one regression would be 151.7 or less, if H0 were true (if the actual coefficient, P, equaled 0)? OLS estimation procedure unbiased If H0 were true StandardError Number of observations Number of parameters Mean[bP] = P = 0 SE[bP] = 47.6 DF = 10  2 = 8 Tails Probability: Probability that a coefficient estimate, bP, resulting from one regression would will lie at least 151.7 from 0, if the actual coefficient, P, equaled 0. t-distribution Mean = 0 SE = 47.6 .0128/2 DF = 8 Tails Probability = .0128 bP .0128 Prob[Results IF H0 True] = = .0064 2 151.7 151.7 151.7 0

  5. H0: P = 0 Cynic is correct: Price has no impact on the quantity demanded H1: P < 0 Cynic is incorrect: A higher price decreases the quantity demanded Prob[Results IF H0 True] = .0064 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]Less Than Significance Level Prob[Results IF H0 True]Greater Than Significance Level Prob[Results IF H0 True] small Prob[Results IF H0 True] large Unlikely that H0 is true Likely that H0 is true Reject H0  Do not reject H0 Would we reject H0 at a 10 percent (.10) significance level? Yes. Would we reject H0 at a 5 percent (.05) significance level? Yes. Would we reject H0 at a 1 percent (.01) significance level? Yes. 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. Does this lend support to the downward sloping demand curve theory? Yes.

  6. Review: One Tailed Tests t-distribution Thus far, we have considered only one tailed tests because the two theories we considered were only concerned with the sign of the coefficient. Mean = 0 SE = .5196 DF = 1 Quiz Score Theory: x > 0 bx = 1.2 .13 To assess the quiz score theory we estimatedthe probability that the coefficient estimate, bx, in one regression would be 1.2 or more, if H0 were true (if the actual coefficient, x, equaled 0). bx 0 1.2 Demand Curve Theory: P < 0 bP = 151.7 t-distribution To assess the demand curve theory we estimatedthe probability that the coefficient estimate, bP, in one regression would be 151.7 or less, if H0 were true (if the actual coefficient, P, equaled 0). Mean = 0 SE = 47.6 .0064 DF = 8 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. bP 151.7 0 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. 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.

  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: PQ = 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

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

  9. Budget Theory of Demand: P = 1.0. EViews Step 1: Collect data, run the regression, and interpret the results 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 variables LogQ = log(GasCons) LogP = log(PriceDollars) 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 . 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.

  10. Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses: 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. H0: P = 1.0 Cynic’s view is correct: Actual price elasticity of demand equals 1.0 H1: 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. 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, P, equals 1.0)? Answer: Prob[Results IF Cynic Correct] or equivalently Prob[Results IF H0 True] Prob[Results IF H0 True] small Prob[Results IF H0 True] large Unlikely that H0 is true Likely that H0 is true Reject H0  Do not reject H0

  11. H0: P = 1.0 Cynic’s view is correct: Actual price elasticity of demand equals 1.0 H1: P ≠ 1.0 Cynic’s view is incorrect: Actual price elasticity of demand does not equal 1.0 Step 4: Use the estimation procedure’s general properties to calculate Prob[Results IF H0 True]. Estimate was .586: What is the probability that the coefficient estimate in one regression would be at least .414 from 1.0, if H0 were true (if the actual coefficient, P, equals 1.0)? OLS estimation procedure unbiased If H0 were true Standarderror Number of observations Number of parameters Mean[bP] = P = 1.0 SE[bP] = .183 DF = 10  2 = 8 Question: Does the tails probability help use here? No. Tails Probability: Probability that a coefficient estimate, bP, resulting from one regression would will lie at least .586 from 0, if the actual coefficient, P, equals 0. t-distribution Mean = 1.00 SE = .183 DF = 8 .027 .027 The tails probability is based on the premise that the coefficient’s actual value equals 0. So, it does not help, since H0: P = 1.0, not 0. bP Lab 9.1a .414 .414 Lab 9.1b Econometrics Lab: 1.414 1.0 .586 Prob[Results IF H0 True] .027 + .027 = .054

  12. An Aside. Hypothesis Testing Using Regression Printouts: Clever Algebraic Manipulations Question: Can we exploit tails probability to calculate the Prob[Results IF H0 True]? Answer: Yes, cleverly define a new coefficient so that H0 can be expressed as the coefficient equaling 0: Clever = P + 1.0 Question: Why do we define Clever this way? Clever = 0 if and only if P = 1.0 LogQ = c + PLogP Since Clever = P + 1.0, P= Clever1.0 LogQ = c + (Clever1.0)LogP LogQ = c + CleverLogPLogP LogQ + LogP = c + CleverLogP LogQPlusLogP = c + CleverLogP where LogQPlusLogP = LogQ + LogP EViews We can now express the null and alternative hypotheses in terms of Clever: H0: P = 1.0  H0: Clever = 0 Actual price elasticity of demand equals 1.0 H1: P1.0  H1: Clever 0 Actual price elasticity of demand does not equal 1.0 We now generate the new variable, LogQPlusLogP. Critical Result:The coefficient estimate equals .414. The coefficient estimate does not equal 0; the estimate is .414 from 0.

  13. H0: P = 1.0 or Clever = 0 Actual price elasticity of demand equals 1.0 H1: P ≠ 1.0 or Clever ≠ 0 Actual price elasticity of demand does not equal 1.0 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, Clever, equals 0)? Answer: Prob[Results IF H0 True]. OLS estimation procedure unbiased If H0 were true StandardError Number of observations Number of parameters Mean[bClever] = Clever = 0 SE[bClever] = .183 DF = 10  2 = 8 Tails Probability: Probability that a coefficient estimate, bClever , resulting from one regression would will lie at least .414 from 0, if the actual coefficient, Clever, equals 0. t-distribution Mean = 0 SE = .183 DF = 8 .0538/2 .0538/2 Tails Probability = .0538 Prob[Results IF H0 True]  .054 bClever .414 .414 NB: Same answer as before. 0 .414

  14. H0: P = 1.0 or Clever = 0 Actual price elasticity of demand equals 1.0 H1: P ≠ 1.0 or Clever ≠ 0 Actual price elasticity of demand does not equal 1.0 Prob[Results IF H0 True]  .054 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]Less Than Significance Level Prob[Results IF H0 True]Greater Than Significance Level Prob[Results IF H0 True] small Prob[Results IF H0 True] large Unlikely that H0 is true Likely that H0 is true Reject H0  Do not reject H0 Prob[Results IF H0 True]  .054 Would we reject H0 at a 10 percent (.10) significance level? Yes. Would we reject H0 at a 5 percent (.05) significance level? No. Would we reject H0 at a 1 percent (.01) significance level? No.

  15. Summary: One Tail Versus Two Tail Tests – Which Is Appropriate? Theory: Coefficient is less than or greater than a specific value (often 0). Theory: Coefficient equals a specific value One tailed test appropriate Two tailed test appropriate Prob[Results IF H0 True] Equals the probability of obtaining results like those we actually got (or even stronger), if H0 were true. Prob[Results IF H0 True] Small Large Reject H0 Do not reject H0

  16. 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 Linear Model:yt = bConst + bxxt + et Coefficient estimate, bx: Estimates the (natural) unit change in y resulting from a one (natural) unit change in x Log Dependent Variable Model: log(yt) = bConst + bxxt + 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 = bConst + bx log(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) = bConst + bx log(xt) + et Coefficient estimate: Estimates the percent change in y resulting from a one percent change in x

  17. Illustration: Wages and Education of Non-College Educated Workers Basic Theory:Additional years of education increases wage rate. EViews Data: Waget = Wage rate (Dollars per hour). HSEduct = Highest high school grade completed (9, 10, 11, or 12) Linear Model: Waget = Const + EHSEduct + et 1 year (natural unit) increase in High School Education Interpretation of Coefficient Estimate Increases Wage by about $1.65 (natural units) Log Dependent Variable Model: LogWaget = Const + EHSEduct + et 1 year (natural unit) increase in High School Education Interpretation of Coefficient Estimate Increases Wage by about 11.4 percent

  18. Log Explanatory Variable Model: Waget = Const + ELogHSEduct + et EViews 1 percent increase in High School Education Interpretation of Coefficient Estimate Increases Wage by about $.17 (natural units) Log-Log (Elasticity) Model: LogWaget = Const + ELogHSEduct + et 1 percent increase in High School Education Interpretation of Coefficient Estimate Increases Wage by about 1.19 percent

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