Wednesday, October 10 Handout: One-Tailed Tests, Two-Tailed Tests, and Logarithms
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1 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: GasCons t + PriceDollars t + e t where GasCons = Quantity of Gasoline Demanded PriceDollars = Price (Chained 1990 Dollars) reflects the change in the quantity demanded resulting from a in the price. The theory suggests that 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: GasCons t U. S. gasoline consumption in year t (millions of gallons per day) PriceDollars t Real price of gasoline in year t (dollars per gallon chained 2000 dollars) Gasoline Gasoline Real Price Consumption Real Price Consumption Year ($ per gallon) (Millions of gals) Year ($ per gallon) (Millions of gals)
2 2 Dependent Variable: GASCONS Included observations: 10 PRICEDOLLARS C 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,, equals 0. Now, we construct the null and alternative hypotheses: H 0 : = 0 Cynic s view is correct: Price has no effect on quantity demanded. H 1 : < 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 What is the probability that the coefficient estimate in one regression would be or less, if H 0 were actually true (if the actual coefficient,, equals 0)? Answer: Prob[Results IF Cynic Correct] or The magnitude of this probability determines whether we accept or reject the null hypothesis: small that H 0 is true large that H 0 is true H 0 H 0
3 3 Step 4: Use the general properties of the estimation procedure, the probability distribution of the estimates, to calculate. Since ordinary least squares Student t-distribution estimation procedure for the Mean = coefficient value is unbiased, the SE = mean of the probability DF = 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. b P OLS estimation Assume H 0 Standard Number of Number of procedure unbiased is true error Observations Parameters Mean[b P ] = = SE[b P ] = DF = = Dependent Variable: GASCONS Included observations: 10 PRICEDOLLARS C Recall that the Prob. column reports the tails probability: Tails Probability: The probability that the coefficient estimate, b P, resulting from one regression would lie at least from, if the actual coefficient,, equals. = 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. Less Than Significance Level Greater Than Significance Level small large Unlikely that H 0 is true Likely that H 0 is true Reject H 0 Do not reject H 0 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 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: Percent Change in X = = = 10 percent We can now generalize this: X X + ΔX Percent Change in X = ΔX X 100 = ΔQ Q 100 ΔP P 100 Substituting for the percent changes = ΔQ P ΔP Q = dq P dp Q Simplifying Taking limits as ΔP approaches 0
5 5 Step 0: Construct a model reflecting the theory to be tested: Constant Price Elasticity Model: Q Price Elasticity of Demand = dq P dp Q 1 P Q 1 P β Const = 1 P = Simplifying. The price elasticity of demand just equals the value of. Taking the derivative of Q with respect to P Substituting β Const for Q. Substituting β Const for Q. Question: What does the budget theory of demand postulate about? P Q = Constant Solving for Q Q = _ Compare this equation to the constant price elasticity demand model: Q. Clearly, β Const = = Answer: The budget theory of demand postulates that the price elasticity of demand equals 1.0. Budget Theory of Demand: = Logarithms allow us to converting a constant price elasticity model into a linear model: Q log(q) = log(β Const ) + log(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 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 6 Critical Result: The coefficient estimate equals. The coefficient estimate does not equal ; the estimate lies from. Theory Evidence 0 Price Elasticity 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: H 0 : = 1.0 Cynic s view is correct: Actual price elasticity of demand equals 1.0 H 1 : 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, b P, in one regression would be at least.414 from 1.0, if H 0 were actually true (if the actual coefficient,, equals 1.0)? Answer: Prob[Results IF Cynic Correct] or
7 7 Step 4: Use the general properties of the estimation procedure, the probability distribution of the estimates, to calculate. 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. Student t-distribution Mean = SE = DF = b b C P OLS estimation Assume H 0 Standard Number of Number of procedure unbiased is true error Observations Parameters Mean[b P ] = = SE[b P ] = DF = = 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, b P, resulting from one regression would lie at least from, if the actual coefficient,, equals. NB: The tails probability is calculated on the premise that the actual value of the coefficient equals. Econometrics Lab: Left Right Tail Tail + =
8 8 Hypothesis Testing: Using Regression Printouts with Clever Algebraic Manipulations Clever Definition: β Clever = = 1.0 if and only if β Clever = 0 Now, recall our equation for the constant price elasticity model: LogQ = c + LogP where LogQ = log(gascons) LogP = log(price) Next, a little algebra: β Clever = + 1.0, = β Clever 1.0. LogQ LogQ LogQ + LogP = c + (β Clever 1.0) LogP = c + β Clever LogP LogP = c + β Clever LogP LogQPlusLogP = c + β Clever LogP Substituting β Clever 1.0 for We can now express the hypotheses in terms of β C. Recall that = 1.0 if and only if β Clever = 0: where LogQPlusLogP = LogQ + LogP H 0 : = 1.0 H 0 : β Clever = 0 Actual price elasticity of demand equals 1.0 H 1 : 1.0 H 1 : β Clever 0 Actual price elasticity of demand does not equal 1.0 Dependent Variable: LOGQPLUSLOGP Included observations: 10 LOGP C 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, b Clever, in one regression would be at least.414 from 0, if H 0 were actually true (if the actual coefficient, β Clever, equals 0)? Answer:
9 9 Next, calculate focusing on β Clever : 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. Student t-distribution Mean = SE = DF = b b C Clever OLS estimation Assume H 0 Standard Number of Number of procedure unbiased is true error Observations Parameters Tails Probability: The probability that the coefficient estimate, b P, resulting from one regression would lie at least from, if the actual coefficient, β Clever, equals. Mean[b Clever ] = = SE[b Clever ] = DF = = Answer: = 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: Less Than Significance Level small Unlikely that H 0 is true Greater Than Significance Level large Likely that H 0 is true Reject H 0 Do not reject H 0 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 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 H 0 : β = c H 1 : β > c Theory: β > c or β < c Probability Distribution Theory: β = c Probability Distribution H 0 : β = c H 1 : β c H 0 : β = c H 1 : β < c c Probability Distribution b = Probability of obtaining results like those we actually got (or even stronger), if H 0 is true Small Large c b c b Reject H 0 Do not reject H 0 Logarithms: A Useful Econometric Tool 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: y t + β x x t + e t Coefficient estimate: Estimates the (natural) unit change in y resulting from a one (natural) unit change in x Log Dependent Variable Model: log(y t ) + β 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 Log Explanatory Variable Model: y t + β x log(x t ) + e t 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(y t ) + β x log(x t ) + e t 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 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. Wage t HSEduc t 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: Wage t + β E HSEduc t + e t 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 Estimated Equation: Wage = HSEduc. Coefficient Interpretation: One _ increase in HSEduc Increases Wage by Log dependent variable model: LogWage t + β E HSEduc t + e t 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 HSEDUC C Estimated Equation: LogWage = HSEduc. Coefficient Interpretation: One _ increase in HSEduc Increases Wage by
12 12 Log explanatory variable model: Wage t + β E LogHSEduc t + e t 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 Estimated Equation: Wage = LogHSEduc. Coefficient Interpretation: One _ increase in HSEduc Increases Wage by Log-log (constant elasticity) model: LogWage t + β E LogHSEduc t + e t 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 LOGHSEDUC C Estimated Equation: LogWage = LogHSEduc. Coefficient Interpretation: One _ increase in HSEduc Increases Wage by
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