Solutions: Monday, October 15

Transcription

1 Amherst College Department of Economics Economics 360 Fall Consider Nebraska petroleum consumption. Solutions: Monday, October 15 Petroleum Consumption Data for Nebraska: Annual time series data of petroleum consumption and prices for Nebraska from 1990 to PetroCons t Consumption of petroleum in year t (1,000 s of gallons) Cpi t Midwest Consumer Price Index in year t ( ) Pop t Nebraska population in year t PriceNom t Nominal price of petroleum in year t (dollars per gallon) Year t Year Generate two new variables from the Nebraska data: PetroConsPC t PriceReal t Per capita consumption of petroleum in year t (gallons) Real price of petroleum in year t (dollars per gallon) Click here to access data [Link to MIT-PetroConsNeb wf1 goes here.] a. Petroleum consumption includes the consumption of all petroleum products: gasoline, fuel oil, etc. Consequently, would you expect petroleum to be a necessity or a luxury? Necessity. b. In view of your answer to part a, would expect the per capita demand for petroleum to be inelastic? Explain. Yes, the demand should be inelastic because the demand for a necessity is inelastic. c. Consequently, why would the numerical value of the real price elasticity of demand be greater than 1? Demand is inelastic whenever the absolute value of the price elasticity of demand is less than 1. But now remember that the numerical value of the price elasticity of demand will be negative since the demand curve is downward sloping. Accordingly, the numerical value of the price elasticity of demand should be greater than 1. d. Apply the hypothesis testing approach that we developed to assess the theory. Calculate Prob[Results IF H 0 True] in two ways: 1) Using the Econometrics Lab. [Link to MIT-TTest 0.1 goes here.] 2) Using the clever definition approach.

2 2 Step 0: Formulate a model reflecting the theory to be tested: Constant price elasticity model: PetroPC = β Const PriceReal β P where β P = Price elasticity of demand Taking natural logarithms of both sides. log(petropc) = log(β Const log(pricereal) LogPetroPC = c + β P LogPriceReal where LogPetroPC = log(petropc) c = log(β Const ) LogPriceReal = log(pricereal) Theory suggests that the absolute value of β P should be less than 1; since β P is negative, theory suggests that β P should be greater than 1. Step 1: Collect data, run the regression, and interpret the estimates. Generate two new variables: LogPetroPC = log(petropc) LogPriceReal = log(pricereal) Dependent variable: LogPetroPC Explanatory variable: LogPriceReal Dependent Variable: LOGPETROPC Included observations: 10 Coefficient Std. Error t-statistic Prob. LOGPRICEREAL C Estimated Equation: EstLogPetroPC = Price Interpretation: We estimate that a 1 percent increase in the price decreases petroleum consumption by.365 percent. That is, the estimate of the price elasticity of demand equals.365. Step 2: Play the cynic, challenge the evidence, and construct the null and alternative hypotheses. Cynic s view: Sure, the coefficient estimate from regression suggests that the demand is inelastic, but this is just the luck of the draw. In fact, the price elasticity of demand is not greater than 1, it equals 1. H 0 : β P = 1 Cynic s view is correct: Actual price elasticity of demand equals 1 H 1 : β P > 1 Cynic s view is incorrect: Demand is inelastic; the elasticity of demand is greater than 1

3 3 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 actual price elasticity of demand equals 1? Specific Question: The regression s coefficient estimate was.365: What is the probability that the coefficient estimate, b P, in one regression would be.365 or more, if H 0 were actually true (if the actual coefficient, β P, equals 1)? Answer: Prob[Results IF Cynic Correct] or Prob[Results IF H 0 True] Step 4: Use the general properties of the estimation procedure, the probability distribution of the estimates, to calculate Prob[Results IF H 0 True]. RegHypoTest-02.eps If the null hypothesis were true, the actual coefficient would equal 1. Since ordinary least squares estimation procedure for the coefficient value is unbiased, the mean of the probability Student t-distribution Mean = 1 SE =.132 DF = distribution of coefficient estimates would be 1. The SE column of the EViews printout provides us with the standard error of the coefficient estimates,.132. The degrees of freedom equal the number of observations, 10, less 2, since we are estimating two parameters, the constant and the coefficient; consequently the degrees of freedom equal 8. OLS estimation Assume H 0 EViews Number of Number of procedure unbiased is true SE column observations parameters.0007 b P Mean[b P ] = β P = 1 SE[b P ] =.132 DF = 10 2 = 8 Econometrics Lab to Calculate Prob[Results IF H 0 True]. [Link to MIT-TTest 0.1 goes here.] Prob[Results IF H 0 True] =.0007

4 4 Clever Algebraic Manipulations to Calculate Prob[Results IF H 0 True]. Alternatively, we can use a clever algebraic manipulation approach by defining β Clever so that it equals 0 when β P equals 1: β Clever = β P + 1 and β P = β Clever 1 Next, consider the model and perform some algebra: LogPetroPC = c + β P LogPriceReal LogPetroPC = c + (β Clever 1)LogPriceReal LogPetroPC + LogPriceReal = c + β Clever LogPriceReal yclever = c + β Clever LogPriceReal where yclever = LogPetroPC + LogPriceReal Expressing the null and alternative hypotheses in terms of β Clever : H 0 : β Clever = 0 β P = 1 Cynic s view is correct: Actual price elasticity of demand equals 1 H 1 : β Clever > 0 β P > 1 Cynic s view is incorrect: Demand is inelastic Dependent Variable: YCLEVER Included observations: 10 Coefficient Std. Error t-statistic Prob. LOGPRICEREAL C RegHypoTest-04.eps Student t-distribution Mean = 1 SE =.132 DF = /2.0013/2 Prob[Results IF H 0 True] = b Clever We have obtained the same value for Prob[Results IF H 0 True]. e. What is your assessment of the theory? Explain. Step 5: Decide on the standard of proof, a significance level Since Prob[Results IF H 0 True] equals.0007, we reject the null hypothesis that demand is unit elastic even at the 1 percent significance level. This supports the theory that demand is inelastic.

5 5 2. Consider the U.S. crude oil supply data. Crude Oil Production Data: Annual time series data of U. S. crude oil production and prices from 1976 to OilProdBarrels t U. S. crude oil productions in year t (thousands of barrels per day) Price t Real well head price of crude oil in year t ( dollars per barrel) Consider the following rather bizarre theory of supply: Theory of Supply: The price elasticity of supply equals.10. a. Apply the hypothesis testing approach that we developed to assess the theory. Calculate the Prob[Results IF H 0 True] using the Econometrics Lab and also using a clever algebraic manipulation. Click here to access data [Link to MIT-OilProd wf1 goes here.] Step 0: Formulate a model reflecting the theory to be tested: The constant price elasticity model is appropriate: Q = β Const Taking natural logarithms of both sides. log(q) = log(β Const log(p) LogQ = c + β P LogP where LogQ = log(q), c = log(β Const ), and LogP = log(p) The theory asserts that β P equals.10. Step 1: Collect data, run the regression, and interpret the estimates. We must generate the two variables: the logarithm of quantity and the logarithm of price: LogQ = log(oilprodbarrels) LogP = log(price) Dependent Variable: LOGQ Included observations: 29 Coefficient Std. Error t-statistic Prob. LOGP C Estimated Equation: EstLogQ = LogP Interpretation: We estimate that a 1 percent increase in the price increases oil production by.2136 percent. That is, the estimate of the price elasticity of supply equals The evidence suggests that the elasticity of supply does not equal.10. More specifically, the estimate is.1136 from.10.

6 6 Step 2: Play the cynic, challenge the evidence, and construct the null and alternative hypotheses. Cynic s view: Sure the coefficient estimate from regression suggests that the price elasticity of supply does not equal.10, but this is just the luck of the draw. In fact, the actual price elasticity of supply equals.10. H 0 : β P =.10 Cynic s view is correct: Actual price elasticity of supply equals.10 H 1 : β P.10 Cynic s view is incorrect: Actual price elasticity of supply does not equal.10 Step 3: Formulate the question to assess the cynic s view. Question for the Cynic: Generic Question: What is the probability of obtaining a result like the one obtained from the regression (or even stronger), if the cynic is correct and actual price elasticity of supply equals.10? Specific Question: The regression s coefficient estimate was.2136: What is the probability that the coefficient estimate, b P, in one regression would be at least.1136 from.10, if H 0 were actually true (if the actual coefficient, β P, equals.10)? Answer: Prob[Results IF Cynic Correct] or Prob[Results IF H 0 True] Step 4: Use the general properties of the estimation procedure, the probability distribution of the estimates, to calculate Prob[Results IF H 0 True]. RegHypoTest-05.eps If the null hypothesis were true, the actual coefficient would equal.10. Since ordinary least squares estimation procedure for the coefficient value is unbiased, the mean of the probability distribution of coefficient estimates would be.10. The SE column of the EViews printout provides us with the standard error of the Student t-distribution Mean =.10 SE =.0695 DF = 27 coefficient estimates. The degrees of freedom equal the number of observations less 2, since we are estimating two parameters, the constant and the coefficient. OLS estimation Assume H 0 EViews Number of Number of procedure unbiased is true SE column observations parameters b P Mean[b P ] = β P =.10 SE[b P ] =.0695 DF = 29 2 = 27

7 7 We can use the Econometrics Lab to compute Prob[Results IF H 0 True]: [Link to MIT-TTest 0.1 goes here.] We shall calculate this probability in two steps: First, calculate the probability of the estimated lying in the right hand tail. Calculate the probability that the estimate lies.1136 or more above.10; that is, the probability that the estimate lies at or above Second, calculate the probability of the estimated lying in the left hand tail. Calculate the probability that the estimate lies.1136 or more below.10; that is, the probability that the estimate lies at or below Left Right Tail Tail Prob[Results IF H 0 True] = Alternatively, we can use a clever algebraic manipulation approach by defining β Clever so that it equals 0 when β P equals.10: β Clever = β P.10 and β P = β Clever +.10 Next, consider the model and perform some algebra: LogQ = c + β P LogP LogQ = c + (β Clever +.10)LogP LogQ.10LogP = c + β Clever LogP yclever = c + β Clever LogPriceReal where yclever = LogQ.10LogP Expressing the null and alternative hypotheses in terms of β Clever : H 0 : β Clever = 0 β P =.10 Cynic s view is correct: Actual price elasticity of supply equals.10 H 1 : β Clever 0 β P.10 Cynic s view is incorrect: : Actual price elasticity of supply does not equal.10 Dependent Variable: YCLEVER Included observations: 29 Coefficient Std. Error t-statistic Prob. LOGP C

8 8 Next, calculate Prob[Results IF H 0 True] focusing on β Clever : RegHypoTest-06.eps Question: What is the probability of obtaining a result like the one calculated from the regression (a coefficient estimate,.1139/2 b Clever, of.1136,.1136 from 0), if the cynic s view and the null hypothesis were correct (that is, if the actual coefficient, β Clever, equals 0)? Student t-distribution Mean = 0 SE =.0695 DF = / b Clever Answer: Prob[Results IF H 0 True] = This is the same answer as before. By a clever algebraic manipulation, we can get EViews to perform the calculations. b. What is your assessment of the theory? Explain. 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: Even at a 10 percent significance level, we do not reject the null hypothesis that the elasticity of supply equals.10. That is, at a 10 percent significance level, the estimate of.2136 is not statistically different from.10.

9 9 3. Consider the following constant elasticity model: Q = β Const I β I ChickP β CP where Q = Quantity of beef demanded P = Price of beef (the good s own price) I = Household income ChickP = Price of chicken a. Show that if β CP = β P, then Q = β Const ( P ChickP )β P I ( ChickP )β I Q = β Const I β I ChickP β CP = β Const I β I ChickP β P β I = β Const I β I ChickP β P ChickP β I P = β Const ( β P )( β I I ) Chick ChickP β I P = β Const ( ChickP )β P I ( ChickP )β I b. If β CP = β P, what happens to the quantity of beef demanded when the price of beef (the good s own price, P), income (I), and the price of chicken (ChickP) all double? P ChickP is unchanged I ChickP is unchanged Q is unchanged c. If β P + β I + β CP = 0, what happens to the quantity of beef demanded when the price of beef (the good s own price, P), income (I), and the price of chicken (ChickP) all double? If β P + β I + β CP = 0, then β CP = β P. In part b, we showed that if β CP = β P, then Q is unchanged when the good s own price, income, and the price of other goods all double.

10 10 4. Again, consider the following constant elasticity model: Q = β Const I β I ChickP β CP What does log(q) equal, where log is the natural logarithm? log(q) = log(β Const log(p) + β I log(i) + β CP 5. Consider the following model: log(q) = log(β Const log(p) + β I log(i) + β CP Let β Clever = β P + β I + β CP. Show that log(q) = log(β Const ) +β P [log(p) ] + β I [log(i) ] + β Clever log(q)= log(β Const log(p) + β I log(i) + β CP Since β Clever = β P + β I + β CP, β CP = β Clever β P ; now, substitute for β CP = log(β Const log(p) + β I log(i) + (β Clever β P ) Multiplying out the last term = log(β Const log(p) + β I log(i) + β Clever β P Rearranging terms = log(β Const log(p) β P + β I log(i) + β Clever Factoring β P from the second and third terms and β I from the fourth and fifth terms = log(β Const [log(p) ] + β I [log(i) ] + β Clever