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1 Amherst College Department of Economics Economics 360 Fall 202 Solutions: Wednesday, September 26. Assume that the standard ordinary least square (OLS) premises are met. Let (x i, y i ) and (, y j ) be the values of the explanatory and dependent variables from two different observations. Let b Slope Slope of the straight line connecting the two points representing these two observations a. What is the algebraic expression for b Slope? b Slope y j y i Consider the simple regression model and two different observations, i and j: y i β Const + β x x i + e i y j β Const + β x + e j b. Using the simple regression model, substitute for y i and y j in the expression for b Slope. (Assume that x i does not equal.) b Slope y j y i β Const + β x + e j (β Const + β x x i + e i ) β Const + β x + e j β Const β x x i e i β x β x x i + e j e i β x ( ) + (e j e i ) β x + e j e i β x + ( e e i ) i c. What does the mean of b Slope s probability distribution, Mean[b Slope ], equal? Mean[b Slope ] Mean[β x + ( e j e i )] β x + Mean[ ( e j e i )] β x + β x + β x Mean[e j e i ] [Mean[e j ] Mean[e i ]]

2 2 d. What does the variance of b Slope s probability distribution, Var[b Slope ], equal? Var[b Slope ] Var[β x + (e j e i )] Var[ ( e j e i )] ( ) 2 Var[(e j e i ] ( ) 2 Var[(e j + ( e i )] ( ) 2 [Var[e j ] + Var[ e ]] i ( ) 2 [Var[e j ] + Var[e ]] i 2 [Var[e] + Var[e]] ( ) 2 ( ) 2 Var[e] 2. Assume that the standard ordinary least square (OLS) premises are met. Consider the Min-Max estimation procedure that we simulated in Econometrics Lab 6.6. Let The actual coefficient equals 2 (β x 2). The variance of the error term s probability distribution equals 00 (Var[e] 00). The sample size equals, and the values of the x s equal: 3, 9,, 2, and 27. Using your answers to exercise : a. What does the mean of the Min-Max estimate s probability distribution equal? b x b Slope for observations and Mean[b Slope for observations and ] β x 2 b. What does the variance of the Min-Max estimate s probability distribution equal? Var[b Slope for observations and ] 2 ( ) 2 Var[e] (27 3) c. Are your answers consistent with the simulations of the Min-Max estimation procedure that we reported in this chapter? Yes

3 3 3. Revisit the U. S. crude oil production data. Crude Oil Production Data: Annual time series data of U. S. crude oil production and prices from 976 to OilProdBarrels t U. S. crude oil productions in year t (thousands of barrels per day) Price t Real price of crude oil in year t (dollars per barrel ) Using statistical software, generate a new variable that expresses crude oil production in thousands of gallons per day rather than thousands of barrels per day. Call the new variable OilProdGallons. Note that there are 42 gallons in barrel. Click here to access data [Link to MIT-OilProd wf goes here.] Getting Started in EViews After opening the file: In the Workfile window: click Genr In the Generate Series by Equation window: enter the formula for the new series: OilProdGallons OilProdBarrels*42 NB: The asterisk, *, is EViews multiplication symbol. Click OK a. Run the following ordinary least squares (OLS) regressions: Dependent variable: OilProdBarrels (crude oil production expressed in barrels) Explanatory variable: Price ) Based on this OilProdBarrels regression, estimate the effect of a $ increase in price on the barrels of oil produced? Dependent Variable: OILPRODBARRELS Included observations: 29 PRICE C Estimated equation: EstOilProdBarrels, Price We estimate that a $ increase in the crude oil price increases crude oil production by 92. thousand barrels per day. 2) Based on your answer to part ), estimate the effect of a $ increase in price on the gallons of oil produced? (Remember that there are 42 gallons in barrel.) Convert 92. barrels into gallons: ,88. We estimate that a $ increase in the crude oil price increases crude oil production by 3,88 thousand gallons per day.

4 4 b. Run the following ordinary least squares (OLS) regressions: Dependent variable: OilProdGallons (crude oil production expressed in gallons) Explanatory variable: Price Dependent Variable: OILPRODGALLONS Included observations: 29 PRICE C Based on this OilProdGallons regression, estimate the effect of a $ increase in price on the gallons of oil produced? Estimated equation: EstOilProdGallons 249, ,88Price We estimate that a $ increase in the price increases crude oil increases crude oil production by 3,88 thousand gallons per day. c. Do the units in which the dependent variable is measure influence the estimate of how the explanatory variable affects the dependent variable? No. 4. Revisit the U. S. gasoline consumption data. Gasoline Consumption Data: Annual time series data U. S. gasoline consumption and prices from 990 to 999. 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) Using statistical software, generate a new variable that expresses the price of gasoline in cents rather than dollars. Call this new variable PriceCents. Click here to access data [Link to MIT-GasolineCons wf goes here.]

5 a. Run the following ordinary least squares (OLS) regressions: Dependent variable: GasCons Explanatory variable: PriceDollars Dependent Variable: GASCONS Method: Least Squares Sample: Included observations: 0 PRICEDOLLARS C ) Based on this PriceDollars regression, estimate the effect of a $ increase in price on the gallons of gasoline demanded? Estimated equation: EstGasCons 6.8.7PriceDollars We estimate that a $ increase in the gasoline price decreases gasoline consumption by.7 million gallons per day. 2) Based on your answer to part ), estimate the effect of a cent increase in price would have on the gallons of gasoline demanded? Since there are 00 cents in dollar, a cent increase in the gasoline price decreases gasoline consumption by an estimated.7 million gallons per day. b. Run the following ordinary least squares (OLS) regressions: Dependent variable: GasCons Explanatory variable: PriceCents Dependent Variable: GASCONS Method: Least Squares Sample: Included observations: 0 PRICECENTS C Based on this PriceCents regression, estimate the effect that a cent increase in price has on the gallons of gasoline demanded? Estimated equation: EstGasCons 6.8.7PriceCents We estimate that a cent increase in the gasoline price decreases gasoline consumption by.7 million gallons per day. c. Do the units in which the explanatory variable is measure influence the estimate of how the explanatory variable affects the dependent variable? No.

6 6. Consider Professor Lord s first quiz. a. Suppose that we know the actual value of the constant and coefficient. More specifically, suppose that the actual value of the constant is 0 and the actual value of the coefficient is 2. Fill in the blanks below to calculate each students error term and the error term squared. Then, compute the sum of squared error terms. β Const 0 β x 2 Student x y 0 + 2x e st Quiz e 2 st Quiz Sum 8 b. In reality, we do not know the actual value of the constant and coefficient. We used the ordinary least squares (OLS) estimation procedure to estimate their values. The estimated constant was 63 and the estimated value of the coefficient was 6. Fill in the blanks below to calculate each student s residual and the residual squared. Then, compute the sum of the squared residuals. b Const 63 b x 6.2 Res y - (b Const + b x x) Student x y Esty x Res st Quiz Res 2 st Quiz Sum 4 c. Compare the sum of squared errors with the sum of squared residuals. The sum of squared residuals is less than the sum of squared errors. d. In general, when applying the ordinary least squares (OLS) estimation procedure could the sum of squared residuals ever exceed the sum of squared errors? No.

7 7 6. Suppose that Student 2 had missed Professor Lord s quiz. Student x y 66 x minutes studied y quiz score a. Plot a scatter diagram of the data. b. What is the equation for the best fitting line? y y.2x + 60 Slope Rise Run y 66 x 6 y 330 6x 30 y 6x y 6 x x x c. What are the residuals for each observation? 0 d. Suppose that the quiz scores were different. For example, suppose that Student received a 70 instead of 66. ) What is the equation for the best fitting line now? y Slope Rise Run y x y 70 x y 70 x y x ) What are the residuals for each observation? 0 x e. Again, suppose that the quiz scores were different. For example, suppose that Student received an 86 instead of 66 or 70. ) What is the equation for the best fitting line now? Slope Rise Run y y.2x y 86 x y 430 x y x + 42 y x + 8.2x ) What are the residuals for each observation? x f. In general, when there are only two observations what will the residuals for the best fitting line equal? 0