Old Exam 3 Solutions

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1 Amherst College Department of Economics Economics 360 Old Exam 3 Solutions 1. (30 points) The EViews workfile diningout.wf1 reports on the income and dining out expenditures of 100 households: INCOME annual income DININGOUT annual dining out expenditures a. Consider the following regression and the plot of the residuals: Regression 1: Dependent Variable: DININGOUT Included observations: 100 Variable Coefficient Std. Error INCOME C Based on this evidence, is the OLS estimation procedure for the income coefficient unbiased, biased up, or biased down? unbiased X biased up biased down. Could the standard error be misleading? Yes X No. Explain. RESID INCOME The evidence suggests that heteroskedasticity is present. As income rises, the variance of the residuals appears to grow. The good news is that the ordinary least squares (OLS) estimation procedure for the coefficient value is still unbiased in the presence of heteroskedasticity. On the other hand, the ordinary least squares (OLS) estimation procedure for the variance of the error term is based on the assumption that the variance of each observation s error term is the same. When heteroskedasticity is present, the assumption is untrue. Consequently, the OLS estimate for the error term variance is based on a false premise and may be misleading. All calculations based on this estimate, including the standard error, may be misleading. From a common sense standpoint, explain why the plot of residuals looks as it does. As a household s income rises, it will have more discretionary income. While some households will choose to spend more dining out, others will spend more on alternative activities. Therefore, it is not surprising to find the presence of heteroskedasticity.

2 2 b. Next, consider the following newly generated variables and regressions using them: SQRINC = 1 INCOME ONEDIVSQRTINC = DININGOUTTIMJESQRTINC = DININGOUT DININGOUTDIVJESQRTINC = DININGOUT INCOMETIMJESQRTINC = INCOME INCOMEDIVJESQRTINC = INCOME = Regression 2: Dependent Variable: DININGOUTTIMSQRTINC INCOMETIMSQRTINC C Regression 3: Dependent Variable: DININGOUTTIMSQRTINC INCOMETIMSQRTINC SQRTINC C Regression 4: Dependent Variable: DININGOUTTIMSQRTINC INCOMETIMSQRTINC SQRTINC Regression 5: Dependent Variable: DININGOUTDIVSQRTINC INCOMEDIVSQRTINC C Regression 6: Dependent Variable: DININGOUTDIVSQRTINC INCOMEDIVSQRTINC ONEDIVSQRTINC C Regression 7: Dependent Variable: DININGOUTDIVSQRTINC INCOMEDIVSQRTINC ONEDIVSQRTINC

3 3 Which regression(s) best estimates the relationship between DININGOUT and INCOME? 7 Explain. Assume that the variance of the error term s probability distribution is proportional to income: Var[e] = INCOME V where V is a constant Consequently, we should divide each variable by the square root of income: DININGOUT = β 1 INCOME + e DININGOUT 1 = β 1 INCOME e + DININGOUT 1 = β 1 + e What is the variance of the new error term: e Var[new error term] = Var[ ] 1 Var[new error term] = INCOME Var[e] = 1 INCOME [INCOME V] = V The variance of the new error term s probability distribution is equal for each observation. Heterosckedasticity is not a problem with the new equation.

4 4 2. (30 points) The EViews workfile entertainment.wf1 reports on the income and entertainment expenditures of one household for each week during the last year: INCOME weekly income ENTERTAINMENT weekly entertainment expenditures a. Consider the following regression and the serial correlation Lagrange multiplier test: Regression 1: Dependent Variable: ENTERTAINMENT INCOME C Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID INCOME C RESID(-1) Based on this evidence, is the OLS estimation procedure for the income coefficient unbiased, biased up, or biased down? unbiased X biased up biased down. Could the standard error be misleading? Yes X No. Explain. The LM test suggests that autocorrelation is present. The good news is that the ordinary least squares (OLS) estimation procedure for the coefficient value is still unbiased in the presence of autocorrelation. On the other hand, the ordinary least squares (OLS) estimation procedure for the variance of the error term is based on the assumption that the error terms are independent. When autocorrelation is present, the assumption is untrue. Consequently, the OLS estimate for the error term variance is based on a false premise and may be misleading. All calculations based on this estimate, including the standard error, may be misleading.

5 5 b. Consider the following regression: Regression 2: Dependent Variable: RES Date: 12/03/01 Time: 19:54 Sample(adjusted): 2 52 Included observations: 51 after adjusting endpoints RESLAG where RES = residual RESLAG = residual lagged one period From a common sense standpoint, how can the sign of the RESLAG coefficient be explained? Perhaps households budget a certain amount of income for entertainment. If a household goes overbudget this week, it would tighten its belt and spend less on entertainment next week. This could explain the negative RESLAG coefficient. c. Next, consider the following newly generated variables and regressions using them: ENTERTAINMENTADJ1 = ENTERTAINMENT ENTERTAINMENT( 1) INCOMEADJ1 = INCOME INCOME( 1) ENTERTAINMENTADJ2 = ENTERTAINMENT ENTERTAINMENT( 1) INCOMEADJ2 = INCOME INCOME( 1) Regression 3: Dependent Variable: ENTERTAINMENTADJ1 INCOMEADJ C Regression 4: Dependent Variable: ENTERTAINMENTADJ1 INCOMEADJ Regression 5: Dependent Variable: ENTERTAINMENTADJ2 INCOMEADJ C Regression 6: Dependent Variable: ENTERTAINMENTADJ2 INCOMEADJ

6 6 Which regression best estimates the relationship between ENTERTAINMENT and INCOME? 3 Explain. Strategy: Algebraically manipulate the original model to derive a new model in which the error terms in the new model are independent: The original model: y t = β 1 x t + e t Autocorrelation model: e t = ρe t 1 + v t v t s are independent Original model for period t: y t = β 1 x t + e t Original model for period t 1: y t 1 = β 1 x t 1 + e t 1 ρy t 1 = ρβ 1 + ρβ 2 x t 1 + ρe t 1 y t = β 1 x t + e t ρy t-1 = ρβ 1 + ρβ 2 x t-1 + ρe t-1 Subtract y t ρy t 1 = β 1 ρβ 1 x t ρβ 2 x t 1 + e t ρe t 1 y t ρy t 1 = β 1 ρβ 1 (x t ρx t 1 ) + e t ρe t 1 y t ρy t 1 = β 1 ρβ 1 (x t ρx t 1 ) + ρe t-1 + v t ρe t 1 (y t ρy t 1 ) = (β 1 ρβ 1 ) (x t ρx t 1 ) + v t v t s are independent; in the new model, we need not worry about autocorrelation. Use the estimated value of ρ from Regression 2, , to generate two new variables: ENTERTAINMENTADJ1 = ENTERTAINMENT ENTERTAINMENT( 1) INCOMEADJ1 = INCOME INCOME( 1) Regression 3 reports the results for these variables.

7 7 3. (40 points) The EViews workfile asparagus.wf1 reports on the quantity of asparagus, the price of asparagus, the price of broccoli, the wages of farm workers, and household disposable income: Regression 1: Dependent Variable: QUANTITYASPARAGUS PRICEASPARAGUS PRICEBROCCOLI DISPINCOME C Regression 2: Dependent Variable: QUANTITYASPARAGUS PRICEASPARAGUS WAGEFARMWORKERS C Regression 3: Dependent Variable: QUANTITYASPARAGUS PRICEBROCCOLI DISPINCOME WAGEFARMWORKERS C Regression 4: Dependent Variable: QUANTITYASPARAGUS PRICEHAT PRICEBROCCOLI DISPINCOME C Regression 5: Dependent Variable: QUANTITYASPARAGUS PRICEHAT WAGEFARMWORKERS C Regression 6: Dependent Variable: PRICEASPARAGUS PRICEBROCCOLI DISPINCOME WAGEFARMWORKERS C

8 8 where QUANTITYASPARAGUS quantity of asparagus PRICEASPARAGUS price of asparagus PRICEBROCCOLI price of broccoli WAGESFARMWORKERS wages of farm workers DISPINCOME disposable income PRICEHAT = PRICEBROCCOLI DISPINCOME WAGESFARMWORKERS a. Which regression(s) provides the OLS estimates of the demand model? 1 Is the OLS estimation procedure for the price coefficient unbiased, biased up, or biased down? unbiased biased up X biased down. Explain. Demand Model: Q D = β D 1 + βd PP + Other Demand Factors + ed e D down e D up P down P up Error terms and x s Positively Correlated OLS Estimation Procedure for Coefficient Value Biased Upward Price P (e D up) P P (e D down) S D (e D up) D D (e D down) Quantity b. Which regression(s) provides the OLS estimates of the supply model? 2 Is the OLS estimation procedure for the price coefficient unbiased, biased up, or biased down? unbiased biased up biased down X. Explain. Supply Model: Q S = β S 1 + βs PP + Other Supply Factors + es e S down e S up P up P down Price P (e S down) P S (e S down) S S (e S up) Error terms and x s Negatively Correlated OLS Estimation Procedure for Coefficient Value Biased Downward P (e S up) D Quantity

9 9 c. Which regression(s) provides the reduced form estimates? 3 and 6 d. Using the reduced form regressions, can you estimate the price coefficient of the demand model? Yes X No If yes, what is it? If no, explain why not. Wage coefficient in Reg 3 Estimate of the Price Coefficient of Demand = Wage coefficient in Reg 6 = = 14.1 e. Using the reduced form regressions, can you estimate the price coefficient of the supply model? Yes No X If yes, what is it?. If no, explain why not. An overidentification problem exists. There are two ways to estimate the price coefficient of supply: DispIncome coefficient in Reg 3 Estimate of the Price Coefficient of Supply = DiscIncome coefficient in Reg 6 = = 10.7 PriceBroc coefficient in Reg 3 Estimate of the Price Coefficient of Supply = PriceBroc coefficient in Reg 6 = = 14.4 f. Define the term unbiased estimation procedure. Unbiased: The estimation procedure does not systematically underestimate or overestimate the actual value: Mean the Estimate s Probability Distribution = Actual Value Loosely speaking (more specifically if the estimate s probability distribution is symmetric), the chances that the estimate will be greater than the actual value equal the chances that the estimate will be less. g. Define the term biased but consistent estimation procedure. Biased But Consistent: As the sample size, the number of observations, grows the magnitude of the bias decreases. More rigorously, As Sample Size Mean the Estimate s Probability Variance the Estimate s Probability Distribution Actual Value Distribution 0

10 10 h. Which regression(s) provides the two-stage least squares estimates? 4 and 5 i. Using the two-stage least squares regressions, can you estimate the price coefficient of the demand model? Yes X No If yes: What is the estimate? Is the estimation procedure biased? Yes X No Is the estimation procedure consistent? Yes X No If no: Explain why you cannot estimate the price coefficient of the demand model. j. Using the two-stage least squares regressions, can you estimate the price coefficient of the supply model? Yes X No If yes: What is the estimate? Is the estimation procedure biased? Yes X No Is the estimation procedure consistent? Yes X _ No If no: Explain why you cannot estimate the price coefficient of the supply model.