University of Maryland Spring Economics 422 Final Examination

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1 Department of Economics John C. Chao University of Maryland Spring 2009 Economics 422 Final Examination This exam contains 4 regular questions and 1 bonus question. The total number of points for the regular questions is 120. The total time allowed for the exam is 120 minutes. There is, of course, the opportunity to gain 25 extra points by completing the bonus question correctly. It is suggested that you allocate to each question the same number of minutes as the number of points that the question is worth; e.g., allocate 25 minutes to a question that is worth 25 points. You should provide explanations for your answers. Right answers with no explanation will receive no credit. On the other hand, wrong answers with thoughtful explanations may receive some credit. Partial credit will be given. GOOD LUCK!

2 QUESTION 1 (30 POINTS) Consider a situation where economic theory suggests that you impose certain restrictions on your estimated multiple regression function. To test the validity of your restrictions, you have your statistical package calculate the corresponding F-statistic. Find the critical value at the 5% and also at the 1% significance level, assuming that the errors of your regression model may not be normally distributed. Comment on whether or not your will reject the null hypothesis at the 5% and the 1% level for each of the following cases: (a) number of observations: 152; number of restrictions tested: 3; realized value of F-statistic: 3.21 (b) number of observations: 1, 732; number of restrictions tested: 7; realized value of F-statistic: 4.92 (c) number of observations: 4000; number of restrictions tested: 5; realized value of F-statistic: QUESTION 2 (25 POINTS) Consider the regression model Y i = β 0 + β 1 D i + u i, i =1,..., n; where Y i represents the wage of individual i and where D i is a binary variable which equals one if the i th individual is female and zero if the i th individual is male. Use the OLS formula for the intercept coefficient to show that β b 0 is the average wage for males. (Hint: for a linear regression model Y i = β 0 + β 1 X i + u i i =1,..., n; the OLS formula for the intercept coefficient is bβ 0 = Y β b 1 X,whereY = n X 1 n Y i, X = n X 1 n X i,andβ b 1 is the OLS estimate for β 1.) i=1 i=1 2

3 QUESTION 3 ( 33 POINTS) Sports economics typically looks at winning percentages of sports teams as one of various outputs, and estimates production functions by analyzing the relationship between twinning percentages and inputs. In Major League Baseball (MLB), the determinants of winning are quality pitching and batting. Data are collected for all 30 MLB teams for the 1999 season. Pitching quality is approximated by Team Earned Run Average" (ERA), and hitting quality by On Base Plus Slugging Percentage" (OPS) Summary of the Distribution of Winning Percentage On Base Plus Slugging Percentage, and Team Earned Run Average for MLB in 1999 Average Std Percentile Dev 10% 25% 40% 50% 60% 75% 90% Team ERA OPS Winning Pct Suppose further that your regression output is: Winpct d = teamera ops, (0.08) (0.008) (0.126) R2 =0.92, SER =0.02; and suppose that in interpreting the estimation and test results for this regression, you do not feel comfortable about assuming that the errors are normally distributed. (a) Interpret the regression. Are the results statistically significant and important? (b) There are two leagues in MLB, the American League (AL) and the National League (NL). One major difference is that the pitcher in the AL does not have to bat. Instead, there is a designated hitter" in the hitting line-up. You are concerned that, as a result, there is adifferent effect of pitching and hitting in the AL from the NL. To test this hypothesis, you allow the AL regression to have a different intercept and different slopes from the NL regression. You therefore create a binary variable for the American League (DAL) and estimate the following specification: Winpct d = DAL teamera (DAL teamera) (0.12) (0.24) (0.008) (0.018) ops (DAL ops), (0.163) (0.160) R 2 = 0.92, SER =0.02. What is the regression for winning percentage in the AL? What is the regression for winning percentage in NL? Next, calculate the t-statistics and say something about the statistical significance of the AL variables at the 5% level. 3

4 (c) Recall from class notes that sequentially testing the significance of slope coefficients is not the same as testing for their significance simultaneously. Hence, suppose that you ask your regression package to calculate the F -statistic that all three coefficients involving the binary variable for AL are zero. Your regression package gives a value of Canyou reject the null hypothesis at the 1% significance level? Should you worry about the small sample size? QUESTION 4 (32 POINTS) Consider analyzing data from the regression Y i = β 0 + β 1 X i + u i. The table below lists OLS estimates of β 1 (i.e., b β 1 ) as well as estimates of the variance of b β 1 (bσ 2 β1 ) for four different data sets. In each case, calculate the p-value for the t statistic for testing the null hypothesis that β 1 =0versus the alternative hypothesis that β 1 6=0. Indicate in which case you would reject the null hypothesis at the 5% significance level. Results for Results for Results for Results for Data Set 1 Data Set 2 Data Set 3 Data Set 4 bβ bσ 2 β

5 BONUS QUESTION (25 POINTS) Consider the system of equations Y i = β 0 + β 1 X i + u i, X i = Y i + Z i, i =1,...,n; where we assume that β 1 6=1and Cov (Z i,u i )=0for every i. It is easy to show (as we have done during the review session) that the reduced form equations for this system are given by Y i = π 0 + π 1 Z i + v i, (1) X i = π 0 + π 2 Z i + v i, (2) where π 0 = β 0,π 1 = β 1,π 2 = 1,andv i = u i. 1 β 1 1 β 1 1 β 1 1 β 1 Now, consider estimating equations (1) and (2) by ordinary least squares (OLS). Let bπ 1 denote the OLS estimator of π 1 in equation (1), and let bπ 2 denote the OLS estimator of π 2 in equation (2). Show that the estimator bβ 1 = bπ 1 bπ 2 is identical to the two-stage least squares (TSLS) estimator of β 1. 5

Econometrics -- Final Exam (Sample)

Econometrics -- Final Exam (Sample) 1) The sample regression line estimated by OLS A) has an intercept that is equal to zero. B) is the same as the population regression line. C) cannot have negative and