ECONOMICS AND ECONOMIC METHODS PRELIM EXAM Statistics and Econometrics August 2007

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1 ECONOMICS AND ECONOMIC METHODS PRELIM EXAM Statistics and Econometrics August 2007 Instructions: Answer all six (6) questions. Point totals are given in parentheses. The parts within each question receive equal weight. You may use a calculator. In estimated equations, standard errors appear in parentheses below coefficients. Any statistical tables that you need are included with the exam, and you should carry out hypotheses tests at the 5% significance level. 1. (9 points) Suppose that X is a continuous random variable having probability density where 8 is a positive parameter. (i) Find the cdf of X. (ii) Show that the median of X is 2('le). (iii) For 8 = 2, verify that the expected value is greater than the median. 2. (1 5 points) Let {Xi : i = 1,..., n) be a random sample from the Poisson(p) distribution, and let be the sample average. Consider estimators of p of the form,h(,b) = a + bx where a,b 2 0. (i) Find the bias of,h(,b) as a function of a and b (and p). For what values of a and b is,h(a,b) unbiased? (ii) Find the mean squared error (MSE) of,h(,b).

2 (iii) If you set b = 1, for what value of a is the MSE minimized? Explain. (iv) If you set a = 0, find the value of by say b*, that minimizes the MSE. What happens to b* as n -t oo. (v) Because you do not know p, you operationalize the choice b* in part (iv) by replacing p with z. Call F* the resulting estimator of p. Can you easily find the bias in G*? Explain. 3. (12 points) State whether you agree or disagree with each statement, and provide brief justification. (i) "If E(YX) = 3 + 2X, then E[(Y - 3-2X)2] 5 E[(Y - a - bq2] for any numbers a and b." (ii) "If {W, : n = ly2,..) is a sequence of strictly positive random variables, that is, W, > 0 for all n, then plim(w,) > 0 whenever the plim exists." (iii) "In multiple regression analysis, a 95% confidence interval based on the t distribution is useless if the errors do not have a normal distribution." (iv) "In a time series regression, the R-squared is useless as a goodness-of-fit measure in the presence of serial correlation." 4. (18 points) The following equations were estimated using 982 elementary schools. The variable score is the average score on a standardized exam, where the maximum score is 100 (average across schools is 59, sd = 15). The variable size is the average fourth-grade class size (with average 24 and ranging from 16 to 30), poverty is the poverty rate (in percent) at the school, and enroll is the school enrollment.

3 A score = size -.46 poverty log(enrol1) (19.12) (.14) (-21) (.93) n = 982, R2 =. 153 n score = size size poverty (20.66) (.44) (.O 18) (-16) n score = size -.48 poverty size -poverty (20.87) (.26) (el71 (.O 1 1) (i) Using equation (4. I), summarize the effect of class size on score, and discuss whether it is statistically and practically significant. (ii) Describe the relationship between score and size from equation (4.2). Provide as complete a description as possible. A picture might help your discussion. (iii) Are the quadratic terms jointly significant in equation (4.2)? (vi) Do size and poverty interact in a significant way? What is the effect of class size for a school with a zero poverty rate? For a school withpoverty = 20? Discuss. (v) Somebody suggests that size should be dropped from equation (4.3) because its t statistic is only about 1.58 in absolute value. Do you think this is a good idea? Explain. (vi) Suppose you obtain the residuals from equation (4.3) and run the regression and obtain a coefficient on log(enrol1) of with t = What do you make of the negative coefficient and its strong statistical significance?

4 5. (1 8 points) In the linear model Y = Xp + U, consider the ridge regression estimator where r 1 0 is a positive number (chosen by us). b, = (X'X + rix)-'x1~, (i) Find E(~,IX) under the Gauss Markov assumptions. For which values of r is B, unbiased? (ii) Find var(b,lx) under the Gauss-Markov assumptions. (iii) Suppose that all GM assumptions hold except that Var(U1X) = diag(o:, a:,..., oi), where = Var(urlX). Now find var(b,lx). Can you conclude that it is smaller (in the matrix sense) than the variance of the OLS estimator? (iv) Now assume that the data are obtained from random sampling, and think of the ridge parameter, r, as depending on the sample size, say r,. What condition on r, is needed for B, to be consistent for p? (v) Assume that the classical linear model assumptions hold (the data need not be from a random sample), and assume that you know the error variance, a2. Suppose you wish to test Ho : p = 0 (that is, all parameters are zero). How would you obtain an exact test using Br? (vi) How does the test in part (v) differ from that based on the OLS estimator? Explain. 6. (9 points) Provide a brief discussion of each of the following pairs of concepts. (i) Generalized Least Squares versus Feasible GLS. (ii) Serial correlation in the errors versus serial correlation in the regressors. (iii) Error variance versus variance inflation factor.

5 7 al Tables Appendix G Statistical Tables ;$!Gg$:! $'p.p: ;J;?-;.!.?>&<,i.-s;. ;:!.. TABLE 6.2 Critical Values of the t Distribution Examples: The 1% critical value for a one-tailed test with 25 df is The 5% critical for a two-tailed test with large (> 120) df is Source: This table was generated using the Statag function invt.

6 Appendix G Statistical Tables 819 TABLE G.3b 5% Critical Values of the F Distribution Example: The 5% critical value for numerator df = 4 and large denominat01 Source: This table was generated using the Statam function invfprob.