ECONOMICS AND ECONOMIC METHODS PRELIM EXAM Statistics and Econometrics August 2013

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1 ECONOMICS AND ECONOMIC METHODS PRELIM EXAM Statistics and Econometrics August 2013 Instructions: Answer all six (6) questions. Point totals for each question are given in parentheses. The parts within each question receive equal weight. You may use a calculator, but only for calculations - not for storage and retrieval of information. Be sure to show enough work so that we can see how you got your answer. 1. (9 points) Consider a population of elementary school children in a large school district, where the population is divided by race for simplicity, "white" and "nonwhite." Suppose it is known that 70% ofthe children are white. Some of the children in the district are eligible for a special education program. (i) Suppose that 900 children participate in the special education program. Let Wbe the number of white children participating. Ifthe probability ofparticipation does not depend on race, what is the expected value of W? (ii) Assuming that participation is independent across children (and that race is not a factor in participating), what is the standard deviation of W? (iii) Suppose that 660 of the participating children are white. Explain the calculation you would make to determine whether this is unrealistically high if program participation does not depend on race.

2 that 2. (9 points) Let ex, Y,Z) be random variables, with (Z) = 3, and Var(Z) 1. Assume (YjX,Z) = 2 +XZ Var(YjX; Z) = 4 Also, assume that Z and X are independent. (i) Show that E(l1X) = 3X. (ii) Find Var(.Y]X). (iii) Suppose you want to predict the outcome on Y. Would you rather use (l1x) or (.Y]x,Z)? Explain. 3. (I8 points) Let X - xponential«(}) with () = (X) > O. (i) Given a random sample of size n, {Xi : i = 1,2,..., n}, what is the maximum likelihood estimator of()? Justify your answer. (ii) Define r = log«(}). What is the MLE of r? Call this estimator y. (iii) Explain whether y has each ofthe following properties: (a) yis unbiased for r. (b) Yis consistent for r. (c)..fo (y - r) has a limiting normal distribution. (iv) What is the asymptotic variance of..fo(y - y)? (v) Construct a statistic for testing the null hypothesis Ho : r = 0 against HI : r > O. (vi) Using the estimator of(} in part (i), construct a statistic for testing Ho : () 1 against HI : () > 1. Is it the same as that in part (v)? 2

3 4. (18 points) Provide a short answer to each of the following questions. (i) In the modelyt =!31 +!32Xt2 +!33Xt !3kXtk + Ufo t = 1,...,n, under the assumption E(utIX) == 0, where X is the matrix of all regressors, is it true that n- I :E: Ut = O? 1 (ii) Let P be the k x 1 vector of OLS coefficients from the regression Yil on I, Xt2,..., Xtk, t = I,...,n and let ~ be the k x I vector of OLS coefficients from the regression YI on 1,X/2,...,Xtk, t = 1,...,n. What is the relationship between the pj and pj,j = 1,2,..., k? (iii) Suppose that in the model Yt == XtP + Ut, t = 1,...,n the matrix of regressors X has full rank k, E(utIX) = 0, and Var(utIX) = 0'2 for all t. However, Cov(Ut, uslx) = 0'2/4 when It - sl = 1 and Cov(Ut, uslx) = 0 when It - sl > 1. Evaluate the following statement: "Under these assumptions, the usual OLS variance estimator &2(X'X)-1 is biased for Var(PIX)." (iv) Suppose in the model Yit =!31 +!32X/2 +!33Xt3 +!34xI4 + Ut the matrix X has full rank and E(utIX) = 0 for all t. The sample correlation between Xt2 and Xt3 is.93 and the sample correlation between Xt4 and X/2 is Evaluate the following statement: "The OLS estimator P4 is probably unbiased for!34, but P3 is almost certainly biased for!33." (v) In the linear model under the Gauss-Markov Assumptions (with an intercept), consider the estimator where WI = II( 1 + x~ X;k)' If P is the OLS estimator, can you say anything about the relationship between Var(~IX) and Var(PIX)? Explain. (vi) In a time series model with random regressors and serial correlation, discuss the pros and cons of OLS versus a feasible GLS procedure such as Cochrane-Orcutt. 3

4 5. (24 points) The Stata output following this question contains regression estimates that study the effect of attending a Catholic high school, eathhs, on a standardized math test taken in 121h grade (mathi2). (i) What percentage of students in the sample attend a Catholic high school? Describe the mean and standard deviation of math12. (ii) Without controlling for any factors, what is the difference in average test scores between those attending a Catholic high school and those not? Is the difference practically large? Is it statistically significant? (iii) Using the regression where mothedue,jathedue, and!famine appear linearly, what is the estimated effect of attending a Catholic high school? How does it compare with the estimate in part (ii)? Explain what is happening. (iv) In the same model from part (iii), interpret the coefficient on!famine. Does the income effect seem large? (v) In the model where!famine appears as a quadratic, would you say that family income no longer has an important effect on math12? Explain. (vi) Does it appear that the effect of attending a Catholic high school depends on the level ofmother's education? (vii) There are two regressions that contain interactions between mother's education with eathhs. How come they give such different estimates ofthe coefficient on eathhs? (viii) In the last regression command, what is the purpose ofthe "robust" qualifier? 4

5 des math12 cathhs female motheduc fatheduc lfaminc storage display value variable name type format label variable label math12 float %9.0g mathematics standardized score cathhs float %9.0g =1 if attended Catholic HS female float %9.0g female student motheduc float %9.0g mother~s ed, continuous variable fatheduc float %9.0g father"s ed, continuous variable lfaminc float %9.0g log of family income sum math12 cathhs female motheduc fatheduc Ifaminc Variable Obs Mean Std. Dev. Min Max math12 I cathhs I female I motheduc I fatheduc I lfaminc I reg math12 cathhs F( I, 7442) Model I Prob > F Residual I R-squared Adj R-squared Total I Root MSE math12 I Coef. Std. Err. t P>I t I [95% Conf. Interval cathhs I _cons I

6 . reg math12 cathhs motheduc fatheduc Ifaminc F( 4, 7439) Model I Prob > F Residual I R-squared Adj R-squared Total I Root MSE math12 I Coef. Std. Err. t P>I t I [95% Conf. Interval cathhs I motheduc I fatheduc I Ifaminc I cons I gen Ifamincsq = lfaminc A 2 reg math12 cathhs motheduc fatheduc lfaminc Ifamincsq F( 5, 7438) Model I Prob > F Residual I R-squared Adj R-squared Total I Root MSE math12 I Coef. Std. Err. t P>ltl [95% Conf. Interval cathhs I motheduc I fatheduc I Ifaminc I Ifamincsq I cons I test Ifaminc Ifamincsq 1) lfaminc = 0 2) lfamincsq 0 F( 2, 7438) Prob > F

7 gen motheduc cathhs = motheduc*cathhs reg math12 cathhs motheduc motheduc cathhs fatheduc l=aminc F( 5, 7438) Model I Prob > F Residual I R-squared Adj R-squared Total I Root MSE math12 I Coef. Std. Err. t P>ltl [95% Conf. Interval cathhs I motheduc I motheduc cathhs I fatheduc I lfaminc I cons I sum motheduc Variable lobs Mean Std. Oev. Min Max motheduc I gen motheduco cathhs = (motheduc )*cathhs reg math12 cathhs motheduc motheduco cathhs fatheduc lfaminc F( 5, 7438) Model I Prob > F Residual I R-squared Adj R-squared Total I Root MSE math12 I Coef. Std. Err. t P>ltl [95% Conf. Interval cathhs I motheduc I motheduco cathhs I fatheduc I Ifaminc I cons I

8 reg math12 cathhs motheduc motheduco cathhs fatheduc Ifaminc, robust Linear regression Number of obs 7444 F( 5, 7438) Prob > F R-squared Root MSE I Robust math12 I Coef. Std. Err. t P>ltl [95% Conf. Interval cathhs I motheduc I motheduco cathhs I fatheduc I Ifaminc I cons I

9 6. (12 points) Let y be an n x 1 vector of dependent variables, with tth entry y t, and let X be an n x k matrix of regressors. Consider the standard linear model written as where Xl is n x kl, X2 is n x k 2, and u is the n x 1 vector of errors. We think that X2 is correlated with u and so we apply instrumental variables. Let Z2 be another n x k2 matrix, and we think E(Z~u) = O. Define the matrix ofinstruments as Z = (XI,Z2) and assume that Z'X is nonsingular (has rank k). The IV estimator is (i) Show that the IV estimators, written in terms of ~l and ~2' solve the set of equations (ii) Use part (i) to show that X'IXI~I + X;X2~2 X'IY I A, "" I Z2XI Z2X2J3 2 = Z2Y (iii) Consider the following procedure to estimate 13 2, (1) Run a matrix regression of Z2 on X I and save the residuals, Z2. (2) Run a matrix regression ofx2 on X I and save the residuals, X2 (3) Use Z2 as instruments for X2 in the equation Call the estimator from step (3) Pz' Show that P 2 = ~2' 9