ECONOMET RICS P RELIM EXAM August 19, 2014 Department of Economics, Michigan State University

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1 ECONOMET RICS P RELIM EXAM August 19, 2014 Department of Economics, Michigan State University Instructions: Answer all ve (5) questions. Be sure to show your work or provide su cient justi cation for your answers. Unless explicitly asked, do not worry about regularity conditions such as existence of moments or di erentiability in parameters. This exam is closed book. You may use a calculator and tables of relevant distributions are provided on additional pages of the exam. 1. (20 points) A sequence of random variables fx n : n = 1; 2; :::g converges in probability to the constant a if for all " > 0 or equivalently: lim P (jx n aj > ") = 0; n!1 lim P (jx n aj ") = 1: n!1 Let fv t : t = 1; 2; :::g be a sequence of independent identically distributed random variables with E(v t ) = and E(jv t j) < 1. Then a law of large numbers holds for the sample average p lim v = p lim n 1 v t = : Let fv t : t = 1; 2; :::g be a sequence of independent identically distributed random variables with E(v t ) = 0 and E(v 2 t ) = 2 < 1. Then a central limit holds for the scaled sample average p nv = n 1=2 v d t! N(0; 2 ): The limits are taken as n! 1. Let y 1 ; y 2 ; :::; y n be a random sample from a normal population with mean and variance 2. Assume that 2 < 1. Let y denote the sample average of the y data. a) First show that y is a consistent estimator of and then calculate lim var p n(y ) : n!1 b) Show that p n(y ) converges in distribution to a mean zero normal random variable. What is the variance of the limiting normal random variable (i.e. asymptotic variance)? Compare your result for the asymptotic variance with the result you obtained in part (a) for lim n!1 var ( p n(y )). 1

2 c) De ne an alternative estimator of as e = y + z n ; where z n is a discrete random variable de ned as z n = 1 with probability n 1 ; z n = 0 with probability 1 n 1 : Assume that z n is independent of the y data. Is e a consistent estimator of? Please provide a detailed argument either way. Next calculate lim var p n(e ) : n!1 d) Compute p lim p nz n and then show that p n(e ) converges in distribution to a mean zero normal random variable. What is the variance of the limiting normal random variable (i.e. asymptotic variance)? Compare your result for the asymptotic variance with the result you obtained in part (c) for lim n!1 var ( p n(e )). Are you surprised? Why or why not? 2

3 2. (30 points) Consider the regression model given by y = X 0 +u where 0 is a k 1 vector of parameters, y and u are n 1 vectors and X is an n k matrix. Let W be an n k matrix. Assume that rank(x) = k, rank(w > X) = k, E(ujX; W) = 0 and var(ujx; W) =E(uu > jx; W) = 2 0I. If the conditions for the Gauss-Markov Theorem hold for a particular estimator of 0, you may use the Gauss-Markov Theorem without having to prove it. a) Let b be the OLS estimator of 0. Derive the bias of b and derive a formula for var( b jx; W). b) Let b IV be the instrumental variables estimator of 0 using W as instruments for X. Derive the bias of b IV and derive a formula for var( b IV jx; W). c) Compare the bias of the two estimators and compare var( b jx; W) with var( b IV jx; W). Is one estimator preferred over the other based on bias and variance? Why or why not? d) Let s 2 = 1 y X n k b > y X b be an estimator of 2 0. Determine whether nor not s 2 is an unbiased estimator of 2 0. e) An alternative estimator of 2 0 is given by s 2 IV = 1 n k y X b > IV y X b IV : Determine whether or not s 2 IV is an unbiased estimator of 2 0: f) Suppose that you found out that W is related to u and because of this E(ujX; W) 6= 0. Suppose that X and u are not related and that E(ujX) = 0 holds. Suppose that the other assumptions continue to hold. How would you adjust your answer to part (c) in this case? You can answer this question qualitatively without deriving explicit formulas for biases and conditional variances. 3

4 3. (20 points) For this question you may assume that the Gauss-Markov assumptions hold and that the population from which the data is sampled is normally distributed. Consider the following regression model to test the rationality of assessments of housing prices. Here we want to test whether the assessed house price is a rational prediction (in the sense that there does not exist systematic over- or under- prediction given available information) for the actual transaction price using data on the following variables: price = house price assess = the assessed housing value (before the house was sold) lotsize = size of the lot, in feet sqf t = square footage; bdrms = number of bedrooms: First, consider a simple regression model price t = assess t + u t ; where the assessment is rational if 1 = 0 and 2 = 1. The OLS estimated model (with values in the parenthesis being usual OLS standard errors) is d price t = 14:45 + 0:975 assess t (16:23) (0:047) n = 92; SSR = 165; 645; Rc 2 = 0:805 (price t assess t ) 2 = 209; 450; ( price d t assess t ) 2 = 58; 400 a) Using a 5% signi cance level, test the hypothesis that H 0 : 1 = 0 against the two-sided alternative. Then, test H 0 : 2 = 1 against the alternative 2 > 1. State the relevant critical values for your test statistics. b) Test the joint hypotheses H 0 : 1 = 0 and 2 = 1 at the 1% signi cance level. State the relevant critical value for your test statistic. c) Using a 5% signi cance level, test H 0 : 3 = 0; 4 = 0; and 5 = 0 in the model (1) price t = assess t + 3 lotsize t + 4 sqft + 5 bdrms t + u t 4

5 where Rc 2 from estimating this model (using the same 92 houses) is Brie y interpret the null hypotheses and the result of your test. Hint: Take the formula for the F-statistic expressed in terms of USSR and RSSR: F = (RSSR USSR) =r ; USSR=(n k) and write it in terms of the R 2 c for the restricted and unrestricted regressions. d) From the model of (1) test the joint hypotheses H 0 : 2 = 0; 3 = 0; 4 = 0; and 5 = 0 at the 5% signi cance level. State the relevant critical value for your test statistic. 4. (10 points) In this question assume that the Gauss-Markov assumptions hold and assume that any regularity conditions needed to justify the use of central limit theorems hold. For the estimation of the regression (2) y t = x 2t + 3 x 3t + u t ; t = 1; : : : ; n; suppose you have the following information: X > X = ; X> Y = ; (y t y) 2 = 200; 100 where X denotes the n 3 matrix stacking n observations of X t = (1; x 2t ; x 3t ), Y is an n 1 vector stacking n observations of y t, and y denotes the sample average of Y. a) Using a 5% signi cance level, test H 0 : 2 0:5 3 = 0:5 against the two-sided alternative. State the relevant critical value for your test statistic. b) Now suppose you have the following information for estimation of the same model (2) using a di erent sample that is independent of the original sample: (3) y t = 1 + 2x 2t + 3x 3t + u t ; t = 1; : : : ; m X > X = ; X> Y = ; X m (y t y) 2 = 100:

6 Combining information from (2) and (3), test the joint hypothesis H 0 : 2 = 2 and 3 = 3 at the 5% signi cance level. State the relevant critical value for your test statistic. Hint: In case you had forgotten, the formula for the F-statistic for testing H 0 : R = r is F = R b > r hrdvar( )R b i 1 > R b r =r; where is the vector of parameters, R and r are known matrices corresponding to the restrictions being tested, r is the number of restrictions being tested and dvar( b ) is an estimate of var( b ). 5. (20 points) Let y t be a binary variable that takes on the values of 0 and 1. Suppose we want to model the probability that y t is equal to 1 conditional on some variable, x t. The simplest model is a linear model given by P (y t = 1jx t ) = x t ; where it trivially holds that P (y t = 0jx t ) = 1 P (y t = 1jx t ). a) Show that E(y t jx t ) = P (y t = 1jx t ) = x t ; b) Using the result from part (a), we can write the regression model where E(u t jx t ) = 0. Derive the formula for y t = x t + u t var(u t jx t ); using the fact that y t is a binary random variable. c) Assuming necessary laws of large numbers hold, will b 2, the OLS estimator of 2, be a consistent estimator? Why or why not? d) Letting 2 = p lim b 2 and assuming that necessary central limit theorems hold, then p n b2 d 2! N(0; V 2 ) where V 2 is the asymptotic variance of b 2. Under what conditions, if any, will V 2 take the form V 2 = 2 0A 22 where A 22 is the lower right element of the 2 2 matrix A = (E(X > t X t )) 1 where X t = [1; x t ]? Please sketch the details of your argument. 6

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