Midterm07 - Department of Economics ECONOMETRICS I Fall...

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ECONOMETRICS I Fall 2007 – Tuesday, Thursday, 1:00 – 2:20 Professor William Greene Phone: 212.998.0876 Office: KMC 7-78 Home page:www.stern.nyu.edu/~wgreene Office Hours: Open Email: wgreene@stern.nyu.edu URL for course web page: www.stern.nyu.edu/~wgreene/Econometrics/Econometrics.htm Midterm 1. In the classical regression model, y i = x i ′ β + ε i the least squares estimator, b LS = ( X X ) -1 X y is unbiased and consistent. The least absolute deviations estimator, b LAD = argmin Σ i | y i - x i ′ β | is consistent, but biased and inefficient (compared to b LS ). On the other hand, b LAD appears to have desirable small sample properties – e.g., a small mean squared error and a tolerable small sample bias. [10] a. Explain the terms unbiased and consistent. Does unbiased imply consistent? Does consistent imply unbiased? Explain. [5] b. Consider an estimator b MIXED that is designed to take advantage of the good properties of both estimators. We compute b MIXED as follows: (1) toss a fair coin. (Probability of HEAD exactly = probability of TAIL = 0.5.) (2) If HEADS, b MIXED = b LS . If TAILS, b MIXED = b LAD . Is b MIXED unbiased? Prove your answer. Is b MIXED consistent? Prove your answer. Department of Economics
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. Suppose (y,x) have a bivariate normal distribution in which E[y] = 0, E[x] = 0, Var[y] = 1, Var[x] = 1, Cov[x,y] = ρ . We have a random sample (y i ,x i ),i = 1,…,n. We are interested in estimating ρ which is the one unknown parameter in this distribution. [5] a. By virtue of the law of large numbers, the sample covariance between x and y can be used to estimate ρ consistently. Explain. [10] b. An alternative approach: We know that E[y|x] = α + β x where β = Cov[x,y]/Var[x] = ρ and α = E[y] - β E[x] = 0. So, y = ρ x + ε . Thus, linear regression of y on x without a constant term consistently estimates ρ . Correct? Explain. What is the asymptotic distribution of this estimator? Explain. How does this estimator differ from the one in part a? [5] c. An alert observer of part b notices that the equation there implies that x = (1/ ρ )y - (1/ ρ ) ε = δ y + δε He therefore suggests that we can also regress x on y to estimate δ , then, by virtue of the Slutsky theorem, obtain a consistent estimator of ρ by taking the reciprocal of the estimator of δ . True or false. Explain. [10] 3 . In a real election case in Pennsylvania, it was alleged that the absentee ballots in a certain state senators race had been tampered with. Orley Ashenfelter (the same Orley Ashenfelter who studied twins in Twinsburg with Alan Krueger) was asked to analyze the data to help the judge decide what to do with the election results. On the basis of a regression of 21 previous elections absentee ballots totals on the corresponding machine ballot totals, Ashenfelter formed a prediction interval for this absentee ballot total and determined that it looked like an outlier (statistically outside the expected range). Detail precisely the computations done for this
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This note was uploaded on 01/05/2012 for the course B 30.3351 taught by Professor Williamgreene during the Fall '11 term at NYU.

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Midterm07 - Department of Economics ECONOMETRICS I Fall...

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