计量经济分析（第六版）答案 Midterm05-answers.doc

Department of Economics Department of Economics ECONOMETRICS I Fall 2005 – Tuesday, Thursday, 1:0 Professor William Greene Phone: 212.998.0876 Office: KMC 7-78 Home page:/~wgreene Office Hours: Open Email: wgreene@ URL for course web page: /~wgreene/Econometrics/Econometrics.htm Midterm 1. In the classical regression model, yi = xi1??1 + xi2??2 + ?i, i = 1,…,n, E[εi|X] = 0, Var[εi|X]=σ2; X1 is K1 variables and X2 is K2 variables. There are two possible estimators of β1, the first K1 coefficients in the “long regression” of y on X1 and X2 and the K1 coefficients in the short regression of y on X1. Let X = [X1,X2]. We will assume that plim[(1/n)X?X] = Q a. [5 points] Assume that plim[(1/n)X1?X2] ≠0. Is either estimator unbiased? Is either estimator consistent? The long regression estimator is unbiased and consistent in all cases. We showed unbiasedness early on – it doesn’t depend on X1?X2. We need plim(1/n)X?? = 0 for consistency, and we have plim[(1/n)X?X] = Q in the problem. The short regression is biased and inconsistent. We showed in class, when you leave variables out of a regression, the estimator is b1 = ?1 + (X1’X1)-1X1’X2 + (X1’X1)X1’? As long as X1?X2 is not zero, the short estimator is biased. As for consistency, even though plim(1/n)X?? would imply plim(1/n)X1?? = 0, the middle term is (divide then multiply by n) is not going to go away. The second term doesn’t go to zero, so the short regression estimator is inconsistent. b. [5 points] Assume that plim[(1/n)X1?X2] = 0. Is either estimator unbiased? Is either estimator consistent? Using the results above, the long regression estimator is still unbiased and consistent. The short regression is still biased. We didn’t assume that X1?X2 = 0. But the bias goes away as X1’X2 goes to zero, so it is consistent. c. [5 points] Explain the difference between consistency and unbiasedness. Does either imply the other? Explain. Done in detail in cl