Download Econometrics Problem Set 5: Consistency and Asymptotic Normality of Estimators and more Assignments Economics in PDF only on Docsity! Economics 203C Moshe Buchinsky Introduction to Econometrics—System Models Department of Economics Spring, 2006 UCLA Problem Set 5 Due: Monday May 22, 2006 Question 1: Consider the Generalized Method of Moments (GMM) estimator defined in Lecture Note 10: bθn = argmin θ∈Θ mn (θ) 0 V −1n mn (θ) , where Vn p−→ V, a non-stochastic non-singular matrix mn (θ) = 1 n nX i=1 ϕ (yi, xi; θ) , and E0 [ϕ (yi, xi; θ0)] = 0. a. Show that bθn is a consistent estimator for θ0. b. Show that bθn is asymptotically normal, that is √n³bθn − θ0´ D−→ N (0, V ). Note: The proofs should follow the same steps as for the proof of consistency for the MLE. In each step make sure to state the exact assumption(s) needed for the statement(s) made. Question 2: Consider the following model: y∗i = z 0 iγ + ui, for i = 1, ...n, where ui conditional on zi has a normal distribution, that is, ui|zi ∼ N ¡ 0, σ2v ¢ . Define yi = ½ 1 if y∗i > 0, 0 otherwise. a. Show that Pr (yi = 1|zi) = Φ (z0iγ/σv). Explain why γ and σv cannot be separately identified. Denote θ = γ/σv, and let θ0 be the population parameter. b. Provide the normalized log likelihood function for θ. c. Provide the first order conditions for bθn, the estimator for θ0. d. Show that the estimator obtained by solving the first order conditions in (c) can be viewed as a Method of Moments estimator. 1 Question 3: Consider the linear model given by yi = x 0 iβ0 + εi E [εi|xi] = 0, for i = 1, ...n, where xi is a K × 1 vector of regressors, that is, xi = (x1i, ..., xKi)0. Define the vector zi = ¡ x0i, x 2 1i, ..., x 2 Ki, x1ix2i, ..., x1ixKi ¢0 , and the following moment functions: ϕ1 (yi, xi;β) = ¡ yi − x0iβ ¢ xi, and ϕ2 (yi, zi;β) = ¡ yi − x0iβ ¢ zi. a. Define the population parameter vector β0. b. Show that when evaluated at the population value β0, E0 [ϕ1 (yi, xi;β0)] = 0, and E0 [ϕ2 (yi, zi;β0)] = 0. c. Define the optimal GMM estimators for β0 based on ϕ1 (yi, xi;β) and ϕ2 (yi, zi;β), saybβ1n, and bβ2n, respectively. d. Provide the asymptotic covariance matrices for bβ1n and bβ2n, from (c). e. Provide a consistent estimator for the asymptotic covariance matrix of bβ2n from (d). Show that the proposed estimator is, in fact, consistent estimator for its population counterpart, using the assumptions put forward in Lecture Note 10. f. Provide the asymptotic distribution for the GMM estimator based on ϕ2 (yi, zi;β) when the weight matrix is Vn = I. Question 4: For this exercise we use the results of question 3 above. In the excel file ps5q4.xls you are provided with the data for this exercise. There are six variables in columns 1 through 6 of the file, corresponding to y, x1, ..., x5, respectively, where x1i = 1 for all i = 1, ..., n. a. The estimator in this part is based on ϕ1 (yi, xi;β) from Question 3. 1. Provide an estimate for β0 and the corresponding standard errors, based on Vn = I. 2. Provide the optimal estimate for β0 and the corresponding standard errors, based on the optimal matrix Vn. 3. Compare the two estimates obtained in (a.1) and (a.2). b. Repeat the exercise in (a) for ϕ2 (yi, zi;β). 2
