Untitled14.385B Problem Set 2 Fall 2007 Due Wednesday, November 28 n1. A kernel density estimator is fˆ h(z) = �i=1 Kh(z − zi)/n, Kh(u) = h−rK(u/h) where r is the dimension of z. Consider z as fixed at some value. Since fˆ h(z) is just a sample average of a function of the data, when the data are i.i.d. we have V ar(fˆ h(z)) = n −1V ar(Kh(z − zi)). a) Give a simple estimator of V ar(fˆ h(z)) based just on the fact that fˆ h(x) is a sample average of a random variable. How could you use this to form a confidence interval for f0(z)? b) Let mˆh(z) = �in =1 yiKh(z − zi)/n. How could you estimate the joint covariance matrix of ˆmh(z) and fˆ h(z) based on them both being sample averages? c) Use part b) and the delta method to form an estimator of the asymptotic variance of the kernel regression estimator ˆgh(z) = mˆh(z)/fˆ h(z) (by treating it as a ratio of sample averages. 2. Consider panel data (yit, xit), (t = 1, 2; i = 1, ..., n), where the data are independent across i. Suppose that the data follow a nonparametric model of the form yit = g0(xit) + αi + ηit, E[ηit|xi1, xi2] = 0. a) If E[αi|xi1, xi2] = 0, how could you estimate g0(xit) from a nonparametric regression of yit on xit for all i and t? b) If E[αi|xi1, xi2] = h(xi1, xi2) � 0, what would E[yi1|xi1, xi2] − E[yi2|xi1, xi2] be? = How could you use this formula and nonparametric estimators of E[yi1|xi1, xi2] and E[yi2|xi1, xi2] to estimate g0(x) up to an additive constant ? c) Consider a series estimator obtained by regressing yi2 − yi1 on approximating func-tions of the form p1(xi2) − p1(xi1), p2(xi2) − p2(xi1), ..., pK (xi2) − pK (xi1). How could you use this series estimator to construct an estimator of g0(x) up to an additive constant? d) Which of the estimators from b) or c) do you prefer, and why? Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 1� 3. Consider the partially linear model E[y|x, w] = xβ0 + h0(w), where x is scalar. a) Show that if V ar(x|w) = 0 for all w then there is a β � β0 and a function h(w)= that is not equal to h0(w) such that E[y|x, w] = xβ + h(w). What does this imply about the ability to consistently estimate β0? b) Consider a series estimator of β obtained by regressing y on x and a K × 1 vector of approximating functions pK (w). Show that this can be written as an OLS regression of y − Eˆ[y|w] on x − Eˆ[x|w], where Eˆ[y|w] and Eˆ[x|w] are series estimators obtained from regressing y and x respectively on pK (w). c) Assume that E[yi 2] and E[xi 2] exist (implying existence of E[E[y|w]2] and E[E[x|w]2]). Also assume that pK (w) has the property that for any function g(w) with E[g(w)2] ex-isting there is γK with limK−→∞ E[{g(w) − pK (w)�γK }2] = 0. Give conditions on the growth rate of K as a function of n such that n n i=1 p{E[yi|wi] − Eˆ[yi|wi]}2/n −→ 0, � i=1 p{E[yi|wi] − Eˆ[yi|wi]}2/n −→ 0. d) Give conditions for consistency of the βˆfrom part b). 4. Let D be a binary variable and consider the nonparametric regression model E[Y |X, D] = α(X) + β(X)D. a) For a given value of δ, what function h(x, δ) minimizes E[{E[Y |X, D] − h(X) − Dδ}2]? b) Find the value of δ that minimizes E[{E[Y |X, D] − h(X) − Dδ}2] over all δ and over all functions h(x). Interpret this as a weighted average of β(X). c) Show that δ also minimizes E[{Y − h(X) − Dδ}2] over all δ and functions h(x). d) Suppose that E[Y |X, D] = α(X) + β(X)D continues to hold. What would be the probability limit of a series estimator of the coefficient of D from regressing Yi on Di and a vector of approximating functions pK (Xi) if K is allowed to grow with n in the way described in question 1) (and the other conditions from question 1) are satisfied)? 5. Using the gasoline demand data from Hausman and Newey (1995), consider a locally linear nonparametric regression estimator of the conditional expectation of the ln(q) given ln(p) and ln(y). a. Using 1983 data and Silverman’s rule of thumb for the bandwidth, plot the estimate of E[ln(q)| ln(p), ln(y)] as a function of ln(p) for two different values of ln(y). Does it appear that E[ln(q)| ln(p), ln(y)] is additively separable in ln(p) and ln(y)? b. Plot the estimate as a function of ln(p) for fixed ln(y) for three different bandwidths, Silverman’s rule, one that is 50 percent larger, and one that is 50 percent smaller. Which do you prefer and why? c. Using the 1985 data repeat a. and b. Do the results appear to depend on which data set is used? Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].