Problem Set 1 MIT 14.385, Fall 2007 Due: Friday, 21 September 2007, in class This problem set emphasizes theory. Next week’s problem set will consist of empirical applica-tions of these results. Prepare very brief, precise answers. Irrelevant, lengthy explanations will be penalized by the means of negative points. State clearly any additional assumptions if needed. You are strongly encouraged to discuss the pset in groups but the final write-up should be individual. The page numbers give a rough indication how detailed your answer should be. For each problem, you can get either the full number of points for a good answer, half the points for an incomplete or only partly correct answer, or no points at all. If you receive less than the full number of points on a given problem, you may hand in a revised answer for that problem one week after you got back the problem set, and get up to 90 percent of the original score for a correct answer. See the last page for some hints. 1. Basic Consistency Exercise #1 [5 points, 1/2 page]: Consider the model yt = β0 + ut where β0 is an unknown scalar parameter and the ut are i.i.d. disturbances with E [ut] = 0 E � ut 2� = β02 True, false or uncertain: computing βˆusing the minimum distance function T � �2 � yt − β β t=1 yields a consistent estimator. Cite as: Victor Chernozhukov, 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� � � � 2. Basic Consistency Exercise #2 [5 points, 1/2 page]: Suppose log yt = β0 + xtβ1 + ut where ut ∼ N 0, σ2 i.i.d. with ut independent of xt. True, false or uncertain: if one implements NLLS (nonlinear least squares) to estimate the model yt = exp (γ0 + xtβ1) + at the resulting estimate of β1 is consistent. 3. Nonlinear Regression with a Transformation of the Dependent Variable [15 points, 1.5 pages]: Consider the empirical relation 3(yt + γ) = δ + xtβ + εt where εt ∼ 0, σ2 i.i.d., xt is a scalar exogenous variable and γ, δ, β are parameters. Evaluate the following statements as true, false or uncertain: (a) NLLS applied to this specification yields consistent estimates for δ and β but not for γ. (b) Nonlinear IV estimation using 1, xt, xt 2 as instruments produces estimates for γ, δ and β that are consistent. 4. Least Absolute Deviation [20 points, 1 page] Consider the model yt = θ1/2 + ut with ut i.i.d. and P (ut ≤ 0) = 12 . ut has a strictly positive density. Assume E [|yt − θ|] < ∞. (a) True, false or uncertain: θ1/2 is the median of yt Cite as: Victor Chernozhukov, 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]. 2� � � (b) True, false or uncertain: The LAD (least absolute deviations) estimator T1 θˆ≡ arg min |yt − θ| θ T t=1 T � � 1 1 ≡ arg min − 1 {yt < θ} (yt − θ) θ T 2 t=1 is a consistent estimator of θ1/2. Hint: use convexity and the fact that � � ∂ ∂ E [|yt − θ|] = E |yt − θ|∂θ ∂θ � � 1 = E 1 {yt < θ} − 2 ∂to check if θ1/2 is the unique optimum of the limit objective function. (Note: ∂θ |yt − θ| is not defined when yt = θ, but for a given θ, P (yt = θ) = 0, so this is unimportant.) (c) True, false or uncertain: OLS is a consistent estimator of θ1/2. (d) True, false or uncertain: (Extra points) The condition E [|yt − θ|] < ∞ is needed for consistency in (b). (e) Now consider a new the error term ˜ut , with P (˜ut ≤ 0) = τ and the model is yt = θτ + ˜ut True, false or uncertain: θτ is the τ-quantile of yt. Now let the τ-quantile estimator be T1 θˆτ ≡ arg min (τ − 1 {yt < θ}) (yt − θ) θ T t=1 Cite as: Victor Chernozhukov, 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]. 3� This is sometimes written as ˆ1 T � + − � θτ ≡ arg min τ (yt − θ) + (1 − τ) (yt − θ)θ T t=1 where x+ = max (x, 0) and x− = − min (x, 0). Be sure you understand why these are equivalent. Note that the median discussed above is a special case. (f) True, false or uncertain: θˆτ is consistent for θ. (g) Why might we ever be interested in θτ for τ �= 12 ? 5. Censored Sample – Top-Coding [25 points, 2.5 pages] Consider the model ∗ y = x �β0 + ε ; ε|x ∼ N � 0, σ02� where y ∗ is the log of income and x are the variables that should predict income, e.g. age, education, ability, etc. Suppose that high incomes are censored for confidentiality reasons. Rather than observing (yi ∗ , xi), we observe (yi ∗ , xi) if yi ∗ < L (L, xi) if yi ∗ ≥ L L is known. Another way to write this is that you only observe yi = min (yi ∗, L) (a) What is the conditional mean of y given x, i.e. E [ y | x ] Note that this is observed y, not the true y ∗ . Cite as: Victor Chernozhukov, 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]. 4(b) Suppose you run an OLS regression on the data you have. Will βˆLS be consistent for β0? Give intuition as to why or why not. (Hint: a sketch will help.) (c) Explain how the expression for the conditional mean you derived in part (a) can be used for Nonlinear Least Squares (NLS) estimation of β0. Is the NLS estimate consistent? A very brief answer suffices. (d) What is the conditional mean of y given x and that y < L, i.e. E [ y | x ; y ∗ < L ] (e) Suppose you run an OLS regression on the non-censored data, i.e. OLS on (yi ∗ , xi) for ∗all i such that yi < L. Will βˆLS be consistent for β0? Give intuition as to why or why not. (Hint: a sketch will help.) (f) What is the (conditional) likelihood function of y given the observed x? Is the MLE consistent (a brief answer suffices)? Give the large sample distribution of the MLE. (g) (Truncated Sample) Now consider the model ∗ y = x �β0 + ε ; ε|x ∼ N � 0, σ02� where y ∗ is the log of income and x are the variables that should predict income, e.g. age, education, ability, etc. Suppose that high