UntitledGMM Estimation and Testing Whitney Newey October 2007 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].Idea: Estimate parameters by setting sample moments to be close to population counterpart. Definitions: β : p × 1 parameter vector, with true value β0. gi(β)= g(wi,β):m × 1 vector of functions of ith data observation wi and paramet Model (or moment restriction): E[gi(β0)] = 0. Definitions: ndef X gˆ(β) = gi(β)/n : Sample averages. i=1 Aˆ: m × m positive semi-definite matrix. GMM ESTIMATOR: βˆ=argmingˆ(β)0Aˆgˆ(β). β 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].GMM ESTIMATOR: βˆ=argmingˆ(β)0Aˆgˆ(β). β Interpretation: Choosing βˆso sample moments are close to zero. ˆFor kgk ˆ=qg0Ag, same as minimizing kgˆ(β) − 0k ˆ.A AWhen m = p,the βˆwith gˆ(βˆ)=0will be the GMM estimator for any AˆWhen m>p then Aˆmatters. Method of MomentsisSpecial Case: Moments : E[yj]=hj(β0), (1 ≤ j ≤ p), Specify moment functions : gi(β)=(yi − h1(β),...,yip − hp(β))0, Estimator :ˆg(βˆ)=0same as yj = hj(βˆ), (1 ≤ j ≤ p). 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].Two-stage Least Squares as GMM: Model : yi = Xi0β + εi,E[Ziεi]=0, (i =1,...,n). Specify moment functions : gi(β)=Zi(yi − Xi0β). nX Sample moments are :ˆg(β)= Zi(yi − Xi0β)/n = Z0(y − Xβ)/n i=1 Z =[Z1,...,Zn]0,X =[X1,...,Xn]0,y =(y1,...,yn)0. Specify distance (weighting) matrix : Aˆ=(Z0Z/n)−1 . GMM estimator is 2SLS : βˆ=argmin[(y − Xβ)0Z/n](Z0Z/n)−1Z0(y − Xβ)/nβ =argmin(y − Xβ)0Z(Z0Z)−1Z0(y − Xβ). β 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].Intertemporal CAPM Nonlinear in parameters. ci consumption at time i, Ri is asset return between i and i+1, α0 is time discount factor, u(c, γ0) utility function; Ii is variables observed at time i; Agent maximizes X∞α−jE[0 u(ci+j,γ0)] j=0 subject to intertemporal budget constraint. 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].First-order conditions are E[Ri · α0 · uc(ci+1,γ0)/uc(ci,γ0)|Ii]=1. Marginal rate of substitution between i and i +1 equals rate of return (in expected value). Let Zi denote a m × 1 vector of variables observable at time i (like lagged con-sumption, lagged returns, and nonlinear functions of these). Moment function is gi(β)= Zi{Ri · α uc(ci+1,γ0)/uc(ci,γ0) − 1}.· Here GMM is nonlinear instrumental variables. Empirical Example: Hansen and Singleton (1982, Econometrica). 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].Dynamic Panel Data It is a simple model that is important starting point for microeconomic (e.g. firm investment) and macroeconomic (e.g. cross-country growth) applications is E[yit|yi,t−1,yi,t−2,...,yi0,αi]=β0yi,t−1+ αi, αi is unobserved individual effect. Microeconomic application is firm investment (with additional covariates). Macroeconomic is cross-country growth equations. Let ηit = yit − E[yit|yi,t−1,...,yi0,αi]. E[yi,t−jηit]=0, (1 ≤ j ≤ t, t =1, ..., T ), E[αiηit]=0, (t =1,...,T ). 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].∆ denote first difference, i.e. ∆yit = yit − yi,t−1, so ∆yit = β0∆yi,t−1+ ∆ηit. Then E[yi,t−j(∆yit − β0∆yi,t−1)] = 0, (2 ≤ j ≤ t, t =1,...,T ). IV moment conditions. Twice (and more) lagged levels of yit canbeusedas instruments for differenced equations. Different instruments for different residuals. Additional moment conditions from orthogonality of αi and ηit.They are E[(yiT − β0yi,T −1)(∆yit − β0∆yi,t−1)] = 0, (t =2, ..., T − 1). These are nonlinear. 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].Combine moment conditions by stacking. Let ⎞⎛ yi0 ⎜⎝ ⎟⎠ g it(β)= . . (∆yit − β∆yi,t−1), (t =2,...,T ) . , yi,t−2 ⎛ ⎞ ∆yi2−.β∆yi1 . . ⎜⎝ ⎟⎠ g iα(β)= .(yiT − βyi,T −1)∆yi,T −1− β∆yi,T −2 These moment functions can be combined as gi(β)=(gi 2(β)0, ..., g iT (β)0,giα(β)0)0. Here there are T (T − 1)/2+(T − 2) moment restrictions. Ahn and Schmidt (1995, Journal of Econometrics) show that the addition of the nonlinear moment condition giα(β) to the IV ones often gives substantial asymptotic efficiency improvements. 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].Arellano and Bond approach also better is small samples: ⎞⎛ git(β)= ⎜⎜⎜⎝ ∆yi,1 ∆yi,2 . . . ∆yi,t−1 ⎟⎟⎟⎠ (yit − βyi,t−1) Assumes that have representation X X∞ ∞=yit atjyi,t−j + btjηi,t−j + Cαi. j=1 j=1 Hahn, Hausman, Kuersteiner approach: Long differences ⎞⎛ gi(β)= ⎜⎜⎜⎝ yi0 yi2 − βyi1 . . . yi,T −1 − βyi,T −2 ⎟⎟( ( )]β⎟− − −y y y y1 0i i1iT i,T −⎠ Has better small sample properties by getting most of the information with fewer moment conditions. 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].IDENTIFICATION: Identification precedes estimation.