GMM Estimation and Testing II 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].Hansen, Heaton, and Yaron (1996): In a Monte Carlo example of consumption CAPM, two-step optimal GMM with with many overidentifying restrictions is bi-ased. Continuously updated GMM estimator (CUE) is much less biased. CUE: Let Ωˆ(β)= P i gi(β)gi(β)0/n. βˆCUE =argmingˆ(β)0Ωˆ(β)−1gˆ(β). β LIML analog. Altonji and Segal (1996): In Monte Carlo examples of minimum distance estimation of variance matrix parameters, two-step optimal GMM with with many overidenti-fying restrictions is biased. GMM with an identity weighting matrix is much less is biased. Give some theory that explains these results. 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].Higher-order Bias and Variance Stochastic Expansion: Estimators that come from smooth (four times continuously differentiable) moment conditions have an expansion of the form n 1/2(βˆ− β0)= ψ˜+ Q1(˜a)/n1/2+ Q2(˜a, ˜ψ, ˜ ψ, ˜ b)/n + Op(1/n3/2), where Q1 and Q2 arelinearineachargument, aresmooth, ψ(z), a(z), b(z), are mean zero random vectors, and ψ˜= P iψ(zi)/n1/2 , a˜ = P ia(zi)/n1/2 , ˜P b = ib(zi)/n1/2 . An approximate bias is given by Bias(βˆ)= E[Q1(˜a)]/n = E[Q1(ψ(zi),a(zi))]/n.ψ, ˜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].Simple example: nX βˆ= r(¯z),r(z) smooth, z¯ = zi/n. i=1 GMM estimator with g(z, β)=: Expand around μ = E[zi];with β0= r(μ): √n(βˆ− β0)= r0(μ)√n(¯z − μ)+r00(μ)n(¯z − μ)2/2√n +r000(μ)n 3/2(¯z − μ)3/6n +[r000(˜z) − r000(μ)]n 3/2(¯z − μ)3/n where z˜ is between z¯ and μ.Here ψ(z)= a(z)= b(z)= z − μ, Q1(ψ, a)= r00(μ)ψa/2. Bias is Bias(βˆ)=a00(μ)Var(zi)/2n. 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].Approximation works quite well in describing how bias depends on number of mo -ment conditions. Breaks down if identification is very, very weak. This is where bias formula for 2SLS comes from. Can also get a variance approximation, though does not work as well. Can interpret as the bias of an approximating distribution, i.e. there is a precise sense in which dropping the higher order terms is OK. Can use this approximation to select moments to minimize higher-order mean square error; Donald and Newey (2001). Can also use it to compare different GMM esitmators, like CUE. 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].Bias of GMM: Recall GMM notation. Let gi = gi(β0),Gi = ∂gi(β0)/∂β, Ω = E[gigi0],G = E[Gi], Σ =(G0Ω−1G)−1,H = ΣG0Ω−1,P = Ω−1− Ω−1GΣG0Ω−1 a =(a1,...,ap),aj ≡ tr(ΣE[∂2 gij(β0)/∂β∂β0])/2 Bias for GMM has three parts: Bias(βˆGMM)=BG + BΩ + BI, BG = −ΣE[Gi0Pgi]/n, BΩ = HE[gigi0Pgi]/n, BI = H(−a + E[GiHgi])/n. 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].Bias(βˆGMM)= BG + BΩ + BI, BG = −ΣE[Gi0Pgi]/n, BΩ = HE[gigi0Pgi]/n, BI = H(−a + E[GiHg Interpretation: BG is bias from estimating G, BΩ is bias from estimating Ω,and BI is bias for GMM estimator with moment functions G0Ω−1gi(β). BG : Comes from correlation of Gi and gi; endogeneity is the source; Example: gi(β0)= Zi(yi − Xi0β),Gi = −ZiXi0,gi = Ziεi; Ω = σε2E[ZiZi0],³ ´ E[Gi0Pgi]= E[Xiε]E[Zi0PZi]=E[Xiε]tr PE[ZiZi0] = E[Xiε]σ−ε 2(m − p), under homoskedasticity. E[Gi0Pgi] nonzero due to correlation of Xi and εi.Grows at same rate as m. C onsistent with large biases in Hansen, Heaton, and Yaron (1996). BΩ : Zero if third moments zero, e.g. E[εi3|Zi]=0in IV setting. Nonzero otherwise, generally grows with m (certainly magnitude of gi0Pgi does). Consistent 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].with biases in Altonji and Segal (1996) (where Gi is constant so BG =0), where gi includes is covariance moment condittions Continuous updating GMM (CUE): βˆCUE =arg mingˆ(β)0Ωˆ(β)−1gˆ(β). β Bias of CUE: Bias(βˆCUE)= BΩ + BI. Eliminates bias due to endogeneity and estimation of Jacobian in optimal linear combination of moments. Explains Hansen, Heaton, and Ya ron (1996). CUE is a Generalized Empirical Likelihood (GEL) estimators. All GEL estimators eliminate BG; some also eliminate BΩ. GEL: For concave function ρ(v) with domain an open interval containing zero. nX βˆGEL =arg minsup ρ(λ0gi(β)). β λi=1 Computation: Concentrate out λ, using analytical derivatives for 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].Special cases of GEL: CUE: Any quadratic ρ(v) (i.e. ρ(v)=A + Bv + Cv2), Empirical Likelihood (EL): For ρ(v)=ln(1 v), −n n nXXX βˆ=arg max ln πi, πi =1, πigi(β)=0. β,π1,...,πn i=1 i=1 i=1 Exponential Tilting (ET): For ρ(v)=− exp(v), n n nβˆ=arg min πi ln πi, πi =1, πigi(β)=0. X β,π1,...,πn i=1 i=1 i=1 Bias for GEL: F or ρj(v)=∂jρ(v)/∂vj,ρj = ρj(0),ρ0=0, ρ1= −1,ρ2= −1, X Bias(βˆGEL)=BI +(1+ ρ3)BΩ. 2For EL, ρ3= ∂3ln(1 − v)/∂v3= −2, so Bias(βˆEL)=BI. X Also, ELishigher-orderefficient. Higher-order variance smaller than direct bias correction. Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT