Least Squares Asymptotics Portlad State University最小二乘估计波特兰州立大学.pptVIP

Least Squares Asymptotics Convergence of Estimators: Review Least Squares Assumptions Least Squares Estimator Asymptotic Distribution Hypothesis Testing Convergence of Estimators Let be an estimator of a parameter vector q based on a sample of size n. is a sequence of random variables. is consistent for q if  Convergence of Estimators A consistent estimator is asymptotically normal ifSuch an estimator is called ?n-consistent. The asymptotic variance is derived from the variance of the limiting distribution of  Convergence of Estimators Delta Method:?n(qn-q) ?d N(0,S) ? ?n[a(qn)-a(q)] ?d N(0, A(q)SA(q)′)where a(.): RK ? Rr has continuous first derivatives with A(q) defined by  Least Squares Assumptions A1: Linearity (Data Generating Process) yi = xi′b + ei (i=1,2,…,n) The stochastic process that generated the finite sample {yi,xi} must be stationary and asymptotic independent: limn?? ?i xixi′/n = E(xi xi′) limn?? ?i xi′yi/n = E(xi′yi) Finite 4th Moments for Regressors:E[(xikxij)2] exists and is finite for all k,j (=1,2,…,K). Least Squares Assumptions A2: Exogeneity E(ei|X) = 0 or E(ei|xi) = 0 E(ei|X) = 0 ? E(ei|xi) = 0 ? E(xikei) = 0. A2’: Weak Exogeneity E(xikei) = 0 for all i=1,2,…,n; k=1,2,…K. It is possible that E(xjkei) ? 0 for some j,k and j?i. Define gi = xiei = [xi1,xi2,…,xiK]ei, thenE(gi) = E(xiei) = 0 What is Var(gi) = E(gigi′) = E(xixi′ei2)? Least Squares Assumptions We assume limn?? ?i xiei/n = E(gi) limn?? ?i xixi′ei2/n = E(gigi′) E(gi) = 0 (A3) Var(gi) = E(gigi′) = Ω is nonsingular CLT states that Least Squares Assumptions A3: Full Rank E(xixi′) = Sxx is nonsingular. Let Q = ?i xixi′/n = X′X/n limn??Q = E(xixi′) = Sxx (A1) Therefore Q is nonsingular (no multicolinearity in the limit). Least Squares Assumptions A4: Spherical Disturbances Homoscedasticity and No Autocorrelation Var(e|X) = E(ee|X) = s2In Var(gi) = E(xixi′ei2) = s2X′X CLT: Least Squares Assumptions A4’: Non-Spherical Disturba