Semiparametric Models and Estimators 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].Semiparametric Models Data: Z1,Z2,... i.i.d. Model: F aset of pdfs. Correct specification: pdf f0 of Zi in F. Parametric model: F = {f(z|θ):θ ∈ Θ}, Θ is finite dimensional. Semiparametric model: F is not finite dimensional but has finite dimensional com -ponents. 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].Ex: Probit, Z =(Y,X), Y ∈ {0, 1}, Pr(Y =1|X)=Φ(X0β0) F = {Φ(x0β)y[1 − Φ(x0β)]1−yh(x):β ∈ B, h is pdf of X} β is parametric component, nonparametric component is h(X). Ex: Linear model Z =(Y, X),E[Y |X]=X0β0; F = {f(z):Ef[Y |X]=X0β}parametric component is β, everything else nonparametric. β depends on f. Probit pdf is an explicit function of a parametric component β and a nonparametric component h(x). Linear model parameters imposes constraints on pdf. 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].Binary Choice with Unknown Disturbance Distribution: Z =(Y, X);Let v(x, β) be a known function, Y =1(Y ∗ > 0),Y ∗ = v(X, β0) − ε, ε independent of X, This equation implies that for the CDF G(t) of ε Pr(Y =1|X)=G(v(X, β0)), Here the parameter is β and everything else is nonparametric. T he model can be written as an explicit function of the parametric component β and two nonpara-metric components. F = . The v(x, β) notation allows location and scale normalization, e.g. v(x, β)= x1+ x02β, x =(x1,x02)0, x1 scalar. Generalization to nonmonotonic G( ).·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].Censored Regression with Unknown Disturbance Distribution: Z = (Y,X), Y =max{0,Y ∗},Y ∗ = X0β0+ ε, med(ε|X)=0; Parameter β, everything else, including distribution of ε, is nonparametric. F = . Binary choice and censored regression are limited dependent variable models. Semi-parametric models are important here because misspecifying the distribution of the disturbances leads to inconsistency of MLE. 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].Partially Linear Regression: Z =(Y,X, W ), E[Y |X, W ]=X0β0+ g0(W ). Parameter β, everything else nonparametric, including additive component of re-gression. Can help with curse of dimensionality, with covariates X entering para-metrically. In Hausman and Newey (1995) W is log income and log price, and X includes about 20 time and location dummies. X may be variable of interest and g0(Z) some covariates, e.g. sample selection. F = . 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].Index Regression: Z =(Y,X), v(x, β) aknown function, E[Y |X]=τ(v(X, β0)), where the function τ( ) is unknown. Binary choice model has E[Y |X]=Pr(Y = ·1|X)=τ(v(X, β0)),with τ( )=G( ). If allow conditional distribution of ε given· ·X to depend (only) on v(X, β0), then binary choice model becomes index model. F = . 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].Semiparametric Estimators Estimators of β0. Two kinds; do and do not require nonpa rametric estimation. Really model specific, but beyond scope to say why. One general kind of estimator: nX βˆ=arg minq(Zi,β)/n, β0 =arg minE[q(Zi,β)], β∈B β∈Bi=1 B set of parameter values. Extremum estimator. Clever choices of q(Z, β) in some semiparametric models. 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].Ex: Censored regression quantiles, Powell (1984, 1986). med(Y ∗|X)= X0β0. max{0,y} is a monotonic transformation, so med(Y |X)= max{0,X0β0}, β0 =arg minE[|Yi − max{0,X0β iβ}|] Sample analog is nX βˆ=arg min|Yi − max{0,X0β iβ}|. i=1 Censored least absolute deviations estimator of Powell (1984). Only requires med(Y ∗|X)= X0β0. Allows for heteroskedasticity. 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].Consistency and Asymptotic Normality of Minimization Estimators Consistency: If i) E[q(Z, β)] has a unique minimum at β0, ii) β0 ∈ B and B is compact; iii) q(Z, β) is continuous at β with probability one; iv) E[supβ∈ B |q(Zi,β)|] < ∞ ; then βˆ=arg minβ∈ B Pin =1 q(Zi,β) p −→ β0. Well known. Allows for q(Z, β) to be discontinuous. Asymptotic Normality: (Van der Vaart, 1995). If βˆpβ0, β0 is in −→theinteriorof B,and i) E[q(Zi,β)] is twice differentiable at β0 with nonsingular Hessian H; ii) there is d(z) such that E[d(Z)2] exists and for all β, β˜∈ B, |q(Z, β˜) − q(Z, β)| ≤ d(Z)kβ˜− βk; iii) with probability one q(Z, β) is differen-tiable at β0 with derivative m(Z),thenfor Σ = E[m(Z)m(Z)0], d√ n(βˆ− β0) −→ N(0,H− 1ΣH− 1). 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].Tests and confidence intervals: Either bootstrap or estimate H−1ΣH−1 Estimating H : Plug into formula for H,or Hˆis a finite difference approximation to second derivative of P q(Zi, βˆ)/n evaluated at βˆwhere differences not too small.