Lecture 9: Quantile Methods 2 1. Equivariance. 2. GMM for Quantiles. 3. Endogenous Models 4. Empirical Examples 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].1. Equivariance to Monotone Transformations. Theorem (Equivariance of Quantiles under Mono-tone Transformations): Let Y = Q(U|X) be the Sko-rohod representation of Y in terms of its conditional quantile function, then for the weakly increasing trans-formation T (·|X), which is possibly X-dependent, we have T (Y |X) ≡ T [Q(U|X)|X] ≡ QT (Y |X)(U|X). Proof: The composi ti on T [Q[·|X]|X] of two mono-tone weakly increasing functions gives a weakly increas-ing monotone function with domain [0, 1]. Thus the composition is the proper quantile function. � Remark. The same is generally not true of the con-ditional expectation function: E[T (Y |X)|X] =� T [E(Y |X)|X]. Affine or location-scale transformations commute with conditional expectations, but more general transforma-tions usually do not. 2 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].Examples. 1. Y = ln W and QY |X(u) = X�β(u), then QW(u|X) = exp(X�β(u)). The same cannot be done generally for mean regression. Many standard duration models specify ln W = x�β + �, � is indep of X where W is a positive random variabl e (duration, capital stock in (S,s) models, wage). Quantile regression allows us to cover and immedia tel y generalize these models. 2. Yb = max[0, Y ] and QY |X(u) = x�β(u), then QYb[u|X] = max[0, x�β(u)]. This is censored quantile regression m odel. Estimation can be done using nonlinear quantile regression (Powell, 1984, JoE). A computationally attractive approximation to Powell’s estimator is given in Chernozhukov and Hong (2002, JASA). 3. Yc = 1{Y > 0} and QY |X(u) = x�β(u), then QYc[u|X] = 1{x�β(u) > 0}. This is the binary quantile regression or maximum score model. Estimation can be done using nonlinear quan-tile regression, known as the maximum score esti m a tor (Manski, 1975, H orow itz, 1997, Kordas, 2005). This estimator is ver y hard to compute. 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].GMM and Bayesian GMM for Q uantiles, cf. Cher-nozhukov and Hong (JoE, 2003) Moment restriction: E[u − 1(Y < X�β(u))|Z] = 0, where Z could differ from X for endogenous cases. Sample analogs of these equations are piecewise-constant functions. 1 Q�(β) = n 2g�(β)TWng�(β) with g�(β) = En[(u − 1(Yi ≤ Xi�β)�Zi] and Wn =1 �En[ZiZi�]�−1 . u(1 − u) Conventional extremum estim a tor is β∗ = arg inf Q�(β), β∈Θ but it is hard, if not impossible, to compute, except on paper. Chernozhukov and Hong (JoE, 2003) propose to use quasi-posterior mean (“Bayesian GMM”): �� e−Q�(β) � βˆ= β dβ. Θ � Θ e−Q�(β�)dβ� 3 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].Integration vi a Markov Chain Monte Carlo. Image by MIT OpenCourseWare. Theorem (Chernozhukov and Hong, 2003): Un-der regularity conditions, quasi-posterior mean and ex-tremum estimators are fi r st-order equivalent: √n(βˆ− β(u)) = √n(β∗− β(u)) + op(1) = (G�W G)−1G�W√ng�(βˆ(u)) + op(1) d −→ N(0, (G�W G)−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].where G = �βE[(u − 1(Yi ≤ Xi�β)�Zi] = EfY (X�β(u)|X, Z)ZX�, and W =1 �EZZ��−1 . u(1 − u) Remarks: 1) Estimation of G can be done analogously to estima-tion of J in QR case. 2) For practical experience with computation, see Cher-nozhukov and Hong (2003). 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. An Endogenous Quantile Regression Model Reference: Chernozhukov and Hansen (2004, 2005, 2006), Chernozhukov and Hong (2003). Empirical Applications: H a usma n and Sidak (2003), Januscewski (2003), others. Model: Y = D�α(U) + X�β(U), D = δ(X, Z , V ), U|Z, X ∼ U ni form(0, 1), u �→ D�α(u) + X�β(u) strictly increasing in u. • D�α(u) + X�β(u) is Structural Quantile Function/ Quantile Treatment R esponse Function Y is the outcome variable, X covariates, D endogenous vari a bl es, Z instruments, U outcome disturbance, V selection disturbance. • Dependence of U and V causes endogeneity. • Independence of U and Z is crucial. 4 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].Example: Demand-Supply ln Qp = α0(U) + α1(U) ln(p),· ln Sp = f(p, Z, U), (1) P ∈ {p : Qp = Sp}, U independent of Z and normalized U(0, 1). U is a demand di sturbance, “level of dema nd”; U is a supply disturbance, “level of supply”; p �→ α0(1/2)+α1(1/2) l n(p) is the median demand curve; p �→ α0(u) + α1(u) ln(p) is the u-quantile demand curve; demand elasticity α1(u) varies with u. Equilibrium quantity Y and price P wil l satisfy ln Y = α0(U) + α1(U) ln P, P = δ(Z, U, U, “sunspots”),� ��V � U is independent of Z. 5 Cite as: Victor Chernozhukov, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT