Lecture 12. Set Estimation and Inference in Moment Condition Models 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].• The notes cover the paper “Estimation and Inference on Parameter Sets in Econometric Models” by C hernozhukov, Hong, and Tamer (Econometrica, 2007). • Consider a population criterion function Q(θ) � 0. An economic model θ ∈ Θ ⊂ Rk passes empirical restrictions if Q(θ) = 0. Denote the set of parameters that pass these restrictions as ΘI. That is, ΘI = {θ ∈ Θ : Q(θ) = 0} = arg min Q(θ). θ∈Θ ΘI will be called the identified set. • Q(θ) typically embodies moment restrictions arising from economic theory and other considerations. In particular, moment inequality and equality restrictions lead to objective functions Q(θ) of GMM type. Concrete examples follow. 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].• Our goal is to provide set estimators Θ�I based on Qn(θ) that are 1) consistent, 2) converge to ΘI at the fastest rate, 3) have confidence interval property, and 4) computationally tractable. • These results extend the classical theory for the case when ΘI is a singleton. 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].Model I: Moment Inequality Problems The moment restrictions are computed with respect to the probability l aw P of the data and take the form EP[mi(θ)] � 0, (1) where mi(θ) = m(θ, wi) is a vector of moment functions parameterized by θ and determined by a vector of real random variables wi. Therefore the set of parameters θ that pass the testable restrictions is given by ΘI = {θ ∈ Θ : EP[mi(θ)] � 0}. It is interesting to comment on the structure of the set ΘI in this model. When the moment functions are linear in parameters, the set ΘI is given by an intersection of linear half-spaces and could be a triangle, trapezoid, or a polyhedron, as in Examples 1 and 2 introduced below . When moment functions are non-linear, the set ΘI is given by an intersection of nonlinear half-spaces 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].which boundaries are defined by nonlinear manifolds. The set ΘI can be characterized as the set of minimizers of the criterion function Q(θ) := �EP[mi(θ)]+W1/2(θ)� 2 , a (2) where W (θ) is a continuous and positive definite for each θ ∈ Θ. Therefore, the inference on ΘI may be based on the empirical analog of Q: Qn(θ) := �En[mi(θ)]+� Wn 1/2(θ)� 2 , En[mi(θ)] = 1 n�mi(θ), (3) n i=1 where Wn(θ) is a uniformly consistent estimate of W (θ). In applications Wn(θ) is often taken to be an identity matrix or chosen to weight the individual empirical moments by estimates of inverses of their individual variances. The modified objective function (θ) = inf (θ�)Q�nQn(θ) −θ�∈Θ Qn5 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].�is another useful analog of Q(θ) for inference. This analog mimics quasi-likelihood ratio statistic more closely, and thus improves power, when infθ�∈Θ Qn(θ�) = 0 in finite samples. 6 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 1: Interval Data. The first simplest example is motivated by missing data problems, where Y is the unobserved real random variable bracketed below by Y1 and above by Y2. Let {(Y1i, Y2i), i = 1, ..., n}, be an i.i. d. seq uence of real random variables with law P on Rd . The parameter of i nterest θ = EP[Y ] is known to satisfy the restriction EP[Y1] � θ � EP[Y2]. The identified set is therefore given by an interval ΘI = {θ : EP[Y1] � θ � EP[Y2]}. This example falls in the moment-inequality framework with moment function mi(θ) = (Y1i − θ, θ − Y2i)� . Therefore, set ΘI can be characterized as the set 7 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].of minimizers of Q(θ) = [mi(θ)]� 2 = (EP[Y1i]−θ)2 +(EP[Y2i]−θ)2 ,�En+ + −with the sample analog Qn(θ) = (En[Y1i] − θ)2 + (En[Y2i] − θ)2 .+ −8 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 2. Interval Outcomes in Regression Models. A regression generalization of the previous basic ex ample is immediate. Suppose a regressor vector Xi is available, and the conditional mean of unobserved Yi is modelled using linear function Xi�θ. The parameters of this function can be bounded using inequality EP[Y1i|Xi] � Xi�θ � EP[Y2i|Xi]. These conditional restrictions can be converted to unconditional ones by considering i nequaliti es EP[Y1iZi] � θ�EP[XiZi] � EP[Y2iZi], where Zi is a vector of positive transformations of Xi, for instance, Zi = {1(Xi ∈ Xj), j = 1, ..., J}, for a suitable collection of regions Xj. The identified set is therefore given by an intersection of linear half-spaces i n Rd . This examples