On Uniform Inference in Nonlinear Models with Endogeneity

This paper explores the uniformity of inference for parameters of interest in nonlinear models with endogeneity. The notion of uniformity is fundamental in these models because due to potential endogeneity, the behavior of standard estimators of these parameters is shown to vary with where they lie in the parameter space. Consequently, uniform inference becomes nonstandard in a fashion that is loosely analogous to inference complications found in the unit root and weak instruments literature, as well as the models recently studied in (Andrews and Cheng 2012a), (Andrews and Cheng 2012b) and (Chen, Ponomareva, and Tamer 2011). We illustrate this point with models widely used in empirical work. Our main example is the standard sample selection model, where the parameter is the intercept term. ((Heckman 1990), (Andrews and Schafgans 1998) and (Lewbel 1997a)). We show that with selection on unobservables, asymptotic theory for this parameter is not standard in terms of there being nonparametric rates and non-gaussian limiting distributions. In contrast if the selection is on observables only, rates and asymptotic distribution are standard, and consequently, there is a discontinuity in the limiting distribution theory for an estimator despite it being uniformly consistent. We show that this discontinuity prevents standard inference procedures from being globally uniformly valid, and this motivates the development of new inference procedures that are proven to be (locally) uniformly valid under drifting parameter asymptotics. Finite sample properties of the new inference procedure is illustrated through a simulation study as well an empirical illustration using the (Mroz 1987) data set. It is also illustrated how the new inference procedure can be useful in other nonlinear models with endogenous variables. JEL Classification: C12,C13,C14,C15