On uniform inference in nonlinear models with endogeneity

This paper explores the uniformity of inference for parameters of interest in nonlinear econometric models with endogeneity. Here the notion of uniformity arises because the behavior of estimators of parameters of interest is shown to vary with where either they or nuisance parameters lie in the parameter space. As a result, 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 (2012), Chen et al. (2014), Han and McCloskey (2019). Our main illustrative example is the standard sample selection model, where the parameter of interest is the intercept term as in Heckman (1990), Andrews and Schafgans (1998) and Lewbel (2007). We show here there is a discontinuity in the limiting distribution for an estimator of this parameter despite it being uniformly consistent. This discontinuity prevents standard inference procedures from being valid, and motivates the development of new methods, for which we establish asymptotic properties. Finite sample properties of the procedure are explored through a simulation study and an empirical illustration using the Mroz (1987) data set as in Newey, Powell, and Walker (1990).