Nonparametric Structural Estimation via Continuous Location Shifts in an Endogenous Regressor

Recent work by Wang and Phillips (2009b, c) has shown that ill posed inverse problems do not arise in nonstationary nonparametric regression and there is no need for nonparametric instrumental variable estimation. Instead, simple Nadaraya Watson nonparametric estimation of a (possibly nonlinear) cointegrating regression equation is consistent with a limiting (mixed) normal distribution irrespective of the endogeneity in the regressor, near integration as well as integration in the regressor, and serial dependence in the regression equation. The present paper shows that some closely related results apply in the case of structural nonparametric regression with independent data when there are continuous location shifts in the regressor. In such cases, location shifts serve as an instrumental variable in tracing out the regression line similar to the random wandering nature of the regressor in a cointegrating regression. Asymptotic theory is given for local level and local linear nonparametric estimators, links with nonstationary cointegrating regression theory and nonparametric IV regression are explored, and extensions to the stationary strong mixing case are given. In contrast to standard nonparametric limit theory, local level and local linear estimators have identical limit distributions, so the local linear approach has no apparent advantage in the present context. Some interesting cases are discovered, which appear to be new in the literature, where nonparametric estimation is consistent whereas parametric regression is inconsistent even when the true (parametric) regression function is known. The methods are further applied to establish a limit theory for nonparametric estimation of structural panel data models with endogenous regressors and individual effects. Some simulation evidence is reported.