Local nonlinear least squares: Using parametric information in nonparametric regression

We introduce a new nonparametric regression estimator that uses prior information on regression shape in the form of a parametric model. In effect, we nonparametrically encompass the parametric model. We obtain estimates of the regression function and its derivatives along with local parameter estimates that can be interpreted from within the parametric model. We establish the uniform consistency and derive the asymptotic distribution of the local parameter estimates and of the corresponding regression and derivative estimates. For estimating the regression function our method has superior performance compared to the usual kernel estimators at or near the parametric model. It is particularly well motivated for binary data using the probit or logit parametric model as a base. We include an application to the Horowitz (1993, Journal of Econometrics 58, 49–70) transport choice dataset.