A note on non-parametric estimation with predicted variables

This article gives the asymptotic properties of non-parametric kernel-based density and regression estimators when one of the variables is predicted. Such variables, also known as 'constructed variables' or 'generated predictors', occur quite frequently in econometric and applied economic analysis. The impact of using predicted rather than observed values on the properties of estimators has been extensively studied in the fully parametric context. The results derived here are applicable to the general situation in which the predictor is estimated using a consistent non-parametric method with standard convergence rates. Therefore, the presented results are, generally speaking, the asymptotics for semi-non-parametric two-step (or plug-in) estimation problems. The case of parametric estimation based on non-parametric predictors is also covered.