A Note on Nonparametric Estimation With Constructed Variables and Generated Regressors

This article gives the asymptotic properties for nonparametric kernel based density and regression estimators when one of the variables, respectively regressors, had to be pre-estimated. Those variables are known as constructed variables or generatedregressors, and their impact on the -nal estimator is well studied in the fully para-metric context. The problem of making inference based on predicted rather than on observed values is quite frequent in econometrics and applied economics. The results are derived in such a way that the pre-estimation steps could be performed by any con-sistent nonparametric or parametric method. The case of parametric estimation with nonparametric predictors is discussed, as well. In most cases it is obvious and mathematically straightforward how to extend the results to semiparametric models or to other nonparametric smoothing methods. We also study the performance of nonparametric estimators with constructed variables by simulations and compare the numerical to our theoretical results