Smooth Varying-Coefficient Nonparametric Models for Qualitative and Quantitative Data∗

We propose a nonparametric varying coefficient method that admits both qualitative and quantitative regressors. The proposed estimator is exceedingly flexible and has a wide range of potential applications including hierarchical (mixed) settings, small area estimation, and so forth. For example, the method provides for semiparametric models in which practitioners can select a parametric functional form for their model but allow the parameters to change in an unrestricted fashion with respect to, say, a qualitative regressor such as group membership or perhaps with respect to a mix of qualitative and quantitative regressors. However, it also provides a range of semiparametric alternatives that other semiparametric methods cannot match, and can deliver a fully nonparametric estimator should one so desire. A data-driven cross-validatory bandwidth selection method is proposed that can handle both the qualitative and quantitative regressors and can also handle the presence of potentially irrelevant regressors, which can result in finite-sample efficiency gains relative to the conventional frequency estimator that is often found in such settings. Theoretical underpinnings including rates of convergence and asymptotic normality are provided. Monte Carlo simulations are undertaken to assess the method’s finite-sample performance relative to the conventional nonparametric frequency estimator, while an empirical application to a seminal dataset is undertaken for illustrative purposes.