To Smooth or Not to Smooth? The Case of Discrete Variables in Nonparametric Regression

In this paper, we consider the nonparametric smoothing technique with both discrete and categorical regressors. In the existing literature, it is generally admitted that it is better to smooth the discrete variables, which is similar to the smoothing technique for continuous regressors but using discrete kernels. However, as we explain in this paper, such approach might lead to a potential problem which is linked to the bandwidth selection for the continuous regressors due to the presence of the discrete regressors. Through the numerical study, we find that in many cases, the performance of the resulting nonparametric regression estimates may deteriorate if the discrete variables are smoothed in the way addressed so far in the literature, and that a fully separate estimation without any smoothing of the discrete variables may provide significantly better results. As a solution, we suggest a simple generalization of the popular approach proposed by Racine and Li [Journal of Econometrics, 2004] to address this problem and to provide estimates with more robust performance. We analyze the problem theoretically, develop the asymptotic theory for the new nonparametric smoothing method and study the finite sample behavior of the proposed generalized approach through extensive Monte-Carlo experiments as well present an empirical illustration.