Generalized nonparametric smoothing with mixed discrete and continuous data

The nonparametric smoothing technique with mixed discrete and continuous regressors is considered. 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, such an 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, it is found that in many cases, the performance of the resulting nonparametric regression estimates may deteriorate if the discrete variables are smoothed in the way previously addressed, and that a fully separate estimation without any smoothing of the discrete variables may provide significantly better results both for bias and variance. As a solution, it is suggested a simple generalization of the nonparametric smoothing technique with both discrete and continuous data to address this problem and to provide estimates with more robust performance. The asymptotic theory for the new nonparametric smoothing method is developed and the finite sample behavior of the proposed generalized approach is studied through extensive Monte-Carlo experiments as well an empirical illustration.