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

In a seminal paper, Racine and Li, (Journal of Econometrics, 2004) introduce a tool which admits discrete and categorical variables as regressors in nonparametric regres- sions. The method is similar to the smoothing techniques for continuous regressors but uses discrete kernels. In the literature, it is generally admitted that it is always better to smooth the discrete variables. In this paper we investigate the potential problem linked to the bandwidths selection for the continuous variable due to the presence of the discrete variables. We find that in some cases, the performance of the resulting regression estimates may be deteriorated by smoothing the discrete variables in the way addressed so far in the literature, and that a fully separate estimation (without any smoothing of the discrete variable) may provide significantly better results, and we explain why this may happen. The problem being posed, we then suggest how to use the Racine and Li approach to overcome these difficulties and to provide estimates with better performances. We investigate through some simulated data sets and by more ex- tensive Monte-Carlo experiments the performances of all the proposed approaches and we find that, as expected, our suggested approach has the best performances. We also briefly illustrate the consequences of these issues on the estimation of the derivatives of the regression. Finally, we exemplify the phenomenon with an empirical illustration. Our main objective is to warn the practitioners of the potential problems posed by smoothing discrete variables by using the so far available softwares and to suggest a safer approach to implement the procedure.