Cross-validated bandwidths and significance testing

The link between the magnitude of a bandwidth and the relevance of the corresponding covariate in a regression has recently garnered theoretical attention. Theory suggests that variables included erroneously in a regression will be automatically removed when bandwidths are selected via cross-validation procedure. However, the connections between the bandwidths of the variables that are smoothed away and the insights from these same variables when properly tested for statistical significance have not been previously studied. This paper proposes a variety of simulation exercises to examine the relative performance of both cross-validated bandwidths and individual and joint tests of significance. We focus on settings where the hypothesis of interest may focus on a single data type (e.g., continuous only) or a mix of discrete and continuous variables. Moreover, we propose an extension of a well-known kernel smoothing significance test to handle mixed data types. Our results suggest that individual tests of significance and variable-specific bandwidths are very close in performance, but joint tests and joint bandwidth recognition produce substantially different results. This underscores the importance of testing for joint significance when one is trying to arrive at the final nonparametric model of interest.