“Measuring Heteroskedasticity in Nonparametric Regressions” by Xiaojun Song
Durham Unversity Business School
This paper aims to introduce new measures that quantify the strength of heteroskedasticity in non-parametric regressions. The measures are based on nonparametric quantile regressions and defined as one minus the ratio of unrestricted and restricted expectations of quantile check loss functions. They can be easily and consistently estimated by replacing the unknown expectations of check loss functions by their nonparametric kernel estimates. We derive a Bahadur-type representation for the nonparametric estimator of the measures of heteroskedasticity. We provide the asymptotic distribution of this estimator, which one can use to build tests for the statistical significance of the measures. We also examine the properties of the latter tests under some local alternatives. Thereafter, we establish the validity of a smoothed local bootstrap that one can use in finite sample settings to perform statistical tests. A Monte Carlo simulation study reveals that the bootstrap-based test has a good finite sample size and power properties for a variety of data generating processes and different sample sizes. Finally, an empirical application is provided to illustrate the importance of the proposed measures of heteroskedasticity.