Testing the heteroscedastic error structure in quantile varying coefficient models

Abstract

Abstract In mean regression the characteristic of interest is the conditional mean of the response given the covariates. In quantile regression the aim is to estimate any quantile of the conditional distribution function. For given covariates, the conditional quantile function fully characterizes the entire conditional distribution function, in contrast to the mean which is just one of its characteristic quantities. Regression quantiles substantially out‐perform the least‐squares estimator for a wide class of non‐Gaussian error distributions. In this article we consider quantile varying coefficient models (VCMs) that are an extension of classical quantile linear regression models, in which one allows the coefficients to depend on other variables. We consider VCMs with various structures for the variance of the errors (the variability function) in order to allow for heteroscedasticity. For longitudinal data, the time ( T ) dependent coefficient functions in the signal and the variability functions are estimated with P‐splines (Penalized B‐splines). Consistency of the proposed estimators is proved. Further, likelihood‐ratio‐type tests are considered for comparing the variability functions. The performance of the testing procedure is illustrated on simulated and real data. The Canadian Journal of Statistics 46: 246–264; 2018 © 2017 Statistical Society of Canada