This chapter provides a practical guide to statistical inference for quantile regression applications. In the earlier chapters, we have described a number of applications of quantile regression and provided various representations of the precision of these estimates; in this chapter and the one to follow, we will describe a variety of inference methods more explicitly. There are several competing approaches to inference in the literature and some guidance will be offered on their advantages and disadvantages. Ideally, of course, we would aspire to provide a finite-sample apparatus for statistical inference about quantile regression like the elegant classical theory of least-squares inference under independently and identically distributed (iid) Gaussian errors. But we must recognize that even in the least-squares theory it is necessary to resort to asymptotic approximations as soon as we depart significantly from idealized Gaussian conditions.

Nevertheless, we will begin by briefly describing what is known about the finite-sample theory of the quantile regression estimator and its connection to the classical theory of inference for the univariate quantiles. The asymptotic theory of inference is introduced with a heuristic discussion of the asymptotic behavior of the ordinary sample quantile; then a brief overview of quantile regression asymptotics is given. A more detailed treatment of the asymptotic theory of quantile regression is deferred to Chapter 4. Several approaches to inference are considered: Wald tests and related problems of direct estimation of the asymptotic covariance matrix, rank tests based on the dual quantile regression process, likelihood-ratio-type tests based on the value of the objective function under null and alternative models, and, finally, several resampling methods are introduced.