Finite-Sample Inference Methods for Quantile Regression Models

Finite-sample inference methods are developed for quantile regression models. The methods are conservative in that (i) they apply to arbitrary sample sizes without the liberal assumption that sample sizes approach infinity, (ii) they apply when the quantiles are partially or set identified, (iii) they allow for the expanding parameter spaces in the sense of Huber-Portnoy, allowing for series expansions, (iv) they apply for extremal quantiles (such as the ones that are < .1 and >.9) without special adjustments and without need for Poisson type approximations, (v) they apply for cases where densities of the response variable are ill-behaved, such as densities that shoot up to infinity or become zero at the quantiles of interest, (vi) when there are ``many” and ``weak” instruments (when instrumentation is needed to correct for endogeneity) , (vii) when parameters are on the ``boundary” in the sense of Andrews, and other cases where conventional inference breaks down or is not credible . The proposed inference methods are analyzed from the decision-theoretic point of view, and the sub-set of optimal procedures is determined. Computational methods are discussed. Conservativeness of the methods is examined, and comparisons with conventional methods are offered