Quantile regression (QR) is an increasingly important empirical tool in economics and other sciences for analysing the impact a set of regressors has on the conditional distribution of an outcome. Extremal QR, or QR applied to the tails, is of interest in many economic and financial applications, such as conditional value at risk, production efficiency, and adjustment bands in (S,s) models. This paper provides feasible inference tools for extremal conditional quantile models that rely on extreme value approximations to the distribution of self-normalized QR statistics. The methods are simple to implement and can be of independent interest even in the univariate (non-regression) case. We illustrate the results with two empirical examples analysing extreme fluctuations of a stock return and extremely low percentiles of live infant birthweight in the range between 250 and 1500 g.

We propose robust methods for inference about the effect of a treatment variable on a scalar outcome in the presence of very many regressors in a model with possibly non-Gaussian and heteroscedastic disturbances. We allow for the number of regressors to be larger than the sample size. To make informative inference feasible, we require the model to be approximately sparse; that is, we require that the effect of confounding factors can be controlled for up to a small approximation error by including a relatively small number of variables whose identities are unknown. The latter condition makes it possible to estimate the treatment effect by selecting approximately the right set of regressors. We develop a novel estimation and uniformly valid inference method for the treatment effect in this setting, called the “post-double-selection†method. The main attractive feature of our method is that it allows for imperfect selection of the controls and provides confidence intervals that are valid uniformly across a large class of models. In contrast, standard post-model selection estimators fail to provide uniform inference even in simple cases with a small, fixed number of controls. Thus, our method resolves the problem of uniform inference after model selection for a large, interesting class of models. We also present a generalization of our method to a fully heterogeneous model with a binary treatment variable. We illustrate the use of the developed methods with numerical simulations and an application that considers the effect of abortion on crime rates.