Self-Normalized Inference in (Quantile, Expected Shortfall) Regressions for Time Series

This paper is the first to propose valid inference tools, based on self-normalization, in time series expected shortfall regressions. In doing so, we propose a novel two-step estimator for expected shortfall regressions which is based on convex optimization in both steps (rendering computation easy) and it only requires minimization of quantile losses and squared error losses (methods for both of which are implemented in every standard statistical computing package). As a corollary, we also derive self-normalized inference tools in time series quantile regressions. Extant methods, based on a bootstrap or direct estimation of the long-run variance, are computationally more involved, require the choice of tuning parameters and have serious size distortions when the regression errors are strongly serially dependent. In contrast, our inference tools only require estimates of the quantile regression parameters that are computed on an expanding window and are correctly sized. Simulations show the advantageous finite-sample properties of our methods. Finally, two applications to stock return predictability and to Growth-at-Risk demonstrate the practical usefulness of the developed inference tools.