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

TL;DR

This paper introduces a self-normalization method for valid inference in time series quantile and expected shortfall regressions, offering a computationally simpler alternative to existing techniques with better size control.

Contribution

It develops a novel self-normalized inference approach for time series regressions of quantiles and expected shortfalls, avoiding complex tuning parameters and bootstrap methods.

Findings

The proposed method provides correctly sized inference in simulations.

Empirical applications demonstrate practical usefulness in finance.

Outperforms traditional methods in serially dependent error scenarios.

Abstract

This paper proposes valid inference tools, based on self-normalization, in time series expected shortfall regressions and, as a corollary, also in quantile regressions. Extant methods for such time series regressions, 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, expected shortfall) regression parameters that are computed on an expanding window, and are correctly sized as we show in simulations. Two empirical applications to stock return predictability and to Growth-at-Risk demonstrate the practical usefulness of the developed inference tools.