Joint value-at-risk and expected shortfall regression for location-scale time series models

Abstract.

This article studies the joint value-at-risk (VaR) and expected shortfall (ES) regression for a wide class of location-scale time series models including autoregressive and moving average models with generalized autoregressive conditional heteroscedasticity errors. In contrast to the quasi-maximum likelihood estimation, we estimate the model parameters with the aim of more accurate VaR and ES estimation. Then, we show consistency and asymptotic normality for parameter estimators under weak regularity conditions. Finally, a simulation study and a real data analysis are shown to illustrate our results.

Keywords:

• Expected shortfall
• value-at-risk
• ARMA-GARCH
• asymptotic normality

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. In this article, VaR is defined as quartiles, and therefore, VaR regression is also known as quantile regression.

2. There are various skewed t distributions used the literatures; in this paper, we use that of Hansen (1994).

3. That is, the conventional method here uses the quasi-maximum likelihood estimate; see Straumann and Mikosch (2006).

4. In this section, all the standardized residuals obtained from the ARMA(1,1)–GARCH(1,1) skewed t model pass the Kolmogorov-Smirnov test. The results suggest that this ARMA–GARCH model is appropriate.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant No.71973133, 72371230) and the Anhui Provincial Natural Science Foundation (Grant No. 2208085J41).