Abstract

This paper proposes efficient estimation methods for Value-at-Risk (VaR) in the framework

of location-scale time series models, including the semi-parametric and parametric composite quantile

regression (CQR). The semi-parametric CQR does not impose any distribution assumptions on the

innovations, while the parametric CQR assumes that the innovations follow some distributions with

explicit and parametric quantile functions. Compared with the quantile regression, the semi-parametric

CQR method improves estimation efficiency by combining data information at multiple quantile levels.

The parametric CQR takes advantage of model flexibility, and can further enhance efficiency in face

of data scarcity when estimating high conditional quantiles. We establish the asymptotic properties

of both CQR methods for location-scale time series models, and particularly for the ARMA-GARCH,

double autoregressive and NAR-GARCH type models.

We also compare both CQR estimators in

estimation efficiency, and compare them with the Gaussian and exponential quasi-maximum likelihood

estimators. Finally, we examine the finite-sample performance of the proposed methods via simulation

studies, and analyze an empirical dataset to illustrate their usefulness in modeling and forecasting VaR

for financial assets.

Information

Preprint No.SS-2024-0167
Manuscript IDSS-2024-0167
Complete AuthorsChaoxu Lei, Qianqian Zhu
Corresponding AuthorsQianqian Zhu
Emailszhu.qianqian@mail.shufe.edu.cn

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Acknowledgments

We are deeply grateful to the Co-Editor, the Associate Editor and two anonymous referees

for their valuable comments that led to the substantial improvement in the quality of this

paper. Zhu’s research was supported by NSFC grants 12001355 and 72373087.

Supplementary Materials

The online Supplementary Material includes all technical details for Sections 2–3, together

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Supplementary materials are available for download.