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PreprintCC BY 4.0arXiv

Estimation and Inference for the Joint Autoregressive Quantile-Expected Shortfall Model

  • Peiyao Cai
  • Kean Ming Tan
  • Xuming He
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Abstract

Expected shortfall is defined as the truncated mean of a random variable that falls below a specified quantile level and is widely recognized as an important risk measure. Motivated by the empirical observation of clustering patterns in financial risks, we consider a joint autoregressive model for both conditional quantile and expected shortfall. Existing estimation methods for such models typically rely on minimizing a nonlinear and nonconvex joint loss function, which is challenging to solve. We introduce a two-step regression framework to estimate the model parameters with theoretical guarantees. We further develop a Portmanteau test for model diagnostics and prove its asymptotic validity for Type I error control. Our empirical results on stock market data indicate that the proposed models effectively capture the clustering patterns and leverage effects on the conditional expected shortfall.

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Cite

BibTeXRIS
BibTeX
@misc{cai2026estimation,
  title = {{Estimation and Inference for the Joint Autoregressive Quantile-Expected Shortfall Model}},
  author = {Peiyao Cai and Kean Ming Tan and Xuming He},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25730},
  eprint = {2610.25730},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25730},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Estimation and Inference for the Joint Autoregressive Quantile-Expected Shortfall Model
AU  - Peiyao Cai
AU  - Kean Ming Tan
AU  - Xuming He
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.25730
UR  - https://arxiv.org/abs/2610.25730
AB  - Expected shortfall is defined as the truncated mean of a random variable that falls below a specified quantile level and is widely recognized as an important risk measure. Motivated by the empirical observation of clustering patterns in financial risks, we consider a joint autoregressive model for both conditional quantile and expected shortfall. Existing estimation methods for such models typically rely on minimizing a nonlinear and nonconvex joint loss function, which is challenging to solve. We introduce a two-step regression framework to estimate the model parameters with theoretical guarantees. We further develop a Portmanteau test for model diagnostics and prove its asymptotic validity for Type I error control. Our empirical results on stock market data indicate that the proposed models effectively capture the clustering patterns and leverage effects on the conditional expected shortfall.
ER  -