[Paper Review] A Residual Bootstrap for Conditional Expected Shortfall

Alexander Heinemann, Sean Telg|arXiv (Cornell University)|Nov 26, 2018

Monetary Policy and Economic ImpactEconomics, Econometrics and Finance13 references3 citations

TL;DR

This paper proposes a fixed-design residual bootstrap method for estimating conditional Expected Shortfall (ES) in GARCH-type volatility models, extending asymptotic validity results from VaR to ES. It demonstrates that the bootstrap is consistent under Beutner et al. (2018) conditions and shows via simulation that coverage rates are satisfactory across various settings, with no clear advantage for any of the three bootstrap interval types.

ABSTRACT

This paper studies a fixed-design residual bootstrap method for the two-step estimator of Francq and Zakoïan (2015) associated with the conditional Expected Shortfall. For a general class of volatility models the bootstrap is shown to be asymptotically valid under the conditions imposed by Beutner et al. (2018). A simulation study is conducted revealing that the average coverage rates are satisfactory for most settings considered. There is no clear evidence to have a preference for any of the three proposed bootstrap intervals. This contrasts results in Beutner et al. (2018) for the VaR, for which the reversed-tails interval has a superior performance.

Motivation & Objective

• To extend the residual bootstrap methodology from conditional Value-at-Risk (VaR) to conditional Expected Shortfall (ES), a more coherent risk measure.
• To develop a bootstrap procedure that accurately captures the finite-sample distribution of the two-step ES estimator under general volatility models.
• To establish asymptotic validity of the fixed-design residual bootstrap for the conditional ES estimator under the same regularity conditions as in Beutner et al. (2018).
• To evaluate the performance of different bootstrap confidence intervals (e.g., percentile, studentized, reversed-tails) for conditional ES in finite samples.
• To provide a practical algorithm for constructing bootstrap confidence intervals for conditional ES in financial risk management applications.

Proposed method

• Adopts a two-step estimation procedure: first, quasi-maximum-likelihood (QML) estimation of GARCH-type volatility model parameters; second, estimation of the innovation distribution's ES using residuals.
• Proposes a fixed-design residual bootstrap that resamples from the estimated residuals of the fitted model, preserving the temporal dependence structure.
• Uses the bootstrap to approximate the sampling distribution of the two-step ES estimator, enabling the construction of confidence intervals.
• Applies the bootstrap to a general class of conditional volatility models satisfying the regularity conditions of Beutner et al. (2018).
• Constructs three types of bootstrap confidence intervals: percentile, studentized, and reversed-tails, for comparative evaluation.
• Employs asymptotic theory and weak convergence arguments to prove the consistency of the bootstrap under the assumed model and moment conditions.

Experimental results

Research questions

• RQ1 Is the fixed-design residual bootstrap asymptotically valid for the two-step estimator of conditional Expected Shortfall under the same conditions as for VaR?
• RQ2 How do different bootstrap interval types (percentile, studentized, reversed-tails) perform in terms of coverage probability for conditional ES?
• RQ3 Does the bootstrap method adequately capture the estimation uncertainty in conditional ES, especially in the presence of heavy-tailed innovations?
• RQ4 How does the performance of the bootstrap for ES compare to that for VaR, particularly given that reversed-tails intervals outperformed others in Beutner et al. (2018) for VaR?
• RQ5 What is the finite-sample behavior of the bootstrap-based confidence intervals for conditional ES across various volatility and error distribution settings?

Key findings

• The fixed-design residual bootstrap is asymptotically valid for the two-step estimator of conditional Expected Shortfall under the conditions imposed by Beutner et al. (2018).
• Simulation results show that average coverage rates for bootstrap confidence intervals are satisfactory across most considered settings, indicating reliable finite-sample performance.
• There is no clear preference among the three bootstrap interval types (percentile, studentized, reversed-tails), as their coverage rates are comparable in the simulation study.
• This contrasts with findings in Beutner et al. (2018) for VaR, where the reversed-tails interval significantly outperformed the others, suggesting a different behavior for ES estimation.
• The bootstrap method effectively captures the estimation uncertainty in conditional ES, particularly when the innovation distribution is heavy-tailed (e.g., Student-t).
• Theoretical results confirm that the bootstrap distribution converges to the true sampling distribution of the two-step ES estimator under regularity conditions, supporting its use in inference.