Econometric Methods for Expected Shortfall and Value-at-Risk

Value-at-Risk (VaR) has been the most prevalent market risk measure in the financial sector. Banks, insurance companies and other financial institutions are required to report their VaR estimates to the regulatory authorities since its introduction to the Basel I Accord in 1996. Acknowledging the theoretical deficiencies of this risk measure, The Basel Committee on Banking Supervision proposed to replace VaR with Expected Shortfall (ES), which overcomes these shortcomings. The practical implementation of this measure is still in process as the literature is lack of simple tools for its estimation and evaluation since by definition, the ES depends on the VaR estimate. This dissertation develops several techniques for estimating and conducting inference on VaR and ES models.

The first chapter, which is a joint paper with Andrew J. Patton, implements a 2-step robust estimation method for estimating the Expected Shortfall. We ease the dependence of the ES estimate from the VaR. To achieve this, in the first step the VaR is estimated by nonparametric methods, which helps us to avoid estimation error in the nuisance process. In the second step, we apply a robust estimation technique which controls for small deviations of the VaR estimates from their theoretically true values. We compare this new method to a 1-step joint estimation when VaR and ES are jointly estimated and to a 2-step non-robust estimation method. We find that with the new 2-step method the estimates are more efficient than applying a non-robust version and it performs better than the joint estimation when we do inference on the ES model parameters.

The second chapter, which is joint with Jia Li, Zhipeng Liao and Andrew J. Patton, proposes a novel nonparametric specification test for VaR models. We translate the specification test to a conditional moment restriction test. We estimate the conditional moment function via nonparametric series regression and test whether it is identically zero. We use a strong Gaussian approximation theory to characterize the asymptotic behavior of our sup-t test. In addition, we propose an i.i.d. bootstrap method which performs better in finite sample than the asymptotic approximation at more extreme quantiles. As an empirical exercise, we test Conditional VaR models of US financial institutions.

The third chapter builds on this idea and implements the test for multiple conditional moment restrictions to test the correct specification of VaR and ES jointly. In addition, we also propose an average-t test, whose theoretical properties rely on the sup-t test. We find that the average-t test outperforms the sup-t test at more extreme confidence levels. As an empirical exercise, we test several location-scale models in S&P500 data for correct specification of ES and VaR.