Two-Step Kernel Estimation of Expected Shortfall for Strong Mixing Time Series

Expected Shortfall (ES) is one of the most popular tools of risk management for financial property, and is an ideal coherent risk measure. In this paper, we discuss the two-step kernel estimator of ES under polynomial decay of strong mixing coefficients of time series. The first step is the kernel estimator of VaR (Value at Risk) and the second step is the kernel estimator of ES. We obtain Bahadur representation of the kernel estimator of ES. Then, we give the mean squares error and the rate of the asymptotic normality.