Expected Shortfall Estimation and Gaussian Inference for Infinite Variance Time Series

We develop methods of non-parametric estimation for the Expected Shortfall of possibly heavy tailed asset returns that leads to asymptotically standard inference. We use a tail-trimming indicator to dampen extremes negligibly, ensuring standard Gaussian inference, and a higher rate of convergence than without trimming when the variance is infinite. Trimming, however, causes bias in small samples and possibly asymptotically when the variance is infinite, so we exploit a rarely used remedy to estimate and utilize the tail mean that is removed by trimming. Since estimating the tail mean involves estimation of tail parameters and therefore an added arbitrary choice of the number of included extreme values, we present weak limit theory for an ES estimator that optimally selects the number of tail observations by making our estimator arbitrarily close to the untrimmed estimator, yet still asymptotically normal. Finally, we apply the new estimators to financial returns data