  Expected Shortfall (ES) is the average return on a risky asset conditional on
the return being below some quantile of its distribution, namely its
Value-at-Risk (VaR). The Basel III Accord, which will be implemented in the
years leading up to 2019, places new attention on ES, but unlike VaR, there is
little existing work on modeling ES. We use recent results from statistical
decision theory to overcome the problem of "elicitability" for ES by jointly
modelling ES and VaR, and propose new dynamic models for these risk measures.
We provide estimation and inference methods for the proposed models, and
confirm via simulation studies that the methods have good finite-sample
properties. We apply these models to daily returns on four international equity
indices, and find the proposed new ES-VaR models outperform forecasts based on
GARCH or rolling window models.
