Bayesian Forecasting for Tail Risk

摘要

This paper evaluates the performances of Value-at-Risk (VaR) and expected shortfall, as well as volatility forecasts in a class of risk models, specifically focusing on GARCH, integrated GARCH, and asymmetric GARCH models (GJR-GARCH, exponential GARCH, and smooth transition GARCH models). Most of the models incorporate four error probability distributions: Gaussian, Student's t, skew Student's t, and generalized error distribution (GED). We employ Bayesian Markov chain Monte Carlo sampling methods for estimation and forecasting. We further present backtesting measures for both VaR and expected shortfall forecasts and implement two loss functions to evaluate volatility forecasts. The empirical results are based on the S&P500 in the U.S. and Japan's Nikkei 225. A VaR forecasting study reveals that at the 1% level the smooth transition model with a second-order logistic function and skew Student's t error compares most favorably in terms of violation rates for both markets. For the volatility predictive abilities, the EGARCH model with GED error is the best model in both markets.