Evaluating Value-at-Risk Models via Quantile Regression

This paper is concerned with evaluating Value-at-Risk estimates. It is well known that using only binary variables, such as whether or not there was an exception, sacrifices too much information. However, most of the specification tests (also called backtests) available in the literature, such as Christofferson (1998) and Engle and Mangenelli (2004) are based on such variables. In this paper we propose a new backtest that does not rely solely on binary variables. It is shown that the new backtest provides a sufficient condtion to assess the finite sample performance of a quantile model whereas the existing ones do not. The proposed methodolgy allows us to identify periods of an increased risk exposure based on a quantile regression model (Koenker and Xiao, 2002). Our theoretical findings are corroborated through a Monte Carlo simulation and an empirical exercise with daily S&P500 time series.