Martingale Tests of Value-at-Risk

We present a general framework for testing the accuracy of Value-at-Risk (VaR) forecasts. The approach is based on the observation that violations – the days on which portfolio losses exceed the VaR – should be unpredictable. Specifically, these violations form a martingale difference sequence. The martingale hypothesis has a long and distinguished history in economics and finance (Durlauf (1991)). And as a result of the extensive toolkit developed in the literature, we are able to cast all existing methods of evaluating VaR under a common umbrella of martingale tests. This immediately suggests several testing strategies. The most obvious is a test of whether any of the autocovariances are nonzero. The standard approach to test for uncorrelatedness is by estimating the sample autocovariances or sample autocorrelations. In particular, we suggest the well-known Box-Ljung test of the violation sequence’s autocorrelation function. The second set of tests is taken from Durlauf (1991). He derives a set of tests of the martingale hypothesis based on the spectral density functions. This approach has several features to commend it. Unlike variance ratio tests, spectral tests have power against any linear alternative of any order. Spectral density tests have power to detect any second moment dynamics. Variance ratio tests are typically not consistent against all such alternatives. In order to assess the performance of the different tests we simulate 5-minute portfolio return data from the Heston (1993) stochastic volatility model. From these simulated data we create daily returns, which are in turn used to calculate (misspecified) daily VaRs based on the historical simulation method. The VaRs are calculated daily for the weekly and biweekly return horizons by (erroneously) scaling the daily VaR by root 5 and root 10 respectively.