Backtesting Value-at-Risk: A Generalized Markov Test

Testing the validity of value‐at‐risk (VaR) forecasts, or backtesting, is an integral part of modern market risk management and regulation. This is often done by applying independence and coverage tests developed by Christoffersen (International Economic Review, 1998; 39(4), 841–862) to so‐called hit‐sequences derived from VaR forecasts and realized losses. However, as pointed out in the literature, these aforementioned tests suffer from low rejection frequencies, or (empirical) power when applied to hit‐sequences derived from simulations matching empirical stylized characteristics of return data. One key observation of the studies is that higher‐order dependence in the hit‐sequences may cause the observed lower power performance. We propose to generalize the backtest framework for VaR forecasts, by extending the original first‐order dependence of Christoffersen to allow for a higher‐ or kth‐order dependence. We provide closed‐form expressions for the tests as well as asymptotic theory. Not only do the generalized tests have power against kth‐order dependence by definition, but also included simulations indicate improved power performance when replicating the aforementioned studies. Further, included simulations show much improved size properties of one of the suggested tests.