Geometric Distribution as Means of Increasing Power in Backtesting VaR

We explore properties of the geometric distribution as means of constructing conditional coverage VaR tests. We study properties of these tests using asymptotic convergence of the test statistics. In this way, we replace Monte Carlo simulated distributions. We provide a unified framework that allows for effective comparison of various procedures. To achieve comparability we modify test statistics and adapt them to the conditional coverage hypothesis. We show that two tests that indirectly use properties of the geometric distribution—the test based on the General Method of Moments and the test based on the Gini coefficient—may be conveniently implemented with the use of known theoretical distributions. We argue that replacing Monte Carlo simulations with these distributions does not pose the risk of overrejecting correct risk models. We also demonstrate their efficiency at detecting incorrect models. We include practical guidelines about significance level and sample size that ensure accurate and efficient testing.