Simple moment-based tests for value-at-risk models and discrete distribution

In this paper, we develop moment-based tests for parametric discrete distributions. Momentbased test techniques are attractive as they provide easy-to-implement test statistics. We propose a general transformation that makes the moments of interest insensitive to the parameter estimation uncertainty. This transformation is valid for some extended families of non-differentiable moments that are of great interest in the case of discrete distributions. Considering the power function under local alternatives, we compare this strategy with the one in which parameter uncertainty is corrected. The special example of backtesting of valueat- risk (VaR) forecasts is treated in detail, and we provide simple moments that have good size and power properties in Monte Carlo experiments. Additional examples considered are discrete counting processes and the geometric distribution. We finally apply our method to backtesting of VaR forecasts derived from a T-GARCH(1,1) model estimated using foreign exchange-rate data.