Backtesting Value-at-Risk using Forecasts for Multiple Horizons, a Comment on the Forecast Rationality Tests of A.J. Patton and A. Timmermann

Patton and Timmermann (2012, 'Forecast Rationality Tests Based on Multi-Horizon Bounds', Journal of Business & Economic Statistics, 30(1) 1-17) propose a set of useful tests for forecast rationality or optimality under squared error loss, including an easily implemented test based on a regression that only involves (long-horizon and short-horizon) forecasts and no observations on the target variable. We propose an extension, a simulation-based procedure that takes into account the presence of errors in parameter estimates. This procedure can also be applied in the field of 'backtesting' models for Value-at-Risk. Applications to simple AR and ARCH time series models show that its power in detecting certain misspecifications is larger than the power of well-known tests for correct Unconditional Coverage and Conditional Coverage.