Comparing predictive accuracy under long memory: With an application to volatility forecasting

This paper extends the popular Diebold-Mariano test to situations when the forecast error loss differential exhibits long memory. It is shown that this situation can arise frequently, since long memory can be transmitted from forecasts and the forecast objective to forecast error loss differentials. The nature of this transmission mainly depends on the (un)biasedness of the forecasts and whether the involved series share common long memory. Further results show that the conventional Diebold-Mariano test is invalidated under these circumstances. Robust statistics based on a memory and autocorrelation consistent estimator and an extended fixed-bandwidth approach are considered. The subsequent Monte Carlo study provides a novel comparison of these robust statistics. As an empirical application, we conduct forecast comparison tests for the realized volatility of the Standard and Poors 500 index among recent extensions of the heterogeneous autoregressive model. While we find that forecasts improve significantly if jumps in the log-price process are considered separately from continuous components, improvements achieved by the inclusion of implied volatility turn out to be insignificant.