This paper extends the popular Diebold-Mariano test to situations when the forecast error loss dierential 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 dierentials. 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 xed-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 nd that forecasts improve signicantly 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 insignicant. Let y 1t and y 2t denote two competing forecasts for the forecast objective series y t and let the loss function of the forecaster be given by g(·) ≥ 0. Forecast errors are dened as e it = y t − y it for i = 1, 2 and the corresponding forecast error loss dierential is denoted by z t = g(e 1t ) − g(e 2t ).By only imposing restrictions on the loss dierential z t , instead of the forecast objective and the forecasts, Diebold and Mariano (1995) test the null hypothesis of equal predictive accuracy, i.e.