Comparing Predictive Accuracy under Long Memory, With an Application to Volatility Forecasting*
将Diebold-Mariano检验扩展到预测误差损失差分存在长记忆的情形，发现传统检验失效，提出稳健统计量，并用S&P500已实现波动率数据验证了考虑跳跃的HAR模型改进显著，而隐含波动率指数改进不显著。
This article extends the popular Diebold–Mariano test for equal predictive accuracy to situations when the forecast error loss differential exhibits long memory. 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 depends on the (un)biasedness of the forecasts and whether the involved series share common long memory. Further theoretical 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 extensive Monte Carlo study provides numerical results on various issues. As empirical applications, we consider recent extensions of the HAR model for the S&P500 realized volatility. While we find that forecasts improve significantly if jumps are considered, improvements achieved by the inclusion of an implied volatility index turn out to be insignificant.