Doubly multiplicative error models with long- and short-run components

We suggest the Doubly Multiplicative Error class of models (DMEM) for modeling and forecasting realized volatility, which combines two components accommodating long-run, respectively, short-run features in the data. Three such models are considered, the Spline-MEM which fits a spline to the slow-moving pattern of volatility, the Component-MEM, which uses daily data for both components, and the MEM-MIDAS, which exploits the logic of MIxed-DAta Sampling (MIDAS) methods. The parameters are estimated by the Generalized Method of Moments (GMM), for which we establish the theoretical properties and the equivalence with the Quasi Maximum Likelihood (QML) estimator under a Gamma assumption. The empirical application involves the S&P 500, NASDAQ, FTSE 100, DAX, Nikkei and Hang Seng indices: irrespective of the market, the DMEM's generally outperform the HAR and other relevant GARCH-type models.