• Long memory in volatility;
  • covariance stationarity;
  • mixture ARCH( ∞ );
  • EM algorithm

We propose a new volatility model, which is called the mixture memory generalized autoregressive conditional heteroskedasticity (MM-GARCH) model. The MM-GARCH model has two mixture components, of which one is a short-memory GARCH and the other is the long-memory fractionally integrated GARCH. The new model, a special ARCH( ∞ ) process with random coefficients, possesses both the properties of long-memory volatility and covariance stationarity. The existence of its stationary solution is discussed. A dynamic mixture of the proposed model is also introduced. Other issues, such as the expectation–maximization algorithm as a parameter estimation procedure, the observed information matrix, which is relevant in calculating the theoretical standard errors, and a model selection criterion, are also investigated. Monte Carlo experiments demonstrate our theoretical findings. Empirical application of the MM-GARCH model to the daily S&P 500 index illustrates its capabilities.