This paper deals with prediction from a general ARMA model with GARCH - in - mean effects. A new method for obtaining multi-period predictions from the ARMA\((r,s)\) - GARCH\((p,q)\) - in - mean model is presented. Coefficients in the presented formula are expressed in terms of distinct roots of the autoregressive polynomials and parameters of the moving average.
The author obtains a closed form expression for the optimal predictor of the conditional mean from the ARMA - GARCH - M model, and gives the infinite moving average representations of the conditional mean and variance. The canonical factorization of the autocovariance generating function of the process and its conditional variance, autocovariance and cross covariance for the process are presented. Also, the minimum mean square error predictor of future values of the squared conditional variance is obtained.