Duality, Forecasting and Selection of Autoregressive Moving Average Models∗

Based on both duality in time between time series processes and lag transformation, we define duality in causality, invertibility for mixed Autoregressive moving average ARMA(p,q) models. We construct expressions, in terms of the parameters of the parmaterized form of ARMA(p,q) models to compare the forecasting efficiency for a given causal/invertible pattern of an arbitrarily primary model relative to the pattern that define the corresponding dual model. The work considered the case when the forecast lead is one period for general univariate ARMA(p,q) as well as for ARMA(1, 1) models when the lead time is more than one period. These expressions are presented in terms of inequalities to serve as criterion for model selection. This study has shown that we need not eliminate noncausal and non-invertible ARMA(p, q) models from consideration if forecasting for more than one period is desired. In essence, we attempt to approach the estimation problem via the relation between a given time series model and its dual model. Numerical and empirical illustrations are reported