Theory and Methods of Statistics

13.6.3 Forecasting an ARMA Series

If {X_t} is mean zero ARMA(p, q), then the method for forecasting X_n+h based on X_n = (X₁, …, X_n)^T combines the methods given for AR and MA models. If the series in invertible, then we can obtain, in principle, the (approximate) best linear predictors of X_n+h, h = 1, 2, …, based on X₁, …, X_n, by approximating {X_t} by an AR(p) process with p = n.

For the moment assume that ε_p = (ε_p+1−q, …, ε_p)^T is known. Then we can obtain ε_p+1, …, ε_n as linear combinations of ε_p and X_n as will be shown below.

Since

$\begin{array}{ll}\hfill X_{n+1}& ={\varphi }_1{X}_n+\cdots +{\varphi }_p{X}_{n+1-p}+{\epsilon }_{n+1}+{\theta }_1{\epsilon }_n+\cdots +{\theta }_q{\epsilon }_{n+1-q}\hfill \\ ={\varphi }_1{X}_n+\cdots +{\varphi }_p{X}_{n+1-p}+{\theta }_1{\epsilon }_n+\cdots +{\theta }_q{\epsilon }_{n+1-q}+{\epsilon }_{n+1},\hfill \end{array}$

the best linear predictor ${\widehat{X}}_{}$