General Linear Process

Definition

A process $\{Z_t\}$ is a general linear process if it can be represented as a weighted linear combination of current and past white noise shocks: $Z_t=a_t+{\psi }_1{a}_{t-1}+{\psi }_2{a}_{t-2}+\cdots ={\sum }_{j=0}^{\infty }{\psi }_j{a}_{t-j}$ where ${\psi }_0=1$ and $\{a_t\}$ is white noise.

Stationarity Condition

For a general linear process to be stationary, the weights must satisfy: ${\sum }_{j=0}^{\infty }{\psi }_j^2<\infty$ This ensures the variance of the process is finite.