Moving Average Process (MA(q))

Definition

A process where the current value is a linear combination of current and past white noise shocks. $Z_t=a_t-{\theta }_1{a}_{t-1}-{\theta }_2{a}_{t-2}-\cdots -{\theta }_q{a}_{t-q}$ where $\{a_t\}$ is white noise. Unlike AR processes, MA(q) processes are always stationary for finite ${\theta }_i$.

Autocorrelation Function

For an MA(q) process, the autocorrelation function (ACF) “cuts off” after lag $q$: ${\rho }_k=0\text{ for }k>q$ For $k\le q$: ${\rho }_k=\frac{-{\theta }_k+{\theta }_1{\theta }_{k+1}+\cdots +{\theta }_{q-k}{\theta }_q}{1+{\theta }_1^2+\cdots +{\theta }_q^2}$ (Note: ${\theta }_0=-1$ in some conventions, signs may vary).