MA(q) Process Model

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.

Tip

$Z_t=a_t-{\sum }_{i=1}^q{\theta }_i{a}_{t-i}$

Stationarity

Unlike AR processes, MA(q) processes are always weakly stationary for finite ${\theta }_i$.

Properties

Property	Expression
Mean	$0$
Autocovariance (${\gamma }_k$)	$\{\begin{array}{ll}(1+{\theta }_1^2+{\theta }_2^2+\cdots +{\theta }_q^2){\sigma }_e^2& k=0\\ -{\theta }_k+{\theta }_1{\theta }_{k+1}+{\theta }_2{\theta }_{k+2}+\cdots +{\theta }_{q-k}{\theta }_q,& k=1,2,\dots ,q\\ 0,& k>q\end{array}$
Autocorrelation (${\rho }_k$) just build from ACF formula	$\{\begin{array}{ll}1,& k=0\\ \frac{-{\theta }_k+{\theta }_1{\theta }_{k+1}+{\theta }_2{\theta }_{k+2}+\cdots +{\theta }_{q-k}{\theta }_q}{1+{\theta }_1^2+{\theta }_2^2+\cdots +{\theta }_q^2},& k=1,2,\dots ,q\\ 0,& k>q\end{array}$