Invertible MA(q) Process

Definition

An MA(q) process is said to be invertible, if the roots of the MA characteristic polynomial lie outside the unit circle.

When invertible, MA(q) can be written as an AR process with infinite order: $a\_t=Z\_t+\pi \_1Z\_t-1+\pi \_2Z\_t-2+\dots$

Inverting MA(1) to AR(∞)

Let $Z\_t=a\_t+\theta a\_t-1$

Then $a\_t=Z\_t-\theta a\_t-1$

Substituting $a\_t-1=Z\_t-1-\theta a\_t-2$, then $a\_t-2=Z\_t-2-\theta a\_t-3$, and so on:

$$\begin{array}{rl}a\_t& =Z\_t-\theta a\_t-1\\ & =Z\_t-\theta (Z\_t-1-\theta a\_t-2)\\ & =Z\_t-\theta Z\_t-1+{\theta }^{2}a\_t-2\\ & =Z\_t-\theta Z\_t-1+{\theta }^2(Z\_t-2-\theta a\_t-3)\\ & =Z\_t-\theta Z\_t-1+{\theta }^{2}Z\_t-2-{\theta }^{3}a\_t-3\\ & =\cdots \\ & =Z\_t-\theta Z\_t-1+{\theta }^{2}Z\_t-2-{\theta }^{3}Z\_t-3+\dots \\ & =\sum \_{j=0}^{\infty }(-\theta {)}^{j}Z\_t-j\end{array}$$

This gives the AR($\infty$) representation: $a\_t=Z\_t-\theta Z\_t-1+{\theta }^{2}Z\_t-2-\cdots =\sum \_{j=0}^{\infty }(-\theta {)}^{j}Z\_t-j$

with $\pi \_j=(-\theta {)}^j$.

Notice that this converges if $|\theta |<1$, since $|\theta {|}^j\to 0$ as $j\to \infty$.