Finding a Wold Representation: An \(m\)th Order Moving Average, \(n\)th Order Autoregression

The following problem is a useful input into solving an interesting class of “signal extraction” problems.

Consider a covariance stationary process \(x_t\) with representation

\[ x_t = \frac{a(L)}{b(L)}u_t \]

where \(u_t\) is a (not necessarily fundamental) white noise and

\[\begin{split} \begin{aligned} b(L) &= (1 - \mu_1 L)\cdots(1-\mu_n L),\quad |\mu_j| < 1\\ a(L) &= (1 - \alpha_1 L)\cdots(1 - \alpha_m L). \end{aligned} \end{split}\]

Note that we assume that the zeros of \(b(z)\) are outside the unit circle, but those of \(a(z)\) are unrestricted. Our problem is to find a Wold moving average representation for \(x_t\).

The solution of this problem is simply

(264)\[x_t = \frac{d(L)}{b(L)}\epsilon_t\]

where \(d(L) = (1 - \lambda_1 L)\cdots(1 - \lambda_m L)\), where \(\lambda_1, \ldots, \lambda_m\) are the zeros of \(a(z) a(z^{-1})\) that do not lie outside the unit circle, and where \(\epsilon_t\) is the fundamental white noise for \(x_t\), with variance \(\sigma_\epsilon^2\) given by

\[ \sigma_\epsilon^2 = \frac{a(1)^2\sigma_u^2}{d(1)^2}. \]

In other words, the denominator polynomial \(b(L)\) is left unaltered while the methods of the preceding section are applied to factor the numerator polynomial. The reader should convince himself that this method delivers an \(\epsilon_t\) process that is a white noise, and that lies in the linear space spanned by \(\{x_t, x_{t-1}, \ldots\}\). This can be done by constructing an argument along the lines of the one in the preceding section, by assuming \(|\lambda_j| < 1\) for \(j = 1, \ldots, m\) and by premultiplying (264) by \(b(L)/d(L)\).