0.1 Moving Average Processes
A time series \(\{X_t\}\) is a moving average process of order \(q\), \(MA(q)\), if \[ X_t = Z_t + \theta_1 Z_{t-1} + \dots + \theta_q Z_{t-q}, \qquad Z_t \sim [[white-noise|WN]](0,\sigma_Z^2). \] Using the backshift operator \(B^kX_t = X_{t-k}\), this is \(X_t = \theta(B)Z_t\) with \(\theta(B) = 1+\theta_1B+\dots+\theta_qB^q\).
Every \(MA(q)\) process is stationary (regardless of the \(\theta_i\)) with ACF \[ \rho_X(k) = \frac{\theta_k + \theta_1\theta_{k+1}+\dots+\theta_{q-k}\theta_q}{1+\theta_1^2+\dots+\theta_q^2}, \quad k=1,\dots,q, \qquad \rho_X(k) = 0,\ k>q. \] This sharp cutoff in the ACF at lag \(q\) is the key diagnostic used to identify the order of an MA model from data.
An \(MA(q)\) model is always causal by construction, and is invertible iff the roots of \(\theta(z)\) lie outside the unit circle, in which case \(Z_t\) can be recovered as a convergent autoregressive-type series in past \(X_t\)’s (an \(MA(\infty)\) process becomes an \(AR(\infty)\) process).
An \(MA(\infty)\) representation \(X_t = \sum_{j=0}^\infty \psi_j Z_{t-j}\) (with \(\sum\psi_j^2<\infty\)) is the generic form guaranteed for any purely non-deterministic stationary process by Wold’s Decomposition Theorem.