6.1 Definition and moments
Definition 6.1.1. Let \(\{Z_t\}\) be a purely random process with mean \(0\) and variance \(\sigma _Z^2\). The process \(\{X_t\}\) is a moving average process of order \(q\), written MA(\(q\)), if \[ X_t = \beta _0Z_t + \beta _1Z_{t-1} + \beta _2Z_{t-2} + \dots + \beta _qZ_{t-q} , \] where the \(\beta _j\) are constants. We normally take \(\beta _0 = 1\) (this fixes the scale of \(Z_t\)).
Each \(X_t\) is a weighted sum of the current shock and the \(q\) previous ones, so a shock affects the series for \(q + 1\) periods and then is forgotten. If the series has mean \(\mu \), the model is \(X_t = \mu + \beta _0Z_t + \dots + \beta _qZ_{t-q}\), which has the same autocovariances.
Theorem 6.1.2. The MA(\(q\)) process has mean \(0\) and autocovariance function \[ \gamma (k) = \begin {cases} \sigma _Z^2\displaystyle \sum _{i=0}^{q-k}\beta _i\beta _{i+k} , & k = 0, 1, \dots , q , \\[2ex] 0 , & k > q , \end {cases} \qquad \gamma (-k) = \gamma (k) . \] In particular \(\Var (X_t) = \sigma _Z^2\sum _{i=0}^q\beta _i^2\). So the process is second-order stationary for every choice of the \(\beta _i\), and its ac.f. is \[ \rho (k) = \frac {\sum _{i=0}^{q-k}\beta _i\beta _{i+k}}{\sum _{i=0}^q\beta _i^2} \quad (k = 1, \dots , q) , \qquad \rho (k) = 0 \quad (k > q) . \]
Proof. \(E(X_t) = \sum _i\beta _iE(Z_{t-i}) = 0\). For \(k \ge 0\), since \(X_t = \sum _i\beta _iZ_{t-i}\) and \(X_{t+k} = \sum _j\beta _jZ_{t+k-j}\), covariance is bilinear and \[ \Cov (X_t, X_{t+k}) = \sum _{i=0}^q\sum _{j=0}^q\beta _i\beta _j\Cov (Z_{t-i}, Z_{t+k-j}) . \] Since \(\Cov (Z_s, Z_u) = \sigma _Z^2\) if \(s = u\) and \(0\) otherwise, only the terms with \(t - i = t + k - j\), that is \(j = i + k\), survive. Such a \(j\) lies in \(0, \dots , q\) only when \(i \le q - k\), which gives the sum; if \(k > q\) there are no such terms. Neither the mean nor \(\gamma (k)\) depends on \(t\), so the process is stationary, and evenness is Theorem 3.3.1(ii). □
The key feature is that the ac.f. of an MA(\(q\)) process cuts off after lag \(q\): it is exactly zero beyond. Two observations more than \(q\) apart share no shock. A sample correlogram that is clearly non-zero up to some lag and negligible afterwards therefore suggests a moving average model of that order.
Solution. Here \(\beta _0 = 1\), \(\beta _1 = 0.5\), \(\beta _2 = -0.3\). By Theorem 6.1.2, \begin {align*} \gamma (0) &= \sigma _Z^2(1 + 0.25 + 0.09) = 1.34\sigma _Z^2 , \\ \gamma (1) &= \sigma _Z^2(\beta _0\beta _1 + \beta _1\beta _2) = \sigma _Z^2(0.5 - 0.15) = 0.35\sigma _Z^2 , \\ \gamma (2) &= \sigma _Z^2\beta _0\beta _2 = -0.3\sigma _Z^2 , \end {align*}
and \(\gamma (k) = 0\) for \(k \ge 3\). So \(\rho (1) = 0.35/1.34 = 0.261\), \(\rho (2) = -0.3/1.34 = -0.224\), and \(\rho (k) = 0\) for \(k \ge 3\).
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