7.1 The identification problem
Example 6.2.2 is an instance of a general fact. Consider the two MA(1) processes \[ \text {A}: \ X_t = Z_t + \beta Z_{t-1} , \qquad \qquad \text {B}: \ X_t = Z_t + \frac 1\beta Z_{t-1} . \] They have the same ac.f., since \[ \frac {1/\beta }{1 + 1/\beta ^2} = \frac {\beta }{\beta ^2 + 1} . \] Thus a moving average process cannot be identified uniquely from its ac.f. To choose between A and B, express each shock in terms of the observations. From A, \(Z_t = X_t - \beta Z_{t-1}\), and substituting repeatedly for the earlier \(Z\)’s, \[ \text {A}: \ Z_t = X_t - \beta X_{t-1} + \beta ^2X_{t-2} - \beta ^3X_{t-3} + \dots , \] and similarly \[ \text {B}: \ Z_t = X_t - \frac 1\beta X_{t-1} + \frac 1{\beta ^2}X_{t-2} - \frac 1{\beta ^3}X_{t-3} + \dots . \] If \(|\beta | < 1\) the series for A converges, while that for B diverges, since its weights grow. So if \(|\beta | < 1\), model A is invertible and model B is not. Imposing invertibility ensures that there is a unique MA process for a given ac.f.
Definition 7.1.1. A process \(\{X_t\}\) is invertible if each shock \(Z_t\) can be written as a convergent weighted sum of the present and past observations, \(Z_t = \sum _{j=0}^\infty \pi _jX_{t-j}\) with \(\sum _j|\pi _j| < \infty \).
Proposition 7.1.2. If \(|\beta | < 1\), the MA(1) process \(X_t = Z_t + \beta Z_{t-1}\) is invertible: with \(S_n = \sum _{j=0}^n(-\beta )^jX_{t-j}\), \[ E\bigl [(Z_t - S_n)^2\bigr ] = \beta ^{2(n+1)}\sigma _Z^2 \to 0 \quad \text {as } n \to \infty . \]
Proof. Substituting \(X_{t-j} = Z_{t-j} + \beta Z_{t-j-1}\), the sum telescopes: \[ S_n = \sum _{j=0}^n(-\beta )^jZ_{t-j} + \sum _{j=0}^n(-\beta )^j\beta Z_{t-j-1} = Z_t - (-\beta )^{n+1}Z_{t-n-1} , \] because the second sum is minus the first sum without its \(j = 0\) term, plus the extra term \(-(-\beta )^{n+1}Z_{t-n-1}\). So \(Z_t - S_n = (-\beta )^{n+1}Z_{t-n-1}\), whose mean square is \(\beta ^{2(n+1)}\sigma _Z^2\), which tends to \(0\) when \(|\beta | < 1\). The weights \(\pi _j = (-\beta )^j\) have \(\sum |\beta |^j = 1/(1 - |\beta |) < \infty \). □
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