Moving-average model

Remarks

Every MA($q$) process is weakly stationary with mean $\mu$. With $\theta_0 = 1$, its autocovariance is $\gamma(h) = \sigma^2 \sum_{j=0}^{q - \abs{h}} \theta_j \theta_{j + \abs{h}}$ for $\abs{h} \le q$ and $\gamma(h) = 0$ for $\abs{h} > q$. So its autocorrelation function vanishes beyond lag $q$. The model is invertible when $\theta$ has no root in the closed unit disk. Then $\varepsilon_t$ is a convergent linear combination of $X_t, X_{t-1}, \dots$, and the process is an autoregressive model of infinite order.

Example: for MA($1$), $X_t = \varepsilon_t + \theta \varepsilon_{t-1}$ has $\gamma(0) = \sigma^2 (1 + \theta^2)$, $\gamma(\pm 1) = \theta \sigma^2$ and $\gamma(h) = 0$ otherwise. Hence $\rho(1) = \theta / (1 + \theta^2)$, which always lies in $[-1/2, 1/2]$.

Non-example: the process $X_t = \sum_{j \ge 0} 0.5^j \varepsilon_{t-j}$ is not an MA($q$) for any finite $q$, because its autocorrelation $0.5^{\abs{h}}$ is nonzero at every lag.

The name has nothing to do with the moving average used to smooth data, although both are weighted sums over a window.