= Moving-average process of order one
{title2=$\operatorname{MA}(1)$}
= MA(1)
{c}
{synonym}
A <moving-average process of order one> has the form
$$
X_t=\mu+\varepsilon_t+\theta\varepsilon_{t-1},\qquad \theta\in\mathbb R,
$$
with centered <white noise> of finite positive <variance> $\sigma^2$. Each disturbance enters the current observation with weight $1$ and the next observation with weight $\theta$, then leaves the filter. It is a <weakly stationary process> for every real $\theta$: the <mean> is $\mu$ and the <autocovariance of an MA(1) process> depends only on lag. <Strong white noise> additionally makes it a <strictly stationary process>. No $|\theta|<1$ restriction is needed for stationarity.
For a worked observation calculation, take $\mu=10$, $\theta=0.7$, and consecutive noise values $\varepsilon_{t-1}=-1$, $\varepsilon_t=2$, $\varepsilon_{t+1}=0$. Then $X_t=11.3$ and $X_{t+1}=11.4$; the same disturbance $2$ appears in both. If $\sigma^2=1$, the <variance> is $1.49$, the lag-one <autocovariance> is $0.7$, and the lag-one <autocorrelation> is $70/149\simeq0.4698$. All higher-lag <autocorrelations> vanish, as shown in <autocorrelation>.
An <MA(1)> is useful for brief dependence caused by overlapping disturbances or measurements. Its name refers to a weighted moving sum of noise; its weights need not be positive or sum to one, and it differs from the data smoother in <seasonal extraction by a centred moving average>. Under <strong white noise>, observations more than one interval apart depend on disjoint independent noise families and are <independent>. Under <weak white noise>, the same cutoff establishes only <uncorrelated random variables>.
<Invertibility of a moving-average model> is a separate question: can the specified driving noise be recovered by a stable linear filter of present and past observations? For an <MA(1)> the answer, with the absolutely summable filter convention, is exactly $|\theta|<1$:
$$
\varepsilon_t=\sum_{j=0}^{\infty}(-\theta)^j(X_{t-j}-\mu).
$$
Thus $\theta=2$ gives a stationary <MA(1)> that is not invertible in this sense. At $\theta=\pm1$ it is still stationary, but the inverse coefficients are not absolutely summable. The case $\theta=0$ is simply shifted <white noise>. See https://www.stat.berkeley.edu/~bartlett/courses/153-fall2010/lectures/5.pdf[Bartlett, lecture 5: MA(1) and stable invertibility].
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