Moving-average process of order one ($\mathrm{MA}(1)$)

A moving-average process of order one has the formwith 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 ${\epsilon }_{t-1}=-1$, ${\epsilon }_t=2$, ${\epsilon }_{t+1}=0$. 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 $70/149\simeq 0.4698$. 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 $|\theta |<1$:Thus $\theta =2$ gives a stationary MA(1) that is not invertible in this sense. At $\theta =\pm 1$ it is still stationary, but the inverse coefficients are not absolutely summable. The case $\theta =0$ is simply shifted white noise. See Bartlett, lecture 5: MA(1) and stable invertibility.

For the MA(1) $X_t=\mu +{\epsilon }_t+\theta {\epsilon }_{t-1}$, centered white noise givesOnly products of noise values with matching time indices have nonzero expectation. At lag zero there are two squared weights, $1$ and ${\theta }^2$; at lag one there is one shared noise term with weight product $\theta$; at higher lags there are none. The calculation needs zero covariance between distinct noise values, not independence. This tridiagonal finite covariance structure makes the finite-sample innovations of an MA(1) process particularly simple.

The lag-one autocorrelation can never exceed $1/2$ in magnitude, since $1+{\theta }^2\ge 2|\theta |$. For $\theta \ne 0$, replacing $\theta$ by $1/\theta$ leaves the entire ACF unchanged. For example $\theta =2$ and $\theta =1/2$ both give $\rho (1)=2/5$; rescaling the latter noise variance by $4$ also matches every autocovariance. Moving-average root reflection explains this ambiguity and why a stable invertibility convention helps identify a representation. Equality of second-order laws need not imply equality of all joint probability distributions unless the processes are Gaussian processes. See Bartlett, lecture 6: equivalent representations and covariance of ARMA processes.