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  "body": "\n## Definition\n\nLet $(\\varepsilon_t)_{t \\in \\Z}$ be a white noise with variance $\\sigma^2 \u003e 0$: the $\\varepsilon_t$ have mean $0$, variance $\\sigma^2$ and are pairwise uncorrelated.\nLet $q \\ge 1$, and let $\\mu, \\theta_1, \\dots, \\theta_q \\in \\R$ with $\\theta_q \\ne 0$.\nA *moving-average model of order $q$*, or MA($q$), is the [[time series|EA3MF4VTEH]] model\n$$X_t = \\mu + \\varepsilon_t + \\theta_1 \\varepsilon_{t-1} + \\cdots + \\theta_q \\varepsilon_{t-q}, \\qquad t \\in \\Z.$$\nIts *moving-average polynomial* is $\\theta(z) = 1 + \\theta_1 z + \\cdots + \\theta_q z^q$.\n\n## Remarks\n\nEvery MA($q$) process is weakly stationary with mean $\\mu$.\nWith $\\theta_0 = 1$, 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 $\\abs{h} \u003e q$.\nSo its [[autocorrelation function|YCZHYY8VE5]] vanishes beyond lag $q$.\nThe model is *invertible* when $\\theta$ has no [[root|019MA1VTR2]] in the closed unit disk.\nThen $\\varepsilon_t$ is a convergent linear combination of $X_t, X_{t-1}, \\dots$, and the process is an [[autoregressive model|2JC96FMGA1]] of infinite order.\n\nExample: 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.\nHence $\\rho(1) = \\theta / (1 + \\theta^2)$, which always lies in $[-1/2, 1/2]$.\n\nNon-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.\n\nThe name has nothing to do with the moving average used to smooth data, although both are weighted sums over a window.\n",
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