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  "body": "\n## Definition\n\nLet $(\\varepsilon_t)_{t \\in \\Z}$ be a white noise with variance $\\sigma^2 \u003e 0$.\nLet $p, q \\ge 0$, and let $\\phi(z) = 1 - \\phi_1 z - \\cdots - \\phi_p z^p$ and $\\theta(z) = 1 + \\theta_1 z + \\cdots + \\theta_q z^q$ be real polynomials with no common [[root|019MA1VTR2]].\nAn *ARMA($p$, $q$) process* is a weakly [[stationary process|DT0QXJE705]] $(X_t)_{t \\in \\Z}$ with mean $\\mu$ such that\n$$X_t - \\mu - \\sum_{i=1}^{p} \\phi_i (X_{t-i} - \\mu) = \\varepsilon_t + \\sum_{j=1}^{q} \\theta_j \\varepsilon_{t-j} \\qquad \\text{for all } t \\in \\Z.$$\nWith the backshift operator $B X_t = X_{t-1}$, this reads $\\phi(B)(X_t - \\mu) = \\theta(B) \\varepsilon_t$.\n\n## Remarks\n\nThe case $q = 0$ is the [[autoregressive model|2JC96FMGA1]] and the case $p = 0$ is the [[moving-average model|PEB9DN0AHA]].\nA stationary solution exists, and is unique, exactly when $\\phi(z) \\ne 0$ for $\\abs{z} = 1$.\nIt is causal, a function of present and past shocks only, when $\\phi$ has no root with $\\abs{z} \\le 1$, and invertible when $\\theta$ has no root with $\\abs{z} \\le 1$.\nThe condition of no common roots excludes redundant parameters.\n\nExample: $X_t = 0.5 X_{t-1} + \\varepsilon_t + 0.3 \\varepsilon_{t-1}$ has $\\phi(z) = 1 - 0.5 z$ with root $2$ and $\\theta(z) = 1 + 0.3 z$ with root $-10/3$.\nBoth roots lie outside the unit disk and they differ, so this is a causal and invertible ARMA($1$, $1$) process.\n\nNon-example: $X_t - 0.5 X_{t-1} = \\varepsilon_t - 0.5 \\varepsilon_{t-1}$ fails the condition on common roots, since both polynomials equal $1 - 0.5 z$.\nIts stationary solution is just $X_t = \\varepsilon_t$, a white noise, so the two parameters are not identifiable.\n",
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