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  "body": "\n## Definition\n\nLet $B$ be the backshift operator, $B X_t = X_{t-1}$, and let $\\nabla = 1 - B$ be the difference operator, so $\\nabla X_t = X_t - X_{t-1}$.\nLet $p, d, q \\ge 0$ be integers.\nA [[time series|EA3MF4VTEH]] $(X_t)$ is an *ARIMA($p$, $d$, $q$) process* if the differenced series $Y_t = \\nabla^d X_t$ is a causal [[ARMA($p$, $q$) process|B1HNX0YT0Z]].\nEquivalently, $\\phi(B) (1 - B)^d X_t = c + \\theta(B) \\varepsilon_t$ for a white noise $(\\varepsilon_t)$, a constant $c$, and polynomials $\\phi$ of degree $p$ and $\\theta$ of degree $q$.\nHere $\\phi(0) = \\theta(0) = 1$, and $\\phi$ has no root in the closed unit disk.\n\n## Remarks\n\nFor $d = 0$ this is a causal ARMA($p$, $q$) process.\nFor $d \\ge 1$ the process is not stationary in general.\nIt is determined by $Y$ only up to $d$ starting values, since $\\nabla^d$ kills polynomials in $t$ of degree less than $d$.\nA series whose $d$-th difference is stationary but whose $(d-1)$-th difference is not is called integrated of order $d$, written I($d$).\n\nExample: the [[random walk|04VZE3FZXF]] $X_t = X_{t-1} + \\varepsilon_t$ is ARIMA($0$, $1$, $0$), because $\\nabla X_t = \\varepsilon_t$ is a white noise, which is ARMA($0$, $0$).\nThe model $X_t = X_{t-1} + 0.5 (X_{t-1} - X_{t-2}) + \\varepsilon_t$ is ARIMA($1$, $1$, $0$), since $\\nabla X_t = 0.5 \\nabla X_{t-1} + \\varepsilon_t$.\n\nNon-example: the explosive series with $X_0 = 0$ and $X_t = 1.5 X_{t-1} + \\varepsilon_t$ for $t \\ge 1$ is not ARIMA for any $d$.\nIts autoregressive polynomial $1 - 1.5 z$ has the root $2/3$ inside the unit disk, so the variance of $X_t$ grows geometrically.\nDifferencing does not remove this explosive root, because $\\nabla^d$ only contributes the root $1$, so $\\nabla^d X_t$ is never stationary.\n\nSeasonal ARIMA models also apply seasonal differences $1 - B^s$ for a period $s$, such as $s = 12$ for monthly data.\n",
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