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  "body": "\n## Definition\n\nLet $(X_t)_{t \\in \\Z}$ be a real weakly [[stationary process|DT0QXJE705]] with mean $\\mu$ and autocovariance $\\gamma$, with $\\gamma(0) \u003e 0$.\nLet $L^2$ be the space of square-integrable random variables.\nFor $t \\in \\Z$ and $h \\ge 2$, let $P_{t,h} Y$ be the [[orthogonal projection|0B8KZRT28M]] in $L^2$ of $Y$ onto the span of $1, X_{t+1}, \\dots, X_{t+h-1}$.\nSo $P_{t,h} Y$ is the best linear predictor of $Y$ from these variables.\nThe *partial autocorrelation function* $\\alpha$ is defined for $h \\ge 1$ by:\n\n1. $\\alpha(1) = \\rho(1)$, where $\\rho$ is the [[autocorrelation function|YCZHYY8VE5]].\n2. For $h \\ge 2$, $\\alpha(h)$ is the [[correlation coefficient|0DNN9B6Q2F]] of $X_{t+h} - P_{t,h} X_{t+h}$ and $X_t - P_{t,h} X_t$.\n\nBy stationarity, $\\alpha(h)$ does not depend on $t$.\n\n## Remarks\n\nEquivalently, $\\alpha(h) = \\phi_{hh}$, the last coefficient of the best linear predictor $\\mu + \\sum_{j=1}^{h} \\phi_{hj} (X_{t+h-j} - \\mu)$ of $X_{t+h}$ from $X_{t+h-1}, \\dots, X_t$, provided the $h \\times h$ matrix $[\\gamma(i - j)]$ is nonsingular.\nThe sample partial autocorrelation is obtained by putting the sample autocovariances into these formulas, for instance by the Durbin–Levinson recursion.\n\nExample: let $X_t = \\phi X_{t-1} + \\varepsilon_t$ be a stationary [[autoregressive model|2JC96FMGA1]] of order $1$ with $\\abs{\\phi} \u003c 1$ and uncorrelated noise $\\varepsilon_t$.\nThen $\\alpha(1) = \\rho(1) = \\phi$.\nFor $h \\ge 2$, the best linear predictor of $X_{t+h}$ from $X_{t+1}, \\dots, X_{t+h-1}$ is $\\phi X_{t+h-1}$, and the residual $\\varepsilon_{t+h}$ is uncorrelated with $X_t - P_{t,h} X_t$, so $\\alpha(h) = 0$.\n\nNon-example: a random walk is not weakly stationary, so clause 1 and clause 2 do not define a function of the lag alone.\n",
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