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  "body": "\n## Statement\n\nLet $(\\varepsilon_t)_{t \\in \\Z}$ be a white noise with variance $\\sigma^2 \u003e 0$.\nLet $(X_t)_{t \\in \\Z}$ be a causal [[autoregressive model|2JC96FMGA1]] of order $p$ with mean $0$,\n$$X_t = \\phi_1 X_{t-1} + \\cdots + \\phi_p X_{t-p} + \\varepsilon_t,$$\nso that $X_t = \\sum_{j \\ge 0} \\psi_j \\varepsilon_{t-j}$ with $\\sum_j \\abs{\\psi_j} \u003c \\infty$.\nLet $\\gamma(h) = E[X_{t+h} X_t]$ be its autocovariance function.\nThen\n$$\\gamma(h) = \\phi_1 \\gamma(h-1) + \\cdots + \\phi_p \\gamma(h-p) \\quad \\text{for } h \\ge 1, \\qquad \\gamma(0) = \\phi_1 \\gamma(1) + \\cdots + \\phi_p \\gamma(p) + \\sigma^2.$$\nIn matrix form, $\\Gamma_p \\phi = \\gamma_p$ and $\\sigma^2 = \\gamma(0) - \\phi^T \\gamma_p$, where $\\Gamma_p = [\\gamma(i - j)]_{i,j=1}^{p}$, $\\phi = (\\phi_1, \\dots, \\phi_p)^T$ and $\\gamma_p = (\\gamma(1), \\dots, \\gamma(p))^T$.\n\n## Remarks\n\nMultiply the defining equation by $X_{t-h}$ and take expectations.\nFor $h \\ge 1$, causality makes $X_{t-h}$ a function of $\\varepsilon_s$ with $s \\le t - h \u003c t$, so $E[\\varepsilon_t X_{t-h}] = 0$, which gives the first equations.\nFor $h = 0$, $E[\\varepsilon_t X_t] = E[\\varepsilon_t^2] = \\sigma^2$, which gives the last one.\nThe matrix $\\Gamma_p$ is a symmetric [[Toeplitz matrix|GKY2WSHE26]], and it is nonsingular because $\\gamma(0) \u003e 0$ and $\\gamma(h) \\to 0$ as $h \\to \\infty$.\nSo the autocovariances $\\gamma(0), \\dots, \\gamma(p)$ determine $\\phi$ and $\\sigma^2$.\n\nExample: for $p = 1$, the equations read $\\gamma(1) = \\phi_1 \\gamma(0)$ and $\\gamma(0) = \\phi_1 \\gamma(1) + \\sigma^2$.\nHence $\\phi_1 = \\rho(1)$ and $\\gamma(0) = \\sigma^2 / (1 - \\phi_1^2)$.\n\nThe *Yule–Walker estimator* replaces $\\gamma$ by the sample autocovariance $\\hat\\gamma$ and solves $\\hat\\Gamma_p \\hat\\phi = \\hat\\gamma_p$.\nCausality is needed: for the non-causal stationary solution of $X_t = 2 X_{t-1} + \\varepsilon_t$, the shock $\\varepsilon_t$ is correlated with the past values $X_{t-h}$ and the first equation fails.\n\n## Explanations\n\n### L1 human\n\nIn an autoregressive model, each value is a weighted sum of earlier values plus fresh noise.\nThe Yule–Walker equations say that the correlations of the series obey the same recursion with the same weights.\nSo once the correlations are known, the weights can be found by solving a small linear system.\n\n### L2 human\n\nThe equations give the simplest estimator of the coefficients of an AR($p$) model, computed from the sample [[autocorrelation function|YCZHYY8VE5]].\nFor a Gaussian causal AR process, this estimator has the same asymptotic distribution as the maximum likelihood estimator.\nThe Durbin–Levinson algorithm solves the Toeplitz system for $p = 1, 2, 3, \\dots$ in turn, and the last coefficient at each order is the partial autocorrelation.\nThe equations are named after Udny Yule and Gilbert Walker.\n",
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