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  "body": "\n## Definition\n\nLet $y_0, y_1, \\dots, y_n$ follow the [[autoregressive model|2JC96FMGA1]] $y_t = \\rho y_{t-1} + u_t$.\nHere the $u_t$ are independent and identically distributed with mean $0$ and variance $\\sigma^2 \u003e 0$.\nWrite it as $\\Delta y_t = \\delta y_{t-1} + u_t$ with $\\Delta y_t = y_t - y_{t-1}$ and $\\delta = \\rho - 1$.\nLet $\\hat\\delta$ be the [[ordinary least squares|YM708KHV2W]] estimate of $\\delta$ in the regression of $\\Delta y_t$ on $y_{t-1}$.\nLet $\\operatorname{se}(\\hat\\delta)$ be its usual standard error, and let $\\tau = \\hat\\delta / \\operatorname{se}(\\hat\\delta)$.\nThe *Dickey–Fuller test* is the [[test|087P03NEND]], based on $\\tau$, of\n$$H_0 \\colon \\delta = 0 \\ (\\text{a unit root}) \\qquad \\text{against} \\qquad H_1 \\colon \\delta \u003c 0 \\ (\\text{stationarity}).$$\nIt rejects $H_0$ when $\\tau$ is below a critical value of the Dickey–Fuller distribution.\n\n## Remarks\n\nUnder $H_0$, the series $y_t$ is a [[random walk|04VZE3FZXF]], an [[ARIMA($0$, $1$, $0$) process|KTDVZP4DF2]], and the usual central limit theory fails.\nIf $y_0 = 0$, then as $n \\to \\infty$\n$$\\tau \\to \\frac{W(1)^2 - 1}{2 \\left( \\int_0^1 W(r)^2 \\, dr \\right)^{1/2}}$$\nin distribution, where $W$ is a standard [[Wiener process|22735FGJV6]].\nThis limit is skewed to the left, so its critical values lie well below those of the standard normal distribution; they are tabulated.\nVariants add a constant, or a constant and a linear trend, to the regression, and each variant has its own limiting distribution and critical values.\nThe *augmented* Dickey–Fuller test adds lagged differences $\\Delta y_{t-1}, \\dots, \\Delta y_{t-k}$ to the regression, to allow for autocorrelated errors.\n\nExample: for a series with $\\hat\\delta = -0.2$ and $\\operatorname{se}(\\hat\\delta) = 0.05$, the statistic is $\\tau = -4$.\nThis lies far in the left tail of the Dickey–Fuller distribution, so the unit root hypothesis is rejected at the usual levels.\n\nNon-example: comparing $\\tau$ with the critical values of the Student $t$ distribution is not the Dickey–Fuller test.\nThat rule rejects a true unit root far too often, because the null distribution of $\\tau$ is not a $t$ distribution.\n",
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