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  "body": "\n## Definition\n\nA real time series is *integrated of order $1$*, written I($1$), if it is not weakly [[stationary|DT0QXJE705]] but its first difference $\\Delta x_t = x_t - x_{t-1}$ is.\nAn example is an [[ARIMA($p$, $1$, $q$) process|KTDVZP4DF2]] whose moving-average polynomial has no root at $1$, such as a random walk.\nLet $x_t = (x_{1,t}, \\dots, x_{k,t})^T$, with $k \\ge 2$, be a vector time series whose components are all I($1$).\nThe series $x_t$ is *cointegrated* if there is a vector $\\beta \\in \\R^k$, $\\beta \\ne 0$, such that the [[linear combination|0V8F3E48E6]] $\\beta^T x_t$ is weakly stationary.\nSuch a $\\beta$ is a *cointegrating vector*.\n\n## Remarks\n\nThe cointegrating vectors, together with $0$, form a linear subspace of $\\R^k$; its dimension $r$, with $0 \\le r \\le k - 1$, is the cointegrating rank.\nMore generally, components that are I($d$) are cointegrated of order ($d$, $b$) when some nonzero combination is I($d - b$) with $b \u003e 0$.\n\nExample: let $w_t$ be a [[random walk|04VZE3FZXF]] and $e_t$ a white noise independent of it, and set $x_{1,t} = w_t$ and $x_{2,t} = 2 w_t + e_t$.\nBoth components are I($1$), and $\\beta = (2, -1)^T$ gives $\\beta^T x_t = -e_t$, which is stationary, so the pair is cointegrated.\n\nNon-example: two independent random walks $w_t$ and $v_t$ with step variance $\\sigma^2 \u003e 0$ are not cointegrated.\nFor $\\beta = (a, b)^T \\ne 0$, the combination $a w_t + b v_t$ is again a random walk with step variance $(a^2 + b^2) \\sigma^2 \u003e 0$, which is not stationary.\n\nIn the Engle–Granger method, $\\beta$ is estimated by regressing one component on the others.\nThen a [[Dickey–Fuller type test|26E0HYRDS8]] is applied to the residuals, with critical values adjusted for the estimation step.\nThe Granger representation theorem links cointegration with error-correction models.\n",
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