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  "body": "\n## Definition\n\nLet $x_1, \\dots, x_n$ be an observed time series.\nThe *Box–Jenkins method* fits an [[ARIMA($p$, $d$, $q$) model|KTDVZP4DF2]] to the series and then uses it to forecast.\nIt repeats the following three stages until the checks of stage 3 are passed:\n\n1. Identification: choose $d$ so that the differenced series $\\nabla^d x_t$ looks stationary. Then choose tentative orders $p$ and $q$ from the sample [[autocorrelation function|YCZHYY8VE5]] and [[partial autocorrelation function|11YRFWD951]] of $\\nabla^d x_t$.\n2. Estimation: estimate the parameters of the [[ARMA($p$, $q$) model|B1HNX0YT0Z]] for the differenced series by [[maximum likelihood|0FREPFKNS7]] or by nonlinear [[least squares|0BYTQ4X1HC]].\n3. Diagnostic checking: test whether the residuals of the fitted model behave like white noise, for instance with a portmanteau test. If they do not, return to stage 1 with a modified model.\n\n## Remarks\n\nThe identification stage uses typical patterns.\nFor an AR($p$) model the partial autocorrelation vanishes beyond lag $p$ while the autocorrelation decays gradually.\nFor an MA($q$) model the autocorrelation vanishes beyond lag $q$ while the partial autocorrelation decays gradually.\n\nExample: suppose the sample autocorrelation of a series decays very slowly.\nSuppose that of its first differences has a single significant value, at lag $1$.\nThen stage 1 suggests $d = 1$, $p = 0$ and $q = 1$, that is ARIMA($0$, $1$, $1$).\n\nNon-example: fitting one fixed AR($1$) model by least squares without examining the residuals is not the Box–Jenkins method, because stage 3 and the loop back to stage 1 are missing.\n\nThe method prefers parsimonious models, with as few parameters as give an adequate fit.\nSeasonal ARIMA models are handled in the same way, with seasonal differences and seasonal orders.\n",
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