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href="#Box–Jenkins_model_identification"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Box–Jenkins model identification</span> </div> </a> <button aria-controls="toc-Box–Jenkins_model_identification-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Box–Jenkins model identification subsection</span> </button> <ul id="toc-Box–Jenkins_model_identification-sublist" class="vector-toc-list"> <li id="toc-Stationarity_and_seasonality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stationarity_and_seasonality"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Stationarity and seasonality</span> </div> </a> <ul id="toc-Stationarity_and_seasonality-sublist" class="vector-toc-list"> <li id="toc-Detecting_stationarity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Detecting_stationarity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Detecting stationarity</span> </div> </a> <ul id="toc-Detecting_stationarity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Detecting_seasonality" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Detecting_seasonality"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Detecting seasonality</span> </div> </a> <ul id="toc-Detecting_seasonality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differencing_to_achieve_stationarity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Differencing_to_achieve_stationarity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Differencing to achieve stationarity</span> </div> </a> <ul id="toc-Differencing_to_achieve_stationarity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Seasonal_differencing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Seasonal_differencing"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.4</span> <span>Seasonal differencing</span> </div> </a> <ul id="toc-Seasonal_differencing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Identify_p_and_q" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identify_p_and_q"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Identify <i>p</i> and <i>q</i></span> </div> </a> <ul id="toc-Identify_p_and_q-sublist" class="vector-toc-list"> <li id="toc-Autocorrelation_and_partial_autocorrelation_plots" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Autocorrelation_and_partial_autocorrelation_plots"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Autocorrelation and partial autocorrelation plots</span> </div> </a> <ul id="toc-Autocorrelation_and_partial_autocorrelation_plots-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Box–Jenkins_model_estimation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Box–Jenkins_model_estimation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Box–Jenkins model estimation</span> </div> </a> <ul id="toc-Box–Jenkins_model_estimation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Box–Jenkins_model_diagnostics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Box–Jenkins_model_diagnostics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Box–Jenkins model diagnostics</span> </div> </a> <button aria-controls="toc-Box–Jenkins_model_diagnostics-sublist" class="cdx-button cdx-button--weight-quiet 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class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> named after the <a href="/wiki/Statistician" title="Statistician">statisticians</a> <a href="/wiki/George_Box" class="mw-redirect" title="George Box">George Box</a> and <a href="/wiki/Gwilym_Jenkins" title="Gwilym Jenkins">Gwilym Jenkins</a>, applies <a href="/wiki/Autoregressive_moving_average" class="mw-redirect" title="Autoregressive moving average">autoregressive moving average</a> (ARMA) or <a href="/wiki/Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">autoregressive integrated moving average</a> (ARIMA) models to find the best fit of a time-series model to past values of a <a href="/wiki/Time_series" title="Time series">time series</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Modeling_approach">Modeling approach</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=1" title="Edit section: Modeling approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The original model uses an iterative three-stage modeling approach: </p> <ol><li><i>Model identification and <a href="/wiki/Model_selection" title="Model selection">model selection</a></i>: making sure that the variables are <a href="/wiki/Stationary_process" title="Stationary process">stationary</a>, identifying <a href="/wiki/Seasonality" title="Seasonality">seasonality</a> in the dependent series (seasonally differencing it if necessary), and using plots of the <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation (ACF)</a> and <a href="/wiki/Partial_autocorrelation" class="mw-redirect" title="Partial autocorrelation">partial autocorrelation (PACF)</a> functions of the dependent time series to decide which (if any) autoregressive or moving average component should be used in the model.</li> <li><i><a href="/wiki/Parameter_estimation" class="mw-redirect" title="Parameter estimation">Parameter estimation</a></i> using computation algorithms to arrive at coefficients that best fit the selected ARIMA model. The most common methods use <a href="/wiki/Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a> or <a href="/wiki/Non-linear_least-squares_estimation" class="mw-redirect" title="Non-linear least-squares estimation">non-linear least-squares estimation</a>.</li> <li><i><a href="/wiki/Statistical_model_validation" title="Statistical model validation">Statistical model checking</a></i> by testing whether the estimated model conforms to the specifications of a stationary univariate process. In particular, the residuals should be independent of each other and constant in mean and variance over time. (Plotting the mean and variance of residuals over time and performing a <a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Ljung–Box test</a> or plotting autocorrelation and partial autocorrelation of the residuals are helpful to identify misspecification.) If the estimation is inadequate, we have to return to step one and attempt to build a better model.</li></ol> <p>The data they used were from a gas furnace. These data are well known as the Box and Jenkins gas furnace data for benchmarking predictive models. </p><p>Commandeur & Koopman (2007, §10.4)<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> argue that the Box–Jenkins approach is fundamentally problematic. The problem arises because in "the economic and social fields, real series are never stationary however much differencing is done". Thus the investigator has to face the question: how close to stationary is close enough? As the authors note, "This is a hard question to answer". The authors further argue that rather than using Box–Jenkins, it is better to use state space methods, as stationarity of the time series is then not required. </p> <div class="mw-heading mw-heading2"><h2 id="Box–Jenkins_model_identification"><span id="Box.E2.80.93Jenkins_model_identification"></span>Box–Jenkins model identification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=2" title="Edit section: Box–Jenkins model identification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Stationarity_and_seasonality">Stationarity and seasonality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=3" title="Edit section: Stationarity and seasonality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first step in developing a Box–Jenkins model is to determine whether the <a href="/wiki/Time_series" title="Time series">time series</a> is <a href="/wiki/Stationary_process" title="Stationary process">stationary</a> and whether there is any significant <a href="/wiki/Seasonality" title="Seasonality">seasonality</a> that needs to be modelled. </p> <div class="mw-heading mw-heading4"><h4 id="Detecting_stationarity">Detecting stationarity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=4" title="Edit section: Detecting stationarity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stationarity can be assessed from a <a href="/wiki/Run_sequence_plot" class="mw-redirect" title="Run sequence plot">run sequence plot</a>. The run sequence plot should show constant location and <a href="/wiki/Scale_(ratio)" title="Scale (ratio)">scale</a>. It can also be detected from an <a href="/wiki/Autocorrelation_plot" class="mw-redirect" title="Autocorrelation plot">autocorrelation plot</a>. Specifically, non-stationarity is often indicated by an autocorrelation plot with very slow decay. One can also utilize a <a href="/wiki/Dickey-Fuller_test" class="mw-redirect" title="Dickey-Fuller test">Dickey-Fuller test</a> or <a href="/wiki/Augmented_Dickey-Fuller_test" class="mw-redirect" title="Augmented Dickey-Fuller test">Augmented Dickey-Fuller test</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Detecting_seasonality">Detecting seasonality</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=5" title="Edit section: Detecting seasonality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Seasonality (or periodicity) can usually be assessed from an autocorrelation plot, a <a href="/wiki/Seasonal_subseries_plot" title="Seasonal subseries plot">seasonal subseries plot</a>, or a <a href="/wiki/Spectral_plot" class="mw-redirect" title="Spectral plot">spectral plot</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Differencing_to_achieve_stationarity">Differencing to achieve stationarity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=6" title="Edit section: Differencing to achieve stationarity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Box and Jenkins recommend the differencing approach to achieve stationarity. However, <a href="/wiki/Curve_fitting" title="Curve fitting">fitting a curve</a> and subtracting the fitted values from the original data can also be used in the context of Box–Jenkins models. </p> <div class="mw-heading mw-heading4"><h4 id="Seasonal_differencing">Seasonal differencing</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=7" title="Edit section: Seasonal differencing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At the model identification stage, the goal is to detect seasonality, if it exists, and to identify the order for the seasonal autoregressive and seasonal moving average terms. For many series, the period is known and a single seasonality term is sufficient. For example, for monthly data one would typically include either a seasonal AR 12 term or a seasonal MA 12 term. For Box–Jenkins models, one does not explicitly remove seasonality before fitting the model. Instead, one includes the order of the seasonal terms in the model specification to the <a href="/wiki/ARIMA" class="mw-redirect" title="ARIMA">ARIMA</a> estimation software. However, it may be helpful to apply a seasonal difference to the data and regenerate the autocorrelation and partial autocorrelation plots. This may help in the model identification of the non-seasonal component of the model. In some cases, the seasonal differencing may remove most or all of the seasonality effect. </p> <div class="mw-heading mw-heading3"><h3 id="Identify_p_and_q">Identify <i>p</i> and <i>q</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=8" title="Edit section: Identify p and q"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Once stationarity and seasonality have been addressed, the next step is to identify the order (i.e. the <i>p</i> and <i>q</i>) of the autoregressive and moving average terms. Different authors have different approaches for identifying <i>p</i> and <i>q</i>. Brockwell and Davis (1991)<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> state "our prime criterion for model selection [among ARMA(p,q) models] will be the AICc", i.e. the <a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">Akaike information criterion</a> with correction. Other authors use the autocorrelation plot and the partial autocorrelation plot, described below. </p> <div class="mw-heading mw-heading4"><h4 id="Autocorrelation_and_partial_autocorrelation_plots">Autocorrelation and partial autocorrelation plots</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=9" title="Edit section: Autocorrelation and partial autocorrelation plots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sample autocorrelation plot and the sample partial autocorrelation plot are compared to the theoretical behavior of these plots when the order is known. </p><p>Specifically, for an <a href="/wiki/AR(1)" class="mw-redirect" title="AR(1)">AR(1)</a> process, the sample autocorrelation function should have an exponentially decreasing appearance. However, higher-order AR processes are often a mixture of exponentially decreasing and damped sinusoidal components. </p><p>For higher-order autoregressive processes, the sample autocorrelation needs to be supplemented with a partial autocorrelation plot. The partial autocorrelation of an AR(<i>p</i>) process becomes zero at lag <i>p</i> + 1 and greater, so we examine the sample partial autocorrelation function to see if there is evidence of a departure from zero. This is usually determined by placing a 95% <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> on the sample partial autocorrelation plot (most software programs that generate sample autocorrelation plots also plot this confidence interval). If the software program does not generate the confidence band, it is approximately <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 2/{\sqrt {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>N</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 2/{\sqrt {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066f521b8d0e9aca5613399293d4db74c4b61501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.133ex; height:3.176ex;" alt="{\displaystyle \pm 2/{\sqrt {N}}}"></span>, with <i>N</i> denoting the sample size. </p><p>The autocorrelation function of a <a href="/wiki/Moving_average_model" class="mw-redirect" title="Moving average model">MA(<i>q</i>)</a> process becomes zero at lag <i>q</i> + 1 and greater, so we examine the sample autocorrelation function to see where it essentially becomes zero. We do this by placing the 95% confidence interval for the sample autocorrelation function on the sample autocorrelation plot. Most software that can generate the autocorrelation plot can also generate this confidence interval. </p><p>The sample partial autocorrelation function is generally not helpful for identifying the order of the moving average process. </p><p>The following table summarizes how one can use the sample <a href="/wiki/Autocorrelation_function" class="mw-redirect" title="Autocorrelation function">autocorrelation function</a> for model identification. </p> <table class="wikitable" style="text-align:left"> <tbody><tr> <th>Shape</th> <th>Indicated Model </th></tr> <tr> <th>Exponential, decaying to zero </th> <td><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregressive model</a>. Use the partial autocorrelation plot to identify the order of the autoregressive model. </td></tr> <tr> <th>Alternating positive and negative, decaying to zero </th> <td>Autoregressive model. Use the partial autocorrelation plot to help identify the order. </td></tr> <tr> <th>One or more spikes, rest are essentially zero (or close to zero) </th> <td><a href="/wiki/Moving_average_model" class="mw-redirect" title="Moving average model">Moving average model</a>, order identified by where plot becomes zero. </td></tr> <tr> <th>Decay, starting after a few lags </th> <td>Mixed autoregressive and moving average (<a href="/wiki/Autoregressive_moving_average_model" class="mw-redirect" title="Autoregressive moving average model">ARMA</a>) model. </td></tr> <tr> <th>All zero or close to zero </th> <td>Data are essentially random. </td></tr> <tr> <th>High values at fixed intervals </th> <td>Include seasonal autoregressive term. </td></tr> <tr> <th>No decay to zero (or it decays extremely slowly) </th> <td>Series is not stationary. </td></tr></tbody></table> <p>Hyndman & Athanasopoulos suggest the following:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>The data may follow an ARIMA(<i>p</i>,<i>d</i>,0) model if the ACF and PACF plots of the differenced data show the following patterns: <ul><li>the ACF is exponentially decaying or sinusoidal;</li> <li>there is a significant spike at lag <i>p</i> in PACF, but none beyond lag <i>p</i>.</li></ul></dd></dl> <dl><dd>The data may follow an ARIMA(0,<i>d</i>,<i>q</i>) model if the ACF and PACF plots of the differenced data show the following patterns: <ul><li>the PACF is exponentially decaying or sinusoidal;</li> <li>there is a significant spike at lag <i>q</i> in ACF, but none beyond lag <i>q</i>.</li></ul></dd></dl> <p>In practice, the sample autocorrelation and partial autocorrelation functions are <a href="/wiki/Random_variable" title="Random variable">random variables</a> and do not give the same picture as the theoretical functions. This makes the model identification more difficult. In particular, mixed models can be particularly difficult to identify. Although experience is helpful, developing good models using these sample plots can involve much trial and error. </p> <div class="mw-heading mw-heading2"><h2 id="Box–Jenkins_model_estimation"><span id="Box.E2.80.93Jenkins_model_estimation"></span>Box–Jenkins model estimation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=10" title="Edit section: Box–Jenkins model estimation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Estimating the parameters for Box–Jenkins models involves numerically approximating the solutions of nonlinear equations. For this reason, it is common to use statistical software designed to handle to the approach – virtually all modern statistical packages feature this capability. The main approaches to fitting Box–Jenkins models are nonlinear least squares and maximum likelihood estimation. Maximum likelihood estimation is generally the preferred technique. The likelihood equations for the full Box–Jenkins model are complicated and are not included here. See (Brockwell and Davis, 1991) for the mathematical details. </p> <div class="mw-heading mw-heading2"><h2 id="Box–Jenkins_model_diagnostics"><span id="Box.E2.80.93Jenkins_model_diagnostics"></span>Box–Jenkins model diagnostics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=11" title="Edit section: Box–Jenkins model diagnostics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Assumptions_for_a_stable_univariate_process">Assumptions for a stable univariate process</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=12" title="Edit section: Assumptions for a stable univariate process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Model diagnostics for Box–Jenkins models is similar to model validation for non-linear least squares fitting. </p><p>That is, the error term <i>A<sub>t</sub></i> is assumed to follow the assumptions for a stationary univariate process. The residuals should be <a href="/wiki/White_noise" title="White noise">white noise</a> (or independent when their distributions are normal) drawings from a fixed distribution with a constant mean and variance. If the Box–Jenkins model is a good model for the data, the residuals should satisfy these assumptions. </p><p>If these assumptions are not satisfied, one needs to fit a more appropriate model. That is, go back to the model identification step and try to develop a better model. Hopefully the analysis of the residuals can provide some clues as to a more appropriate model. </p><p>One way to assess if the residuals from the Box–Jenkins model follow the assumptions is to generate <a href="/wiki/Statistical_graphics" title="Statistical graphics">statistical graphics</a> (including an autocorrelation plot) of the residuals. One could also look at the value of the <a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Box–Ljung statistic</a>. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBoxJenkins1970" class="citation book cs1">Box, George; Jenkins, Gwilym (1970). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/timeseriesanalys0000boxg"><i>Time Series Analysis: Forecasting and Control</i></a></span>. San Francisco: Holden-Day.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time+Series+Analysis%3A+Forecasting+and+Control&rft.place=San+Francisco&rft.pub=Holden-Day&rft.date=1970&rft.aulast=Box&rft.aufirst=George&rft.au=Jenkins%2C+Gwilym&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftimeseriesanalys0000boxg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABox%E2%80%93Jenkins+method" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCommandeurKoopman2007" class="citation book cs1">Commandeur, J. J. F.; Koopman, S. J. (2007). <i>Introduction to State Space Time Series Analysis</i>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+State+Space+Time+Series+Analysis&rft.pub=Oxford+University+Press&rft.date=2007&rft.aulast=Commandeur&rft.aufirst=J.+J.+F.&rft.au=Koopman%2C+S.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABox%E2%80%93Jenkins+method" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrockwellDavis1991" class="citation book cs1">Brockwell, Peter J.; Davis, Richard A. (1991). <i>Time Series: Theory and Methods</i>. Springer-Verlag. p. 273. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991tstm.book.....B">1991tstm.book.....B</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Time+Series%3A+Theory+and+Methods&rft.pages=273&rft.pub=Springer-Verlag&rft.date=1991&rft_id=info%3Abibcode%2F1991tstm.book.....B&rft.aulast=Brockwell&rft.aufirst=Peter+J.&rft.au=Davis%2C+Richard+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABox%E2%80%93Jenkins+method" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHyndmanAthanasopoulos" class="citation book cs1">Hyndman, Rob J; Athanasopoulos, George. <a rel="nofollow" class="external text" href="https://www.otexts.org/fpp/8/5"><i>Forecasting: principles and practice</i></a><span class="reference-accessdate">. Retrieved <span class="nowrap">18 May</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Forecasting%3A+principles+and+practice&rft.aulast=Hyndman&rft.aufirst=Rob+J&rft.au=Athanasopoulos%2C+George&rft_id=https%3A%2F%2Fwww.otexts.org%2Ffpp%2F8%2F5&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABox%E2%80%93Jenkins+method" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeveridgeOickle1994" class="citation cs2">Beveridge, S.; Oickle, C. (1994), "Comparison of Box–Jenkins and objective methods for determining the order of a non-seasonal ARMA model", <i><a href="/wiki/Journal_of_Forecasting" title="Journal of Forecasting">Journal of Forecasting</a></i>, <b>13</b> (5): 419–434, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Ffor.3980130502">10.1002/for.3980130502</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Forecasting&rft.atitle=Comparison+of+Box%E2%80%93Jenkins+and+objective+methods+for+determining+the+order+of+a+non-seasonal+ARMA+model&rft.volume=13&rft.issue=5&rft.pages=419-434&rft.date=1994&rft_id=info%3Adoi%2F10.1002%2Ffor.3980130502&rft.aulast=Beveridge&rft.aufirst=S.&rft.au=Oickle%2C+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABox%E2%80%93Jenkins+method" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPankratz1983" class="citation cs2">Pankratz, Alan (1983), <i>Forecasting with Univariate Box–Jenkins Models: Concepts and Cases</i>, <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Forecasting+with+Univariate+Box%E2%80%93Jenkins+Models%3A+Concepts+and+Cases&rft.pub=John+Wiley+%26+Sons&rft.date=1983&rft.aulast=Pankratz&rft.aufirst=Alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABox%E2%80%93Jenkins+method" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Box%E2%80%93Jenkins_method&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070318000551/http://statistik.mathematik.uni-wuerzburg.de/timeseries/">A First Course on Time Series Analysis</a> – an open source book on time series analysis with SAS (Chapter 7)</li> <li><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc445.htm">Box–Jenkins models</a> in the Engineering Statistics Handbook of <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a></li> <li><a rel="nofollow" class="external text" href="http://robjhyndman.com/papers/BoxJenkins.pdf">Box–Jenkins modelling</a> by Rob J Hyndman</li> <li><a rel="nofollow" class="external text" href="http://support.sas.com/resources/papers/proceedings13/454-2013.pdf">The Box–Jenkins methodology for time series models</a> by Theresa Hoang Diem Ngo</li></ul> <p><span class="noviewer" typeof="mw:File"><span><img alt="Public Domain" 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