{ "id": "eprdzkd86z", "slug": "box-jenkins-method", "title": "Box–Jenkins method", "updated": "2026-10-21", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_stochastic_processes", "label": "Stochastic processes", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_stochastic_processes" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Non-geographic · 1946 to 2000: Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" } ] } ], "excerpt": "The Box–Jenkins method is an iterative procedure in time series analysis for building ARIMA models, described by George Box and Gwilym Jenkins in their 1970 book Time Series Analysis: Forecasting and Control.", "snippet": "The Box–Jenkins method is an iterative procedure in time series analysis for building ARIMA models, described by George Box and Gwilym Jenkins in their 1970 book Time Series Analysis: Forecasting and Control.", "node": "physical.math.statistics.stats_stochastic_processes", "markdown": "# Box–Jenkins method\n\nThe Box–Jenkins method is an iterative procedure in time series analysis for building [autoregressive integrated moving average](https://www.edgepedia.org/autoregressive-integrated-moving-average) (ARIMA) models: it identifies a model class from the data, estimates its parameters, checks the residuals, and produces forecasts, with further uses in control and intervention analysis. It was described in the book *Time Series Analysis: Forecasting and Control*, published by Holden-Day in San Francisco.[1](https://garfield.library.upenn.edu/classics1989/A1989AV48500001.pdf) The method's output is a fitted ARIMA model together with optimal forecasts.[2](https://www.wiley.com/en-us/Time+Series+Analysis:+Forecasting+and+Control,+5th+Edition-p-9781118675021)\n\n| Key fact | Detail |\n|---|---|\n| What it produces | A fitted ARIMA(p,d,q) model, diagnostic checks, and forecasts[3](https://numdam.org/item/RO_1977__11_2_129_0.pdf) |\n| Core cycle | Three iterative stages: identification, estimation, diagnostic checking[4](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/The_Box-Jenkins_Method.pdf) |\n| Model form | \\( \\phi_{p}(B)(1-B)^{d} X_{t} = \\theta_{0} + \\theta_{q}(B) a_{t} \\), with \\( B^{j} x_{t} = x_{t-j} \\)[3](https://numdam.org/item/RO_1977__11_2_129_0.pdf) |\n| Minimum data | At least 40 observations by one account, at least 50 by another; longer for seasonal models[5](https://numdam.org/item/RO_1977__11_1_3_0.pdf) • [4](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/The_Box-Jenkins_Method.pdf) |\n| Order identification | ACF and PACF patterns distinguish AR, MA, and mixed ARMA models[5](https://numdam.org/item/RO_1977__11_1_3_0.pdf) |\n| Main diagnostic | Residual white-noise test comparing a Q statistic with chi-square values for \\( M - p - q \\) degrees of freedom, \\( M \\geq 20 \\)[3](https://numdam.org/item/RO_1977__11_2_129_0.pdf) |\n| Modern practice | Automatic order selection by information criteria (auto.arima and equivalents)[6](https://www.mdpi.com/1999-5903/15/8/255) |\n\n## How it works\n\nThe method models a stationary time series as an ARIMA(p,d,q) process. The general non-seasonal form is\n\n\\[ \\phi_{p}(B)(1-B)^{d} X_{t} = \\theta_{0} + \\theta_{q}(B) a_{t} \\]\n\nwhere \\( p \\) is the autoregressive degree, \\( d \\) the differencing degree, \\( q \\) the moving-average degree, and \\( B \\) the backshift operator, defined by \\( B^{j} x_{t} = x_{t-j} \\); the differenced series \\( w_{t} = (1-B)^{d} X_{t} \\) is stationary for integer \\( d \\).[3](https://numdam.org/item/RO_1977__11_2_129_0.pdf) The Box–Jenkins model assumes the series is stationary, and differencing a non-stationary series one or more times to achieve stationarity (statistical properties that don't change over time) is what produces an ARIMA model, with the \"I\" standing for \"Integrated\".[7](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc445.htm)\n\nStationarity and invertibility are ensured when all roots of \\( \\phi_{p}(B) = 0 \\) and \\( \\theta_{q}(B) = 0 \\) lie outside the unit circle, which guarantees the specified model is uniquely representative[3](https://numdam.org/item/RO_1977__11_2_129_0.pdf); equivalently, for ARMA-type models the roots of the polynomial \\( A(z) = 1 - a_{1} z - \\cdots - a_{p} z^{p} \\) must lie outside the unit circle in the z-plane.[8](https://www.mdpi.com/1099-4300/27/3/279)\n\nThe reason for iterating rather than fitting a fixed model is that the correct orders \\( p \\), \\( d \\), and \\( q \\) are not known in advance. ARIMA processes form a rich class of models, so it is usually possible to find a process that adequately describes the data[9](https://robjhyndman.com/papers/BoxJenkins.pdf), but which one requires cycles of tentative specification, fitting, and checking until the diagnostics show no improvement.[4](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/The_Box-Jenkins_Method.pdf)\n\n## How it is done\n\nThe original procedure is an iterative three-stage process of model selection, parameter estimation, and model checking; recent explanations add a preliminary data-preparation stage and a final model-application stage.[9](https://robjhyndman.com/papers/BoxJenkins.pdf)\n\n1. **Identification.** Transform the series if needed (logarithm or power transformation) and difference it \\( d \\) times to stationarity.[10](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_2005_C_BoxJenkinsMethod.pdf) • [7](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc445.htm) Then inspect the sample autocorrelation function (ACF) and partial autocorrelation function (PACF).[5](https://numdam.org/item/RO_1977__11_1_3_0.pdf)\n2. **Reading the ACF and PACF.** The PACF of an AR(p) process becomes zero at lag \\( p+1 \\) and greater, so the sample PACF is examined against a 95% confidence interval for departures from zero.[11](https://itl.nist.gov/div898/handbook/pmc/section4/pmc446.htm) If the partial autocorrelations cut off after a few lags, the last lag with a large value is the estimated \\( p \\)[4](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/The_Box-Jenkins_Method.pdf); a PACF cutting off indicates an AR model, an ACF cutting off indicates an MA(q) model, and if neither cuts off an [ARMA model](https://www.edgepedia.org/autoregressive-moving-average-model) is inferred.[5](https://numdam.org/item/RO_1977__11_1_3_0.pdf) Sample autocorrelations determine \\( q \\) for an MA(q) model.[10](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_2005_C_BoxJenkinsMethod.pdf)\n3. **Estimation.** Parameters are estimated by maximum likelihood[12](https://www.nber.org/system/files/chapters/c4329/c4329.pdf), with backcasting used in estimation.[4](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/The_Box-Jenkins_Method.pdf)\n4. **Diagnostic checking.** The residuals should behave as white noise: a Q statistic is compared with chi-square values for \\( M - p - q \\) degrees of freedom, with \\( M \\geq 20 \\).[3](https://numdam.org/item/RO_1977__11_2_129_0.pdf)\n5. **Iterate.** Final selection may compare candidate models using t-statistics, Durbin–Watson statistics, residual correlation behavior, and predictive success under the principle of parsimony.[13](https://estima.com/webhelp/topics/boxjenkins.html)\n\nFor ARMA-type models the one-step-ahead forecast is \\( \\hat{y}_{t+1} = a_{1} y_{t} + \\cdots + a_{p} y_{t-p+1} \\), and higher-order forecasts follow by reiterating this equation.[8](https://www.mdpi.com/1099-4300/27/3/279)\n\n## Origin\n\nThe book integrated existing knowledge and developed a coherent, versatile three-stage iterative cycle for time series identification, estimation, and verification, known as the Box–Jenkins approach[14](https://robjhyndman.com/papers/ijf25.pdf), and it popularized the use of ARMA models.[15](https://autobox.com/makridakis.pdf) It built on earlier work: authors including Bartlett, Durbin, Hannan, Jenkins, Kendall, Quenouille, and Wald had studied the properties of ARMA processes, their autocorrelation functions, fitting methods, and adequacy tests before the Box–Jenkins systematization.[12](https://www.nber.org/system/files/chapters/c4329/c4329.pdf) • [14](https://robjhyndman.com/papers/ijf25.pdf)\n\n## Variants\n\n**Seasonal models.** SARIMA(p,d,q)(P,D,Q)ₘ adds a second, seasonal set of AR, I, and MA terms at lag \\( m \\) (12 for monthly data, 7 for daily data with a weekly pattern).[16](https://www.forecaststudio.com.mx/blog/arima-sarima-models) Seasonal models need longer series than the non-seasonal case.[5](https://numdam.org/item/RO_1977__11_1_3_0.pdf)\n\n**Intervention analysis.** Intervention analysis extended the framework to address questions of whether, given a known intervention, there is evidence that a change in the series of the kind expected actually occurred.[3](https://numdam.org/item/RO_1977__11_2_129_0.pdf)\n\n**Fractional differencing.** Hosking's 1984 paper in *Water Resources Research*, *Modeling persistence in hydrological time series using fractional differencing*, proposed fractionally differenced ARIMA processes as a more flexible way of simultaneously modeling the long-term and short-term behavior of a time series, covering the \\( d > 0 \\) (long memory) and short memory cases.[17](https://doi.org/10.1029/wr020i012p01898)\n\n**Automatic order selection.** The auto.arima function in R is a popular heuristic based on a step-wise algorithm that starts with a small set of basic ARIMA models, selects the minimum-AIC one, and iterates over variations until no lower AIC is found.[6](https://www.mdpi.com/1999-5903/15/8/255)\n\n## Applications\n\nThe book's five stated application areas are forecasting, determining the transfer function of a system, modeling the effects of intervention events, developing multivariate dynamic models, and designing simple control schemes.[2](https://www.wiley.com/en-us/Time+Series+Analysis:+Forecasting+and+Control,+5th+Edition-p-9781118675021) In hydrology, fractionally differenced ARIMA processes were proposed for modeling persistence in hydrological time series.[17](https://doi.org/10.1029/wr020i012p01898)\n\n## Limitations and alternatives\n\n**Sample size.** The method requires medium-to-long series: one account sets the minimum at 40 observations, longer for seasonal models[5](https://numdam.org/item/RO_1977__11_1_3_0.pdf), another at least 50[4](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/The_Box-Jenkins_Method.pdf), and Granger and Newbold recommend at least 40–50, noting the method requires more data than exponential smoothing.[13](https://estima.com/webhelp/topics/boxjenkins.html)\n\n**Forecast horizon and data regime.** ARIMA and its variants (SARIMA, ARIMAX) perform well for short-term forecasts, but performance is severely degraded for long-term predictions.[6](https://www.mdpi.com/1999-5903/15/8/255) ARIMA can outperform machine-learning counterparts when the dataset has a limited range of values or a limited time span, because deep learning models require large amounts of data to train effectively.[6](https://www.mdpi.com/1999-5903/15/8/255)\n\n**Competition evidence.** In the M3 competition (3003 series), ARIMA with order selection based on Box–Jenkins methodology performed fine but could not beat its competitors, with Theta outperforming all other methods.[18](https://openforecast.org/2024/03/14/the-role-of-m-competitions-in-forecasting/) A key competition lesson is that ARIMA is effective but the manual Box–Jenkins methodology may not be practical, and using information criteria for order selection is a better approach[18](https://openforecast.org/2024/03/14/the-role-of-m-competitions-in-forecasting/); benchmark studies had already found that many objective selection criteria provide structures equal or superior to the time-consuming Box–Jenkins method.[19](https://onlinelibrary.wiley.com/doi/10.1002/for.3980130502)\n\n**A caveat on information criteria.** Using AIC to decide between different orders of differencing is technically invalid, since one data point is lost with each order of differencing; the Auto Arima algorithm therefore uses a unit root test to select the order of differencing and uses AIC only for the AR and MA orders.[20](https://skforecast.org/latest/user_guides/forecasting-statistical-models)\n\n**Current tooling.** The stepwise selection algorithm is outlined in Hyndman and Khandakar (2008)[21](https://github.com/sktime/sktime/blob/main/sktime/forecasting/arima/%5Fpmdarima.py), and the R forecast package's auto.arima uses that algorithm except that the default method for selecting seasonal differences is now based on an estimate of seasonal strength (Wang, Smith and Hyndman, 2006).[22](https://www.rdocumentation.org/packages/forecast/versions/9.0.0/topics/auto.arima) Published comparisons do not settle how the method compares specifically with state-space/Kalman or GARCH alternatives, nor its standing against deep-learning forecasters such as [N-BEATS](https://www.edgepedia.org/n-beats) or TFT.\n\n## References\n\n1. [Citation Classic: Box G E P & Jenkins G M. Time series analysis: forecasting and control. San Francisco, CA: Holden-Day. (1970)](https://garfield.library.upenn.edu/classics1989/A1989AV48500001.pdf)\n2. [Time Series Analysis: Forecasting and Control, 5th Edition (Wiley)](https://www.wiley.com/en-us/Time+Series+Analysis:+Forecasting+and+Control,+5th+Edition-p-9781118675021)\n3. [The Box-Jenkins approach to time series analysis and forecasting: principles and applications (RAIRO 1977)](https://numdam.org/item/RO_1977__11_2_129_0.pdf)\n4. [The Box-Jenkins Method (NCSS statistical software documentation)](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/The_Box-Jenkins_Method.pdf)\n5. [The Box-Jenkins approach to time series analysis (RAIRO 1977, part 1)](https://numdam.org/item/RO_1977__11_1_3_0.pdf)\n6. [A Review of ARIMA vs. Machine Learning Approaches for Time Series Forecasting in Data Driven Networks](https://www.mdpi.com/1999-5903/15/8/255)\n7. [6.4.4.5. Box-Jenkins Models (NIST/SEMATECH e-Handbook)](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc445.htm)\n8. [Selected Topics in Time Series Forecasting: Statistical Models vs. Machine Learning](https://www.mdpi.com/1099-4300/27/3/279)\n9. [Box-Jenkins modelling (Hyndman)](https://robjhyndman.com/papers/BoxJenkins.pdf)\n10. [Specification of ARIMA Models by the Box-Jenkins Method (Dufour)](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_2005_C_BoxJenkinsMethod.pdf)\n11. [6.4.4.6. Box-Jenkins Model Identification (NIST/SEMATECH e-Handbook)](https://itl.nist.gov/div898/handbook/pmc/section4/pmc446.htm)\n12. [Analysis and Modeling of Seasonal Time Series (NBER chapter, by Box)](https://www.nber.org/system/files/chapters/c4329/c4329.pdf)\n13. [Box-Jenkins (ARIMA) Modeling (Estima/RATS documentation)](https://estima.com/webhelp/topics/boxjenkins.html)\n14. [25 Years of Time Series Forecasting (Hyndman, International Journal of Forecasting)](https://robjhyndman.com/papers/ijf25.pdf)\n15. [Makridakis et al., The M3-Competition related material (hosted PDF)](https://autobox.com/makridakis.pdf)\n16. [ARIMA and SARIMA models: the Box-Jenkins method for time series forecasting | Forecast Studio](https://www.forecaststudio.com.mx/blog/arima-sarima-models)\n17. [J. R. M. Hosking (1984). Modeling persistence in hydrological time series using fractional differencing. Water Resources Research.](https://doi.org/10.1029/wr020i012p01898)\n18. [The role of M competitions in forecasting (OpenForecast, 2024)](https://openforecast.org/2024/03/14/the-role-of-m-competitions-in-forecasting/)\n19. [A Comparison of Box–Jenkins and objective methods for determining the order of a non-seasonal ARMA Model](https://onlinelibrary.wiley.com/doi/10.1002/for.3980130502)\n20. [ForecasterStats - Skforecast Docs](https://skforecast.org/latest/user_guides/forecasting-statistical-models)\n21. [sktime/forecasting/arima/_pmdarima.py](https://github.com/sktime/sktime/blob/main/sktime/forecasting/arima/%5Fpmdarima.py)\n22. [auto.arima function - RDocumentation](https://www.rdocumentation.org/packages/forecast/versions/9.0.0/topics/auto.arima)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Stochastic processes*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Oct 15, 2026; Oct 21, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/box-jenkins-method", "markdown_url": "https://www.edgepedia.org/box-jenkins-method.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Box–Jenkins method\", Edgepedia (EdgeChat), https://www.edgepedia.org/box-jenkins-method. Edgepedia Community License 1.0.", "credit_md": "\"[Box–Jenkins method](https://www.edgepedia.org/box-jenkins-method)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/box-jenkins-method](https://www.edgepedia.org/box-jenkins-method). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Box–Jenkins method\", Edgepedia (EdgeChat), https://www.edgepedia.org/box-jenkins-method. Edgepedia Community License 1.0.", "speakable": "The Box–Jenkins method is an iterative procedure in time series analysis for building ARIMA models, described by George Box and Gwilym Jenkins in their 1970 book Time Series Analysis: Forecasting and Control." }