Box–Jenkins method: Difference between revisions

Revision as of 22:24, 13 April 2022

Revision as of 03:47, 10 July 2022

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{{Short description|Method to find best fit of a time-series model}}

In [[time series analysis]], the '''Box–Jenkins method,'''<ref>{{cite book |lastlast1=Box |firstfirst1=George |last2=Jenkins |first2=Gwilym |year=1970 |title=Time Series Analysis: Forecasting and Control |url=https://archive.org/details/timeseriesanalys0000boxg |url-access=registration |___location=San Francisco |publisher=Holden-Day }}</ref> named after the [[statistician]]s [[George Box]] and [[Gwilym Jenkins]], applies [[autoregressive moving average]] (ARMA) or [[autoregressive integrated moving average]] (ARIMA) models to find the best fit of a time-series model to past values of a [[time series]].

==Modeling approach==

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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.

Commandeur & Koopman (2007, §10.4)<ref>{{cite book |lastlast1=Commandeur |firstfirst1=J. J. F. |last2=Koopman |first2=S. J. |year=2007 |title=Introduction to State Space Time Series Analysis |publisher=[[Oxford University Press]] }}</ref> 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.

==Box–Jenkins model identification==

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===Identify ''p'' and ''q''===

Once stationarity and seasonality have been addressed, the next step is to identify the order (i.e. the ''p'' and ''q'') of the autoregressive and moving average terms. Different authors have different approaches for identifying ''p'' and ''q''. Brockwell and Davis (1991)<ref>{{cite book |lastlast1=Brockwell |firstfirst1=Peter J. |last2=Davis |first2=Richard A. |year=1991 |title=Time Series: Theory and Methods |publisher=Springer-Verlag |page=273|bibcode=1991tstm.book.....B }}</ref> state "our prime criterion for model selection [among ARMA(p,q) models] will be the AICc", i.e. the [[Akaike information criterion]] with correction. Other authors use the autocorrelation plot and the partial autocorrelation plot, described below.

====Autocorrelation and partial autocorrelation plots====

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Hyndman & Athanasopoulos suggest the following:<ref>{{cite webbook|last1=Hyndman|first1=Rob J|last2=Athanasopoulos|first2=George|title=Forecasting: principles and practice|url=https://www.otexts.org/fpp/8/5|access-date=18 May 2015}}</ref>

:The data may follow an ARIMA(''p'',''d'',0) model if the ACF and PACF plots of the differenced data show the following patterns:

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==Further reading==

* {{citation | title= Comparison of Box–Jenkins and objective methods for determining the order of a non-seasonal ARMA model | author1-first= S. | author1-last= Beveridge | author2-first= C. | author2-last= Oickle | journal= [[Journal of Forecasting]] | year= 1994 | volume= 13 | issue= 5 | pages= 419–434 | doi= 10.1002/for.3980130502}}

* {{citation |last=Pankratz |first=Alan |year=1983 |title=Forecasting with Univariate Box–Jenkins Models: Concepts and Cases |publisher= [[John Wiley & Sons]] }}