Box–Jenkins method: Difference between revisions

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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 |last1=Box |first1=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==

The original model uses an iterative three-stage modeling approach:

#''[[Model identification]] and [[model selection]]'': making sure that the variables are [[stationary process|stationary]], identifying [[seasonality]] in the dependent series (seasonally differencing it if necessary), and using plots of the [[autocorrelation|autocorrelation (ACF)]] and [[partial autocorrelation|partial autocorrelation (PACF)]] functions of the dependent time series to decide which (if any) autoregressive or moving average component should be used in the model.

#''[[Parameter estimation]]'' using computation algorithms to arrive at coefficients that best fit the selected ARIMA model. The most common methods use [[maximum likelihood estimation]] or [[non-linear least-squares estimation]].

#''[[Statistical model validation|Statistical model checking]]'' 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 [[Ljung–Box test]] 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.

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====Detecting stationarity====

Stationarity can be assessed from a [[run sequence plot]]. The run sequence plot should show constant ___location and [[Scale (ratio)|scale]]. It can also be detected from an [[autocorrelation plot]]. Specifically, non-stationarity is often indicated by an autocorrelation plot with very slow decay. One can also utilize a [[Dickey-Fuller test]] or [[Augmented Dickey-Fuller test]].

====Detecting seasonality====