Box–Jenkins

In time series analysis, the Box–Jenkins method, named after the statisticians 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.

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

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) 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.

The first step in developing a Box–Jenkins model is to determine if the time series is stationary and if there is any significant seasonality that needs to be modelled.

Stationarity can be assessed from a run sequence plot. The run sequence plot should show constant location and scale. It can also be detected from an . Specifically, non-stationarity is often indicated by an autocorrelation plot with very slow decay.

Seasonality (or periodicity) can usually be assessed from an autocorrelation plot, a seasonal subseries plot, or a spectral plot.

Box and Jenkins recommend the differencing approach to achieve stationarity. However, fitting a curve and subtracting the fitted values from the original data can also be used in the context of Box–Jenkins models.

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