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.
Modeling approach
The original model uses an iterative three-stage modeling approach:
Model identification and model selection: making sure that the variables are stationary, identifying seasonality in the dependent series (seasonally differencing it if necessary), and using plots of the autocorrelation (ACF) and 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 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. 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.
Box–Jenkins model identification
Stationarity and seasonality
The first step in developing a Box–Jenkins model is to determine whether the time series is stationary and whether there is any significant seasonality that needs to be modelled.
Detecting stationarity
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 autocorrelation plot. Specifically, non-stationarity is often indicated by an autocorrelation plot with very slow decay.
Detecting seasonality
Seasonality (or periodicity) can usually be assessed from an autocorrelation plot, a seasonal subseries plot, or a spectral plot.
Differencing to achieve stationarity
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.
Seasonal differencing
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 ARIMA 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.
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) 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.