BoxJenkins

Box-Jenkins is a systematic methodology for identifying, estimating, and checking ARIMA time series models for forecasting. Developed by George Box and Gwilym M. Jenkins, the approach was popularized in their 1970 book Time Series Analysis: Forecasting and Control. The method emphasizes a four-step cycle: model identification, parameter estimation, diagnostic checking, and forecasting, iterating as needed to improve adequacy.

Model identification focuses on making the series approximately stationary, using differencing to remove trends and seasonality as required. The order of the autoregressive part p, the degree of differencing d, and the moving-average part q are chosen by examining autocorrelation function (ACF) and partial autocorrelation function (PACF) plots and information criteria (AIC/BIC).

Parameter estimation uses maximum likelihood or least squares to fit the chosen ARIMA(p,d,q) model. Diagnostic checking evaluates whether residuals behave like white noise using plots and tests such as the Ljung-Box test, and may include examining residual normality and homoscedasticity. If diagnostics fail, the model is revised (e.g., different p, d, q, or a seasonal component).

Once a satisfactory model is obtained, it can be used to generate forecasts and prediction intervals. The Box-Jenkins framework also supports extensions to seasonal data (SARIMA) and, with additional components, to exogenous inputs (ARIMAX).

Limitations include sensitivity to model misspecification, reliance on linear dynamics, and assumption of stationarity after differencing. The method remains influential for structured time series analysis, but modern approaches may combine Box-Jenkins with more flexible techniques for nonlinear or regime-switching behavior.