In time series analysis, the Box–Jenkins method,[1] 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 problem arises because in "the economic and social fields, real series are never stationary however much differencing is done".
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.
Instead, one includes the order of the seasonal terms in the model specification to the ARIMA estimation software.
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.
Specifically, for an AR(1) process, the sample autocorrelation function should have an exponentially decreasing appearance.
However, higher-order AR processes are often a mixture of exponentially decreasing and damped sinusoidal components.
The sample partial autocorrelation function is generally not helpful for identifying the order of the moving average process.
Although experience is helpful, developing good models using these sample plots can involve much trial and error.
Estimating the parameters for Box–Jenkins models involves numerically approximating the solutions of nonlinear equations.
The main approaches to fitting Box–Jenkins models are nonlinear least squares and maximum likelihood estimation.
This article incorporates public domain material from the National Institute of Standards and Technology