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Box Jenkins Model Definition
Box-Jenkins Model make use of autoregressive moving average or Autoregressive integrated moving average models to find the best fit of a time series to past values of this time series. This is used in order to make forecasting and the original model uses an iterative three-stage modeling approach as follows:
Stage 1: Model identification and model selection make sure that the variables are stationary process identifying seasonality in the dependent series and making use of the plots of the autocorrelation as well as partial autocorrelation. These are functions of the dependent time series to decide which autoregressive or moving average component should be used in the model.
Stage 2: Parameter estimation is performed using computation algorithms to arrive at coefficients which best fit the selected ARIMA model and the most common methods use maximum likelihood estimation or non linear least squares estimation.
Stage 3: Statistical model validation by testing whether the estimated model conforms to the specifications of a stationary univariate process and so the residuals should be independent of each other and constant in mean and variance over time. Plotting the mean as well as 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 and if the estimation is inadequate it is required to go to step one and attempt to build a better model.
The data they used were from a gas furnace and so these data are well known as the Box and Jenkins gas furnace data for benchmarking predictive models.
The first step in developing a Box–Jenkins model is to determine if the time series is stationary process and if there is any significant seasonality that is to be modeled. Stationarity can be determined from a run sequence plot where the run sequence plot should show constant location as well as a scale. It can also be detected from an autocorrelation plot where the non-stationarity is often specifically indicated by an autocorrelation plot with very slow decay. Seasonality can usually be assessed from an autocorrelation plot or a seasonal subseries plot as well as from a Spectral plot. Box and Jenkins suggest the differencing approach to achieve stationarity where the curve fitting and subtracting the fitted values from the original data can also be used in the context of Box–Jenkins models.
