The model is updated as needed, and the diagnostic checking stage ends when all of the criteria for an acceptable model are met. The final step is to generate the forecast values. The user can also elect to have this process start with an examination of the original time series. This may be necessary for those cases where the series is overwhelmingly influenced by outlier variables. We now present a summary of the mathematical properties underlying this procedure.
Outliers can occur in many ways.
They may be the result of a gross error, for example, a recording or transcript error. They may also occur by the effect of some exogenous intervention. The reader should note that the residual outlier analysis as conducted in the course of diagnostic checking is an AO type. Also note that AO and IO models are relatable. In other words, a single IO model is equivalent to a potentially infinite AO model and vice versa.