Model Specification

We have developed a large class of parametric models for both stationary and nonstationary time series—the ARIMA models. We now begin our study and implementation of statistical inference for such models. The subjects of the next three chapters, respectively, are:

• 1. how to choose appropriate values forp, d, and q for a given series;

• 2. how to estimate the parameters of a specific ARIMA(p,d,q) model;

• 3. how to check on the appropriateness of the fitted model and improve it if needed.

Our overall strategy will first be to decide on reasonable—but tentative—values for p, d, and q. Having done so, we shall estimate the φ’s, θ’s, and σe for that model in the most efficient way. Finally, we shall look critically at the fitted model thus obtained to check its adequacy, in much the same way that we did in Section 3.6 on page 42. If the model appears inadequate in some way, we consider the nature of the inadequacy to help us select another model. We proceed to estimate that new model and check it for adequacy.

With a few iterations of this model-building strategy, we hope to arrive at the best possible model for a given series. The book by George E. P. Box and G. M. Jenkins (1976) so popularized this technique that many authors call the procedure the “Box- Jenkins method.” We begin by continuing our investigation of the properties of the sample autocorrelation function.