As stated in the beginning, given a set of time series data we would like to determine the underlying mechanism that generated the series. In other words, our goal is to identify a model that can "explain" the observed properties of the series. If we assume that after appropriate transformations the series is governed by an ARMA type of model, model identification amounts to selecting the orders of an ARMA model.

In general, selecting a model (model identification), estimating the parameters of the selected model (parameter estimation), and checking the validity of the estimated model (diagnostic checking) are closely related and interdependent steps in modeling a time series. For example, some order selection criteria use the estimated noise variance obtained in the step of parameter estimation, and to estimate model parameters we must first know the model. Other parameter estimation methods combine the order selection and parameter estimation. Often we may need to first choose a preliminary model, and then estimate the parameters and do some diagnostic checks to see if the selected model is in fact appropriate. If not, the model has to be modified and the whole procedure repeated. We may need to iterate a few times to obtain a satisfactory model. None of the criteria and procedures are guaranteed to lead to the "correct" model for finite data sets. Experience and judgment form necessary ingredients in the recipe for time series modeling.

In this section we concentrate on model identification. Since the correlation function is the most telling property of a time series, we first look at how to estimate it and then use the estimated correlation function to deduce the possible models for the series. Other order selection methods will also be introduced. Parameter estimation and diagnostic checking are discussed in Section 1.6.