Time Series: Theory and Methods pp 273-329 | Cite as

# Model Building and Forecasting with ARIMA Processes

## Abstract

In this chapter we shall examine the problem of selecting an appropriate model for a given set of observations n{*X* _{ t }, *t* = 1,…, n} data (a) exhibits no apparent deviations from stationarity and (b) has a rapidly decreasing autocorrelation function, we shall seek a suitable **ARMA** process to represent the mean-corrected data. If not, then we shall first look for a transformation of the data which generates a new series with the properties (a) and (b). This can frequently be achieved by differencing, leading us to consider the class of **ARIMA** (autoregressive-integrated moving average) processes which is introduced in Section 9.1. Once the data has been suitably transformed, the problem becomes one of finding a satisfactory **ARMA**(p, *q*) model, and in particular of choosing (or identifying) *p* and *q*. The sample autocorrelation and partial autocorrelation functions and the preliminary estimators \({\hat \phi _m}\)
and \({\hat \theta _m}\)
of Sections 8.2 and 8.3 can provide useful guidance in this choice. However our prime criterion for model selection will be the AICC, a modified version of Akaike’s AIC, which is discussed in Section 9.3. According to this criterion we compute maximum likelihood estimators of **φ**, **θ** and σ^{2} for a variety of competing *p* and *q* values and choose the fitted model with smallest AICC value. Other techniques, in particular those which use the *R* and *S* arrays of Gray et al. (1978), are discussed in the recent survey of model identification by de Gooijer et al. (1985).

## Keywords

Model Selection Autocorrelation Function Model Building Model Identification Recent Survey## Preview

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