# CHAPTER *2*

*STATIONARY TIME-SERIES MODELS*

The theory of linear difference equations can be extended to allow the forcing process {*x*_{t}} to be stochastic. This class of linear stochastic difference equations underlies much of the theory of time-series econometrics. Especially important is the Box–Jenkins (1976) methodology for estimating time-series models of the form

Such models are called autoregressive integrated moving-average (ARIMA) time-series models. This chapter has three aims:

- Present the theory of stochastic linear difference equations and consider the time-series properties of stationary ARIMA models. A stationary ARIMA model is called an autoregressive moving-average (ARMA) model. It is shown that the stability conditions described in the previous chapter are necessary conditions for stationarity.
- Develop the tools used in estimating ARMA models. Especially useful are the autocorrelation and partial autocorrelation functions. It is shown how the Box–Jenkins methodology relies on these tools to estimate an ARMA model from sample data.
- Consider various test statistics to check for model adequacy. Several examples of estimated ARMA models are analyzed in detail. It is shown how a properly estimated model can be used for forecasting.

## 1. STOCHASTIC DIFFERENCE EQUATION MODELS ...