Box and Jenkins: Modelling Seasonal Time Series and Transfer Function Analysis

Abstract

As was developed in some detail in Chapter 6, the Box-Jenkins approach to modelling time series revolves around the ARMA process

$$$$

which has an eventual forecast function that is the solution to the difference equation $$, where B is understood to operate on l (cf. §6.39). Box and Jenkins (1970, chapter 9) argued that, to be able to represent seasonal behaviour, the forecast function would need to trace out a periodic pattern. This could be achieved by allowing the autoregressive operator φ(B) to consist of a mixture of sines and cosines, possibly mixed with polynomial terms to allow for changes in the level of x _t and changes in the seasonal pattern. For example, a forecast function containing a sine wave with a 12-month period, which is adaptive in both phase and amplitude, will satisfy the difference equation

$$$$

The operator $$ has roots of exp (± i2π/12) on the unit circle and is thus homogeneously non-stationary. Box and Jenkins pointed out, however, that periodic behaviour would not necessarily be represented parsimoniously by mixtures of sines and cosines.