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

  • Terence C. Mills
Part of the Palgrave Advanced Texts in Econometrics book series (PATEC)


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.


Transfer Function Impulse Response Noise Model Step Response Transfer Function Model 
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Copyright information

© Terence C. Mills 2013

Authors and Affiliations

  • Terence C. Mills
    • 1
  1. 1.Loughborough UniversityUK

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