A Very British Affair pp 343-393 | Cite as

# Granger: Long Memory, Fractional Differencing, Spurious Regressions and Co-integration

Chapter

## Abstract

Granger’s research on bilinear models had left him dissatisfied as he did not feel that they, or indeed many other forms of non-linear models, were of much practical use. He was also struck by the limitations imposed by ARIMA(

*p*,*d*,*q*) models on the behaviour of the ACF, which declines either geometrically, when*d*= 0, or linearly when*d*= 1. In Granger (1979) he suggested the possibility of models displaying*long memory*, taking his cue from the water resources literature (as surveyed by Lawrence and Kottegoda, 1977) and, in particular, the models proposed by Mandelbrot and Van Ness (1968). Rather than having a spectrum taking the form*ω*^{−2}for small frequencies*ω*, as would be the case for a process that required first differencing for it to be rendered stationary (see §**10.2**below), such models would have spectra proportional to*ω*^{−}^{ α }for 0 <*α*< 2. If long memory models should prove useful then ‘ordinary integer differencing is inappropriate, yet the series would have an infinite variance and its correlogram would suggest differencing according to the Box-Jenkins rules. If such series arise in practice, they could be of considerable importance and “fractional differencing” should become a standard component of analysis’ (Granger, 1979, page 251). An example given by Granger of a long-memory process was the infinite moving average where

*a*_{ t }is white noise, although he presented no analysis of it.## Keywords

Random Walk Error Correction Unit Root Test Error Correction Term Absolute Return
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Terence C. Mills 2013