Long memory and fractional differencing: Revisiting Clive W.J. Granger's contributions and further developments

Abstract:

In 1980, Sir Clive W. J. Granger discovered the fractional differencing operator and its fundamental properties in discrete-time mathematics, which sparked an enormous literature concerning the fractionally integrated autoregressive moving average models. Fractionally integrated models capture a type of long memory and have useful theoretical properties, although scientists can find them difficult to estimate or intuitively interpret. His introductory papers from 1980, one of which with Roselyne Joyeux, show his early and deep understanding of this subject by showing that familiar short memory processes can produce long memory effects under certain conditions. Moreover, fractional differencing advanced our understanding of cointegration and the properties of traditional Dickey-Fuller tests, and motivated the development of new unit-root tests against fractional alternatives. This article honors his significant contributions by identifying key areas of research he inspired and surveying recent developments in them.