Fractionally diﬀerenced process
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In the last two decades there has been considerable interest in time series models with longer “memory” than those of the autoregressive moving average (ARMA) type. By long-memory it is meant that the autocovariance function γk of the process has a much slower decaying rate than those of the usual stationary time series models. For instance, one way to achieve longer memory is to allow
∑∞ k=−∞ |γk| to be di-
vergent. Long-memory models appear in economics, ﬁnance, hydrology, and climatology. For example, in economics, Granger (1980b) has shown that long-memory models can arise from aggregating simple dynamic micro-relationships. More recently, Ding, Granger, and Engle (1993) and Granger, Spear and Ding (2000) suggested that the absolute returns of daily data for a number of ﬁnancial series exhibit the long memory property. Booth, Kaen and Koveos (1982) and Cheung (1993) suggested that long memory structure may be present in some exchange rate series. Cheung and Lai (1993) studied purchasing power parity using the long memory concept. Baillie (1996) gave a comprehensive review on ﬁnancial applications of long memory time series. In climatology, tree-ring width variation in trees are being used to backcast climatological patterns several hundreds of years before the ﬁrst scientiﬁc record (LaMarche, 1974). In hydrology, long-memory time series models have long been a subject of interest and are closely related with the Hurst phenomenon, (Lawrance and Kottegoda, 1977; Hipel and McLeod, 1978). See Beran (1994) for more examples.