Luis A. Gil-Alana and Javier Hualde 441the majority of countries, the fractional differencing parameter lies between 0 and
1, and it seems to be considerably smaller forex postreal rates than forex anterates.
10.2.2.4 Applications to stock markets
Long memory analysis was first conducted in stock return series in Greene and
Fielitz (1977). They report evidence of persistence in daily US stock return series
using R/S methods. However, Aydogan and Booth (1988) concluded that there was
no significant evidence of long memory in common stock returns. Lo (1991) used
the modified R/S method, along with spectral regression methods, and found no
evidence of long memory in stock returns. Many other authors have found little
or no evidence of long memory in stock markets (see, for example, Hiemstra and
Jones, 1997). On the other hand, Crato (1994), Cheung and Lai (1995), Barkoulas
and Baum (1996), Barkoulas, Baum, and Travlos (2000), Sadique and Silvapulle
(2001), Henry (2002), Tolvi (2003) and Gil-Alana (2006) are among those who find
evidence of long memory in monthly, weekly, and daily stock market returns.
Several papers use the Standard & Poor (S&P) 500 index over a long span of daily
observations. Granger and Ding (1995a, 1995b) focus on power transformations
of the absolute value of the returns (which they use as a proxy for volatility). They
estimate a long memory process to study persistence in volatility, and establish
some stylized facts concerning the temporal and distributional properties of abso-
lute returns. However, in a related study, Granger and Ding (1996) find that the
parameters of the long memory model vary considerably from one sub-series to
the next. The issue of fractional integration with structural breaks in stock markets
has been examined in Mikosch and Starica (2000) and Granger and Hyung (2004).
Stochastic volatility models using fractional integration have been implemented
in Crato and de Lima (1994), Bollerslev and Mikkelsen (1996), Ding and Granger
(1996), Breidt, Crato, and de Lima (1997, 1998), Arteche (2004) and Baillieet al.
(2007).
10.2.2.5 Applications to geophysics and other sciences
Fractional integration has also been applied in many other areas. Examples include
meteorology (Haslett and Raftery, 1989; Bloomfield, 1992; Hussain and Elbergali,
1999; Gil-Alana, 2005a, 2008b); etternet (and internet) traffic traces (Abry and
Veitch, 1998; Karagiannis, Molle and Faloutsos, 2004); hydrology (Montanari,
Rosso and Taqqu, 1997, 2000; Rao and Bhattacharya, 1999; Wanget al., 2005;
Wanget al., 2007); and political sciences (Box-Steffensmeier and Smith, 1996, 1998;
Byers, Davidson and Peel, 1997, 2000; Dolado, Gonzalo and Mayoral, 2003).
10.2.2.6 Applications using seasonal and cyclical FI models
This review has so far focused exclusively on models with the pole or singular-
ity in the spectrum occurring at the zero frequency. In this sub-section we briefly
review the empirical literature on seasonal and cyclical fractional models. Start-
ing with the seasonal model in (10.4), Porter-Hudak (1990) applied a seasonally
fractionally integrated model to quarterly US monetary aggregates, concluding