Luis A. Gil-Alana and Javier Hualde 439series have a typical shape where by the spectral density increases dramatically as
the frequency approaches zero. However, differencing the data frequently leads
to overdifferencing at the zero frequency. Some 15 years later, Robinson (1978)
and Granger (1980) showed that aggregation could be a source of fractional inte-
gration. Since then, fractional processes have been widely employed to describe
the dynamics of many time series. Given the vast amount of empirical work, this
section is divided into various sub-sections according to the different nature of the
series under examination.
10.2.2.1 Applications to macroeconomics
Diebold and Rudebusch (1989) and Sowell (1992b) analyzed US quarterly post-war
real output data and obtained estimates ofdbelow unity. Nevertheless, their results
were in line with Rudebusch’s (1993) and Christiano and Eichembaum’s (1990)
conclusions in the sense that their confidence intervals fordincluded the unit
root and, in Sowell’s case, also the trend-stationaryI( 0 )representation. Haubrich
and Lo (1991) examined US real output using R/S techniques and found little evi-
dence of long-range dependence in the business cycle. Using Bayesian techniques,
Koopet al.(1997) also examined real US GNP and found some evidence of long
memory, although their results also reflected model uncertainty. Michelacci and
Zaffaroni (2000) showed that GDP per capita in 16 OECD countries exhibited long
memory. Mayoral (2006) also finds evidence of fractional integration in real GNP
and GNP per capita in the US, showing that their results are robust to the presence
of structural breaks in the deterministic components.^7
Long memory in inflation rates is another topic that has been widely examined
in the empirical literature. Much of the evidence supports the view that infla-
tion is fractionally integrated with a differencing parameter that is significantly
different from zero or unity. For US monthly data, Backus and Zin (1993) found a
fractional degree of integration. They argue that aggregation across agents with het-
erogeneous beliefs results in long memory in inflation. Hassler (1993) and Delgado
and Robinson (1994) provide strong evidence of long memory in the Swiss and
Spanish inflation rates respectively. Baillie, Chung and Tieslau (1996) examined
monthly post-World War II CPI inflation in ten countries, and found evidence of
long memory with mean-reverting (with smaller memory than one) behavior in all
countries except Japan. Similar evidence was found in Hassler and Wolters (1995)
and Baum, Barkoulas and Caglayan (1999). In the context of structural breaks, Bos,
Franses and Ooms (1999, 2001) examined inflation in the G7 countries, finding
that long memory is quite resistant to level shifts, although, for a few inflation
rates, they found that the evidence for long memory disappeared. Evidence of
long memory behavior in the conditional mean of inflation is found in Baillie,
Chung and Tieslau (1996) and Baillie, Han and Kwon (2002). Other recent papers
relating long memory and structural breaks in inflation rates are Gadea, Sabate and
Serrano (2004), Franses, Hyung and Penn (2006) and Gil-Alana (2008a), and fore-
casting issues are examined in Franses and Ooms (1997) and Barkoulas and Baum
(2006).