436 Fractional Integration and Cointegration
10.2 Fractional integration
10.2.1 Concept and modelization
The idea of fractional integration was introduced by Granger and Joyeux (1980),
Granger (1980, 1981) and Hosking (1981), though Adenstedt (1974) and Taqqu
(1975) earlier showed an awareness of its representation. Assuming thatxt is
given by equation (10.1), the first point we have to deal with is the treatment
of pre-sample observations. In short memory contexts (that is,d=0), different
initial value conventions lead to parameter estimates which typically share the
same first-order asymptotic properties but have different finite-sample properties.
In empirical work zeros or the sample mean often initiate the series, with early
observations being thrown away. Different conventions have also been followed
in non-stationary series with an autoregressive unit root. In stationary fractional
processes, infinitely many pre-sample values have to be chosen, so the potential
divergence between rival methods and parameter estimates is greater, though first-
order asymptotic properties are again robust. In fractional contexts, two definitions
of fractional integration have been employed. From equation (10.1), ford< 1 /2:
xt=−dut,t=0,±1,...,where= 1 −L. For integerda≥0:
zat=−dax#t, t=0,±1,...,is called a Type I(d+da)process, where the # superscript attached to a scalar
or vector sequence has the meaningw#t =wt 1 (t> 0 ),1(·)being the indicator
function. Similarly:
zbt=−da−du#t, t=0,±1,...,is called a Type II (d+da)process. Whend=0,x#t=u#tand hencezat=zbt, so both
definitions are equivalent in non-fractional contexts.^2 Note that the polynomial
on the left-hand side of (10.1) can be expanded as:
( 1 −L)d=∑∞j= 0(
d
j)
(− 1 )jLj= 1 −dL+
d(d− 1 )
2
L^2 −....Thus, ifdis an integer value,xtwill be a function of a finite number of past
observations, while ifdis real,xtdepends upon values of the time series far in the
past. The higher the value ofd, the higher the level of association between the
observations.
There exist several sources that might produceI(d)processes, aggregation being
the usual argument. Robinson (1978) and Granger (1980) showed that fractionally
integrated data could arise as a result of aggregation when: (i) data are aggregated
across heterogeneous autoregressive (AR) processes, and (ii) data involving hetero-
geneous dynamic relationships at the individual level are then aggregated to form
the time series. Cioczek-Georges and Mandelbrot (1995), Taqqu, Willinger and