Palgrave Handbook of Econometrics: Applied Econometrics

Luis A. Gil-Alana and Javier Hualde 435

Until the 1980s a standard approach was to impose a deterministic (linear or
quadratic) function of time, thus assuming that the residuals from the regres-
sion model were stationary. Later on, and especially after the seminal work of
Nelson and Plosser (1982), there was a general agreement that the non-stationary
component of most series was stochastic, and unit roots (or first differences) were
commonly adopted. However, the unit root is merely one particular model to
describe such behavior. In fact, the number of differences required to get to sta-
tionarity may not necessarily be an integer value but any point in the real line.^1 In
such a case, the process is said to be fractionally integrated orI(d). TheI(d)models
belong to a wider class of processes called long memory, which we can define in
either the time or the frequency domains.
Let us consider a zero mean process{xt,t=0,±1,...}withγu=E(xtxt+u). The
time domain definition of long memory states that:

∑∞

u=−∞

∣∣

γu

∣∣

=∞.

Now, assuming thatxthas an absolutely continuous spectral distribution, so that
it has spectral density function:

f(λ)=

1

2 π

⎛

⎝γ 0 + 2

∑∞

u= 1

γucos(λu)

⎞

⎠,

the frequency domain definition of long memory states that the spectral density
function is unbounded at some frequency in the interval[0,π). Most of the empir-
ical literature has concentrated on the case where the singularity, or pole, in the
spectrum takes place at the zero frequency. This is the standard case ofI(d)models
of the form:

( 1 −L)dxt=ut, t=0,±1,..., (10.1)

whereLis the lag operator(Lxt=xt− 1 )andutisI( 0 ). However, fractional inte-
gration may also occur at some other frequencies away from zero, as in the case of
seasonal/cyclical models.
In the multivariate case, the natural extension of fractional integration is the
concept of fractional cointegration. Though the original idea of cointegration, as
espoused by Engle and Granger (1987), allows for fractional orders of integration,
all the empirical work carried out during the 1990s was restricted to the case of
integer degrees of differencing. Only in recent years have fractional values also
been taken into account.
In this chapter we review fractional integration and cointegration, placing special
emphasis on the latter concept, which has recently emerged in the time series lit-
erature. We also present an empirical application using some of the most novel
techniques in this area. The outline of the chapter is as follows. Section 10.2
concentrates on fractional integration and some of its most recent developments.
Section 10.3 deals with fractional cointegration, while section 10.4 is devoted to
an empirical example.