Palgrave Handbook of Econometrics: Applied Econometrics

Luis A. Gil-Alana and Javier Hualde 437

Sherman (1997) and Chambers (1998) also use aggregation to motivate long mem-
ory processes, while Parke (1999) uses a closely related discrete time error duration
model. More recently, Diebold and Inoue (2001) proposed another source of long
memory based on regime-switching models.^3
Note thatutin (10.1) may also include some type of weak dependence structure:
for example, a stationary and invertible autoregressive moving average (ARMA)
process of the form:

φ(L)ut=θ(L)εt, t=0,±1,...,

whereεtis an independent and identically distributed (i.i.d.) sequence. Thus,
whend< 1 /2,xtin (10.1) can be written as:

φ(L)( 1 −L)dxt=θ(L)εt, t=0,±1,..., (10.2)

which is usually called an autoregressive fractionally integrated moving average
(ARFIMA) process. Sowell (1992a) analyzed the exact maximum likelihood (ML)
estimator of the parameters of the ARFIMA model (10.2) in the time domain, using a
recursive procedure that allows quick evaluation of the likelihood function, which
is given by:

( 2 π)−n/^2 ||−^1 /^2 exp

(

−

1

2

X′n−^1 Xn

)

,

whereXn=(x 1 ,x 2 ,...,xn)′andXn∼N(0,). Other parametric methods of esti-
matingdin the frequency domain were proposed, among others, by Fox and
Taqqu (1986) and Dahlhaus (1989). Small sample properties of these and other
estimators were examined in Smith, Taylor and Yadav (1997) and Hauser (1999).
In the former, several semiparametric procedures were compared with Sowell’s
(1992a) ML estimation method, finding that Sowell’s procedure outperforms the
semiparametric ones in terms of bias and mean square error. Hauser also compares
Sowell’s procedure with others based on the exact and the Whittle likelihood func-
tion in the time and the frequency domain, and shows that Sowell’s procedure
dominates the others in the case of fractionally integrated models. A semipara-
metric frequency domain estimator is the log-periodogram estimator proposed by
Geweke and Porter-Hudak (1983).^4 Other parametric and semiparametric methods
have been proposed: see, for example, Robinson (1994a, 1995a, 1995b), Tanaka
(1999), Velasco (1999a, 1999b), and Phillips and Shimotsu (2004, 2005).^5
So far we have focused on the case where the singularity occurs at the zero
frequency. Let us consider now the following process:

( 1 −2 coswrL+L^2 )dxt=ut, t=0,±1,..., (10.3)

wherewris a real value equal to 2πr/n, withr=n/c. In this context, ifd>0, the
process is also fractionally integrated, although the pole (unboundedness) in the
spectrum now occurs at a (cyclical) frequencyλ=0, andcwill be an indicator of
the number of periods per cycle. These processes were introduced by Gray, Yhang
and Woodward (1989, 1994), who showed that the polynomial in (10.3) can be