Palgrave Handbook of Econometrics: Applied Econometrics

438 Fractional Integration and Cointegration

expressed in terms of the Gegenbauer polynomial such that, for alld=0, and
settingμ=coswr:

( 1 − 2 μL+L^2 )−d=

∑∞

j= 0

Cj,d(μ)Lj,

where:

Cj,d(μ)=

∑j

k= 0

(− 1 )k(d)j−k( 2 μ)j−^2 k

k!(j− 2 k)!

; (d)j=

(d+j)

(d)

,

and(x)is the Gamma function. Alternatively, we can use the recursive formula
C0,d(μ)=1,C1,d(μ)= 2 μd, and:

Cj,d(μ)= 2 μ

(

d− 1

j

+ 1

)

Cj−1,d(μ)−

(

2

d− 1

j

+ 1

)

Cj−2,d(μ), j=2, 3,....

(See, for instance, Magnus, Oberhettinger and Soni, 1966, or Rainville, 1960, for
further details on Gegenbauer polynomials.) Gray, Yhang and Woodward (1989)
showed thatxtin (10.3) is stationary ifd<0.5 for|μ=coswr|<1 and ifd<0.25
for|μ|=1. Lobato and Robinson (1998) proposed a semiparametric approach for
testing this type of model, and Dalla and Hidalgo (2005) suggested a parametric
test where the unbounded frequency in the spectrum is assumed to be unknown.^6
As mentioned above, these processes are characterized by an unbounded spectral
density function at a single frequency. There may be cases, however, where the
spectrum is unbounded at several frequencies simultaneously, the most typical
corresponding to a seasonal process. We can consider a model of the form:

( 1 −Ls)dxt=ut, t=0,±1,..., (10.4)

wheresrefers to the number of time periods per year (that is,s=4 with quarterly
data,s=12 with monthly). Similarly to (10.1), the (seasonal) fractional polynomial
above can be expressed as:

( 1 −Ls)d=

∑∞

j= 0

(

d

j

)

(− 1 )jLjs= 1 −dLs+

d(d− 1 )

2

L^2 s−...,

for all reald, so thatdbecomes crucial for describing the degree of seasonal per-
sistence. The notion of fractional Gaussian noise with seasonality was initially
suggested by Abrahams and Dempster (1979) and Jonas (1981), and extended
to a Bayesian framework by Carlin, Dempster and Jonas (1985) and Carlin and

Dempster (1989). Note that if, for example,s=4, the polynomial( 1 −L^4 )can be
decomposed into( 1 −L)( 1 +L)( 1 +L^2 ), which is thus a case of multiple poles in
the spectrum (at zero,π, andπ/ 2 ( 3 π/ 2 )ofa2πcycle), all of them, according to
(10.4), with the same order of integrationd.

10.2.2 Empirical evidence of fractional integration

The empirical literature on fractional integration is large. In the 1960s, Granger
(1966) and Adelman (1965) had pointed out that most aggregate economic time