Palgrave Handbook of Econometrics: Applied Econometrics

450 Fractional Integration and Cointegration

a higher rate of convergence is achieved and the limiting distribution is normal,
which is natural given that now the deterministic component inxtis dominant.^11
In the multivariate setting (10.11), by considering the spectral density ofutto
be a nonparametric function, Hualde and Robinson (2006) propose an extension
of the estimators of Robinson and Hualde (2003), allowing for the simultaneous
presence of strong and weak cointegrating relations. Because of the generality of the
framework, the representation of the estimators and asymptotic results are rather
involved but, essentially, the same properties as in Robinson and Hualde (2003) are
achieved by the estimators of the cointegrating parameters in strong cointegrating
relations, whereas the estimators of parameters in weak cointegrating relations are
asymptotically normal, but with a slower rate of convergence than the parametric
one (

√
n) given bym^1 /^2 (n/m)δp−γ, whereδp,γdenote the common integration
order of the observables and that of the cointegrating error of the particular weak
cointegrating relation, respectively. This result leads to Wald statistics for testing
linear restrictions among the elements of!in (10.11) having a standard null chi-
squared limit distribution, irrespective of the type of cointegrating relations present
in the model.
In the different setting of Chen and Hurvich (2003a), whose focus is on estimat-
ing the space of cointegration and not cointegrating regressions, then, on assuming
the cointegrating rank isr, this space is estimated by the eigenvectors correspond-
ing to thersmallest eigenvalues of the averaged tapered periodogram matrix ofyt.
The intuition behind this result is the following. Noting (10.12):

∑m

j= 1

ReIy

(

λj

)

=A

∑m

j= 1

ReIx

(

λj

)

A′+A

∑m

j= 1

ReIxu

(

λj

)

B′+B

∑m

j= 1

ReIux

(

λj

)

A′

+B

∑m

j= 1

ReIu

(

λj

)

B′.

Given thatd>du, the right-hand side of the above equation is dominated by the
first term, and setting conditions ensuring that
∑m
j= 1 ReIx(λj)is positive definite
with probability approaching one (basically meaning that there is no cointegra-
tion among the elements ofxt), then, with probability approaching one, the
cointegrating space (Ker(A′))is the space of eigenvectors ofA

∑m

j= 1 ReIx(λj)A

′, with

corresponding eigenvalues equal to zero. Finally, Chen and Hurvich (2006), in
the more general setting (10.13), estimate separately each cointegrating sub-space
using appropriate sets of eigenvectors of an averaged periodogram matrix of tapered
observations.

10.3.3 Evidence of fractional cointegration

Since the early 1990s fractional cointegration has attracted the attention of many
empirical researchers working in different fields. We detail below some of the most
relevant applications.