Palgrave Handbook of Econometrics: Applied Econometrics

Luis A. Gil-Alana and Javier Hualde 445

also a particular case of the fractional setting proposed by Jeganathan (1999, 2001).
This multivariate modelization is a straightforward generalization of the bivariate
setting, given that the time series involved still depend on two integration orders,γ
andδ. A richer structure is proposed by Hualde and Robinson (2006), who consider
the model:

!zt=−^1 (δ)u#t, (10.11)

where δ =

(

δ 1 ,...,δp

)′

, zt is ap×1 vector observable process, (δ) =

diag

(

δ^1 ,...,δp

)

andutis ap×1 zero mean covariance stationary unobserv-

able process. The presence of the various integration orders inδposes additional
difficulties and, unlessδand!are restricted, (10.11) does not ensure cointegration
and identification. Hualde and Robinson (2006) impose restrictions (simplifying
matters substantially in an already complicated setting) which ensure identifiabil-
ity and imply thatztisI

(

δp

)

(δpis assumed to be the largest fractional order among

those inδ). The restrictions also imply that there arercointegrating relations
among the elements ofzt, and allow the integration orders of thercointegrating
errors to vary, unlike in previous models of multivariate fractional cointegration.
In a different setting, Chen and Hurvich (2003a) model aq×1 time series,zt, such
that its (p− 1 )th difference (wherepis an integer),yt, is a covariance stationary
process with common memory parameterd∈

(

−p+0.5, 0.5

)

. They assume thatyt
has the common-components representation:

yt=Axt+But, (10.12)

where, for 1≤r≤q−1,A,Bareq×

(

q−r

)
andq×runknown deterministic
matrices of ranksq−randr, respectively, andxt,ut, are

(

q−r

)
×1 andr×1,
unknown unobserved processes with memories( danddu, respectively, wheredu∈
−p+0.5, 0.5

)
<d. Basically, these conditions imply thatytis cointegrated, with
cointegrating space identified by the null space ofA′(Ker(A′)), noting that, ifα∈
Ker(A′), thenα′yt=α′But, which has at most memorydu. Chen and Hurvich
(2006) further enrich this setting by means of a common components model in
which the components have different memory parameters, while still allowing the
q×1 vector of observed series to have just one common memory parameter. Thus,
using notation similar to Chen and Hurvich (2003a), they set:

yt=A 0 ut(^0 )+A 1 u(t^1 )+···+Asu(ts), (10.13)

where, fork=1,...,s, theAkareq×akunknown deterministic matrices with

a 0 =q−r,
∑s
j= 1 aj=r, theu

(k)
t areak×1 unobservable processes with memory
dk, such that−p+0.5<ds<···<d 0 <0.5, and all rows ofA 0 are non-zero.
This setting ensures that all the components inythave common memoryd 0 , the
cointegrating rank isr, such that 1≤r<q, and there arescointegrating sub-
spaces, with 1≤s≤r. In this framework, Chen and Hurvich (2006) defines
different cointegrating sub-spacesBk,k=1,...,s, with the main characteristic
being that, ifβ∈Bk,β′Al=0,l=0,...,k−1, andβ′Ak=0.