Palgrave Handbook of Econometrics: Applied Econometrics

Luis A. Gil-Alana and Javier Hualde 449

is generated by a vector autoregressive (VAR) process:

ut=

∑p

j= 1

Bjut−j+εt,

and defining:

Zt

(

c,d

)

=

(

xt(c),xt

(

d

)

,w′t− 1

(

c,d

)

,...,w′t−p

(

c,d

))′

,

wherewt

(

c,d

)

=

(

xt(c),xt

(

d

)

,yt(c)

)′
, they analyzed the behavior of the estima-
torsˆν(γ,δ),ν(ˆγˆ,δ)ˆ, whereγˆ,δˆare corresponding estimators ofγ,δ, andˆν

(

c,d

)

=

i′ 1 G

(

c,d

)− 1

g

(

c,d

)

. Here,i 1 =(1, 0,...,0)′:

G

(

c,d

)

=Q

1

n

∑n

t=p+ 1

Zt

(

c,d

)

Zt′

(

c,d

)

Q′, g

(

c,d

)

=Q

1

n

∑n

t=p+ 1

Zt

(

c,d

)

yt(c),

whereQcaters for the possibility of prior zero restrictions on theBj’s. As in
Robinson and Hualde (2003), this method is based on a GLS-type of correction.
Hualde and Robinson (2007) showed that both estimators are

√
n-consistent and
asymptotically normal, but with different asymptotic variance, which in the case
ofν(ˆγˆ,δ)ˆ depends on howγ,δ, are estimated. Here, it is important to note that, in
order to get a

√

n-consistentν(ˆγˆ,δ)ˆ, it is essential thatγ,δ, are

√
n-consistently esti-
mated, and Hualde and Robinson (2007) proposed a feasible method of estimation
for the case whereBjis upper-triangular for allj=1,...,p.
Robinson and Iacone (2005) consider the data-generating process (10.7)–(10.8),
and deal with estimation ofνand of those elements inμ 1 j,j=1,...,p 1 ,μ 2 j,
j=1,...,p 2 , whose associated deterministic trends are not dominated (in the pre-
cise way defined in Robinson and Iacone, 2005) by stochastic components. They
consider three different estimators of the parameters of interest. First, they ana-
lyze the properties of (10.16), concluding that when the deterministic term in
(10.7) dominates the stochastic and deterministic components inxt,νˆOLSis not
even consistent. Otherwise, consistency is retained, and the authors offer a very
detailed analysis of the different possibilities involved. The second scenario refers
to estimating by OLSνandμ 1 in (10.7), whereμ 1 collects theμ 1 j’s associated with
the deterministic terms not dominated byu 1 t(−γ)(which are the only ones which
could be consistently estimated). Here, due to the accounting of the deterministic
components, the estimate ofνis always consistent, with asymptotic properties very
dependent on

(

γ,δ,φ 2 ∀

)
, whereφ 2 ∀is the maximum value of thoseφ 2 j’s for which
μ 2 j=0. Finally, they also consider GLS estimation, taking into account the deter-
ministic terms in (10.7)–(10.8), thus extending the time and frequency domain
estimators of Robinson and Hualde (2003), also forδ−γ>0.5. Under identi-
cal regularity conditions, whenδ>φ 2 ∀, the estimate ofνhas identical asymptotic
properties to that of Robinson and Hualde (2003); whenδ=φ 2 ∀, the rate of conver-
gence remains the same, but the limiting distribution changes, whereas ifδ<φ 2 ∀,