Palgrave Handbook of Econometrics: Applied Econometrics

448 Fractional Integration and Cointegration

estimator of Phillips and Hansen (1990), assuming knowledge ofγandδ. Kim
and Phillips (2002) considered an alternative extension to FM-OLS, and analyzed
its relationship to Gaussian ML estimation in (10.9)–(10.10), assuming parametric

autocorrelation in

(

v( 1 γt),v( 2 δt)

)′

. Jeganathan (1999, 2001) considered ML estimation
of (10.9)–(10.10) assuming knowledge of the distribution of the innovations and
also ofγandδ, although he did include some discussion of their estimation (as
also did Kim and Phillips, 2002).
In model (10.5)–(10.6), and assuming the bivariate processuthas a parametric
spectral densityf(λ)=f(λ;θ), whereθis an unknown vector of short memory
parameters, Robinson and Hualde (2003), based on generalized least squares (GLS)
type corrections, propose time and frequency domain methods to estimate opti-
mally (under Gaussianity)νwhenδ−γ>0.5 (denoted strong cointegration). For
simplicity, we just present the frequency domain approach, which is asymptoti-
cally equivalent (to first-order properties) to that of the time domain, and which
will be applied to empirical data in section 10.4. Denoting:

zt(c,d)=(yt(c),xt(d))′, ζ=(1, 0)′, p(λ;h)=ζ′f(λ;h)−^1 ,

a(c,d,h)=

∑n

j= 1

p(λj;h)wx(c)(−λj)wz(c,d)(λj), q(λ;h)=ζ′f(λ;h)−^1 ζ,

b(c,d)=

∑n

j= 1

q(λj;h)Ix(c)(λj),

and defining:

νˆ

(

c,d,h

)

=

a

(

c,d,h

)

b

(

c,h

) ,

they considered five different estimators, given by:

ν(γˆ ,δ,θ),ν(γˆ ,δ,θ)ˆ,ν(γˆ ,δˆ,θ)ˆ,ν(ˆγˆ,δ,θ)ˆ,ν(ˆγˆ,δˆ,θ)ˆ, (10.18)

whereγˆ,δˆ,θˆare corresponding estimators of the nuisance parametersγ,δ,θ. The
estimators in (10.18) reflect different knowledge about the structure of the model,
the first being in general infeasible, the second assuming just knowledge of the
integration orders (as was done previously in the standard cointegrating literature),
whereas the last estimator represents the most realistic situation. Under regularity
conditions,^10 Robinson and Hualde (2003) showed that any of the estimators in
(10.18) isnδ−γ-consistent with identical mixed-Gaussian asymptotic distributions,
leading to Wald tests on the parameterν:

W(γ,δ,θ),W(γ,δ,θ)ˆ,W(γ,δˆ,θ)ˆ,W(γˆ,δ,θ)ˆ,W(γˆ,δˆ,θ)ˆ, (10.19)

whereW(c,d,h)=b(c,h){ˆν(c,d,h)− 1 }^2 , with a chi-squared limit.
Hualde and Robinson (2007) propose an estimator ofνin (10.5)–(10.6) under
the more adverse situationδ−γ<0.5 (denoted weak cointegration). Assumingut