Palgrave Handbook of Econometrics: Applied Econometrics

702 Panel Methods to Test for Unit Roots and Cointegration

dynamic model necessitates the estimation of a substantial number of parameters.
Clearly, it has to be mentioned that accurate estimation of the long-run variances
for the nonparametric methods requires sufficiently long time series.
Open issues are allowing for a factor structure in the error processes of the
VAR models to allow for cross-sectional dependencies, and considering structural
change in the deterministic components.
Without imposing any cross-sectional homogeneity assumption, the cross-
sectionally independent panel VAR DGP considered is given inerror correction
formby:

Yi,t=C 1 i+C 2 it+αiβ

′

iYi,t− 1 +

∑pi

j= 1

ijYi,t−j+wi,t, (13.30)

withYi,t=(yi,t,xi,t)′∈Rm,m=l+1,C 1 i,C 2 i∈Rm,αi,βi∈Rm×kiwith full rank

ki,ij ∈Rm×mandwitare cross-sectionally independentm-dimensional white-
noise processes with covariance matricesi>0.^30 To ensure that all the processes
described by (13.30) are (up to the deterministic components)I(1) processes, the

matricesα

′

i⊥iβi⊥have to be invertible, whereαi⊥∈R

m×(m−ki),β

i⊥∈R

m×(m−ki)

are full rank matrices such thatα

′

i⊥αi=0,β

′

i⊥βi=0 andi=Im−

∑pi− 1

j=i ij. One

possible choice is given byαi⊥=Im−αi(α

′

iαi)

− (^1) α′
iand similarly forβi. Under these
assumptions the column space of the matrixβiis theki-dimensional cointegrating
space of uniti.
In the VAR cointegration literature the following five specifications of the deter-
ministic components are usually discussed. Case 1 is without any deterministic
components. In case 2 restricted intercepts of the form C 1 i = αiτiare con-
tained in the cointegrating space whereas in case 3 unrestricted intercepts C 1 ithat
induce linear time trends in Yi,tare included. In case 4 unrestricted intercepts and
restricted trend coefficientsC 2 i=αiκiare included, which allows for linear trends
in both the data and the cointegrating relationships. Finally, in case 5 unrestricted
intercepts and trend coefficients are included, which leads to quadratic determin-
istic trends in Yi,t. For a detailed discussion of these specifications concerning the
deterministic variables see Johansen (1995, sec. 5.7). This monograph also includes
a very detailed discussion of the statistical analysis, that is, parameter estimation
via reduced rank regression, testing for the cointegrating rank as well as hypothesis
testing. Therefore we do not include a description of this widely-used method here.
In the following two sub-sections we discuss two approaches, due to Larsson,
Lyhagen and Löthgren (2001) and Breitung (2005), to test for cointegration in panel
VAR models. In common with the single-equation approaches described above,
both these methods putat leastthe restriction of a cross-sectionally homogeneous
cointegrating space, that is,βi=β, in place. Again the argument in favor of such a
restriction is that in order to apply panel methods fruitfully, some of the coefficients
have to be considered cross-sectionally identical. Since in cointegration analysis the
main focus is on the cointegrating relationships, it is natural to assume identical co-
integrating spaces and allow for individual specific short-run dynamics. This is