Palgrave Handbook of Econometrics: Applied Econometrics

680 Panel Methods to Test for Unit Roots and Cointegration

As in the unit root discussion, before dealing with the general problem, let us
consider a set of simplifications that help to illustrate many of the issues involved.
Suppose we start, as in section 13.2, by switching off the factor dependence struc-
ture and assuming that both the deterministic processes and the cointegrating
vector are unbroken. Then a very simple version of the system reduces to:

yi,t=Di,t,m+x′i,tβi+ui,t, m=1, 2, 3 (13.18′)

( 1 −L)xi,t=νi,t, (13.22′)

where the indexmagain describes the usual specifications of the deterministic
component.Di,t,mcan either be empty (that is, contain no deterministic terms),
m=1, or have a constant,m=2, or a constant and a linear trend,m=3.
Let us further assume that the vectorsxi,t, and thereforeβi andνi,t, arel-
dimensional and that, ifyi,tandxi,tare cointegrated, theuniquecointegrating

vector is given by(1,−βi′)′. Assume further that, when cointegration prevails, the
processesei,t=(ui,t,νi′,t)′are cross-sectionallyindependentstationary autoregres-
sive moving average (ARMA) processes. The ARMA assumption is stronger than
we need but serves well for the purposes of illustration. In particular we are able
to appeal, under this ARMA assumption, to the existence of a finite long-run
covariance matrix forei,t, given by:

i=

[

ω^2 u,i uν,i

′uν,i ν,i

]

.

ν,iis taken to be of full rank, which excludes cointegration amongst the variables
xi,t. We also need to define the conditional long-run variance as:

ωu^2 .ν,i=ω^2 u,i−uν,i−ν,^1 i′uν,i,

and the matrix:

%i=

∑∞

j= 0

E(ei,te′i,t−j),

partitioned conformably withi. Finally, let us takeβi =β∀iif cointegration
exists. This is an assumption that can be generalized easily, as discussed by Pedroni
(2004).
If there is no cointegration, equation (13.18′)is a spurious relationship withyi,t
being anI(1) process not cointegrated withxi,t. In this case we can assume, analo-

gously to the ARMA assumption above, thatyi,t=u
sp
i,tand thate

sp

i,t=

(

u

sp

i,t,v

′

i,t

)′

are cross-sectionally independent stationary ARMA processes with full rank long-
run covariance matrices. Of course, this includes the case of independence ofyi,t
andxi,t. The superscript “sp” here is chosen to indicate the spurious regression
nature of the relationship in the case of no cointegration.