716 Panel Methods to Test for Unit Roots and Cointegration
innovation sequences. Under some further simplifying assumptions, the long-run
covariance matrix ofut,say, is given by:
=′F+diag(
ωi)
,withF∈Rr×rdenoting the long-run covariance matrix of the first difference of
the factors anddiag(ωi)collecting the long-run variances ofeitfori=1,...,N.
As is well known, cointegration prevails when the long-run covariance matrix has
reduced rank. Starting with the idiosyncratic components, we know that the long-
run variance ofei,tis equal to 0 whenei,tis itself already stationary. The first part
ofhas at most rankr. A cointegrating relationship has to be contained in the
left kernel of both terms above. These observations allow us to immediately study
the cointegration properties ofut.
Assume, without loss of generality, that the first 0≤r 1 ≤rfactors are integrated
but not cointegrated and the remaining ones are stationary. Also assume that the
units are ordered in such a way that the first 0≤N 1 ≤Ncoordinates ofetare sta-
tionary and the remaining ones integrated. With this set-up, cointegration prevails
if, inβ′ut, the firstr 1 factors and the lowerN−N 1 coordinates ofetare annihi-
lated. Partition the factors asFt=
[
(Ft^1 )′,(F^2 t)′]′
, withF^1 t ∈Rr^1 ,F^2 t ∈Rr−r^1 and=
[
′
1 ,′
2]′
accordingly. Then the cointegrating space is given by the intersec-tion of the orthogonal-complement of
′
1 and[
IN 1 ,0′
(N−N 1 )×N 1]′
to wipe out the
two non-stationary contributions to the vectorut. Without further assumptions
on the various spaces the cointegrating rank cannot be determined.
Note in addition that the identification of the factors forN→∞rests upon the
assumption that lim
N→∞
1
N∑N
i= 1 πiπ′
i=>0, which does not, determine the ranks
of the loading matrices for finite cross-sectional dimension and hence does not,
in particular, determine the number of linearly independent integrated common
factors for finiteN.
However, some simple observations can be made. First, cointegration can only
occur if some of the idiosyncratic components are stationary, since it can only
occur between series with stationary idiosyncratic components.
Second, if all common factors are stationary, the dimension of the cross-unit
cointegrating space is zero.
Third, to allow for a bit more detailed analysis of the cointegrating space assume
now that, for the given cross-sectional dimensionN, it holds thatrk
(
1)
=r 1 .We
already know from the first observation that we only need to consider the firstN 1
series
(
u1,t,...,uN 1 ,t)′
corresponding to the stationary idiosyncratic components
to analyze cointegration. Amongst these series the cointegrating space is given by
the left kernel of:
⎡
⎢
⎢⎢
⎣
(
π^11)′..
(.
πN^11)′⎤
⎥
⎥⎥
⎦
∈RN^1 ×r^1. (B.2)