Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 635

that lim
j→∞

E

(

yi,t+j−at

)

=μifor alli=1,...,N. Then the economies are said to

converge. Convergence is called conditional if not allμ 1 ,...,μNare equal to 0 and
is called absolute if allμ 1 ,...,μNare equal to 0.

This definition of convergencepotentiallyleads to several interesting implications
for non-stationary panel analysis since it presumes the existence of one joint trend
processat(which is typically unobservable and is related to technology), such
that the limits of the expected values of the deviations from this trend exist and
are constant. Given that income is often found to be a unit root non-stationary
process, this allows us to reduce or specialize the general formulation given in the
definition above to anI(1) context.
Thus, we consider henceforth the income series in theNcountries to be jointly
described by a vectorI(1) process.^3 In this case the EK definition of convergence
implies that the deviations from the trend processatare asymptotically station-
ary. It implies furthermore that the cross-section members will not be independent
of each other, given their relation to the single common trendat. Clearly, if we
assume that the income series areI(1), but the deviations fromatare stationary,
this implies thatatis also anI(1) process. Thus the panel exhibits within this
framework cross-sectional dependence via the stochastic component of the com-
mon trendat. Cross-sectional dependence is discussed in detail in Appendix B.
We can already foresee, given that we formulate the discussion here within anI(1)
modeling framework, that the specific formulation of the convergence definition
of EK has strong cointegration implications which we detail next.
If we keep, for the moment, the cross-sectional dimension as fixed, the joint vec-

tor processyt=

(

y1,t,...,yN,t

)′
is – under appropriate assumptions – also jointly an
I(1) vector process and thus has a Granger-type representation, where for simplicity
we abstract from detailing the initial values and their effects, given by:
⎡
⎢⎢
⎣

y1,t

..

.

yN,t

⎤

⎥⎥

⎦=

⎡

⎢⎢

⎣

C 1

..

.

CN

⎤

⎥⎥

⎦ηt+

⎡

⎢⎢

⎣

D 1

..

.

DN

⎤

⎥⎥

⎦Tt+c

∗(L)ε

t, (13.1)

withηt∈Rrther≥1 linearly independent common stochastic trends,Tt∈Rsthe
deterministic components of the data-generation process (DGP) andc∗(L)εtthe
stationary part, withLdenoting the backward shift operator, that is,L

(

xt

)

( t∈Z=
xt− 1

)

t∈Zandc

∗(L)=∑∞

j= 0 c

∗

jL

jsuch thatc∗(L)ε
tis a stationary process.
In particular, in anI(1) setting, the EK definition implies that the cointegrating
space for theN-dimensional vector of income series is of dimensionN−1. This
follows immediately from the fact that any pair-wise difference of income between
countries in which the unit root processatis annihilated is stationary. Note here
that this already implies that all non-constant deterministic components are also
annihilated by taking pair-wise differences. Consequently, the definition not only
specifies the dimension of the cointegrating space but also fully determines the
space itself (that is, a basis of the cointegrating space). Note, furthermore, that
by the same argument it is also true that the deviations of any individual income