Palgrave Handbook of Econometrics: Applied Econometrics

636 Panel Methods to Test for Unit Roots and Cointegration

series from the cross-sectional average are stationary, that is, any of theNseries

yi,t−ytwithyt=N^1

∑N
i= 1 yi,tis also stationary, with mean 0 in the case of abso-
lute convergence. Thus, the definition of convergence in theI(1) context of EK is
given by:

Definition (I(1) framework): If each income seriesyi,tis described by anI(1) process,
but all seriesyi,t−ytare stationary, the set ofNeconomies is said to converge.
Convergence is said to be absolute if the means of all the seriesyi,t−ytare equal
to 0 and relative otherwise. The economies are said to diverge if all seriesyi,t−yt
are non-stationary.^4

Given our Granger representation in (13.1) above, we may ask about the restrictions
imposed by this representation, that is, under what conditions are all seriesyi,t−yt
stationary? Computing the cross-sectional average, we obtain:

yt=
1
N

∑N

i= 1

ci,1η1,t+···+

1

N

∑N

i= 1

ci,rηr,t+

1

N

∑N

i= 1

di,1T1,t+···+

1

N

∑N

i= 1

di,sTs,t+c∗(L)εt,

(13.2)

where we use the notationCi=

[

ci,1,...,ci,r

]

andDi=

[

di,1,...,di,s

]

, andc∗(L)εt=

1

N

∑N

i= 1 c

∗
i(L)εi,tis stationary.
Now, in order to have convergence in the EK sense, each of the deviations of the
income series from the cross-sectional average has to be stationary. This necessi-
tates that all stochastic trends have to be annihilated – as well as all non-constant
deterministic components. Consequently, for alli=1,...,N, it has to hold that:

ci,k−

1

N

∑N

i= 1

ci,k=0, (13.3)

which implies thatci,k=ckfor alli=1,...,Nand for allk=1,...,r. This in turn

implies that only one common stochasticI(1) trend, given byηt=

∑r
k= 1 ckηk,t,
is present and identifiable in the data.^5 This is, of course, not a surprise given
that the EK definition implies the presence ofN−1 cointegrating relationships
(hence of only one stochastic trend), which is furthermore loaded with the same
weight into each series. Similar arguments apply to all non-constant elements in
the deterministic component: the coefficient to each non-constant deterministic
variable has to be equal for each series; thus, for example, if present in the data,
all linear trend slopes need to be identical for relative convergence to prevail. For
absolute convergence to hold, in addition, all intercepts need to coincide.
Thus, we see that the EK definition of convergence imposes a lot of structure and
hence testable restrictions on the panel of time series. Furthermore, as mentioned
previously, it also implies that there is long-run cross-sectional dependence in the
panel as defined in Appendix B.
It is worth observing at this stage that if one had enough observations to perform
a multivariate time series analysis, in relation to the considerations above, one