Anindya Banerjee and Martin Wagner 679Single-equation methods, however, offer some advantages, since they allow us to
consider – paralleling much of the developments in panel unit root analysis – both
cross-sectional dependence via factor models and structural changes in the deter-
ministic components. None of these two aspects has yet been studied in system
methods for panel cointegration analysis.
This section starts with a general formulation of the single equation panel coin-
tegration set-up and then continues with discussing tests for cointegration that
abstract from cross-sectional dependence and structural change. Structural change
is considered next in tests for cointegration, following which allowing for cross-
sectional dependence is also added to the testing structure. In section 13.3.1.6 we
discuss single equation estimators of the cointegrating vector, for the situation
without cross-sectional dependence or structural change. Section 13.3.2 then dis-
cusses testing for cointegration as well as estimation of the cointegrating spaces in
panel VAR models, under the assumptions of cross-sectional independence and no
structural change. Empirical examples are again used throughout to illustrate the
techniques.
13.3.1 Single equation analysis of cointegration
Paralleling the set-up of the DGP for studying the unit-root problem given by
(13.5)–(13.7) above, we may consider describing the general set-up of testing for
cointegration in panels:
yi,t=Di,t+x′i,tβi,t+ui,t (13.18)ui,t=πi′Ft+ei,t (13.19)
( 1 −L)Ft=C(L)ηt (13.20)
(
1 −φiL)
ei,t=Hi(L)εi,t (13.21)
( 1 −L)xi,t=νi,t, (13.22)where the major difference to the set-up considered in (13.5)–(13.7) is the presence
of additional regressorsxi,tin (13.22) that are potentially related to the dependent
variableyi,tvia a cointegrating relationship. Note that in general one can consider,
as in (13.18), the cointegrating relationship to be both individual specific and time
varying.
It is important to note that cointegration in the usual sense occurs if the error
processui,tin (13.18) is stationary. However, this is not the only possible form
of relationship in which one could be interested. One could, for example, also
consider cointegration after taking out the common factors,^25 and this is the addi-
tional aspect that the cross-sectional dimension and the modeling of cross-sectional
dependence brings into play.
Short-run dependence across the units may, under appropriate circumstances
and assumptions, be dealt with as before by considering the variance–covariance
structure of the error processes.