Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 637

would not need to resort to panel techniques. However, this is typically not the
case and we therefore turn more specifically to the panel (unit root) implications of
the convergence definition. In the case of convergence, each of the seriesyi,t−yt
is stationary and divergence has been defined by EK as (unit root) non-stationarity
of all theNseriesyi,t−yt. Thus, the null hypothesis of convergence can be tested
by a panel stationarity test, as presented, for example, in section 13.2.1.4. More
often, however, the null hypothesis of divergence is tested against the alternative of
convergence by using panel unit root tests that are specified against the alternative
that all series are stationary. Under appropriate assumptions on the DGP, this can
be done within panel Dickey–Fuller-type regressions of the form (with individual-
specific autoregressive orderspi):



(

yi,t−yt

)

=δi+ρi

(

yi,t− 1 −yt− 1

)

+

∑pi

k= 1

φi,k

(

yi,t−k−yt−k

)

+ui,t, (13.4)

withui,tdenoting here the residual process of the autoregression. When the cross-
sectionally demeaned data are described by an autoregression of the corresponding
order, the processesui,t are white-noise processes. Note here that EK, despite
testing the null hypothesis of divergence, proposed such testing in a Dickey–Fuller-
type regression including only intercepts and no other deterministic components,
which are in fact not restricted under the divergence hypothesis by EK. Thus,
in effect, what EK proposed was to test the null hypothesis of divergence with
respect to the stochastic trend whilst allowing only for individual specific inter-
cepts and linear trends. This restriction of the deterministic components needs
to be investigated statistically, not least to prevent misspecification of the panel
Dickey–Fuller-type regression above.
With the specified restriction, the null hypothesis of divergence is given by
H 0 :ρi=0 for alli=1,...,Nagainst the alternative hypothesisHA:ρi < 0
for alli=1,...,N, where, as is usual in unit root testing, we really consider only
stationary alternatives, which imposes restrictions onρiunder the alternative that
depend upon the unknown coefficientsφi,kin case the data are really described
by an autoregression andui,tis white noise. In general there are no simple links
between the innovationsεtfrom the Granger representation above and the univari-
ate regression errorsui,t, with the latter being a function of the former. Generally,
unless very strong restrictions are imposed, the errorsui,twill be cross-sectionally
dependent. Issues are, however, even more problematic when testing the null
hypothesis of divergence. Under divergence, when there are no restrictions on
the stochastic trends present in the panel, the deviations from the cross-sectional
averages will also in general be linked by stochastic trends, and these deviations can
thus be cross-unit cointegrated.^6 Thus, the discussion here highlights strongly that
in general we will need to consider panel unit root (and to a certain extent panel
stationarity) tests that allow for cross-sectional dependencies, where in particular
cross-sectional dependencies may enter via the errors or may arise via common
factors. This is a theme to which we shall return in detail during the course of our
chapter.