638 Panel Methods to Test for Unit Roots and Cointegration
Clearly, if the panel is characterized by several independent stochastic trends,
simple cross-sectional demeaning will not be able to remove all these non-
stationary components. Allowing for such a situation – bearing in mind that a full
systems analysis will typically not be feasible due to data limitations – is one of the
major motivations for considering so-called factor models in the non-stationary
panel literature. Factor models, by assuming that the data are generated by the
sum of two components, common factors and idiosyncratic components, allow for
modeling by putting restrictions on the spectrum of the jointly stationary process
yt.^7
To explain further, let us consider the simplest case in relation to our conver-
gence discussion. Ignoring deterministic components for simplicity, assume that
each income series can be written asyi,t=at+νi,t, where the processesatandνi,tare
all independent, and both components are either stationary orI(1) processes. Due
to the assumption thatatandνi,tare independent, all seriesyi,tare also either sta-
tionary or integrated. The deviations from the cross-sectional averages are given by
ν ̃i,t=νi,t−N^1
∑N
j= 1 νj,t=(
1 −N^1)
νi,t−N^1∑N
j=1,j =iνj,t. Now, in the case of conver-
gence, the second term in this equation converges to 0 (under appropriate technical
conditions); whereas in the case of divergence, the second term does not converge
to 0. This shows that even if we start with cross-sectionally independent processes
νi,t, the processesv ̃i,tdescribing the deviations from the cross-sectional averages
are asymptotically independent asN→∞only when convergence prevails, but
need not be even asymptotically independent in the case of divergence.
The above example shows that, in general, if one is confronted with unob-
served factors, cross-sectional averaging will not necessarily lead to asymptotically
cross-sectionally independent series by studying the deviations from the cross-
sectional averages. In our example above, due to the unobserved factoratbeing
loadedin all series with the same weight, considering the deviations from the
cross-sectional averages in fact allows elimination of this common factor. How-
ever, dependencies are introduced via the seriesν ̃i,tthat persist when the series
are integrated. These remarks all hold similarly in cases where more than one
common factor is considered or the factors do not enter all series with the same
weights.
Therefore, when focusing only on the set of jointI(1) processes that can be
characterized by factor models, the need for appropriate estimation and inference
techniques arises. Such techniques should deliver, again formulated for the con-
vergence example considered here, the following: first, they need to construct
consistent estimates of the common factoratas well as tests for stationarity (respec-
tively unit root) behavior of this series. Second, the procedures need to establish
panel unit root tests for the de-factored series,yi,t−aˆt, where the circumflex
indicates that only estimates of the unobservable factor are available. With these
tools, convergence in the sense of EK can be tested by establishing first thatatis an
integrated process, and all seriesν ̃i,tare stationary. Essentially these tools, allowing
for multiple factors and heterogeneous loadings, have been developed by Bai and
Ng (2004).