656 Panel Methods to Test for Unit Roots and Cointegration
this issue, which complicates the use of first-generation panel unit root tests on
estimated de-factored data, is not considered carefully enough in the literature.
Bai and Ng derive the asymptotic behavior of tests paralleling those of Choi
(2001) and Maddala and Wu (1999), henceforth referred to asBNNandBNχ 2 , under
the assumption of cross-sectional independence of the idiosyncratic components.
The Choi (2001)-type test is constructed as follows. Denote thep-value associated
with the ADF test on the residual series from theith unit,ˆei,t,i=1, 2,...,N,by
pˆe(i). Then the following result:
BNN=− 2∑N
i= 1logpˆe(i)− 2 N
√
4 N⇒N(0, 1),can be established by careful analysis, paralleling the type of panel unit root test
proposed by Choi (2001) for cross-sectionally independent panels. The test statistic
BNχ 2 is, of course, given by− 2
∑N
i= 1 logpeˆ(i)∼χ
2
( 2 N).
Note that simple sequential limit theory with firstT→∞followed byN→∞,
as used in the panel unit root tests for cross-sectionally independent data, does
not apply here. This stems from the fact that consistent estimation of factors and
loadings is based on joint limit theory with minimum rate restrictions for both
dimensions of the panel, which implies that the observedeˆi,tare, in general, not
cross-sectionally independent for finite samples even under the assumption that
theei,tare independent. The panel unit root test result is established both when the
factors are computed after differencing (intercept only case) and after differencing
and demeaning (intercept and trend), and requires that the idiosyncratic terms be
independent acrossi. It should be noted, however, that the main results of the
Bai and Ng (2004) paper (consistent estimation of the factors and loadings and
testing for common trends in the common factors) allow for weak cross-sectional
correlation of the idiosyncratic errors.
Two general observations should be made of the Bai and Ng method. First, it
is sufficient that at least one integrated factor be present for all theyi,tseries to
have unit roots, if this factor is loaded into all series. Bai and Ng (2004) call this
integration or non-stationarity due to “a pervasive source.” The integration prop-
erties of the idiosyncratic components, however, feed, under the assumption of
their cross-sectional independence, uniquely into each series. That is, if all com-
mon factors are stationary the seriesyi,thas a unit root if and only ifei,thas a unit
root. This is referred to as a “series specific source” and may be thought to be a
measure of the existence or otherwise of a unit root in a given unit of a panel once
account has been taken of all the common trends driving the data (and describing
the dependence). The judgment on which is the more important source depends
upon the phenomenon being studied – that is, whether each series has its “own
root” or whether it comes from the same factor(s).
Second, the importance of the Bai and Ng (2004) analysis is in showing that, by
applying the method of principal components to first-differenced data, it is possible
to obtain consistent estimators of the factors and the idiosyncratic terms regardless