Anindya Banerjee and Martin Wagner 663integrated panels needs to take account of shifts in the deterministic parts of the
processes, since breaks in series may lead (as in Perron, 1989) to spurious findings
of a unit root. The analysis can, in principle, also be extended to allow for breaks
in the factors or in the loading coefficients on the factors, but this takes us beyond
the scope of the current discussion.
13.2.3 Relaxing structural stability – Bai and Carrion-i-Silvestre
(2007)
The restrictions underlying (13.5)–(13.7) above, with specification of the breaks
given in (13.9), are relatively mild. This model formulation allows for a very
general specification of the testing framework, which allows not only for cross-
sectional dependence via common factors but also for structural instability. As
in the assumptions underlying the Bai and Ng model (given in section 13.2.2.2)
the factor loadings need to be identifiable; there are conditions on the short- and
long-run variance ofFtand theεi,tprocesses are assumed to be weakly serially
correlated but cross-sectionally independent (as given by the set of conditions (iii)
forεi,tin section 13.2.2.2).^20 Under certain assumptions, the machinery of feasible
GLS-type corrections could be employed for cases where the errors are thought to
be correlated in the short-run, but within the broader problem we wish to address
in this section, this issue remains a somewhat mild technicality. Finally, some
restrictions on the initial conditions are also needed.
The key part of the testing strategy of Bai and Carrion-i-Silvestre (2007) is
based on constructing so-called modified Sargan–Bhargava statistics (henceforth
MSB), due to Sargan and Bhargava (1983) and Stock (1999). The unit-specific MSB
statistics are based on computing:
MSBi=T−^2∑T
t= 1eˆ^2 i,tωˆi^2, (13.13)where the denominator is a measure of the long-run variance ofei.t.
The most important aspect of the analysis is therefore to extract consistent esti-
mates ofei,t(denotedˆei,t)in the presence of common factors and structural breaks,
and to test these series for the presence of unit roots. In addition to the different
specifications for the breaks which may be considered (to allow for breaks only
in the intercept or also to allow for changes in the trend slope), a further crucial
distinction among the methods is to deal with the cases where the break date(s)
are known versus cases where they need to be estimated consistently (for example,
by means of the algorithms considered in Bai and Perron, 1998).^21 We touch upon
all these issues in turn.
Under the assumption of cross-sectional independence of the idiosyncratic com-
ponents, the final form of the test statistic proposed by Bai and Carrion-i-Silvestre
takes the form of pooling the individualMSBistatistics, using correction terms
for the mean and variance so that the limiting distribution is standard normal.
An alternative strategy involves poolingp-values in the spirit of Maddala and Wu