Anindya Banerjee and Martin Wagner 721- The efficacy of information criteria such as AIC and BIC in dealing with serial correlation
is an issue of some debate. Simulation evidence suggests that, due to the choice of very
low lag lengths, especially with BIC, serial correlation remains. - Phillips and Moon (1999) contains an excellent discussion concerning sequential (that is,
firstTto infinity followed byNto infinity) versus joint (that is,TandNtend to infinity
simultaneously, potentially with some restrictions on the divergence rates), as well as
the relationships between the different asymptotic concepts. Appendix C of our chapter
discusses some of these issues. - Hlouskova and Wagner (2006) discuss the consequences of assuming, instead of constant
covariance, a Toeplitz structure for the covariance matrix which corresponds to a geomet-
rically declining correlation with distance. This same idea is pursued in more detail in
Baltagi, Bresson and Pirotte (2007). For simplicity, since it serves to make the substantive
point, in our illustration this geometrical decline is absent. - Hlouskova and Wagner (2006) present a larger experimental design and also consider
more tests. - This is a simplifying assumption adopted here, and is stronger than is needed for the Bai
and Ng (2004) results to hold. Bai and Ng allow for weak cross-correlation in the errors,
“weak in the sense that the column sum of the error covariance matrix remains bounded”
(Bai and Ng, 2004, p. 1131). This leads to what is termed an approximate factor model. - Considering the factor loadings to be stochastic appears to be more of mathematical
rather than practical value, given that for each loading it is only one realization that
generates the data. - If the number of factorsris unknown, it must be estimated consistently using, for exam-
ple, the Bai and Ng information criteria. The factors themselves are estimated via principal
components, as described above in the discussion of the Bai and Ng (2004) method. - The model discussed here does not allow for a linear trend. Section 3 of Moon and Perron
(2007) extends the analysis to allow for linear trends. - Moon and Perron (2004) derive results for the distribution of the estimator under local-
to-unity alternatives for the root. They also need a further restriction on the rate of
convergence ofNandTto infinity in order to obtain consistent estimatesωˆ^2 e,φˆ^4 eandλˆNe.
Assumption 10 (p. 91) of their paper gives this asa=lim inf(N,T→∞)logT/logN>1.
The parameterais related to the speed ofN/Ttending to zero. - As noted in note 15, cross-sectional independence of theei,tprocesses, via the cross-
sectional independence of theεi,tprocesses, is not needed for estimation and inference
concerning the factors. We assume it here (and earlier) for simplicity – since this allows
for the construction of pooled panel tests for the idiosyncratic terms. - The problem is formulated in terms of obtaining consistent estimates of thebreak fractions
(as discussed below.) This allows the derivations to deal withT→∞in order to provide
large-sample results. If instead the break date were fixed, the problem would become
degenerate in cases whereTwas large. - The Bai and Carrion-i-Silvestre result presented above, as well as those discussed below,
rely upon limiting arguments not fully discussed either by these authors or in the seminal
Bai and Ng (2004) paper. A major problem is posed by the fact that, for finiteN, the de-
factored observations and, consequently, the test statistics based upon these, are not
cross-sectionally independent, since only the product of the estimated loadings and the
estimated factors are subtracted. Cross-sectional independence requires bothNandTto
go to infinity to have consistent factors (and loadings). Therefore, one cannot simply
appeal to sequential limit theory to derive the asymptotic distribution forMSB(i)above
by lettingTtend to infinity first (to derive thep-values) and then derive the asymptotic
distribution ofBCNby lettingN“tend to infinity again.” A similar issue arises with the
derivation of theZstatistic above. In fact, similar problems plague all of the rapidly
growing panel unit root literature based on de-factored observations that fails to take
into account that joint limit theory for (in finite samples) cross-sectionally dependent