Anindya Banerjee and Martin Wagner 717The assumption of full rank of 1 does not imply that the above sub-matrix com-
posed of the firstN 1 columns of 1 also has rank equal tor 1. Clearly, it necessarily
has rank smaller thanr 1 ifN 1 <r 1 , but can also have reduced rank otherwise. A
reduced rank,s 1 say, of the matrix in (B.2) implies that only this smaller numbers 1
of common trends is distinguishable within the firstN 1 coordinates ofut. The co-
integrating space is given by the orthogonal-complement inRN^1 of the space
spanned by the matrix (B.2), which is of dimensionN 1 – s 1 in our discussion. The
cross-unit cointegrating space can be determined as in the definition above, again
by projecting on the orthogonal-complement of the stacked individual specific
cointegrating spaces (β-in the notation introduced above). In our univariate con-
text individual specific cointegration means that a seriesui,t, for somei=1,...,N 1 ,
is stationary, which happens when the correspondingλ^1 i =0. In this case the
corresponding entry inβ-is set equal to 1. Note that, due to possible rank reduc-
tion in (B.2), for this to happen it is not necessary for the conditionπi^1 =0to
hold, although it is of course sufficient. The above analysis also applies when all
idiosyncratic components are stationary, in which caseN 1 =N.
Altogether the discussion highlights the price that has to be paid, in terms of
very restricted structures of the cointegrating spaces, when reducing model com-
plexity (to be precise the modeling of cross-sectional dependence) by resorting to
(approximate) factor models.
Clearly, the limitations discussed above are also present when using factor mod-
els in a multivariate panel context, either by using factor DGPs as error processes
in single equation cointegration analysis (as in section 13.3.1.4) or when con-
sidering systems inference procedures with a factor structure in the stochastic
component.
13.7 Appendix C: Limiting concepts for integrated panels
One major difference to classical microeconometric panels is that for time series
panel applications the assumption of cross-sectional independence is often unten-
able. For many applications from the realms of international macroeconomics or
finance, dependence of the variables across countries appears to be the norm rather
than the exception.
The first generation of the literature, by which we label all methods (that is, tests
and estimation procedures) that are based on the assumption of cross-sectional
independence, has made this strong assumption of cross-sectional independence
not because of its empirical validity but because of methodological simplicity.
The assumption of cross-sectional independence facilitates many parts of the
theoretical analysis considerably.
To illustrate the relative simplicity that the cross-sectional independence assump-
tion – in conjunction with sequential limit theory whereN→∞afterT→∞–
consider the following simple example. Assume that for each cross-section member
the data are generated according toyit=ρiyit− 1 +εit, that is, by an autoregres-
sion of order 1 without any deterministic components and withεita white-noise