722 Panel Methods to Test for Unit Roots and Cointegration
quantities has to be employed. Such limit theory is, to the best of the authors’ knowledge,
very sparse at best.
- For completeness we have also performed the computations including only those coun-
tries where no structural breaks are detected or with a shorter panel that excludes the
structural breaks. The latter experiment, however, suffers from the poor performance of
panel unit root and cointegration tests for short panels, compare again Hlouskova and
Wagner (2006) and Wagner and Hlouskova (2007). The potentially poor performance
notwithstanding, the additionally obtained results, available upon request, are highly
similar in terms of findings and conclusions to the results for the full country set and full
period panel discussed in the chapter.
- As in testing for unit roots in panels, the role of the null and alternative hypotheses can
be reversed in the construction of the tests – that is, the null hypothesis can be taken
to be that a cointegration vector exists while the alternative hypothesis is that of no
cointegration. The limitations nevertheless remain.
- The analogous discussion with respect to unit roots is contained in section 13.2.2.2.
- There are other ways in which one can think of introducing instability, in particular
through instability in the factors themselves or in the factor loadingsπi. We choose not
to address this issue here, since the general problem is already over-detailed.
27.ηtin (13.20) may be taken to be independent ofνi,t.
- Remember that in sections 13.2.2.2 and 13.2.3.1 we used the notationy ̃i,tto denote first
differences whereas we use this notation here to denote the deviations from the cross-
sectional averages. Some overlap in notation appears to be unavoidable but we hope that
this does not lead to any confusion.
- Note that Phillips and Moon (1999) is the only paper that formulates explicitly stochas-
tic assumptions on the underlying DGP that ensure the existence of the required
cross-sectional limits by introducing so-called stochastic linear processes, in which the
coefficients describing the DGP are cross-sectionally independent random variables with
certain properties. This implies that one can, under appropriate assumptions, determine
the required limits by laws of large numbers. Clearly the case of deterministic coefficients
with corresponding assumptions is nested in this framework.
- Again some overlap in notation occurs since the indexmhas been used before to indi-
cate the three main specifications of the deterministic components in unit root and
single equation and cointegrating testing whereas here and in Appendix B it denotes
the dimension of the multivariate time series in the panel data.
- The authors consider also a specific joint limit where the rate conditionN
1 / 2
T →0is
put in place. For this specific joint limit, a Lindeberg-type condition is required since
the cross-section specific building blocks of the panel test statistic are not i.i.d. for finite
values ofT, whereas onceThas passed to infinity, the time series building blocks of the
test statistics are identically distributed in the cross-section dimension.
- Note that the restriction to the same dimensionmfor each panel member is not required
for the discussion of the cointegration properties.
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