678 Panel Methods to Test for Unit Roots and Cointegration
moments) relatively simple sequential central limit theory provides the asymp-
totic normality of these densities. As soon as the assumption of cross-sectional
dependence is lifted, however, issues become much more intricate, as out-
lined in the highly stylized discussion in Appendix C. Note that cross-sectional
dependence of the data often requires the use of joint limit theory, as, for exam-
ple, in the work of Bai and Ng (2004). More generally, the joint limit theory
applied in factor models, where consistent estimation of the factors requires
the cross-sectional dimension to tend to infinity, has implications for all unit
root and cointegration test procedures that use de-factored observations. This
is an issue that we believe deserves more attention in the literature.
(f) The simple ways in which factor structures are utilized and the intuitive manner
in which the tests are constructed.
(g) The natural ways in which structural breaks are incorporated.
All of these features are important to the development and use of this methodology,
and further work, theoretical, simulation-based and empirical, to investigate the
efficacy of the various testing strategies proposed is still largely necessary.
Many of (a)–(g) carry over when it is not a unit root hypothesis that we are
interested in testing, but a hypothesis of cointegration among variables of interest,
and it is to a consideration of these methods to which we now turn. We could
think, for example, of PPP, where we might be interested in seeing co-movement
between foreign and domestic prices expressed in the same currency; or exchange
rate pass-through, where one could look at how changes in exchange rates are
transmitted to the price of imported goods; or the Feldstein–Horioka puzzle where
the apparent co-movement of savings and investments (which runs contrary to
commonly held beliefs on the consequences of quasi-perfect capital markets); or,
of course, the growth literature and issues of convergence discussed earlier.
13.3 Cointegration analysis in non-stationary panels
The majority of cointegration analysis in multivariate time series panels is con-
ducted within the single equation set-up, in which them-dimensional time series
Yi,tare separated intoYi,t=
[
yi,t,x′i,t]′
, where subsequentlyyi,tis the dependent
variable andxi,tare the regressors. This approach is subject to the same limita-
tions as Engle and Granger (1987) single-equation cointegration analysis in the
time series case. The most important restriction is that the analysis is limited to
situations in which there is either no cointegration inYi,t(under the null hypoth-
esis of the “no cointegration” tests) or only one cointegrating relationship (under
the alternative of the “no cointegration tests”).^24 This assumption is in general,
withxi,ta multivariate vector of regressor variables, not easily sustainable, except
perhaps in special cases. Clearly, methods that allow for higher-dimensional cointe-
grating spaces are therefore also relevant in the panel cointegration context. Such
methods have until now been based on panel extensions of VAR cointegration
analysis and are discussed in section 13.3.2.