642 Panel Methods to Test for Unit Roots and Cointegration
denote the date of thejth break in the intercept and thekth break in the trend slope
for theith unit. As in the empirical example in section 13.3.1.5, the specification
can be simplified by settingli=mi= 1 ∀i=1, 2,...,N, but the theory is available
for the general case (see, for example, Bai and Carrion-i-Silvestre, 2007).
Cross-sectional dependence
The matrixFtcollects the common effects that are present across the cross-section
dimension. The non-stationarity (or integration) ofFtwill mean that all the units
in the panel have common non-stationary (or integrated) components entering
into each individual unityi,twith loadings of magnitudeπi. Within the context of
the Evans and Karras (1996) example, theπi’s are all equal to one, and the single
common factor is concentrated out of the problem by cross-sectional demeaning
- that is, by constructing the variableyi,t−N^1
∑N
i= 1 yi,t, so that the focus then
lies on testing the null hypothesis of a unit root in the regression model given by
(13.4) above. As mentioned above, however, the simple cross-sectional averaging
in general introduces correlations in the error terms describing the deviations from
the cross-sectional averages.
As alluded to already in the convergence example, the presence of one or more
common stochastic factors with heterogeneous loadings necessitates new testing
and estimation strategies. One of these approaches, which is given in the work
of Bai and Ng (2004) and Pesaran (2006), considers a general alternative to fac-
tor extraction by allowing cross-sectional averages (asN→∞)to approximate
the effects of the unobserved or latent common factors. In this way, the need to
estimate the factors and their loadings is avoided. The Pesaran approach, in the
context of testing for unit roots, is discussed in section 13.2.2.3.
A further very interesting approach has been introduced in Pesaran, Schuerman
and Weiner (2004) and Garrattet al.(2006) who introduce the concept of global
vector autoregressions (GVARs), which consists of specifying VAR models for each
cross-sectional member of the panel, including specifically constructed averages of
the other cross-section members’ variables as exogenous variables. This approach,
feasible whenever the data permit VAR modeling for each cross-section member,
is a parsimonious way of combining the information of all cross-section members.
The construction of the weighted cross-sectional averages of the other countries’
variables is a key issue in this modeling approach. One example considered by
the authors is to use trade shares as weights. Obtaining further understanding of
appropriate weighting schemes, which in general will depend upon the applica-
tion considered, as well as the performance of these methods, is an important task.
Under certain assumptions the joint system can be solved for based on the individ-
ual specific VAR models with exogenous variables. Given that a recent book-length
discussion of this approach by Garrattet al. (2006) is available, we abstain from
including this approach here.
Cross-sectional dependence can also emerge through correlations among theei,t
via theεi,tvariables across the unitsi, as in models of spatial correlation considered
in a paper by Baltagi, Bresson and Pirotte (2007) or in purchasing power parity
examples such as O’Connell (1998) discussed below. Note that we discuss in detail