Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 641

ranks ofC(1) the processFtis a cointegratedI(1) process with the corresponding
number of cointegrating relationships and common trends.
One of the key issues to consider is that both the common part of the processes
(modeled via the factors) and the idiosyncratic parts are allowed to be integrated
or stationary. This implies that determining the time series properties of theyi,t
series requires not only consistent estimation but also inference on the properties
of the constituent parts. We may, for example, think of these processes as being
integrated “unconditionally” if either the factors or the idiosyncratic parts (or both)
are integrated of order one, while conditional on taking account of the factors (that
is, the dependence across the units of the panel), theyi,t−ˆπi′Fˆtseries may be tested
for integration or stationarity. The circumflexes above the factor loadings and the
factors indicate that, for testing, estimates of these unobserved quantities have to
be obtained. Note also that when we talk about stationarity of series, we consider
stationarity of the stochastic part of the series, whilst also allowing for the presence
of deterministic componentsDi,t.
In general, cross-sectional dependence can arise through the unit-specific
idiosyncratic terms as well as through the common factors. Both of these chan-
nels in general may lead to cointegration between the cross-section members. To
make the discussion more precise, we define the concepts of short- and long-
run cross-sectional dependence, as well as of cross-unit cointegration, in detail
in Appendix B. There we also discuss in some detail the cointegration implications
of the (approximate) factor model. For example, the papers of Banerjee, Mar-
cellino and Osbat (2004, 2005) have highlighted that failing to account properly for
cross-unit cointegration may severely distort panel cointegration testing. Further
simulation evidence in this respect is contained in Wagner and Hlouskova (2007).
Formulating the discussion in terms of (13.5)–(13.7) above allows us to high-
light specific important aspects of the theory of testing for unit roots in panels,
summarized as follows.

Deterministics and breaks
For example, a typical formulation of the deterministic part of the process in unit
root and cointegration analysis is to consider:

Di,t=μi+δit,i=1, 2,...,N, (13.8)

where theiindex denotes unit specificity of the deterministic components. This
formulation allows for deterministic unit specific intercepts and trend growth rates,
but a generalization allows for breaks in intercept and trend. For example, allowing
for up tolibreaks in the intercept andmibreaks in trend in unitigives:

Di,t=μi+δit+

∑li

j= 1

θi,jDUi,j,t+

∑mi

k= 1

γi,kDTi∗,k,t,i=1, 2,...,N, (13.9)

where the dummy variables are defined asDUi,j,t=1 fort>Tai,jand 0 elsewhere,

andDTi∗,k,t=(t−Tbi,k)fort>Tbi,kand 0 elsewhere. In other words,Tai,jandTbi,k