640 Panel Methods to Test for Unit Roots and Cointegration
13.2 Unit root analysis in non-stationary panels
In order to highlight the potential advantages that panel data offer, some appropri-
ate assumptions have to be made. Without any restrictions non-stationary panel
data, or, to be more precise, integrated panel data, can be written as:
yi,t=Di,t+ui,t,whereDi,tdenotes the deterministic part of the process generating the data,ui,t
is the stochastic part,i=1,...,Nis the cross-sectional index andt=1,...,Tis
the time index. Now, without any restrictions on the joint stochastic behavior of
theui,tseries, no gains from pooling the data (that is, the cross-sectional and time
series information), for example by constructing a test statistic, can be expected.
Essentially, for the approach to be valuable, parsimonious representations of the
joint DGP of the processesui,tare required. The most parsimonious is given by
assuming that the stochastic componentsui,tare cross-sectionally independent,
which is – especially for macroeconometric questions – too strong an assumption.
Thus, in order to allow for cross-sectional dependence in a parsimonious way,
so-called (approximate) factor models have gained popularity. These model the
individual seriesui,tas the sum of two components, namely some common factors
Ftand an idiosyncratic componentei,t, thus arriving atui,t=πi′Ft+ei,twithπi
denoting the so-called factor loadings. In this set-up the panel unit root testing
problem can formulated for the following data-generating process:^8
yi,t=Di,t+πi′Ft+ei,t (13.5)( 1 −L)Ft=C(L)ηt (13.6)
( 1 −φiL)ei,t=Hi(L)εi,t, (13.7)observed fort=1, 2,...,T,i=1, 2,...,N, withC(L)=
∑∞
j= 0 CjLjandH
i(L)=
∑∞
j= 0 Hi,jL
j. In the general case,F
tis anr×1 vector of “common factors,” generated
by some multivariate white noise processηt, which accounts for the dependence
that exists across the units of the panel. The unit-specific idiosyncratic termsei,t
are sometimes considered to be cross-sectionally independent, under the assump-
tion that all cross-sectional dependencies are captured by the common factors. In
case the idiosyncratic components are not assumed to be cross-sectionally indepen-
dent, the above model is known as an approximate factor model. For the statistical
analysis of the model as outlined above, appropriate assumptions have to be made
both for identification as well as to establish the asymptotic behavior of estimators
and test statistics. We discuss one set of possible assumptions when describing the
method of Bai and Ng (2004) in section 13.2.2.2.
The properties of the functionsC(L) andHi(L) (see the detailed assumptions
in section 13.2.2.2) determine the time series properties of theFandeseries. For
example, ifC(1)=0,Ftwill beI(0). If, on the other hand,C(1) is of full rank,Ftis an
I(1) process composed ofrlinearly independent stochastic trends. For intermediate