Anindya Banerjee and Martin Wagner 643in Appendix B the implications of correlation in the variablesei,t, which also can
lead to long-run cross-sectional dependencies.
In the simplest case of models (13.5)–(13.7), we could think of switching off
cross-sectional dependence by settingπi=0 for all the units, specifying that the
εi,tare independent overi, and allowing specifications for the deterministic process
to be given by (13.8) (that is, without breaks).^9 Whenφi=1, and henceρi=0,
the seriesyi,tcontains a unit root.
13.2.1 Tests without cross-sectional dependence or structural breaks
Several tests, depending on the specification of the null and alternative hypotheses,
have been developed to test for unit roots in panels.
13.2.1.1 Levin, Lin and Chu (2002)
As discussed in the many excellent descriptions in the literature (see, for example,
Hlouskova and Wagner, 2006), the test is based on running augmented Dickey–
Fuller regressions:
yi,t=μi+δit+ρiyi,t− 1 +∑pik= 1φi,kyi,t−k+νi,t,i=1, 2,...,N;t=pi+2,...,T.(13.10)
Here we denote byνi,tthe error process corresponding to the autoregressive speci-
fication to which the ADF set-up corresponds. Only if the data are really generated
by autoregressions of orderspi+1 will theνi,tbe white-noise processes. In practice
this implies that lag length selection will be an issue (see below). Three sub-cases
concerning the specification of the deterministic component are considered by
Levin, Lin and Chu (2002) (henceforth referred to as LLC):
- no deterministic terms;
- intercept only;
- intercept and linear trend.
We index the specification of these three deterministic components withm=
1, 2, 3. The null hypothesis of the LLC test isH 0 :ρi= 0 ∀i=1, 2,...,Nagainst the
homogeneous alternative hypothesisHA:ρi=ρ< 0 ∀i=1, 2,...,N.^10 This formu-
lation of the null and alternative hypotheses allows for the construction of pooled
tests (once appropriate corrections are made for the cross-sectional heterogeneity
arising from other features of the DGP). Pooling may be made in both the within
and between dimensions and gives rise to the tests (LLC and IPS, respectively)
described below.
As mentioned above, selection of the lag length in (13.10) is an issue. In case the
data are not described by finite-order autoregressive processes, the lag lengths have
to increase as a function of theT-dimension of the panel (as was first studied by
Said and Dickey, 1984, in the time series context) and LLC propose specifically that
pi(T)grows at rateTκ,0<κ<0.25. Careful lag length selection is necessary to