Anindya Banerjee and Martin Wagner 685We focus here on the parametric version of the pooledt-test corrected paramet-
rically (given the good properties of this test identified by Wagner and Hlouskova,
2007), but the theoretical and simulation analysis can be repeated for any of the
tests developed by Pedroni discussed above or in the related literature.
Let us begin, as in the previous section, with the following flexible representation
of the model:
yi,t=Di,t+x′i,tβi,t+ui,t (13.18′′)( 1 −L)xi,t=νi,t, (13.22′′)where the further generalization at this stage is to allow for theDi,tandβi,tterms
to be broken, although for the moment the factor structure remains switched off.
For simplicity, the restriction (in relation to the Bai and Carrion-i-Silvestre, 2007,
paper) imposed here is that there is only one break allowed for, and that the breaks
in intercept and/or trend and/or cointegrating coefficients (under the alternative)
all occur at the same, possibly unknown, time period. It is important to note,
however, that the timing of these breaks is allowed to vary across the units. These
breaks are specified in six ways, constituting six different sub-models nested within
(13.18′′).
Start with the general functional form for the deterministic termDi,t:
Di,t=μi+δit+θiDUi,t+γiDTi∗,t,where:
DUi,t= 0 ∀t≤Tb,i
= 1 ∀t>Tb,i,and:
DTi∗,t= 0 ∀t≤Tb,i
=(t−Tb,i)∀t>Tb,i.Note thatTb,idenotes the time of the break for theith unit andλi=TTb,i, which
is the fraction of the sample at which the break occurs in theith unit, remains
constant asT→∞and belongs to a closed sub-set of (0,1). The time-varying
cointegrating vector is specified as a function of time so that:
βi,t=βi+bi·DUi,t,whereDUi,tis as defined above.
With the allowed breaks, at least six different model specifications may be con-
sidered and we list these below. The list is not exhaustive, but probably includes
the most relevant specifications, especially for the empirical applications that we
would have in mind.
Model 1: Constant term with a change in intercept but stable cointegrating vector:
yi,t=μi+θiDUi,t+x′i,tβi+ui,t. (13.24)