Palgrave Handbook of Econometrics: Applied Econometrics

686 Panel Methods to Test for Unit Roots and Cointegration

Model 2: Time trend with a change in intercept but stable cointegrating vector:

yi,t=μi+δit+θiDUi,t+x′i,tβi+ui,t. (13.25)

Model 3: Time trend with change in both intercept and trend but stable cointegrat-
ing vector:
yi,t=μi+δit+θiDUi,t+γiDTi∗,t+x′i,tβi+ui,t. (13.26)

Model 4: Constant term with change in both intercept and changing cointegrating
vector:
yi,t=μi+δit+θiDUi,t+x′i,tβi,t+ui,t. (13.27)

Model 5: Time trend with change in intercept and changing cointegrating vector
(the slope of the trend does not change):

yi,t=μi+δit+θiDUi,t+x′i,tβi+ui,t. (13.28)

Model 6: The intercept, the trend slope and the cointegrating vector all change:

yi,t=μi+δit+θiDUi,t+γiDTi∗,t+x′i,tβi,t+ui,t. (13.29)

Under any of these specifications, Banerjee and Carrion-i-Silvestre (2007) propose
methods for testing the null hypothesis of no cointegration against the alternative
hypothesis of cointegration, with corresponding breaks.
The Banerjee and Carrion-i-Silvestre (2007) proposal is conceptually extremely
simple. The analysis may be specialized to the case where theλi(break fractions)
are assumed to be known but, since this is perhaps not a likely scenario, we present
the more general analysis where the break dates are assumed to be unknown. The
tests for cointegration in panels with structural breaks are based on time series
cointegration tests developed by Gregory and Hansen (1996) and the panel cointe-
gration tests of Pedroni (1999, 2004) discussed above. Testing entails several steps.
First, one of the models (depending upon the precise empirical motivation) given
in (13.24)–(13.29) above is estimated by OLS for each unitiof the panel. Since this
requires us to specify the break fractionλi, and we assume here that that break
dates are unknown, the models must be estimated for each unit for every possible
choice of break fraction within a bounded interval, taken here to be the interval
%=[0.15, 0.85]. This gives rise to a set of ADF regressions, derived from the
particular choice ofλi.
Next, for a given choice ofλi, and for a particular choice of model, residuals
uˆi,t(λi)are extracted for each uniti. These residuals are then used to estimate the
augmented Dickey–Fuller-type regression given by:

uˆi,t(λi)=ρiuˆi,t− 1 (λi)+

∑k

j= 1

φi,juˆi,t−j(λi)+εi,t.

Since under the null hypothesisρi= 0 ∀i, the next step is to computetρi= 0 (λi)for
each of these ADF regressions and to take as the estimate of the break-point/fraction