684 Panel Methods to Test for Unit Roots and Cointegration
has geometrically declining weights – that is, the(i,j)th element of the matrixis
given byω|i−j|)and where there is also cross-unit cointegration.
They report that, among the seven Pedroni single-equation tests for the null
hypothesis of no cointegration, the two parametric tests (based on estimating ADF
regressions) are the ones that have the best size properties. The remaining five
are severely undersized and also have low power, especially for small values ofT.
The authors conjecture that this may be due to the use of asymptotic correction
terms (used to standardize the test statistics) when finite sample correction terms
or bootstrapped values may be more beneficial.
However, the conservative properties of the parametric tests also imply that these
are the ones least affected by the presence of short-run cross-sectional correlation or
cross-unit cointegration of the form considered by Wagner and Hlouskova (2007).
Systems tests for cointegration, discussed in section 13.3.2, are in fact outperformed
by Pedroni-type tests (although the latter are not applicable in the presence of
multiple cointegrating vectors). There is a tendency to overestimate the rank of
the cointegrating space (that is, the number of cointegrating vectors), and size and
power distortions occur not only whenTis relatively small but also when theN-
dimension is large (because of the difficulty with dealing with high-dimensional
systems alluded to earlier.) The use of finite sample corrections or bootstrapped
values is found to be efficacious here, as also reported in earlier work by Banerjee,
Marcellino and Osbat (2004).
Banerjee and Carrion-i-Silvestre (2007) consider extensions of the Pedroni tests to
allow for structural breaks, discussed in the following sub-section, and for structural
breaks and cross-sectional dependence, discussed in section 13.3.1.4 thereafter.
Structural breaks are allowed to occur in both the deterministic components and
the cointegrating vector.
13.3.1.3 Allowing for structural breaks in the Pedroni tests
As mentioned, the considered structural changes may take the form of breaks in
the deterministic processes and/or in the slopes of the cointegrating coefficientsβ.
In the next sub-section, we also allow for cross-sectional dependence via a factor
structure.
In this sub-section, we focus on the consequences of relaxing the assumption
that the deterministic processes are not broken. Referring back to (13.18), under
the null hypothesisyi,tandxi,tare not cointegrated, that is,ui,tis anI(1) process.
Under the alternative hypothesis,ui,tis stationary but either the deterministic
terms or the cointegrating vectors are time-variant (defined more precisely below),
so that, while cointegration exists under the alternative, it does so in the presence
of instabilities.
That such a situation is not of pure academic interest becomes evident both
from considering the simulation results in Banerjee and Carrion-i-Silvestre (2007),
and from the empirical example presented below, where the presence of structural
breaks is crucial for establishing the presence of cointegration. The presence of
structural breaks severely undermines the size and power properties of the tests for
cointegration, especially when the break occurs in the deterministic trend of the
processes.