Information Technology Reference
In-Depth Information
18.6.1 Phillips-Ouliaris Cointegration Test
The unit root tests based on analysis of residuals were introduced by Phillips [25].
In particular, in his study Phillips first considered two statistics Z α and Z t for testing
the null of no cointegration in time series.
Because many unit root tests, constructed before 1987, were founded on the as-
sumption that the errors in the regression are independent with common variance
(which is rarely met in practice), Phillips wanted to relax the rather strict condition
that the time series are driven by independent identically distributed innovations. In
other words, he wanted to develop the testing procedures based on the least squares
regression estimation and the associated regression t statistic, which would allow
for rather general weakly dependent and heterogeneously distributed sequence of
error terms.
The properties of asymptotic distributions of residual-based tests for the presence
of cointegration in multiple time series were thoroughly investigated by Phillips and
Ouliaris [26]. The characteristic feature of these tests is that they utilize the residuals
computed from regressions among the univariate components of multivariate series.
The residual-based procedures developed by Phillips and Ouliaris are designed to
test the null of no cointegration by means of testing the null hypothesis of the unit
root presence in the residuals against the alternative of a root that lies inside the
complex unit circle. The hypothesis H 0 of the absence of cointegration is rejected,
if the null of a unit root in the residuals is rejected. In the nutshell, the procedures
are simply residual-based unit root tests.
As noted in [26], the residual-based unit root tests are asymptotically similar, and
can be represented via the standard Brownian motion. Moreover, the ADF and Z t
tests are proved to be asymptotically equivalent. However, these two tests are not
as powerful as the test based on statistic Z α , because it was shown by Phillips and
Ouliaris [26] that the rate of divergence under cointegration assumption is slower
for the ADF and Z t than other tests, such as the Z α -statistic test. The later test (i.e.,
the cointegration test based on Z α ) is also widely known as the Phillips-Ouliaris
cointegration test .
It is noteworthy that the null hypothesis for the Phillips-Ouliaris test is that of
no cointegration (instead of cointegration). This formulation is chosen because of
some major pitfalls found in procedures that are designed to test the null of cointe-
gration in multiple time series. These defects (discussed in more detail in [26]) are
significant enough to be a strong argument against the indiscriminate use of the test
formulations based on the null of cointegration, and to support the continuing use
of residual based unit root tests.
Consider the K -dimensional vector autoregressive process Y
(
t
)
. Let us parti-
)
(
)=(
,
=
(
)
tion Y
t
U t
V t
into the univariate component U t
Y 1
t
and the ( K
1)-
)) .
The residuals are determined by fitting linear cointegrating regression:
=(
(
) ,...,
(
dimensional V t
Y 2
t
Y K
t
U
(
t
)=
cV
(
t
)+ ξ t ,
t
=
1
,
2
,...
.
(18.35)
Search WWH ::




Custom Search