Information Technology Reference
In-Depth Information
Residual-based tests are formulated to test the null hypothesis that the multiple time
series
Y
are not cointegrated using the
scalar
unit root tests, such as the ADF test,
which are applied to the residuals
(
t
)
in (18.35)
In [26], the ADF test as well as two additional tests
Z
α
and
Z
t
, developed earlier
by Phillips [25], were applied to check for the presence of a unit root in the residuals
ξ
t
. In order to perform the unit root test, we fit an AR(1) model to
ξ
t
,
t
=
1
,
2
,...
ξ
t
,
t
=
1
,
2
,...
according to
ξ
t
=
αξ
t
−
1
+
ρ
t
,
t
=
1
,
2
,...
.
(18.36)
Then the statistic
Z
α
in Phillips-Ouliaris test is defined as follows:
s
Tl
−
s
2
1
2
·
ρ
(
α
−
Z
α
=
T
1
)
−
1
,
(18.37)
1
T
2
t
2
t
ξ
∑
=
2
−
whereas the
Z
t
statistic is given by the following formula:
T
t
=
2
ξ
1
2
s
Tl
−
s
2
ρ
·
(
α
−
)
s
Tl
−
1
1
2
·
2
Z
t
=
2
,
(18.38)
s
Tl
T
2
∑
1
t
−
1
t
2
ξ
=
2
t
−
where
T
t
=
1
ρ
1
T
s
2
2
t
ρ
=
,
(18.39)
T
t
=
1
ρ
T
s
=
1
w
sl
T
∑
1
T
2
T
s
Tl
=
2
t
+
t
=
s
+
1
ρ
t
ρ
t
−
s
,
(18.40)
s
w
sl
=
1
−
1
.
(18.41)
l
+
Note that
s
2
ρ
2
ρ
and
s
Tl
are consistent estimators for the variance
σ
of
ρ
t
and the partial
lim
T
→
∞
E
T
S
T
, where
S
T
=
∑
2
t
sum variance
σ
=
ξ
t
is the partial sum of the
=
1
error terms in (18.36).
The critical values for
Z
α
and
Z
t
statistics can be found in [26] (Tables I and II).
Phillips and Ouliaris tabulated the values for cointegrating regressions with at most 5
explanatory variables. Some estimates of the critical values for the Phillips-Ouliaris
test (
Z
α
) are listed in Table 18.7.
Table 18.7: Critical values of the asymptotic distributions of the
Z
α
statistic for test-
ing the null of no cointegration (Phillips-Ouliaris demeaned, reproduced from [26]).
Parameter
n
(
n
=
K
−
1
)
represents the number of explanatory variables
n
90%
95%
99%
1
−
17
.
0390
−
20
.
4935
−
28
.
3218
2
−
22
.
1948
−
26
.
0943
−
34
.
1686
3
−
27
.
5846
−
32
.
0615
−
41
.
1348
4
−
32
.
7382
−
37
.
1508
−
47
.
5118
5
−
37
.
0074
−
41
.
9388
−
52
.
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