relationship between them. We use the static equation between two I(1) variables which now may possibly be cointegrated: = +
(8)
Where
is monthly index of S&P 50 0
is monthly index of SSE
If and are not cointegrated then there is no possible value of the parameters can be stationary. If they are cointegrated however, then there is a single value for the two parameters such that the linear combination − ( + ) is stationary. If a set of variables are cointegrated, then the residuals from a static regression will be stationary. If not, then the residuals will be integrated. The unit root Durbin Watson test can be used to test cointegration in the residuals from a static regression and is described in Sargan and Bhargava (1983). 6 and such that
ii. Testing cointegration
0 0
: :
= (1)
= (0) Here we could again employ the Dickey Fuller test again to determine the order of integration of whereas we would use an alternative method which is Durbin Watson test.
Durbin Watson test statistic:
n
n
n
n
n
t
2
2
2
e e )
e
e
ee
ee
(
t
t
t
t
t t 1
t t 1
1
1
t
t
t
t
2
2
n 2
2
2
d
2
22
n
n
n
n
2
2
2
2
e
e
e
e
e
t
t
t
t
t
t
t
t
t
t
1
1
1
1
1
, the sample error terms, also known as residuals, could be simply computed. Let and be possible estimated of the regression parameters and , and fit the regression line to describe the relationship between expected value of the dependent variable and the independent variable. ̂ = + (9) The difference between the observed and predicted values of the dependent variable is
̂ =
=
−
− −
Under the null hypothesis that
is a random walk and that = 0 , so there is no
cointegration, and
becomes a random walk with theoretical first order
6 Pierse, R. . (no date) Lecture 8: Nonstationality, Unit Roots and Cointegration . .
10
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