Statistical Tests:
Two methods can be used to test the serial correlation. They are:
(a) Durbin-Watson test:
Let be the correlation between and , for example, as
correlation between and .
We want to determine whether there exists correlation between the observations. That is, testing vs .
Note: the other hypothesis equivalent to this above hypothesis is
’s are uncorrelated vs ,
where and is independent of and .
That is,
’s are uncorrelated vs ’s are autoregressive residuals with log 1.
Durbin-Watson statistic:
The Durbin-Watson statistic for testing
vs
is
.
Properties of Durbin-Watson statistic:
l :
Since , thus
.
Therefore,
.
l (very strong positive correlation), .
(very strong negative correlation), .
(no correlation), .
[Heuristic Justification:]
Suppose and
as is small compared to . Thus,
.
Therefore,
, ,
Primary Durbin-Watson Test:
(I)
where and are some critical values which can be found in table 7.1.
(Draper & Smith, pp. 184~192).
Note: in table 7.1, sample size, number of covariates.
(II)
(III)
Example:
Suppose the following are the residuals from the model
,
/ / / / ... / /0.12 / 0.66 / 0.72 / -0.32 / … / -0.64 / -0.68
Then,
Then, if we want to test , we obtain
(by table 7.1). Since , we reject . That is, we conclude there exist serial correlation in residuals.
Note: the inconclusive feature of the tests above is not attractive.
A Simplified, Approximate Durbin-Watson Test:
(I)
(II)
(III)
(b) Run test:
Motivating Example:
Suppose we have the following 5 residuals:
-1.5 / -2.1 / 0.4 / -0.7 / -0.6 / 1.8The signs of the above residuals are
(- -) (+) (- -) (+),
total 4 runs. Intuitively, a very small number of runs imply that the residuals might have positive serial correlation, for example, for only one run, (+ + + ...)
. On the other hand, a very large number of runs (a very large number of sign switches) imply the residuals might have negative serial correlation, for example, (+) (-) (+) (-) (+) (-)....
For 6 residuals, suppose there are 2 positive residuals and 4 negative residuals. Then, the following sign arrangements are possible:
Arrangements / Number of runs(+ +) (- - - -) / 2
(+) (-) (+) (- - -) / 4
(+) (- -) (+) (- -) / 4
(+) (- - -) (+) (-) / 4
(+) (- - - -) (+) / 3
(-) (+ +) (- - -) / 3
(-) (+) (-) (+) (- -) / 5
(-) (+) (- -) (+) (-) / 5
(-) (+) (- - -) (+) / 4
(- -) (+ +) (- -) / 3
(- -) (+) (-) (+) (-) / 5
(- -) (+) (- -) (+) / 4
(- - -) (+ +) (-) / 3
(- - -) (+) (-) (+) / 4
(- - - -) (+ +) / 2
There are totally combinations. The distribution of runs is
Runs / 2 / 3 / 4 / 5Frequency / 2 / 4 / 6 / 3
Empirical probability / / / /
Cumulative Empirical Probability / / / / 1
As we want to know if too few runs occur with , then, the hypothesis is
.
If we have 6 observations, then 6 residuals could be obtained. Suppose the number of runs of 6 residuals is 2. Thus, we would conclude too few runs since
.
On the other hand, if the number of runs of the 6 residuals is greater than 2, then we would not reject . As we want to know if too many runs occur with , then, the hypothesis is
.
Suppose the number of runs of 6 residuals is 5. Thus, we would conclude too many runs since
.
On the other hand, if the number of runs of the 6 residuals is smaller than 5, then we would not reject .
Run Test:
Let be the sample size, be the number of positive residuals, be the number of negative residuals and be the number of runs.
(I) Small sample size, :
The p-values for the run test can be found in tables 7.5 and 7.6 (pp. 196~197).
Example:
Suppose we fit a regression model. 20 residuals, 10 positive and 10 negative, were obtained. Suppose the runs of signs are 5. Is this an unusually small number at level??
[solutions:]
(by table 7.5). We conclude there are too few runs.
(I) Large sample size, :
As the sample size is large, it is convenient to use a normal approximation, , where and . Thus,
as the residuals are uncorrelated and testing for too few runs.
as testing for too many runs.
(I)
(II)
(III)
Example:
Suppose we fit a regression model. 27 residuals, 15 positive and 12 negative, were obtained. Suppose the runs of signs are 7. Does the arrangement of signs appear to have “too few runs”?
[solution:]
.
too few runs!!
6