1. Problems with the Durbin-Watson test
It assumes a specific form of serial correlation and may fail
to detect other forms.
2. It may not generate a definite answer if the statistic falls in
the uncertain range.
3. It becomes unreliable when there are lagged endogenous
variables included as regressors in the model.

2. Durbin’s h test
If the model contains a lagged endogenous variable then we
can use Durbin’s h test
The test statistic is defined as:
In large samples this is distributed as N(0,1).

3. The Breusch-Godfrey test
For this test we first estimate the model and generate a set
of residuals:
We then estimate an auxiliary model in which we regress the
residuals on the original regressors and their own lagged values:
We then carry out either an F-test or a Chi-squared test for
the joint significance of the lagged residuals in the above
equation

4. The Box-Ljung Test
This is a test for general serial correlation up to order m.
The test statistic is:
Under the null hypothesis that the errors are not serially
correlated, this statistic is distributed as Chi-squared with
m degrees of freedom.
Tests of this type, in which the serial correlation can be of a
very general type, are sometimes referred to as portmanteau
tests.

5. Implications of serial correlation
The implications of serial correlation depend on what causes it
in the first place.
Serial correlation can be the result of:
Error dynamics – in this case our model has the correct set of
RHS variables but the error is serially correlated.
2. Omitted variables – the model specified has left out a RHS
variable which is itself serially correlated.

6. Error dynamics
If serial correlation is due to error dynamics then OLS can
be shown to be unbiased.
However, OLS will be inefficient since the GM assumptions
no longer hold and it is possible to find an estimator with
lower variance.
If the error exhibits positive autocorrelation and the x variable
is also positively autocorrelated then the standard error of the
OLS estimator will be biased downwards.
This last property means that we may over-estimate the
statistical significance of the RHS variable(s).

7. ‘Correcting’ for Serial Correlation
Most regression packages offer mechanical procedures which
‘correct’ for the presence of serial correlation. For example, if
we have:
We can use this equation to simultaneously estimate the slope
coefficient and the autoregressive parameter.
Estimation in this case is by non-linear least squares.

8. ‘Quasi-Differencing’ Methods
We have already seen that the model with an AR(1) error can
be written:
Suppose we have an estimate of the AR parameter, then we can
write:
This is the quasi-differenced form of the model.
Methods such as the Cochrane-Orcutt and Hildreth-Lu methods
use an iterative process to estimate the unknown parameters.

9. Example: regression of log consumption on log GDP

10. Common Factor Restrictions
Some econometricians have argued against the use of mechanical
corrections for autocorrelation on the grounds that it imposes an
untested restriction. Suppose we have:
The correction for autocorrelation can be obtained by imposing:
It is argued that we should test this restriction before we apply
mechanical corrections for the presence of autocorrelation.

11. Omitted Variables
If we omit a variable from a regression equation that should be
included, and that variable is itself serially correlated, the
result will be a serially correlated error.
In this case OLS will be biased, inefficient and the standard
errors of the coefficients will be unreliable.
There is little we can do in this case other than go back and
respecify the model from the start.
Note that if this is the cause of serial correlation then inference
based on ‘corrections’ for serial correlation becomes unreliable.