1. Serial Correlation
Serial correlation is a problem associated with time-series
data.
It occurs when the errors of the regression model are
correlated with their own past values.
Serial correlation by itself does not mean that OLS will
be biased.
The most common effect of serial correlation is to bias
downwards the standard errors of the OLS estimator
(though this is not always the case).

2. Definition of Serial Correlation
Consider the regression model
The errors of this model are said to be serially correlated if
Since this is not consistent with the second Gauss-Markov
assumption, it follows that OLS is no longer BLUE.

3. Serial correlation and autocorrelation
Serial correlation is a general term used to describe any
situation in which the error terms are not completely independent.
Autocorrelation is a particular type of serial correlation in which
the error terms are a function of their own past values.
Both of these equations describe serial correlation but only the
first describes autocorrelation. The second equation describes a
moving average error which is a different type of serial correlation.

4. Orders of serial correlation
In general the order of serial correlation refers to the maximum
lag on the right hand side of the equation describing the error
term. For example:
is a fourth-order autocorrelation process.
The order of the serial correlation process is often related to the
frequency of the data. For example, models estimated with
quarterly data often have errors which exhibit fourth-order
autocorrelation.

5. Detection of serial correlation
Suppose we have a regression model of the form
We wish to test for the presence of serial correlation
in the residuals which are defined as:

6. Formal methods for the detection of serial correlation are
based on the sample autocorrelations. These are defined as:
The sample correlogram gives a visual guide to the structure
of the sample autocorrelations. This can be used as a diagnostic
tool for a regression model.

7. The shape of the correlogram depends on the type of serial
correlation. For example consider the case of first order positive
autocorrelation:

8. The Durbin-Watson test
The Durbin-Watson test provides a more formal test for the
presence of first-order autocorrelation.
Consider the following model:
We wish to test:

9. Consider the test statistic
If the sample size is large then
and therefore

10. Under the null hypothesis that there is no autocorrelation we
have
E(DW)=2
If there is positive autocorrelation then E(DW)<2.
If there is negative autocorrelation then E(DW)>2
Note that DW is bounded between 0 and 4.

11. Interpreting the Durbin-Watson Test