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).

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.

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.

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.

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:

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.

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

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:

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

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.

Interpreting the Durbin-Watson Test