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.

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).

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

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.

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.

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).

‘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.

‘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.

Example: regression of log consumption on log GDP

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.

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.