Likelihood ratio test of no cross-correlation


Test statistic that the covariance is diagonal.


The null hypothesis is that the shock covariance matrix is diagonal, and so all correlations are 0. In this case, there are no gains to using GLS estimation in the system estimator.

When the null is rejected, there should be efficiency gains to using GLS as long the regressors are not common to all models.

The LR test statistic is defined as

\[LR=n\left[\sum_{i=1}^{k}\log\hat{\sigma}_i^2 -\log\left|\hat{\Sigma}\right|\right]\]

where \(\hat{\sigma}_i^2\) is the sample residual variance for series i and \(\hat{\Sigma}\) is the residual covariance. n is the sample size. It has an asymptotic \(\chi^2_{k(k-1)/2}\) distribution. The asymptotic distribution of the likelihood ratio test requires homoskedasticity. See [1] for details.



Greene, William H. Econometric analysis. Pearson Education, 2003.

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