# Testing for multiple conditional heteroskedasticity in multivariate regression

Briefly, I am looking for an extension of the Breusch-Pagan test to the case of multivariate dependent variables. The scalar form of the test, with multiple regressors, assumes the model $$\begin{split} y &= \vec{\beta}^{\top} \vec{x} + \epsilon,\\ \epsilon^2 &= h(\vec{z}^{\top}\vec{\alpha}), \end{split}$$ where the first element of $\vec{z}$ is identically one, and $h$ is some known function with first and second derivatives. The test is of the null hypothesis $$H_0: \alpha_2 = \alpha_3 = \ldots = \alpha_p = 0.$$

I am curious about the case where one is performing multivariate multiple linear regression, with vector $\vec{y}$. Are there known tests for this?

• Recalling that in econometrics multivariate linear regressions are called seemingly unrelated regression models, Google Scholar searches immediately become bounteous. – tchakravarty Jun 12 '14 at 22:26