In this paper, we propose several finite-sample specification tests for multivariate linear regressions (MLR). We focus on tests for serial dependence and ARCH effects with possibly non-Gaussian errors. The tests are based on properly standardized multivariate residuals to ensure invariance to error covariances. The procedures proposed provide: (i) exact variants of standard multivariate portmanteau tests for serial correlation as well as ARCH effects, and (ii) exact versions of the diagnostics presented by Shanken (1990) which are based on combining univariate specification tests. Specifically, we combine tests across equations using a Monte Carlo (MC) test method so that Bonferroni-type bounds can be avoided. The procedures considered are evaluated in a simulation experiment: the latter shows that standard asymptotic procedures suffer from serious size problems, while the MC tests suggested display excellent size and power properties, even when the sample size is small relative to the number of equations, with normal or Student-t errors. The tests proposed are applied to the Fama-French three-factor model. Our findings suggest that the i.i.d. error assumption provides an acceptable working framework once we allow for non-Gaussian errors within 5?year sub-periods, whereas temporal instabilities clearly plague the full-sample dataset.
