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Truncated multivariate distributions arise extensively in econometricmodelling when non-negative random variables are intrinsic to thedata-generation process. More broadly, truncated multivariate distributionshave appeared in censored and truncated regression models, simultaneousequations modelling, multivariate regression, and applications going back tothe now-classic papers of Amemiya (1974) and Heckman (1976). In someapplications of truncated multivariate distributions, there arises the problemof characterizing the distribution through correlation and independenceproperties of sub-vectors. In this paper, we characterize the truncatedmultivariate normal random vectors for which two complementary sub-vectors aremutually independent. Further, we characterize the multivariate truncatedelliptical distributions, proving that if two complementary sub-vectors aremutually independent then the distribution of the joint vector is truncatedmultivariate normal, as is the distribution of each sub-vector. As anapplication, we apply the independence criterion to test the hypothesis ofindependence of the entrance examination scores and subsequent course averagesachieved by a sample of university students; to do so, we verify the regularityconditions underpinning a classical theorem of Wilks on the asymptotic nulldistribution of the likelihood ratio test statistic.

Independence Properties of the Truncated Multivariate Elliptical Distributions