  Truncated multivariate distributions arise extensively in econometric
modelling when non-negative random variables are intrinsic to the
data-generation process. More broadly, truncated multivariate distributions
have appeared in censored and truncated regression models, simultaneous
equations modelling, multivariate regression, and applications going back to
the now-classic papers of Amemiya (1974) and Heckman (1976). In some
applications of truncated multivariate distributions, there arises the problem
of characterizing the distribution through correlation and independence
properties of sub-vectors. In this paper, we characterize the truncated
multivariate normal random vectors for which two complementary sub-vectors are
mutually independent. Further, we characterize the multivariate truncated
elliptical distributions, proving that if two complementary sub-vectors are
mutually independent then the distribution of the joint vector is truncated
multivariate normal, as is the distribution of each sub-vector. As an
application, we apply the independence criterion to test the hypothesis of
independence of the entrance examination scores and subsequent course averages
achieved by a sample of university students; to do so, we verify the regularity
conditions underpinning a classical theorem of Wilks on the asymptotic null
distribution of the likelihood ratio test statistic.
