# Independence Properties of the Truncated Multivariate Elliptical Distributions

## TL;DR

This paper characterizes the independence properties of truncated multivariate elliptical distributions, especially the truncated multivariate normal, and applies these findings to test independence in educational data.

## Contribution

It provides a novel characterization of independence in truncated multivariate elliptical distributions, extending known results for the normal case.

## Key findings

- Mutual independence of sub-vectors implies the joint distribution is truncated multivariate normal.
- The paper verifies regularity conditions for applying Wilks' theorem in a practical test.
- Application to educational data demonstrates the usefulness of the independence criterion.

## Abstract

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.