# On the joint asymptotic distribution of the restricted estimators in multivariate regression model

## TL;DR

This paper establishes the joint asymptotic distribution of restricted and unrestricted estimators in multivariate regression, analyzing their performance under various restrictions and providing insights into their relative efficiency.

## Contribution

It generalizes the joint asymptotic normality results for estimators in multivariate regression and compares their risks under different restrictions.

## Key findings

- Restricted estimators perform better near the restriction.
- Unrestricted estimators outperform restricted ones far from the restriction.
- The paper derives the asymptotic distributional risk for the estimators.

## Abstract

The main Theorem of Jain et al.[Jain, K., Singh, S., and Sharma, S. (2011), Re- stricted estimation in multivariate measurement error regression model; JMVA, 102, 2, 264-280] is established in its full generality. Namely, we derive the joint asymp- totic normality of the unrestricted estimator (UE) and the restricted estimators of the matrix of the regression coefficients. The derived result holds under the hypothesized restriction as well as under the sequence of alternative restrictions. In addition, we establish Asymptotic Distributional Risk for the estimators and compare their relative performance. It is established that near the restriction, the restricted estimators (REs) perform better than the UE. But the REs perform worse than the unrestricted estimator when one moves far away from the restriction.