Abstract
Abstract Linear regression models are foundation of current statistical theory and have been a prominent object of study in statistical data analysis and inference. A special class of linear regression models is called the seemingly unrelated regression models (SURMs) which allow correlated observations between different regression equations. In this article, we present a general approach to SURMs under some general assumptions, including establishing closed-form expressions of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of all unknown parameters in the models, establishing necessary and sufficient conditions for a family of equalities of the predictors and estimators under the single models and the combined model to hold. Some fundamental and valuable properties of the BLUPs and BLUEs under the SURM are also presented.
Highlights
Linear regression models are foundation of current statistical theory and have been a prominent object of study in statistical data analysis and inference
We present a general approach to seemingly unrelated regression models (SURMs) under some general assumptions, including establishing closed-form expressions of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of all unknown parameters in the models, establishing necessary and su cient conditions for a family of equalities of the predictors and estimators under the single models and the combined model to hold
We consider a SURM of the form: L : y =X β +ε, (1.1)
Summary
Linear regression models are foundation of current statistical theory and have been a prominent object of study in statistical data analysis and inference. A special class of linear regression models is called the seemingly unrelated regression model (SURM) which allows correlated observations between regression equations. Where yi ∈ Rni× are vectors of observable response variables, Xi ∈ Rni×pi are known matrices of arbitrary ranks, βi ∈ Rpi× are xed but unknown vectors, i = , , ε ∈ Rn × and ε ∈ Rn × are random error vectors satisfying.
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