System of Regression Equation

Seemingly Unrelated Regressions

Many small “unrelated regressions” can be one $Y=X\beta +U$ estimated by OLS

Contemporaneous correlation

Correlation only happens when $t=s$

Special product: 一个○里面一个 X $⨂$

$Var(U)={\sum }_{M\times M}⨂I_{T\times T}$

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${\hat{\beta }}_{GLS}=(X^{\prime }{\Omega }^{-1}X{)}^{-1}{X}^{\prime }{\Omega }^{-1}y$

${\Omega }^{-1}=(\sum ⨂I{)}^{-1}={\sum }^{-1}⨂I$

Comparison of Seemingly Unrelated Regressions and Ordinary Least Squares

Simultaneous Equations Model also knowns as SEM

Identification:

Order Condition: $K-k\ge g-1$

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Reference