Conditional Prediction And Unbiasedness in Structural Equations

Cover Conditional Prediction And Unbiasedness in Structural Equations
Conditional Prediction And Unbiasedness in Structural Equations
G M Kaufman
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■* P) = J(e22 ^ 422^ " -^' ^°" §1 2 " =12 =22' ^^'^ ^ little algebra shows that ill = ill. 2 ^ Si. 2 i22 Si. 2- Consequently J(|^2 " Si. 2^ = '^2!^ ^^ J(£ -> e_ ) = 1. Multiplying all four Jacobians together prcves the lemma.
We can now find the joint density of (P, R^_2' ill. 2' =22^ ^^°^ ^^^^ °^ (P, |) by substituting in (3. 1) and multiplying by the Jacobian I422 1 • ^'^^^^ rewrite - 6 - tr H e = tr {H^^ e^^ + H^^ hi ^ S21 ^12 + ^22 hl^ = ^^ Sll S1I. 2 ^ ^^ 222 hi ^ ^^ SllfSl. 2-SnSl2J|22L^.
... 2-S'l^l2^' by writing out tr H e, substituting using the definitions of fi„ in terms of the H.. , 1 and the assertions 1, 2, and 3 of the previous section follow directly.

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