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f2009_ex1sol - STA 6207 Exam 1 Fall 2009 PRINT Name All...

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STA 6207 – Exam 1 – Fall 2009 PRINT Name _____________________ All Questions are based on the following 2 regression models, where SIMPLE REGRESSION refers to the case where p =1, and X is of full column rank (no linear dependencies among predictor variables). ( 29 ( 29 I 0 ε β X ε Y 2 2 1 1 0 , ~ 1 ' ' : 2 Model , 0 ~ ,..., 1 : 1 Model σ σ ε ε β β β N p p n NID n i X X Y i i ip p i i × × + = = + + + + = Given: ( 29 ( 29 ( 29 ( 29 Y Y Y μ AV AY Y A Ax x Ax x' a x x a ' ' symmetric) ( 2 ' + = = = tr E d d d d Cochran’s Theorem: Suppose Y is distributed as follows with nonsingular matrix V: ( 29 ( 29 μ A Y A Y AV V V μ Y ' 2 1 : parameter ity Noncentral (b) and ) ( df (a) : with central - non d distribute is 1 ' 1. : idempotent is if n the , ~ 2 2 2 2 σ χ σ σ = = = r n r N _________________________________________________________________________________________ _ 1. For model 2, derive the least squares estimate for β ( 29 ( 29 ( 29 ( 29 ( 29 Y X X X β 0 Y X X β β X X Y X β 2Y Y Y β β X β 2Y Y Y X β Y X β Y Y Y Y Y T 1 T ^ T T T T T T T T T T T T T T T T T T T T - = = - = + - = + - = + - = + - - = - - = 2 2 : ) on hat" " put the you now , 0 (Setting 2 2 0 d dQ d d d dQ Q 2.
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