Dr. Hackney STA Solutions pg 185

Dr. Hackney STA Solutions pg 185 - 2 S 2 /S xx = S 2 xy /S...

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Second Edition 11-13 with d i = ( x i - ¯ x ) S xx X j c j d j = - ¯ x S xx x i X j d 2 j = 1 S xx , and substituting these values shows Cov(ˆ ± i , ˆ β ) = 0. 11.32 Write the models as 3 y i = α + βx i + ± i y i = α 0 + β 0 ( x i - ¯ x ) + ± i = α 0 + β 0 z i + ± i . a. Since ¯ z = 0, ˆ β = ( x i - ¯ x )( y i - ¯ y ) ( x i - ¯ x ) 2 = z i ( y i - ¯ y ) z 2 i = ˆ β 0 . b. ˆ α = ¯ y - ˆ β ¯ x, ˆ α 0 = ¯ y - ˆ β 0 ¯ z = ¯ y since ¯ z = 0. ˆ α 0 n( α + β ¯ z,σ 2 /n ) = n( α,σ 2 /n ) . c. Write ˆ α 0 = X 1 n y i ˆ β 0 = X ± z i z 2 i ² y i . Then Cov(ˆ α, ˆ β ) = - σ 2 X 1 n ± z i z 2 i ² = 0 , since z i = 0. 11.33 a. From (11.23.25), β = ρ ( σ Y X ), so β = 0 if and only if ρ = 0 (since we assume that the variances are positive). b. Start from the display following (11.3.35). We have
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Unformatted text preview: 2 S 2 /S xx = S 2 xy /S xx RSS/ ( n-2) = ( n-2) S 2 xy ( S yy-S 2 xy /S xx ) S xx = ( n-2) S 2 xy ( S yy S xx-S 2 xy ) , and dividing top and bottom by S yy S xx nishes the proof. c. From (11.3.33) if = 0 (equivalently = 0), then / ( S/ S xx ) = n-2 r/ 1-r 2 has a t n-2 distribution....
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