Dr. Hackney STA Solutions pg 185

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

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Second Edition 11-13 with d i = ( x i - ¯ x ) S xx j c j d j = - ¯ x S xx x i 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 + β ( x i - ¯ x ) + i = α + β 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 . b. ˆ α = ¯ y - ˆ β ¯ x, ˆ α = ¯ y - ˆ β ¯ z = ¯ y since ¯ z = 0. ˆ α n( α + β ¯ z, σ 2 /n ) = n( α, σ 2 /n ) . c. Write ˆ α = 1 n y i ˆ β = z i z 2 i y i . Then Cov(ˆ α, ˆ β ) = - σ 2 1 n z i z 2 i = 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: ˆ β 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 finishes the proof. c. From (11.3.33) if ρ = 0 (equivalently β = 0), then ˆ β/ ( S/ √ S xx ) = √ n-2 r/ √ 1-r 2 has a t n-2 distribution....
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