Confidence Interval / C.I.

题目

Confidence Interval / C.I.
Dear all,
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Plz provide me with some ideas of solving this problem.
Thx a million :)
Consider the linear model
Y1 = β1 + 3β2 + ε1
Y2 = 2β1 - β2 + ε2
Y3 = 3β1 - 4β2 + ε3,
where εi are i.i.d N ( 0,σ2 ) variables,i = 1,2,3.
Derive the F-statistic for testing H:β1 = β2,
which can be written as C'β = d,
with C' = (1,-1),β = (β1,β2)' and d=0.
Y1 = 6,Y2 = 4,Y3 =10.
a.Obtain the β hat and I-P,
verify that C' (β hat) - d = ( 6Y1 + 33Y2 + 57Y3 ) / 243,
C' (X'X)-1 C = 2/27,∂2 = y' (I-P) y.
b.Explain how you will find the 95% C.I.for β1 - β2?
( No calculations necessary)
Reference:
Title:A First Course in Linear Model Theory
Author:Nalim Ravishankar & Dipak K.Dey
Publisher:Chapman & Hall / CRC
ISBN:9781584882476
.
It is a college course for senior or graduate student.
R u sure someone could give us a satisfactory answer?
If so,3x in advanve!

答案

a.Obtain the β hat and I-P,
verify that C' (β hat) - d = ( 6Y1 + 33Y2 + 57Y3 ) / 243,
C' (X'X)-1 C = 2/27,∂2 = y' (I-P) y.
b.Explain how you will find the 95% C.I.for β1 - β2?
( No calculations necessary)
Reference:
Title:A First Course in Linear Model Theory
Author:Nalim Ravishankar & Dipak K.Dey
Publisher:Chapman & Hall / CRC