358 Chapter 9: Regression
Using Equation 9.3.1 along with the relationship
A=
∑n
i= 1
Yi
n
−Bx
shows thatAcan also be expressed as a linear combination of the independent normal
random variablesYi,i=1,...,n and is thus also normally distributed. Its mean is
obtained from
E[A]=
∑n
i= 1
E[Yi]
n
−xE[B]
=
∑n
i= 1
(α+βxi)
n
−xβ
=α+βx−xβ
=α
ThusAis also an unbiased estimator. The variance ofAis computed by first expressing
Aas a linear combination of theYi. The result (whose details are left as an exercise) is that
Var(A)=
σ^2
∑n
i= 1
xi^2
n
(n
∑
i= 1
xi^2 −nx^2
) (9.3.3)
The quantitiesYi−A−Bxi,i=1,...,n, which represent the differences between the
actual responses (that is, theYi) and their least squares estimators (that is,A+Bxi) are
called theresiduals. The sum of squares of the residuals
SSR=
∑n
i= 1
(Yi−A−Bxi)^2
can be utilized to estimate the unknown error varianceσ^2. Indeed, it can be shown that
SSR
σ^2
∼χn^2 − 2
That is,SSR/σ^2 has a chi-square distribution withn−2 degrees of freedom, which implies
that
E
[
SSR
σ^2
]
=n− 2