1)
(20 points) The last lessons have spent a lot of time describing the slope and intercept terms
(and their variances) of the one-variable sample regression function.
We also know that for
any particular value of the independent variable (call it X
0
), that the predicted value of Y
0
is
ˆ
ˆ
ˆ
Y
X
.
(This is sometimes called a “point prediction.”)
a)
(10) Prove that
0
ˆ
Y
is an unbiased estimator of E[Y
0
|X
0
].
0
0
1
0
0
1
0
0
1
0

b)
(10) Derive the formula for the variance of
0
ˆ
Y
.
Show at least two steps in this derivation.
a.
Hint 1: You are looking for
0
0
1
0
ˆ
ˆ
ˆ
(
)
(
)
Var Y
Var
X
.
This is the variance of a
sum of two random variables.
What is the general formula for such a sum? (Go
back to week 2 lectures, if you need a reminder.)
Use that formula now.