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
.
This is the variance of a sum of two random variables.
What is
the general formula for such a sum? (Go back to the beginning of week 2, if you need a reminder.)
Use that formula
now.
ˆ
ˆ
ˆ
(
)
(
)
Var Y
Var
X

3)
(20 points, 4 points for each piece.) Compare the following two regressions:
i.
0
1
ˆ
ˆ
i
i
i
Y
X
e
ii.
0
1
ˆ
ˆ
(3
)
i
i
i
Y
X
e

Since SST is obviously unchanged (Y hasn’t been touched), *THEREFORE, THE VALUE OF R
2
WILL BE
UNCHANGED.
Similarly, since SSR will also be unchanged between the two regressions,
2
ˆ
S
will also be unchanged.
1
1
1
2
2
ˆ
ˆ
2
2*
2
2
2*
2
ˆ
ˆ
ˆ
ˆ
ˆ
2*
2
2
2
(
1)
1
1
(
1)
(
1)3
9 (
1)
9
X
X
X
X
S
S
n
S
S
S
S
S
S
n
S
n
S
n
S
*In the last line, recall if you multiply a variable by the factor b, the variance increases by the factor b
2
.
So Sx
2*
changes to 3
2
Sx
2
.
FACTORING THIS OUT, WE SEE THAT THE VARIANCE OF THE SLOPE WILL SHRINK BY
A FACTOR OF 9.
(Standard deviation falls by factor of 3)