STAT 100B HW 5 Due Friday 5pm
Problem 1:
For the simplest regression model
y
i
=
βx
i
+
i
,
i
= 1
, ..., n
, suppose
x
i
are fixed, and
E[
i
] = 0, Var[
i
] =
σ
2
, and
i
are independent. Consider the estimator
ˆ
β
=
∑
n
i
=1
w
i
y
i
/
∑
n
i
=1
w
i
x
i
,
where
w
i
may depends on
x
i
, but not on
y
i
.
(1) Find E[
ˆ
β
] and Var[
ˆ
β
].
(2) Show that Var[
ˆ
β
] is minimized when
w
i
∝
x
i
. Calculate the minimum.
Problem 2:
For the simple regression
y
i
=
β
0
+
β
1
x
i
+
i
,
i
= 1
, ..., n
, define the sample variance
of (
x
i
, i
= 1
, ..., n
) as
V
x
=
∑
n
i
=1
˜
x
2
i
/n
, define the sample variance of (
y
i
, i
= 1
, ..., n
) as
V
y
=
∑
n
i
=1
˜
y
2
i
/n
, and define the sample covariance between (
x
i
, y
i
, i
= 1
, ..., n
) as
C
xy
=
∑
n
i
=1
˜
x
i
˜
y
i
/n
,
where ˜
x
i
=
x
i

¯
x
, and ˜
y
i
=
y
i

¯
y
. Define the sample correlation as
ρ
xy
=
C
xy
/
(
√
V
x
p
V
y
).
(1) Show that
ρ
xy
can be interpreted as cosine of an angle between two vectors.
(2) Let
ˆ
β
1
be the least squares estimate of
β
1
. Show that
ˆ
β
1
=
C
xy
/V
x
=
ρ
xy
p
V
y
/
√
V
x
.
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 Winter '11
 Wu
 Regression Analysis, Variance, Yi, regression model Yi, simple regression yi, xi yi /n

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