Stat 331  Assignment 1  Winter 2012
Due on Friday June 1st by 5pm in my office (M3 3112).
1. In some cases, it may be more realistic to consider a simple linear regression model
through the origin:
y
i
=
βx
i
+
i
, i
= 1
,
2
, . . . , n
where
i
iid
∼
N
(0
, σ
2
)
(a) Find an expression for the least squares estimate of
β
and denote it as
ˆ
β
.
(b) Show that
ˆ
β
is an unbiased estimator of
β
and find an expression for
V
(
ˆ
β
)
.
(c) Let
e
i
=
y
i

ˆ
βx
i
denote the estimated residual.
i. Show that
∑
n
i
=1
e
i
x
i
= 0
.
ii. Is it true that
∑
n
i
=1
e
i
= 0
?
iii. Hence, can you conclude that sample correlation between the estimated resid
uals and the predictor variable equals zero? That is, do we have
∑
n
i
=1
(
e
i

¯
e
)(
x
i

¯
x
)
p
∑
n
i
=1
(
e
i

¯
e
)
2
p
∑
n
i
=1
(
x
i

¯
x
)
2
= 0?
(d) Provide an expression for the least squares estimate of
σ
2
.
2. Consider the simple linear regression model:
y
i
=
β
0
+
β
1
x
i
+
i
, i
= 1
,
2
, . . . , n
where
i
iid
∼
N
(0
, σ
2
)
Let
e
i
=
y
i

ˆ
β
0

ˆ
β
1
x
i
. In this question, we would like to show that
ˆ
σ
2
=
∑
n
i
=1
e
2
i
n

2
is an
unbiased estimator of
σ
2
.
(a) Show that
∑
n
i
=1
e
2
i
=
s
yy

s
2
xy
s
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