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120
APPENDIX C
SOLUTIONS TO PROBLEMS
C.1
(i) This is just a special case of what we covered in the text, with
n
= 4:
E(
Y
) =
μ
and
Var(
Y
) =
σ
2
/4.
(ii) E(
W
) = E(
Y
1
)/8 + E(
Y
2
)/8 + E(
Y
3
)/4 + E(
Y
4
)/2 =
μ
[(1/8) + (1/8) + (1/4) + (1/2)] =
μ
(1 +
1 + 2 + 4)/8 =
μ
, which shows that
W
is unbiased.
Because the
Y
i
are independent,
Var(
W
) = Var(
Y
1
)/64 + Var(
Y
2
)/64 + Var(
Y
3
)/16 + Var(
Y
4
)/4
=
σ
2
[(1/64) + (1/64) + (4/64) + (16/64)]
=
σ
2
(22/64)
=
σ
2
(11/32).
(iii) Because 11/32 > 8/32 = 1/4, Var(
W
) > Var(
Y
) for any
σ
2
> 0, so
Y
is preferred to
W
because each is unbiased.
C.3
(i) E(
W
1
) = [(
n
– 1)/
n
]E(
Y
) = [(
n
– 1)/
n
]
μ
, and so Bias(
W
1
) = [(
n
– 1)/
n
]
μ
–
μ
= –
μ
/
n
.
Similarly, E(
W
2
) = E(
Y
)/2 =
μ
/2, and so Bias(
W
2
) =
μ
/2 –
μ
= –
μ
/2.
The bias in
W
1
tends to
zero as
n
→
∞
, while the bias in
W
2
is –
μ
/2 for all
n
.
This is an important difference.
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 Spring '11
 Suzuki
 Econometrics, Normal Distribution, Standard Deviation, Variance, var

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