ECON4003: Practical Assignment
Part I
(a) 𝑥𝑖 − 𝜀 = 𝛼𝑌𝑖 E
𝑥𝑖 − 𝜀
𝑌𝑖 =
𝛼
𝑥𝑖 𝜀𝑖
𝑌𝑖 = −
𝛼 𝛼
𝑥𝑖 𝜀𝑖
− = 𝛽𝑜 + 𝛽1 𝑥𝑖 + 𝑢𝑖
𝛼 𝛼
𝑥𝑖 𝜀𝑖
𝑢𝑖 = − − 𝛽0 − 𝛽1 𝑥𝑖
𝛼 𝛼
1
𝜀𝑖
= ( − 𝛽1 ) 𝑥𝑖 − − 𝛽0
𝛼
𝛼
1 − 𝛼𝛽1
𝜀𝑖
=(
) 𝑥𝑖 − − 𝛽0
𝛼
𝛼
1 − 𝛼𝛽1
𝜀𝑖
=
𝑥𝑖 − 𝛽0 −
𝛼
𝛼
1−𝛼𝛽1
(b) 𝐸(𝑢𝑖 |𝑋) = 𝐸 (
𝛼
𝜀
𝑋𝑖 − 𝛽0 − 𝛼𝑖 |𝑋)
1 − 𝛼𝛽
𝜖
= 𝐸 ( 𝛼 1 𝑋𝑖 |𝑋) − 𝐸(𝛽0|𝑋) − 𝐸 ( 𝛼𝑖 |𝑋)
1 − 𝛼𝛽1
1
(𝐸(𝑋𝑖 |𝑋)) − 𝐸(𝛽0 |𝑋) − 𝐸(𝜀𝑖 |𝑋)
𝛼
𝛼
1 − 𝛼𝛽1
1
=
𝑋𝑖 − 𝛽0 − × 0
𝛼
𝛼
1 − 𝛼𝛽1
=
𝑋𝑖 − 𝛽0
𝛼
=
1
(c) Assumption SR.3 confirms that the sample outcomes of 𝑋𝑖 , 𝑖 = 1, … , 𝑛 , must take at
least two different values
𝑛
̂1 = ∑ 𝑤𝑖 𝑌𝑖
𝛽
𝑖=1
𝑛
= ∑ 𝑤𝑖 (𝛽0 + 𝛽1 𝑋𝑖 + 𝑢𝑖 )
𝑖=1
This substitution is possible by inputting the true model expression derived from the
linearity assumption
𝑛
𝑛
𝑛
𝛽̂1 = 𝛽0 ∑ 𝑤𝑖 + 𝛽1 ∑ 𝑤𝑖 𝑋𝑖 + ∑ 𝑤𝑖 𝑢𝑖
𝑖=1
𝑖=1
𝑖=1
𝑛
= 𝛽1 + ∑ 𝑤𝑖 𝑢𝑖
𝑖=1
since
𝑛
∑ 𝑤 = 0,
𝑖=1
𝑛
∑ 𝑤𝑋𝑖 = 1
𝑖=1
Taking the expectations of 𝛽̂1 conditional on the sample values of regressor X, where
𝑤𝑖 is treated as non-random in this case since it is a function only of X
𝑛
𝐸(𝛽̂1|𝑋) = 𝐸(𝛽1 + ∑ 𝑤𝑖 𝑢𝑖 |𝑋)
𝑖=1
𝑛
= 𝐸(𝛽1 |𝑋) + 𝐸(∑ 𝑤𝑖 𝑢𝑖 |𝑋)
𝑖=1
𝑛
= 𝛽1 + ∑ 𝑤𝑖 𝐸(𝑢𝑖 |𝑋)
𝑖=1
2
𝛽1 is a constant and thus is its own
expected value and 𝑤𝑖 can be
removed from the conditional
expectation expression as it is non
random
3
(d)
4