Econometrics And Risk Management. - Analytics - Assignment

Homework 1 solution for MSA 8200 spring 2017
1. a. Show that E(βˆ|X) = β if X ′X is nonsingular and the zero conditional mean assumption
E(u|x) = 0 is satisfied.
b. In addition to the assumptions from part a, assume that V ar(u|x) = σ2. Show that
V ar(βˆ|X) = σ2(X ′X)−1.
Solution:
a. The OLS estimator is
βˆ = (X ′X)−1(X ′y) = (X ′X)−1[X ′(Xβ + u)] = β + (X ′X)−1(X ′u)
So we have:
E(βˆ|X) = β + (X ′X)−1X ′E(u|X) = β + (X ′X)−1X ′0 = β.
b.
V ar(βˆ|X) = V ar [(X ′X)−1(X ′u)|X] since β is not random.
= (X ′X)−1X ′V ar(u|X) [(X ′X)−1X ′]′
= (X ′X)−1X ′σ2
[
(X ′X)−1X ′
]′
= σ2(X ′X)−1X ′
[
(X ′X)−1X ′
]′
= σ2(X ′X)−1
2. Show that the estimator Bˆ , N−1
∑N
i=1 uˆ
2
ixix
′
i is consistent for B = E(u
2xx′) by showing that
N−1
∑N
i=1 uˆ
2
ixix
′
i = N
−1∑N
i=1 u
2
ixix
′
i + op(1). Assume that all necessary expectations exist and are
finite. (Hint: Write uˆ2i = u
2
i − 2x′iui(βˆ − β) + [x′i(βˆ − β)]2, and use the facts that sample averages
are Op(1) when expectations exist and that βˆ − β = op(1). )
Solution: To show Bˆ is consistent for B is to show Bˆ−B = op(1). Plug in the definition of Bˆ and
B we get:
Bˆ −B = 1
N
N∑
i=1
uˆi
2xix
′
i − E(u2xx′)
=
1
N
N∑
i=1
u2ixix
′
i −
2
N
N∑
i=1
uix
′
i(βˆ − β)xix′i +
1
N
N∑
i=1
[x′i(βˆ − β)]2xix′i − E(u2xx′)
= − 2
N
N∑
i=1
uix
′
i(βˆ − β)xix′i +
1
N
N∑
i=1
[x′i(βˆ − β)]2xix′i + op(1).
1
1
N
N∑
i=1
uix
′
i(βˆ − β)xix′i =
1
N
N∑
i=1
uixix
′
i
K∑
k=1
xik(βˆk − βk) =
K∑
k=1
(βˆk − βk) 1
N
N∑
i=1
uixikxix
′
i
Since βˆk − βk = op(1) and 1N
∑N
i=1 uixikxix
′
i = Op(1), we have
1
N
∑N
i=1 uix
′
i(βˆ − β)xix′i = op(1).
1
N
N∑
i=1
[x′i(βˆ − β)]2xix′i =
1
N
N∑
i=1
[
K∑
k=1
xik(βˆk − βk)]2xix′i
=
1
N
N∑
i=1
[
K∑
k=1
K∑
j=1
xijxik(βˆj − βj)(βˆk − βk)]xix′i
=
K∑
k=1
K∑
j=1
(βˆj − βj)(βˆk − βk) 1
N
N∑
i=1
xijxikxix
′
i
Since (βˆj−βj)(βˆk−βk) = op(1) and 1N
∑N
i=1 xijxikxix
′
i = Op(1), we have
1
N
∑N
i=1[x
′
i(βˆ−β)]2xix′i =
op(1) ·Op(1) = op(1). And in summary, we have Bˆ−B = op(1), which means Bˆ is consistent for B.
3. Consider estimating the effect of personal computer ownership, as represented by a binary
variable, PC, on college GPA, colGPA. With data on SAT scores and high school GPA you
postulate the model
ColGPA = β0 + β1hsGPA+ β2SAT + β3PC + u
a. Why might u and PC be positively correlated?
b. If the given equation is estimated by OLS using a random sample of college students, is βˆ3
likely to have an upward...