1105 words - 5 pages

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...

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