Problem Set #2 Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
A simple random variable
a) The minimal sample spaces are:
x cfw_0, 1
y cfw_0, 1, 2
b) Let the joint PDF be dened by f (X1 , X2 , Y ) = Pr(x1 = X1 x2 = X2 y = Y ). Since the sample
Problem Set #1 Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
Sets
1. (a) is true.
2. (a) is true. The correct statements in parts 2 and 3 of this question are known as DeMorgans laws.
3. (b) is true.
4. (b) is true.
2
Functions: Limits
1
Pick
Exam #1 Answer Key
Economics 435: Quantitative Methods
Spring 2006
1
Starter questions (20 points)
a)
cov(ax, b + cx + dy) = ac var(x) + ad cov(x, y)
b)
Yes.
Yes.
It is 1 =
0.23+0.77
2
= 0.5.
c) Suppose that E(u|x) = 0. The law of iterated expectations
Midterm Answer Key
Economics 435: Quantitative Methods
Fall 2009
1
Properties of expected values
a) We start with the denition, and then use a little algebra along with the linearity of expectations theorem:
cov(x, y)
= E (x E(x) (y E(y)
= E (xy xE(y) yE(
Problem Set #2: Probability (revised 9/16/2011)
Economics 435: Quantitative Methods
Fall 2011
1
A simple random variable
Suppose you flip a coin twice. Let:
x1
= I(first coin is heads)
x2
= I(second coin is heads)
y
= x1 + x2
where I is the indicator1 fun
Problem Set #6 : LDV Models and Regression in R
Economics 435: Quantitative Methods
Fall 2011
1
Public sector unionization and size of government: Part III
We continue our analysis of public sector unionization and the size of government, by estimating so
Problem Set #10: Specification and Data Problems
Economics 435: Quantitative Methods
Fall 2011
1
Proxy variables
An important policy question in labor economics is the extent to which young people from lower-income
families are credit constrained in their
Problem Set #9: Instrumental Variables (revised 10/17/2011)
Economics 435: Quantitative Methods
Fall 2011
1
Public sector unionization and the size of government: Part VI
This time we will use an instrumental variables approach to accounting for the possi
Problem Set #3: Statistics
Economics 435: Quantitative Methods
Fall 2011
1
Averages
Let x1 , x2 , . . . , xn be a random sample of size n on the random variable x. Let = E(x) and 2 = var(x).
Let a1 , a2 , . . . , an be a sequence of n numbers.
a) Let:
Wn
Problem Set #5: The MLR Model
Economics 435: Quantitative Methods
Fall 2011
1
Consequences of leaving out a quadratic term
Suppose that:
E(y|x) = 0 + 1 x + 2 x2
and you estimate the (misspecified) regression:
y = 0 + 1 x + u
where you also assume that SLR
Problem Set #1: Math Review
Economics 435: Quantitative Methods
Fall 2011
1
Sets
Let A, B, C be sets, let A B denote the intersection of A and B, let A B denote the union of A and B.
Let Ac be the complement of A (i.e., Ac = cfw_x : x
/ A), and let be th
Problem Set #8: Panel Data (updated 11/2/2011)
Economics 435: Quantitative Methods
Fall 2011
Please use WebCT to turn in both the Word document and R code for question (1) and turn in your answer
to question (2) in class.
1
Public sector unionization and
Problem Set #4: Using R and other tools
Economics 435: Quantitative Methods
Fall 2011
Some academics and politicians have argued that public sector unions can be a powerful political force
in increasing the size of government. Public sector employees have
Problem Set #7: Causality
Economics 435: Quantitative Methods
Fall 2011
1
Public sector unionization and the size of government: Part IV
a) Fix any errors in the Tables 1-3 and Figures 1-3.
b) Complete Section 3 and Section 4.2 of the paper. Things to kee
Problem Set #4 Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
Public sector unionization and size of government: Part I
Please see the file http:/www.sfu.ca/~bkrauth/econ435/restricted/psu1.pdf. The R code used to generate these results is ava
Problem Set #6 Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
Public sector unionization and size of government: Part III
Please see the file http:/www.sfu.ca/~bkrauth/econ435/restricted/psu3.pdf. The R code used to generate these results is a
Exam #2 Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
Short answers
a) Hopefully youve done this a few times before, but here it is again:
E(u) = E(E(u|x)
= E(0)
(law of iterated expectations)
(given)
= 0
E(xu) = E(E(xu|x)
(law of iterated ex
Exam #1 Answer Key
Economics 435 Spring 2002
1
The lazy professor (12 pts)
a) Let actual understanding be u. The variance of u s is higher than the variance of u s. Ill
also accept a claim that s is not a consistent estimate of u, because the number of q
Problem Set #5 Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
Consequences of leaving out a quadratic term
a) Let
= y E(y|x). Applying the law of large numbers and Slutskys theorem, we have:
1
cov(x, y)
var(x)
cov(x, y)
=
var(x)
cov(x, 0 + 1 x
Problem Set #3 Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
Averages
a) The condition is a1 + a2 + + an = 1.
b) The answer is:
var(Wn ) = (a2 + a2 + + a2 ) 2
1
2
n
c) If Wn is an unbiased estimator then
12
1
(a1 + a2 + + an )2
=
= a2 + a2 +
Problem Set #10 Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
Proxy variables
a)
cov(school, f aminc)
var(f aminc)
cov(0 + 1 f aminc + 2 ability + u, f aminc)
=
var(f aminc)
cov(ability, f aminc)
= 1 + 2
var(f aminc)
A
plim 1
=
b) I would ima
Exam #1 Answer Key
Economics 435: Quantitative Methods
Spring 2006
1
Fitted values and residuals (33 points)
a) No it isnt.
plim yi
plim (0 + 1 xi )
= plim 0 + plim 1 plim xi
= 0 + 1 xi
= yi u i = yi
=
b) Yes it is:
plim ui
= plim (yi yi )
= plim yi plim
Exam #1 Answer Key
Economics 435: Quantitative Methods
Spring 2005
1
Regression with standardized variables
a)
0 = 0
b)
2
R2 = 1
c)
corr(, x) = 1
y
d) Well, since we have shown that 1 = corr(, y ), and all correlations are between 1 and +1, the answer
Problem Set #7 Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
Public sector unionization and the size of government: Part IV
Every student will do this question dierently, so there are many correct answers here. If I see any common
mistakes or
Problem Set #8 Answer Key
Economics 435: Quantitative Methods
Fall 2011
Please use WebCT to turn in both the Word document and R code for question (1) and turn in your answer
to question (2) in class.
1
Public sector unionization and the size of governmen
Midterm Answer Key
Economics 435: Quantitative Methods
Fall 2011
1
Warmup
a) Yes. A proof is not required, but here it is:
E(n )
x
=
=
=
=
=
=
b) As usual, var(n ) =
x
2
n .
1
n
E
1
n
1
n
n
xi
i=1
n
E (xi )
i=1
n
(E I (i is even) + O I (i is odd)
i=1
E
1
Midterm Answer Key
Economics 435: Quantitative Methods
Fall 2010
1
Cluster samples
a) Yes.
E()
x
= E
=
=
=
1
nS
1
nS
1
nS
1
nS
n
S
xis
s=1 i=1
n
S
E(xis )
s=1 i=1
n
S
E( + vs + uis )
s=1 i=1
n
S
s=1 i=1
1
(Sn)
nS
=
=
b)
var(xis )
=
var( + vs + uis )
=
va
Exam #2 Answer Key
Economics 435: Quantitative Methods
Spring 2005
1
Some short-answer questions
a) You need 2 IVs, and it is ne if you have more.
b) The one that jumps to my mind is that parents who only have the resources to send one son to university
w
Exam #2 Answer Key
Economics 435 Spring 2004
1
A few issues in research design
a) You cannot. The reason is that while you have a panel data set, you are estimating the eect of one variable at a particular point in time on another variable at a particular