Steven N. Durlauf
Economics 704
Fall 2016
Lecture Notes 2: Decisions and Data
In these notes, I describe some basic ideas in decision theory under
uncertainty. These ideas provide links between econometric analysis and policy
evaluation.
Basic decision th

Econ 312/702 - Graduate Midterm1 Solution (Q3-Q4)
Professor: Noah Williams, TA: Tomoaki Kotera & Phai Phongthiengtham
Note: Question 3 and 4 of the undergraduate exam have the exact same setup but with less
sub questions. The grading rubric is also more l

Noah Williams
Department of Economics
University of Wisconsin
Economics 312
Macroeconomics
Spring 2014
Midterm Examination Solutions1
FOR UNDERGRADUATE STUDENTS ONLY
Instructions: This is a 75 minute examination worth 100 total points. Question 1 is worth

Econ 708 Discussion Handout 1
Yue Li
January 21, 2017
1. Probability Space
Denition:
Let
F
be a collection of subsets of
1.
F
2.
A F = Ac F
3. If
cfw_Ai
Denition:
is a sequence of sets in
Let
(, F)
F,
then
, F
is a
-algebra
(eld) if
Ai F
i
be a measure

Econ 312/702 - Midterm I Solution
Professor: Noah Williams, TA: Tomoaki Kotera & Phai Phongthiengtham
1
(a)1 The equation for k in this case is (note that =
1
holds2 ):
N
k = syt kt
kt = syt kt nyt kt = (s n)kt kt .
Nt
1
1 . Accordingly,
In the steady st

Main Results
Economics 705
Spring 2017
Instrumental Variables
For i = 1, . . . , n,
yi = x0i + i
and there is an L 1 column vector zi , called instrumental variables.
Assumptions
(IV 0) (yi , x0i , zi0 )i=1,.,n is an i.i.d. sequence
(IV 1) E(zi zi0 ) is f

Econ 312/702 - Problem Set 2 Solution
Professor: Noah Williams, TA: Junhyong Kim & Tomoaki Kotera
1
(a) The equation for
k
in this case is (note that
=
1
holds):
N
k = syt kt
kt = syt kt nyt kt = (s n)kt kt .
Nt
In the steady state (denoted by
and
n
c =

Econ 312/702 - Problem Set 2 Solution
Professor: Noah Williams, TA: Junhyong Kim & Tomoaki Kotera
1
(a) The equation for
k
in this case is (note that
=
1
holds):
N
k = syt kt
kt = syt kt nyt kt = (s n)kt kt .
Nt
In the steady state (denoted by
and
n
c =

Main Results
Economics 705
Spring 2017
Linear Regression
For i = 1, . . . , n,
yi = x0i + i
where yi and i are scalar, and xi and are k 1 column vectors.
Assumptions
(OLS 0) (yi , x0i )i=1,.,n is an i.i.d. sequence
(OLS 1) E(xi x0i ) is finite and nonsing

Economics 704
Problem Set 3
Due Friday, October 14 in discussion
THEORY
1. Consider the following choices of tax rate . You know the state of the
world is either 1 or 2 , which will yield the following levels of revenue
loss. Compute the max loss and max

Proof of ex-ante dependent
I pick X1 and X2 and prove that they are dependent. If X1 and X2 are independent, we have
P (X1 a \ X2 b) = P (X1 a)P (X2 b)
The ex-ante (we dont know value of ) joint cumulative density distribution of X1 and X2 is
F (a, b)
P (

Economics 704
Steven N. Durlauf
Fall 2016
Lecture Notes 4: Linear Regression
In these notes I provide a population-level interpretation of linear
regressions. By this, I mean that I discuss the relationships between random
variables rather than the relati

Economics 704
Steven N. Durlauf
Fall 2016
Lecture Notes 3. Linear Space Theory
Linear econometric models are ubiquitous in theoretical and empirical
work. These notes describe the mathematical foundations of these models,
which come from linear space theo

Economics 704
Problem Set 2
Due Friday, Sept. 30 in discussion section
THEORY
1. Consider two versions of a stochastic process xt that can take on values 1
or -1.
(a) xt+1 = xt
(b) xt+1 = xt
For both, you know that
(
1
with probability p
x0 =
1 with proba

Economics 704
Problem Set 2
Due Friday, Sept. 30 in discussion section
THEORY
1. Consider two versions of a stochastic process xt that can take on values 1
or -1.
(a) xt+1 = xt
(b) xt+1 = xt
For both, you know that
(
1
with probability p
x0 =
1 with proba

Exchangeability
At the beginning of time,
generated: xt N 0,
2
.
is drawn from a N 0,
2
distribution. A time series is then
1. Is P (x2 2 [ 1, 1]) = P (x2 2 [ 1, 1] |x1 2 [ 1, 1])?
2. Are the random variables cfw_xt independently and identically distribu

Economics 704
Problem Set 1
Due Friday, Sept. 23 in discussion
THEORY
1. Consider the following stochastic process, cfw_xt for t = 1, 2, 3, 4, 5. Describe the full sample space of this
process and describe its support. In other words, what are the concei

Economics 704
Problem Set 1 Solutions
Due Friday, Sept. 23 in discussion
THEORY
1. The full sample space of cfw_x is the set of all sequences where each
element x is either 1 or -1. The support is equal to the sample space,
since any of these sequences i

Econ 704 Econometrics I
Discussion notes 4
Posterior probability densities
Let us have a perior belief about the probability distribution function is p() and observations x with the
likelihood p(x|), then the posterior probability is defined as
p(|x) =
p(

Econ 312/702 - Week 5
Professor: Noah Williams, TA: Junhyong Kim & Tomoaki Kotera
Solution for review questions
1. This is the question 4 in problem set 2.
2. Consider a variation of the Solow model where the savings rate is variable instead of constant.

clear all
set more off
set obs 10000
set seed 2015
gen x = rnormal()
gen n = rnormal()
gen e = rnormal()
*a)
local alph = .25
local beta = .75
local gamm = .5
gen z = `gamm'*x + n
gen y = `alph'*x + `beta'*z + e
*b)
reg y
local
*c)
reg y
local
x z
coefb =

Econ 704 Econometrics I
Discussion notes 2
1
Random variables
1.1 Denition
A random variable X is a mapping from the sample space to the real line R. The uncertainty in the
random variable derives from the uncertainty of the outcome , so given every membe

Econ 700 Midterm 2
Dr. Hansen
November 9th, 2015
Answer the following questions to the best of your ability. Do not provide
irrelevant information. If a question seems ambiguous to you, state the most
reasonable assumptions to resolve the ambiguity and pr

Economics 704
Steven N. Durlauf
Fall 2016
Lecture Notes 1: Probability Theory
In these notes I review some basic concepts in probability theory. Part of
my purpose is to demystify the use of these terms, which appear too often in
economics papers.
Economi

Econ 700 Final Exam
Dr. Hansen
December 19th, 2015
Answer the following questions to the best of your ability. Do not provide
irrelevant information. If a question seems ambiguous to you, state the most
reasonable assumptions to resolve the ambiguity and

Economics 704
Steven N. Durlauf
Fall 2016
Lecture Notes 1: Probability Theory
In these notes I review some basic concepts in probability theory. Part of
my purpose is to demystify the use of these terms, which appear too often in
economics papers.
Economi

Economics 704
Steven N. Durlauf
Fall 2016
Econometrics 1
Syllabus
This course is the first in a three part masters-level introduction to econometrics. The
focus of this course is the nature of linear statistical models of socioeconomic data. The
course is

Econ 700 Final Exam
Dr. Hansen
December 19th, 2015
Answer the following questions to the best of your ability. Do not provide
irrelevant information. If a question seems ambiguous to you, state the most
reasonable assumptions to resolve the ambiguity and