University of Southern California
Econ 513
Spring 20122013
Prof. Sakata
Solution of Problem Set 2
7. By denition,
Cov[X, Y ] = E[(X E[X ])(Y E[Y ])].
Because
(X E[X ])(Y E[Y ]) = X (Y E[Y ]) E[X ](Y E[Y ]),
it follows by the linearity of the expectation t

University of Southern California
Econ 513
Spring 20122013
Prof. Sakata
Solution of Problem Set 1
1.
(i) We prove the result by contradiction. Suppose that P () = 0. Then we have by axiom (a) that
P () > 0. Notice that S , , , , . . . are mutually exclusi

Economics 513, Test Midterm Problem 1 To study the relation between a dependent variable y and K - 1 independent variables x2 , . . . , xK-1 we specify a multiple linear regression model. (i) Give the equation for the inexact linear relation between

University of Southern California
Econ 513
Spring 20122013
Prof. Sakata
Solution of Problem Set 3
13. Because any linear function of X1 and X2 is normally distributed, Y is normally distributed. Thus, it
suces to nd the mean and variance of Y to determine

Department of Economics, USC, Fall 2016
Econ 513, Practice of Econometrics
1
Instructor and TA
Instructor:
Geert Ridder, KAP 306A
E-mail: ridder@usc.edu
Office hours: Thu 12-1pm or by appointment.
TA
Ida Johnsson
E-mail: ida.b.johnsson@gmail.com
Office ho

USC, Fall 2016, Economics 513
Lecture 8: Instrumental variables
Endogenous and exogenous independent variables
In lecture 7 we found that if relevant independent variables that are correlated
with included independent variables are omitted, then the OLS e

USC, Fall 2016, Economics 513
Lecture 4: The Classical Linear Regression Model II
The CLR model
The Classical Linear Regression model is a set of assumptions under which the
OLS estimator is optimal.
Re-label the variables x1 , . . . , xK1 as x2 , . . . ,

Econ 513: Practice of Econometrics
Department of Economics, USC, Fall 2014
Problem Set 1; September 11, 2014
Instructions
This problem set has 3 pages. Return this problem set by uploading it to Blackboard before
Thursday, September 18, 2014, 4:00pm (i.e.

University of Southern California
Econ 513
Fall 20122013
Prof. Sakata
Problem Set 2
Due on Wednesday, September 12
7. Let X and Y be random variables. Prove that Cov[X, Y ] = E[X(Y E[Y ])].
8. Let X and Y be discretely distributed random variables such th

Linear probability model
3 Denition A linear regression model with a binary dependent variable is called a linear probability model.
Suppose that a binary random variable
Y
and random variables
X1 , X2 ,
.,
Xk
are related through
Y = 0 + 1 X1 + + k Xk + u

16.
Experiments and Quasi-Experiments
Experiments and quasi-experiments are often used in program evaluation, which studies the eect of a
program, policy, or some other intervention (treatment). E.g.:
What's the eect on college attendance of making low-co

Imposing a few more regularity conditions on
g,
we can prove that for each
g (x, h) =
1 Denition Let
(h)x exp(h)
.
x!
be positive real number. If a random variable
P [W = w] =
then the distribution of
W
w exp()
,
w!
W
satises that
w cfw_0 N,
is called the

Ordered Response Models
Let
y
be a latent variable which is related to the choice among
0,
1,
y = 2,
.
.
.
m 1,
if
if
if
m alternatives through the following manner:
y 0,
0 < y 1 ,
1 < y 2 ,
(S.7)
.
.
.
if
y > m1 .
Ordered probit model
Model
y
as being co

ESTIMATING AVERAGE TREATMENT EFFECTS:
INTRODUCTION
Jeff Wooldridge
Michigan State University
BGSE/IZA Course in Microeconometrics
July 2009
1. Introduction
2. Basic Concepts
1
1. Introduction
What kinds of questions can we answer using a modern approach

1
Midterm, March 9 2010
Answer questions concisely. Only give derivations if I ask for them.
Problem 1 (10 points each) We have data on a cross-section of randomly selected firms in
the US. For each firm we observe the value of production and labor input

USC, Fall 2016, Economics 513
Lecture 3: The Classical Linear Regression Model I
Fitting linear relations and mathematical statistics
OLS is a method that gives the best fitting linear relation between a
dependent variable and a set of independent variab

University of Southern California
Econ 513
Fall 20122013
Prof. Sakata
Problem Set 6
Due on Wednesday, October 10
32. Let Y denote the GPA of a student in a term, and X how many hours of private tutoring the student
has in the term (which is zero, if the s

University of Southern California
Econ 513
Fall 20122013
Prof. Sakata
Solution of Problem Set 5
27.
(i) The OLS estimator is given by
= n1
n
X
e
i=1
xi x0i
1
e e
n1
Pn
i=1
= n1
xi x0i
i=1
n
X
i=1
By premultiplying both sides of this equality by n1
n
X

University of Southern California
Econ 513
Fall 20122013
Prof. Sakata
Problem Set 5
Due on Wednesday, October 3
(0 , 1 , . . . , k )0 denote the OLS estimator in
27. Suppose that Assumption MLR.RS holds. Let
0
regression of yi on xi (1, xi1 , . . . , xi

University of Southern California
Econ 513
Fall 20122013
Prof. Sakata
Problem Set 1
Due on Wednesday, September 5
1. Consider a random experiment with the sample space S.
(i) Using the axioms of probability, show that P () = 0. (Hint: You might want to gi

University of Southern California
Econ 513
Fall 20122013
Prof. Sakata
Solution of Problem Set 1
1.
(i) We prove the result by contradiction. Suppose that P () 6= 0. Then we have by axiom (a) that
P () > 0. Notice that S, , , , . . . are mutually exclusive

Appendix D
Summary of Matrix Algebra
his appendix summarizes the matrix algebra concepts, including the algebra of
probability, needed for the study of multiple linear regression models using ma-
trices in Appendix E. None of this material i

USC, Fall 2016, Economics 513
Lecture 7A: Using regression analysis to measure gender discrimination.
A simple regression model for gender differences in wages
If we have data on a sample of individuals who report their wage W and their
gender D (with D =

USC, Fall 2016, Economics 513
Lecture 1: Empirical economic relations
What is econometrics?
Econometrics is concerned with the measurement of economic relations.
So we need to know
What is an economic relation and why do we want to measure it?
How do we

USC, Fall 2016, Economics 513
Lecture 2: Some properties of the OLS solution
Properties of the OLS solution
We fit a linear relation between a dependent variable y and K independent
variables x1 , . . . , xK . The relation also has an intercept. The obser

USC, Spring 2016, Economics 513
Lecture 7: Inference in the multiple linear regression model: Bootstrap.
Finding the sampling distribution of the OLS estimator
Until now we have found two methods to obtain the sampling distribution of
the OLS estimator (a

USC, Fall 2016, Economics 513
Lecture 5: Inference in the CLR model
The CLR model as a random experiment
The CLR assumptions specify a random experiment. In this experiment
1. The nK matrix X is drawn from some distribution such that rank(X) =
K.
2. The n

USC, Fall 2016, Economics 513
Lecture 6: Inference in the multiple linear regression model: Asymptotics.
In lecture 5 we were able to derive the exact sampling distribution of the OLS
estimator and related statistics, so that we obtained confidence interv

15.
Limited Dependent Variable Models
Discrete Choice Models
We have already discussed the binary response/choice models. What if we have more than two alternatives
in choice made by individuals?
Multinomial probit model
Suppose that there are
m
m 1,
alte

Goodness of t
R2
is probably not satisfactory in the probit analysis, as discussed earlier in the section on the linear
probability model. Two measures for goodness of t are well known.
Fraction correctly predicted: Consider the rule that predicts 1 if th

Problem Set I, Economics 513, USC, Fall 2016
Problem Set I
Problem 1 For this problem you will have to use the data set nls.asc which are available on
the website for the course. There are 930 observations on nine variables in this data set,
luwe (log wee

University of Southern California
Econ 513
Fall 20122013
Prof. Sakata
Solution of Problem Set 4
19.
(i) A 95% confidence interval for the mean test score is
Y 1.96se(Y ),
where Y is the sample average of the test score, 1.96 is the 97.5th percentile of th

University of Southern California
Econ 513
Fall 20122013
Prof. Sakata
Solution of Problem Set 3
15.
(i) Because Y is binary, its distribution is a Bernoulli distribution. Because the fraction p of the population
were born in the city, the probability of s

University of Southern California
Econ 513
Fall 20122013
Prof. Sakata
Solution of Problem Set 6
32. The hours of tutoring is not the only determinant of the GPA. The students preparation for the courses
in the current semester are likely to affect the GPA

University of Southern California
Econ 513
Fall 20122013
Prof. Sakata
Problem Set 4
Due on Wednesday, September 26
19. Using the California Test Score Data Set (caschool.dta), answer the following questions. Attach a Stata
output, where you use Stata. In