Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Denition of Econometrics and goal of this course
Econometrics is the development of statistical methods for estimating economic relationships, testing eco
Problem Set 5
ECON 4720 Fall 2015
Due Monday, September 26
1. According to a particular economist, the production of a certain commodity y only needs
two inputs, labor L and capital K. however, the productivity of each factor also depends upon some
unobse
Department of Economics University of Virginia Ofce: Monroe Hall 208A MW 2:00pm - 3:15pm, Monroe Hall 116
Wayne-Roy Gayle e-mail: [email protected] Phone: 434-243-4336 Ofce Hours: TH 2:00pm - 3:00pm
ECON4720: Introductory Econometrics
Spring 2011
Prerequi
ECON 4720
Introductory Econometrics
Wayne-Roy Gayle
Fall 2013
Sample Questions
1. Consider the linear model
y = x + u,
satisfying OLS1 - OLS5. Suppose you have a sample size of n observations. You
dene the following estimator for
= (X AX )1 X AY,
where
Econ 4720
Section 1
Properties of Sums
n
xi stands for the sum x1 + x2 + . . . + xn1 + xn .
i=1
Some useful properties of sums are:
N
a = Na
i=1
n
n
axi = a
xi
i=1
i=1
n
n
n
(xi + yi ) =
i=1
n
xi +
i=1
n
m
xi
xi yj =
n
yj
j=1
n
m
n
m
m
n
xi yij =
i=1 j=1
Econ 4720
Section 2
Exercise 1. Let Y be a Bernoulli random variable with success probability P (Y = 1) = p, and let
Y1 , ., Yn be iid draws from this distribution. Let p be the fraction of successes (1s) in this sample.
a. Show that p = Y .
b. Show that
Econ 4720
Section 3
Exercise 1. Let A be a square 2X2 matrix
1
4
3
5
a. Find det(A).
b. Find the inverse of A
c. Find the eigenvalues and eigenvectors of A
Exercise 2. Suppose that b = (1, 0)T . Consider matrix A from exercise 1. Find vector x that
minimi
Econ 4720
Section 4
Exercise 1 (Sample Exam Question) As a soon to be college graduate, you are interested in
estimating the salary you can expect to receive once you graduate from UVa. Although you believe
you are an above average student, you decide tha
Econ 4720
Section 5
Exercise 1
Suppose that a researcher, using data on class size (CS) and average test scores from 100 third-grade
classes, estimates the OLS regression
T estScore = 520.4 5.82 CS
(20.4)
(2.21)
a. A classroom has 22 students. What is the
Econ 4720
Section 6
Exercise 1 (Sample Exam Problem)
You have collected data on the prices of diamonds at an online retailer (you can assume its a random
sample). In particular, you have the price ( price) in U.S. dollars and the weight in carats (carats)
Problem Set 1
ECON 4720 Fall 2015
Suggested Solutions
1. (a) 38.2 percent of the presidents were Republicans and were in the White House
while Republicans controlled the House of Representatives. 44.2 percent of all
presidents were Democrats.
(b) E (X) =
Problem Set 2
ECON 4720 Fall 2015
Suggested Solutions
1. In the setup of the problem, we are told that Y is distributed as a binomial with
parameter p (i.e., for a given p, Y has a binomial distribution). We then know that
E (Y | p) = np, where n is the n
Problem Set 3
ECON 4720 Fall 2015
Suggested Solutions
1. From the denition of covariance:
This is identical to:
i=1
(Xi X)Yi (Xi X)Y
(1)
(Yi Y )Xi (Yi Y )X
(2)
i=1
and
XY =
(Xi X)(Yi Y )
n
1
n
XY =
n
1
n
XY =
1
n
n
i=1
Equation (1) can be rewritten
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Limited Dependent Variable (LDV) Models
1.1 Introduction
An LDV is a dependent variable whose range of values is substantively restricted. For example, a
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Models of Self-selectivity (Sample Selection Models)
1.1 Introduction
Sample selection models are designed to deal with data that are generated by individ
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Geometry of the Least Squares Estimator
1.1 Introduction
We start with the model given by:
yi = xi1 1 + + xik k + ui ,
i = 1, , n
or equivalently
yi = xi
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Normal Distribution Theory
So far we have established conditions under which the OLS estimator is BLUE. We also derived an
unbiased estimator for the vari
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 The Basic Linear Model; Identication and Finite Sample
Results
So far, we have not considered any model with specic conditions. Our results pertained only
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Asymptotics Properties of the OLS Estimator
The goal of this chapter is to investigate key properties of the OLS estimator as the sample size
grows withou
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Asymptotics
In the previous chapter, we assumed that the conditional distribution of the population regression
residual given the regressors is normal. Th
A Geometric Representation of the Frisch-Waugh-Lovell Theorem
Walter Sosa Escudero
Documento de Trabajo Nro. 29
Marzo 2001
A Geometric Representation of the
Frisch-Waugh-Lovell Theorem
Walter Sosa Escuderoy
Universidad Nacional de La Plata
March 21, 2001
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Heteroskedasticity
We continue to consider the linear regression model
y = x + u.
Recall OLS5, which states that u has the same variance given any value o
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 More on Time Series Models
1.1 Introduction
In the previous chapter, we discussed estimation and testing of time series models under the classical assumpt
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Accounting for Heteroskedasticity and Autocorrelation
1.1 Model and Assumptions
The model relating the T sample observations is given by
y = X + u ,
T 1
T
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Other Sampling Schemes
So far we have seen two data structures; cross section and time series data sets. In this chapter, we
discuss some other popular da
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 Instrumental Variables
1.1 Introduction
Our model is given by
y = x + u.
We maintain that E[u] = 0. However, as discussed before, it may be the case that
Wayne-Roy Gayle
ECON 472: Introductory Econometrics
Department of Economics
University of Virginia
1 The Generalized Method of Moments (GMM) in the Linear
Regression Model
1.1 The Method of Moments (MoM)
The MoM is one of the oldest estimation methods in
Econ 4720
Section 2
Exercise 1. Let A be a square 2X2 matrix
1
4
3
5
a. Find det(A).
b. Find the inverse of A
c. Find the eigenvalues and eigenvectors of A
Exercise 2. Suppose that b = (1, 0)T . Consider matrix A from exercise 1. Find vector x that
minimi