ECON 501 (Fall 2008)
Introduction to the Mathematical and Statistical Foundations of Econometrics
Instructor
Professor Herman J. Bierens
Office: 510 Kern
Tel.: 865-4921
E-mail: hbierens@psu.edu
Office hours: Wednesday 2-4 PM, and by appointment.
Teaching assistant
Haiqing Xu
Office: 404 Kern
E-mail: hux100@psu.edu
Office hours: Wednesday 4-6 PM
Time and place
Tuesday & Thursday 9:45-11:00 AM in 101 Walker
Objectives and grading
The objective of this course is to prepare the first year Ph.D. students in
economics for the study of econometrics, by providing a rigorous introduction to
probability and measure theory and mathematical statistics.
Each week a number of exercises from the textbook will be assigned as homework.
The final grade will be determined by the homework (20%), a written
closed book midterm exam (40%), and a written closed book final exam (40%). The
final exam will cover the material in the mid-term exam as well. If you score
higher on the final exam than on the midterm exam, the latter score will be
ignored, and the final exam will count for 80% of the final grade, provided
that you have done all your homework.
As to the homework, the two lowest scores will be ignored for the grade.
If you do not turn in a homework, the score for this homework assignment will be zero.
Textbook
- Bierens, H.J. (2004),
Introduction to the Mathematical and Statistical
Foundations of Econometrics,
Cambridge University Press (2007 paperback edition).
- Bierens, H.J. (2008),
Errata and improvements to "Introduction to the
Mathematical and Statistical Foundations of Econometrics"
(Updated: December 12, 2008)
The first six chapters will be covered, except the appendixes to these chapters.
I will assume that you are familiar with the contents of the appendixes:
- Review of Linear Algebra
- Miscellaneous Mathematics
- A Brief Review of Complex Analysis
If not, you have to read and digest these appendixes yourself!
Topics
- Probability and Measure
- The Texas lotto
1.1 Introduction
1.2 Binomial numbers
1.3 Sample space
1.4 Algebras and sigma-algebras of events
1.5 Probability measure
- Quality control
2.1 Sampling without replacement
2.3 Sampling with replacement
2.4 Limits of the hypergeometric and binomial
probabilities
- Why do we need sigma-algebras of events?
- Properties of algebras and sigma-algebras
- Properties of probability measures
- The uniform probability measure
- Outer measure
- Lebesgue measure and Lebesgue integral
- Random variables and their distributions
- Density functions
- Conditional probability, Bayes' rule, and
independence
- Borel Measurability, Integration, and Mathematical Expectations
- Introduction
- Borel measurability
- Integrals of Borel measurable functions with
respect to a probability measure
- General measurability, and integrals of random
variables with respect to probability measures
- Mathematical expectation
See Bierens (2008)
for a better proof of Theorem 2.18
- Some useful inequalities involving
mathematical expectations
- Expectations of products of independent random
variables
- Moment generating functions and characteristic
functions
- Conditional Expectations
- Introduction
- Properties of conditional expectations
- Conditional probability measures and conditional independence
- Distributions and Transformations
- Discrete distributions
- Transformations of discrete random variables and vectors
- Transformations of absolutely continuous random variables
- Transformations of absolutely continuous random vectors
- The normal distribution
- Distributions related to the normal distribution
- The Multivariate Normal Distribution
- Expectation and variance of random vectors
- The multivariate normal distribution
- Conditional distributions of multivariate
normal random variables
- Independence of linear and quadratic
transformations of multivariate normal random
variables
- Distribution of quadratic forms of
multivariate normal random variables
- Applications to statistical inference under normality
- Modes of Convergence
- Introduction
- Convergence in probability and the weak law of
large numbers.
See Bierens (2008)
for an addition to the proof of Theorem 6.5
- Almost sure convergence, and the strong law of
large numbers
- The uniform law of large numbers and its
applications
- Convergence in distribution
- Convergence of characteristic functions
- The central limit theorem
- Asymptotic normality of M-estimators
Homework assignments
- Chapter 1, Section 1.11:
- Exercises 1, 3, 4, 5, 7-13, due Thursday September 11
- Exercises 14, 15, 16, due Thursday September 18
- Exercises 17, 18, 19, 21, due Thursday September 25
- Exercises 22, 23, due Thursday October 2
- Chapter 2, Section 2.9:
- Exercises 1-6, due Thursday October 2
- Exercises 7-14, 17, 18, due Thursday October 16
- Exercises 20, 22, 29, 30, 31, 32, due Thursday October 23
- Chapter 3, Section 3.6:
- Exercises 4, 5, 8, 10, 11, due Thursday October 30
- Chapter 4, Section 4.9:
- Exercises 7, 9, 10, 11, 13, 20, 21, due Thursday November 6
- Appendix I, Section I.18, pp. 280-282
- Exercises 1, 6, 8-14, due Thursday November 13
- Chapter 5, Section 5.8:
- Exercises 1, 2, 5, 6, 8, due Thursday November 20
- Chapter 6, Section 6.11:
- Exercises 1, 2, 3, and an additional exercise,
due Thursday December 4
- Exercises 4, 5, 6, 8, 9, and an additional exercise,
due Thursday December 11
Exams
- Midterm: Thursday October 9, 9:45-11 AM in 101 Walker
- Final:
Wednesday, December 17, 12:20 PM to 2:10PM,
in 403 Kern for the students with last name starting with A to L, and
in 413 Kern for the students with last name starting with Q to Z
Disability Message
The Pennsylvania State University encourages qualified persons with
disabilities to participate in its programs and activities. If you anticipate
needing any type of accommodation in this course or have questions about
physical access, please tell the instructor as soon as possible.
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