ECON 501 (Fall 2008)

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

1. Bierens, H.J. (2004), Introduction to the Mathematical and Statistical Foundations of Econometrics, Cambridge University Press (2007 paperback edition).
2. 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

Topics

1. Probability and Measure
   1. 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
   2. Quality control
      2.1 Sampling without replacement
      2.3 Sampling with replacement
      2.4 Limits of the hypergeometric and binomial probabilities
   3. Why do we need sigma-algebras of events?
   4. Properties of algebras and sigma-algebras
   5. Properties of probability measures
   6. The uniform probability measure
   7. Outer measure
   8. Lebesgue measure and Lebesgue integral
   9. Random variables and their distributions
   10. Density functions
   11. Conditional probability, Bayes' rule, and independence
2. Borel Measurability, Integration, and Mathematical Expectations
   1. Introduction
   2. Borel measurability
   3. Integrals of Borel measurable functions with respect to a probability measure
   4. General measurability, and integrals of random variables with respect to probability measures
   5. Mathematical expectation
      See Bierens (2008) for a better proof of Theorem 2.18
   6. Some useful inequalities involving mathematical expectations
   7. Expectations of products of independent random variables
   8. Moment generating functions and characteristic functions
3. Conditional Expectations
   1. Introduction
   2. Properties of conditional expectations
   3. Conditional probability measures and conditional independence
4. Distributions and Transformations
   1. Discrete distributions
   2. Transformations of discrete random variables and vectors
   3. Transformations of absolutely continuous random variables
   4. Transformations of absolutely continuous random vectors
   5. The normal distribution
   6. Distributions related to the normal distribution
5. The Multivariate Normal Distribution
   1. Expectation and variance of random vectors
   2. The multivariate normal distribution
   3. Conditional distributions of multivariate normal random variables
   4. Independence of linear and quadratic transformations of multivariate normal random variables
   5. Distribution of quadratic forms of multivariate normal random variables
   6. Applications to statistical inference under normality
6. Modes of Convergence
   1. Introduction
   2. Convergence in probability and the weak law of large numbers.
      See Bierens (2008) for an addition to the proof of Theorem 6.5
   3. Almost sure convergence, and the strong law of large numbers
   4. The uniform law of large numbers and its applications
   5. Convergence in distribution
   6. Convergence of characteristic functions
   7. The central limit theorem
   8. 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.