ECON 501: Econometrics I --- Herman J. Bierens

ECON 501: Econometrics I, Fall 2009

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.
Extra office hours:
Monday October 26, from 10 AM to 12 PM and 1 PM to 5 PM
Monday December 14, from 10 AM to 12 PM and 1 PM to 5 PM

Teaching assistant

Yao Luo
Office: 407 Kern
E-mail: yul162@psu.edu
Office hours: Wednesday 4-6 PM

Time and place

Tuesday and Thursday from 9:45 AM to 11:00 AM in 111 Borland

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. (2009), Errata and improvements to "Introduction to the Mathematical and Statistical Foundations of Econometrics". (Updated: October 21, 2009)

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
1. The Texas lotto (See also BINOMIAL.PDF for the Pennsylvania lottery)
  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 (See also INDEPENDENCE.PDF)

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 (2009) 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

Conditional Expectations
1. Introduction
2. Properties of conditional expectations
3. Conditional probability measures and conditional independence ( See also CONDITIONAL_EXPECTATIONS.PDF)

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

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

Modes of Convergence
1. Introduction
2. Convergence in probability and the weak law of large numbers.
  See Bierens (2009) 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

Remark: To see the current assignment, you may need to refresh this web page first.

Chapter 1, Section 1.11:
- Exercises 1, 3, 4, 5, 7-13
- Exercises 14, 15
- Exercises 16-21
Chapter 2, Section 2.9:
- Exercises 1-7
- Exercises 8-14
  Hints:
  - Exercise 8: You need Theorem 2.11 to prove part (b) of Theorem 2.9.
  - Exercise 9: You may assume that all the integrals involved are finite. Then use Theorems 2.6, 2.10 and/or 2.11 and the results for the simple cases.
  - Exercise 10: Again you may assume that all the integrals involved are finite.
- Exercises 17, 18, 22, 26, 27, 29, 30, 31
Chapter 3, Section 3.6:
- Exercises 4, 5, 8, 10, 11
Chapter 4, Section 4.9:
- Exercises 7, 9, 10, 11, 13, 20, 21, due Thursday November 5
  Hint: In exercise 9, assume in first instance that F(x) is strictly monotonic.
Appendix I, Section I.18, pp. 280-282
- Exercises 1, 6, 8-14
Chapter 5, Section 5.8:
- Exercises 1, 2, 5, 6, 8
Chapter 6, Section 6.11:
- Exercises 1, 2, 3, and an additional exercise
- Exercises 4, 5, 6, 8, 9, and an additional exercise

Exams

Midterm: Tuesday October 27, 9:45-11 AM in 111 Borland
The midterm will cover Chapters 1, 2 and 3, without the appendixes to these chapters and the proof of Theorem 2.18.
Final: Tuesday, December 15 from 10:10 AM to 12:00 PM in 213 BUCKHOUT

Final Exam Conflict

Students have the right to file for a final exam conflict, so that their final exam will be held on another day and/or time than the officially scheduled day and time. However, that implies that I have to make two different final exams, which puts an unduly burden on me. Therefore, before you file for a final exam conflict, talk to me first!

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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