ECON 501 (Fall 2007)

Instructor

T.A./Grader

Time and place:

Tuesday and 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 mid-term 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 mid-term exam, the latter score will be ignored, and the final exam will count for 80% of the final grade, provided that you have done 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 (paperback edition).

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

1. Review of Linear Algebra
2. Miscellaneous Mathematics
3. A Brief Review of Complex Analysis

Topics

• 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. Lebesgue measure and Lebesgue integral
  8. Random variables and their distributions
  9. Density functions
  10. Conditional probability, Bayes' rule, and independence
• 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 ERRATA2.PDF 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
  4. Conditioning on increasing sigma-algebras
  5. Conditional expectations as the best forecast schemes
• Distributions and Transformations
  1. Discrete distributions
     1.1 The hypergeometric distribution
     1.2 The binomial distribution
     1.3 The Poisson distribution
     1.4 The negative binomial distribution
  2. Transformations of discrete random 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 ERRATA2.PDF 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. Stochastic boundedness, tightness, and the O_p and o_p notations

Homework assignments

• Section 1.11
  • Exercises 1, 3, 4, 5, 11 and 12 (Due Thursday September 13)
  • Exercises 13, 14, 15 and 16 (Due Thursday September 20)
  • Exercise 21 (Due Thursday September 27)
• Section 2.9
  • Exercises 1 through 7 (Due Thursday October 4)
  • Exercises 8 through 14 (Due Thursday October 11)
  • Exercises 17 through 23, and 29, 30, 31 (due Thursday October 18)
• Section 3.6
  • Exercises 3, 4, 10 and 11 (Due Thursday November 1)
• Section 4.9
  • Exercises 7, 9, 10, 11, 13, 19, 20 and 21 (Due Thursday November 8)
• Section 5.8
  • Excercises 1 through 7 (Due Thursday November 29)
• Section 6.11
  • Exercises 1 and 2, and an additional exercise (Due Thursday December 6)
  • Exercises 4, 5, 6, 8, 9 (Due Thursday December 13)

Exams

• Midterm: Tuesday October 23
• Final: Tuesday, December 18, 8-9:50 AM in 109 BOUCKE

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.