Introduction to Measure Theory and Asymptotics Applied to Econometrics

Herman J. Bierens

Pennsylvania State University, USA

July 4-29, 2005

Objective

The purpose of this mini-course is to enable the students to read, understand, and apply the newest developments in theoretical econometrics. These developments are quite often cast in measure theoretical terms, and the properties of estimators are derived using asymptotic convergence results. Therefore, the focus of this course will be on the measure-theoretical foundations of econometrics, and on asymptotic theory of nonlinear M-estimators. M-estimators are derived from maximizing or minimizing an objective function, for example (non-)linear regression and maximum likelihood estimators.

Textbook

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

Prerequisite level

It will be assumed that the audience is familiar with statistics at the level of, for example,

• Hogg, R. V., and A.T. Craig, Introduction to Mathematical Statistics, Macmillan,

• Greene, W. H., Econometric Analysis, Macmillan.

Topics

1. PROBABILITY AND MEASURE
   (Bierens 2004, Chapter 1)
   • The probability space
     • Sample space
     • Algebras and sigma-algebras of events
     • Probability measure
   • Why do we need sigma-algebras of events?
   • Properties of algebras and sigma-algebras
   • Properties of probability measures
   • The uniform probability measure
   • Lebesgue measure and Lebesgue integral
   • Random variables and their distributions
2. BOREL MEASURABILITY, INTEGRATION, AND MATHEMATICAL EXPECTATIONS
   (Bierens 2004, Chapter 2)
   • 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
   • Some useful inequalities involving mathematical expectations
   • Moment generating functions and characteristic functions
3. CONDITIONAL EXPECTATIONS
   (Bierens 2004, Chapter 3)
   • Measure-theoretical definition of conditional expectation
   • Properties of conditional expectations
4. MODES OF CONVERGENCE
   (Bierens 2004, Chapter 6)
   • Introduction
   • Convergence in probability and the weak law of large numbers
   • Almost sure convergence, and the strong law of large numbers
   • The uniform law of large numbers
   • Convergence in distribution
   • Convergence of characteristic functions
   • The central limit theorem
   • Consistency and asymptotic normality of M-estimators
     (in particular, nonlinear regression and maximum likelihood estimators)
5. DEPENDENT LAWS OF LARGE NUMBERS AND THE MARTINGALE DIFFERENCE CENTRAL LIMIT THEOREM
   (Bierens 2004, Chapter 7)
   • Stationarity
   • The Wold decomposition
   • Weak laws of large numbers for stationary processes
   • Martingales and martingale differences
   • The martingale difference central limit theorem
   • Consistency and asymptotic normality of M-estimators for time series models