Introduction to the Mathematical and Statistical Foundations of Econometrics

This book is intended for use in a rigorous introductory PhD level course in econometrics.

Herman J. Bierens (Author)

9780521834315, Cambridge University Press

Hardback, published 20 December 2004

344 pages, 19 b/w illus. 12 tables
22.9 x 15.2 x 2.4 cm, 0.68 kg

'The objective of this book is to use it as an introductory text for a Ph.D. level course in Econometrics. … Appendixes are self contained with review which are easy to learn and understand. As a whole, I consider this book as unique and self-contained and it will be a great resource for researchers in the area of Econometrics.' Zentralblatt MATH

This book is intended for use in a rigorous introductory PhD level course in econometrics, or in a field course in econometric theory. It covers the measure-theoretical foundation of probability theory, the multivariate normal distribution with its application to classical linear regression analysis, various laws of large numbers, central limit theorems and related results for independent random variables as well as for stationary time series, with applications to asymptotic inference of M-estimators, and maximum likelihood theory. Some chapters have their own appendices containing the more advanced topics and/or difficult proofs. Moreover, there are three appendices with material that is supposed to be known. Appendix I contains a comprehensive review of linear algebra, including all the proofs. Appendix II reviews a variety of mathematical topics and concepts that are used throughout the main text, and Appendix III reviews complex analysis. Therefore, this book is uniquely self-contained.

Part I. Probability and Measure: 1. The Texas lotto
2. Quality control
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 measures
7. Lebesque measure and Lebesque integral
8. Random variables and their distributions
9. Density functions
10. Conditional probability, Bayes's rule, and independence
11. Exercises: A. Common structure of the proofs of Theorems 6 and 10, B. Extension of an outer measure to a probability measure
Part II. Borel Measurability, Integration and Mathematical Expectations: 12. Introduction
13. Borel measurability
14. Integral of Borel measurable functions with respect to a probability measure
15. General measurability and integrals of random variables with respect to probability measures
16. Mathematical expectation
17. Some useful inequalities involving mathematical expectations
18. Expectations of products of independent random variables
19. Moment generating functions and characteristic functions
20. Exercises: A. Uniqueness of characteristic functions
Part III. Conditional Expectations: 21. Introduction
22. Properties of conditional expectations
23. Conditional probability measures and conditional independence
24. Conditioning on increasing sigma-algebras
25. Conditional expectations as the best forecast schemes
26. Exercises
A. Proof of theorem 22
Part IV. Distributions and Transformations: 27. Discrete distributions
28. Transformations of discrete random vectors
29. Transformations of absolutely continuous random variables
30. Transformations of absolutely continuous random vectors
31. The normal distribution
32. Distributions related to the normal distribution
33. The uniform distribution and its relation to the standard normal distribution
34. The gamma distribution
35. Exercises: A. Tedious derivations
B. Proof of theorem 29
Part V. The Multivariate Normal Distribution and its Application to Statistical Inference: 36. Expectation and variance of random vectors
37. The multivariate normal distribution
38. Conditional distributions of multivariate normal random variables
39. Independence of linear and quadratic transformations of multivariate normal random variables
40. Distribution of quadratic forms of multivariate normal random variables
41. Applications to statistical inference under normality
42. Applications to regression analysis
43. Exercises
A. Proof of theorem 43
Part VI. Modes of Convergence: 44. Introduction
45. Convergence in probability and the weak law of large numbers
46. Almost sure convergence, and the strong law of large numbers
47. The uniform law of large numbers and its applications
48. Convergence in distribution
49. Convergence of characteristic functions
50. The central limit theorem
51. Stochastic boundedness, tightness, and the Op and op-notations
52. Asymptotic normality of M-estimators
53. Hypotheses testing
54. Exercises: A. Proof of the uniform weak law of large numbers
B. Almost sure convergence and strong laws of large numbers
C. Convergence of characteristic functions and distributions
Part VII. Dependent Laws of Large Numbers and Central Limit Theorems: 55. Stationary and the world decomposition
56. Weak laws of large numbers for stationary processes
57. Mixing conditions
58. Uniform weak laws of large numbers
59. Dependent central limit theorems
60. Exercises: A. Hilbert spaces
Part VIII. Maximum Likelihood Theory
61. Introduction
62. Likelihood functions
63. Examples
64. Asymptotic properties if ML estimators
65. Testing parameter restrictions
66. Exercises.

Subject Areas: Applied mathematics [PBW], Economic statistics [KCHS], Econometrics [KCH]