Introduction to the Mathematical and Statistical Foundations of Econometrics
Part of Themes in Modern Econometrics
 Author: Herman J. Bierens, Pennsylvania State University
 Date Published: December 2004
 availability: Available
 format: Paperback
 isbn: 9780521542241
Paperback

The focus of this book is on clarifying the mathematical and statistical foundations of econometrics. Therefore, the text provides all the proofs, or at least motivations if proofs are too complicated, of the mathematical and statistical results necessary for understanding modern econometric theory. In this respect, it differs from other econometrics textbooks.
Read more Rigorous and comprehensive overview of the mathematical and statistical foundations of econometrics
 The focus is on understanding 'why' rather than 'how', therefore all the proofs are provided
 Appendices contain enough advanced material to make the book suitable for a specialty course in econometric theory
Reviews & endorsements
"Overall, this is an excellent textbook. It offers a unique perspective different from the standard approach in the mainstream textbooks. It encourages the mastering of fundamental concepts and theoretical perspectives at a formal level geared to develop a 'mathematical mind'. It will prove valuable not only for graduate students in econometrics and econometric theory but also as a reference to all researchers in modern economics, econometrics, statistics and financial econometrics."  Economic Record
See more reviews"One outstanding virtue of Bierens' book is the inclusion of a large number of proofs. Some are in the text, and some are relegated to chapter appendices, but in any case, these are an essential ingredient of any such text.... Taken as a whole, this book can be seen as a rather personal compendium of things that Professor Beirens regards as important for students to know. It would be difficult indeed to fit more bits of knowledge useful to the apprentice econometrician into a book of this compass. It represents both an outstanding investment for the graduate student and an item that many researchers and practitioners will find invaluable for reference."  Econometric Reviews
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×Product details
 Date Published: December 2004
 format: Paperback
 isbn: 9780521542241
 length: 344 pages
 dimensions: 229 x 152 x 20 mm
 weight: 0.51kg
 contains: 19 b/w illus. 12 tables
 availability: Available
Table of Contents
Part I. Probability and Measure:
1. The Texas lotto
2. Quality control
3. Why do we need sigmaalgebras of events?
4. Properties of algebras and sigmaalgebras
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 sigmaalgebras
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 opnotations
52. Asymptotic normality of Mestimators
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.Instructors have used or reviewed this title for the following courses
 Introduction to Econometrics I
 Statistics for Economists
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