Introduction to the Mathematical and Statistical Foundations of Econometrics

ISBN-13: 9780521542241 / Angielski / Miękka / 2004 / 344 str.

Herman J Bierens

Opis

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.

Recenzje

'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

Spis treści

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.