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Introduction to the Mathematical and Statistical Foundations of Econometrics



Part of Themes in Modern Econometrics

  • Date Published: March 2005
  • availability: Available
  • format: Paperback
  • isbn: 9780521542241

£ 34.99

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About the Authors
  • 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.

    • 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
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    Reviews & endorsements

    '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

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    Product details

    • Date Published: March 2005
    • 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 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.

  • Instructors have used or reviewed this title for the following courses

    • Introduction to Econometrics I
    • Statistics for Economists
  • Author

    Herman J. Bierens, Pennsylvania State University
    Herman J. Bierens is Professor of Economics at the Pennsylvania State University and part-time Professor of Econometrics at Tilburg University, The Netherlands. He is Associate Editor of the Journal of Econometrics and Econometric Reviews, and has been an Associate Editor of Econometrica. Professor Bierens has written two monographs, Robust Methods and Asymptotic Theory in Nonlinear Econometrics and Topics in Advanced Econometrics Cambridge University Press 1994), as well as numerous journal articles. His current research interests are model (mis)specification analysis in econometrics and its application in empirical research, time series econometrics, and the econometric analysis of dynamic stochastic general equilibrium models.

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