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Introduction to the Mathematical and Statistical Foundations of Econometrics : Themes in Modern Econometrics - Herman J. Bierens

Introduction to the Mathematical and Statistical Foundations of Econometrics

Themes in Modern Econometrics


Published: 20th December 2004
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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.

'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

Probability and Measure
The Texas lotto
Quality control
Why do we need sigma-algebras of events?
Properties of algebras and sigma-algebras
Properties of probability measures
The uniform probability measures
Lebesque measure and Lebesque integral
Random variables and their distributions
Density functions
Conditional probability, Bayes's rule, and independence
Exercises: A. Common structure of the proofs of Theorems 6 and 10, B. Extension of an outer measure to a probability measure
Borel Measurability, Integration and Mathematical Expectations
Borel measurability
Integral 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
Expectations of products of independent random variables
Moment generating functions and characteristic functions
Exercises: A. Uniqueness of characteristic functions
Conditional Expectations
Properties of conditional expectations
Conditional probability measures and conditional independence
Conditioning on increasing sigma-algebras
Conditional expectations as the best forecast schemes
Exercises: A. Proof of theorem 22
Distributions and Transformations
Discrete distributions
Transformations of discrete random vectors
Transformations of absolutely continuous random variables
Transformations of absolutely continuous random vectors
The normal distribution
Distributions related to the normal distribution
The uniform distribution and its relation to the standard normal distribution
The gamma distribution
Exercises: A. Tedious derivations, B. Proof of theorem 29
The Multivariate Normal Distribution and its Application to Statistical Inference
Expectation and variance of random vectors
The multivariate normal distribution
Conditional distributions of multivariate normal random variables
Independence of linear and quadratic transformations of multivariate normal random variables
Distribution of quadratic forms of multivariate normal random variables
Applications to statistical inference under normality
Applications to regression analysis
Exercises: A. Proof of theorem 43
Modes of Convergence
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 and its applications
Convergence in distribution
Convergence of characteristic functions
The central limit theorem
Stochastic boundedness, tightness, and the Op and op-notations
Asymptotic normality of M-estimators
Hypotheses testing
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
Dependent Laws of Large Numbers and Central Limit Theorems
Stationary and the world decomposition
Weak laws of large numbers for stationary processes
Mixing conditions
Uniform weak laws of large numbers
Dependent central limit theorems
Exercises: A. Hilbert spaces
Maximum Likelihood Theory
Likelihood functions
Asymptotic properties if ML estimators
Testing parameter restrictions
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780521834315
ISBN-10: 0521834317
Series: Themes in Modern Econometrics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 344
Published: 20th December 2004
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 2.54
Weight (kg): 0.61