Topics in Advanced Econometrics: Probability Foundations / Edition 1

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Editorial Reviews

The mathematical problems encountered in nonlinear econometrics are not easily treated by the analytical methods most commonly acquired by students, who have learned about probability and inference through the use of density functions. Few students enter graduate programs in economics prepared to approach probability theory in measure theoretic terms. Dhrymes economics, Columbia U. addresses the need for stronger analytic skills in this text, appropriate for graduate students and professionals in nonstandard contemporary econometrics. Annotation c. Book News, Inc., Portland, OR
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Product Details

  • ISBN-13: 9780387971780
  • Publisher: Springer New York
  • Publication date: 12/6/1989
  • Edition description: 1989
  • Edition number: 1
  • Pages: 379
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.88 (d)

Table of Contents

1 Mathematical Foundations.- 1.1 Introduction.- 1.2 Sets and Set Operations.- 1.3 Limits of Sequences.- 1.4 Measurable Spaces, Algebras, and Sets.- 1.5 Measures and Probability Measures.- 1.5.1 Measures and Measurable Functions.- 1.6 Integration.- 1.6.1 Miscellaneous Convergence Results.- 1.7 Extensions to Abstract Spaces.- 1.8 Miscellaneous Concepts.- 2 Foundations of Probability.- 2.1 Discrete Models.- 2.2 General Probability Models.- 2.2.1 The Measurable Space (Rn, Bn, Rn).- 2.2.2 Specification of Probability Measures.- 2.2.3 Fubini’s Theorem and Miscellaneous Results.- 2.3 Random Variables.- 2.3.1 Generalities.- 2.3.2 Random Elements.- 2.3.3 Moments of Random Variables and Miscellaneous Inequalities.- 2.4 Conditional Probability.- 2.4.1 Conditional Probability in Discrete Models.- 2.4.2 Conditional Probability in Continuous Models.- 2.4.3 Independence.- 3 Convergence of Sequences I.- 3.1 Convergence a.c. and in Probability.- 3.1.1 Definitions and Preliminaries.- 3.1.2 Characterization of Convergence a.c. and Convergence in Probability.- 3.2 Laws of Large Numbers.- 3.3 Convergence in Distribution.- 3.4 Convergence in Mean of Order p.- 3.5 Relations among Convergence Modes.- 3.6 Uniform Integrability and Convergence.- 3.7 Criteria for the SLLN.- 3.7.1 Sequences of Independent Random Variables.- 3.7.2 Sequences of Uncorrelated Random Variables.- 4 Convergence of Sequences II.- 4.1 Introduction.- 4.2 Properties of Random Elements.- 4.3 Base and Separability.- 4.4 Distributional Aspects of R.E.- 4.4.1 Independence for Random Elements.- 4.4.2 Distributions of Random Elements.- 4.4.3 Moments of Random Elements.- 4.4.4 Uncorrelated Random Elements.- 4.5 Laws of Large Numbers for R.E.- 4.5.1 Preliminaries.- 4.5.2 WLLN and SLLN for R.E.- 4.6 Convergence in Probability for R.E.- 4.7 Weak Convergence.- 4.7.1 Preliminaries.- 4.7.2 Properties of Measures.- 4.7.3 Determining Classes.- 4.7.4 Weak Convergence in Product Space.- 4.8 Convergence in Distribution for R.E.- 4.8.1 Convergence of Transformed Sequences of R.E.- 4.9 Characteristic Functions.- 4.10 CLT for Independent Random Variables.- 4.10.1 Preliminaries.- 4.10.2 Characteristic Functions for Normal Variables.- 4.10.3 Convergence in Probability and Characteristic Functions.- 4.10.4 CLT for i.i.d. Random Variables.- 4.10.5 CLT and the Lindeberg Condition.- 5 Dependent Sequences.- 5.1 Preliminaries.- 5.2 Definition of Martingale Sequences.- 5.3 Basic Properties of Martingales.- 5.4 Square Integrable Sequences.- 5.5 Stopping Times.- 5.6 Upcrossings.- 5.7 Martingale Convergence.- 5.8 Convergence Sets.- 5.9 WLLN and SLLN for Martingales.- 5.10 Martingale CLT.- 5.11 Mixing and Stationary Sequences.- 5.11.1 Preliminaries and Definitions.- 5.11.2 Measure Preserving Transformations.- 5.12 Ergodic Theory.- 5.13 Convergence and Ergodicity.- 5.14 Stationary Sequences and Ergodicity.- 5.14.1 Preliminaries.- 5.14.2 Convergence and Strict Stationarity.- 5.14.3 Convergence and Covariance Stationarity.- 5.15 Miscellaneous Results and Examples.

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