Topics in Advanced Econometrics

For sometime now, I felt that the evolution of the literature of econo metrics had mandated a higher level of mathematical proficiency. This is particularly evident beyond the level of the general linear model (GLM) and the general linear structural econometric model (GLSEM). The problems one encounters in nonlinear econometrics are not easily amenable to treatment by the analytical methods one typically acquires, when one learns about probability and inference through the use of den sity functions. Even in standard traditional topics, one is often compelled to resort to heuristics; for example, it is difficult to prove central limit theorems for nonidentically distributed or martingale sequences, solely by the use of characteristic functions. Yet such proofs are essential, even in only moderately sophisticated classroom exposition. Unfortunately, relatively few students enter a graduate economics de partment ready to tackle probability theory in measure theoretic terms. The present volume has grown out of the need to lay the foundation for such discussions. The motivating forces were, chiefly, (a) the frustration one encounters in attempting to communicate certain concepts to stu dents wholly in analytic terms; and (b) the unwillingness of the typical student to sit through several courses in mathematics departments, in order to acquire the requisite background.

Klappentext

This textbook is intended for graduate students and professionals who have an interest in linear and nonlinear simultaneous equation models. These models arise in a great many settings in econometrics. The author's aim is to present a readable account, starting from an introduction to the general linear structural econometric model. From there, the book covers the identification problem, maximum likelihood methods, two and three stage least square methods, the general nonlinear model, and more advanced topics such as the general nonlinear simultaneous equations model. The reader is assumed to have a basic background in probability theory but otherwise this account is self-contained.

Inhalt
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.- 1.6 Integration.- 1.7 Extensions to Abstract Spaces.- 1.8 Miscellaneous Concepts.- 2 Foundations of Probability.- 2.1 Discrete Models.- 2.2 General Probability Models.- 2.3 Random Variables.- 2.4 Conditional Probability.- 3 Convergence of Sequences I.- 3.1 Convergence a.c. and 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.- 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.- 4.5 Laws of Large Numbers for R.E.- 4.6 Convergence in Probability for R.E.- 4.7 Weak Convergence.- 4.8 Convergence in Distribution for R.E.- 4.9 Characteristic Functions.- 4.10 CLT for Independent Random Variables.- 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.- 5.12 Ergodic Theory.- 5.13 Convergence and Ergodicity.- 5.14 Stationary Sequences and Ergodicity.- 5.15 Miscellaneous Results and Examples.