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.

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.