Probability theory and statistical inference : econometric modelling with observational data

This major textbook from a distinguished econometrician is intended for students taking introductory courses in probability theory and statistical inference. No prior knowledge other than a basic familiarity with descriptive statistics is assumed. The primary objective of this book is to establish the framework for the empirical modelling of observational (non-experimental) data. This framework known as 'Probabilistic Reduction' is formulated with a view to accommodating the peculiarities of observational (as opposed to experimental) data in a unifying and logically coherent way. Probability Theory and Statistical Inference differs from traditional textbooks in so far as it emphasizes concepts, ideas, notions and procedures which are appropriate for modelling observational data. Aimed at students at second-year undergraduate level and above studying econometrics and economics, this textbook will also be useful for students in other disciplines which make extensive use of observational data, including finance, biology, sociology and psychology and climatology. Title ......Page 3 Copyright ......Page 4 Contents......Page 5 Preface ......Page 10 Acknowledgments ......Page 23 Symbols ......Page 25 Acronyms ......Page 27 1.1 Introduction ......Page 28 1.2 Stochastic phenomena, a preliminary view ......Page 30 1.3 Chance regularity and statistical models ......Page 40 1.4 Statistical adequacy ......Page 43 1.5 Statistical versus theory information ......Page 46 1.6 Observed data ......Page 47 1.7 Looking ahead ......Page 56 1.8 Exercises ......Page 57 2.1 Introduction ......Page 58 2.2 Simple statistical model: a preliminary view ......Page 60 2.3 Probability theory: an introduction ......Page 66 2.4 Random experiments ......Page 69 2.5 Formalizing condition [a]: the outcomes set ......Page 72 2.6 Formalizing condition [b]: events and probabilities ......Page 75 2.7 Formalizing condition [c]: random trials ......Page 96 2.8 Statistical space ......Page 100 2.9 A look forward ......Page 101 2.10 Exercises ......Page 102 3.1 Introduction ......Page 104 3.2 The notion of a simple random sample ......Page 105 3.3 The general notion of a random variable ......Page 112 3.4 The cumulative distribution and density functions......Page 116 3.5 From a probability space to a probability model ......Page 124 3.6 Parameters and moments ......Page 131 3.7 Moments ......Page 136 3.8 Inequalities ......Page 158 3.9 Summary ......Page 159 3.10 Exercises ......Page 160 Appendix A Univariate probability models ......Page 162 4.1 Introduction ......Page 172 4.2 Joint distributions ......Page 174 4.3 Marginal distributions ......Page 182 4.4 Conditional distributions ......Page 185 4.5 Independence ......Page 194 4.6 Identical distributions ......Page 198 4.7 A simple statistical model in empirical modeling: a preliminary view ......Page 202 4.8 Ordered random samples ......Page 208 4.10 Exercises ......Page 211 Appendix B Bivariate distributions ......Page 212 5.1 Introduction ......Page 217 5.2 Early developments ......Page 220 5.3 Graphical displays: a t-plot ......Page 222 5.4 Assessing distribution assumptions ......Page 224 5.5 Independence and the t-plot ......Page 239 5.6 Homogeneity and the t-plot ......Page 244 5.7 The empirical cdf and related graphs ......Page 256 5.8 Generating pseudo-random numbers ......Page 281 5.9 Summary ......Page 285 5.10 Exercises ......Page 286 6.1 Introduction ......Page 287 6.2 Non-random sample: a preliminary view ......Page 290 6.3 Dependence between two random variables: joint distributions ......Page 296 6.4 Dependence between two random variables: moments ......Page 299 6.5 Dependence and the measurement system ......Page 309 6.6 Joint distributions and dependence ......Page 317 6.7 From probabilistic concepts to observed data ......Page 336 6.8 What comes next? ......Page 357 6.9 Exercises ......Page 362 7.1 Introduction ......Page 364 7.2 Conditioning and regression ......Page 366 7.3 Reduction and stochastic conditioning ......Page 383 7.4 Weak exogeneity ......Page 393 7.5 The notion of a statistical generating mechanism (GM) ......Page 395 7.6 The biometric tradition in statistics ......Page 404 7.8 Exercises ......Page 424 8.1 Introduction ......Page 427 8.2 The notion of a stochastic process ......Page 430 8.3 Stochastic processes: a preliminary view ......Page 437 8.4 Dependence restrictions ......Page 447 8.5 Homogeneity restrictions ......Page 453 8.6 "Building block" stochastic processes ......Page 458 8.7 Markov processes ......Page 460 8.8 Random walk processes ......Page 462 8.9 Martingale processes ......Page 465 8.10 Gaussian processes ......Page 471 8.11 Point processes ......Page 485 8.12 Exercises ......Page 487 9.1 Introduction to limit theorems ......Page 489 9.2 Tracing the roots of limit theorems ......Page 492 9.3 The Weak Law of Large Numbers ......Page 496 9.4 The Strong Law of Large Numbers ......Page 503 9.5 The Law of Iterated Logarithm ......Page 508 9.6 The Central Limit Theorem ......Page 509 9.7 Extending the limit theorems ......Page 518 9.8 Functional Central Limit Theorem ......Page 522 9.9 Modes of convergence ......Page 530 9.11 Exercises ......Page 537 10.1 Introduction ......Page 539 10.2 Interpretations of probability ......Page 541 10.3 Attempts to build a bridge between probability and observed data......Page 547 10.4 Toward a tentative bridge ......Page 555 10.5 The probabilistic approach to specification ......Page 568 10.6 Parametric versus non-parametric models ......Page 573 10.8 Exercises ......Page 583 11.1 Introduction ......Page 585 11.2 An introduction to the classical approach ......Page 586 11.3 The classical versus the Bayesian approach ......Page 595 11.4 Experimental versus observational data ......Page 597 11.5 Neglected facets of statistical inference ......Page 602 11.6 Sampling distributions ......Page 605 11.7 Functions of random variables ......Page 611 11.8 Computer intensive techniques for approximating sampling distributions ......Page 621 11.9 Exercises ......Page 627 12.1 Introduction ......Page 629 12.2 Defining an estimator ......Page 630 12.3 Finite sample properties ......Page 634 12.4 Asymptotic properties ......Page 642 12.5 The simple Normal model ......Page 648 12.6 Sufficient statistics and optimal estimators ......Page 654 12.8 Exercises ......Page 662 13.1 Introduction ......Page 664 13.2 Moment matching principle ......Page 666 13.3 The least-squares method ......Page 675 13.4 The method of moments ......Page 681 13.5 The maximum likelihood method ......Page 686 13.6 Exercises ......Page 705 14.1 Introduction ......Page 708 14.2 Leading up to the Fisher approach ......Page 709 14.3 The Neyman-Pearson framework ......Page 719 14.4 Asymptotic test procedures ......Page 740 14.5 Fisher versus Neyman-Pearson ......Page 747 14.7 Exercises ......Page 754 15.1 Introduction ......Page 756 15.2 Misspecification testing: formulating the problem ......Page 760 15.3 A smorgasbord of misspecification tests ......Page 766 15.4 The probabilistic reduction approach and misspecification ......Page 780 15.5 Empirical examples ......Page 792 15.6 Conclusion ......Page 810 15.7 Exercises ......Page 811 References ......Page 814 Index ......Page 833 The Primary Objective Of This Text Is To Establish The Framework For The Empirical Modelling Of Observational (non-experimental) Data. The Book Is Aimed At Students Taking Introductory Courses In Proabilitity Theory And Statistical Inference. 1. An Introduction To Empirical Modeling -- 2. Probability Theory : A Modeling Framework -- 3. The Notion Of A Probability Model -- App. A. Univariate Probability Models -- 4. The Notion Of A Random Sample -- App. B. Bivariate Distributions -- 5. Probabilistic Concepts And Real Data -- 6. The Notion Of Non-random Sample -- 7. Regression And Related Notions -- 8. Stochastic Processes -- 9. Limit Theorems -- 10. From Probability Theory To Statistical Inference -- 11. An Introduction To Statistical Inference -- 12. Estimation I: Properties Of Estimators -- 13. Estimation Ii: Methods Of Estimation -- 14. Hypothesis Testing -- 15. Misspecification Testing. Aris Spanos. Includes Bibliographical References (p. 787-805) And Index.

This major new textbook is intended for students taking introductory courses in probability theory and statistical inference. The primary objective of this book is to establish the framework for the empirical modeling of observational (nonexperimental) data. The text is extremely student friendly, with pathways designed for semester usage, and although aimed primarily at students at second-year undergraduate level and above studying econometrics and economics, Probability Theory and Statistical Inference will also be useful for students in other disciplines that make extensive use of observational data, including finance, biology, sociology and psychology.

This major new textbook is intended for students taking introductory courses in Probability Theory and Statistical Inference. The text is extremely student-friendly, with pathways designed for semester usage, and although aimed primarily at students of econometrics and economics, will have considerable utility for courses in all disciplines using observational data