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Full text available at: http://dx.doi.org/10.1561/0800000004 Information and Entropy Econometrics — A Review and Synthesis Full text available at: http://dx.doi.org/10.1561/0800000004 Information and Entropy Econometrics — A Review and Synthesis Amos Golan Department of Economics American University 4400 Massachusetts Avenue NW Washington DC 20016-8029 USA agolan@american.edu Boston – Delft Full text available at: http://dx.doi.org/10.1561/0800000004 Foundations and Trends in Econometrics Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 USA Tel. +1-781-985-4510 www.nowpublishers.com sales@nowpublishers.com Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 The preferred citation for this publication is A. Golan, Information and Entropy Econometrics — A Review and Synthesis, Foundations and Trends in Economet- rics, vol 2, no 1–2, pp 1–145, 2006 ISBN: 978-1-60198-104-2 c 2008 A. Golan All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. Photocopying. In the USA: This journal is registered at the Copyright Clearance Cen- ter, Inc., 222 Rosewood Drive, Danvers, MA 01923. Authorization to photocopy items for internal or personal use, or the internal or personal use of speciﬁc clients, is granted by now Publishers Inc for users registered with the Copyright Clearance Center (CCC). The ‘services’ for users can be found on the internet at: www.copyright.com For those organizations that have been granted a photocopy license, a separate system of payment has been arranged. Authorization does not extend to other kinds of copy- ing, such as that for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. 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Greene Department of Economics New York Univeristy 44 West Fourth Street, 7–78 New York, NY 10012 USA wgreene@stern.nyu.edu Editors Manuel Arellano, CEMFI Spain Wiji Arulampalam, University of Warwick Orley Ashenfelter, Princeton University Jushan Bai, NYU Badi Baltagi, Syracuse University Anil Bera, University of Illinois Tim Bollerslev, Duke University David Brownstone, UC Irvine Xiaohong Chen, NYU Steven Durlauf, University of Wisconsin Amos Golan, American University Bill Griﬃths, University of Melbourne James Heckman, University of Chicago Jan Kiviet, University of Amsterdam Gary Koop, Leicester University Michael Lechner, University of St. Gallen Lung-Fei Lee, Ohio State University Larry Marsh, Notre Dame University James MacKinnon, Queens University Bruce McCullough, Drexel University Jeﬀ Simonoﬀ, NYU Joseph Terza, University of Florida Ken Train, UC Berkeley Pravin Travedi, Indiana University Adonis Yatchew, University of Toronto Full text available at: http://dx.doi.org/10.1561/0800000004 Editorial Scope Foundations and Trends in Econometrics will publish survey and tutorial articles in the following topics: • Identiﬁcation • Modeling Non-linear Time Series • Model Choice and Speciﬁcation • Unit Roots Analysis • Cointegration • Non-linear Regression Models • Latent Variable Models • Simultaneous Equation Models • Qualitative Response Models • Estimation Frameworks • Hypothesis Testing • Biased Estimation • Interactions-based Models • Computational Problems • Duration Models • Microeconometrics • Financial Econometrics • Treatment Modeling • Measurement Error in Survey • Discrete Choice Modeling Data • Models for Count Data • Productivity Measurement and Analysis • Duration Models • Semiparametric and • Limited Dependent Variables Nonparametric Estimation • Panel Data • Bootstrap Methods • Dynamic Speciﬁcation • Nonstationary Time Series • Inference and Causality • Robust Estimation • Continuous Time Stochastic Models Information for Librarians Foundations and Trends in Econometrics, 2006, Volume 2, 2 issues. ISSN paper version 1551-3076. ISSN online version 1551-3084. Also available as a combined paper and online subscription. Full text available at: http://dx.doi.org/10.1561/0800000004 Foundations and Trends in Econometrics Vol. 2, Nos. 1–2 (2006) 1–145 c 2008 A. Golan DOI: 10.1561/0800000004 Information and Entropy Econometrics — A Review and Synthesis Amos Golan Department of Economics, American University, 4400 Massachusetts Avenue, NW Washington, DC 20016-8029, USA, agolan@american.edu This work is dedicated to Marge and George Judge Abstract The overall objectives of this review and synthesis are to study the basics of information-theoretic methods in econometrics, to exam- ine the connecting theme among these methods, and to provide a more detailed summary and synthesis of the sub-class of methods that treat the observed sample moments as stochastic. Within the above objectives, this review focuses on studying the inter-connection between information theory, estimation, and inference. To achieve these objectives, it provides a detailed survey of information-theoretic concepts and quantities used within econometrics. It also illustrates Part of this study was done during the senior fellowship at the Institute of Advanced Study and the faculty of statistics of the University of Bologna. The author thanks the Institute and the University for their Support. Detailed Comments from Douglas Miller and Mirko Degli Esposti and an anonymous reviewer on an earlier version of this manuscript, and comments from Essie Maasoumi on earlier sections of this survey, are greatly appreciated. The author also thanks Doug Miller for pointing some unknown references. Finally, the author thanks the Editor — Bill Greene — for all his helpful comments and suggestions throughout the process of composing this review. Full text available at: http://dx.doi.org/10.1561/0800000004 the use of these concepts and quantities within the subﬁeld of information and entropy econometrics while paying special attention to the interpretation of these quantities. The relationships between information-theoretic estimators and traditional estimators are dis- cussed throughout the survey. This synthesis shows that in many cases information-theoretic concepts can be incorporated within the tradi- tional likelihood approach and provide additional insights into the data processing and the resulting inference. Keywords: Empirical likelihood; entropy, generalized entropy; informa- tion; information theoretic estimation methods; likelihood; maximum entropy; stochastic moments. JEL codes: C13, C14, C49, C51 Full text available at: http://dx.doi.org/10.1561/0800000004 Preface This review and synthesis is concerned with information and entropy econometrics (IEE). The overall objective is to summarize the basics of information-theoretic methods in econometrics and the connecting theme among these methods. The sub-class of methods that treat the observed sample moments as stochastic is discussed in greater detail. Within the above objective, we restrict our attention to study the inter- connection between information theory, estimation, and inference. We provide a detailed survey of information-theoretic concepts and quan- tities used within econometrics and then show how these quantities are used within IEE. We pay special attention for the interpretation of these quantities and for describing the relationships between information- theoretic estimators and traditional estimators. In Section 1, an introductory statement and detailed objectives are provided. Section 2 provides a historical background of IEE. Section 3 surveys some of the basic quantities and concepts of information the- ory. This survey is restricted to those concepts that are employed within econometrics and that are used within that survey. As many of these concepts may not be familiar to many econometricians and economists, a large number of examples are provided. The concepts discussed ix Full text available at: http://dx.doi.org/10.1561/0800000004 x Preface include entropy, divergence measures, generalized entropy (known also as Cressie Read function), errors and entropy, asymptotic theory, and stochastic processes. However, it is emphasized that this is not a survey of information theory. A less formal discussion providing interpretation of information, uncertainty, entropy and ignorance, as viewed by sci- entists across disciplines, is provided at the beginning of that section. In Section 4, we discuss the classical maximum entropy (ME) princi- ple (both the primal constrained model and the dual concentrated and unconstrained model) that is used for estimating under-determined, zero-moment problems. The basic quantities discussed in Section 3, are revisited again in connection with the ME principle. In Section 5, we discuss the motivation for information-theoretic (IT) estimators and then formulate the generic IT estimator as a constrained optimization problem. This generic estimator encompasses all the estimators within the class of IT estimators. The rest of this section describes the basics of speciﬁc members of the IT class of estimators. These members com- pose the sub-class of methods that incorporate the moment restrictions within the generic IT-estimator as (pure) zero moments’ conditions, and include the empirical likelihood, the generalized empirical likeli- hood, the generalized method of moments and the Bayesian method of moments. The connection between each one of these methods, the basics of information theory and the maximum entropy principle is discussed. In Section 6, we provide a thorough discussion of the other sub-class of IT estimators: the one that views the sample moments as stochastic. This sub-class is also known as the generalized maximum entropy. The relevant statistics and information measures are summa- rized and connected to quantities studied earlier in the survey. We conclude with a simple simulated example. In Section 7, we provide a synthesis of likelihood, ME and other IT estimators, via an example. We study the interconnections among these estimators and show that though coming from diﬀerent philosophies they are deeply rooted in each other, and understanding that interconnection allows us to under- stand our data better. In Section 8, we summarize related topics within IEE that are not discussed in this survey. Readers of this survey need basic knowledge of econometrics, but do not need prior knowledge of information theory. Those who are familiar Full text available at: http://dx.doi.org/10.1561/0800000004 Preface xi with the concepts of IT should skip Section 3, except Section 3.4 which is necessary for the next few sections. Those who are familiar with the ME principle can skip parts of Section 4, but may want to read the example in Section 4.7. The survey is self contained and interested readers can replicate all results and examples provided. No detailed proofs are provided, though the logic behind some less familiar argu- ments is provided. Whenever necessary the readers are referred to the relevant literature. This survey may beneﬁt researchers who wish to have a fast introduc- tion to the basics of IEE and to acquire the basic tools necessary for using and understanding these methods. The survey will also bene- ﬁt applied researchers who wish to learn improved new methods, and applications, for extracting information from noisy and limited data and for learning from these data. Full text available at: http://dx.doi.org/10.1561/0800000004 Contents 1 Introductory Statement, Motivation, and Objective 1 2 Historical Perspective 7 3 Information and Entropy — Background, Deﬁnitions, and Examples 15 3.1 Information and Entropy — The Basics 15 3.2 Discussion 26 3.3 Multiple Random Variables, Dependency, and Joint Entropy 28 3.4 Generalized Entropy 32 3.5 Axioms 34 3.6 Errors and Entropy 36 3.7 Asymptotics 39 3.8 Stochastic Process and Entropy Rate 45 3.9 Continuous Random Variables 46 4 The Classical Maximum Entropy Principle 49 4.1 The Basic Problem and Solution 49 4.2 Duality — The Concentrated Model 51 xiii Full text available at: http://dx.doi.org/10.1561/0800000004 xiv Contents 4.3 The Entropy Concentration Theorem 55 4.4 Information Processing Rules and Eﬃciency 57 4.5 Entropy — Variance Relationship 58 4.6 Discussion 59 4.7 Example 61 5 Information-Theoretic Methods of Estimation — I 63 5.1 Background 63 5.2 The Generic IT Model 64 5.3 Empirical Likelihood 67 5.4 Generalized Method of Moments — A Brief Discussion 72 5.5 Bayesian Method of Moments — A Brief Discussion 81 5.6 Discussion 83 6 Information-Theoretic Methods of Estimation — II 85 6.1 Generalized Maximum Entropy — Basics 85 6.2 GME — Extensions: Adding Constraints 93 6.3 GME — Entropy Concentration Theorem and Large Deviations 94 6.4 GME — Inference and Diagnostics 96 6.5 GME — Further Interpretations and Motivation 99 6.6 Numerical Example 103 6.7 Summary 105 7 IT, Likelihood and Inference 107 7.1 The Basic Problem — Background 107 7.2 The IT-ME Solution 109 7.3 The Maximum Likelihood Solution 111 7.4 The Generalized Case — Stochastic Moments 112 7.5 Inference and Diagnostics 114 Full text available at: http://dx.doi.org/10.1561/0800000004 Contents xv 7.6 IT and Likelihood 117 7.7 Conditional, Time Series Markov Process 118 8 Concluding Remarks and Related Work Not Surveyed 131 References 137 Full text available at: http://dx.doi.org/10.1561/0800000004 Introductory Statement, Motivation, and Objective All learning, information gathering and information processing, is based on limited knowledge, both a priori and data, from which a larger “truth” must be inferred. To learn about the true state of the world that generated the observed data, we use statistical models that repre- sent these outcomes as functions of unobserved structural parameters, parameters of priors and other sampling distributions, as well as com- plete probability distributions. Since we will never know the true state of the world, we generally focus, in statistical sciences, on recovering information about the complete probability distribution, which repre- sents the ultimate truth in our model. Therefore, all estimation and inference problems are translations of limited information about the probability density function (pdf) toward a greater knowledge of that pdf. However, if we knew all the details of the true mechanism then we would not need to resort to the use of probability distributions to capture the perceived uncertainty in outcomes that results from our ignorance of the true underlying mechanism that controls the event of interest. Information theory quantities, concepts, and methods provide a uni- ﬁed set of tools for organizing this learning process. They provide a 1 Full text available at: http://dx.doi.org/10.1561/0800000004 2 Introductory Statement, Motivation, and Objective discipline that at once exposes more clearly what the existing methods do, and how we might better accomplish the main goal of scientiﬁc learning. This review ﬁrst studies the basic quantities of information theory and their relationships to data analysis and information pro- cessing, and then uses these quantities to summarize (and understand the connection among) the improved methods of estimation and data processing that compose the class of entropy and information-theoretic methods. Within that class, the review concentrates on methods that use conditional and unconditional stochastic moments. It seems natural to start by asking what is information, and what is the relationship between information and econometric, or statistical analysis. Consider, for example, Shakespeare’s “Hamlet,” Dostoevsky’s “The Brothers Karamazov,” your favorite poem, or the US Consti- tution. Now think of some economic data describing the demand for education, or survey data arising from pre-election polls. Now consider a certain speech pattern or communication among individuals. Now imagine you are looking at a photo or an image. The image can be sharp or blurry. The survey data may be easy to understand or extremely noisy. The US Constitution is still being studied and analyzed daily with many interpretations for the same text, and your favorite poem, as short as it may be, may speak a whole world to you, while disliked by others. Each of these examples can be characterized by the amount of information it contains or by the way this information is conveyed or understood by the observer — the analyst, the reader. But what is information? What is the relationship between information and econo- metric analysis? How can we eﬃciently extract information from noisy and evolving complex observed economic data? How can we guarantee that only the relevant information is extracted? How can we assess that information? The study of these questions is the subject of this survey and synthesis. This survey discusses the concept of information as it relates to econometric and statistical analyses of data. The meaning of “informa- tion” will be studied and related to the basics of Information Theory (IT) as is viewed by economists and researchers who are engaged in deciphering information from the (often complex and evolving) data, Full text available at: http://dx.doi.org/10.1561/0800000004 while taking into account what they know about the underlying process that generated these data, their beliefs about the (economic) system under investigation, and nothing else. In other words, the researcher wishes to extract the available information from the data, but wants to do it with minimal a priori assumptions. For example, consider the fol- lowing problem taken from Jaynes’s famous Brandeis lectures (1963). We know the empirical mean value (ﬁrst moment) of, say one million tosses of a six-sided die. With that information the researcher wishes to predict the probability that in the next throw of the die we will observe the value 1, 2, 3, 4, 5 or 6. The researcher also knows that the probability is proper (sum of the probabilities is one). Thus, in that case, there are six values to predict (six unknown values) and two observed (known) values: the mean and the sum of the probabilities. As such, there are more unknown quantities than known quantities, meaning there are inﬁnitely many probability distributions that sum up to one and satisfy the observed mean. In somewhat more general terms, consider the problem of estimating an unknown discrete prob- ability distribution from a ﬁnite and possibly noisy set of observed (sample) moments. These moments (and the fact that the distribu- tion is proper — summing up to one) are the only available informa- tion. Regardless of the level of noise in these observed moments, if the dimension of the unknown distribution is larger than the number of observed moments, there are inﬁnitely many proper probability distri- butions satisfying this information (the moments). Such a problem is called an under-determined problem. Which one of the inﬁnitely many solutions should one use? In all the IEE methods, the one solution cho- sen is based on an information criterion that is related to Shannon’s information measure — entropy. When analyzing a linear regression, a jointly determined system of equations, a ﬁrst-order Markov model, a speech pattern, a blurry image, or even a certain text, if the researcher wants to understand the data but without imposing a certain structure that may be incon- sistent with the (unknown) truth, the problem may become inherently under-determined. The criterion used to select the desired solution is an information criterion which connects statistical estimation and infer- ence with the foundations of IT. This connection provides us with an Full text available at: http://dx.doi.org/10.1561/0800000004 4 Introductory Statement, Motivation, and Objective IT perspective of econometric analyses and reveals the deep connection among these “seemingly distinct” disciplines. This connection gives us the additional tools for a better understanding of our limited data, and for linking our theories with real observed data. In fact, information theory and data analyses are the major thread connecting most of the scientiﬁc studies trying to understand the true state of the world with the available, yet limited and often noisy, information. Within the econometrics and statistical literature the family of IT estimators composes the heart of IEE. It includes the Empirical (and Generalized Empirical) Likelihood, the Generalized Method of Moments, the Bayesian Method of Moments and the Generalized Max- imum Entropy among others. In all of these cases the objective is to extract the available information from the data with minimal assump- tions on the data generating process and on the likelihood structure. The logic for using minimal assumptions in the IEE class of estimators is that the commonly observed data sets in the social sciences are often small, the data may be non-experimental noisy data, the data may be arising from a badly designed experiment, and the need to work with nonlinear (macro) economic models where the maximum likelihood esti- mator is unattractive as it is not robust to the underlying (unknown) distribution. Therefore, (i) such data may be ill-behaved leading to an ill-posed and/or ill-conditioned (not full rank) problem, or (ii) the underlying economic model does not specify the complete distribution of the data, but the economic model allows us to place restrictions on this (unknown) distribution in the form of population moment condi- tions that provide information on the parameters of the model. For these estimation problems and/or small and non-experimental data it seems logical to estimate the unknown parameters with minimum a priori assumptions on the data generation process, or with minimal assumptions on the likelihood function. Without a pre-speciﬁed likeli- hood, other non maximum likelihood methods must be used in order to extract the available information from the data. Many of these methods are members of the class of Information-Theoretic (IT) methods. 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