Bayesian econometrics (advances in econometrics)

Trivedi, ‘‘A Bayesian Analysis of the OPES Modelwith a Nonparametric Component: An Application to Dental Insurance andDental Care,’’ is a good example of how Bayesian methods are increas

ADVANCES IN ECONOMETRICS

Series Editors: Thomas B Fomby and R Carter Hill

Recent Volumes:

Volume 15: Nonstationary Panels, Panel Cointegration,

and Dynamic Panels, Edited by Badi Baltagi Volume 16: Econometric Models in Marketing, Edited

by P H Franses and A L Montgomery Volume 17: Maximum Likelihood Estimation of

Misspecified Models: Twenty Years Later, Edited by Thomas B Fomby and

R Carter Hill Volume 18: Spatial and Spatiotemporal Econometrics,

Edited by J P LeSage and R Kelley Pace Volume 19: Applications of Artificial Intelligence in

Finance and Economics, Edited by

J M Binner, G Kendall and S H Chen Volume 20A: Econometric Analysis of Financial and

Economic Time Series, Edited by Dek Terrell and Thomas B Fomby Volume 20B: Econometric Analysis of Financial and

Economic Time Series, Edited by Thomas B Fomby and Dek Terrell Volume 21: Modelling and Evaluating Treatment Effects

in Econometrics, Edited by Daniel L met, Jeffrey A Smith and Edward J Vytlacil Volume 22: Econometrics and Risk Management,

Milli-Edited by Thomas B Fomby, Knut Solna and Jean-Pierre Fouque

Department of Economics, Louisiana State University

United Kingdom – North America – Japan

India – Malaysia – China

JAI Press is an imprint of Emerald Group Publishing Limited

Howard House, Wagon Lane, Bingley BD16 1WA, UK

First edition 2008

Copyright r 2008 Emerald Group Publishing Limited

Reprints and permission service

Contact: booksandseries@emeraldinsight.com

No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA

by The Copyright Clearance Center No responsibility is accepted for the accuracy of information contained in the text, illustrations or advertisements The opinions expressed

in these chapters are not necessarily those of the Editor or the publisher.

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN: 978-1-84855-308-8

ISSN: 0731-9053 (Series)

Awarded in recognition of Emerald’s production department’s adherence to quality systems and processes when preparing scholarly

Michael K Andersson Sveriges Riksbank, Stockholm, Sweden

Crete, Rethymno, Greece

Kowloon, Hong Kong, China

University, Taiwan

University, St Louis, MO

and Econometrics, University of Sydney, NSW, Australia

Financial Markets, Board of Governors

of the Federal Reserve System, Washington, DC

Connecticut, Storrs, CT

Leicester, Leicester, UK

and Econometrics, University of Sydney, NSW, Australia

Riksbank, Stockholm, Sweden

California, Irvine, CAix

William Griffiths Department of Economics, University of

Melbourne, Vic., Australia

University, Ames, IA

Melbourne, Vic., Australia

California, Irvine, CA

University, Ames, IA

University, O¨rebo, Sweden

Australian School of Business, University of New South Wales, Sydney, Australia

Strathclyde, Glasgow, UK

Strathclyde, Glasgow, UK

University of New York, Binghamton, NY

California, Irvine, CA

Studies (GRIPS), Tokyo, Japan

Universite´ de Montre´al, CIREQ, Canada

Statistics, University of Tampere, Tampere, Finland

LIST OF CONTRIBUTORSx

Jani Luoto School of Business and Economics,

University of Jyva¨skyla¨, Jyva¨skyla¨, Finland

Universite´ de Montre´al, CIREQ and CIRANO, Montre´al, QC, Canada

University of the West Indies, Mona, Kingston, Jamaica

South Florida, Tampa, FL

Queensland, Brisbane, Australia

Queensland, Brisbane, Australia

Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands

University of Science and Technology, Kowloon, Hong Kong

Queensland, Brisbane, Australia

Connecticut, Storrs, CT

State University, Baton Rouge, LA

University, West Lafayette, IN

Indiana University, Bloomington, IN

Efthymios G Tsionas Department of Economics, Athens

University of Economics and Business, Athens, Greece

Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands

University of Sydney, NSW, Australia

of Chicago, Chicago, IL

LIST OF CONTRIBUTORSxii

Siddhartha Chib, William Griffiths, Gary Koop and Dek Terrell

ABSTRACTBayesian Econometrics is a volume in the series Advances in Econometricsthat illustrates the scope and diversity of modern Bayesian econometricapplications, reviews some recent advances in Bayesian econometrics, andhighlights many of the characteristics of Bayesian inference andcomputations This first paper in the volume is the Editors’ introduction

in which we summarize the contributions of each of the papers

1 INTRODUCTION

In 1996 two volumes of Advances in Econometrics were devoted to Bayesianeconometrics One was on computational methods and applications and theother on time-series applications This was a time when Markov chain MonteCarlo (MCMC) techniques, which have revolutionized applications ofBayesian econometrics, had started to take hold The adaptability of MCMC

to problems previously considered too difficult was generating a revival ofinterest in the Bayesian paradigm Now, 12 years later, it is time for anotherAdvancesvolume on Bayesian econometrics Use of Bayesian techniques has

Bayesian Econometrics

Advances in Econometrics, Volume 23, 3–9

Copyright r 2008 by Emerald Group Publishing Limited

All rights of reproduction in any form reserved

ISSN: 0731-9053/doi: 10.1016/S0731-9053(08)23021-5

3

become widespread across all areas of empirical economics Previouslyintractable problems are being solved and more flexible models are beingintroduced The purpose of this volume is to illustrate today’s scope anddiversity of Bayesian econometric applications, to review some of the recentadvances, and to highlight various aspects of Bayesian inference andcomputations.

The book is divided into three parts In addition to this introduction, Part Icontains papers by Arnold Zellner, and by Paolo Giordani and Robert Kohn

In his paper ‘‘Bayesian Econometrics: Past, Present, and Future,’’ ArnoldZellner reviews problems faced by the Federal Reserve System, as described

by its former chairman, Alan Greenspan, and links these problems to asummary of past and current Bayesian activity Some key contributions to thedevelopment of Bayesian econometrics are highlighted Future researchdirections are discussed with a view to improving current econometricmodels, methods, and applications of them

The other paper in Part I is a general one on a computational strategy forimproving MCMC Under the title ‘‘Bayesian Inference using AdaptiveSampling,’’ Paolo Giordani and Robert Kohn discuss simulation-basedBayesian inference methods that draw on information from previous samples

to build the proposal distributions in a given family of distributions Thearticle covers approaches along these lines and the intuition behind some ofthe theory for proving that the procedures work They also discuss strategiesfor making adaptive sampling more effective and provide illustrations forvariable selection in the linear regression model and for time-series modelssubject to interventions

2 MICROECONOMETRIC MODELING

Part II of the book, entitled ‘‘Microeconometric Modeling’’ containsapplications that use cross-section or panel data The paper by Murat K.Munkin and Pravin K Trivedi, ‘‘A Bayesian Analysis of the OPES Modelwith a Nonparametric Component: An Application to Dental Insurance andDental Care,’’ is a good example of how Bayesian methods are increasinglybeing used in important empirical work The empirical focus is on the impact

of dental insurance on the use of dental services Addressing this issue iscomplicated by the potential endogeneity of insurance uptake and the factthat insurance uptake may depend on explanatory variables in a nonlinearfashion The authors develop an appropriate model which addresses boththese issues and carry out an empirical analysis which finds strong evidence

SIDDHARTHA CHIB ET AL.4

that having dental insurance encourages use of dentists, but also of adverseselection into the insured state.

MCMC simulation techniques are particularly powerful in discrete-datamodels with latent variable representations In their paper ‘‘Fitting andComparison of Models for Multivariate Ordinal Outcomes,’’ Ivan Jeliazkov,Jennifer Graves, and Mark Kutzbach review several alternative modelingand identification schemes for ordinal data models and evaluate how eachaids or hampers estimation using MCMC Model comparison via marginallikelihoods and an analysis of the effects of covariates on category probabili-ties is considered for each parameterization The methods are applied toexamples in educational attainment, voter opinions, and consumers’ reliance

on alternative sources of medical information

In ‘‘Intra-Household Allocation and Consumption of WIC-ApprovedFoods: A Bayesian Approach,’’ Ariun Ishdorj, Helen H Jensen, and JustinTobias consider the Special Supplemental Nutrition Program for Women,Infants, and Children (WIC) that aims to provide food, nutrition education,and other services to at-risk, low-income children and pregnant, breastfeed-ing, and postpartum women They assess the extent to which the WICprogram improves the nutritional outcomes of WIC families as a whole,including the targeted and nontargeted individuals within the household.This question is considered under the possibility that participation in theprogram (which is voluntary) is endogenous They develop an appropriatetreatment–response model and conclude that WIC participation does notlead to increased levels of calcium intake from milk

A second paper that illustrates the use of Bayesian techniques for analyzingtreatment–response problems is that by Siddhartha Chib and Liana Jacobi

In their paper ‘‘Causal Effects from Panel Data in Randomized Experimentswith Partial Compliance,’’ the authors describe how to calculate the causalimpacts from a training program when noncompliance exists in the trainingarm Two primary models are considered, with one model including arandom effects specification Prior elicitation is carefully done by simulatingfrom a prior predictive density on outcomes, using a hold out sample.Estimation and model comparison are considered in detail The methods areemployed to assess the impact of a job training program on mental healthscores

Basic equilibrium job search models often yield wage densities that do notaccord well with empirical regularities When extensions to basic models aremade and analyzed using kernel-smoothed nonparametric forms, it is difficult

to assess these extensions via model comparisons In ‘‘Parametric andNonparametric Inference in Equilibrium Job Search Models,’’ Gary Koop

develops Bayesian parametric and nonparametric methods that are able to those in the existing non-Bayesian literature He then shows howBayesian methods can be used to compare the different parametric andnonparametric equilibrium search models in a statistically rigorous sense.

compar-In the paper ‘‘Do Subsidies Drive Productivity? A Cross-Country Analysis

of Nordic Dairy Farms,’’ Nadine McCloud and Subal C Kumbhakardevelop a Bayesian hierarchical model of farm production which allows forthe calculation of input productivity, efficiency, and technical change Thekey research questions relate to whether and how these are influenced bysubsidies Using a large panel of Nordic dairy farms, they find that subsidiesdrive productivity through technical efficiency and input elasticities,although the magnitude of these effects differs across countries

The richness of available data and the scope for building flexible modelsmakes marketing a popular area for Bayesian applications In ‘‘Semipara-metric Bayesian Estimation of Random Coefficients Discrete ChoiceModels,’’ Sylvie Tchumtchoua and Dipak K Dey propose a semiparametricBayesian framework for the analysis of random coefficients discrete choicemodels that can be applied to both individual as well as aggregate data.Heterogeneity is modeled using a Dirichlet process prior which (importantly)varies with consumer characteristics through covariates The authors employ

a MCMC algorithm for fitting their model, and illustrate the methodologyusing a household level panel dataset of peanut butter purchases, andsupermarket chain level data for 31 ready-to-eat breakfast cereals brands.When diffuse priors are used to estimate simultaneous equation models,the resulting posterior density can possess infinite asymptotes at points oflocal nonidentification Kleibergen and Zivot (2003) introduced a prior toovercome this problem in the context of a restricted reduced formspecification, and investigated the relationship between the resultingBayesian estimators and their classical counterparts Arto Luoma and JaniLuoto, in their paper ‘‘Bayesian Two-Stage Regression with ParametricHeteroscedasticity,’’ extend the analysis of Kleibergen and Zivot to asimultaneous equation model with unequal error variances They apply theirtechniques to a cross-country Cobb–Douglas production function

3 TIME-SERIES MODELINGPart III of the volume is devoted to models and applications that use time-series data The first paper in this part is ‘‘Bayesian Near-Boundary Analysis

in Basic Macroeconomic Time-Series Models’’ by Michiel D de Pooter,

SIDDHARTHA CHIB ET AL.6

Francesco Ravazzolo, Rene Segers, and Herman K van Dijk The boundaryissues considered by these authors are similar to that encountered by ArtoLuoma and Jani Luoto in their paper There are a number of models wherethe use of particular types of noninformative priors can lead to improperposterior densities with estimation breaking down at boundary values ofparameters The circumstances under which such problems arise, and howthe problems can be solved using regularizing or truncated priors, areexamined in detail by de Pooter et al in the context of dynamic linearregression models, autoregressive and error correction models, instrumentalvariable models, variance component models, and state space models.Analytical, graphical, and empirical results using U.S macroeconomic dataare presented.

In his paper ‘‘Forecasting in Vector Autoregressions with ManyPredictors,’’ Dimitris Korobilis introduces Bayesian model selection methods

in a VAR setting, focusing on the problem of drawing inferences from adataset with a very large number of potential predictors A stochastic searchvariable selection algorithm is used to implement Bayesian model selection

An empirical application using 124 potential predictors to forecast eight U.S.macroeconomic variables is included to demonstrate the methodology.Results indicate an improvement in forecasting accuracy over modelselection based on the Bayesian Information Criteria

In ‘‘Bayesian Inference in a Cointegrating Panel Data Model,’’ GaryKoop, Robert Leon-Gonzalez, and Rodney Strachan focus on cointegration

in the context of a cointegrating panel data model Their approach allowsboth short-run dynamics and the cointegrating rank to vary across cross-sectional units In addition to an uninformative prior, they propose aninformative prior with ‘‘soft homogeneity’’ restrictions This informativeprior can be used to include information from economic theory that cross-sectional units are likely to share the same cointegrating rank without forcingthat assumption on the data Empirical applications using simulated dataand a long-run model for bilateral exchange rates are used to demonstratethe methodology

Cointegration is also considered by Deborah Gefang who develops tests ofpurchasing power parity (PPP) within an exponential smooth transition(ESVECM) framework The Bayesian approach offers a substantialmethodological advantage in this application because the Gibbs samplingscheme is not affected by the multi-mode problem created by nuisanceparameters Results based on Bayesian model averaging and Bayesian modelselection find evidence that PPP holds between the United States and each ofthe remaining G7 countries

‘‘Bayesian Forecast Combination for VAR Models’’ by Michael K.Andersson and Sune Karlsson addresses the issue of how to forecast avariable (or variables) of interest (e.g., GDP) when there is uncertainty aboutthe dimension of the VAR and uncertainty about which set of explanatoryvariables should be used This uncertainty leads to a huge set of models Theauthors do model averaging over the resulting high-dimensional model spaceusing predictive likelihoods as weights For forecast horizons greater thanone, the predictive likelihoods will not have analytical forms and the authorsdevelop a simulation method for estimating them An empirical analysisinvolving U.S GDP shows the benefits of their approach.

In ‘‘Bayesian Inference on Time-Varying Proportions,’’ William J.McCausland and Brahim Lgui derive a highly efficient algorithm forsimulating the states in state space models where the dependent variables areproportions The authors argue in favor of a model which is parameterizedsuch that the measurement equation has the proportions (conditional on thestates) following a Dirichlet distribution, but the state equation is a standardlinear Gaussian one The authors develop a Metropolis–Hastings algorithmwhich draws states as a block from a multivariate Gaussian proposaldistribution Extensive empirical evidence indicates that their approachworks well and, in particular, is very efficient

Christopher J O’Donnell and Vanessa Rayner use Bayesian methodology

to impose inequality restrictions on ARCH and GARCH models in theirpaper ‘‘Imposing Stationarity Constraints on the Parameters of ARCH andGARCH Models.’’ Bayesian model averaging is used to resolve uncertaintywith regard to model selection The authors apply the methodology to datafrom the London Metals Exchange and find that results are generallyinsensitive to the imposition of inequality restrictions

In ‘‘Bayesian Model Selection for Heteroskedastic Models,’’ Cathy W S.Chen, Richard Gerlach, and Mike K P So discuss Bayesian model selectionfor a wide variety of financial volatility models that exhibit asymmetries (e.g.,threshold GARCH models) Model selection problems are complicated bythe fact that there are many contending models and marginal likelihoodcalculation can be difficult They discuss this problem in an empiricalapplication involving daily data from three Asian stock markets andcalculate the empirical support for their competing models

Using a scale mixture of uniform densities representation of the Student-tdensity, S T Boris Choy, Wai-yin Wan, and Chun-man Chan provide aBayesian analysis of a Student-t stochastic volatility model in ‘‘BayesianStudent-t Stochastic Volatility Models via Scale Mixtures.’’ They develop aGibbs sampler for their model and show how their approach can be extended

SIDDHARTHA CHIB ET AL.8

to the important class of Student-t stochastic volatility models with leverage.The different models are fit to returns on exchange rates of the Australiandollar against 10 currencies.

In ‘‘Bayesian Analysis of the Consumption CAPM,’’ Veni Arakelian andEfthymios G Tsionas show thatLabadie’s (1989)solution to the CAPM can

be applied to obtain a closed form solution and to provide a traditionaleconometric interpretation They then apply Bayesian inference to bothsimulated data and theMehra and Prescott (1985)dataset Results generallyconform to theory, but also reveal asymmetric marginal densities for keyparameters The asymmetry suggests that techniques such as generalizedmethod of moments, which rely on asymptotical approximations, may beunreliable

BAYESIAN ECONOMETRICS: PAST, PRESENT, AND FUTURE

Arnold Zellner

ABSTRACT

After briefly reviewing the past history of Bayesian econometrics and AlanGreenspan’s (2004) recent description of his use of Bayesian methods inmanaging policy-making risk, some of the issues and needs that hementions are discussed and linked to past and present Bayesianeconometric research Then a review of some recent Bayesian econometricresearch and needs is presented Finally, some thoughts are presented thatrelate to the future of Bayesian econometrics

1 INTRODUCTION

In the first two sentences of her paper, ‘‘Bayesian Econometrics, The FirstTwenty Years,’’ Qin (1996) wrote, ‘‘Bayesian econometrics has been acontroversial area in the development of econometric methodology Althoughthe Bayesian approach has been constantly dismissed by many mainstreameconometricians for its subjectivism, Bayesian methods have been adoptedwidely in current econometric research’’ (p 500) This was written more than

10 years ago Now more mainstream econometricians and many others haveadopted the Bayesian approach and are using it to solve a broad range of

Bayesian Econometrics

Advances in Econometrics, Volume 23, 11–60

Copyright r 2008 by Emerald Group Publishing Limited

All rights of reproduction in any form reserved

ISSN: 0731-9053/doi: 10.1016/S0731-9053(08)23001-X

11

econometric problems in line with my forecast inZellner (1974), ‘‘Further, itmust be recognized that the B approach is in a stage of rapid developmentwith work going ahead on many new problems and applications While this isrecognized, it does not seem overly risky to conclude that the B approach,which already has had some impact on econometric work, will have a muchmore powerful influence in the next few years’’ (p 54).

See also,Zellner (1981, 1988b, 1991, 2006)for more on the past, present,and future of Bayesian econometrics in which it is emphasized that alleconometricians use and misuse prior information, subjectively, objectively,

or otherwise And it has been pointed out that Bayesian econometricianslearn using an explicit model, Bayes’ Theorem that allows prior information

to be employed in a formal and reproducible manner whereas non-Bayesianeconometricians learn in an informal, subjective manner For empiricalevidence on the rapid growth of Bayesian publications over the years ineconomics and other fields that will be discussed below see Poirier (1989,

1992, 2004) and Poirier (1991) for an interesting set of Bayesian empiricalpapers dealing with problems in economics and finance

In the early 1990s, both the International Society for Bayesian Analysis(http://www.bayesian.org) and the Section on Bayesian Statistical Science ofthe American Statistical Association (http://www.amstat.org) were formedand have been very active and successful in encouraging the growth ofBayesian theoretical and applied research and publications Similarly, theNBER-NSF Seminar on Bayesian Inference in Econometrics and Statistics(SBIES) that commenced operation in 1970, has been effective for many years

in sponsoring research meetings, publishing a number of Bayesian books andactively supporting the creation of ISBA and SBSS in the early 1990s InBerry, Chaloner, and Geweke (1996), some history of the SBIES and a largenumber of Bayesian research papers are presented Also, under the currentleadership of Sid Chib, very productive meetings of this seminar in 2004 and

2005 have been held that were organized by him and John Geweke In August

2006, the European–Japanese Bayesian Workshop held a meeting in Viennaorganized by Wolfgang Polasek that had a very interesting program In 2005,the Indian Bayesian Society and the Indian Bayesian Chapter of ISBA had aninternational Bayesian meeting at Varanasi with many of the paperspresented that have appeared in a conference volume In September 2006, aBayesian research meeting was held at the Royal Bank of Sweden, organized

by Mattias Villani that attracted leading Bayesian econometricians from allover the world to present reports on their current work on Bayesianeconometric methodology And now, this Advances in Econometrics volumefeatures additional valuable Bayesian econometric research And last, but not

least, the International Society for Bayesian Analysis has commencedpublication of an online Bayesian journal called Bayesian Analysis; seehttp://www.bayesian.org for more information about this journal with

R Kass the founding editor and listings of articles for several years thatare downloadable These and many more Bayesian activities that have takenplace over the years attest to the growth and vitality of Bayesian analysis inmany sciences, industries, and governments worldwide

1.1 An Example of Bayesian Monetary Policy-Making

As an example of extremely important work involving the use of Bayesianmethodology and analysis, Alan Greenspan, former Chairman of the U.S.Federal Reserve System presented an invited paper, ‘‘Risk and Uncertainty inMonetary Policy’’ at the 2004 Meeting of the American EconomicAssociation that was published in the American Economic Review in 2004along with very knowledgeable discussion by Martin Feldstein, HarvardProfessor of Economics and President of the National Bureau of EconomicResearch, Mervyn King of the Bank of England, and Professor Janet L.Yellen of the Haas School of Business, University of California, Berkeley.The paper is notable in that it presents a comprehensive description of theways in which he approached and solved monetary policy problems ‘‘ fromthe perspective of someone who has been in the policy trenches’’ (p 33).Greenspan’s account should be of interest to Bayesians econometriciansand many others since he states, ‘‘In essence, the risk management approach

to policymaking is an application of Bayesian decision-making’’ (p 37) Inaddition, he writes, ‘‘Our problem is not, as is sometimes alleged, thecomplexity of our policy-making process, but the far greater complexity of aworld economy whose underlying linkages appear to be continuously evol-ving Our response to that continuous evolution has been disciplined by theBayesian type of decision-making in which we have been engaged’’ (p 39).Feldstein (2004), after providing an excellent review of Greenspan’ssuccessful policy-making in the past wrote, ‘‘Chairman Greenspan empha-sized that dealing with uncertainty is the essence of making monetary policy(see also Feldstein, 2002) The key to what he called the risk-managementapproach to monetary policy is the Bayesian theory of decision-making’’(p 42) After providing a brief, knowledgeable description of Bayesiandecision theory, Feldstein provides the following example to illustrate a case

of asymmetric loss in connection with a person making a decision whether tocarry an umbrella when the probability of rain is not high ‘‘If he carries the

umbrella and it does not rain, he is mildly inconvenienced But if he does notcarry the umbrella and it rains, he will suffer getting wet A good Bayesianfinds himself carrying an umbrella on many days when it does not rain Thepolicy actions of the past year were very much in this spirit The Fed cut theinterest rate to 1 percent to prevent the low-probability outcome of spiralingdeflation because it regarded that outcome as potentially very damagingwhile the alternative possible outcome of a rise of the inflation rate from 1.5percent to 2.5 percent was deemed less damaging and more easily reversed’’(p 42).

Mervyn King of the Bank of England commented knowingly about modelquality and policy-making, ‘‘Greenspan suggests that the risk-managementapproach is an application of Bayesian decision-making when there isuncertainty about the true model of the economy Policy that is optimal inone particular model of the economy may not be ‘robust’ across a class ofother models In fact, it may lead to a very bad outcome should analternative model turn out to be true Of course, although such anapproach is sensible, it is still vulnerable to policymakers giving excessiveweight to misleading models of the economy But, in the end, there is noescaping the need to make judgments about which models are more plausiblethan others’’ (pp 42–43) These are indeed very thoughtful remarks aboutproblems of model uncertainty in making policy but do not recognize thatjust as with Feldstein’s umbrella example above, a Bayesian analysis canutilize posterior probabilities associated with alternative models that reflectthe quality of past performance that have been shown to be useful inproducing useful combined forecasts and probably will be helpful in dealingwith model uncertainty in policy-making

1.2 Greenspan’s Policy-Making Problems

Below, I list and label important problems that Greenspan mentioned inconnection with his successful policy-making over the years that reveal hisdeep understanding of both obvious and very subtle problems associatedwith model-building, economic analyses, forecasting, and policy-making

1 Structural changes: For example, ‘‘ increased political support forstable prices, globalization which unleashed powerful new forces ofcompetition, and an acceleration of productivity which at least for a timeheld down cost pressures’’ (p 33) ‘‘I believe that we at the Fed, to ourcredit, did gradually come to recognize the structural economic changesthat we were living through and accordingly altered our understanding

of the key parameters of the economic system and our policy stance But as we lived through it, there was much uncertainty about theevolving structure of the economy and about the influence of monetarypolicy’’ (p 33).

2 Forecasting: ‘‘In recognition of the lag in monetary policy’s impact oneconomic activity, a preemptive response to the potential for buildinginflationary pressures was made an important feature of policy As aconsequence, this approach elevated forecasting to an even moreprominent place in policy deliberations’’ (p 33)

3 Unintended consequences: ‘‘Perhaps the greatest irony of the past decade

is that the gradually unfolding success against inflation may well havecontributed to the stock price bubble of the latter part of the1990s The sharp rise in stock prices and their subsequent fall were,thus, an especial challenge to the Federal Reserve’’ (p 35)

‘‘The notion that a well-timed incremental tightening could have beencalibrated to prevent the late 1990s bubble while preserving economicstability is almost surely an illusion Instead of trying to contain aputative bubble by drastic actions with largely unpredictable conse-quences, we chose to focus on policies to mitigate the fallout when itoccurs and, hopefully, ease the transition to the next expansion’’ (p 36)

4 Uncertainty: ‘‘The Federal Reserve’s experiences over the past twodecades make it clear that uncertainty is not just a pervasive feature ofthe monetary landscape; it is the defining characteristic of thatlandscape The term ‘‘uncertainty’’ is meant here to encompass both

‘Knightian uncertainty,’ in which the probability distribution ofoutcomes is unknown, and ‘risk,’ in which uncertainty of outcomes isdelimited by a known probability distribution In practice, one is neverquite sure what type of uncertainty one is dealing with in real time, and itmay be best to think of a continuum ranging from well-defined risks tothe truly unknown’’ (pp 36–37)

5 Risk management: ‘‘As a consequence, the conduct of monetary policy inthe United States has come to involve, at its core, crucial elements of riskmanagement This conceptual framework emphasizes understanding asmuch as possible the many sources of risk and uncertainty thatpolicymakers face, quantifying those risks, when possible, and assessingcosts associated with each of the risks In essence, the risk-managementapproach to monetary policymaking is an application of Bayesiandecision-making’’ (p 37)

6 Objectives: ‘‘This [risk management] framework also entails devising, inlight of those risks, a strategy for policy directed at maximizing the

probabilities of achieving over time our goals of price stability and themaximum sustainable economic growth that we associate with it’’ (p 37).

7 Expert opinion: ‘‘In designing strategies to meet our policy objectives, wehave drawn on the work of analysts, both inside and outside the Fed,who over the past half century have devoted much effort to improvingour understanding of the economy and its monetary transmissionmechanism’’ (p 37)

8 Model uncertainty: ‘‘A critical result [of efforts to improve ourunderstanding of the economy and its monetary transmission mechan-ism] has been the identification of a relatively small set of keyrelationships that, taken together, provide a useful approximation ofour economy’s dynamics Such an approximation underlies the statisticalmodels that we at the Federal Reserve employ to assess the likelyinfluence of our policy decisions

However, despite extensive efforts to capture and quantify what weperceive as the key macroeconomic relationships, our knowledge aboutmany of the important linkages is far from complete and, in all likelihoodwill always remain so Every model, no matter how detailed or how welldesigned, conceptually and empirically, is a vastly simplified representa-tion of the world that we experience with all its intricacies on a day-to-day basis’’ (p 37)

9 Loss structures: ‘‘Given our inevitably incomplete knowledge about keystructural aspects of an ever-changing economy and the sometimesasymmetric costs or benefits of particular outcomes, a central bankneeds to consider not only the most likely future path for the economy,but also the distribution of possible outcomes about that path Thedecision-makers then need to reach a judgment about the probabilities,costs and benefits of the various possible outcomes under alternativechoices for policy’’ (p 37)

10 Robustness of policy: ‘‘In general, different policies will exhibit differentdegrees of robustness with respect to the true underlying structure of theeconomy’’ (p 37)

11 Cost–benefit analysis: ‘‘As this episode illustrates, policy practitionersoperating under a risk-management paradigm may, at times, be led toundertake actions intended to provide insurance against [low prob-ability] especially adverse outcomes The product of a low-probability event and a potentially severe outcome was judged a moreserious threat to economic performance than the higher inflation thatmight ensue in the more probable scenario’’ (p 37)

12 Knightian uncertainty: ‘‘When confronted with uncertainty, especiallyKnightian uncertainty, human beings invariably attempt to disengagefrom medium- to long-term commitments in favor of safety andliquidity Because economies, of necessity, are net long (that is, have netreal assets) attempts to flee these assets causes prices of equity assets tofall, in some cases dramatically The immediate response on the part

of the central bank to such financial implosions must be to inject largequantities of liquidity ’’ (p 38)

13 Parameters (fixed- and time-varying): ‘‘The economic world in which wefunction is best described by a structure whose parameters arecontinuously changing We often fit simple models [with fixedparameters] only because we cannot estimate a continuously changingset of parameters without vastly more observations than are currentlyavailable to us’’ (p 38)

14 Multiple risks: ‘‘In pursuing a risk-management approach to policy, wemust confront the fact that only a limited number of risks can bequantified with any confidence Policy makers often have to act, orchoose not to act, even though we may not fully understand the fullrange of possible outcomes, let alone each possible outcome’s like-lihood As a result, risk management often involves significant judgment

as we evaluate the risks of different events and the probability that ouractions will alter those risks’’ (p 38)

15 Policy rules: ‘‘For such judgment [mentioned above], policymakers haveneeded to reach beyond models to broader, though less mathematicallyprecise, hypotheses about how the world works For example, inferencesabout how market participants and, hence, the economy might respond

to a monetary policy initiative may need to be drawn from evidenceabout past behavior during a period only roughly comparable to thecurrent situation

Some critics have argued that such an approach to policy is tooundisciplined – judgmental, seemingly discretionary, and difficult toexplain The Federal Reserve, they conclude, should attempt to be moreformal in its operations by tying its actions, solely, on the weakerparadigm, largely, to the prescriptions of a simple policy rule Indeed,rules that relate the setting of the federal funds rate to the deviations ofoutput and inflation from their respective targets, in some configurations,

do seem to capture the broad contours of what we did over the pastdecade and a half And the prescriptions of formal rules can, in fact,serve as helpful adjuncts to policy, as many of the proponents of these

rules have suggested But at crucial points, like those of our recent policyhistory (the stock market crash of 1987, the crises of 1997–1998, and theevents that followed September, 2001), simple rules will be inadequate aseither descriptions or prescriptions for policy Moreover, such rulessuffer from much of the same fixed-coefficient difficulties we have withour large-scale models’’ (pp 38–39).

16 Forecasting: ‘‘While all, no doubt, would prefer that it were otherwise,there is no way to dismiss what has to be obvious to every monetarypolicymaker The success of monetary policy depends importantly onthe quality of forecasting The ability to gauge risks implies somejudgment about how current economic imbalances will ultimately playout Thus, both econometric and qualitative models need to becontinually tested’’ (p 39)

17 Monetary policy: ‘‘In practice, most central banks, at least those notbound by an exchange-rate peg, behave in roughly the same way Theyseek price stability as their long term goal and, accounting for the lag inmonetary policy, calibrate the setting of the policy rate accordingly .All banks ease when economic conditions ease and tighten wheneconomic conditions tighten, even if in differing degrees, regardless ofwhether they are guided by formal or informal inflation targets’’ (p 39)

18 Uncontrolled outcomes and targets: ‘‘Most prominent is the appropriaterole of asset prices in policy In addition to the narrower issue ofproduct price stability, asset prices will remain high on the researchagenda of central banks for years to come There is little disputethat the prices of stocks, bonds, homes, real estate, and exchange ratesaffect GDP But most central banks have chosen, at least to date, toview asset prices not as targets of policy, but as economic variables to beconsidered through the prism of the policy’s ultimate objective’’ (p 40)

19 Performance rating: ‘‘We were fortunate to have worked in aparticularly favorable structural and political environment But we trustthat monetary policy has meaningfully contributed to the impressiveperformance of our economy in recent decades’’ (p 40) Furtherevaluation of current monetary policies dealing with the 2007–2008credit crisis is an important issue

1.3 Greenspan’s Problems and Econometric Research

It is of interest to relate Greenspan’s problem areas to current and pastBayesian econometric research In econometric research, along with other

scientific research, three main areas of activity have been recognized,namely, deduction, induction, and reduction, seeJeffreys (1957, 1939[1998])and Zellner (1985, pp 3–10and 1996, Chapter 1) for discussions of thesetopics and references to the huge literature on the definitions and otheraspects of these research areas Briefly, deduction involves use of logic andmathematics to prove propositions given certain assumptions Inductioninvolves development and use of measurement, description, estimation,testing, prediction, and decision-making procedures, while reductioninvolves creating new models and methods that are helpful in explainingthe past, predicting as yet unobserved outcomes at various places and/ortimes and in solving private and public decision problems.

While much more can be and has been said about deduction, induction,and reduction, most will agree about the difficulty of producing good new orimproved models that work well in explanation, prediction, and decision-making However, as we improve our understanding of these three areas andtheir interrelations in past and current work and engage in more empiricalpredictive and other testing of alternative models and methods, testing that

is much needed in evaluation of alternative macroeconomic models, asemphasized by Christ (1951, 1975), Fair (1992), and many others, morerapid progress will undoubtedly result

A categorization of Greenspan’s problems by their nature is shown inTable 1

It is seen that many of Greenspan’s problems have a deductive ortheoretical aspect to them but, as recognized in the literature, deductionalone is inadequate for scientific work for a variety of reasons, perhaps bestsummarized by the old adage, ‘‘Logical proof does not imply completecertainty of outcomes,’’ as widely appreciated in the philosophical literatureand elsewhere Perhaps, the most striking aspect of Table 1 is the largenumber of entries in category III, reduction Economic theorists, econome-tricians, and others have to get busy producing new models and methods thatare effective in helping to solve former Chairman Greenspan’s and nowChairman Bernanke’s problems SeeHadamard (1945) for the results of a

Table 1 Tabulation of Greenspan’s Problems Listed Above

Categories Problem Numbers

(I) Deduction 3, 4, 6, 9, 10, 11, 12, 13, 14, 16, 17, 19 (II) Induction 2, 3, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19 (III) Reduction 1, 4, 8, 12, 13, 15, 16, 18

survey of mathematicians that provides information on how major throughs in mathematics occurred and tips on how to create new theories inmathematics that may also be helpful in reductive econometric work asdiscussed inZellner (1985, pp 8–10) Also, inZellner and Palm (2004)somemethods for creating new econometric models and checking old econometricmodels and applications of them by a number of researchers are presentedthat may be helpful in the production of new econometric models thatperform well in explanation, prediction, and policy-making More will besaid about this reductive problem area below.

break-1.4 Overview of Paper

With this in the way of an introduction to some current problems facing us,

inSections 2 and 3we shall review some early and recent work in Bayesianeconometrics and relate it to some of the problems mentioned by ChairmanGreenspan and consider future possible developments in Bayesian econo-metrics inSection 4

2 THE PAST

2.1 Early Bayesian Econometrics

As is the case with many others who commenced study of econometrics inthe 1950s, in my graduate econometrics courses at the University ofCalifornia at Berkeley there was no mention of Bayesian topics except in agame theory course that I took with David Blackwell (who many years laterintroduced an elementary Bayesian statistics course at Berkeley usingBerry’s(1996) text) Also, there was no mention of Bayesian analysis inTintner’s(1952)popular text or in most Cowles Commission publications Although,

in Klein’s Textbook of Econometrics (1953, p 62) some discussion ofBayesian decision theory along with a reservation about prior distributionsappeared that he apparently abandoned later in an invited paper, ‘‘WhitherEconometrics?’’ published in JASA in whichKlein (1971)wrote, ‘‘Bayesianmethods attempt to treat a priori information in a systematic way As a pureand passive forecaster of econometric methodology I can see a great deal offuture research effort being channeled in that direction Systematic ways ofintroducing a priori information are to be desired’’ (p 420) Also Theil’s(1978)econometrics text included a chapter titled, ‘‘Bayesian Inference and

Rational Random Behavior’’ in which he explained Bayes’ theorem andprovided some interesting applications of it However, he expressed strongreservations about improper prior distributions and also wrote, ‘‘TheBayesian approach is itself a matter of considerable controversy This is notsurprising, given that the approach takes a fundamentally different view ofthe nature of the parameters by treating them as random variables’’ (p 254).There is no question but that Klein’s forecast regarding futureeconometric methodology, presented above, has been quite accurate Muchpast and current Bayesian research is indeed focused on how to formulateand use prior distributions and models that incorporate ‘‘a prioriinformation’’ in analyses of a wide range of estimation, prediction, andcontrol problems with applications in many fields using fixed and randomparameter models See the early papers byDre`ze (1962),Rothenberg (1963),andZellner (1965)presented at the first World Congress of the EconometricSociety in Rome, 1965 for some Bayesian results for analyzing the importantsimultaneous equations model In my paper, I presented some numericalintegration results, obtained using ‘‘old-fashioned’’ numerical integrationmethods that were of great interest to Malinvaudwhose well-known 1964(translated from French into English in 1966) econometrics text, along withmany others, made no mention of Bayes and Bayesian methods, nor of theearly Bayesian papers thatQin (1996)cites: ‘‘The early 1960s saw pioneeringBayesian applications in econometrics These included published works byFisher (1962), Hildreth (1963), Tiao and Zellner (1964, 1965) and Zellnerand Tiao (1964) and unpublished works byDre`ze (1962) andRothenberg(1963)’’ (pp 503–504) See alsoChetty (1968)for an early Bayesian analysis

of macroeconomic models introduced by Haavelmo

In spite of these and a number of other theoretical and applied Bayesianpublications that appeared in the 1960s and early 1970s, in the 1974,completely revised, 2nd edition of Klein’s Textbook Of Econometrics, hewrote:

Bayes’ theorem gives a logical method of making probability inferences if the a priori probabilities are known They seldom are known and this is the objection to the use of this theorem for most problems of statistical inference A major contribution of decision function theory [that he ably describes in this chapter of his book] is to show the relation

of various inferences to Bayes’ type solutions The beauty of the theory is that it includes hypothesis testing and estimation methods as special cases of a more general approach to inference (p 64)

It is clear that Klein, along with Dre`ze, Leamer, and some other metricians, had a deep understanding of the decision theoretic approach toBayesian statistical inference that Ramsey, Savage, Friedman, Raiffa,

Schlaifer, and others had developed However, he along with many otherswas either unfamiliar with or possibly dissatisfied with the early Bayesianeconometric work, cited above, and with Jeffreys’ important Bayesian books,The Theory of Probabilitythat first appeared in 1939 with new versions in

1948, 1961, 1967, and 1998 and his Scientific Method that first appeared in

1931 followed by later editions in 1937 and 1957 This work provided hisnatural science colleagues and those in other disciplines an effective methodfor learning from their data and experience by use of Bayesian procedures forestimation, testing, and prediction that he illustrated in many applications Inparticular, he put forward an approach to inference using diffuse ornoninformative priors to represent ‘‘knowing little’’ or ‘‘ignorance’’ that heused in solving many central statistical problems facing scientists and notedthat his approach and results could be used to good advantage in decision-making Most importantly, Jeffreys’ work involved not just theory, includingamong other things, a new general method for generating diffuse, invariantpriors, new Bayesian statistical testing methods, and an axiom system forscientific induction, but also analyses of many important applied problems.See papers in Zellner (1980) for more information about Jeffreys and histheoretical and applied research Thus, while some did have a good under-standing of Bayesian theory, there was limited use of Bayesian techniques inthe 1960s and 1970s

2.2 The Cowles Commission Activities

Modern econometrics had its beginnings in the 1940s by workers of theCowles Commission at the University of Chicago, strongly influenced byHaavelmo’s (1944) monograph, ‘‘The Probability Approach in Econo-metrics’’; see Christ (1994),Epstein (1987),Morgan (1990), andQin (1993)for more details.Marschak (1950, 1953)provided introductory chapters forthe influential Cowles Commission volumes 10 and 14 in which he discussedgeneral econometric methodology, including deductive, inductive, andreductive issues, in a most knowledgeable and thoughtful manner What isnot emphasized enough in the historical works, cited above, in my opinion, isthat the situation at the Cowles Commission was not a case of peaceful,innovative research devoted to carrying out the research programs sothoughtfully described in Haavelmo’s, Koopmans’, and Marschak’s works,but was literally a battle between Cowles Commission researchers andFriedman, and others who questioned the usefulness of the elaboratesimultaneous equations models and complicated, non-Bayesian statistical

methods considered and developed by Haavelmo, Koopmans, Klein,Anderson, Rubin, and others.Christ (1951) performed forecasting tests ofone of Klein’s models and found that it did not perform very well and asimilar conclusion was reached vis-a`-vis one of Tinbergen’s early models,results discussed byFriedman (1951), who regarded Tinbergen’s model as avaluable source of hypotheses for future work and Klein’s model as requiringsome disaggregation but thought that we lacked the knowledge about theeconomic dynamics of an economy needed to produce a successfuldisaggregated model Further, there was much ‘‘reductive’’ controversyabout the Keynesian and monetarist approaches to modeling economies.Work by Friedman and Meiselman (1963) compared simple one equation

‘‘reduced form’’ Keynesian and monetarist multiplier models in empiricalanalyses with the finding that the monetarist model was strongly favored bydata for many historical periods except for the period including the GreatDepression of the 1930s when both models worked about equally well Laterthese results were confirmed by Geisel (1975) using posterior odds in apioneering application of Bayesian methods to evaluate alternative models.Also, in my doctoral dissertation and inZellner (1957)and later inZellner,Huang, and Chau (1965), using U.S quarterly data, it was found empiricallythat a Pigou or monetarist real balance effect on consumer spending, muchemphasized in the monetarist literature, exists and is important particularlywith respect to the effects of changes in consumer money holdings onconsumer spending on durable goods and services That these importantmonetarist real balance effects were not included in Cowles Commissionmodels, that tended to be Keynesian in general, was another point ofcontention

Last, and perhaps most important, the large, dynamic simultaneousequations models that the Cowles Commission researchers and othersproduced using much prior information, were implemented using non-Bayesian estimation and testing procedures Many contained a large number

of nonlinear stochastic difference equations, were very complicated and notvery well understood by the model-builders and others See Adelman andAdelman (1959) and Zellner and Peck (1973)for some results illustratingthese points produced by use of simulation experiments with one of Klein’sand Goldberger’s early models of the U.S economy and the FederalReserve-MIT-PENN model of the U.S economy containing about 170equations, some of them nonlinear, that revealed a number of unusualfeatures of these two models These issues relating to the complexity ofCowles Commission type models and their failure to explain the past andforecast well, as compared to simple random walk and other time series

benchmark models, were central in very heated exchanges during the CowlesCommission days and beyond As Greenspan notes above, having modelsthat forecast well and are helpful in explaining the past is most important.The above reductive inference issues involved important disagreementswith respect to the relative virtues of simplicity and complexity in buildingmodels, with Friedman and many of his colleagues favoring starting simplyand complicating if necessary while Klein and others appear to have takenthe position that the world is complicated and thus we need complicatedmodels Even today, this range of issues is still being debated by Sims,Hendry, Diebold, Lu¨tkepohl, Pagan, myself, and many others What was notexplicitly recognized is that if there is a need for capturing much detail,models that are large and sophisticatedly simple can be constructed ratherthan large and complicated models and that the former will be moreunderstandable and will probably work better in explanation and prediction.Also, these issues of simplicity and complexity in model-building in sciencehave been considered at length in the literature; see, for example, the papersand references inZellner, Kuezenkampf, and McAleer (2001), presented anddiscussed at a conference on simplicity and complexity in the Netherlands Itseems that over the years many more have come to favor KISS, Keep itSophisticatedly Simple, in model formulation, in part based on esthetics butmore importantly on empirical results that indicate that sophisticatedlysimple models often work better in forecasting and explanation thancomplicated models In this connection, see the quotation fromGreenspan’s(2004)paper in point number 7 above regarding his successful use of simplemodels and the results of a survey of economics Nobel Prize winners inZellner et al (2001) that showed that almost all of them favored keepingmodels sophisticatedly simple and starting simply and complicating ifnecessary.

2.3 Bayesian Decision Theoretic Approach

In addition, in important articles Friedman and Savage (1948, 1952)developed a decision theoretic Bayesian statistical approach that combinedeconomic utility theory and statistical theory, deeply appreciated by Klein(1953, 1971), and many others that is very different from the sampling theorystatistical estimation, testing, prediction, and decision-making approachesutilized by researchers at the Cowles Commission Thus, while Haavelmo andothers did emphasize and produce a probabilistic approach to econometrics(without, however, thoroughly defining and defending their concept of

probability), their implementation of it was quite different from that favored

by Friedman, Savage, and others, namely the decision theoretic approachmentioned above in the quotations from Klein’s text and invited JASA paper.Also, the econometric models that they developed were found not to forecastvery well, were not easy to justify in terms of alternative macroeconomictheories and found to be very complicated with dynamic properties that werenot well understood by many, including the model-builders All of this wasfuel for prolonged ‘‘discussions’’ and debates held during the CowlesCommission days at Chicago

The work on Bayesian decision theory by Savage, Friedman, and othersled to a very impressive book byRaiffa and Schlaifer (1961), on methods andapplications of Bayesian decision theory that influenced Martin Feldstein,who worked with Raiffa and Schlaifer at Harvard Most of this work went

on recognizing earlier theoretical Bayesian research by Bayes (1763),Edgeworth (seeBowley (1928) for a description of Edgeworth’s impressiveBayesian research results), de Finetti (1974), Savage (1954), and Ramsey(1931)but in large part overlookingJeffreys’ (1939 [1998]) book, Theory ofProbability, and his earlier and later work Note that physicists do not likethe word ‘‘statistics’’ and thus use ‘‘probability theory’’ instead Also, in theinfluential Cowles Commission Monographs 10, Statistical Inference inDynamic Economic Models, edited by T C.Koopmans (1950)and 14, Studies

in Econometric Method, edited by Hood and Koopmans (1953), includingarticles by leading statisticians, T W Anderson, H Hotelling, H Rubin, A.Wald et al., there is not a single reference to Bayes’ theorem and Jeffreys’important work on Bayesian philosophy, estimation, testing, and othertopics Nor are there any references to the decision theoretic work of Savage,Friedman, and others

2.4 The Jeffreys’ Approach

I was introduced to Jeffreys’ work when participating in a University ofWisconsin Department of Statistics seminar devoted to a reading of Jeffreys’book, Theory of Probability, in the early 1960s along with George Box,George Tiao, Irwin Guttman, Norman Draper, Mervyn Stone, and otherstatisticians Each week at the seminar meeting one of the participants wouldreport on a chapter of Jeffreys’ book followed by questions, answers, anddiscussion I reported on one of the most difficult chapters of the book, thechapter on hypothesis testing, a topic that is even quite controversial today;see Berger (2003), Ziliak and McCloskey (2004), and accompanying

published comments on both papers that deal with issues of whether to test

or not and if so, how and comparisons of various approaches to testing It isindeed surprising that after all these years that there should be suchcontroversy about an important, widely used inference technique, namelytesting, with most current Bayesians agreeing that a posterior odds approach,along the lines suggested by Jeffreys, is favored SeePress (2003)who wrote

at the end of his chapter on Bayesian hypothesis testing, ‘‘The Bayesian(Jeffreys) approach is now the preferred method of comparing scientifictheories Richard Feynman suggests that to compare theories (in physics)one should use the Bayesian approach’’ (p 230)

This extensive and thorough reading and discussion of Jeffreys’ bookprovided much material on the philosophy of science, including guidance onissues in deductive inference (namely, an axiom system for systematiclearning from data and experience), reductive inference (his and Wrinch’s

‘‘simplicity postulate’’ and a quantitative measure of the simplicity ofmodels), and inductive inference, his Bayesian procedures for analyzing andlearning from data that are applicable in all areas of science In addition tothese foundational issues, he provided important critiques of the Fisherian,Neyman-Pearson, and other approaches to statistical inference, that led me

to the conclusion that it would be worthwhile to undertake a program ofresearch to evaluate these alternative approaches, theoretically and in appliedeconometric studies as well as to evaluate his positions on simplicity versuscomplexity, how to learn from data, and how to formulate good models thatare successful in explaining the past, predicting the future, and in makingdecisions This emphasis on all three areas, deduction, induction, andreduction is a unique feature of Jeffreys’ work and was not emphasizedenough in early Bayesian and non-Bayesian econometric research After thisseminar experience, I determined to try to remedy the situation in my futureresearch to improve learning, model-building, estimation, testing, prediction,and decision procedures in econometrics

As stated at the end of Chapter 1 of Zellner (1971), ‘‘Since, in the past,most econometricians have employed non-Bayesian techniques in theirwork, it is useful and interesting to compare Bayesian and non-Bayesiananalyses of a range of models and problems In the following chapters thiscomparative approach is pursued, since, as Anscombe (1961) remarkedsome years ago about the state of statistics, ‘A just appreciation of thesituation can only be had by studying the orthodox and the Bayesianapproaches to a variety of statistical problems, noting what each one doesand how well it does it.’ ’’ (p 11) Over the years many of my graduatestudents, fellow researchers, and I found Anscombe’s approach very fruitful

and much to our liking Running Bayesian versus non-Bayesian ‘‘horseraces’’ is fun and scientific too.

2.5 The NBER-NSF Seminar on Bayesian Inference in Econometrics

In my research program beginning in the early 1960s and to the present, asnoted above, I pursued Anscombe’s approach by developing and comparingBayesian and non-Bayesian solutions to a number of theoretical and appliedmodel formulation, estimation, testing, prediction, and decision problemsdrawing on Bayesian research results derived with the help of colleagues andgraduate students These competitive ‘‘horse races’’ were of interest to bothBayesians and non-Bayesians since many wished to learn how to be moreeffective in their econometric work And this approach that concentrated onrelative performance and not on deep, drawn-out philosophical discussions,not only appealed to many but was also generally recognized as being animportant part of a scientific approach to the evaluation of alternative,econometric methods, and approaches

In the late 1960s, it occurred to me that it would be useful to set up aseminar group that would meet semiannually to discuss Bayesian econometricresearch Thus, I recommended to the National Bureau of EconomicResearch board members that a Seminar on Bayesian Inference in Econo-metrics be established Fortunately, my recommendation was approved andthe NBER-NSF Seminar on Bayesian Inference in Econometrics wasestablished that held its first meeting at the University of Chicago in 1970.Since this meeting and later ones involved both econometricians andstatisticians, the name was changed to Seminar on Bayesian Inference inEconometrics and Statistics (SBIES) The seminar met two times a year from

1970 to 1996; seeBerry et al (1996)for a listing of SBIES meetings and somehistorical information about its activities and accomplishments In particular,they write in their Preface, ‘‘For many years the seminar was unique inproviding a sizable audience of receptive yet scientifically critical Bayesians towhom investigators could bring new research These meetings have witnessedthe conception or birth of many path breaking ideas later published inscientific journals, and the death of a few others Both processes havecontributed to the continued progress and vitality of Bayesian econometricsand statistics’’ (p xvii) Currently, the SBIES is operating under the direction

of Sid Chib and has had two very successful recent meetings, one arranged byChib in 2004 and the other by Geweke in 2005

An overview of new, early Bayesian econometric research in the 1970s can

be obtained by viewing the programs of the early meetings of the SBIES, the

titles and authors of Savage Award winning and honorable mentionBayesian doctoral dissertations presented in Fienberg and Zellner (1975)and in the International Society for Bayesian Analysis (ISBA) web page,and the titles of books published in the SBIES sponsored series, Studies inBayesian Econometrics and Statistics, published by North-Holland It is thecase that research reported at the early meetings of the SBIES and in theSavage Award theses and North-Holland volumes covered a wide range oftopics and was not limited to just research on the ‘‘standard’’ simultaneousequations model Indeed, there were important sessions of the meetings and

a volume devoted to economic theory and Bayesian analysis Further, as thetitles of the volumes indicate, there was much interest in the comparativeevaluation of not only Bayesian and non-Bayesian approaches, but alsoalternative Bayesian approaches, in particular those of Savage, Jeffreys,and de Finettti as well as the Jaynes’ Bayes-max-entropy approach of theBayes–Maxent Workshop group with which our SBIES group maintainedclose ties As an example of this close interaction, inJaynes’ (1986) paper,published in the SBIES volume honoring de Finetti, he generalized the deFinetti representation theorem in a significant way and showed how thetheorem has been used in analyses of physical systems As stated in thePreface of this volume, ‘‘Professor Jaynes discusses the extended de Finettirepresentation theorem for both finite and infinite exchangeable sequences.The discussion and the proof are simple and concise Jaynes also gives anexample of the profound impact of this representation theorem in statisticalmechanics’’ (p 3) Thus the deductive part of de Finetti’s Bayesian approachwas linked to important results in physics Also, Jaynes, who was fond

of solving paradoxes, published a 1980 article, ‘‘Marginalization andPrior Probabilities’’ inZellner (1980) in which he showed that the Dawid–Stone–Zidek ‘‘marginalization paradox,’’ widely interpreted to reveal adefect in Bayesian methods, was based on a ‘‘subtle failure to takeaccount of all the relevant information’’ (p 43) See also the response ofDawid, Stone, and Zidek (1980)followed by Jaynes’ rejoinder

In a similar vein,Leamer (1986)in his de Finetti volume paper providedvaluable analysis linking the de Finetti representation of speculative markets

to economists’ analyses of such markets As stated in the Preface, ‘‘ProfessorLeamer critically examines de Finetti’s demonstration of subjectiveprobabilities being equivalent to prices of lottery tickets which are

‘coherent.’ Leamer suggests that most exchanges of lottery ticketsinvolve bid-ask spreads and thus we need to elicit upper and lowerprobabilities (intervals of probabilities), since these can be elicited economic-ally and can be more stable than sharp probabilities He uses several models

for intervals of probabilities to explore the accuracy and economy of variouselicitation games At least in an economic context, Leamer presents aninteresting point of view which needs further investigation’’ (p 5).

These are but two examples of how deductive or axiom systems underlyingBayesian analysis have been subjected to critical analysis in the early stages

of the development of Bayesian econometrics Other axiom systems, forexample, Jeffreys’ and Savage’s have undergone similar close scrutiny, seeMachina (2004) for a fascinating attempt to provide an axiom system forinductive inference that takes account of both elements of Jeffreys’ axiomsystem and ‘‘utility-based’’ axiom systems, such as those of Savage and deFinetti How to represent good or optimal learning behavior and decision-making behavior in a reasonable and effective way by producing a unified set

of axioms indicating how individuals ‘‘should’’ behave so that they learncoherently and effectively and make good decisions is indeed a greatchallenge Be that as it may, the important foundational work of de Finetti,Jeffreys, Savage, and other Bayesians has contributed substantially toimproving our learning and decision-making capabilities and is reflected inmany early papers in Bayesian Econometrics

The programs of the first six meetings of the NBER-NSF BayesianSeminar, 1970–1972, published in Fienberg and Zellner (1975) reveal thebreadth of early Bayesian econometric research that included research in theareas of reduction, deduction, and induction For example, the session titlesfor the first meeting were: ‘‘Bayesian Methods for Comparing and ChoosingAmong Models,’’ ‘‘Bayesian Analyses of Regression and SimultaneousEquation Models,’’ and ‘‘Bayesian Adaptive Control Problems,’’ with papersanalyzing a wide range of problems There was a desire to show that theBayesian approach produced better estimation, testing, model comparison,prediction, and decision procedures Note for example, the papers by E C.Prescott on ‘‘The Multi-Period Control Problems Under Uncertainty’’ and

H Woods Bowman, ‘‘Bayesian Control Theory and the St Louis Model,’’

on very important decision problems, the sort that Greenspan and otherpolicy-makers face

In later meetings of the SBIES, research results on a broad range of topicsare also reported including one session on ‘‘Bayesian Methods in EconomicTheory’’ including a paper by Richard M Cyert, an economist and formerpresident of Carnegie-Mellon University and the famous statistician Morris

M DeGroot, ‘‘Analysis of Cooperation and Learning in a Duoply Context.’’And at the third meeting at Harvard University, arranged by EdwardLeamer, there were papers on Bayes–Stein estimation, pre-testing, identifica-tion in probability, and a Bayesian computer program In addition, we had

the pleasure of listening to Leonard J Savage talk on ‘‘Elicitation of PersonalProbabilities and Expectations.’’ Also, at the fifth and sixth meetings of theSeminar a number of Bayesian papers on a variety of topics were presentedand discussed including stimulating papers by E Leamer, ‘‘Multicollearity:

A Bayesian Interpretation;’’ C Sims, ‘‘Post-Data Model Construction asEstimation with Infinitely Many Parameters;’’ S Grossman, ‘‘A BayesianApproach to Static, Stochastic Equilibria;’’ and G C Chow, ‘‘Effect ofUncertainty on Optimal Control Policies.’’ In a later paper,Grossman (1975)introduced a self-fulfilling, rational expectations distribution of price ratherthan the usual Muthian assumption that an anticipated price is equal to itsconditional mean Grossman utilized the distribution of price, given the data,model and prior information, rather than just its conditional mean inrepresenting price expectations, a very novel and ingenious idea

Last, it is noteworthy that at the 6th meeting at the University ofWisconsin in Madison, May 4–5, 1973, there was a Panel Discussion onBayesian Inference in Econometrics with G E P Box, J H Dre`ze, and

S Geisser as the discussants, who provided very interesting and usefulcomments that were much appreciated All recognized the need for goodmodels that perform well and suggested procedures for checking currentmodels These cogent remarks were of particular interest to me since FranzPalm and I had just recently become engaged in research on how to buildgood dynamic econometric models that resulted in our early papers,Zellnerand Palm (1974, 1975) that have been reprinted along with many otherpapers using the Structural Econometric Time Series Analysis Approach inZellner and Palm (2004) In the 1974 paper, the approach was applied toevaluate dynamic variants of a small Keynesian model put forward andestimated by Haavelmo and in the second paper to evaluate variants of adynamic monetary model formulated by Friedman In both cases, we foundthe two initial models to be inadequate and in the second case, we elaboratedthe model to produce a variant that appeared to be compatible with theinformation in the data by use of various diagnostic checks, etc Over theyears, the approach has been developed further, augmented by use ofBayesian predictive testing of models’ point and turning point predictiveperformance and is thought to be helpful in building good dynamiceconometric models

Those attending the SBIES meetings appreciated the work of manydifferent types of Bayesians and some non-Bayesians, including the famousstatistician George Barnard who has worked on a broad range of problems,was familiar with many statistical approaches and contributed insightful andconstructive comments at several of our meetings When the proposal to

produce a volume in his honor came up, some one said, ‘‘We can’t do that,he’s not a Bayesian.’’ Then another person responded, ‘‘That doesn’t matter.He’s a great guy!’’ After that we voted in favor of producing a volumehonoring that ‘‘great guy’’ George Barnard in which likelihood and Bayesianmethods were discussed and compared Apparently, Barnard was hesitant tointroduce formal priors in his analyses but was willing to introduce

‘‘weights’’ in his ‘‘weighted likelihood’’ approach When I asked himwhether the weights could be interpreted as probabilities, he responded, ‘‘Inyour Bayesian framework they may be probabilities but in mine they areweights.’’ And when he wrote to me inquiring about why a Bayesian groupwould publish a volume in honor of him, a ‘‘likelihood advocate,’’ I wrote tohim explaining our discussion described above He responded in a letterdated, January 31, 1997 as follows:

I had wondered why I should have been judged fit for a Bayesian Festschrift; though in showing back in 1946 that the proper approach to sampling inspection was Bayesian perhaps I did help to draw Bayesian theory back into the limelight And I’m proud, too,

of being the first to tell Jimmie Savage about the likelihood principle (as an obvious consequence of his Bayesian approach) He said he wished he’d thought of it himself.

In these early meetings and subsequent meetings, there were not onlyeconometricians in attendance but also statisticians, general economists, and

a few from the natural and other social sciences This diversity tended toproduce rather interesting productive results at times For example, when EdJaynes, a well-known physicist, who had an interest in various systems,including economic systems, requested and read a current graduatemacroeconomics text that I recommended, he mentioned that models likethose in the text would not work in physics since they do not take account ofturbulence That is, he pointed out that the macroeconomic models in thetext give the same results whether there is 6% unemployed with no hiring andfiring or 6% unemployed with a lot of hiring and firing Also, his comments

on paradoxes, mentioned above, and impressive work on maximum entropyand information theoretic techniques and their applications, see Jaynes(2003), livened up some of our meetings

Last, with respect to the important influence that the seminar exerted, theBayesian econometrician Leamer (1978) wrote in the Preface of his well-known book, ‘‘Many of my ideas have been influenced by attendees at thesemi-annual Seminars in Bayesian Inference in Econometrics Arnold Zellnerespecially should be mentioned He and Jacques Dre`ze were the first to carrythe Bayesian fasces into the econometrics arena Another attendee at theseseminars (and a Harvard colleague), John Pratt, has had a significant

influence on my thoughts Though they may wish to deny it, I havediscovered kindred souls in the form of James Dickey and ThomasRothenberg’’ (pp viii–ix).

All of the above indicates that Bayesian econometricians in the earlyperiod were very concerned with and working on a wide range of topics andintent upon showing that Bayesian methods do indeed solve inductiveproblems better than non-Bayesian methods do For example, optimalBayesian estimates for parameters of many models, including the simulta-neous equations model, were derived that have been shown to have bettersampling properties than leading non-Bayesian estimators; seeDiebold andLamb (1997)for an example involving estimation of a key parameter in thefamous Nerlove agricultural supply model for which empirical workers werehaving difficulty in obtaining good estimates using maximum likelihood(ML) and other techniques while unaware that these procedures producedestimators having sampling densities with thick tails and sometimes bimodalthat produced many outlying estimates in applications, a general property oflimited information maximum likelihood (LIML) estimators for simulta-neous equation models’ parameters in small samples They showed thatrather simple Bayesian methods led to a complete posterior distribution forthe key supply parameter and an optimal point estimate that has very goodsampling properties relative to those of widely used sampling theoryestimators that many empirical econometricians had been using for manyyears See also Monte Carlo evidence from a number of studies summarized

inZellner (1998)that demonstrate the good sampling properties of Bayesianestimators vis-a`-vis those of popular non-Bayesian estimators, for example2SLS, OLS, LIML, Fuller’s modified LIML, etc

Another very important result was that of Stein (1956, 1962), who usedBayesian methods to produce his famous shrinkage estimators thatuniformly dominate least squares, ML, and diffuse prior Bayesian estimatorsrelative to quadratic loss for the multimean, regression, and other modelsunder fairly broad conditions Given the importance of Stein’s results,Walter Vandaele and I spent most of a summer trying to understand Stein’sresults and their relations to traditional ML and Bayesian results, withresults reported in our 1975 Savage volume paper Also, we reviewedLindley’s and others’ alternative methods for deriving shrinkage estimators.And in later work on forecasting annual growth rates for 18 industrializedcountries using single equation third order autoregressive relations contain-ing lagged leading indicator variables, it was found that shrinkage,particularly when combined with joint estimation of the 18 countries’relations led to reasonably good forecasting results in experiments conducted

during the 1980s and 1990s and reported in papers in Zellner and Palm(2004).

At a Bayesian Valencia meeting in the late 1980s, I heard Jose´ Quintanareport on his Bayesian portfolio analysis work in which he and his colleaguesemployed dynamic equations for individual stock prices and fitted them one

by one by single equation Bayesian techniques I commented that it might beuseful to fit the equations jointly using Bayesian SUR and Stein shrinkagetechniques He very kindly accepted my suggestion and on his return to WallStreet implemented it in his impressive computer program, along with the use

of time-varying coefficients and a time-varying error term covariance matrix

to derive optimal Bayesian portfolios, month by month, building on theBayesian portfolio work of H Markowitz, R Litterman, S Brown,

P Jorion, V K Chetty, A Zellner, and others discussed in a volume,edited byBawa, Brown, and Klein (1979) Use of shrinkage, joint estimation,and sequential optimization techniques improved the performance ofQuintana’s investment firm’s rates of return, as reported in a series ofpapers during the 1990s published in the annual proceedings volumes of theASA’s Section on Bayesian Statistical Science available atwww.amstat.org.See, among other striking papers, ‘‘Global Asset Allocation: StretchingReturns by Shrinking Forecasts’’ byQuintana, Chopra, and Putnam (1995),Putnam and Quintana (1995), and Quintana, Putnam, and Wilford (1997).There is no question but that Bayesian portfolio analysis on Wall Street andelsewhere has been helpful in dealing with risk and possibly uncertainty.Perhaps as some state-space engineers claim, allowing parameters to be timevarying helps models to adapt quickly to unforeseen structural changes, aproblem of great concern to Alan Greenspan, as noted above

2.6 Dynamic Model Formulation and Effective Forecasting and

Decision-Making

As regards the very difficult area of producing dynamic multivariate timeseries models that work well in explanation, prediction, and decision-making,there was much disagreement among Bayesians and others about how toproceed Also, there was very little methodological guidance on this difficultproblem in the Cowles Commission and in the Bayesian and non-Bayesianeconometric literature of the 1960s and 1970s Some wished to push aheadwith the old Cowles Commission structural equation modeling approachwhile others, particularly the ‘‘Minnesota crew,’’ Sims, Litterman, and othersturned to ‘‘a theoretical’’ vector autoregressions (VARs), models that were

advertised as involving little prior misinformation from economic theory andelsewhere, first unrestricted and then restricted by the clever Litterman

‘‘Minnesota’’ prior, called Bayesian VARs or BVARs These early VARs,some involving seven macroeconomic variables with 6 lags on each, givingeach equation of the system a relation involving 42 input variables and anintercept that were criticized by many for being over-parameterized, as well

as the newly developed Cowles Commission type models did not forecast aswell as some naive benchmark models (e.g., random walk or simple Box-Jenkins time series models); see Christ (1975), Cooper (1972), Litterman(1986a, 1986b), McNees (1986), Nelson (1972), and Nelson and Plosser(1982)for results relating to the quality of alternative models’ point forecasts.Further, in 2001, when I visited the Bank of England and talked with theresearch staff, they informed me that their experiments with BVARs did notproduce very good forecasts More recently,Adolfson, Laseen, Linde, andVillani (2005)reported on the relative forecasting performance of an openeconomy dynamic stochastic general equilibrium (DSGE) model for theEuro area, implemented using Bayesian methods, vis-a`-vis a wide range ofreduced form forecasting models such as VARs, BVARs, univariate randomwalks, and naive forecasts based on the means of most recent data Theyfound that the DSGE model performed well relative to competing models,particularly at horizons four to eight quarters ahead and state that a possiblereason for this is that the DSGE model has a richer theoretical structure thatprobably has an impact on forecasts in the long-run, where historicalpatterns captured in the VAR systems can lead to more erroneous forecasts,

at least without a prior on the steady state In addition, many forecastersseem to be in general agreement that all models tended to perform poorly inforecasting economic turning points, that is, downturns and upturns ineconomic activity

Perhaps one of the most cogent, early evaluations of the a theoretical, timeseries approach via VARs, etc., was provided in the comments by Klein(1975)on a paper by Sargent and Sims, ‘‘If they do not introduce some moreaspects of system structure, both from economic theory and knowledge ofeconomic institutions, for restricting the parametric specifications of theirmodels, I am afraid that all is lost All the problems of collinearity, shortage

of degrees of freedom, and structural change will so confound theirinterpretation of their results that we shall not know what to make of them

In this respect, I find their approach to be disappointingly retrogressive andcontrary to the main stream of econometric analysis’’ (p 203) Here we findKlein, an experienced model-builder calling for the wise use of priorinformation in building models Indeed, there is a great need for combining

prior knowledge and data in a fruitful way to produce good models, asrecognized by many, including Leamer (1978), Litterman (1986a, 1986b),Phillips (1991),Zellner and Palm (1974, 2004), and many others.

Similarly, the somewhat a theoretical ‘‘General-to-Specific’’ (gets) building strategy of Hendry and others does not appear as yet to haveproduced models that work well in explanation, prediction and policy-making Note, however, that in a recent paper,Lu¨tkepohl (2007)writes, afterpointing out that few actually employ a gets approach, ‘‘ I have arguedthat the leading approach used for modeling multiple time series is a spec[specific to general] approach It has theoretical and practical advantages inparticular if cointegrated variables are involved In fact, a bottom-upapproach to multiple time series analysis that starts from analyzingindividual variables or small groups of variables and uses the results fromthat analysis to construct a larger overall model has a long tradition (see, e.g.Zellner & Palm, 1974;Wallis, 1977, for early contributions to this literatureandMin & Zellner, 1993andZellner & Palm, 2004, for more recent relatedwork)’’ (p 323)

model-In recent work, described in Zellner and Chen (2001), Zellner andIsrailevich (2005), and Kim (2006), it is shown how the ‘‘bottom-upapproach’’ has been applied to produce disaggregated, dynamic macro-econometric models that incorporate prior information from economictheory and other sources and have provided encouraging performance insome forecasting tests and simulation experiments

This model-building activity is part of reductive inference, and as statedearlier, reductive inference procedures are not well-defined and remain to alarge extent an art What apparently is needed and may be emerging is auseful model-building strategy that makes use of data, prior information,economic theory, institutional knowledge, and mathematical, computersimulation and statistical procedures in a logical sequential fashion toproduce models that perform well

2.7 The Bayesian Learning Model

With respect to the Bayesian learning model, Bayes’ theorem, it wasrecognized by many that the theorem, as with all theorems, is based oncertain assumptions, in particular ‘‘the law of insufficient reason,’’ that maynot be satisfied in all circumstances; see Jeffreys (1939 [1998], pp 24–25),Stigler (1986), and Zellner (2004) for discussion and additional references.For example, the product rule of probability that is used by many to prove

Bayes’ theorem is based on the assumption that the probability of drawing aparticular element of a set containing n elements, is 1/n, the same for allelements, an assumption that is supposed to reflect ignorance and no a prioriprejudice, according to Jeffreys and others Jeffreys (1939 [1998], pp 23–25)was so concerned about the general validity of this assumption that heintroduced the product rule as an axiom rather than as a theorem in hisaxiom system.

This concern about the usual proof of Bayes’ theorem, led me in my 1988paper (Zellner, 1988a) to put forward an information theoretic ‘‘optimizationapproach’’ for deriving optimal learning models, including the Bayesianlearning model, Bayes’ theorem When standard inputs, the information in aprior density and a likelihood function, and outputs, the information in apost data density for the parameters and a marginal density for theparameters are employed along with the Gibbs–Shannon informationmeasure, minimization of the difference between the output and inputinformation with respect to the form of the output density for theparameters, the optimal solution turned out to be the result provided byBayes’ theorem, namely, take the posterior pdf ¼ (prior pdf) (likelihoodfunction)/(marginal density for the observations) and when this is done,input information ¼ output information and thus the information proces-sing rule, Bayes’ theorem is 100% efficient See comments on this result bythe discussants of my paper, Jaynes, Hill, Kullback, and Bernardo and myresponse Fortunately, the commentators were rather positive about myapproach and results and suggested that it can serve as a basis for muchadditional fruitful work, since it provides an optimization approach forderiving optimal information processing or learning rules, including Bayes’theorem, that are very useful in practice and provide additional theoreticaljustification for the Bayesian learning model, Bayes’ theorem

In later work, I have used the optimization approach along with a variety

of differing inputs to produce a range of optimal information processingrules, all of them 100% efficient See Table 2for some results that involveusing just a likelihood function as an input, or inputting weighted ordiscounted likelihood functions and quality adjusted priors, called ‘‘powerpriors’’ in the literature The optimal output post data densities for theparameters are operational and optimal variants of Bayes’ theorem that havebeen employed in a number of applied studies Now it is the case that wehave a set of optimal information processing rules ‘‘on the shelf,’’ for use,many of which have been shown to be effective in applied studies; forreferences see my invited ASA 2003 meetings paper, ‘‘Some Aspects of theHistory of Bayesian Information Processing,’’ inGolan and Kitamura (2007)