Bayesian econometrics (advances in econometrics)

ADVANCES IN ECONOMETRICS
Series Editors: Thomas B. Fomby and R. Carter Hill
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Econometric Models in Marketing, Edited
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Maximum Likelihood Estimation of
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Econometric Analysis of Financial and
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ADVANCES IN ECONOMETRICS

VOLUME 23

BAYESIAN
ECONOMETRICS

EDITED BY

SIDDHARTHA CHIB
Olin Business School, Washington University

WILLIAM GRIFFITHS
Department of Economics, University of Melbourne

GARY KOOP
Department of Economics, University of Strathclyde

DEK TERRELL
Department of Economics, Louisiana State University

United Kingdom – North America – Japan
India – Malaysia – China


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LIST OF CONTRIBUTORS
Michael K. Andersson

Sveriges Riksbank, Stockholm, Sweden

Veni Arakelian

Department of Economics, University of
Crete, Rethymno, Greece

Chun-man Chan

Hong Kong Community College,
Kowloon, Hong Kong, China

Cathy W. S. Chen

Department of Statistics, Feng Chia
University, Taiwan

Siddhartha Chib

Olin Business School, Washington
University, St. Louis, MO

S. T. Boris Choy

Discipline of Operations Management
and Econometrics, University of Sydney,
NSW, Australia

Michiel de Pooter

Division of International Finance,
Financial Markets, Board of Governors
of the Federal Reserve System,
Washington, DC

Dipak K. Dey

Department of Statistics, University of
Connecticut, Storrs, CT

Deborah Gefang

Department of Economics, University of

Leicester, Leicester, UK

Richard Gerlach

Discipline of Operations Management
and Econometrics, University of Sydney,
NSW, Australia

Paolo Giordani

Research Department, Sveriges
Riksbank, Stockholm, Sweden

Jennifer Graves

Department of Economics, University of
California, Irvine, CA
ix


x

LIST OF CONTRIBUTORS

William Griffiths

Department of Economics, University of
Melbourne, Vic., Australia

Ariun Ishdorj

Department of Economics, Iowa State
University, Ames, IA

Liana Jacobi

Department of Economics, University of
Melbourne, Vic., Australia

Ivan Jeliazkov

Department of Economics, University of
California, Irvine, CA

Helen H. Jensen

Department of Economics, Iowa State
University, Ames, IA
Swedish Business School, O¨rebo
University, O¨rebo, Sweden

Sune Karlsson
Robert Kohn

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

Gary Koop

Department of Economics, University of
Strathclyde, Glasgow, UK

Dimitris Korobilis

Department of Economics, University of
Strathclyde, Glasgow, UK

Subal C. Kumbhakar

Department of Economics, State
University of New York, Binghamton,
NY

Mark Kutzbach

Department of Economics, University of
California, Irvine, CA

Roberto Leon-Gonzalez

National Graduate Institute for Policy
Studies (GRIPS), Tokyo, Japan
De´partement de Sciences E´conomiques,
Universite´ de Montre´al, CIREQ,
Canada

Brahim Lgui

Arto Luoma

Department of Mathematics and
Statistics, University of Tampere,
Tampere, Finland


xi

List of Contributors

Jani Luoto

William J. McCausland

School of Business and Economics,
University of Jyva¨skyla¨, Jyva¨skyla¨,
Finland
De´partement de Sciences E´conomiques,
Universite´ de Montre´al, CIREQ and
CIRANO, Montre´al, QC, Canada

Nadine McCloud

Department of Economics, The
University of the West Indies, Mona,
Kingston, Jamaica

Murat K. Munkin

Department of Economics, University of
South Florida, Tampa, FL

Christopher J. O’Donnell

School of Economics, University of
Queensland, Brisbane, Australia

Francesco Ravazzolo

Norges Bank, Oslo, Norway

Vanessa Rayner

School of Economics, University of
Queensland, Brisbane, Australia

Rene Segers

Tinbergen Institute and Econometric
Institute, Erasmus University
Rotterdam, Rotterdam, The
Netherlands

Mike K. P. So

Department of ISOM, Hong Kong
University of Science and Technology,
Kowloon, Hong Kong

Rodney Strachan

School of Economics, The University of
Queensland, Brisbane, Australia

Sylvie Tchumtchoua

Department of Statistics, University of
Connecticut, Storrs, CT

Dek Terrell

Department of Economics, Louisiana
State University, Baton Rouge, LA

Justin Tobias

Department of Economics, Purdue
University, West Lafayette, IN

Pravin K. Trivedi

Department of Economics, Wylie Hall,
Indiana University, Bloomington, IN


xii

LIST OF CONTRIBUTORS

Efthymios G. Tsionas

Department of Economics, Athens
University of Economics and Business,
Athens, Greece

Herman K. van Dijk

Tinbergen Institute and Econometric
Institute, Erasmus University
Rotterdam, Rotterdam, The
Netherlands

Wai-yin Wan

School of Mathematics and Statistics,
University of Sydney, NSW, Australia

Arnold Zellner

Graduate School of Business, University
of Chicago, Chicago, IL


BAYESIAN ECONOMETRICS: AN
INTRODUCTION
Siddhartha Chib, William Griffiths, Gary Koop and
Dek Terrell
ABSTRACT
Bayesian Econometrics is a volume in the series Advances in Econometrics

that illustrates the scope and diversity of modern Bayesian econometric
applications, reviews some recent advances in Bayesian econometrics, and
highlights many of the characteristics of Bayesian inference and
computations. 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 Bayesian
econometrics. One was on computational methods and applications and the
other on time-series applications. This was a time when Markov chain Monte
Carlo (MCMC) techniques, which have revolutionized applications of
Bayesian econometrics, had started to take hold. The adaptability of MCMC
to problems previously considered too difficult was generating a revival of
interest in the Bayesian paradigm. Now, 12 years later, it is time for another
Advances volume 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


4

SIDDHARTHA CHIB ET AL.

become widespread across all areas of empirical economics. Previously
intractable problems are being solved and more flexible models are being

introduced. The purpose of this volume is to illustrate today’s scope and
diversity of Bayesian econometric applications, to review some of the recent
advances, and to highlight various aspects of Bayesian inference and
computations.
The book is divided into three parts. In addition to this introduction, Part I
contains papers by Arnold Zellner, and by Paolo Giordani and Robert Kohn.
In his paper ‘‘Bayesian Econometrics: Past, Present, and Future,’’ Arnold
Zellner reviews problems faced by the Federal Reserve System, as described
by its former chairman, Alan Greenspan, and links these problems to a
summary of past and current Bayesian activity. Some key contributions to the
development of Bayesian econometrics are highlighted. Future research
directions are discussed with a view to improving current econometric
models, methods, and applications of them.
The other paper in Part I is a general one on a computational strategy for
improving MCMC. Under the title ‘‘Bayesian Inference using Adaptive
Sampling,’’ Paolo Giordani and Robert Kohn discuss simulation-based
Bayesian inference methods that draw on information from previous samples
to build the proposal distributions in a given family of distributions. The
article covers approaches along these lines and the intuition behind some of
the theory for proving that the procedures work. They also discuss strategies
for making adaptive sampling more effective and provide illustrations for
variable selection in the linear regression model and for time-series models
subject to interventions.

2. MICROECONOMETRIC MODELING
Part II of the book, entitled ‘‘Microeconometric Modeling’’ contains
applications that use cross-section or panel data. The paper by Murat K.
Munkin and Pravin K. Trivedi, ‘‘A Bayesian Analysis of the OPES Model
with a Nonparametric Component: An Application to Dental Insurance and
Dental Care,’’ is a good example of how Bayesian methods are increasingly

being used in important empirical work. The empirical focus is on the impact
of dental insurance on the use of dental services. Addressing this issue is
complicated by the potential endogeneity of insurance uptake and the fact
that insurance uptake may depend on explanatory variables in a nonlinear
fashion. The authors develop an appropriate model which addresses both
these issues and carry out an empirical analysis which finds strong evidence


Bayesian Econometrics: An Introduction

5

that having dental insurance encourages use of dentists, but also of adverse
selection into the insured state.
MCMC simulation techniques are particularly powerful in discrete-data
models with latent variable representations. In their paper ‘‘Fitting and
Comparison of Models for Multivariate Ordinal Outcomes,’’ Ivan Jeliazkov,
Jennifer Graves, and Mark Kutzbach review several alternative modeling
and identification schemes for ordinal data models and evaluate how each
aids or hampers estimation using MCMC. Model comparison via marginal
likelihoods and an analysis of the effects of covariates on category probabilities is considered for each parameterization. The methods are applied to
examples in educational attainment, voter opinions, and consumers’ reliance
on alternative sources of medical information.
In ‘‘Intra-Household Allocation and Consumption of WIC-Approved
Foods: A Bayesian Approach,’’ Ariun Ishdorj, Helen H. Jensen, and Justin
Tobias 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, breastfeeding, and postpartum women. They assess the extent to which the WIC
program 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 the
program (which is voluntary) is endogenous. They develop an appropriate
treatment–response model and conclude that WIC participation does not
lead to increased levels of calcium intake from milk.
A second paper that illustrates the use of Bayesian techniques for analyzing
treatment–response problems is that by Siddhartha Chib and Liana Jacobi.
In their paper ‘‘Causal Effects from Panel Data in Randomized Experiments
with Partial Compliance,’’ the authors describe how to calculate the causal
impacts from a training program when noncompliance exists in the training
arm. Two primary models are considered, with one model including a
random effects specification. Prior elicitation is carefully done by simulating
from a prior predictive density on outcomes, using a hold out sample.
Estimation and model comparison are considered in detail. The methods are
employed to assess the impact of a job training program on mental health
scores.
Basic equilibrium job search models often yield wage densities that do not
accord well with empirical regularities. When extensions to basic models are
made and analyzed using kernel-smoothed nonparametric forms, it is difficult
to assess these extensions via model comparisons. In ‘‘Parametric and
Nonparametric Inference in Equilibrium Job Search Models,’’ Gary Koop


6

SIDDHARTHA CHIB ET AL.

develops Bayesian parametric and nonparametric methods that are comparable to those in the existing non-Bayesian literature. He then shows how
Bayesian methods can be used to compare the different parametric and
nonparametric equilibrium search models in a statistically rigorous sense.
In the paper ‘‘Do Subsidies Drive Productivity? A Cross-Country Analysis

of Nordic Dairy Farms,’’ Nadine McCloud and Subal C. Kumbhakar
develop a Bayesian hierarchical model of farm production which allows for
the calculation of input productivity, efficiency, and technical change. The
key research questions relate to whether and how these are influenced by
subsidies. Using a large panel of Nordic dairy farms, they find that subsidies
drive 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 models
makes marketing a popular area for Bayesian applications. In ‘‘Semiparametric Bayesian Estimation of Random Coefficients Discrete Choice
Models,’’ Sylvie Tchumtchoua and Dipak K. Dey propose a semiparametric
Bayesian framework for the analysis of random coefficients discrete choice
models 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 methodology
using a household level panel dataset of peanut butter purchases, and
supermarket 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 of
local nonidentification. Kleibergen and Zivot (2003) introduced a prior to
overcome this problem in the context of a restricted reduced form
specification, and investigated the relationship between the resulting
Bayesian estimators and their classical counterparts. Arto Luoma and Jani
Luoto, in their paper ‘‘Bayesian Two-Stage Regression with Parametric
Heteroscedasticity,’’ extend the analysis of Kleibergen and Zivot to a
simultaneous equation model with unequal error variances. They apply their
techniques to a cross-country Cobb–Douglas production function.

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

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


Bayesian Econometrics: An Introduction

7

Francesco Ravazzolo, Rene Segers, and Herman K. van Dijk. The boundary
issues considered by these authors are similar to that encountered by Arto
Luoma and Jani Luoto in their paper. There are a number of models where
the use of particular types of noninformative priors can lead to improper
posterior densities with estimation breaking down at boundary values of
parameters. The circumstances under which such problems arise, and how
the problems can be solved using regularizing or truncated priors, are
examined in detail by de Pooter et al. in the context of dynamic linear
regression models, autoregressive and error correction models, instrumental
variable models, variance component models, and state space models.
Analytical, graphical, and empirical results using U.S. macroeconomic data
are presented.
In his paper ‘‘Forecasting in Vector Autoregressions with Many
Predictors,’’ Dimitris Korobilis introduces Bayesian model selection methods
in a VAR setting, focusing on the problem of drawing inferences from a
dataset with a very large number of potential predictors. A stochastic search
variable 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 model
selection based on the Bayesian Information Criteria.
In ‘‘Bayesian Inference in a Cointegrating Panel Data Model,’’ Gary
Koop, Robert Leon-Gonzalez, and Rodney Strachan focus on cointegration

in the context of a cointegrating panel data model. Their approach allows
both short-run dynamics and the cointegrating rank to vary across crosssectional units. In addition to an uninformative prior, they propose an
informative prior with ‘‘soft homogeneity’’ restrictions. This informative
prior can be used to include information from economic theory that crosssectional units are likely to share the same cointegrating rank without forcing
that assumption on the data. Empirical applications using simulated data
and a long-run model for bilateral exchange rates are used to demonstrate
the methodology.
Cointegration is also considered by Deborah Gefang who develops tests of
purchasing power parity (PPP) within an exponential smooth transition
(ESVECM) framework. The Bayesian approach offers a substantial
methodological advantage in this application because the Gibbs sampling
scheme is not affected by the multi-mode problem created by nuisance
parameters. Results based on Bayesian model averaging and Bayesian model
selection find evidence that PPP holds between the United States and each of
the remaining G7 countries.


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SIDDHARTHA CHIB ET AL.

‘‘Bayesian Forecast Combination for VAR Models’’ by Michael K.
Andersson and Sune Karlsson addresses the issue of how to forecast a
variable (or variables) of interest (e.g., GDP) when there is uncertainty about
the dimension of the VAR and uncertainty about which set of explanatory
variables should be used. This uncertainty leads to a huge set of models. The
authors do model averaging over the resulting high-dimensional model space
using predictive likelihoods as weights. For forecast horizons greater than
one, the predictive likelihoods will not have analytical forms and the authors
develop a simulation method for estimating them. An empirical analysis

involving 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 for
simulating the states in state space models where the dependent variables are
proportions. The authors argue in favor of a model which is parameterized
such that the measurement equation has the proportions (conditional on the
states) following a Dirichlet distribution, but the state equation is a standard
linear Gaussian one. The authors develop a Metropolis–Hastings algorithm
which draws states as a block from a multivariate Gaussian proposal
distribution. Extensive empirical evidence indicates that their approach
works 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 their
paper ‘‘Imposing Stationarity Constraints on the Parameters of ARCH and
GARCH Models.’’ Bayesian model averaging is used to resolve uncertainty
with regard to model selection. The authors apply the methodology to data
from the London Metals Exchange and find that results are generally
insensitive 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 selection
for a wide variety of financial volatility models that exhibit asymmetries (e.g.,
threshold GARCH models). Model selection problems are complicated by
the fact that there are many contending models and marginal likelihood
calculation can be difficult. They discuss this problem in an empirical
application involving daily data from three Asian stock markets and
calculate the empirical support for their competing models.
Using a scale mixture of uniform densities representation of the Student-t
density, S. T. Boris Choy, Wai-yin Wan, and Chun-man Chan provide a
Bayesian analysis of a Student-t stochastic volatility model in ‘‘Bayesian
Student-t Stochastic Volatility Models via Scale Mixtures.’’ They develop a

Gibbs sampler for their model and show how their approach can be extended


Bayesian Econometrics: An Introduction

9

to the important class of Student-t stochastic volatility models with leverage.
The different models are fit to returns on exchange rates of the Australian
dollar against 10 currencies.
In ‘‘Bayesian Analysis of the Consumption CAPM,’’ Veni Arakelian and
Efthymios G. Tsionas show that Labadie’s (1989) solution to the CAPM can
be applied to obtain a closed form solution and to provide a traditional
econometric interpretation. They then apply Bayesian inference to both
simulated data and the Mehra and Prescott (1985) dataset. Results generally
conform to theory, but also reveal asymmetric marginal densities for key
parameters. The asymmetry suggests that techniques such as generalized
method of moments, which rely on asymptotical approximations, may be
unreliable.

REFERENCES
Kleibergen, F., & Zivot, E. (2003). Bayesian and classical approaches to instrumental variable
regression. Journal of Econometrics, 114, 29–72.
Labadie, P. (1989). Stochastic inflation and the equity premium. Journal of Monetary
Economics, 24, 195–205.
Mehra, R., & Prescott, E. C. (1985). The equity premium: A puzzle. Journal of Monetary
Economics, 15, 145–162.


BAYESIAN ECONOMETRICS: PAST,

PRESENT, AND FUTURE
Arnold Zellner
ABSTRACT
After briefly reviewing the past history of Bayesian econometrics and Alan
Greenspan’s (2004) recent description of his use of Bayesian methods in
managing policy-making risk, some of the issues and needs that he
mentions are discussed and linked to past and present Bayesian
econometric research. Then a review of some recent Bayesian econometric
research and needs is presented. Finally, some thoughts are presented that
relate to the future of Bayesian econometrics.

1. INTRODUCTION
In the first two sentences of her paper, ‘‘Bayesian Econometrics, The First
Twenty Years,’’ Qin (1996) wrote, ‘‘Bayesian econometrics has been a
controversial area in the development of econometric methodology. Although
the Bayesian approach has been constantly dismissed by many mainstream
econometricians for its subjectivism, Bayesian methods have been adopted
widely in current econometric research’’ (p. 500). This was written more than
10 years ago. Now more mainstream econometricians and many others have
adopted 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

12

ARNOLD ZELLNER

econometric problems in line with my forecast in Zellner (1974), ‘‘Further, it
must be recognized that the B approach is in a stage of rapid development
with work going ahead on many new problems and applications. While this is
recognized, it does not seem overly risky to conclude that the B approach,
which already has had some impact on econometric work, will have a much
more 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 all
econometricians use and misuse prior information, subjectively, objectively,
or otherwise. And it has been pointed out that Bayesian econometricians
learn using an explicit model, Bayes’ Theorem that allows prior information
to be employed in a formal and reproducible manner whereas non-Bayesian
econometricians learn in an informal, subjective manner. For empirical
evidence on the rapid growth of Bayesian publications over the years in
economics and other fields that will be discussed below see Poirier (1989,
1992, 2004) and Poirier (1991) for an interesting set of Bayesian empirical
papers dealing with problems in economics and finance.
In the early 1990s, both the International Society for Bayesian Analysis
() and the Section on Bayesian Statistical Science of
the American Statistical Association () were formed
and have been very active and successful in encouraging the growth of
Bayesian theoretical and applied research and publications. Similarly, the
NBER-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 and
actively supporting the creation of ISBA and SBSS in the early 1990s. In

Berry, Chaloner, and Geweke (1996), some history of the SBIES and a large
number of Bayesian research papers are presented. Also, under the current
leadership 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 Vienna
organized by Wolfgang Polasek that had a very interesting program. In 2005,
the Indian Bayesian Society and the Indian Bayesian Chapter of ISBA had an
international Bayesian meeting at Varanasi with many of the papers
presented that have appeared in a conference volume. In September 2006, a
Bayesian research meeting was held at the Royal Bank of Sweden, organized
by Mattias Villani that attracted leading Bayesian econometricians from all
over the world to present reports on their current work on Bayesian
econometric methodology. And now, this Advances in Econometrics volume
features additional valuable Bayesian econometric research. And last, but not


Bayesian Econometrics: Past, Present, and Future

13

least, the International Society for Bayesian Analysis has commenced
publication of an online Bayesian journal called Bayesian Analysis; see
for more information about this journal with
R. Kass the founding editor and listings of articles for several years that
are downloadable. These and many more Bayesian activities that have taken
place over the years attest to the growth and vitality of Bayesian analysis in
many 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 Bayesian

methodology and analysis, Alan Greenspan, former Chairman of the U.S.
Federal Reserve System presented an invited paper, ‘‘Risk and Uncertainty in
Monetary Policy’’ at the 2004 Meeting of the American Economic
Association that was published in the American Economic Review in 2004
along with very knowledgeable discussion by Martin Feldstein, Harvard
Professor of Economics and President of the National Bureau of Economic
Research, 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 the
ways in which he approached and solved monetary policy problems ‘‘ . . . from
the perspective of someone who has been in the policy trenches’’ (p. 33).
Greenspan’s account should be of interest to Bayesians econometricians
and many others since he states, ‘‘In essence, the risk management approach
to policymaking is an application of Bayesian decision-making’’ (p. 37). In
addition, he writes, ‘‘Our problem is not, as is sometimes alleged, the
complexity of our policy-making process, but the far greater complexity of a
world economy whose underlying linkages appear to be continuously evolving. Our response to that continuous evolution has been disciplined by the
Bayesian type of decision-making in which we have been engaged’’ (p. 39).
Feldstein (2004), after providing an excellent review of Greenspan’s
successful policy-making in the past wrote, ‘‘Chairman Greenspan emphasized that dealing with uncertainty is the essence of making monetary policy
(see also Feldstein, 2002). The key to what he called the risk-management
approach to monetary policy is the Bayesian theory of decision-making’’
(p. 42). After providing a brief, knowledgeable description of Bayesian
decision theory, Feldstein provides the following example to illustrate a case
of asymmetric loss in connection with a person making a decision whether to
carry an umbrella when the probability of rain is not high. ‘‘If he carries the


14

ARNOLD ZELLNER

umbrella and it does not rain, he is mildly inconvenienced. But if he does not
carry the umbrella and it rains, he will suffer getting wet. A good Bayesian
finds himself carrying an umbrella on many days when it does not rain. The
policy actions of the past year were very much in this spirit. The Fed cut the
interest rate to 1 percent to prevent the low-probability outcome of spiraling
deflation because it regarded that outcome as potentially very damaging
while the alternative possible outcome of a rise of the inflation rate from 1.5
percent to 2.5 percent was deemed less damaging and more easily reversed’’
(p. 42).
Mervyn King of the Bank of England commented knowingly about model
quality and policy-making, ‘‘Greenspan suggests that the risk-management
approach is an application of Bayesian decision-making when there is
uncertainty about the true model of the economy. Policy that is optimal in
one particular model of the economy may not be ‘robust’ across a class of
other models. In fact, it may lead to a very bad outcome should an
alternative model turn out to be true . . . Of course, although such an
approach is sensible, it is still vulnerable to policymakers giving excessive
weight to misleading models of the economy. . . . But, in the end, there is no
escaping the need to make judgments about which models are more plausible
than others’’ (pp. 42–43). These are indeed very thoughtful remarks about
problems of model uncertainty in making policy but do not recognize that
just as with Feldstein’s umbrella example above, a Bayesian analysis can
utilize posterior probabilities associated with alternative models that reflect
the quality of past performance that have been shown to be useful in
producing useful combined forecasts and probably will be helpful in dealing
with model uncertainty in policy-making.
1.2. Greenspan’s Policy-Making Problems
Below, I list and label important problems that Greenspan mentioned in

connection with his successful policy-making over the years that reveal his
deep understanding of both obvious and very subtle problems associated
with model-building, economic analyses, forecasting, and policy-making.
1. Structural changes: For example, ‘‘ . . . increased political support for
stable prices, globalization which unleashed powerful new forces of
competition, and an acceleration of productivity which at least for a time
held down cost pressures’’ (p. 33). ‘‘I believe that we at the Fed, to our
credit, did gradually come to recognize the structural economic changes
that we were living through and accordingly altered our understanding


Bayesian Econometrics: Past, Present, and Future

2.

3.

4.

5.

6.

15

of the key parameters of the economic system and our policy stance . . . .
But as we lived through it, there was much uncertainty about the
evolving structure of the economy and about the influence of monetary
policy’’ (p. 33).
Forecasting: ‘‘In recognition of the lag in monetary policy’s impact on

economic activity, a preemptive response to the potential for building
inflationary pressures was made an important feature of policy. As a
consequence, this approach elevated forecasting to an even more
prominent place in policy deliberations’’ (p. 33).
Unintended consequences: ‘‘Perhaps the greatest irony of the past decade
is that the gradually unfolding success against inflation may well have
contributed to the stock price bubble of the latter part of the
1990s . . . 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 been
calibrated to prevent the late 1990s bubble while preserving economic
stability is almost surely an illusion. Instead of trying to contain a
putative bubble by drastic actions with largely unpredictable consequences, we chose . . . to focus on policies to mitigate the fallout when it
occurs and, hopefully, ease the transition to the next expansion’’ (p. 36).
Uncertainty: ‘‘The Federal Reserve’s experiences over the past two
decades make it clear that uncertainty is not just a pervasive feature of
the monetary landscape; it is the defining characteristic of that
landscape. The term ‘‘uncertainty’’ is meant here to encompass both
‘Knightian uncertainty,’ in which the probability distribution of
outcomes is unknown, and ‘risk,’ in which uncertainty of outcomes is
delimited by a known probability distribution. In practice, one is never
quite sure what type of uncertainty one is dealing with in real time, and it
may be best to think of a continuum ranging from well-defined risks to
the truly unknown’’ (pp. 36–37).
Risk management: ‘‘As a consequence, the conduct of monetary policy in
the United States has come to involve, at its core, crucial elements of risk
management. This conceptual framework emphasizes understanding as
much as possible the many sources of risk and uncertainty that
policymakers face, quantifying those risks, when possible, and assessing
costs associated with each of the risks. In essence, the risk-management

approach to monetary policymaking is an application of Bayesian
decision-making’’ (p. 37).
Objectives: ‘‘This [risk management] framework also entails devising, in
light of those risks, a strategy for policy directed at maximizing the


16

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8.

9.

10.

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ARNOLD ZELLNER

probabilities of achieving over time our goals of price stability and the
maximum sustainable economic growth that we associate with it’’ (p. 37).
Expert opinion: ‘‘In designing strategies to meet our policy objectives, we
have drawn on the work of analysts, both inside and outside the Fed,
who over the past half century have devoted much effort to improving
our understanding of the economy and its monetary transmission
mechanism’’ (p. 37).
Model uncertainty: ‘‘A critical result [of efforts to improve our
understanding of the economy and its monetary transmission mechanism] has been the identification of a relatively small set of key
relationships that, taken together, provide a useful approximation of

our economy’s dynamics. Such an approximation underlies the statistical
models that we at the Federal Reserve employ to assess the likely
influence of our policy decisions.
However, despite extensive efforts to capture and quantify what we
perceive as the key macroeconomic relationships, our knowledge about
many of the important linkages is far from complete and, in all likelihood
will always remain so. Every model, no matter how detailed or how well
designed, conceptually and empirically, is a vastly simplified representation of the world that we experience with all its intricacies on a day-today basis’’ (p. 37).
Loss structures: ‘‘Given our inevitably incomplete knowledge about key
structural aspects of an ever-changing economy and the sometimes
asymmetric costs or benefits of particular outcomes, a central bank
needs to consider not only the most likely future path for the economy,
but also the distribution of possible outcomes about that path. The
decision-makers then need to reach a judgment about the probabilities,
costs and benefits of the various possible outcomes under alternative
choices for policy’’ (p. 37).
Robustness of policy: ‘‘In general, different policies will exhibit different
degrees of robustness with respect to the true underlying structure of the
economy’’ (p. 37).
Cost–benefit analysis: ‘‘As this episode illustrates, policy practitioners
operating under a risk-management paradigm may, at times, be led to
undertake actions intended to provide insurance against [low probability] especially adverse outcomes . . . . The product of a lowprobability event and a potentially severe outcome was judged a more
serious threat to economic performance than the higher inflation that
might ensue in the more probable scenario’’ (p. 37).


Bayesian Econometrics: Past, Present, and Future

17

12. Knightian uncertainty: ‘‘When confronted with uncertainty, especially
Knightian uncertainty, human beings invariably attempt to disengage
from medium- to long-term commitments in favor of safety and
liquidity. Because economies, of necessity, are net long (that is, have net
real assets) attempts to flee these assets causes prices of equity assets to
fall, in some cases dramatically . . . The immediate response on the part
of the central bank to such financial implosions must be to inject large
quantities of liquidity . . . ’’ (p. 38).
13. Parameters (fixed- and time-varying): ‘‘The economic world in which we
function is best described by a structure whose parameters are
continuously changing. . . . We often fit simple models [with fixed
parameters] only because we cannot estimate a continuously changing
set of parameters without vastly more observations than are currently
available to us’’ (p. 38).
14. Multiple risks: ‘‘In pursuing a risk-management approach to policy, we
must confront the fact that only a limited number of risks can be
quantified with any confidence . . . . Policy makers often have to act, or
choose not to act, even though we may not fully understand the full
range of possible outcomes, let alone each possible outcome’s likelihood. As a result, risk management often involves significant judgment
as we evaluate the risks of different events and the probability that our
actions will alter those risks’’ (p. 38).
15. Policy rules: ‘‘For such judgment [mentioned above], policymakers have
needed to reach beyond models to broader, though less mathematically
precise, hypotheses about how the world works. For example, inferences
about how market participants and, hence, the economy might respond
to a monetary policy initiative may need to be drawn from evidence
about past behavior during a period only roughly comparable to the
current situation.
Some critics have argued that such an approach to policy is too
undisciplined – judgmental, seemingly discretionary, and difficult to

explain. The Federal Reserve, they conclude, should attempt to be more
formal in its operations by tying its actions, solely, on the weaker
paradigm, largely, to the prescriptions of a simple policy rule. Indeed,
rules that relate the setting of the federal funds rate to the deviations of
output and inflation from their respective targets, in some configurations,
do seem to capture the broad contours of what we did over the past
decade 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


18

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17.

18.

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ARNOLD ZELLNER

rules have suggested. But at crucial points, like those of our recent policy
history (the stock market crash of 1987, the crises of 1997–1998, and the
events that followed September, 2001), simple rules will be inadequate as
either descriptions or prescriptions for policy. Moreover, such rules
suffer from much of the same fixed-coefficient difficulties we have with
our large-scale models’’ (pp. 38–39).
Forecasting: ‘‘While all, no doubt, would prefer that it were otherwise,
there is no way to dismiss what has to be obvious to every monetary

policymaker. The success of monetary policy depends importantly on
the quality of forecasting. The ability to gauge risks implies some
judgment about how current economic imbalances will ultimately play
out . . . . Thus, both econometric and qualitative models need to be
continually tested’’ (p. 39).
Monetary policy: ‘‘In practice, most central banks, at least those not
bound by an exchange-rate peg, behave in roughly the same way. They
seek price stability as their long term goal and, accounting for the lag in
monetary policy, calibrate the setting of the policy rate accordingly. . . .
All banks ease when economic conditions ease and tighten when
economic conditions tighten, even if in differing degrees, regardless of
whether they are guided by formal or informal inflation targets’’ (p. 39).
Uncontrolled outcomes and targets: ‘‘Most prominent is the appropriate
role of asset prices in policy. In addition to the narrower issue of
product price stability, asset prices will remain high on the research
agenda of central banks for years to come. . . . There is little dispute
that the prices of stocks, bonds, homes, real estate, and exchange rates
affect GDP. But most central banks have chosen, at least to date, to
view asset prices not as targets of policy, but as economic variables to be
considered through the prism of the policy’s ultimate objective’’ (p. 40).
Performance rating: ‘‘We were fortunate . . . to have worked in a
particularly favorable structural and political environment. But we trust
that monetary policy has meaningfully contributed to the impressive
performance of our economy in recent decades’’ (p. 40). Further
evaluation of current monetary policies dealing with the 2007–2008
credit 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 past
Bayesian econometric research. In econometric research, along with other

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Bayesian Econometrics: Past, Present, and Future

scientific research, three main areas of activity have been recognized,
namely, deduction, induction, and reduction, see Jeffreys (1957, 1939 [1998])
and Zellner (1985, pp. 3–10 and 1996, Chapter 1) for discussions of these
topics and references to the huge literature on the definitions and other
aspects of these research areas. Briefly, deduction involves use of logic and
mathematics to prove propositions given certain assumptions. Induction
involves development and use of measurement, description, estimation,
testing, prediction, and decision-making procedures, while reduction
involves creating new models and methods that are helpful in explaining
the past, predicting as yet unobserved outcomes at various places and/or
times 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 or
improved models that work well in explanation, prediction, and decisionmaking. However, as we improve our understanding of these three areas and
their interrelations in past and current work and engage in more empirical
predictive and other testing of alternative models and methods, testing that
is much needed in evaluation of alternative macroeconomic models, as
emphasized by Christ (1951, 1975), Fair (1992), and many others, more
rapid progress will undoubtedly result.
A categorization of Greenspan’s problems by their nature is shown in
Table 1.
It is seen that many of Greenspan’s problems have a deductive or
theoretical aspect to them but, as recognized in the literature, deduction
alone is inadequate for scientific work for a variety of reasons, perhaps best

summarized by the old adage, ‘‘Logical proof does not imply complete
certainty of outcomes,’’ as widely appreciated in the philosophical literature
and elsewhere. Perhaps, the most striking aspect of Table 1 is the large
number of entries in category III, reduction. Economic theorists, econometricians, and others have to get busy producing new models and methods that
are effective in helping to solve former Chairman Greenspan’s and now
Chairman Bernanke’s problems. See Hadamard (1945) for the results of a
Table 1.
Categories
(I) Deduction
(II) Induction
(III) Reduction

Tabulation of Greenspan’s Problems Listed Above.
Problem Numbers
3, 4, 6, 9, 10, 11, 12, 13, 14, 16, 17, 19
2, 3, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19
1, 4, 8, 12, 13, 15, 16, 18


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ARNOLD ZELLNER

survey of mathematicians that provides information on how major breakthroughs in mathematics occurred and tips on how to create new theories in
mathematics that may also be helpful in reductive econometric work as
discussed in Zellner (1985, pp. 8–10). Also, in Zellner and Palm (2004) some
methods for creating new econometric models and checking old econometric
models and applications of them by a number of researchers are presented
that may be helpful in the production of new econometric models that
perform well in explanation, prediction, and policy-making. More will be

said about this reductive problem area below.

1.4. Overview of Paper
With this in the way of an introduction to some current problems facing us,
in Sections 2 and 3 we shall review some early and recent work in Bayesian
econometrics and relate it to some of the problems mentioned by Chairman
Greenspan and consider future possible developments in Bayesian econometrics in Section 4.

2. THE PAST
2.1. Early Bayesian Econometrics
As is the case with many others who commenced study of econometrics in
the 1950s, in my graduate econometrics courses at the University of
California at Berkeley there was no mention of Bayesian topics except in a
game theory course that I took with David Blackwell (who many years later
introduced an elementary Bayesian statistics course at Berkeley using Berry’s
(1996) text). Also, there was no mention of Bayesian analysis in Tintner’s
(1952) popular text or in most Cowles Commission publications. Although,
in Klein’s Textbook of Econometrics (1953, p. 62) some discussion of
Bayesian decision theory along with a reservation about prior distributions
appeared that he apparently abandoned later in an invited paper, ‘‘Whither
Econometrics?’’ published in JASA in which Klein (1971) wrote, ‘‘Bayesian
methods attempt to treat a priori information in a systematic way. As a pure
and passive forecaster of econometric methodology I can see a great deal of
future research effort being channeled in that direction. Systematic ways of
introducing a priori information are to be desired’’ (p. 420). Also Theil’s
(1978) econometrics text included a chapter titled, ‘‘Bayesian Inference and


Bayesian Econometrics: Past, Present, and Future

21

Rational Random Behavior’’ in which he explained Bayes’ theorem and
provided some interesting applications of it. However, he expressed strong
reservations about improper prior distributions and also wrote, ‘‘The
Bayesian approach is itself a matter of considerable controversy. This is not
surprising, given that the approach takes a fundamentally different view of
the nature of the parameters by treating them as random variables’’ (p. 254).
There is no question but that Klein’s forecast regarding future
econometric methodology, presented above, has been quite accurate. Much
past and current Bayesian research is indeed focused on how to formulate
and use prior distributions and models that incorporate ‘‘a priori
information’’ in analyses of a wide range of estimation, prediction, and
control problems with applications in many fields using fixed and random
parameter models. See the early papers by Dre`ze (1962), Rothenberg (1963),
and Zellner (1965) presented at the first World Congress of the Econometric
Society in Rome, 1965 for some Bayesian results for analyzing the important
simultaneous equations model. In my paper, I presented some numerical
integration results, obtained using ‘‘old-fashioned’’ numerical integration
methods that were of great interest to Malinvaud whose well-known 1964
(translated from French into English in 1966) econometrics text, along with
many others, made no mention of Bayes and Bayesian methods, nor of the
early Bayesian papers that Qin (1996) cites: ‘‘The early 1960s saw pioneering
Bayesian applications in econometrics. These included published works by
Fisher (1962), Hildreth (1963), Tiao and Zellner (1964, 1965) and Zellner
and Tiao (1964) and unpublished works by Dre`ze (1962) and Rothenberg
(1963)’’ (pp. 503–504). See also Chetty (1968) for an early Bayesian analysis
of macroeconomic models introduced by Haavelmo.
In spite of these and a number of other theoretical and applied Bayesian
publications that appeared in the 1960s and early 1970s, in the 1974,

completely revised, 2nd edition of Klein’s Textbook Of Econometrics, he
wrote:
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 econometricians, had a deep understanding of the decision theoretic approach to
Bayesian statistical inference that Ramsey, Savage, Friedman, Raiffa,