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We discuss bayesian inferential procedures within the family of instrumental variables regression models and focus on two issues: existence conditions for posterior moments of the parameters of interest under a flat prior and the potential of direct monte carlo (dmc) approaches for efficient evaluation of such possibly highly non-elliptical posteriors. We show that, for the general case of m endogenous variables under a flat prior, posterior moments of order r exist for the coefficients reflecting the endogenous regressors’ effect on the dependent variable, if the number of instruments is greater than m +r, even though there is an issue of local non-identification that causes non-elliptical shapes of the posterior. This stresses the need for efficient monte carlo integration methods. We introduce an extension of dmc that incorporates an acceptance-rejection sampling step within dmc. This acceptance-rejection within direct monte carlo (ardmc) method has the attractive property that the generated random drawings are independent, which greatly helps the fast convergence of simulation results, and which facilitates the evaluation of the numerical accuracy. The speed of ardmc can be easily further improved by making use of parallelized computation using multiple core machines or computer clusters. We note that ardmc is an analogue to the well-known “metropolis-hastings within gibbs” sampling in the sense that one ‘more difficult’ step is used within an ‘easier’ simulation method. We compare the ardmc approach with the gibbs sampler using simulated data and two empirical data sets, involving the settler mortality instrument of acemoglu et al. (2001 acemoglu , d. Johnson , s. Robinson , j. A. ( 2001 ). The colonial origins of comparative development: an empirical investigation . American economic review 91 ( 5 ): 1369 – 1401 .[crossref], [web of science ®], , [google scholar]) and father's education's instrument used by hoogerheide et al. (2012a hoogerheide , l. F. Block , j. H. Thurik , a. R. (2012a). Family background variables as instruments for education in income regressions: a bayesian analysis. Economics of education review 31(5):515–523.[crossref], [web of science ®], , [google scholar]). Even without making use of parallelized computation, an efficiency gain is observed both under strong and weak instruments, where the gain can be enormous in the latter case.

Using a disaggregated Marshallian Macroeconomic Model (MMM-DA), this paper investigates how the adoption of a set of 'free market reforms' may affect the economic growth rate of South Africa. Accounting for possible side effects mainly on the budget deficit, our findings suggest that the institution of the proposed policy reforms would yield a substantial growth in the aggregate annual real GDP. The resulting GDP growth rate could range from 5.3 percent to 9.8 percent depending on which variant of the reform policies is implemented.

This working paper is the first draft of an overview and commentary on the papers to appear in a Macroeconomic Dynamics Special Issue on Measurement with Theory. The included papers in the special issue are part of a larger initiative to promote "measurement with theory" in economics and planned to appear as special issues of other journals. A later revised draft of this initial commentary is planned to appear as the introduction to the Macroeconomic Dynamics special issue, expected to appear in 2010.

This paper introduces this special issue of Economic Modelling which marks the significant contribution made by P.A.V.B Swamy to econometrics. His work spans almost 50 years and has been both innovative and challenging. The many distinguished authors who have contributed to this volume attests to the high regard in which he is held in the profession.

Computationally efficient methods for Bayesian analysis of seemingly unrelated regression (SUR) models are described and applied that involve the use of a direct Monte Carlo (DMC) approach to calculate Bayesian estimation and prediction results using diffuse or informative priors. This DMC approach is employed to compute Bayesian marginal posterior densities, moments, intervals and other quantities, using data simulated from known models and also using data from an empirical example involving firms’ sales. The results obtained by the DMC approach are compared to those yielded by the use of a Markov Chain Monte Carlo (MCMC) approach. It is concluded from these comparisons that the DMC approach is worthwhile and applicable to many SUR and other problems.

Bayesian shrinkage a la Stein and others can improve estimation of individual parameters and forecasts of individual future outcomes. In this paper the issue of the impact of shrinkage on the estimation of sums or totals of individual parameters and of individual outcomes is analyzed. Quadratic and “balanced” loss functions will be employed. The latter are a linear combination of “goodness of fit” and “precision of estimation” loss functions. Several examples will be analyzed in detail to illustrate general principles.

A description of computationally efficient methods for the Bayesian analysis of Student- t seemingly unrelated regression (SUR) models with unknown degrees of freedom is given. The method combines a direct Monte Carlo (DMC) approach with an importance sampling procedure to calculate Bayesian estimation and prediction results using a diffuse prior. This approach is employed to compute Bayesian posterior densities for the parameters, as well as predictive densities for future values of variables and the associated moments, intervals and other quantities that are useful to both forecasters and others. The results obtained using our approach are compared to those yielded by the use of DMC for a standard normal SUR model.

S. James Press's many contributions to statistical research, lecturing, mentoring students, the statistics profession, etc., are summarized. Then some new developments in Bayesian analysis are described and remarks on the future of Bayesian analysis are presented.

S. James Press's many contributions to statistical research, lecturing, mentoring students, the statistics profession, etc. are summarized. Then some new developments in Bayesian analysis are described and remarks on the future of Bayesian analysis are presented.

For many years, traditional Bayesian (TB) and information theoretic (IT) procedures for learning from data were viewed as distinctly different approaches. Derivations of the TB and IT learning models are reviewed and compared. Then the synthesis by Zellner in 1988 of the TB and IT learning models and generalizations of them are described along with descriptions of selected applications. Included are learning procedures that do not require use of likelihood functions and/or priors. Works by leading Bayesians and information theorists are cited and related to TB/IT issues.

In this paper the usual product rule of probability theory is generalized by relaxing the assumption that elements of sets are equally likely to be drawn. The need for such a generalization has been noted by Jeffreys [1998. Theory of Probability, 3rd ed, reprinted in Oxford classics Series, Oxford University Press (1st ed.,1939), Oxford. pp. 24–25], among others, in his work on an axiom system for scientific learning from data utilizing Bayes's Theorem. It is shown that by allowing probabilities of elements to be drawn to be different, generalized forms of the product rule and Bayes's Theorem are obtained that reduce to the usual product rule and Bayes's Theorem under certain assumptions that may be satisfactory in many cases encountered in practice in which the principle of insufficient reason is inadequate. Also, in comparing alternative hypotheses, allowing the prior odds to be random rather than fixed provides a useful generalization of the standard posterior odds.

In this paper, the philosophy and objectives of econometrics are discussed. The roles of induction, deduction and reduction in economic research are explained. Further, the roles of sophisticated simplicity and predictive performance in model building are described. Many examples, drawn from the work of leading scientists, are provided to illustrate general points. In addition, the work that has been done leading to the formulation of a disaggregate Marshallian Macroeconomic Model is briefly described. It is concluded that greater emphasis in teaching to explain the roles of deduction, induction and reduction in economic research would be very beneficial in terms of producing more valuable and useful research results. Also, development and use of many more sophisticatedly simple models and further use of Bayesian inference and decision techniques will do much to promote more rapid progress in economic science.

In this progress report, we first indicate the origins and early development of the Marshallian macroeconomic model (MMM) and briefly review some of our past empirical forecasting experiments with the model. Then we present recently developed one-sector, two-sector and n -sector models of an economy that can be employed to explain past experience, predict future outcomes and analyze policy problems. The results of simulation experiments with various versions of the model are provided to illustrate some of its dynamic properties that include “chaotic” features. Last, we present comments on planned future work with the model.

In this progress report, we first indicate the origins and early development of the Marshallian Macroeconomic Model and briefly review some of our past empirical forecasting experiments with the model. Then we present recently developed one-sector, two-sector and n-sector models of an economy that can be employed to explain past experience, predict future outcomes, and analyze policy problems. The results of simulation experiments with various versions of the model are provided to illustrate some of its dynamic properties that include “chaotic” features. Last, we present comments on planned future work with the model.

A graduate program has to decide how many offers of admission to make to achieve a targeted number of acceptances for its entering class given uncertainty about how many will accept such offers. We first structure and solve selection problems under the assumption that probabilities associated with uncertain outcomes have assumed values using symmetric and asymmetric loss functions and show sensitivity of solutions to errors in these assumed values. Then we show how these problems can be solved when these probabilities are heterogeneous and have to be estimated from sample data.

Recent results in information theory, see Soofi (1996; 2001) for a review, include derivations of optimal information processing rules, including Bayes' theorem, for learning from data based on minimizing a criterion functional, namely output information minus input information as shown in Zellner (1988; 1991; 1997; 2002). Herein, solution post data densities for parameters are obtained and studied for cases in which the input information is that in (1) a likelihood function and a prior density; (2) only a likelihood function; and (3) neither a prior nor a likelihood function but only input information in the form of post data moments of parameters, as in the Bayesian method of moments approach. Then it is shown how optimal output densities can be employed to obtain predictive densities and optimal, finite sample structural coefficient estimates using three alternative loss functions. Such optimal estimates are compared with usual estimates, e.g., maximum likelihood, two‐stage least squares, ordi...

Science involves learning from data. Herein this process of learning or information processing is considered within the context of optimal information processing, as in Zellner (1988,1991,1997). Information criterion functionals are formulated and optimized to provide optimal information processing rules, one of which is Bayes’ theorem. By varying the inputs and using alternative side conditions, various optimal information processing rules are derived and evaluated. Generally output information = input information for these rules and thus they are 100% efficient learning rules. When different weights or costs are associated with alternative inputs, “anchoring” like effects, much emphasized in the psychological literature are the results of optimal information processing procedures. Further, dynamic information processing results are reviewed and extensions noted. Last, some implications of the information processing approach for learning from data will be discussed.

In this article we extend previous BMOM results by showing how information about a variance parameter and its relation to regression coefficients produces a rich class of postdata densities for regression parameters. Prediction and model selection techniques are also described. We also discuss the well-documented link between cross-entropy and the average log odds and then use this criterion in an experiment to compare results obtained from BMOM and Bayes approaches using data generated from known models.

In this note the results of an experiment to determine the effects of aggregation and disaggregation in forecasting the median growth rate of eighteen industrialized countries' annual output (GDP) growth rates are reported.

The three papers presented at this session involve interesting and useful applications of the maxent and Bayesian approaches to the analysis of important problems. Foster and White (FW) use Bayesian and maxent techniques to analyze an option pricing problem, Golan, Moretti, and Perloff (GMP) employ maxent in the analysis of ajob choice problem involving the use of a sample selection model, and LaFrance (LF) analyzes an important food nutrition problem using a Bayesian methodof-moments maxent procedure. The papers illustrate the fact that Bayesian and maxent procedures are useful in these and other applications. Further, GMP and LF have compared alternative maxent and other approaches in their papers in a scientific effort to appraise their relative merits. The authors are to be

After discussing the need for good finite sample estimation procedures for simultaneous equations models and showing the inadequacies of asymptotically justified estimators, it is shown how the Bayesian method of moments (BMOM) provides an exact, finite sample analysis of unrestricted reduced form systems. Then optimal, finite sample estimates of structural coefficients are derived using three standard loss functions and they are compared to traditional Bayesian optimal estimates. Monte Carlo experimental evidence from four studies on the relative performance of Bayesian and non-Bayesian estimators is reviewed with the finding that the performance of Bayesian estimators is better.

We consider generalized production functions, introduced in Zellner and Revankar (1969), for output y=g(f) where g is a monotonic function and f is a homogeneous production function. For various choices of the scale elasticity or returns to scale as a function of output, differential equations are solved to determine the associated forms of the monotonic transformation, g(f). Then by choice of the form of f, the elasticity of substitution, constant or variable, is determined. In this way, we have produced and generalized a number of homothetic production functions, some already in the literature. Also, we have derived and studied their associated cost functions to determine how their shapes are affected by various choices of the scale elasticity and substitution elasticity functions. In general, we require that the returns to scale function be a monotonically decreasing function of output and that associated average cost functions be U- or L-shaped with a unique minimum. We also represent production functions in polar coordinates and show how this representation simplifies study of production functions' properties. Using data for the US transportation equipment industry, maximum likelihood and Bayesian methods are employed to estimate many different generalized production functions and their associated average cost functions. In accord with results in the literature, it is found that the scale elasticities decline with output and that average cost curves are U- or L-shaped with unique minima. © 1998 John Wiley & Sons, Ltd.

Formulation of models for observations and prior densities for their parameters is an important activity in many sciences. In the present paper, after a discussion of this area of activity, entropy-based methods are employed to derive many central econometric and statistical models and noninformative and informative prior densities for their parameters in an explicit, reproducible manner. Examples are provided to illustrate the general procedures. In particular, maxent is employed to produce linear and nonlinear regression and autoregression models, hierarchical models, time-varying parameter models, etc. Then maximal data information prior (MDIP) densities for hyperparameters, common parameters in different likelihood functions, multinomial parameters, etc., are derived. Also the MDIP approach is utilized to produce prior odds for alternative hypotheses or models.

In this article salient aspects of the past, present and future of econometrics are considered. These include a resume of past key developments in econometric modeling, inference and uses of econometrics. Further, some comments are made relating to various statistical inference procedures, techniques of model formulation and the relations of theory and application. It is concluded that a stronger interaction between theory and application will do much to promote further progress in econometrics in the future.

This article presents new operational convergence criteria for the Gibbs sampler (GS) and related procedures that are useful for determining whether they not only have converged but also have converged to provide reliable results. Three GS convergence criteria are presented and applied: the difference convergence criterion (DC2), the ratio convergence criterion (RC2), and the anchored ratio convergence criterion (ARC2). Their uses and properties are discussed and examples are analyzed to illustrate their application.

We show how several models with moving average errors can be easily rewritten as models with autoregressive errors, thereby simplifying inference.

Using several variants of a Marshallian Macroeconomic Model (MMM), see Zellner and Israilevich (2005) and Ngoie and Zellner (2012), this paper investigates how various tax rate reductions may help stimulate the U.S. economy while not adversely affecting aggregate U.S. debt. Variants of our MMM that are shown to fit past data and to perform well in forecasting experiments are employed to evaluate the effects of alternative tax policies. Using quarterly data, our one-sector MMM has been able to predict the 2008 downturn and the 2009Q3 upturn of the U.S. economy. Among other results, this study, using transfer and impulse response functions associated with our MMM, finds that permanent 5 percentage points cut in the personal income and corporate profits tax rates will cause the U.S. real GDP growth rate to rise by 3.0 percentage points with a standard error of 0.6 percentage points. Also, while this policy change leads to positive growth of the government sector, its share of total real GDP is slightly reduced. This is understandable since short run effects of tax cuts include the transfer of tax revenue from the government to the private sector. The private sector is allowed to manage a larger portion of its revenue while government is forced to cut public spending on social programs with little growth enhancing effects. This broadens private economic activities overall. Further, these tax rate policy changes stimulate the growth of the federal tax base considerably which helps to reduce annual budget deficits and the federal debt.

This paper analyzes the production process of scientific outputs and its implications on the U.S. economy using variants of a disaggregated Marshallian Macroeconomic Model (MMM). Federal spending on scientific activities produces innovation which we measure using the number of patents awarded. Additionally, this study makes use of the Bass diffusion model to investigate how innovative patents generate new products that attract new firms in existing sectors of the U.S. economy. Firms are assumed to be Bayesian learners while forming expectations about product prices. Using a set of policy simulations, this research provides measured information on how selected science policies may affect sectoral growth of the U.S. economy. Moreover, issues such as bifurcation pertaining to dynamic models are thoroughly addressed in this paper. Among others, our findings suggest that federal spending on applied research has larger shortrun growth enhancement effects than spending on development or basic research. The return of current federal spending on applied research depends largely on past spending on basic research, something that is well captured through the lag structure imposed in our model. Recipients of federal grants for basic research often lay foundation for outstanding applied research.

We discuss Bayesian inferential procedures within the family of instrumental variables regression models and focus on two issues: existence conditions for posterior moments of the parameters of interest under a flat prior and the potential of Direct Monte Carlo (DMC) approaches for efficient evaluation of such possibly highly onelliptical posteriors. We show that, for the general case of m endogenous variables under a flat prior, posterior moments of order r exist for the coefficients reflecting the endogenous regressors’ effect on the dependent variable, if the number of instruments is greater than m r, even though there is an issue of local non-identification that causes non-elliptical shapes of the posterior. This stresses the need for efficient Monte Carlo integration methods. We introduce an extension of DMC that incorporates an acceptance-rejection sampling step within DMC. This Acceptance-Rejection within Direct Monte Carlo (ARDMC) method has the attractive property that the generated random drawings are independent, which greatly helps the fast convergence of simulation results, and which facilitates the evaluation of the numerical accuracy. The speed of ARDMC can be easily further improved by making use of parallelized computation using multiple core machines or computer clusters. We note that ARDMC is an analogue to the well-known 'Metropolis-Hastings within Gibbs' sampling in the sense that one 'more difficult' step is used within an 'easier' simulation method. We compare the ARDMC approach with the Gibbs sampler using simulated data and two empirical data sets, involving the settler mortality instrument of Acemoglu et al. (2001) and father's education's instrument used by Hoogerheide et al. (2012a). Even without making use of parallelized computation, an efficiency gain is observed both under strong and weak instruments, where the gain can be enormous in the latter case.

In this chapter an account of our experiences in modeling and forecasting the annual output growth rates of 18 industrialized countries is presented. A s tructural e conometric m odeling, t ime- s eries a nalysis (SEMTSA) approach is described and contrasted with other approaches. Theoretical and applied results relating to variable and model selection and point and turning-point forecasting are discussed. A summary of results and directions for future research concludes the chapter. Introduction In this chapter we shall provide an account of some of our experiences in modeling, forecasting, and interpreting time series data. Since the literature on these topics is so extensive, a comprehensive survey would require one or probably more volumes. Thus, we have decided to describe our approach, the experience that we have had with it, and its relation to a part of the statistical and econometric time series literature. Obtaining good macroeconomic and microeconomic and other time series models is important since they are useful in explanation, prediction, and policy-making or control. The basic issue that is addressed in this chapter is how to produce such good time series models. Our SEMTSA approach (see, e.g., Zellner and Palm 1974, 1975; Plosser 1976, 1978; Zellner 1979, 1984, 1991; Palm 1983; Wallis 1983; Webb 1985; Manas-Anton 1986; Hong 1989; Min 1991) will be briefly compared to several other approaches that have emerged in the literature. Rather than just present theoretical procedures that may be useful in producing good models, an account will be given of both theoretical and applied results in what follows.

The purpose of this paper is to consider estimation of the mean time to failure using loss functions that reflect both of fit and precision of estimation. We show how this can be done using balanced loss functions (BLF) of the type introduced in Zellner (1994) and weighted balanced loss function (WBLF) introduced in this paper. Optimal point estimates relative to BLF and WBLF are shown to be a compromise between usual Bayesian and non-Bayesian estimates. Using diffuse and informative priors, posterior expected losses associated with alternative estimates are evaluated and compared.

See also the publication in 'Econometric Reviews' , 2014, 33(1-2), 3-35. A Direct Monte Carlo (DMC) approach is introduced for posterior simulation in theInstrumental Variables (IV) model with one possibly endogenous regressor, multipleinstruments and Gaussian errors under a flat prior. This DMC method can also beapplied in an IV model (with one or multiple instruments) under an informativeprior for the endogenous regressor's effect. This DMC approach can not be appliedto more complex IV models or Simultaneous Equations Models with multiple endogenous regressors. An Approximate DMC (ADMC) approach is introduced thatmakes use of the proposed Hybrid Mixture Sampling (HMS) method, which facilitates Metropolis-Hastings (MH) or Importance Sampling from a proper marginalposterior density with highly non-elliptical shapes that tend to infinity for a pointof singularity. After one has simulated from the irregularly shaped marginal distri-bution using the HMS method, one easily samples the other parameters from theirconditional Student-t and Inverse-Wishart posteriors. An example illustrates theclose approximation and high MH acceptance rate. While using a simple candidatedistribution such as the Student-t may lead to an infinite variance of ImportanceSampling weights. The choice between the IV model and a simple linear model un-der the restriction of exogeneity may be based on predictive likelihoods, for whichthe efficient simulation of all model parameters may be quite useful. In future workthe ADMC approach may be extended to more extensive IV models such as IV withnon-Gaussian errors, panel IV, or probit/logit IV.

Bayesian methods for combining models and their forecasts are developed and applied using various time series models which yield forecasts of output growth rates for eighteen countries, 1974–87. Optimal Bayesian combining procedures involve use of posterior odds for alternative models which are derived and computed for several fixed parameter vs. time-varying parameter models. These odds shed light on the parameter constancy issue and are also used in predictive tests to decide whether or not combine forecasts of alternative models since, as indicated in the paper, it is not always optimal to combine forecasts. Finally, various Bayesian pooling techniques and non-Bayesian forecast combining techniques are described and their performance in forecasting one-year-ahead output growth rates of eighteen countries, 1974–87, is reported.

Computationally efficient simulation methods for hierarchical Bayesian analysis of the seemingly unrelated regression (SUR) and simultaneous equations models (SEM) are proposed and applied. These methods combine a direct Monte Carlo (DMC) approach and an importance sampling procedure to calculate Bayesian estimation and prediction results, namely, Bayesian posterior densities for parameters, predictive densities for future values of variables and associated moments, intervals and other quantities. The results obtained by our approach are compared to those yielded by use of MCMC techniques. Finally, we show that our algorithm can be applied to the Bayesian analysis of state space models.

This paper addresses issues such as: Does it always pay to combine individual forecasts of a variable? Should one combine an unbiased forecast with one that is heavily biased? Should one use optimal weights as suggested by Bates and Granger over twenty years ago? A simple model which accounts for the main features of individual forecasts is put forward. Bayesian analysis of the model using noninformative and informative prior probability densities is provided which extends and generalizes results obtained by Winkler (1981) and compared with non-Bayesian methods of combining forecasts relying explicitly on a statistical model for the individual forecasts. It is shown that in some instances it is sensible to use a simple average of individual forecasts instead of using Bates and Granger type weights. Finally, model uncertainty is considered and the issue of combining different models for individual forecasts is addressed.

(1992). Statistics, Science and Public Policy. Journal of the American Statistical Association: Vol. 87, No. 417, pp. 1-6.