APPLICATIONS IN BAYESIAN ANALYSIS Arnold Zellner university of Chicago 1. Introduction The rapid growth of Bayesian econometrics ment and consumption data. Production function and statistics since the 1950s has involved many models have been analyzed from the Bayesian applications of Bayesian inference and decision point of view by Sankar (1969), Sankar and techniques to a wide range of problems and the Chetty (1969), Zellner and Richard (1973), and development of Bayesian computer programs--see, Rossi (1980, 1984). Tsurumi (1976) and Tsurumi e.g., Press (1980). This experience in applying and Tsurumi (1981) used Bayesian techniques to Bayesian techniques has indicated that Bayesian analyze structural change problems. Reynolds solutions to applied problems are as good or bet­ (1980) developed and applied Bayesian estimation ter than non-Bayesian solutions. Indeed, it has and testing procedures in an analysis of survey been shown that many non-Bayesian results can be data relating to health status, income and other obtained by Bayesian methods under special as­ variables. Harrison and Stevens (1976), Harri­ sumptions. However, these special assumptions son, West and Migon (1984), Litterman (1980), are often unsatisfactory and thus full Bayesian Doan, Litterman and Sims (1984), and Highfield solutions, which relax them, are preferable. (1984) have developed and applied Bayesian fore­ Further, as is well known, the Bayesian approach casting models. Merton (1980), used Bayesian permits the flexible and formal use of prior in­ estimation procedures in a study of stock market formation in obtaining solutions to applied in­ data. Akaike and Ishiguro (1983) have developed ference and decision problems, a feature of the a Bayesian approach and a computer program, BAYSEA, Bayesian approach which is particularly valuable for seasonal analysis and adjustment. Morris when sample information is limited. These and (1983) has discussed and referenced many appli­ other features of the Bayesian approach will be cations of empirical Bayes' procedures. Pres­ discussed further in what follows. cott (1975), Bowman and Laporte (1975), Harkema The plan of the paper is as follows. In (1975), Zellner (1971, Ch.ll), and Zellner and Section 2 references will be provided to a number Geisel (1968) have compared Bayesian and non­ of applications of Bayesian analyst's. Some gen­ Bayesian solutions to several control problems. eral features of Bayesian applied studies will be Swamy (1980) and Swamy and Mehta (1983) develop­ discussed in Section 3. In Section 4, a few ex­ ed and applied Bayesian solutions to the "under­ plicit applied results are discussed to illus­ sized sample" problem. Monahan (1983) has trate several points made in Section 3. Conclud­ developed and applied Bayesian model selection, ing remarks are provided in Section 5. estimation and prediction procedures for Box­ 2. Applications of Bayesian Analysis Jenkins' ARMA time series models. Wright (1983) used Bayesian methods to solve an overhead cost It should be appreciated that Bayesian solu­ allocation problem. Miller (1980) reviewed tions to many standard inference and decision Bayesian applications in actuarial statistics. problems and some applications are included in Many applications of Bayesian analysis are the following general works: Jeffreys (1957, reported in Kanjii (1983). Brown (1976), Bawa, 1967), Good (1950, 1965), Savage (1954), Lindley Brown and Klein (1979) and Jorion (1983) devel­ (1965, 1971), DeGroot (1970), Zellner (1971, oped and applied Bayesian procedures for port­ 1984), Leamer (1978), Box and Tiao (1973), Berger folio choice problems. For further references, (1980), Jaynes (1980), and Broeme1ing (1985). see the many entries under the key words "Bayes" Also the following volumes of collected papers and "Bayesian" in annual volumes of the ASA/IMS contain many valuable theoretical and applied Current Index to Statistics. contributions: Fienberg and Zellner (1975), The references cited above are just a sam­ Bernardo et al. (1980), Zellner (1980), and Goel ple of applied Bayesian studies. In all of and Zellner (1985). them, application of relatively simple Bayesian As regards some specific, relatively recent procedures have been found to yield rather good interesting applications of Bayesian analysis, results. In a number of these studies, explicit Geisel (1970, 1975) used Bayesian prediction comparisons of Bayesian and non-Bayesian solu­ procedures and posterior odds to compare the tions are presented with the general finding relative performance of simple Keynesian and that Bayesian solutions are as good or better Quantity of Money models. Peck (1974) utilized than non-Bayesian solutions. For example, Lit­ Bayesian estimation techniques in an analysis of terman (1980) showed that forecasts of U.S. investment behavior of firms in the electric quarterly macroeconomic variables obtained from utility industry. varian (1975) developed and his Bayesian vector autoregressive model are applied Bayesian methods for solving real estate better than those provided by univariate Box­ tax assessment problems. Flood and Garber Jenkins' models and by two large-scale econo­ (1980a,b) and Baxter (1983) applied Bayesian metric models. In work on portfolio choice methods in the study of several monetary reforms. problems and control problems, Bayesian solu­ Evans (1978) employed posterior odds to compare tions have been shown to be superior to certain­ alternative models of the German hyperinflation. ty equivalence solutions. In Swamy (1980) and Cooley and LeRoy (1981), Shiller (1973), Zellner Swamy and Mehta (1983), Bayesian solutions to and Geisel (1970), and Zellner and Williams the "undersized sample" problem were shown to :;. (1973) employed Bayesian methods in studies of be as good or better than non-Bayesian solutions. time series models for U.S. money demand, invest- In Monte Carlo experiments, Bayesian estimators 923 have been shown to have better sampling proper­ wide range of models including univariate and ties than non-Bayesian estimators--see e.g., multivariate regression models, time series Thornber (1967), Lee, Judge and Zellner (1975), models, logit models, etc. The posterior pdf Fomby and Guilkey (1978), Griffiths and Dao p(8ID) can be employed to make inferences about (1980), Surekha and Griffiths (1983), and Park elements of 8. In some problems analytical (1982) . integration can be employed to evaluate c, the In summary, it is the case that Bayesian normalizing constant in (1), moments of the methods have been shown to be very effective in elements of 8, and to make probability state­ solving applied problems in many areas of study ments about the elements of e, e.g. pr(8i > 010) including economics and business, other social or Pr(o<ej<lID), etc. For problems in which sciences, the natural and biological sciences, integrals cannot be evaluated analytically, etc. Given this record of successful applica­ numerical integration procedures can be employed tion of Bayesian methods, metaphysical arguments --see, e.g., zellner and Rossi (1984) for appli­ about whether to use Bayesian methods are now of cations of this approach and a discussion and minor importance. Currently interest centers on comparison of several numerical integration pro­ determining the extent to which use of Bayesian cedures. Also, analytical and/or numerical methods, which are described in most current integration procedures have been employed to statistics and econometrics textbooks, can pro­ obtain marginal posterior pdfs for subsets or vide better solutions to applied problems. To individual parameters. That is if 8' (8i8;Z)' gain perspective on this issue, in the next sec­ the marginal posterior pdf for 81 is tion some salient features of the use of Bayes­ P(81ID) "Jp(8 ,8)D)d8 (3) ian methods in applied studies will be discussed. 1 2 3. Salient Features of Bayesian Methods " Jp(81IS2,D)p(82ID)d82 In applied estimation problems, Bayesians where the integration is performed over the use Bayes' Theorem to obtain exact finite sample region for 8 2 , This is an extremely useful way results. As is well known, the inputs to Bayes' of getting rid of nuisance parameters. Note Theorem are a likelihood function, denoted by from the second line of (3) that the integration p(yI8,I ) where y is an Observation vector, 8 is can be viewed as an averaging of conditional O a parameter vector and IO denotes initial or posterior pdfs p(81Ie2,D) with the weighting prior information and a prior probability den­ function being the marginal posterior pdf for sity function (pdf), TI(8IIO)' Then application 8 2 , P(82ID). Also, the conditional pdf of Bayes' Theorem yields P(81IS2,D) has been employed in many studies to determine how sensitive inferences about 8 are (1) 1 to assumptions about possible values of e 2 • whare p{8ID) is the posterior pdf with D~ (y,I )' As regards pOint estimation, if we have a O the sample and prior information and c is a nor­ convex loss function L(e,e), where § is some malizing constant given by c-l=!Tf(8IIO) x estimate, then the Bayesian estimate, say DB is p{yI8,IO)d8 with the integration performed over obtained as the solution to the following prob­ the parameter space G. More succinctly, (1) can lem: be expressed as mine EL(8,8) = mine !LC8,8)p(8ID)d8. (4) Posterior pdf 0: (Prior pdf) x (Likelihood function) For example, if L(8,e) = (8-8) 'Q{8-8), with Q a (2) given positive definite symmetric matrix, the value of 8 which minimizes expected loss is where "0:" denotes "is proportional to." The posterior pdf, p(8ID) incorporates both prior BB=E{8ID), the posterior mean. For a scalar parameter, use of Le8 18-81, an absolute and sample information. The prior information ,8) = error loss function, yields the optimal point enters not only via the prior pdf Tf(81Iol but also in formulating the likelihood function. estimate, BB = the median of the posterior pdf. For other loss functions, the problem in (4) can Further (1) can be viewed as a transformation from the prior pdf, which reflects our initial be solved analytically or numerically to obtain the optimal Bayesian estimate.