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 924 this result are the same as those for the asymp Theorem yieldS the following expression for the totic normality of the MLE. However, when the posterior odds, denoted by K12 . observations are stochastically dependent, as in K12 (K / K 2) [Pi (yIH )/p (yIH) J (7) time series problems, Heyde and Johnstone (1979) l l 2 show the conditions needed for the asymptotic (Prior odds) x (Bayes' Factor) normality of a posterior pdf are simpler and more robust than those needed to prove the as where the Bayes' Factor, PI (yIHl )/P2(yl-H2) is the ratio of the pdfs for y under HI and under ymptotic normality of the MLE. Thus. Bayesians have asymptotic results available wh1ch can be H2' If HI and H2 are both simple hypotheses, then the Bayes' Factor is identical to the like employed as approximations in finite.sam~les .. The quality of the asymptotic approx1mat10ns 1S lihood ratio. If Hl and H2 are composite hy potheses involving free parameters 8 and 8 , an issue, however, in both Bayesian and non 1 2 respectively, then the pdf's in (7) are given by Bayesian approaches. By computing exact poste rior pdfs, this issue is avoided in Bayesian p. (yIH.) =ip. (yIS.,H.)p(s.IH.)d8. i=1,2 (8) 1 1 1 1 1 1 1 1 analyses. See Zellner and Rossi (1984) for com parisons of exact and asymptotic results for where Pi(yI8i ,Hi) is the conditional pdf for y logit models which indicate that asymptotic given ei and Hi and p(8iIHi) is the prior pdf approximations are not very accurate in small for e i given Hi' When the pdfs in (8) are in to moderate sized samples. serted in the Bayes' Factor in (7), it is seen With respect to Bayesian prediction proce that it becomes a ratio of averaged likelihood dures, let Yf denote a vector of as yet unob functions. served observations with pdf P(Yf1e), where e Bayes' Factors have been computed for a is a vector of parameters. The predictive pdf wide range of alternative models and hypotheses- for Yf, given D == (y,10)' the sample and prior see e.g. Jeffreys (1967), Geisel (1975), Leamer information is (1978), Monahan (1983), Leamer (1978), Rossi (5) (1980, 1984), and Zellner (1971, 1984). Jeffreys (1967), who is a pioneer in this area has deriv where p(eID) is the posterior pdf for e given in ed and applied Bayes'Factors for evaluating hy (1). Thus the predictive pdf incorporates both potheses about meanS, variances, correlation sample and prior information and can readily be coefficients, binomial parameters, etc. Geisel employed to make inferences regarding the pos (1975), Leamer (1978), Zellner (1971, 1978), and sible values of the elements of Yf, e.g. a number of others have considered use of odds pr(a 925 Also posterior odds have been shown to have very consumption and income data, Zellner and Moulton good sampling properties--see e.g. discussion and (1984) , Varian's (1975) real estate assessment references in Zellner (1984). problem, fuld a Bayesian minimum expected loss Above posterior odds were discussed without (MELD) estimate for structural econometric indicating how they can be used to make decisions. models' parameters. In each case, salient If losses associated with errors in choosing hy features of the Bayesian approach will be empha potheses are specified and if posterior proba sized. bilities relating to alternative hypotheses have In Zellner and Moulton (1984), the hypoth been computed, then one can act so as to mini esis that permanent consumption (c) is propor mize expected loss in choice of a model or hy tional to permanent income (y) was investigated pothesis, a procedure which is discussed at using Houthakker's data for 26 countries. Ear length in the literature. For example, if there lier theoretical analysis suggested a possible are equal losses associated with choosing HI departUre from proportionality at very low in when H2 is appropriate and with choosing H2 when come and very high income levels. After noting HI is appropriate, minimizing expected loss will that the use of the linear. relation ci = aO + a l y i involVe choosing HI if K12 > 1 and H2 if K12 < L + £i and the log-log relatlon log ci = So + 13110g Extension to cases in which asymmetric loss Yi + Vi and the condition on permanent consump tion and income 0< c < Yi involve a truncation structures are appropriate is direct--see e.g. i DeGroot (1970), Berger (1980), Leamer (1978), of the dependent variable which affects infer and Zellner (1971). ence results, it was decided to use the follow When choice between ur among alternative ing logit-transformation model (LTM): hypotheses or models is difficult, they can be (10) carried along to be investigated with additional data. Further, predictions from several differ where Wi ::; ci/Y., the consumption income ratio. ent models can be combined using their respec If Y1 = 0, the 60nsumption income ratio does not tive posterior probabilities, a standard BaYes depend on Yi' If Yl < 0, then ci/Yi -+1 as Yi -+0. ian procedure which is optimal relative to Assuming that the ui's are normally and quadratic loss--see Geisel (1975) for an appli independently distributed with zero means and cation of this procedure. Further, if we have common unknown variance, posterior distributions two hypotheses, HI : 6 =: 60 and H2 : 6 ¢ eO where 60 for Yl were computed using diffuse and informa is a given value, with posterior probabilities tive priors. When a diffUse prior for YOI Yl PI and I-PI respectively a~d if Wx have ~ qua and a was employed, it was found that the poste dratic loss fUnction, L(6,e) = (6-6) 'Q(6-6), with rior pdf for Yl was concentrated over negative Q pds, the value of § that minimizes posterior values with posterior mean -.215 and posterior expected loss is standard deviation .115. From the marginal (9) Student-t posterior pdf for Yl, the posterior e* = P l 6 0 + (I-Pl)62 probability that Y1 < 0 was computed to be .969, 8 + [l/(l+K ) 1(6 -8 ) which is very large. With such a negative value 0 12 2 0 for Yl, ci/Yi for low income countries was com where 8 posterior mean of under H2' and 2 e puted to be in the vicinity of .95 while for Kl2 = PIleI-PI) is the posterior odds for HI vs. high income countries it was about .91. Note H • From the first line of (9), the optimal 2 that the value .95 implies a savings rate of 5% point estimate relative to quadratic loss is a while that of .91 yields a much higher savings simple average of 6 , the value oK 6 under HI 0 rate of 9%. The conclusion that Yl is probably and the posterior mean under H , 6 with the 2 2 negative was studied further by taking account posterior probabilities, PI a~d I-PI as weights. of several outlying points, varying the prior The second line of (9) shows 8* to be a "shrink distribution, studying the impact of influential age" estimate with shrinkage factor 1/(1+KI2)' points, and pursuing Bayesian residual analysis Thus there is an intimate relation between and several other diagnostic checks. These cal "averaging over hypotheses" and shrinkage esti culations were performed using the computer pro mates. gram BRAP which provided output conveniently and To summarize with respect to evaluation of relatively inexpensively. The analysis illus hypotheses, Bayesian posterior odds analysis en trates the operational properties of the Bayes ables investigators to evaluate nested and non ian approach in the analysis of a data set which nested hypotheses and models utilizing sample has attracted considerable attention in the eco and prior information conveniently. Prior odds nomics profession. get converted into posterior odds by use of (7). In Varian's (1975) study, he used the well Posterior distributions for parameters and mar known regression approach to tax assessment. ginal distributions of the observations, Pl(yjH ) l This involves a multiple regression model link and P2(y1H2) as shown in (8) are available. ing market prices of houses which were sold in Further, posterior probabilities are useful in a particular year (Yi) to house characteristics obtaining estimates, as shown above, and also in averaging predictions from alternative models. (xi)' that is Yi =xii3+ui' i=1,2, ... ,n or in matrix notation, y = + u. This relation is Applications of posterior odds analysis have been xS usually fitted by least squares and used to pre made in analyzing a wide range of problems with dict the prices of homes which have not been rather good results. sold in the given year given their characteris 4. Discussion of Selected Problems tics. As is well known, use of the least squares prediction is optimal relative to a In this section several specific problems will be briefly discussed, namely an analysis of symmetric, squared error loss function and a 926 diffuse prior distribution for the regression elements of II = (TIl: III) : parameters. Varian formulated and employed (15) several informative prior distributions in the XTT I "" xIIIYl + XISI analysis of his data as well as loosening the assumption of a symmetric, squared error loss where Yl and Sl are ml x 1 and kl x 1 vectors of parameters and ~Xl is an n x kl sub-matrix of X, function. If an over-prediction of a house's that is X = (Xl: X ). using (14) and (15), the market price, leading to an over-assessment for O system can be expressed as: tax purposes is considered more serious than under-prediction by an equal amount, then an (16) asymmetric loss function is needed. With the error of prediction given by Yf-Yf' where Yf is a point prediction and Yf is the unobserved house price being predicted, Varian's LINEX loss func a multivariate non-linear regression model in tion is given by volving the parameters III' Yl' and Sl which we wish to estimate. If we write the restrictions in (15) as with a,b,c > O. See Varian's paper for plots of (17) this function for which L(O) = 0, L' (0) = 0, and L"(O) > O. Also the LINEX loss function rises where Zl = (XIII: Xl) and 6i = (YiSi), a useful almost exponentially for Yf-y f > 0 and almost loss function, denoted by L is linearly for YrY f < O. L = (XTT - Z ( ) , (XlT - 21 ~l) (18) Using the LINEX loss function and a normal I 1 1 I predictive pdf for Yf with mean m and variance (6 -~ ) 'z'z (0 -~ ) v, derived from his Bayesian regression analysis 1 1 III 1 of Y = xS + u using data for San Mateo county in where ~l is some estimate of 6 and in going California, Varian derived the point prediction 1 from the first line of (18) to the second, (17) that minimizes expected LINEX loss. It is was employed. It is seen that L reflects the Yf = m - av/2 + (l/a) 5/,n (c/ab) (12) extent to which the restrictions fail to hold when 6 is inserted for 6 and also is in the Note that Y~ departs from the mean m of the pre 1 1 form of a generalized quadratic loss function. dictive pdf because of the asymmetry of the Now L depends On IT = (TTl: IT ), the coefficient LINEX loss function. Only if a is small and l matrix in (13). If we have a posterior pdf for c = ab will be close to m which is the optimal y¥ IT that has finite first and second moments, we point forecast for a symmetric, squared error can use it to compute expected loss, loss function. See Varian's paper for calcula tions showing the effects of using rather EL = ElT'X'XTT - 26'EZ'XTT + 6'EZ'Z 6 Yf 1 1 III 1 III than m as a point prediction of Yf' The last problem to be discussed in this which can easily be minimized with respect to 61 section is a difficult one, namely estimation of to provide the Bayesian MELO estimate, 8t, coefficients in linear structural econometric == (EZiZl) -lE2iXTTI' (19) models. Currently, most sampling theory esti 8i mators such as 2SLS, LIML, 3SLS, FIML, etc. are This is a very simple, general solution to a given asymptotic justifications. While much rather difficult problem. In the references Bayesian research has been done to provide com cited earlier, explicit forms for 8f are derived, plete posterior distributions for structural applied and their properties are studied. See parameters--see e.g., Dreze and Richard in also Swamy (1980) and Swamy and Mehta (1983). Zellner (1980) and the references contained in it, here just a simple, Bayesian minimum expect 5. Concluding Remarks ed loss (MELD) approach for generating point From what has been presented above, it is estimates will be described. See Zellner (1978), the case that there is evidence of more wide Zellner and Park (1979) and Park (1982) for spread use of Bayesian methods in econometrics further results and applications. and statistics and that results obtained have Given an n x m matrix of observations on m been rather good. Some general principles of dependent or endogenous variables Y generated Bayesian analysis have been reviewed to indi by a multivariate regression or reduced form cate their generality and usefulness. A few system, specific analyses have been considered to indi Y=xII+v (13) cate the operational features of Bayesian analy sis. Overall, the conclusions that I suggest where X is a given n,x k matrix of rank k, II is a are (1) Given adVances in computer techniques k x m regression or reduced form coefficient and numerical integration techniques, very flex matrix, and V is an n x m matrix of errors or ible and useful Bayesian methods are currently reduced form disturbances. Now partition Y = available. (2) The review of a number of applied (YI: Y ) and write I Bayesian studies has indicated that the Bayesian (Y : Y ) = X(TT : III) + (vI: VI) (14) approach, carefully applied, produces very good I l I results. (3) Wise use of Bayesian procedures where Yl, TTl and VI are column vectors of order can lead to improved econometric and statistical nXl, kxl and nxl respectively. In structural solutions to estimation, testing, prediction, econometric models, subject matter considera design, control and policy problems. tions lead to the following restrictions on the 927 References Akaike, H. and M. Ishiguro (1983), "comparative Leonard J. Savage, Amsterdam: North-Holland, Study of the x~ll and BAY SEA Procedures of 227 256. Seasonal Adjustment," in A. Zellner (ed.), Goel, P.K. and A. Zellner (eds.) (1985), Bayes Applied Time Series Analysis of Economic Data, ian Inference and Decision Techniques with Washington, DC: GPO, 17 53 (with discussion). Applications: Essays in Honor of Bruno de Bawa, V.S., Brown, S.J. and R.W. Klein (1979), Finetti, Amsterdam: North Holland (in press) Estimation Risk and Optimal Portfolio Choice, Good, I.J. (1950), Probability and the Weighing Amsterdam: North Holland. of Evidence, London: C. Griffin and Co. Baxter, M. (1983) t "The Role of Expectations in Good, I.J. (1965), The Estimation of Probabili Stabilization Policy t" presented at the Econo ties, Cambridge, MA: MIT press. metric Society meeting, Pisa, Italy. Griffiths, W.E. and D. Dao (1980), "A Note on Berger, J.O. (1980), Statistical Decision Theory, the Bayesian Estimator in an Autocorrelated New York: Springer-Verlag. Error Model," Journal of Econometrics, 12, Bernardo, J.M., DeGroot, M.H., Lindley, D.V., 389-392. and A.F .M. Smith (eds.) (1980), Bayesian Sta Harkema, R. (1975), "An Analytical Comparison tistics: Proceedings of the First International of Certainty Equivalence and Sequential up Meeting Held in Valencia (Spain), May 28 to dating," Journal of the American Statistical June 2, 1979, valencia, Spain: University Press. Association, 70, 348-350. Bowman, H.W. and A.M. Laporte (1975), "Stochas Harrison, P.J. and C.F. Stevens (1976), "Bayes tic Optimization in Recursive Equation Systems ian Forecasting," Journal of the Royal Statis with Random Parameters with an Application to tical Society, B, 38, 205-228. Control of the Money Supply," in S.E. Fienberg Harrison, P.J., West~M. and H.S. Migon (1984), and A. Zellner (eds.), Studies in Bayesian "Dynamic Generalized Linear Models and Bayes Econometrics and Statistics in Honor of Leonard ian Forecasting," Theory and Methods Invited J. Savage, Amsterdam: North-Holland, 427-440. Paper, presented at ASA Meeting, to be pub Box, G.E.P. and G.C. Tiao (1973), Bayesian Infer lished in the Journal of the American Statis ence in Statistical Analysis, Reading, MA: tical Association. Addison Wesley. Heyde, C.C. and I.M. Johnstone (1979), "OnAsymp Broemeling, L.D. (1985), Bayesian Analysis of totic Posterior Normality for Stochastic Pro Linear Models, New York: Marcel Dekker, Inc. cesses," Journal of the Rpyal Statistical Brown, S.J. (1976), Optimal Portfolio Choice Society, B, 41, 184 189. Under Uncertainty, Ph.D. Thesis, Graduate Highfield, R.A-:-(l984), "Forecasting with Bayes School of Business, u. of Chicago. ian state Space Models," ms., Graduate School Cooley, T.F. and S.F. LeRoy (1981), "Identifica of Business, U. of Chicago. tion and Estimation of Money Demand," American Jaynes, E.T. (1980), Papers on probability, Sta Economic Review, 71, 825-844. tistics and Statistical Physics, edited by DeGroot, M. (1970) ,-Optimal Statistical Deci R.D. Rosenkrantz, Dordrecht, Holland: De sions, New York: McGraw Hill Book Co. Reidel Publishing Co. Doan, T., Litterman, R.B. and C.A. Sims (1984), Jeffreys, Sir Harold (1957), Scientific Infer "Forecasting and Conditional Projection using ence, 2nd ed., Cambridge: Cambridge U. Press. Realistic Prior Distributions," Econometric Jeffreys, Sir Harold (1967), Theory of Probabil Reviews, 3, 1-144. ity, 3rd rev. ed., London: Oxford u. Press. Evans, P. (1978), "Time Series Analysis of the Jorion, P. (1983), Portfolio Analysis of Inter German Hyperinflation," International Economic national Equity Investments: Bayes Stein Esti Review, 19, 195-209. mators for the Mean and Implications for Risk Fienberg, ~E. and A. Zellner (eds.) (1975), Reduction, Ph.D. Thesis, Graduate School of Studies in Bayesian Econometrics and Statistics Business, U. of Chicago. in Honor of Leonard J. Savage, Amsterdam: Kanjii, G.K. (ed.) (1983), "Proceedings of the North Holland. Annual Conference on Practical Bayesian Sta Flood, R. and P. Garber (1980a), "An Economic tistics," Journal of the Institute of Statis Theory of Monetary Reform," Journal of Politi ticians, 32, 1 278. cal Economy, 88, 24-58. Leamer, E.E-:-(1978), Specification Searches, New Flood, R. and P-:-Garber (1980b), "Process Con York: John Wiley & Sons, Inc. sistency and Monetary Reform: Further Evidence Lee, T.C., Judge, G.G. and A. Zellner (1975), and Implications," ms., Dept. of Economics, U. Estimating the Parameters of the Markov Proba of Rochester and U. of Virginia. bility Model from Aggregate Time Series Data, Fomby, T.B. and O.K. Guilkey (1978), "On Choos 2nd ed., Amsterdam: North-Holland. ing the Optimal Level of Significance for the Lindley, D.V. (1965), Introduction to Probabil Durbin-watson Test and the Bayesian Alterna ity and Statistics from a Bayesian Viewpoint tive," Journal of Econometrics, 8, 203-213. (2 vols.), Cambridge: Cambridge U. Press. Geisel, M.S. (1970), Comparing and Choosing Lindley, D.V. (1971), Bayesian Statistics, A. Among Parametric Statistical Models, Ph.D. Review, Philadelphia: Soc. Industrial and Ap Thesis, Graduate School of Business, U. of plied Mathematics. Chicago. Litterman, R.B. (1980), "A Bayesian Procedure Geisel, M.S. (1975), "Bayesian Comparison of for Forecasting with Vector Autoregressions," Simple Macroeconomic Models," in S.E. Fienberg ms., Dept. of Economics, MIT. and A. Zellner (eds.), Studies in Bayesian Merton, R.C. (1980), "On Estimating the Expected Econometrics and Statistics in Honor of Return on the Market: An Exploratory Investi- 928 gation," Journal of Financial Economics, .§., Thornber, H. (1967), "Finite Sample Monte Carlo 323-361. Studies: An Autoregressive Illustration," Miller, R. (1980), "Actuarial Applications of Journal of the American Statistical Associa Bayesian Statistics," in A. Zellner, Bayesian tion, 62, 801-808. Analysis in Econometrics and Statistics: Essays Tsurumi,H. (1976), "A Bayesian Test of the in Honor of Harold Jeffreys, Amsterdam: North Product Life Cycle Hypothesis Applied to Holland, 197-212. Japanese Crude Steel Production, 11 Journal of Monahan, J.F. (1983), "Fully Bayesian Analysis Econometrics, 4, 371-392. of ARIMA Time Series Models," Journal of Econo Tsurumi, H. and Y. Tsurumi (1981), "U.S.-Japan metrics, 21, 307-332. Automobile Trade: A Bayesian Test of a Product Morris, C.N-:-(1983), "Parametric Empirical Bayes Life Cycle," ms., Bureau of Economic Research, Inference: Theory and Applications," Journal of Rutgers u. the American Statistical Association, ~, 47- Varian, H.R. (1975), "A Bayesian Approach to 54. Real Estate Assessment," in S.E. Fienberg and Park, S.-B. (1982), "Some Sampling Properties of A. Zellner (eds.), Studies in Bayesian Econo Minimum Expected Loss (MELO) Estimators of metrics and Statistics in Honor of Leonard J. Structural Coefficients," Journal of Economet Savage, Amsterdam: North Holland, 195 208. rics, 18, 295-312. Wright, R.L. (1983), "Measuring the Precision of Pe~S.C=- (1974), "Alternative Investment Models Statistical Cost Allocations," Journal of for Firms in the Electric Utilities Industry," Business and Economic Statistics_, !, 93-100. Bell Journal of Economics and Management Sci Zellner, A. (1971), An Introduction to Bayesian ence, ~, 420-548. Inference in Econometrics, New York: John Prescott, E.C. (1975), "Adaptive Decision Rules Wiley. for Macroeconomic Planning," in S.E. Fienberg Zellner, A. (1978), "Estimation of Functions of and A. zellner (eds.), Studies in Bayesian population Means and Regression Coefficients Econometrics and Statistics in Honor of Including Structural Coefficients: A Minimum Leonard J. Savage, Amsterdam: North Holland, "Expected wss (MELO) Approach," Journal of 427-440. Econometrics, 8, 127-158. Press, S.J. (1980), "Bayesian Computer programs," Zellner, A. (ed.) (1980), Bayesian Analysis of in A. Zellner (ed.), Bayesian Analysis in Econometrics and Statistics: Essays in Honor Econometrics and Statistics: Essays in Honor of Harold Jeffreys, Amsterdam: North Holland. of Harold Jeffreys, Amsterdam: North-Holland, Zellner, A. (1984), Basic Issues in Econometrics, 429-442. Chicago: U. of Chicago Press. Reynolds, R.A. (1980), A Study of the Relation Zellner, A. and M.S. Geisel (1968), "Sensitivity ship between Health and Income, Ph.D. Thesis, of Control to Uncertainty and Form of the Cri Dept. of Economics, U. of Chicago. terion Function," in D.G. Watts (ed.), The Rossi, P.E. (1980), "Testing Hypotheses in Multi Future of Statistics, New York: AcademiC-Press, variate Regression: Bayes vs. Non-Bayes Pro Inc., 269-289. cedures," ms., Graduate School of Business, U. Zellner, A. and H.S. Geisel (1970), "Analysis of of Chicago. Distributed Lag Models with Applications to Rossi, P.E. (1984), Specification and Analysis Consumption Function Estimation," Econometrica, of Econometric Production Models, Ph.D. Thesis, ~, 865-888. Graduate School of Business, U. of Chicago. Zellner, A. and B.R. Moulton (1984), "Bayesian Sankar, U. (1969), Elasticities of Substitution Regression Diagnostics with Applications to and Returns to Scale in Indian Manufacturing International Consumption and Income Data," to Industries, Ph.D. Thesis, Dept. of Economics, appear in Journal of Econometrics' Annals of U. of Wisconsin, Madison. Appl~ed Econometrics. Sankar, U. and V.K. Chetty (1969), "On Estima Zellner, A. and S .-B. Park (1979), "Minimum Ex tion of the CES Production Function," Review pected Loss (MELO) Estimators for Functions of of Economic Studies, 36. Parameters and Structural Coefficients of Savage, L.J. (1954), The-Foundations of Statis Econometric Models," Journal of the American tics, New York: John Wiley. Statistical Association, 74, 185-193. Shiller, R.J. (1973), "A Distributed Lag Estima Zellner, A. and J.-F. Richard (1973), "Use of tor Derived from Smoothness Priors," Econo Prior Information in the Analysis and Estima metrica, 41, 775-788. tion of Cobb-Douglas Production Function Surekha, K. and W.E. Griffiths (1983), "A Monte Models," International Economic Review, 14, Carlo Comparison of Some Bayesian and Sampling 107-119. Theory Estimators in Two Heteroscedastic Error Zellner, A. and P.E. Rossi (1984), "Bayesian Models," ms., Indian Statistical Institute, Analysis of Dichotomous Quantal Response New Delhi and U. of New England, Armidale, Models," Journal of Econometrics, 25, 365-393. Australia. Zellner, A. and A.D. Williams (1973)~"Bayesian Swamy, P .A. V.B. (1980), "A Comparison of Esti Analysis of the Federal Reserve-MIT-Penn mators for Undersized Samples," Journal of Model's Almon Lag Consumption Function," Econometrics, 14, 161-181. Journal of Econometrics, !, 267-299. Swamy, P.A.V.E. and J.S. Mehta (1983), "Further Results on Zellner's Minimum Expected Loss and Full Information Maximum Likelihood Estimators for Undersized Samples," Journal of Business and Economic Statistics, !, 154-162. 929