Further sampling properties of previously derived (Zellner 1975) MELO estimates for reciprocals and ratios of parameters and for structural coefficients of econometric models are presented. Approximate small-sample moments and risk functions of MEW estimators for these problems are derived and compared with those of other well-known estimators. The exact finite sample distribution of the MEW reciprocal mean estimator is derived and compared with that of the maximum likelihood estimator. Properties of alternative structural coefficient estimators are compared with those of the MEW estimator both theoretically and in terms of two applications.KEY WORDS: Bayes estimators; Ratio estimation; Simultaneous equations estimation j Maximum likelihood estimators.well-known structural econometric models, Klein's Model I and the Girshick-Haavelmo food model.
FINITE SAMPLE PROPERTIES OF MELO
ESTIMATORS FOR RECIPROCALSWe assume that the observations Yi, i = 1, 2, ... , n, are independently drawn from a normal population with mean /.I (/.I~0) and variance u 2 • Our problem is to estimate 8 = 1//.1. As is well-known, the ML estimator for 8, @ Journal of the American Statistical Association March 1979, Volume 74, Number 365 Theory and Methods Section where E,. == posterior expectation operator and jl and varu are the posterior mean and variance of /.I, respectively. If u 2 = u o 2 , a known value, and a diffuse prior pdf for /.I is employed in the normal mean problem, then jl = fi, var/.l = u o 2 /n, and from (2.2), (2.2) (2.1) (2.3) e(1 + 2 2 ) , {j = 1/'ii , A 1 1 8 = ------fj 1 + uN(nfj2) where fi = L~=l Yi/n, does not possess finite moments and has infinite risk relative to quadratic and many other loss functions. As pointed out in Zellner (1978), a MELO estimate of 8 relative to L.(O, 0) = [(0 -0)/OJ2 (1 -/.10)2 is A 1 ( 1 ) O-E E 2 _ _ -,./.I/,.fJ. -_ 1 + / 2 ' /.I var/.l il where e 2 = uo 2/n and 2 = fi/e has a N(/.I/e, 1) sampling pdf. Thus, the MELO estimator in (2.3) is equal to the ML estimator, 1/fi, multiplied by a "shrinking factor" 2 2 / (1 + 2 2 ) . Note that 2 = n1fi/u o is a sampling theory test statistic for testing /.I = 0; and also the reciprocal of the coefficient of variation of the posterior pdf for /.I. As pointed out in Zellner (1978), 101~te and thus 0 possesses finite moments of all orders and finite risk relative to L.(8, 0) and many other loss functions. Also, when 121 < 1, the posterior pdf for 8 shows pronouncedbimodality and special treatment of this problem within the MELO estimation framework is required-see the Appendix of Zellner (1978).