Bayesian Full Information Analysis of Simultaneous Equation by L. Bauwens

By L. Bauwens

In their overview of the "Bayesian research of simultaneous equation systems", Dr~ze and Richard (1983) - hereafter DR - exhibit the next perspective in regards to the current country of improvement of the Bayesian complete info research of such sys­ tems i) the tactic permits "a versatile specification of the past density, together with good outlined noninformative past measures"; ii) it yields "exact finite pattern posterior and predictive densities". even if, they demand additional advancements in order that those densities might be eval­ uated via 'numerical equipment, utilizing an built-in software program packa~e. as a result, they suggest using a Monte Carlo process, because van Dijk and Kloek (1980) have confirmed that "the integrations might be performed and the way they're done". during this monograph, we clarify how we give a contribution to accomplish the advancements urged by way of Dr~ze and Richard. A simple thought is to take advantage of recognized homes of the porterior density of the param­ eters of the structural shape to layout the significance services, i. e. approximations of the posterior density, which are wanted for organizing the integrations.

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Proof. I. 3); consequently, we know the kernel of the marginal density. ) of p(o) 1. ) = 1. v. IP 1 ,1 (0. 1. 6) p. ) = 1. 1. , 1. 1. i. e. the reciproaa'l of the, integrating 1. 1. , h p (0. 10';') ~ T* ~ I 1. 1. 1. ,. 1. 1. 1. ,. - fl. ) 1. 1. is obtained from B by de'leting the i-1;h ao'lumn; flI is obtained from fl by de'leting the i-1;h aoz,umn and the R,i aorresponding rows; The presence of sence of -~\)* BI Si ' Xi' Hi are funations of s*, fl*, M* defined in Proof. I Pi (or) k (or) in the kernel of is justified in Appendix B.!.

Th 5 models covering a variety of situations, as summarized in the following table : TABLE 1 Main featuPes of models used 8BM Johnston Klein EX W nuaber of ... ) ... ) identitiee structural coefficients Cl) ... coefl. of endogenous variables C1a) ... coeff. 1 ) +1 +1 +1 "I =1 C1r) DUllber of observations CT) prior references of previous Bayesian atudie. e diffuse infomative diffuse KYO (1978) VOK (1980) Bauwens & d 'Alcantara CO) Morales (1971) VDK (1981, Richard (1973) (*) 1982 b) (1981) we thank Jean-Paul Lambert for allowing us to use this model.

Clearly, each approach has some degree of arbitrariness: the value of vo' the ranges of the parameters. or In any case, it would be wise to check the sensitivity 44 of the posterior results with respect to these values, if the model were to be used for decision purposes. Table 4 contains the results, in the same presentation as Table 2. Fi~ures 5 to 7 (see Appendix C) show the marginal importance functions and posterior densities of ai' a2 and Y2 , and the corresponding posterior distribution functions.

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