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Bayesian Analysis of Haavelmo's Models V. K. Chetty Econometrica, Vol. 36, No. 3/4. (Jul. - Oct., 1968), pp. 582-602. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28196807%2F10%2936%3A3%2F4%3C582%3ABAOHM%3E2.0.CO%3B2-H Econometrica is currently published by The Econometric Society. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/econosoc.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Wed Oct 10 07:08:45 2007 Econotnetricm. Vol. 36, No. 3-4 (July-October, 1968) BAYESIAN ANALYSIS O F HAAVELMO'S MODELS1 In this paper, the exact posterior distributions of the parameters of Haavelmo's model I is derived for locally uniform prior distributions. Marginal distrlbutions of the parameters have been obtained for Haavelmo's data. Then the predictive probability density of the model is derived for given values of the exogenous variable, investment. In order to check some of the specifying assumptions, the model IS expanded and analyzed under the assumption that the error terms are generated by a first order autoregressive scheme. Exact finlte sample results are obtained and the posterior distributions are computed for Haavelmo's data. Conditional distributions of the parameters of the model are computed for given values of the autocorrelation parameter. p. in order to assess the effects of departures from our specifying assumptions. Another specifying assumption that is examined concerns the exogenous nature of in~estment.For this. Haavelmo's model 11. in Ivhich investment is assumed to be endogenous. is used. Posterlor d~strlbutionsof the parameters of the model are computed for this model. The ~ensiti\enessof the inference about the parameters of the model to the assumption that investment is exogeneous is studied by computing various conditional distributions for model 11. It is seen that this assumption IS very crucial for Haavelmo's data. F111all).ti$(-,diffcrcnt prior distrlbutions reflect~ng1.0 different \ i t a s about In\estlncnt are introduced. The posterlor distributions of the same parameter are then used to determ~ne how one's prior belief is modified by the sample information. 1. INTRODUCTION IN RECENT YEARS. the Bayesian approach has been used to analyze the robustness of specifying assumptions in stochastic models. Box and Tiao [3] used this approach to assess the effects of a departure from normality in the comparison of variances. Zellner and Tiao [18]. using the Bayesian approach, analyzed the effects of departures from serial independence of the error terms in a multiple regression model on inferences about the model's parameters. In this paper, we make use of Bayesian methods similar to those used in the cited studies to analyze the robustness of assumptions in two simple simultaneous equation econometric models. The exact finite sample posterior distributions of the parameters are derived for given prior distributions of the parameters for the two models. Compared with some models of actual economies, the models that we analyze are very simple. The two models are chosen for convenience in experimenting with the Bayesian techniques, rather than for their inherent economic merits. The plan of the paper is as follows. In Section 2 we state Haavelmo's first model and its specifying assumptions, and review the sampling theory analysis of the model. In Section 3 we analyze the model from the Bayesian point of view and get ' This is a revlsed verslon of a chapter of the author's thesis submitted to the University of Wisconsin [4]. The author wishes to acknowledge Professors A Zellner and Jacques Dr?re and the referees for their valuable comments. BAYESIAN ANALYSIS 583 the exact posterior distribution of the parameters. In Section 4 the predictive density of the model is derived. Specification analysis of the model is done in Section 5. In Section 6 Haavelmo's second model is analyzed from the Bayesian point of vie\{. In Section 7 informative prior distributions for a parameter are introduced for the second model in order to study how the sample modifies the investigator's prior belief. 2. FIRST MODEL A N D SPECIFYING ASSUMPTIONS Haavelmo's first model [9] consists o f a consumption equation and an identity, namely, ( t = l , 2,..., T ) , where c, represents consumers' expenditures in constant dollars per capita, y, is disposable income in constant dollars per capita, and :, is investment expenditures in constant dollars per capita. In this model, investment is assumed to be exogenous. The u's are structural disturbance termsand r and P are structural coefficients. Regarding the disturbance terms, the following assumptions are made: (a) the u's are normally distributed; (b) E(u,) =O and E(u:) = a 2 for every value of t ; c , ) = O for s f 0 : (d) the time series :, ( t = 1, 2. . .., T ) is exogenous in relation to c, and y,. Condition (d) is fulfilled if either (d.1)the sequence :, is a sequence of given numbers, or (d.2) each z , is a random variable that is stochastically independent of u,. The reduced form equation for c, is Haavelmo [9] has shown that a consistent estimate of a can be obtained from the least squares estimate of r,l(l - r) derived from equation (2.3). Since this is a just-identified model, all sampling theory estimation procedures including maximum likelihood, two-stage least squares, three-stage least squares, etc., will lead to the same consistent estimate of r , namely, B = ki (k + 1), where F = (1, T)CT=,c, and Z = (1, T)C:=, 2,. The exact sampling distributions of the least squares and the maximum likelihood estimators of a for this model have been derived by Bergstrom [2] for the special case where the :'s are assumed to be nonstochastic. V. K . CHETTY 3. BAYESIAN ANALYSIS OF THE MODEL The likelihood function for the model's parameters a, fl, and o is given by Regarding the prior probability density functions for the parameters r, fl, and o, we assume (following Jeffreys [lo], Savage [12], Geisser and Cornfield [8], and Tiao and Zellner [14]) that p(a, fl, o) zc a- ', O < r < We believe that our prior beliefs are adequately represented by these pdf's. The analysis of the model, however, can be easily done using different prior distributions. Now, the posterior distribution of the parameters is given by Since we are interested in making inferences about a, we need its marginal distribution. On integrating out fl from (3.2), we have, where \ V ~ = C , - Xand J , G = I:=,wr/'T.Now integrating out a , we have where Ts2 = I:= (cr- By, - fl)2, M = Iy: - ((Iyr)2/'T),and B and fl are the least squares coefficients computed by regressing c, on y,. Rothenberg [ll] obtained this posterior distribution of r when analyzing this modei. In order to evaluate the normalizing factor, we have to perform the following integrat~on: where k=hl;Ts2. In actual analysis, the normalized density function can be easily obtained using the numerical integration methods described in the Appendix. The posterior distribution of r is derived for U.S. data for the period 192941. The data used are given in Table I. Numerical integration methods have been used to evaluate the normalized density and its moments. The mean value of a is found to be ,6597. This can be used as an estimate of a. The mean of the posterior distribution is an optimal point estimate, if the quadratic loss function L(a. B) = (a- ?)2is used. The marginal posterior distributions of r and fl are shown in Fig. I . Here and below we use the symbol p to denote pdf's generally and not one specific pdf. The argument of the function p identifies the particular distribution being considered. 585 BAYESIAN ANALYSIS TABLE l Year Disposable income (dollars per capita). deflated Consumer expenditures (dollars per c a p ~ t a ) , deflated 2 , =I,, - c, Gross investment (dollars per capita), deflated Source of d a t a . Haavelmo [9]. The variance of the distribution of r is found to be ,0043. Since we have the complete distribution of the parameters, we can compute the probability that the parameter lies in any particular interval. Such an interval is called the Bayesian confidence interval. The sampling theory estimate of r for the same data is found to be ,674, with a variance of ,005. .58 .62 .66 a .70 .74 .78 106 108 110 112 114 I16 P E(P-€ ( 0 ) 1 3 =71.3034 E(a)= 0 . 6 6 0 E ( Q - E ( Q ) ) 0.001 ~= E ( Q - E ( a ) I 2 =0 . 0 0 4 € ( a - ~ ( a ) ) ~ = 0 . 0 0 0 2 € ( ~ - € ( ~ ) ) ~ = 1 2 9 . €6 (4~2- ~ ( ~ ) ) ~ = 4 9 2 8 3 . E ( P ) =111.589 FI(,L'KEI.--Marginal posterior distributions of a and /( for Model I V . K. CHETTY 4. PREDICTIVE DENSITY On many occasions we will be interested in making inferences about observations that are as yet unobserved, that is, future observations. In the Bayesian approach, the probability density function for the as yet unobserved observations can be derived and is known as the predictive pdf. Here we derive the predictive pdf for the investment cT+,-the consumption expenditures in period T + 1-given expenditures zT+,. The joint posterior distribution of cT+,, cc, P, and a is, P(c,+,,cc,fl,oldata) = p(cT+llr,P,~,data)~(~,P,oldata) Now and x exp Hence /- r - (1- rl7 p(cr+ l,cc,fl,aldata) x -- a T+2 + Let M', = C, - sc (1 - r)z,. Then. Hence (1 - r ) T L p(cT+l.sc.fl,oldata) x 7 3 5 - 2a2 BAYESIAN ANALYSIS Integrating out 8, we get exp (- lT+l 1I ( l - ~ ) ~ w: 2a2 I=l - - -j 1 T+l T+l I=1 I \t.,J 11. Integrating out a, we get Substituting for w,, where ? = ( 1 : T ) C L C , and Z = (l:T)C:=, z,. From (4.1)it is apparent that the predictive density for c T + given r , is a univariate student t distribution. The predictive pdf for c T + ,can be obtained by integrating out u using numerical methods. The predictive pdf for c T + ,has been computed assuming z T + ,= 110 and is shown in Figure 2.As an estimate of the consumption expenditure in period T + 1, the mean of the predictive pdf, namely 561.49, can be used. where E T + , is For a loss function of the type L ( c T + C T + = ( c T + - C T + the predictive value of c T + the mean of the predictive pdf will be optimal in the sense of minimizing the expected loss. Also we note that the variance of the predictive pdf is 600.314 and hence the distribution has a wide spread. This indicates that c T + ,cannot be predicted with a great degree of precision. This distribution can be used for analyzing decision and control problems. For example, i f j T + represents a target value, as defined by Tinbergen [15], for yT+ we can determine the distribution of d T + = y T + -\;,+, from p(cT+,, ~ l d a t a i)n (4.1),since d T + ,= j T + , =cT+,+zT+,-\;,+,. The joint distribution of d T + ,and u is obtained by substituting d T + ,- z T + , j T + for c T + in (4.1).The marginal distribution of d T + ,can then be obtained by integrating out x ~isingnumerical methods. Further, if we have a loss function which depends upon yT+ -.Fr+ it seems possible to determine the z T + that minimizes the mathematical expectation of the loss function. It is interesting to compare this approach to prediction with the sampling theory approach. In the sampling theory framework, the reduced form equation for c,, namely (2.31, can be used to predict the future consumption. say c T + for a given ,, ,, ,) ,, , , , ,, , + , ,, ,, 588 V . K . CHETTY ,. value of z T + The forecast which minimizes the expected value of the loss function is ?,+ =(?;I -8)2,+ ,, where d is the of the type L ( c T +,,?,+ ,) = ( c T + -?,+ least squares estimator of x in the regression (2.3).The sampling theory prediction for c T +,, when z T + , = 110, is 565.865 with a variance of forecast 998.56. It can be , FI(;UKE2.-Predictive , density for c,, , for Model I . seen that in this approach, the prior information about r , namely O< u < 1, is not incorporated. This prior information accounts for the substantial reduction of the variance (from 999 to 6 0 0 ) when the Bayesian approach is used. 5. SPECIFICATION ANALYSIS OF THE MODEL One of the assumptions we have made about the error terms is that they are serially independent. In time series econometric models. however, the error terms are often found to be serially dependent. In order to assess the effect of departures from specifying assumptions about the error terms on inferences made about the model's coefficients, we analyze the model by assuming that the error terms are generated by the first order autoregressive process, namely, 589 BAYESIAN ANALYSIS (5.1) u,=pu,,+~,, -1<p<1, ( t = 1, ..., T). The approach used here is similar to the one used by Zellner and Tiao [IS]. It is assumed that the 6,'s are normally and independently distributed with zero means and common variance a2.From (2.1) and (5.1) we obtain (5.2) ct= ~ c t - 1 + x ( \ ; , - p y , - , ) + ( 1 - p ) P + ~ , (t = 1, ..., T ) . We assume that yo is fixed and known. We assume the following prior distributions : p(3, P, p, a ) cc a- O< r < 1, and - 1 < p < 1. The likelihood function for the model is ', Combining the likelihood function with the prior distributions, we get the distribution of the parameters : _I Let w t = c t - p c , , - ~ ( y ~ - p y ~and - ~ )E=C:=, w,/T. Then Integrating out P, Integrating out a, The normalizing constant can be evaluated by bivariate numerical integration over 3 and p. The marginal distribution of u can be obtained by integrating out p numerically. The conditional distribution p(ulp, data) provides inferences about x for an assumed value of p. If the conditional distribution is sensitive to changes in the value of p, then it is clear that assuming that p equals some fixed value-say p =0, corresponding to the assumption that the errors are independent-could lead to a posterior distribution of u far different from the one that can be obtained from (5.4). The posterior distributions of 3 and p for the data given in Table I are shown in Figure 3. The mean of the posterior distribution of x is ,5273 as compared with the mean of x, namely ,6597, obtained under the assumption of zero correlation among the error terms. This difference in E(uldata) reflects the fact that E(p1data) is very different from 0. The difference seems to be considerable in terms of the multiplier 1 (1 - x), viz., 2.1 18 versus 2.924. It is also seen from Figures 1 and 3 that the spread of the marginal distribution of u is sensitive to the specification about p. If p is assumed to be zero. the marginal posterior distribution of a has a variance of p(a 1 d a t a ) I p ( pi d a t a ) F r c ; r - ~3. r M a r g i n a l posterior distributions of 1 and p for Model I. 0.004, as compared with a variance of ,0105 uhen p is not assumed to be zero. The variance in the latter case is about 2.6 times the variance in the former case. To assess the effect of departures from the specifying assumptions about p, u e have also computed conditional distributions of x for various values of p uhich are plotted in Figure 4. The conditional posterior distribution of r, given p =0, is similar to p(r) shown in Figure 1. These two are not exactly the same since p(xlp=O) is computed using the sample of size 12 while the other is based on the BAYESIAN ANALYSIS 59 1 sample of size 13. This figure shous that the mean and the spread of the distribution are quite sensitive to such changes. Thus an inappropriate assumption about p can vitally affect the mean and the spread of the distribution. From Figure 3, it is seen that the marginal posterior distribution of y is highly concentrated in the range .8 to 1. This clearly shows that p cannot be assumed to be zero. In this case, assuming that p = 1 should lead to a posterior distribution of r which is very similar to that of the marginal posterior distribution of 3 in Figure 3, because the posterior distribution of p is highly concentrated in the range .8 to 1. FI(;I.RE4.- Conditional posterior distributions of r for varlous p for Model I This is shown by the fact that the conditional distribution of u when p = 1 in Figure 4 is very similar to the marginal distribution of x in Figure 3. Thus, in this case, using first differences of the observations, which corresponds to the assumption p = I . will lead to a posterior distribution of x that is similar to the marginal posterior distribution of u in Figure 3. Another assumption that should be examined is that investment is an exogenous variable. In some theories, (e.g.. Schumpeter's theory of innovations) investment is considered as an exogenous variable. In other theories, the notion of induced investment is introduced; investment is assumed to be partly autonomous---its 592 Y. K . CHETTY main determinants being such external factors as growth of population, new inventions, etc., or past values of such variables as profit, sales, etc.-and partly induced. If the latter theories are true, then the specifying assumption that investment is exogenous is not valid. Departures from our specifying assumptions about investment may affect our inference about the parameters of the model to a great extent. This can be seen from the controversy between Friedman and Meiselman [6],Ando and Modigliani [I], and others. Friedman and Meiselman, using simple macro-models which are very similar to Haavelmo's model I, come to the conclusion that the Keynesian income-expenditures approach to the prediction of national income is almost completely useless [6. p. 1871 when compared to the quantity theory approach. Ando and Modigliani and De Prano and Mayer [5] point out that one of the shortcomings of Friedman and Meiselman's approach is that investment is specified to be autonomous, whereas, in fact, it is partly induced ; they contend that such shortcomings in the Friedman and Meiselman procedure make the results of their elaborate battery of tests essentially worthless. In order to see the effects of departures from the specifying assumption about investment on our inference about the model's parameters, we turn to Haavelmo's expanded model [9]. It may be noted that the following analysis is not intended to shed any particular light on the Friedman-Meiselman controversy, but merely to illustrate the general problem. Haavelmo [9] assumes that investment, z,, is the difference between gross investment, u,,and gross business savings, r,. Gross investment here is defined as gross private capital formation plus government net deficit. This part of investment is exogenous. Gross business savings are defined as the sum of depreciation and depletion charges, capital outlay charged to current expenses, income credited to other business reserves, and corporate savings minus revaluation of business inventories. Hence z, = x , - r,, t = 1 .. . T. The gross business savings is assumed to be a function of what might be called gross disposable income, c, + x,. Hence the expanded model is : + B + u, , (6.1) c, = xy, (6.2) r,=p(c,+x,)+v+r,, (6.3) y,= c , + x , - r , , (6.4) x , is an autonomous variable ( t = l , 2 , ..., T ) . The variables c, J; r, x are all as defined before, u, and c, are structural disturbance terms, and a, fl, p, v are structural coefficients. We note that if p = O and o,,=Eu, r , = 0, the model in (6.1)-(6.4) reduces to the first model (2.1)-(2.2). When p= 1 the 593 BAYESlAN ANALYSIS model reduces to one in which income is measured with error and has constant mean. We shall inbestigate how sensitive our inferences are to departures from our assumptions, namely p=O and a,, =O Regarding the error terms, we assume that E ( u r )= E ( c , ) = 0 , ~ ( c r f =at, ) E ( 1 f )= a f , E ( ~ ' , e , ~ , ) = E ( u , u , - , ) = 0for ~ # 0 and , E(u,t.,)=a,,. We do not assume that u, and c, are independent. The likelihood function for the model's parameters is: (6.5) l(x,fl,~i,v,o~,o~,a,,.)-Jc IX-'lTi2jl - a ( 1 -p)IT exp [ - i t r V' V C - ' i where V = (u, c ) , a T x 2 matrix of disturbance terms and Regarding the prior distributions for the parameters, we assume that On combining the prior pdf's with the likelihood function in ( 6 . 9 , we obtain the following pdf for the model's parameters: Integrating with respect to X ', using the properties of Wishart's distribution, we obtain the posterior pdf for the structural coefficients: Except for the factor j1 -cr(l- p ) ) I T , the form of (6.7) is completely analogous to the posterior distribution of coefficients in a multivariate regression model analysed by Tiao and Zellner [13]. On integrating out fi and v, the posterior pdf for a and 11 is found to be where B and fi are least squares coefficients obtained by regressing el on ,:, and rr on c,+s,, respectively ; M,.,.=CT=, ( y , - . ~ ) a~nd the si', i, j= 1, 2 are elements of the inverse of the following matrix : ' Geisser [7] has po~ntedout that tll~sprlor distribution for C ' may be justified by various rules, e.g., invariance. conjugate families, stable estimation, etc. 594 V. K . CHET I Y where p and D are least squares coefficients obtained in the regressions mentioned above. The posterior distribution of a and p would be a bivariate t centered at the least squares quantities, but for the factor [I - a(l - , ~ i ) ] ~Also . it can be seen that if the covariance between the contemporaneous error terms, namely, a,, = E(utz-,), t = 1, . . ., T, is assumed to be zero. the posterior distribution of x and LL reduces to (6.9) p(a, ~i 1 o,, = 0. data) [I-41 -dl' -- I --- + [TS$ (a-&)2 - '[Ts: + ( p - p-) where T Ts$ = 2 (c, -Byt)' - 1=l p(alu,, = 0, d a t a ) pis I a,, = 0, d a t a ) FI(;I:RE 5.- Marginal posterior distributions of r for Model I 1 2jVf2],7 1 ) 2 BAYESIAN ANALYSIS 595 and 9, ,Li are as defined before. From the posterior joint distributions of a and p, we can compute the conditional posterior pdf for r, the marginal propensity to consume, for given values of p, and these pdf's can be used to find out how sensitive our inferences about x are to what is assumed about investment's dependence on income in (6.2). Using the data for the variables given in Table I, the posterior distributions in (6.8)and (6.9) are analyzed using bivariate numerical integration techniques. The marginal distributions of Y and are shown in Figures 5,6, and 7 for the two cases a,, # 0 and a,, = 0. I t is ~nterestingto compare the marginal posterior distribution of r obtained from the ~ccondmodel with that ofthe first model.Ascan be observed from Figures p ( p l u,,= 0 , d a t a ) Frc,r RE 6 . Marginal posterlor d i s t r ~ b i ~ t ~olo nji tor Model 11 1 and 5. the mean of the distribution a for the first model is a little less than that obtained for the second model. There is considerable difference among the dispersions of the distributions of x obtained for the two models. The variance of the marginal posterior distribution of x obtained from the first model is about three times the variance of the distributions of cr that is obtained from the second model. .4s can be seen from Figures 6 and 7, the posterior distribution of LL is very sensitive to the specification of a,,. In the case wheno,,.#O, distribution of ~i is highly concentrated around p = 16. The distribution of x does not appear to be very sensitive to the specification of a,,..In Figure 8. the conditional distributions of a for various values of LL are shown, when a,,. is assumed to be zero. It can be seen that these conditional distributions are somewhat sensitive to the specification of p. 596 V . K . CHETTY The conditional distributions of ci for various values of p, in the case when a,, is not assumed to be zero, are shown in Figure 9. It can be easily seen that both the mean and the spread of the distribution change remarkably and hence the posterior distribution of ci is highly sensitive to the specification of p. This means that our inference about ci, the marginal propensity to consume, is highly dependent upon FIGURE 7.--Marginal posterior distribution of p for Model I1 our assumption about the investment variable. The contours of the joint density function for the cases,a,, = Oand a,,# 0 are shown in Figures 10 and 11,respectively. Our analysis clearly shows that specifications about the nature of the variables (namely endogenous or exogenous) and about the correlation between the contemporaneous error terms condition our inference about the parameters of the model to a remarkable extent. Our analysis also shows how Bayesian methods can be conveniently used to assess the effects of departures from specifying assumptions. BAYESIAN ANALYSIS alp. m u " = 0 , d a t a ) FIGURF 8.-Conditional alp, mu, posterior distributions of cc for various p for Model 11 when a,, = O . $0, data) F ~ G L R9 .FC o n d 1 t i o n a l posterlor distributions o f r for various p for Model 11 when a,, # 0 . V. K . CHETTY BAYESIAN ANALYSIS 7. INFORMATIVE PRIOR DISTRIBUTIONS The choice between the two models we discussed depends upon the investigator's belief about the value of p, that is, whether he regards investment to be exogenous or not. In order to investigate how the prior information about p is modified by the sample information, different prior distributions are used for p and the posterior distributions of p are obtained. One prior distribution used for p ( p Iu,, = 0, d a t a ) 15t .05 .I .I5 .2 .25 .3 Ir Prior Distribution: p ( p ) a p 0 . 5 ( I - p ) 2 7 . 5 , ~ < p < l , mean = . 0 5 , variance =.0015 Prior D ~ s t r i b u t i o n :p ( p ) = p 2 ( 1 - p ) 2 , O < p < I , mean = 0.5, variance =.0357 FIGURE12.-Posterior distributions of p for different priors when cr,, =O. p is p(p) clc p0.5(1--,u)~'5 , 0 < p< 1, which has mean .05 and has very small probabilities beyond p = .15. This can represent the prior knowledge of an investigator who strongly believes that investment is exogenous. Another prior distribution used for p is p ( p )x p 2 ( 1 -p)', O< p< 1, which has mean 0.5 and is symmetrically distributed around the mean in the interval ( 0 , l ) .The posterior distributions of p 600 V. K. CHETTY are shown in Figures 12 and 13 for the cases a,,=O and a,,#O respectively. Also it is to be noted that the marginal posterior distributions of p, when p is assumed a priori to be distributed uniformly in the interval (O,l), are shown in Figures 6 and 7, for the two specifications about a,,.. In the case when a,, is assumed to be zero, it is seen from Figures 6 and 12 that p ( p I uuv = 0, d a t a ) Prior Distribution: ~ ( , u ) ~ , u . ~ ( I - , u ) ~ ~O. <~ p, < I p(~Iu,,, .I # 0, d a t a ) .2 .3 .4 . p Prior Distribution: p ( p ) a p 2 ( ~ - p 1 2 , O < p < I FIGURE 13.-Posterior distributions of p for different priors when u,,,.ZO the investigator's belief that p is nearly zero is modified by the sample to some extent. In the case when a,,, is not assumed to be zero. it can be seen from Figures 7 and 13 that the sample modifies the prior belief very markedly. The posterior distribution of p is very sharp and is concentrated around p= .156, ir'respective of the nature of the prior distribution. Columbia University BAYESIAN ANALYSIS APPENDIX In order to integrate a function, say y=f (x) numerically, Simpson's rule was used. T o use this rule, the interval of integration was subdivided into 100 subintervals, of equal length, h, using 101 points. The value of the function at each one of the points was evaluated. If these values are y o , y,, ..., y,,,, the value of the integral is given by When the function to he integrated. the rangeofintegration.and the numberofs~~hintervalsarespecified, the program evaluates the v, and computes A. Also the program prints out ?,/A, the ordinates of the normalized density function, and x, and its first four moments about the mean. For evaluating a double integral, this rule was used twice. REFERENCES [I] ANDO,ALBERT,AND FRANCO MODIGLIANI: "The Relative Stability of Monetary Velocity and the Investment Multiplier," American Economic Reciew, 55 (1965) 693-728. [2] BERGSTROM, A. R.: "The Exact Sampling Distributions of Least Squares and Maximum Likelihood Estimators of the Marginal Propensity to Consume," Econometrics, 30 (1962), 480-490. [3] Box, G . E. P., A N D G . C. TIAO:"A Further Look at Robustness Via Bayes' Theorem," Bio- metrika, 49 (1962), 419-433. [4] CHETTY. V. K . : "Bayesian Analysis of Some Simultaneous Equation Models and Specification Errors," unpublished doctoral dissertation submitted to the Univers~tyof Wisconsin. 1966. A N D THOMAS "Tests of the Relative Importance of Autonomous L51 DE PRANO,MICHAEL, MAYER: Expenditures and Money," American Economic Review, 55 (1965), 729-752. [6] FRIEDMAN, MILTON,A N D D. MEISELMAN: "The Relative Stability of Monetary Velocity and the Investment Multiplier in the United States, 1897-1958," in Commission on Money and Credit, Stabi1i:ation Policies, Englewood Cliffs (1963), 165-288. [7] GEISSER, S.: "Bayesian Estimation in Multivariate Analysis," University of Wisconsin Mathematics Research Center Report No. 503. published in the Annrrlc o f Muthematic.c and Statistics. 60 (1965), 15C159. [8] GEISSER, S.,A N D J. A. CORNFIELD: "Posterior Distr~butionsfor Multivariate Normal Parameters." Journalof the Ro,vrrl Stutist~calSocietl,, Ser~esB, 25 (1963). 368-376. [9] HAAVELMO, T . : "Methods of Measuring the Marginal Propensity to Consume," Journal of American Statistical Association, 42 (1947), 105-122. Reprinted as Ch. 4 in Wm. C. Hood and T. C. Koopman's Studies in Econometric Method, New York, John Wiley and Sons, Inc., 1953. [lo] JEFFREYS, H . : Theory of Probabil~ty(3rd ed.), Oxford: Clarendon Press (1961). T. J . : "A Bayesian Analysis of Simultaneous Equation Systems," Report 6315, [ l l ] ROTHENBERG, Econometric Institute, Netherlands School of Economics, Rotterdam, 1963. [12] SAVAGE, L. 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Published in Journal of the American Srarisrical Association, 60 (1965), 608-616. [IS] ZELLNER, A,, A N D G. C. TIAO:"Bayesian Analysis of the Regression Model with Auto Correlated Errors," Journal of the American Sratistical Association, 59 (1964) 763-778. http://www.jstor.org LINKED CITATIONS - Page 1 of 2 - You have printed the following article: Bayesian Analysis of Haavelmo's Models V. K. Chetty Econometrica, Vol. 36, No. 3/4. (Jul. - Oct., 1968), pp. 582-602. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28196807%2F10%2936%3A3%2F4%3C582%3ABAOHM%3E2.0.CO%3B2-H This article references the following linked citations. If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR. References 1 The Relative Stability of Monetary Velocity and the Investment Multiplier Albert Ando; Franco Modigliani The American Economic Review, Vol. 55, No. 4. (Sep., 1965), pp. 693-728. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28196509%2955%3A4%3C693%3ATRSOMV%3E2.0.CO%3B2-S 2 The Exact Sampling Distributions of Least Squares and Maximum Likelihood Estimators of the Marginal Propensity to Consume A. R. Bergstrom Econometrica, Vol. 30, No. 3. (Jul., 1962), pp. 480-490. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28196207%2930%3A3%3C480%3ATESDOL%3E2.0.CO%3B2-7 3 A Further Look at Robustness Via Bayes's Theorem G. E. P. Box; G. C. Tiao Biometrika, Vol. 49, No. 3/4. (Dec., 1962), pp. 419-432. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28196212%2949%3A3%2F4%3C419%3AAFLARV%3E2.0.CO%3B2-1 14 Bayes's Theorem and the Use of Prior Knowledge in Regression Analysis George C. Tiao; Arnold Zellner Biometrika, Vol. 51, No. 1/2. (Jun., 1964), pp. 219-230. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28196406%2951%3A1%2F2%3C219%3ABTATUO%3E2.0.CO%3B2-Z NOTE: The reference numbering from the original has been maintained in this citation list. http://www.jstor.org LINKED CITATIONS - Page 2 of 2 - 17 Prediction and Decision Problems in Regression Models from the Bayesian Point of View Arnold Zellner; V. Karuppan Chetty Journal of the American Statistical Association, Vol. 60, No. 310. (Jun., 1965), pp. 608-616. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28196506%2960%3A310%3C608%3APADPIR%3E2.0.CO%3B2-6 18 Bayesian Analysis of the Regression Model With Autocorrelated Errors Arnold Zellner; George C. Tiao Journal of the American Statistical Association, Vol. 59, No. 307. (Sep., 1964), pp. 763-778. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28196409%2959%3A307%3C763%3ABAOTRM%3E2.0.CO%3B2-A NOTE: The reference numbering from the original has been maintained in this citation list.