Download Bayesian Analysis of Haavelmo`s Models V. K. Chetty Econometrica

Bayesian Analysis of Haavelmo's Models
V. K. Chetty
Econometrica, Vol. 36, No. 3/4. (Jul. - Oct., 1968), pp. 582-602.
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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.
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[7] GEISSER,
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[8] GEISSER,
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[9] HAAVELMO,
T . : "Methods of Measuring the Marginal Propensity to Consume," Journal of
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[lo] JEFFREYS,
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[ l l ] ROTHENBERG,
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[12] SAVAGE,
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[13] TIAO,G . C.. AND A. ZELLNER:
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[14] TIAO.G . C.. A N D A. ZELLNER:
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[IS] TINBERGEN,
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[16] ZELLNER,
A . : "Bayesian Inference and Simultaneous Equation Econometric Models," paper
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1965.
602 V . K. CHETTY
[17] ZELLNER,
A,, AND V. K. CHETTY:"Prediction and Decision Problems in Regression Models
from the Bayesian Point of View," Systems Formulation and Methodology Workshop Paper
6403, University of Wisconsin, 1964. 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.
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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.
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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
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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
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