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El Colegio de México. Resumen: ... través de la maximización de una funcional producen dis- tribuciones ... de Good-Bernardo-Zellner en varios modelos económicos. Abstract: ...... In some cases, however, it might only be possible to analyze ...
ON INFORMATION, PRIORS, ECONOMETRICS, AND ECONOMIC MODELING

Centro

Francisco Venegas-Martínez de Investigación y Docencia Económicas

Instituto

Enrique de A l b a Tecnológico Autónomo de México

Manuel Ordorica-Mellado* El Colegio de México

Resumen:

Se busca reconciliar aquellos métodos inferenciales que a través de l a maximización de una funcional producen distribuciones a p r i o r i no-informativas e informativas. E n particular, las distribuciones a p r i o r i de E v i d e n c i a M i n i max (Good, 1968), las de Máxima Información de los Datos (Zellner, 1971) y las de Referencia (Bernardo, 1979) son vistas como casos especiales de la maximización de un criterio más general. Bajo un enfoque unificador se presentan las distribuciones a p r i o r i de Good-Bemardo-Zellner, que aplicamos en varios métodos de inferencia Bayesiana útiles en investigación económica. Asimismo, utilizamos las distribuciones de Good-Bernardo-Zellner en varios modelos económicos.

Abstract:

T h i s paper attempts to reconcile a l l inferential methods w h i c h by m a x i m i z i n g a criterion functional produce n o n i n f o r m a t i v e and i n f o r m a t i v e priors. In particular, G o o d ' s (1968) M i n i m a x Evidence Priors, MEP, Zellner's (1971) M a x i m a l D a t a Information Priors, MDIP, and Bernardo's (1979) Reference Priors, RP, are seen as special cases o f m a x i m i z i n g a more general criterion functional. In a unifying approach Good-Bernardo-Zellner priors are introduced and applied to a number o f Bayesian inference procedures w h i c h are useful i n economic research, such as the K a l m a n Filter and the N o r m a l Linear M o d e l . W e also use the G o o d Bernardo-Zellner distributions in several economic models.

* We are indebted to Arnold Zellner, José M . Bernardo, Jim Berger, George C. Tiao, Manuel Mendoza, and David Mayer for valuable comments and suggestions on earlier drafts of this paper. The authors bear sole responsibility for opinions and errors.

E E c o , 14, 1, 1999

53

54

ESTUDIOS E C O N Ó M I C O S

1. Introduction The distinctive task i n Bayesian analysis of deriving priors so that the inferential content of the data is minimally affected i n the posterior distribution, has been of great interest for more than 200 years since the early work of Bayes (1763). M o r e current approaches to this problem, based on the maximization of a specific criterion functional, have been suggested by G o o d (1969), Zellner (1971) and Bernardo (1979). W h e n modeling economic systems or conducting empirical research, prior information from previous research or from our knowledge of economic theory is always available. In either case, the estimates of the parameters of a regression model or the estimates o f the time-varying parameters of a state-space model can usually be improved by incorporating any information about the parameters beyond that contained in the sample. In this work, we provide a broad class of priors that are likely to be useful in a variety of situations i n economic modeling. The principle of m a x i m u m invariantized negative cross-entropy is i n troduced in Good's (1969) minimax evidence method of deriving priors. There, the initial density is taken as the square root of Fisher's information. Zellner (1971) presents, for the first time, a method to obtain priors through the maximization of the t o t a l information about the parameters provided by independent replications of an experiment (prior average information in the data minus the information i n the prior). Bernardo (1979) proposed a procedure to produce reference priors by maximizing the expected information about the parameters provided by independent replications of an experiment (average information in the posterior minus the information in the prior). A l l of the above methods have certain advantages: i ) W h i l e Zellner's method is based on an exact finite sample criterion functional, Good's approach uses a limiting criterion functional, and Bernardo's procedure is based on asymptotic results. In Bernardo's proposal a reference prior (posterior) is defined as the limit of a sequence of priors (posteriors) that maximize finite-sample criteria. M a n y reference prior algorithms have been developed i n a pragmatic approach in which results are most important. See, for instance, Berger, Bernardo and M e n d o z a (1989), and Berger and Bernardo (1989), (1992a), (1992b), Bernardo and Smith (1994), and Bernardo and Ramon (1997). i i ) The criterion functional used by Bernardo is cross-entropy, which satisfies a number of remarkable properties; in particular, it is invariant

ECONOMIC MODELING

55

with respect to one-to-one transformations of the parameters (Lindley, 1956). In contrast, the total information functional employed by Zellner is invariant only for the location-scale family and under linear transformations o f the parameters. Additional side conditions are needed to generate in variance under more general transformations. Hi) The way in which these methods have been tested is by seeing how w e l l they perform in particular examples. The evaluation is often based on contrasting the derived priors with Jeffreys' (1961), usually improper, priors which are somewhat arbitrary and inconsistent. In fact, there are cases in which one can strongly recommend avoiding Jeffreys' priors. See, for instance: B o x and Tiao (1973), p. 314; A k a i k e (1978), p. 58; and Berger and Bernardo (1992a), p. 37. In this paper, we attempt to reconcile all inferential methods that produce n o n - i n f o r m a t i v e and i n f o r m a t i v e priors. In our unifying approach, M i n i m a x Evidence Priors (Good, 1968 and 1969), M a x i m a l Data Information Priors (Zellner, 1971, 1977, 1991, 1993, and 1995) and Reference Priors (Bernardo, 1979 and 1996) are seen as special cases of maximizing an indexed criterion functional. Hence, properties of the derived priors w i l l depend on the choice of indexes from a wide range of possibilities, instead of on a few personal points of view with ad h o c modifications. In the spirit of A k a i k e (1978) and Smith (1979), we can say that this w i l l look more like Mathematics than Psychology — w i t h o u t denigrating the importance of the latter in the Bayesian framework. This unifying approach w i l l enable us to explore a vast range of possibilities for constructing priors. Needless to say, a good choice w i l l depend on the specific characteristics of the problem we are concerned with. It is worthwhile mentioning that our general method extends Soofi's (1994) pyramid in a natural way by adding more vertices and including their convex hull. This paper is organized as follows. In section 2, we w i l l introduce an indexed family of information functionals. In section 3 we w i l l state a relationship between Bernardo's (1979) criterion functional and some members of the indexed family, on the basis of asymptotic normality. In section 4, we w i l l study a Bayesian inference problem associated with convex combinations of relevant members of the proposed indexed family. Here, we w i l l introduce the Good-Bernardo-Zellner priors and their c o n t r o l l e d versions as solutions to the problem of maximizing discounted entropy. We w i l l pay special attention to the existence and uniqueness of the solution to the corresponding optimization problems. In section 5, we

ESTUDIOS E C O N Ó M I C O S

56

w i l l study the Good-Bernardo-Zellner priors as K a i m a n Filtering priors. In section 6, we w i l l apply Good-Bernardo-Zellner priors to the normal linear model, In section 7, we apply Good-Bernardo-Zellner priors to a variety of situations in economic modeling. Finally, in section 8, we present conclusions, acknowledge limitations, and make suggestions for future research.

2. A n Indexed Family of Information Functionals In this section, we define an indexed family of information functionals and study some distinguished members. For the sake of simplicity, we w i l l remain in the single parameter case. Suppose that we wish to make inferences about an unknown parameter 6 6 0 c 1R of a distribution P , from which an observation, say, X , is 9

available. Assume that /> has density/(JC I 9) (Radon-Nikodym derivative) e

with respect to some fixed dominating a - f i n i t e measure X on 1R for all 6 e 0 c JR. That is, d P / d X = f ( x I 9) for all 9 e 0 c 1R and thus P ( A ) d

= J f ( x I Q)dX(x) A

Q

for all Borel sets A e JR.

The Bayesian approach starts with a prior density, 7t(9), to describe

initial knowledge about the values of the parameter, 9. We w i l l assume that 71(9) is a density with respect to some a-finite measure u on IR. Once a prior distribution has been prescribed, then the information about the parameter provided by the data, x is used to modify the initial knowledge, via Bayes'

t h e o r e m , to o b t a i n a p o s t e r i o r d i s t r i b u t i o n o f 9, namely,

/(9 I x ) «c/(jt I 9)7i(9) for every in JC e IR (We use/generically to represent densities). The normalized posterior distribution is then used to make inferences about 9. Let us define an infinite system of nesting functionals (cf.

Venegas-

Martinez,1997): K

«. 8

( 7 t )

=

J "(6)0(^(8). m

where G ( / ( 9 ) , F { Q ) , y, a , 8) =

7- « , 8 ) 4 i ( 9 ) ,

(2.1)

ECONOMIC MODELING

57

0 < Y < l , a e {0, l } , 8 e {0, l } , a n d

I ( e ) =

J

±

lo f( \Q)\f(x\Q)dMx) g

X

(2.2)

is Fisher's information about 9 provided by an observation X with density f ( x I 9), and RQ) =

if(x\Q)logf(x\Q)dMx)

(2.3)

is the negative Shannon's information o f f ( x I 9), provided 1(9) and R 8 ) exist. I n the case that n independent observations of X are drawn from P , say, ( X X , . . . , X ) , then 1(9) and F(6) w i l l still stand for the average Fisher's information and the average negative Shannon's information of f ( x I 9), respectively. It is not unusual to deal with indexed functionals in inference problems about a distribution; see G o o d (1968). In particular, note that for the location parameter family g

V

2

n

/ ( ^ ! 9 ) = / ( x - 9 ) , 9 e IR, with the properties j [ f ' ( x ) ] / f ( x ) d k ( x ) < ~ and j f ( x ) l o g f ( x ) d k ( x ) < where X. = u. stands for the Lebesgue measure, both 1(9) and R 9 ) are constant. Observe also that the scale parameter f a m i l y / ( * I 9) = (1/9)/(JC/8), 9 > 0, with the above properties, satisfies the following relationship between F(9) and 1(9): 2

P(6) = y l o g I ( 9 ) +constant.

(2.4)

Throughout this paper, we w i l l be concerned with the following i n dexed family: A = conv[{V

y a 5

(n)}]

= convex hull of the closure of the family

iK, , (7t)}. a

5

We readily identify a number of distinguished members of A : ( i ) Criterion for M a x i m u m Entropy Priors,

MAXENTP:

K o . . ( « ) = - J«(e)io K(e)4i(9), g

58

ESTUDIOS E C O N Ó M I C O S

w h i c h is just Shannon's information measure of a density jt(6), or Jaynes' (1957) criterion functional to derive maximum entropy priors. Notice also that (2.3) can be rewritten i n a simpler way as F(9) = - V ( f ( x \ 0)). ( i i ) Criterion for M i n i m a x Evidence Priors, M E P : Q

Q

1

which is Good's invariantized negative cross-entropy, taking as initial density p ( Q ) = C [1(6)] j"[I(0)]

1/2

1/2

with

C={j[I(Q)]

provided

d x(Q)r ,

l / 2

l

i

that

4 i ( 9 ) < oo. We can also write (2.5) as:

i, i W - K o, i ( « ) = J « ( 6 ) log [i(e)]' ^(0).

K

/2

( H i ) Criterion for M a x i m a l Data Information Priors, V

0

0

0

(7t) = j j f ( x ) f ( 0 \ x )

MDIP:

log ^ diiiQ) dX(x), 71(9)

' '

(2.6)

(2.7)

which is Zellner's criterion functional. Here, as usual, f ( e \ ) /~

{ l o g i/„}~ ,

variables

where

=

satisfies

n

n

(3.2)

J

= j r„(co)W (co)4i(co)

U

61

sup J

= 0,

(3.3)

\ f (%\Q)dv(\)d\i(Q)

(3.4)

WogU \dP n

n > l Hogt/„l>E

/>{£, e A , 9 e B ] = ¡n(B) for all A

G

lR a n d B n

G

0.

77*en, a i n —» °°, «¿ = K , , , ( « ) - M>, o,i 1

dhJQ)

7t(9) 4i(9) 1

= F(9), h J - °°) = 0, hJ°°) = V °'

n n

2

2

dg (Q) — = a d d ) , g (7t(9) d u ( 9 ) * k

t

a

n

(rc) - V A n ) < °°, °' n n

o

a

°°) = 0, g.(°o) < oo, jfc = 0, 1, 2,

S k

i

s

S k

where a s 1 = a . T h e n , a n e c e s s a r y c o n d i t i o n f o r 7t*(9) t o be an o p t i m a l c o n t r o l is g i v e n by Q

P,W

TC*(9) - [1(G)] 2 e x p { p ( 9 ) F ( 6 ) + X y 9 ) a ( 9 ) } , 2

t

(4.7)

*=o wrtere p.(9) = p . e , j = 1 , 2 , a n d X 0

v9

c o - s t a t e ' v a r i a b l e s a s s o c i a t e d with

m = X^e™, k = 0, 1

the state

j are the

v a r i a b l e s h ( Q ) , j = 1, 2, a n d

ECONOMIC MODELING

69

g ( Q ) , k= 0,\, s, respectively. F u r t h e r m o r e , the constants p. , j = 1, 2, a n d X , k = 0, 1, s can be c o m p u t e d f r o m the f o l l o w i n g n o n - l i n e a r system ofs + 3 equations: Q

1 + log «,(oo) = log{J l o g [ I ( e ) F m ( p 2

10)

p

1 + l o g n (oo) •= l o g j j R 6 ) m ( p 2

1 + log g

k

m

1 0 >

J ^ , X , . . . . X ; 6)41(6)}, 1 0

p

2 Q

= log {Ja (9)m(p , p

M

t

10

J 0

,X , \ m

, A^,, X

2 0

¿ = 0 , 1 , 2,

Q

, X

1

0

s

, 6)4i(8)},

Q

, X

j

0

; 8)4i(8)},

j ,

where ^10.

«(p.o. p . V 20

0

) = f [iW]

2

e » p

R

9

)

*HI

A o

°"

( e )

u= 1

• '

5. Kalman Filtering Priors In this section, we w i l l study Good-Bernardo-Zellner priors as K a l m a n Filtering priors (Kalman 1960, and K a l m a n and B u c y 1961). W e w i l l continue to work with the single parameter case, and focus our attention on the location parameter family. Let Y Y ,Y be a set o f indirect measurements, from a polling system or a sample survey, o f an unobserved state variable ft. The objective is to make inferences about ft. T h e relationship between Y and ft is specified by the measurement or observation equation: v

2

t

t

y =A ft +e, t

f

f

(5.1)

where A * 0 is k n o w n , a n d £, is the observation error distributed as W(0, 0 ) with a k n o w n . Note that the main difference between the measurement equation and the linear model is that, i n the former, the coefficient ft changes with time. Furthermore, we suppose that ft is driven by a first order autoregressive process, that is, (

2

2

ESTUDIOS E C O N Ó M I C O S

70

(5.2)

p > Z , p _ , + !!,_„ (

where Z * 0 is k n o w n , and x\ ~ N ( 0 , a ) 1

with o

known. In what follows,

2

e,, and n are^ndependent random variables. W e

we will'assume that p

(

could state nonlinear versions o f (5!l) and (5.2), but this would not make any essential difference i n the subsequent analysis. Suppose now that at time t = 0, supplementary information is given by P and a , the mean and variance o f p respectively. That is, 2

0

0

J

TC(P )rfp =l, 0

0

(5.3)

J_P TC(Po)dP = P , 0

0

0

(P -P )2TC(p )dp = O .

f

0

0

0

2

0

In this case, the Good-Bernardo-Zellner prior is given by rc;(p ) * [ I ( p ) ] * / 2 e x p { ( l - 1. 2

Consider a vector of independent and identically distributed normal random variables (X,, X , X ) with common and known variance o satisfying 2

E(X )

= a e + a

k

2

n

H

i

k

2

6 + . . . + a 2

k

m

Q

m

,

*=1,2,

n

where A = ( a . ) is a matrix of known coefficients for which ( A A ) ~ T

(6.1) 1

exists.

Let X and 9 stand for the column vectors of variables X and paramek

ters 6,, respectively.

Then (6.1) can be written in matrix notation as,

E(X) = A B . In this case, we have = (^z")"

7 2

exp{-

ll£ - A9II }, 2

(6.2)

where £ = ( x x , x ) . Since a has been assumed known, only the location parameter is unknown. The analogue of (2.2) is now given by the matrix: v

2

n

2

ECONOMIC MODELING

TT-tog/(jtl6)

ae, =

l o g / ( A r l e )

73

f(x\B)d\(x) A < 1,1 < m

-hA A T

(6.3)

and so det[I (9)] is constant.which implies that the Good-Bernardo-Zellner prior distribution

TC'O), describing

a situation o f vague information o n 9,

must be ajocally uniform prior distribution. L e t 9 be the least squares estimate for 9. Then it is known that A A Q = A X , E ( 9 ) = 9, and Var (9) = a { A A ) ~ . Noting from equation (6.2) T

T

2

T

l

that

/(§6):

(

1 2TCG

V

^

2

2

e x p { - ~ ~ ~ (ll£ - A9II + ( A A ( Q - 9), 9 - 9 ))}, 2a 2

T

2

2

J

and applying Bayes' theorem, we get as the posterior distribution of 9 i

m

'

i

i

/(9I£) = ( 2 n ) ~ ( d e t [ ^ A A ] ) H x p { - \ ( \ A A { Q - 9), 9 - 9 )}. 2

T

T

If supplementary information about the mean, c, and the variance-covariance matrix, D , is now incorporated, then the (informative) Good-Bernardo-Zellner prior is given by

n'Aß) = (2rc)" (det[Z>])~ e x p { - j { D ~ \ Q - c), 0 - c ) } . 2

2

The posterior distribution is now / ( 9 l £ ) = (27i)"T(det[S])2 x exp {- \ < B [ 9 - ( ( D B ) ^ c + ^ ((DB)- c + i

whereß^D-'+^jA )!. 7

J

a

B~ A AQ)] 0

-B- A AQ))}, ,

2

T

l

T

ESTUDIOS E C O N Ó M I C O S

74

7. Good-Bernardo-Zellner Priors in Economic Modeling In this section we apply Good-Bernardo-Zellner priors to a variety of situations i n economic modeling. E x a m p l e 7.1 Let us examine the behavior of an individual who learns about the parameters of her/his utility function under inflation. If we think of the parameters as random variables, then the information gained from experience (consumption) is incorporated into a prior distribution. Once a prior is available, the agent makes consumption decisions. To illustrate this process, we shall borrow some ideas from C a l v o (1986). Let us consider a small open economy with a single infinitely-lived consumer in a world with a single perishable consumption good. Suppose that the good is freely traded, and its domestic price level, P , is determined by the purchasing power parity condition, namely P = P * E , where P* is the foreign-currency price of the good, and E is the nominal'exchange'rate. Throughout the paper, we w i l l assume, for the sake of simplicity, that P* is equal to 1. We also assume that the exchange-rate initial value, E , 'is known and equal to 1. The expected utility function of a representative individual at the present, t = 0, has the following separable form:

V=f\fu{c ;Q)e- dL(Q)dQ t

n

(7.1)

where « ( c ; 6) is the utility of consumption; c is consumption; 6 > 0 is a parameter related to the utility index; r is the subjective rate of discount; rc(9) is a prior distribution describing initial knowledge of 6 coming from the experience of the consumer before t = 0 (the present). f

Let us assume that: 1) the representative individual has perfect foresight of the inflation rate so P/P, = q = q , that is, she/he accurately perceives the rate at which inflation is proceeding, the value P(0) is assumed to be k n o w n , 2) there are no barriers to free trade, 3) the international interest rate is equal to r, 4) capital mobility is perfect. If i is the nominal interest rate then r = i + q . Denoting income and government lump-sum transfers by y and g respectively, we can write the consumer's budget constraint, at time / = 0, as e

e

t

t

ECONOMIC MODELING

«o + l

(7.2)

(y + g , ) e - " d t = \ ( c . + i m ) e - ' d t , t

0

75

r

0

where for the sake of simplicity we have chosen y = y = constant. The con;

sumer holds two assets: cash balances, m = M / P ,

t

t

where M is the nominal

t

t

stock of money; and an international bond, £ . The bond pays a constant interest rate r (i.e., pays r units of the consumption good per unit of time). Thus, the consumer's wealth, a is defined by f

a =m t

(7.3)

+ k ,

t

t

where a is exogenously determined. Furthermore, we suppose that the rest Q

of the world does not hold domestic currency. Consider a cash-in-advance constraint of the Clower-Lucas-Feenstra form, m > etc,, where c, is consumption, and a > 0 is the time that money t

must be held to finance consumption. G i v e n that i > 0, the cash-in-advance constraint w i l l hold with equality, m, = ac, For

(7.4)

the sake of concreteness, let us suppose that u(c \ 8) = t

Plainly, u > 0 and u c

c c

e~ . 9c

< 0. Moreover, let us assume that there is supplemen-

tary information about 6 > 0 in terms of the mean value E[8] = l / X . We also assume that 1(8) and F(9)

are constant, i.e., before supplementary

information becomes available, initial knowledge is vague. In such a case, following Proposition 4.1, the Good-Bernardo-Zellner prior is given by TC*(8) = e" , 8 > 0, and (7.1) can be written as x e

V=r\f-e-^ o [o J

J

+

^dd] e

J

r t

dt =

fo - ( -c V+ Xl

J

f

r

'dt.

(7.5)

t

In m a x i m i z i n g (7.5) subject to (7.2) the first-order condition for an interior solution is:

1

( c + X)

=X(\

+ ai),

(7.6)

where X is the Lagrange multiplier associated with (7.2).We assume a government budget constraint of the form

76

ESTUDIOS E C O N Ó M I C O S

f

g.e-"dt = b + \

(7.7)

(m + qm,)e- 0 , then the Good-Bernardo-Zellner prior is 1 —i-101 K*(Q)= — e « , 2a

ECONOMIC MODELING

77

w h i c h is a Laplace distribution. ( i i ) If

k(9) = e

k(e)=*

ä = - - + ß , ß e 1R a

and

e

ä, = « T ( l + - ) a p

2

where K is Euler's constant, then TC*(8) = c x e ^ - ^ e x p } - e a G u m b e l (or extreme value) distribution. ( H i ) If fl,(9)

a(fl

- P ' } , which is

«, = 1

W> 9

a (0) = 9

and

2

a (9) = log9

5 = |,-a>0,ß>0 2

a = \|/(a)-logP

3

3

where7

is the usual indicator function and, as before, V|/(ct) is the p r i

function, then TC*(9) = ^

( P 6 ) " pe " , which is a G a m m a distribution a

1

p e

(or Erlang distribution, i f a is a positive integer). ( i v ) If äj = 1

a (9) = eP, p > 0 [ and

ä, =

2

fl,(6) = log9

2

-

-,a>0 a K log a p ß

where K is Euler's constant, then TC*(9) = a P 9 ~ V distribution. p

0

* , which'is a Weibull

E x a m p l e 7.2 We w i l l develop Good-Bernardo-Zellner interval estimates to test convergence o f rational expectations. Consider a simple macroeconomic

model

ECONOMIC MODELING

79

W e may also write ft as ft = X \ where x , is any martingale, that is, x is any stochastic process that satisfies E{x

l

+

l

\I,)=x

r

Therefore, there are infinitely many divergent forward rational expectations solutions. Convergence w i l l require ft = 0 for all t. N o t e now that from successive substitution of (7.13) into (7.14), we can show that E , m _ = p/'m, _ = 0, 1 , a n d therefore (7.14) becomes t + j

x

_g- p

pm,_,

V;_

+

1 - 7 ( 1 - P)

K

'

6

1-Y

There are many stochastic processes (bubbles) consistent with (7.15), for instance, Xft — with p r o b a b i l i t y q, 0 < q < 1, q 0 with p r o b a b i l i t y 1 - q,

P\+i

or ft

= Xft + ri ,

+1

(7.17)

(

where the TI/S are independent Gaussian variables with mean zero and variance a . 2

We suppose that the ft's are unobserved location parameters satisfying (7.17). We also assume that there is supplementary information i n terms of the two first moments on the i n i t i a l (3 , namely £{|30}=P0 and £{p2} =o2+ pg. Then, according to Proposition 4.1, the Good-BernardoZellner prior compatible with such a information is N ( % , c ) . We suppose that the random variables p , e and r\, are independent. Hence, under normally distributed errors, the rational expectations system is given by 0

2

0

(

ft^ft^+Tl,.,,

'~' -5-|Q y 1 - 7 ( 1 - p) p m

P l

P i

I

V

'

1-y'

80

ESTUDIOS E C O N Ó M I C O S

or equivalent^, i n terms of (5.1) and (5.2), m 1 -7(1 - p )

where ypo

e, ~ N(0, a ) , and a = 2

-|2 v

2

[1 - 7 ( 1 - P ) ] ( l - 7 ) To test the common assumption o f convergence with available data on p , m,, y, 5, p, and y, and under normally distributed errors we use equations t

(5.7), (5.9) and (5.10) with univariate error terms. In such a case, the posterior distribution of p, 11,_, is N(P,, a ) , where 2

p^G^P^j+O-e,)

p + S y -

1 -7(1 - p ) '

CT = ( l - e ) a , 2 (

2

f

The null hypothesis to be tested is H : p > 0 for all t > 1. Proceeding recursively and starting off at f = 1, we reject H i f a f appears for w h i c h P, = 0 does not lie within a highest posterior density interval with a given uniform significance level a , namely (p, O , P, + c r ) where, as usual, P { Z > ) = a / 2 a n d Z ~ \/(0, 1). 0

0

Z a / 2

Z

a

/

Z a / 2

t

(

1

Finally, we w i l l apply Good-Bernardo-Zellner priors to consumption decisions under uncertain inflation. W e assume that there is a large number o f identical consumers, each o f whom makes consumption decisions in T- 1 periods (f = 0, 1 , T - 1), and has the following budget constraint: w _ M t

x

t

= w _ M _, +g,_ +y,_ t

x

t

l

l

-

c_, t

l

(7.18)

ECONOMIC MODELING

t = 1,

T, M

given,

> 0

0

M

T

81

> 0 ,

where M is the stock of currency owned at the beginning of period r, w is f

(

the value of the currency measured in goods at t (the reciprocal of the price level), g stands for government lump-sum transfers at t, y is real income at t

t

t, and c 'is consumption at t. Equation (7.18) can be rewritten, i n terms of the inflation rate

"t-1

. 1

71, =

as ,)«,_,

(1 + 7 t ) m , = ( l +TC,_ (

+ y,_, - c , _ , - n _ ,/«,_,, f

(7.19)

r,

i = 1,

where m = w M represents money balances and the last term on the right(

f

(

hand side stands for depreciation of money balances from inflation. Note, however that the above budget constraint requires additional information on w_

and w . T

Private agents have no knowledge of w _

w

p

, w

Q

r

and therefore,

they do not know the inflation rate, n . However, we assume they have t

partial information on the distribution of moments, say, E{ w_,} By

= w _ , and E{w _,} 2

w _

v

in terms of the first two

= a l , + w _,. 2

using Proposition 4.1, with 1(9) and F(9)

constant ( i.e., before

supplementary information becomes available, initial knowledge is vague), we find that the Good-Bernardo-Zellner prior compatible with the available information for w _ , is N ( w _ , , & ,). Therefore, w_

i

M = (1 + 7 t ) m ~ W(w_ 0

0

0

X

M , a . 0

2

X

M ). 2

O f course, we assume that vv_, > 0. Suppose also that private agents are capable of making indirect measurements , 7t,, of 7t , according to the rule r

(1 +7C )m = ( l + 7 t ) m , + £,, f

(

t=l,...,T,

(7.20)

82

ESTUDIOS E C O N Ó M I C O S

where m is a constant target chosen by the monetary authority at t = 1. W e assume that the observation errors, e, are independent normal random variables with mean zero, variance a and E{w_ ,£ } = 0. 2

(

The representative individual's objective is" to maximize, at the present (f = 0), his total expected utility of consumption over T-\

periods,

namely,

(7.21)

Note that, for simplicity, no discount factor has been included in the overall utility, and money services provide no utility. The utility function is expressed as the quadratic function u(c ) = a t

l

C

t

- ^ c l

Here, a a > 0, and the ratio a / a Note that « ( 0 ) = u ( 2 a / a ) = 0, u ( c ) > 0 c, > 2 a / a , u ' ( c , ) > 0 for 0 < c, < a / a , salvage value is chosen as v ( w _ M ) = v

2

2

2

2

2

r

X

(7.22)

t = 0,...,T-\.

T

determines the level of satiation. for 0 < c, < 2 a / a , u ( c , ) < 0 for and u { c ) < 0 for c, > a / a . The ( a / 2 ) [ w _ ,M ] . 2

2

2

T

T

2

We assume that the income of the individual fluctuates randomly around his income satiation level following

V f

= fi+Ti , f

Ti, ~ AJ(0, a ) , 2

t = 0,...,T-\,

(7.23)

where the n ' s are independent endowment shocks satisfying E{e r\} = 0 for allf,j,andE{ _ Ti }=0. W

I