DISCUSSION PAPER

-

~ ~ o ~ l a s s i c a l Econometr ics

P a r t TWO: Mot iva t ion and S y n t h e s i s

b Y

Michael J. H a r t l e y

December 1 9 8 1

Development Research Department Economics and Research S t a f f

World Bank

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- B .

The views p resen ted h e r e a r e t h o s e of t h e a u t h o r , and they should n o t b e i n t e r p r e t e d a s r e f l e c t i n g t h o s e of t h e World Bank

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N e o c l c s s i c a l E c o n o m e t r i c s :

Pnrt: Two: Y o t i v a t i o n a n d S y n t h e s i s

Y

* ? l i chae l J . H a r t l e y

Dcvelo~ment . Economics Depar tment The rJorld Bank*"

Wash ing t ry , D. C.

December 1981

+ T i ~ i s p a p e r h ~ s becn p r e p a r e d at. t h e i n v i t a t i c r ~ of t h e Amer ican S t a t i s t i c a l - \ s s o c i n t l o n (ASA) f o r p r e s e n t a t i o n a t a memor ia l s e s s i o n of i t s 1982 Xnnual ! lcct ings i n honor of my f a t h e r , H. 0. H a r t l e y (HgH), a P a s t P r e s i d e n t of t h e ASA. The i i l f l u e n c e of his work o n t h i s pape r is i ~ n p l ; ~ i t . I would l i k e t-o t h a n k t h e ASA f o r o r g a n i z i n g t h i s s c s s l o n ,

"Approac t~es f o r !lodeling t h e 'Rea l World' a l a 110H" , , ~ n d Dr. Paul B ~ c m c r , s e s s i o n cha i rman and a Eormer s t u d e n t of IIOH, f o r t h ~ i n v i t n t t . o n I'o p a r t i c i p a t e .

** The views e x p r e s s e d i n t h i s p a p e r r e p r e s e n t t h o s e of t h e a u t h o r a n d , i n no way s h o u l d b c @ ; ~ s s o c i n t e d w i t h t h o s e of ' t h c I-Jorld Rank. The

k nu t h o r d o e s , however , wish t o acknowtcdge h e l p f u l d i s c u s s i o l l s w i t h 6 c c r t a i n c o l l c n g u c s I n t h e Bank, p a r t . i c u l ; \ r l y H o l l i s Chcncry , Ardy

S C , L :-, 1 . !I 71 ! * , - 7 . - 1 \ 7 7 ,.,i$ y - . ' 4 : * l 7 * - ' I , I . .

.- , > > ,

~'clter..;'- .lntl Fred I r v y . S p c c i n l t h a n k s a r e due t o my c o l l e a g u e a n d ? c o l l a h r n t . o r , E r i c Swanson, who r e a d and commented on e a r l i e r d r , ~ € t . % and tr;, Adam Ynrmolinsky, who h a s made mc r e a l i z e c e r t a i n of -

1 i c n t . i o n s . F i n a l l y , I must e x p r e s s n s p e c i a l d e b t t o b Y a r c e l ie Pagnno, R i c l~; l r t l C,hi;ind r and P a u l Samuelson f o r s p a r i n g t.he - t i n e from t h e i r b~i.;y s c h c d i ~ l e s t o s e r v e a s "sounding-bo<irds" f o r t h e s e i d e n s .

Abstract

The paper proposes a new approach to econometric model$ng and I

parameter estimation which permits the construction of Macro-ecpnometric

models from general Microeconomic-f oundations . It provides, inter-alia Full 7'

I

Information haximum Llkclihood estimation methods which permit the following :

I

I

I

(a) a statistical solution to the (Exact or Consistent) Aggregation Problem, I

I I

(b) a new approach to the problem of Parameter ~dentificat~,n,

(c) a systematic approach to the problem of Model selection, I I

(d) obviates the need for the use of the Duality Theory of Cost and Profit functions by working solely on Primal problems,

(e) treats the spectrum of perfectly competitive to monopolistic/monopsonistic market structures, I

I

(f) applies, in principle, to both partial and general equilibrium models, and I

(g) obtains the structural form of "limited dependent varisble" econometric estimation methods directly from neoclassical (cons trained optimization) behavioral models . I

I

--all in the context of well-behaved, but essentially unrestricked, functional C b e I

I

foms, subject to a general set of "rationality postulates" reg)arding the I I

bchavlnt of individual s c q r o a i c 7:ntts. These are a l l ievelc?ed tn the c o u r s e - I

I

of Parts One to Four of the paper. I

I

Part Two: Motivation and Synthesis

I . Introduction pp. 1-8

11. Spec i f ic Issues pp. 8-85

A . The Implicit Function Theorem (9-21)

B . The Model Se lect ion Problem (21-28)

C . The Ident i f icat ion P r o b l ~ (28-45)

D . Uncertainty, Expected Values and Certainty Equivalence ( 46-54)

E . The E-M Algorithm (55-60)

Neor ! a s s i c a l Econometrics : - - -. - 1 / Par t Tbo: Motivation and Synthesis -

Michael J. Hart ley The World Bank

I. In t roduct ion:

I n Par t One I have: (1) provided a d e s c r i p t i o n of t h e c l a s s of

opt imizat ion problems t y p i c a l l y being addressed i n economics, i n p a r t i c u l a r ,

and, by analogy, i n the f i e l d of (experimental/non-experimental) behaviora l

sc iences , i n genera l ; and (2) t r i e d t o provide a d e s c r i p t i o n of the proposed I

approach t o what we have termed Neoc1ass:Lcal Econometrics ( a t t h i s junc ture ,

i n the s p e c i f i c context of unconstrained opt imiza t ion problems involving a

"well-behaved" ob jec t ive func t ion defined over Euclidean J- space).

I The methoc: being advocated would appear t o apply equal ly w e l l to

I o the r behavioral s c i ences . I would suggest t h a t t h e key d i s t i n c t i o n between

I the behaviora l , a s opposed t o C& p h y s i c a l sc iences is the following:

k. " In the ~ h y s i c a l sc iences t h e ob jec t ive is t o d iscover t h e

I , . u n d ~ r l ; i t q g ' T a w s s: na tc rc 5y ::,Lrnc' , : > ~ + r ~ : ~ : . 3 7 7 0: r b . e ~ r :;a:ilral - -3

manif@stations--i .e. , i n terms of v a r i a b l e s r e f e r r i n g t o observable o r - - t a n g i k e - phys ica l c h a r a c t e r i s t i c s . In the behavioral s c i ences , the "laws OF - s o c i a l behavioru-- subject t o t he evolu t ion (over time) of a s e t of " technical '

constrair . ts" and " s t a t e s of natureu--are observed a s "mediated by" o r

" f i l t e r e d through" the behavior of i nd iv idua l human agents . In order t o

1/ In t h i s and subsequent P a r t s , the equat ion numbers, ( i . j ) , w i l l r e f e r ta - equat ion j wi th in Pa r t 1.

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" s u f f i c i e n t l y large ' ' number of d i f f e r e n t exogenously determined c o n d i t i o n s

( s t a t e s of n a t u r e , e t c . ) , then t h e i n f o r m a t i o n provided by t h e sample o f

observed responses w i l l permit a "keen observer" t o i n f e r t h e s t r u c t u r e of t h e

i n d i v i d u a l ' s maximand provided t h a t t h e i n d i v i d u a l behaves r a t i o n a l l y . I n

t h i s connec t ion , it should be obvious t h a t t h e problem of how t o h a n d l e

s t r u c t u r a l change over time, e t c . , is of fundamental importance (and a k i n t o , I t h e Heisenberg Uncer ta in ty P r i n c i p l e of p h y s i c s ) . I

It i s f o r p r e c i s e l y t h i s r eason t h a t I would a r g u e t h a t

macroeconomics, as p r e s e n t l y p r a c t i c e d , i s bankrupt . 'Ihe " t y p i q a l p r a c t i c e N

invo lves p o s t u l a t i n g a t h e o r y about i n d i v i d u a l a g e n t s ' behavior--i.e., f i r m s ,

households , banks, governments ( i f endogenous), etc.--and, i n l i g h t of t h e I

f a c t t h a t t h e so- ca l led Aggregation Problem has been i m p o s s i b l e t o s o l v e

p r a c t i c e , a s opposed t o i n theory , t o a rgue by ana logy t h a t t h e mic ro t h e o r y

" c a r r i e s over" e x a c t l y a s an h y p o t h e s i s aboa t "macro-behavior." As e v e r y

s t u d e n t i n Economic P r i n c i p l e s ( s e c , e .g ., Samuelson [ 79 1) knows, t h i s

argument may i n v o l v e a F a l l a c y of Composition ( o r Aggregat ion) Problem.

Indeed, a s t h o s e who have sought t o t a c k l e t h e Aggregat ion Problem -

a n a l y t i c a l l y have discovered-- see, e .g ., May [ 6 3 , 64 1, Green [ 23 1 , etc .- r l

L - the only f u n c t i o n which appears t o admit " exac t aggregations '-- i .e . , a

f u n c t i o n , d e f i n e d a t t h e l e v e l of t h e i n d i v i d u a l (economic) qtit , quch tb.at .. t h e same f u n c t i o n a l form is .preserved under t h e summation op&ation a c r o s s

&. uni t s- - is t h e l i n e a r f u n c t i o n wi th i d e n t i c a l " s lope c o e f f i c i e L t s ' ' a c r o s s a l l -

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i n d i v i d u a l sa- s e e , e .g ., Green [ 23 ] . Thus, one is l e f t w i t h t h e s o- c a l l e d

Aggregation Problem ( o r Paradox): i . e . , one is Eorced t o t h e p r o p o s i t i o n t h a t

t h e " r e a l world" i s " l i n e a r- w i t h- i d e n t i c a l s l o p e - c o e f f i c i e n t s " a c r o s s a l l

i n d i v i d u a l a g e n t s , i n o r d e r t h a t the impl ied "macro- rela t ions" are l o g i c a l l y

i n f e r r a b l e from "micro- foundat ions ." any or,^ w i t h any s e n s e would d e s p a i r at

t h i r 'deduction .I1 The " r ea l world" must be d i f f e r e n t than one Sn which a l l

i nd iv idua l agents have e s s e n t i a l l y i d e n t i c a l quadrat ic- forms fo$ o b j e c t i v e I I

func t ions , and, thus , l i n e a r "decis ion ru les" (i .e ., demand "culjves ,I1 supply - I

"curves ,I1 e tc .) . The " r e a l world" simply must be more complex. Rut, i f i t ~

is, then w e a r e stymied, i n t h a t it is then impossible (o r s o it( would appear ) I

t o r e so lve t h e Aggregation Problem. I n such a s t a t e of a f f a i r s , ~ ve a r e then

forced t o the p o s i t i o n t h a t t h e r e must be a "divorce" between t 4 e p r a c t i c e of I

Microeconomics and Macrosconomics--and , hence, t he p re sen t bankduptcy of

"macro-economics a s a s c i ence .I1 I would con jec tu re t h a t , i f Sa uelson R [ 78 1, I would not be w r i t i n g t h i s paper today. In s h o r t , eve4 s i n c e t h e

I

"Keynesian Revolution ,'I we, a s economists , have been " led down t d e (wrong) I

had ;.roceeded "one s t e p beyond" equa t ion ( 2 ) on page 259 of h i s

garden pa th .* I ~

Foundations

I I

The s t a r t l i n g f a c t is t h a t t h e so- cal led Aggregation Fioblem - d i s i n t e g r a t e s the moment we note t h a t :

* I * ~ where x is a (vec to r o f ) macroeconomic agg rega t e ( s ) and x i s a ( v e c t o r ---. . A L -i- I -- I

o f ) "optimal" dec i s ion values def ined , r e l a t i v e t o the maintaindd hypo thes i s - I I

.7 . ; + - % j b , ,.<., T! .: - ( ~ V D ~ ) + ~ D _ S ~ S \ bv !-he f n ~ l i ~ f ~ F l ~ n r t i _ o n ?), ' ) r : : , ? r ; > . . ~ _ ? . ' , ---- -- --L- --_-- lj--.----I __-I_-'-::-

func t ion , (1.3). Perhaps, the r n ~ t ex t r ao rd ina ry aspect of t h i s , puzzle is the - I

ques t ion of 3 t h i s simple proposit*n has escaped n o t i c e . I flave no I I

I I e xp lana t icn! , I ~

There a r e some f u r t h e r s t a r t l i n g imp l i ca t i ons of t h i s p r o p o s i t i o n . I I

Indeed, I submit t h a t the major i ty of the so- cal led "major prob4ems" i n I

economics today have a r i s e n a s a d i r e c t consequence of t h i s i s s d e . kt. me I

i l l u s t r a c e with two c o r o l l a r y p ropos i t i ons : I I ~ I

(A) The b a s i c con t rove r sy between t h e t h r e e "Cambridge Schools," which

has r e s u l t e d i n a n e n t i r e f i e l d c a l l e d C a p i t a l Theory, is t h e problem of I

adding up machines of d i f f e r e n t types , v i n t a g e s , e t c . , t o o b t a i h "Aggregate

C a p i t a l Stock," which. i n t u r n , is embedded w i t h i n a n "Aggregate Product ion

Funct ion .I1 Controversy has swi r led around t h e i s s u e of whether such concep t s

e x i s t ( i n theory o r i n p r a c t i c e ) , and whether t hey are meaningfhl i n t h e sense

of Samuelson [ 7 8 1. C l e a r l y , they do no t e x i s t and a r e no t meaningful

concepts-- except a s c rude approximations t o a complex wor ld . That is f a r f rom

s a t i s f a c t o r y i n a macro-economic sc ience!

(B) L i t e r a l l y hundreds of books and papers have been w r i t t e n on t h e

problem of e s t i m a t i n g so- ca l led "Demand and Supply Funct ions ," def ined a s

aggrega tes over a set of p a r t i c i p a n t s w i t h i n a "market .I1 While a "aarket"

c l e a r l y e x i s t s , t h e Demand and Supply func t i ons are s imply e x p l i c i t f u n c t i o n s , N

de f ined by equa t i ons (1.2), (1.3), (1.5) and t h e r e l a t i o n x = C x . a i=1 i.

Such ful.:ctions would seem t o e x i s t i f and on ly i f t h e Aggregatipn Problem is

so lved . But, i f " exact aggregat ion" r e q u i r e s l i n e a r i t y , and i f t he " r e a l

world" impl ies t h a t t h e e x p l i c i t f u n c t i o n , (1.3), is no t l i n e a r , then it

fo l lows t h a t Demand and Supply Funct ions do no t e x i s t . k & h e r , , u t i l i t y and -- - L

prodhci ion f u n c t i o n s , e t c ., exis;, and a l l t h e paramete rs r e q u i ~ e d t o

o p e r a t i o n a l l y d e f i n e t h e concepts of "Demand and -- Supply Functions" are

con ta ined w i t h i n t h e s e primary behav io r a l / ta.chnologica1 s t r u c t u i a l forms, and;

C

t h u s , t h e impl ied o r d e r i v a t i v e Demand and Supply Func t i ons , f equ i r i ng both an I .

e x p l i c i t s o l u t i o n t o :he i m p l i c i t f u n c t i o n s ( t h e " f i r s t - o r d e r cdndi t ions" ) and

a s o l u t i o n t o t h e exac t Aggregation Problem, do not e x i s t . It $allows t h a t

t h e so- cal led " S t r u c t u r a l Form" of t h e " s tandard Macro-Econornet~~Lc Model" i s

no t a S t r u c t u r a l Form a t a l l . Rather , it is a De r iva t i ve Form, r e p r e s e n t i n g a

"crude approximation" t o an aggrega te v e r s i o n of t h e Walras ian [1 87 ] g e n e r a l

e q u i l i b r i u m model- - suitably g e n e r a l i z e d t o admit i m p e r f e c t competition--which

a t t e m p t s t o f i n e s s e t h e f a c t t h a t t h e s o l u t i o n s t o t h e I m p l i c i t ~ J n c t i o n

Theorem and Exact Aggregatlo; a r c no t a n a l y t i c ~ l l y p o s s i b l e . I I I I

There is one branch of economic r e s e a r c h , which has bee! l a r g e l y I

s corned o r ignored b j t h e a c a d e n i c p r o f e s s i o n of economis t s . It d o n s i s t s of I 1

a t t e m p t i n g t o conduct c o n t r o l l e d " Skinner ian" exper iments on r a t s t o a t t e m p t I

I

t o unders tand t h e " u t i l i t y func t ion" of r a t s v i t h r e s p e c t t i le cho4ce s e t of I

v a r i o u s s o f t d r i n k s . Ihe purpose is t o e s t i m a t e t h e r a t ' s lf2r?mand curve" f o r , I I

s ay , Cuca Cola . R e l a t i v e p r i c e s of t h e v a r i o u s goods a r e impl ic i t ' , , i n t h e I

I

s e n s e t h a t a g iven declsion-- say, whether o r n 3 t t o push a bar--id rewarded by I

d i f f e r e n t amounts of t h e v a r i o u s goods. Thus, by r e s t r i c t i n g t h e n u m b e r of I

"bar-pushes" p e r u n i t of t ime ( t h e r e b y i:=posing a budget o r incorn4 c o n s t r a i n t ) l

and by e x p e r i m e n t a l l y v a r y i n g t h e amounts of "rewards" p e r bar-pudh over t i m e , l

one may "deduce" t h e f o m of t h e r a t s demand curve , and, hence, tde i m ~ l i e d I I

s t r u c t u r e of t h e rat's u t i l i t y funct ion- - at l e a s t w i t h r e s p e c t t o t h i s c l a s s

I once had occas&m t o t o u r t h i s exper imenta l l a b o r a t o r $ i n t h e I

Economics Department a t Texas A and M U n i v e r s i t y whi le on a combihed 9 e

I

lecturelparental-visit-home. I must c o n f e s s t h a t I thought t h e r d s e a r c h t o be

r a the r n e c u l i a r , if n_ot downr ight smus'_n~--see ,?. , v a.L- . Jm~*wF*-d .-

I

[ 48 1. Now, 'f am n o t s o s u r e . C l e a r l y , t h e problem--how i o g e n e r a l i z e I * I

from t h e behavior of j a t r t o t h a t of humans--is a major u n r e s o l v a b l e , and - I .. I

s h a r e d w i t h much of exper imenta l psychology (where t h e c t i l i t y f u 4 c t i o n

I

appears i n t h e broader c o n t e x t of g e n e r a l human ( r a t ) b e h a v i o r ) . However, 1

I

1

- I

1

11 I s h a l l a t t e m p t t o s e t up t h e " econometrics" of t h i s problem 4s a n - i l l u s t r a t i v e example i n P a r t I h r e e .

t h e methodology, i m p l i c i t i n t h e approach, i s s u g g e s t i v e of a p o s s i b l e pLt9

t h a t economists might w i s e l y fo l low. b

While c o n t r o l l e d experiment:^ a r e one approach, t h e obvious znzlogglr

t o such a n approach i s t h e use of consumer a t t i t u d e s u r v e y s i n a noa-

t v ~ e r i m e n t a l economic environment. Perhaps t h e major proLlem w i t h a t t i t a d % m a l

o r o t h e r consumer su rxeys is t h a t t h e c o n d i t i o n s of e x p e r i m e n t a l c o n t r o l (the

so- cal led " c e t e r i s parib:lsU assumption i n t h e economists ' jargon) a r e n o t

m e t . A s a r e s u l t , p o l l s , su rveys , etc., must be t r e a t e d i n t h e same way that

w e t r e a t o t h e r non- experimental data--i .e. , w e must e x p l i c i t l y c o n t r o l f o r a

"changing economic / soc ia l environment'' by embeddirg t h e survey- response

" dec i s ions" w i t h i n a socio-economic model. This o f f e r s a n approach to both

" d i r e c t o b e r v a t i o n a t t h e household- level on t h e d e c i s i o n process ," w h i l e axso

c o n t r o l l i n g f o r e x t e r n a l c o n d i t i o n s . Indeed, i t s u g g e s t s t h a t t h e r e s u r g e n c e

of i n t e r e s t i n t h e micro-economics of household behavior ( a l a Becker [3] ,

Lehrer and Nerlove [ 5 b , 571 , Seckman [40 , 411, Deaton and Muellbaucr [ l l l ) i fs

not misp laced . The "new household economics" is s u r e l y t h e b a s i s f o r

advancing our knowledge of both micro and macro behav ior . A s i m i l a r a r g u m e m

a p p l i e s t o firms-- see, e .g ., Westphal .and Rhee [go] f o r an extreme v e r s i o n - * =

In s h o r t , I am s u g g e s t i n g t h a t t h e d i r e c t i o n whic5 economic r e s e a r c h

e x h i b i t genu ine ly meaningful theoreins abou t i n d i v i d u a l economic a g e n t s and - *

t h e i r maoacroeconomic o B a g g r e g a t i v e consequences, is t o b u i l d macrb n o d e s f-

micro f o u n d a t i o n s . I s h a l l propose some e x p l i c i t s u g g e s t i o n s a s t o how t h i s --

may be conducted l a t e r i n t h i s paper . For t h e p r e s e n t i t s u f f i c e s t o n o t e

t h a t , a p a r t f r ~ m t h e e c o n ~ , , i i c c o s t s of d a t a c o l l e c t i o n and p r o c e s s i n g and t k

n a t u r e o r purpose of t h e i n v e s t i g a t i o n at hand--i.e., do we wish a comple te

g e n e r a l e q u i l i b r i u m econometr ic model o r a p a r t i a l e q u i l i b r i u m r e p r e s e n t a t i a n

of a p a r t i c u l a r "sec tor , " t h e r e is no s u b s t i t u t e f o r cons t ruc t ing the macro

model " d i r e c t l y from t h e bottom up."

This i s not. n e c e s s a r i l y cu r r en t gene ra l p r a c t i c e . There is a "from- I I

the-top-down" school--as exemplified by the attachment of inpu t- output models 1

t o macroeconometric models i n order t o d i saggrega te t h e macra-level& of

aggregate demand and input requirements by sector- - see Duesenberry,Fromm, I

Klein and I(uh [ 15 ]--which proposes t he oppos i t e . I n c o n t r a s t , w e have t h e I I

approach of Orcut t [ 71 I , involving the bu i ld ing s p of a macro s imbla t ion

model of t h e e n t i r e economy from e x p l i c i t models of mic ro- un i t s . Thdeed, t h e I

present approach being proposed is, i n essence , a statistical/econc$netric I

implementation of t h e Orcut t v i s ion , where 5ehav io ra l parameters atje

e x p l i c i t l y es t imated by F ~ 1 1 Information Maximum Likelihood (FIML) methods I

using a su i tab ly- s ized random/s t ra t i f ied scmple of i n d i v i d u a l un i t$ , and where I

the s i z e of t t e rando: sample w i th in each s t r a t i f i c a t i o n- c e l l is t i i l o r e d ( a s - 1/ i n c l a s s i c a l sampling theory) t o t h e des i r ed p rec i s ion of estimate?,

I

Spec i f i c I s sues

As a mrans of both motivat ing t h e proposed approach t o t h e s tudy of . I

I

Neoclass ica l Econometrics ( see Par t one, subsec t ion 1I .D . ) and exhibiti1.g

* r L ~ r z , - 3 - ~ e ~ ; ~ a . ~ . - m s o f *>? ' ' 2 ~ x 7 I <lea , "

'r Wo by proceeding, s e r i a t i m tl-rough the

* C

t heo re t i ca l%ssues : - ( a j The Imp l i c i t Function Theorem

fol lowing sequence

I

I

l/ It should a l s o be c l e a r t h a t , by analogy with t he s t a t i s t i c a l : theory of - c t r a t i f i e d random sampling, t h a t i n o rde r t o so lve " the Aggreqatioa Problem w e a l s o need a tatis is tical f raqe ," providing t h e s i z e of the "ce l l " i n t he populat ion, r e l a t i v e t o the s i z e of t he sample drawn from

' + rn each cell--and, hence, the need f o r decennia l censuses of households and f i r m s . I

I

I

condit ionsl l - -principal ly cont inuous dtfLerentiability--there e x i s t s a s e t of J

func t ions ,

* such t h a t any element ,>f t he vec to r of J opt imal dec i s ion v a r i a b l e s , zi., can

be expressed a s func t ions of t h e remaining v a r i a b l e s , z and 8. -i - A s Lancaster [ 53 ] s t a t e s 'I.. . t h i s cons&&tutes the main r e s u l t of

t h e Tiuplicit Funct ion Theorem, and g ives t he warrant f o r t h e e l imina t ion cjf

v a r i a b l e s between equat ions o t h e r than l i n e a r equat ions."

Suppose r:e de f ine t he J + K + M v a r i a b l e s ,

#

' z' 8'1 + w is a mapping f r o u one " r e a l space" i n t o another : Thus. ki., -i.* - -i J+K+M i n t o R ~ + ~ + ~ . I f w e denote t h e Jx(K+M) ma t r ix , i.e., from R

2 * & x : m Q ~ ~ i ~ , by I$i and t h e JU matr ix , [ " ~ i m ] *

a x i m awi2n , by MZi, where

- aLyim n=1, ..., J ) , a n d & -9 is the a r b i t r a r y (m,n) element of M* (rrL=l, . . . , J

a x i m &i2n a. 2 i -

and n = l , . . . ,K+M& then the Jacnbian mat r ix of the mapping is e a s i l y seett t o - I,

I *

have t h e form,

* * I

where I is t h e i d e n t i t y m a t r i x of o r d e r J. Thus, d e t M = d e t M 2 i i and t h e J i * *

Jacob ian does riot v a n i s h i f M is nonsingular--whatever t h e rank o M . 2 i f 1i

* There fo re , provided d e t M,i(0) = 0 f o r a l l 0 E Oi , where O denote4 t h e - - i

* f e a s i b l e parameter s p a c e , and g iven z.., t h e n M i s i n v e r t i b l e , and ( t h e i I

I

1 / e x i s t a and is un ique .- i n v e r s e mapping, w - , , I , i-

Let 0 deno te t h e i n t e r s e c t i o n of t h e f e a s i b l e and a d m i s s i d l e

pa ramete r s p a c e s , Oi, and GO, r e s p e c t i v e l y , - a s s o c i a t e d w i t h each i n d i v i d u a l , I

i , i n t h e random ( o r s t r a t i f i e d ) sample, i - e . , l e t I ~

where O d e n o t e t h e " t rue" parameter s p a c e . Thus, O d e n o t e s t h e dodain w i t h i n 0 I

I

t h e a d m i s s i b l e parameter spac::, 00, such t h a t , f o r a l l i n d i v i d u a l s , ~ i=l , . . . ,N, I

w i t h i n t h e sample ( g i v e n t h e i r a s s o c i a t e d i z v a l u e s ) , we a r e Wadanteed i- I

I shor t : t h e domain, Q,is t h e s u b s e t of t h e a d m i s s i b l e dr t r u e s p a c e

I

t h a t t h e r e e x i s t s a unique s o l u t i o n t o t h e i r s e p a r a t e o p t i m i z a t i o n

'It t h e r e f o r e f o l l o w s t h a t , provided - 0 ~ 0 , i t is p o s s i b l e t o de te rmine

v a l u e s f o r t h e complete s e t of i n d i v i d u a l maximizat ion problems

c h a r a c . t e r i z e t h e behav io r of a l l i n d i v i d u a l s w i t h i n o u r ( s u b ) sample.

where t h e r e is a non- vanishing Jacob ian m a t r i x o v e r a l l i n d i v i d u a l s i n t h e - sample, and , hence, p r o v i d e s an e x i s t e & e theorem f o r t h e s e t of o p l i m a l

I

problems.

unique

presumed t o

In

* d e c i s i o n s , 1 N under t h e S i n t a i n e d h y p o t h e s i s .

d m

1/ It s h o u l d , however, be s t r e s s e d t h a t t h i s is a " l o c a l " ( a s oppoQed t o - " g l o b ~ l " ) r e s u l t , and ho lds w i t h i n some neighborhood of t h e p a r q i c u l a r 0 a t hand. This shou ld s e r v e . a s a warn ing a s t o t h e importance o t t h e i n i t i a l parameter v e c t o r , e(O), i n problems where t h e i m p l i c i t (2 .1 ) have m u l t i p l e f e a s i b i e so iu t ions- - see below.

It fol lows t h a t i f ~ E O , one is guaranteed t.he e x i s t e n c e of unique

demand/supply equa t ions , r ep re sen t ing t h e consequences of op t imiz ing I behavior

on t h e p a r t of i nd iv idua l agents . While t h i s r e s u l t may be of some i n t e r e s t I

f o r ' heo re t i c i ans , t he re a r e c e r t a i n f u r t h e r p r a c t i c a l imp l i ca t i ons d o r t h I I

pursuing. I

F i r s t , i f t he e x i s t e n c e cond i t i ons (i l . , non-vanishing ad obi an I I l

over t h e s e t of - 9cO a r e s a t i s f i e d f o r a p a r t i c u l a r choice of 8, the '"xistence I

theorem" simply t e l l s us t h a t unique opt imal dec i s ion l e v e l s ex is t , but it I

does - not t e l l us how t o f i n d them. Indeed, on ly i n h i g h l y spec i a l i zhd ca se s I

is it poss ib l e t o e x p l i c i t l y so lve t he set of i m p l i c i t f unc t ions , (2!1) f o r an

e x ~ l i c i t s o l u t i o n of t he form (2.2) by a n a l y t i c methods. E.g., i f t$e I

o b j e c t i v e func t ion happens t o be a q u a d r a t i c form i n t he E ~ . and ( s ee I

subsequent d i s cus s ion of t h i s s p e c i a l case i n subsec t ions I1 .C and 11 .D) , t hen I *

t h e dec i s ion func t ions f o r t h e opt imal x a r e l i n e a r i n z . In a l b o s t a l l -i -i

I

problems of more genera l i n t e r e s t , i t w i l l not be pos s ib l e t o o b t a i n t h e I * I

e x p l i c i t f unc t iona l form of t h e x I*' However, i n a v a s t number of s k t u a t i o n s , I I

i t w i l l be pos s ib l e t o o b t a i n t he unique numerical va lues f o r t h e opi imal I - I

I

s o l u t i o n , given the form of t he o b j e c t i v e func t ion , t h e set of state^ I

v a r i a b l e s , %* , and t h e f e a s i b l e domain, 0. Indeed, it should be apbreciated ' ' I

t h a t t h i s e x a c t l y what is proposed i n t he a lgo r i t hm of sub- sect ion 1~1.~ of I -

Par t One.

In l b h t of Ekis d i s cus s ion t h e n a t u r a l ques t i on t o a s k is^: "What s I = I can w e say i n genera l about O?" Several po in t s seem worthy of n m e . F i r s t ,

I I

given the d e f i n i t i o n of 0, a s the i n t e r s e c t i o n of t h e " f ea s ib l e " and^ I I

"admissible" parameter spaces a s soc i a t ed wi th t h e set of i n d i v i d u a l ~ opt imiza t ion problems, i t is l o g i c a l l y p o s s i b l e t h a t t he i n t e r s e c t i d n is t h e

I I

n u l l set . Suppose t h i s is the ca se . What does i t imply? C lea r ly , t h a t

(given s u i t a b l e r e g u l a r i t y cond i t i ons ) f o r every pos s ib l e choice of - 8, - a t l - I

l e a s t one of the op t imiza t ion problems a l l e g e d t o being "solved" by the I

i n d i v i d u a l agents v i o l a t e s the maintained hypothesis : I.e., given

func t iona l form of the optimand, the opt imal dec is ion of a t l e a s t one -

i nd iv idua l w i th in t he ( s t r a t i f i e d ) random sample is i n f e a s i b l e . 0 e might I~ t he re f o r e conclude t h a t such observa t ions , provided the number of such

I

i n s t ances is " s u f f i c i e n t l y small", a r e " o u t l i e r observationsw--and t h u s should

be d i scarded from the sample being examined under the maintained h;r~pothesis.

These observa t ions simply don ' t f i t t he model, and t h i s may be r e c t i f i e d by

r e so lv ing the o r i g i n a l problem with a smaller sample s i z e .

A l t e rna t ive ly , p a r t i c u l a r l y i f the number " o u t l i e r" of ob e r v a t i o n s s i n a p a r t i c u l a r p robles (with assoc ia ted maintained hypothes i s ) is large," one might ques t ion the v a l i d i t y of the maintained hypothes i s . 1 s h 11 term a t h i s the O u t l i e r Problem. Indeed, t h i s might suggest t o the i n v e s t i g a t o r t h a t l a s i n g l e maintained hypothes i s , spanning a l l i nd iv idua l s w i t h i n t he o r i g i n a l

sample, i s an ove r s imp l i f i ca t i on of the " r ea l worldw--i.e ., e i t h e r s e v e r a l

(pos s ib ly d i f f e r e n t ) o b j e c t i v e s a r e concurren. t ly being pursued by d f f e r e n t k members of the sample, o r t h a t the same func t iona l form of t he ob je t i v e i s c being pursued by var ious subgroups w i th in t h e sample, bu t t h a t t h e r

r

d i f f e r e n t parameter va lues wi th in such o b j e c t i v e func t ions . ~ be l i eve , a p r i o r i , t h a t our research o b j e c t i v e is t o d i s cove r a s i n l e

maintained hypothesis t h a t has universa) : a p p l i c a b i l i t f t o a l l mzrnbe s of a

given popula t ion , t h i s sugges ts a s u i t a b l e "disaggregat ion" of t h e o r i g i n a l

" s ing l e op t imiza t ion problem" i n t o more than one op t imiza t ion "sub- i

problems ." The ques t i on , which must then faced , i s : "How t o r e a s s i g n ~ i n d i v i d u a l s wi th in the sample t o the app rop r i a t e sub-problem?" Thii can be

done e i t h e r on the b a s i s of a p r i o r i c l a s s i f i c a t i o n of i n d i v i d u a l s t o t h e

s e p a r a t e hypotheses, o r by c l a s s i c a l s t a t i s t i c a l methods, o r by adbpt ing a I

Bayesian view, 1, which t h e " p r io r b e l i e f s" of t h e i n v e s t i g a t o r a r e I

sys tematized i n ,:he form of a " p r io r d i s t r i b u t i o n ." I am no t symp1athet$c t o

t h i s l a s t approach--see Hart ley [29] . I s h a l l , however, elaborate1 on each of ~ t he se a t an i n t u i t i v e l e v e l l a t e r . An a l t e r n a t i v e " solu t ion" t o tbe

I

occurrence of t h i s event is t h a t t he i n v e s t i g a t o r may be a b l e t o e bed t h e C va r ious proposed a l t e r n a t i v e hypotheses--along wi th t h e o r i g i n a l i n t a i n e d i hypothesis--within a "super o b j e c t i v e func t ion ," which con ta in s e&h of t h e

former a s s p e c i a l c a se s . An e x p l i c i t d i s cus s ion of t he se i s s u e s , In the

context of the Model Se l ec t i on Problem, w i l l be t r e a t e d i n t h e fo l iowing

sul : e c t i o n .

The second poss ib l e i n t e r p r e t a t i o n of the O u t l i e r ~ r o b l e b - - i .e ., t h e ~ same o b j e c t i v e func t ion , but d i f f e r e n t parameter values-- is more t r i c k y . If

one wishes t o r e t a i n t h e no t ion t h a t parameters a r e , indeed, cons t

no t , t h e m ~ e l v e s , random v a r i a b l e s (see, e .g ., the l i t e r a t u r e on t h

Random Coef f i c i en t s Model f o r t he converse--Swamy [83] , Cooley and P r e s c o t t

f 9 ] , e tc . ) , it sugges ts t h a t t h e r e is a f law i n the o r i g i n a l s p e c i k i c a t i o n of

I

t he o b j e c t i v e function--whence c e r t a i n key s t a t e v a r i a b l e s have befn c

- * :o + r e o r Z ldcns and h h i r r s oocn m y be presumed co f o i i ov t h e i r r e s p e c t i v e - '5 - i nd iv idua l s e l f - i n t e r e s t s , but t h e i r u t i l i t y func t ions d i f f e r i n - a sys t ema t i c '? I

I,

~ manner according t o r ace , and t h a t such 'd i f fe rences a r e manif e s t cd i n

I*

excluded. E .g., suppose, f o r purposes of p resen t d i s cus s ion , t h a t

I

d i f f e r e n t parameter va lues wi th in t h e i r r e spec t ive u t i l i t y functiohs--both I

i n t h e

being of the same f u n c t i o n a l form. Suppose, f u r t h e r , t h a t a s a coCsequenre of

such " di f fe rences i n t a s t e , " i t is i n f a c t the case t h a t t h e r e i s

I " in t e r ac t ion" between t h e s t a t e v a r i a b l e ' r ace ' and t h e dec i s ion v r i a b l e ' t h e a ~

number of Cadi l lacs purchased i n a given year .'--see, e .g . , Tobin's [ 8 5 ]

o r i g i n a l con t r ibu t ion t o t h i s problem, and t h e more r ecen t t reatmedt by Dubin I

and McFadden [14] on developing econometric models f o r purchases of1 consumer I

durables . I f , under t he o r i g i n a l maintained hypothes is , t h e model & a i l e d t o I

account f o r t h i s i n t e r a c t i o n , then one would conceivably encounter Lases where I

t he O u t l i e r Problem obtained f o r s i t u a t i o n s i n which the random samble

contained both Black and White members; o r , i f no t , t he l i ke l ihood i t h which Y the o r i g i n a l maintained hypotheses "explains" t h e observed sample of d a t a

i

w i l l , presumably, be i n f e r i o r t o the l i ke l ihood value under a r ev i s ion of t h e I

maintained hypothesis-+ .e., one i n which t h e underlying u t i l i t y fu4c t ion I I

e x p l i c i t l y captures the abnve type of i n t e r a c t i o n . I I I

I n e i t h e r ca se , t h e " solut ion" t o t h e so- cal led O u t l i e r P oblem 4 suggests a r e s p e c i f i c a t i o n of the o r i g i n a l maintained hypothesis--e t h e r t o a 4 I s i n g l e , expanded, but more " r e a l i s t i c " form, o r t o a model i n c h d i n $ s o r e

I i

d isaggregat ion i n the var ious poss ib le maintained hypotheses. ~ h u s , i n my I

view, t he f a c t t h a t t he condi t ions required f o r a unique s o l u t i o n t{ t h e I

Impl ic i t Function Theorem may riot be m e t i n a p a r t i c u l a r instance-- .e., a 4 vanishing Jacobian f o r a t l e a s t one sample member f o r any choice of ' the

parameter vec tor , 8, is, indeed a 'oehefi t t o t he pu r su i t of economic a s a n -- I empir ical science--provided, of course , t h a t t he " rea l world" does nbt prodnrr -

I I

such in s t ances . In s.&rt, a vanishin: ~ a c o b i a n - a t every po in t , 8, vk th in d.

- I

0 is , i n f a c t , a "signal" t o the i n v e s t i g a t o r t h a t t he re is somethi 0 *

with the o r i g i n a l spedlf icat ion- - see aubsec t ion I I B which fol lows: ~ I

We have a l r eady explored the case i n which O is the nul l- s

one hopes, w i l l r a r e l y -occur .when " carefu l thought" is given t o form l a t i o n of

the o r i g i n a l malntalned hypothes is . The next ques t ion r e l a t e s t o thh choice I

of " s t a r t i n g values," ?(O), t o i n l t i a t e t he a lgor i thm of Subsection 11.c i n

Par t One. Clear ly , t h e i n i t i a l choice of - 0(O), must be f e a s i b l e , i .e ., it

must permit numerical s o l u t i o n of the complete sample of i n d i v i d u a l

maximization problems ( l . 2 ) , f o r 1x1, . . . , N . We have n o t , a s yet , provided

any o t h e r method f o r choosing - .O(0), s h o r t of ' ' t r i a l and e r r o r" seaich--but may

be guided by e s t i s a t e s from o the r previous ~ t u d i e s . However, t h e kequirement

of f s a s i b i l i t y f o r - 0('), is a sine-qua-rion f o r the completion of skep 0.m.l i n

the proposed a lgor i thm. , I

Suppose, t h e r e f o r e , t h a t by " t r i a l and e r r o r" w e l o c a t e h " f ea s ib l e "

i n i t i a l parameter po in t , - 0(') . Clear ly - 0") c0, i .e . , - 0") belongs t o some I

f e a s i b l e and connected subspace of t he admiss ib le parameter space, . Three ' Oo

ques t ions a r i s e :

( a ) Can we always r e s t r i c t - 0'") a r & subsequent i t e r a t i o n , b=1,2, . . . , I

t o remain wi th in a f e a s i b l e subspace, 01

(b) Can w e be assured t h a t , given any i n i t i a l va lue , - O ( " ) ~ O , t h e

a lgor i thm of subsec t ion C, P a r t One, w i l l produce a l o c a l mixlmum of

* * t he concentrated log- likel ihood func t ion , L ( - 0), given t h a t t h e

I

i n i t i a l choice of - 0(O) r e s t r i c t s us t o a f e a s i b l e domain, ~b I I

( c ) Is the l o c a l maximum obtained i n (b) a g loba l maximum--anh hence - t h e

maximum-likelihood es t imator? I

We s h a l l a t tempt t o provide answers t o each of t he se ques t i ons a t an

i n t u i t i v e l e v e l , without regard f o r e s t a b i i s h i n g a set of r e g u l a r i t y .* I & I

condi t ions" on Q($ .,%. ,i). In s h o r t , our procedure w i l l be t o app roacg

B I s t he se i s s u e s a s a "numerical ana lys t ," but guided by both t h e foundation- s f

I

t he L n p l i c i t Function Theorem, and the f a c t t h a t 01.r proposed a lgo i i t hm embeds

t he i n d i v i d u a l ' s maximization problems w i t h i n t h e conf ines of a coocent ra ted

log- likel ihood func t ion .

I

Question ( a ) r equ i r e s t h a t we be a b l e tc empi r i ca l ly "{dentifyl' o r I

I I locatel 'L/ the b o u n d 3 of the f e a s i b l e subspace, 0, a s soc i a t ed - f e a s i b l e i n i t i a l parameter vec tor , - 8(0), and by w i t h i n - any

i t e r a t i o n , n, and given t h a t the p r e v i ~ u s i t e r a t i o n ' s parameter 1 po in t , 2(n), was f e a s i b l e , be a b l e t o guarantee t h a t t h e a l g o r i t

subsec t ion 1 I .C of P a r t One w i l l produce a cu r r en t i t e r a t e , - 8(11+ I, such t h a t

t h r e e condi t ions hold:

* * ( i ) ~**(e("+')) - > 5 ( - 8'")) f o r a s u i t a b l e choice of A

(it) 8 ("+') belongs t o some f e a s i b l e subspace 0 of El0.

** ( i i i ) L (8) - is uniformly bounded from above f o r a l l - 8cElO.

The f i r s t condi t ion r e q u i r e s t h a t t h e sequence of va lues f o r t h e

l i ke l ihood func t ion is monotone inc reas ing . The second condi t iod is t h a t a t

each i t e r a t i o n , n , wi th in t h e sequence, {2(n) : n=O, 1, . . .), t h parameter 4 po in t , - 8("), remains i n s i d e the boundary of a f e a s i b l e parameter subspace O.

Certain prel iminary poin ts should be noted. F i r s t , it

compact (i .e ., closed and bounded) space and contained wi th in the) e n t i r e

is probably a

the union of every f e a s i b l e parameter subse t , 0, is, i t s e l f , a

abmissible parameter space, El0; o r , 9 - ' s rna t ive ly , c+,, may, i t se l

I t

' lcorrect'l assumption (and not simply a mathematical convenience) t h a t : e i t h e r

subset of a

1 i m compact; o r f i n a l l y , f o r every element, 8 , of 8,

3 7 0 L 5 = - - m

*F - . Provided t h a t any one of these t h r e e assumptions is

Zollows t h a t , s t a r t i n g f r o m a n y i n i t i a l f e a s i b l e parameter p o i n t , I 0 (Or w i t h i n I - Y * - * I

1/ "Locate," i n the sense t h a t the a lgor i thm must - p a r t i c u l a r s t e p i n any i t e r a t i o n "s t rays" i n t o ou t s ide of 8 , i n which case it must "back up .'I.

- 18 - I

scme feasible subset, 0, and by restricting the movenent from one paradeter I

-7

point, - O(n), to the next, to feasible movements, the maximum Eclidean .--- distance between any two points, - O'n) and - O(m) i l , our A function o

( 1.29)), is uniformly bounded Ll Hence, since each possible 0 is, its if, e contained within a compact subspace, and the union of all possible 0's is ' contained within a compact subspace, it follows that ~ l c h feasible alglrithm

step must be of finite and uniformly bounded length. Thus, provided tbat we I

restrict all admissible steps in the algorithm to also be feasible (in1 the

* * of L (0) is guaranteed.

sense of permitting a unique solution to the equations (2 1 relative to the

The question, (c), cannot, in general, be established. Indc d, most

numerical optimization algorithms can guarantee only local optimality. 1 Two - points are, however, worth mention in this connection. First, by try

various different, but feasible (relative to some O) initial paramete

estimates, spanning a wide region in the admissible parameter space,

current - 0cO under the maintained hypotheses), convergence to a local maximum

algorithm is, indeed, the FIML estimate (i.e., the maximum maximorum -

6

provide increasing (subjective) confidence that any 0 so obtained via -

* * of L (0) for 0~0). Second, in certain specific pammetrizations, global - - -

our

concavity of both the log-likelihood and the objective function under the a I I

maintained hypothesis, {Q(Z~.,&~.,~)}, can be established 6

-- -.-LL - in such qLnstnncos, cu~y 9 I s the l ~ ~ i a u e FTML e s t l a a t p . One such examp e I s a i

r 11 I .e ., the algorithm recursion-formula (1.27) implies that the only -

admissible movements in the parameter space, if connected by stra:.ght lines, gives a piece-wise linear "pattern diagramw--a special being the familiar "hemstitch" pattern of a gradient alg%rithm upwa'rd to the "mounta&ztopl' through the sbsmess route" of an inclined valley (see, e .g, Goldfeld and Wandt [19] ), which can be "covered" compact set .

instance prxeeding

by a

Thus, t he I m p l i c i t Functlon Theorem not only motivates our

algori thm, but, under very genera l t o p o l ~ g i c a l assumptions, provideb t h e b a s i s I

(through the a b l l i t y t o monitor the choice of each 8'") t o ensure f e a s i b i l i t y t

with respec t t o both the "outer r e s t r i c t i o n s , ' ' 00, and t h e " inner ~ r e s t r i c t i o n s , " 0) t o ensure conwrgence t o a l o c a l maximum. 1

The i n t u i t i o n behind s t a t i n g t h e assumption t h a t "...the Lnion of I

a l l poss ib l e f e a s i b l e , bu t possibly disconnected ( i n t h e t o p o l o g i c ~ l sense)

subspaces is, i t s e l f , a subspace of a compact subspace of 0 0 "' " is t h a t , a t

t h i s l e v e l of g e n e r a l i t y , t he re is no way t o preclude t h e p o s s i b i l

a lgori thm w i l l "jump," on occasion, from one f e a s i b l e connected su

another , ( i .e . , - 8'") and - 8 ("'I) may belong t o d i f f e r e n t , d i s j o i n t {eas ib i e

subspaces). This doesn't matter provided w e a r e a b l e t o establish^ 1 / propos i t ions ( i ) , ( i i ) and (fii.1-

But, f o r s u i t a b l y small 6 > 0 i n (1.20), and due t o t h e f a c t t h a t

l i m L**(n,*) = a ~ * * ( ) -

6+0 - as -

" I L**( . ~("'l)) - > L**( - 0(")), with equa l i t y a t convergence (whence

whereas F") is, by cons t ruc t ion , p o s i t i v e d e f i n i t e a t each i t e r a t i o n , i t

0 * *

- = - ~ ( ~ ' l ) ) . Convergence t o , a t l e a h t , a !ocal maximum of- L I:),

f o ~ l o w s t h a t a s t e p , - 0'") + B(~+'), can always be found sgch t ~ a t -

with 0 r e s t r i c t e d t o f e a s i b l e parameter subspaces w i th in - - --

I

guaranteed, provided that . ; ( i i i ) can be e s t ab l i shed f o r t h e , L - *

I - I

I / In c e r t a i n i n s t ances (e .g . , t he "markets- in-disequilibrium" mo - cond i t i on , ( i i i ) w i l l be v io l a t ed i n t h z t t he l i k e l i h o o d funct unbounded over t h e e n t i r e parameter soace. I n t h i s c a s e , t he / 'admissible1' parameter space, g0, must be r e s t r i c t e d- s e e Hart ley 1321--to knclude a

2 neighborhood of t h e poin t , a t O , cZ0) = (30, 0, 0). .--

I I

I - L l - I

(concentrated) log- l i re l ihood a t hand. E .g . , f o r a l l 8rO Qhe log - l ike l ihooe - - 0'

funct ion may be uniformly bounded from above. If so , then wd have e x h i b i t e d a 1 mnotone increas ing sequence, { - a(") I, defined over a compact subse t , w i t h a I ** uniform upper bound on I, (g(n)), and hence t h e sequence con

Thus, i n terms of t h e proposed a l g o r i t k , we have i s t a b l i s h e d : 1 - -- Theorem: If 8") is feasib:e ( i .e., s a t i s f i e s 0 G), and Bn

defined such t h a t L----( - 8 1 daca) is uniformly bounded over a l l $ B ~ , then

s u i t a b l e choices of 6 > C ( i n (20)1, A(") > 0 ( i n 27)) and E I> 0 i n Step n&

can always be found such t h a t tho a lgor i thm of subsec t ion I I ~ C of P a r t Orme 2s I ** I I

waran teed t o converge t o a l o c a l maximum of L ( - 81 daza). - I

B. Model Se lec t ion

Let us, f o r t h e purposes of present d i scuss ion , assu e t n a t t h e r e 3 s

no a r i o r i - b a s i s (Bayesian or o the r ) t o a s s ign a p a r t i c u l a r bservat ion, i,

ta one of H poss ib le maintained hypotheses. Each "maintained hypothesis," %

h-l, 2, . . ., H, i n t h i s c o n t r z t , is defined a s a "well-behaveb" o b j e c t i v e

* 8 ), and, poss ib ly , subjec t t o 3. s e t of o n s t r a i n t s ( s e e funct ion , Qh(zi :zi a ,+- b l a t e r ) on the domain of the Cz I, where Q is a p a r t i c u l a r f n c t i o n a l fow,

-4 h

parametrized v i a 0 Suppose t h a t we can def ine the "super-o -he

funct ion" a s the convex combination of the (concentrated) log r l i k e l i h o o c

funct ions assoc ia ted wlth each of t he d i s t i n c t maintained hypith&ses, ia., in

I terms of the nota t ion of subsection I 1 . B of P a r t One def ine I

*

where the {%:h=l. 2, . . . , Y) a r e unknown parameters ( i n add i t iod t o I

t h e C8 :h=1 ,2 , . . .,HI), such t h a t f o r each h, fi I

and

whereas

We note that the (8 } may h2ve "cross-likelihood restrictions," -h

cases in which all 8 are identical, -h

8 = 8 = ... = 8 = 8 (say), -1 -2 -E -

or in which there are certain eiemerits, 11 S h , in common--

and 1 .-

- %h

- (say), h=l, . . ., H. I

'hen, we may simply interpret the 'funknowx~t' constant parameters, 14). as .the

probability that a random drawing from the population of individ*al agents

behaves according to maintained hypothesis, %." Thus, - the

** r $ :h=l, ..., H} may be viewed as the conditional

L

likelihood, given maintained hypotinesis, H,; and the "super objdctive - I

function," L**, of (2.6) is the unconditional - <concentrated) - 1 0 ~ ~ ~ l r a l l h m d -- ..

function relative to the finite set of maintained hypotheses, { 1 . A -z Ir -

It follows that ior any particular member, I#,. of the let of I * m

maintained hypotheses. the' proposed algorithm of subsect ion I1 .C ' Part C91e t applies to obtain (conditional) FIML estimates uf &,h=1,2, . . . %is

11 We shall illustrate such situations later in the context of pecific - neoclassical models of the firm/household. $ I

I

approach t o Model Se l ec t ion i n t h e context of a f i n i t e , denumerable

ua in ta ined neoc la s s i ca l hypotheses appears t o be new. It permits not

under genera l r e g u l a r i t y cond i t i ons , t h e l i k e l i h o o d r a t i o test of t h

hypothesis

Ho = a

s e t of

on ly a

d i r e c t test of the "super hypothes is ," t h a t one and only one of the

t r u e ( r e l a t i v e t o the p a r t i c u l a r s e t of H-1 a l t e r n a t i v e s ) , but a l s o

of d i r e c t es t imat ion of t h e p r o b a b i l i t y a s soc i a t ed with each \ ,h=1,.

the "super hypothesis" is t h a t a l l H,, are poss ib l e . In t h e former - -

versus t h e s p e c i f i c a l t e r n a t i v e ,

{I$,) i s - - g method

. . , H, -If - s: . tuat ion,

w i l l be a uniformly most powerful test.

The non- classical approaches t o t h e Model Se l ec t ion Proble

using continuous Bayesian p r i o r s i n p lace of t h e {a,, 1, o r " point p r i

which the x's a r e chosen a priozi-should fo l low s t a n d i r d argimenta ion , i f

these a r e t he preference of t h e i n v e s t i g a t o r or suggested by t h e p a r t i c u l a r I 11 problem a t hand-

One f i n a l point of i n t e r p r e t a t i o n seems worth mentioning. Suppose

t h a t the index, i, denotes both a cross- sect ion observa t ion , j ( i ) , w i h i a a . - I p a r t l c u l a r yea r , t ( i ) , i.e., is+ (j( i ) , t(i1). In t h i s ca se we have b i 11 s t r u c t u r a l approach t o time-ser:.es/cross-sec2ion modelling," which

I l l u s t r a t e d i n t e r n s of t h e £0110-ding problem:

1/ Indeed, the notion of "point p r io r " f o r t h e parameters. (&.%I ). su - t h a t t he re is a l e g i t i m a t e r o l e f o r Bayesian approaches wi th in C1 (F i she r i an ) S t a t i s t i c a l Est imation Theory--i.e., a s genera t ing i n

(0) parameter vec to r s g and a('), t o commence ( C l a s s i c a l ) Maximum Likelihood Estimators-- see Hart ley [ 29 I .

:ges ts l s s i c a l .t i a l

"Suppose t h a t w e are i n t e r e s t e d i n e s t i m a t i n g the pa rabe t e r s of a

product ion func t ion f o r a r e p r e s e n t a t i v e rubber farmer i n S r i La ka . Suppose C f u r t h e r t h a t w e a r e w i l l i n g t o p o s t u l a t e t h a t t h e r e a r e t h r e e ad

maintained hypotheses regard ing the "mechanism" by which t h e inp t (and l1 output ) dec is ions are made :'I ~

Maintained Hypothesis HI: ''Rubber i s a very c t>mpl ica~ed c rop t o - c u l t i v a t e and h a r v e s t . A s a r e s u l t , agronomists, s o i l s c i e n t i s t b , b i o l o g i s t s

and o t h e r s p e c i a l i s t s i n tbe sub jec t have s tud i ed the crop under

of experimental and non-experimental c o n d i t i o n s , and

c o l l e c t i v e wisdom i n t h e form of a s e t of "recommended

a c r e under c u l t i v a t i o n , and with adjustment f o r t h e

cond i t i ons ( s o i l type , e l eva t ion , expected weather

producer s e l e c t s the p a r t i c u l a r "recipe" f o r rubber c u l t i v a t i o n , ( and

ha rves t i ng app rop r i a t e t o h i s environmental conditions." ~ Maintained Hypothesis H2: - "Producers select t h e i r v a r i b l e i n p u t b - l e v d s s o a s t o maximize ( t h e presen t va lue of t h e expected f u t u e streazu 05) t

p r o f i t s with full-knowledge of the production- function (and wi th a paramet r ic

r u l e t c . def ine t h e i r expec ta t ions a s t o t h e f u t u r e t r a j e c t o r y of a l l s t a t e -- I variable.1 over some planning horizon as a func t ion af cu r r en t . and p a s t

I e in format ion) ."

Maintained Hypothesis H3: "Producers vary t h e leveIbs;l, - --

i n p u t s ( c u y e n t dec i s ion v a r i a b l e s ) i n response t o c u r r e n t (and L

f u t u r e ) st*e v a r i a b l e s . However, t he se dec i s ions are no t neces a r i l y -0pt3mal -

Ir w ( i n t h e sen<e of maximizing some ob jec t ive- - pro f i t s , o~: tput , e tc l ) . F i n a l l y ,

. I

the l e v e l of output is determined r ecu r s ive ly , given t h e product on f u n c t i o n ,

c u r r e n t arid pas t s t a t e v a r i a b l e s , and c u r r e n t and p a s t v a r i a b l e nputs." i

* itnder HI, t he x ' 6 a r e determined exogenously accord i

-i.

prescr ibed recipe--a so- cal led " engineering approach" t o rubber {u l t i va t ion ,

which makes no adjustment f o r changes i n t he r e l a t i v e p r i c e s of he tt i n g r e d i e n t s , d e s p i t e t h e f a c t t h a t s u b s t i t u t i o n - s i t h i n and acrods time

periods-- is techni-cal ly f e a s i b l e . Under H2 w e have t h e t r a d i t i o 1 economic 4 p r o f i t maxisizing model-the s t a t i c ve r s ion of which has been de c r ibed i n 4 subsec t ion I1 .B of Par t One (with Q i being prof i t s ) . F i n a l l y , udder H3 We

have a n hypothesis somewhere "between" H1 and H2, i n t h e sense t a t d e p a r t u r e s h from recommended p r a c t i c e a r i s e i n response t o p r e v a i l i n g s t a t e

such adjustment is not "economically optimal." The s t r u c t u r e of

under Hg is i d e n t i c a l t o the recurs ive model of dl--except t h a t a more

" sophis t ica ted" dec i s ion r u l e is employed- Indeed, Hg is e s s e n t i l l y our I.

according t o H H o r Hq, as well a s under an unknown convex-com i n a t i o n of 1' 2 - t

approach (b) i n s e c t i o n I o f ' Par t One.

These th ree maintained hypotheses i l l u s t r a t e how one

a problem of Model Se l ec t ion a s descr ibed above. One can es t imato

Wrameters of the product ion func t ion embedded w i t h i n Q(x z 8) + .'-i-'-

a l l t h r e e , where the d a t a "decide" t h e r e l a t i v e l i ke l ihood (msrgihal

might encounter

t h e

of (1.4)

p r o b a b i l i t i e s ) a s soc i a t ed with each i n t he populat ion a t l a r g e .

I Suppose now t h a t dhe has t h e a d d i t i o n a l complicat ion t h t over t h h

sample per iod , t(i ) r { l ,2, . . . ,TI, each of H I , 5 o r H is pursued, sub jec t to s 3 ".

t=1,2, . . . ,T, the underlying o b j e c t i v e function--and a s s o c i a t e a padameter

- changing set of i n s t i t u t i o n a l (or o t h e r ) r e s t r i c t i o n s . Thusjqfor

li

b vec to r , - - a t e presumed not t o change, and, moreover,

each

I the populat ion propor t ions , ah, adhering t o I$, vary wi th t . The lset of

I

c o n s t r a i n t s (not yet t r e a t e d e x p l i c i t l y i n our d i scuss ion s o f a r ) w i l l , I

e a c h H h , h = , . , , a s w e l l a s each t i m e per iod , t=1,2, ..., T is pobsible and, -

I

however, vary with t--and, hence, a d i f f e r e n t log- likel ihood func t ion under

indeed, es t imablz i n s o f a r a s t he parameter v e c t o r , 8, w i t h i n t h e same, I -

product ion func t ion under each I&,, is , t h e r e f o r e , common t o each. I

To i l l u s t r a t e t h i s po in t , it s u f f i c e s t o no te t h a t over t h e per iod I

from 1900, t h e f lrst year of bignif i c a n t rubber c u l t i v a t i o n i n S r i =&a (nee

Ceylon), u n t i l t h e p re sen t , rubber producers have been ope ra t i ng , v a r ous ly ,

under t h e fo l lowing types of c o n s t r a i n t s :

( I ) I a rge ly u n r e s t r i c t e d i n s t i t u t i o n s ,

( 11) Voluntary export cor i t rols ( t h e Stevenson Scheme),

1 ( i i i ) Car te l- es tab l i shed expor t quotas i n t h e aggrega te ( t h e I n t e r n a t Tonal

Rubber R e s t r i c t i o n Agreement).

( i v ) Phys ica l import c o n s t r a i n t s on imported i npu t s t o t he produ t i o n 4 process ( e .g ., f e r t i l i z e r sho r t ages during World War 11) , ~

(v) Subsidized input p r l c e s ( e .g ., f e r t i l i z e r , under t h e Rubber

Replantirig Subsidy Scheme of 1953),

( v i i i )

Subsidies i n kind (government supply of p l an t ing m a t e r i a l s )

Loans a t below-market i n t e r e s t r a t e s ( a vo1untary smallhold

r e p l a n t i n g s t imulus , t r i e d unsuccess fu l ly i n 1949) ,

Government c o n t r o l of permits t o new p l an t o r r e p l a n t rubbe# ( s i n c e

National, ization of a l l l a r g e > 50 a c r e s ) e s t a t e s (implemented over

t h e per iod from 1972 t o 1975),

i s adopted, is t o de f ine t h e e x p l i c i t form of t h e (concent ra ted) l o

l i ke l i hood func t ion a s soc i a t ed with each maintained-hypothesis/tlme-'?eriod

combination--each being a func t ion of t he rlme behaviora l I

e t c . - Thus, the "econometrician's problem, once thel$pproach suggested i n :his paper

- 27 -

parameters i, whereas the "operat ions researcher ' s problem" is t d provide a n - I

a lgor i thm t o so lve t h e "determinis t i c vers ion" of t h e op t imizer ' s problem,

z and 0 ( a s w e l l a s pos s ib l e given Hh, 1 , Qh(~i . ,z i . , B), ~ i o , - i . - I

i n t h e c o n s t r a i n t process-- see l a t e r ) . This requi red i n t e r f a c e b tween

s t r u c t u r e and behaviora l conten t i n t o "economic time-series analyslis," a s I

econometricians and ope ra t i ons r e sea rche r s is long overdue. Furtl-er,

approach suggests an apparen t ly novel procedure t o i n t roduce economic

opposed t o the r a t h e r mechanical use of e x i s t i n g time-series mode & (such a s

t h e u n c r i t i c a l use of t h e s o- c a l l e l Box-Jenkins approach [ 6 1,

Analysis of Economic Time-Series, Granger and Hatanaka [ 22 1,

t h e

In tegra ted Moving Average (ARIMA) processes-- see, e .go, Sims [ 81 ] e t c .) i n I -

i n s t ances where t he re is c l e a r l y a " s t r u c t u r a l change" over t i m e- s, e.g., i n a I

t he i n t e r e s t i n g , but non- st ruc tura l polynomial s p l i n e model of Poi L

I F i n s l l y , one may permit t h e x's, themselves, t o be func

time--say, r e s t r i c t e d t o t he c l a s s of s t e p func t l?ns over t i m e ( v i b

app rop r i a t e use of "dummy var iab les" ) , o r even s t a t e v a r i a b l e s , a s

1 i n z y, ( Z . a ) , where t he func t ion , -i. - 41, may r e f e r t o the mult inomial

I choice p r o b a b i l i t i e s ( b g i t , ' h o b f F ; StC.9 over a f i n i t e set of ~ orderedlunordered a l t e r n a t i v e s . An example of t h e l a s t of t he se

problem of es t imar ing of the e f f e c t over tfme of the government

s e r v i c e (suirabl:? measured a s an element of z ) on t h e p r o b a b i l i t --i -

re* ommended p r a c t k e s (HI) a r e being fol lowed, a s opposed t o a t tem C i su$cessful ( H ~ ) o r unsuccess fu l (H3)--to maximize prof i t s , with

6

p a b a b i l i t y a2 and 1 - al - a r e s p e c t i v e l y , and where i nd iv idua l 2'

I

throogh some known sequence of ( i ) - ( x ) , e t c . , I n t e r m i t t e n t l y ove t t i m e , and

I

i n t he contex t of . a l l of the dynamic technologica l and economic ~ I I

c h a r a c t e r i s t i c s of a pe renn ia l crop such a s rubber . Indeed, i t wal/ t h i s

s p e c i f i c problem--estimation of the " a g r i c u l t u r a l supply funct ion" f o r S r i

Lankan rubberL/--which suggested t h i s p a r t i c u l a r paper .

C . The I d e n t i f i c a t i o n Problem:

The c l a s s i c d e f i n i t i o n of i d e n t i f i c a t i o n of a parameter

vec tor , s, due t o Wald 186, Lema 51, is the fol lowing:

"8 is i d e n t i f i e d i n OO i f , f o r any o t h e r 8 + 8 wi th 8r84, 4 - -0' -

with p r o b a b i l i t y one, where the measure is taken r e l a t i v e denotes the j o i n t dens i ty func t ion of the j o i n t l y dependen variables ' '

4 - I :L *' * - applicable i n t he present case with G m = [ Q ~ S * ] being the i t h l o w of Y,

and having a m u l t i v a r i a t e normal dens i ty def ined by ~

L T I B of P a r t One. , t ~ e

and with n being the ith row of JI def ined by (1.10) and (1.11) i n --I

* m 8 e r customary paramet r iza t ions of the o b j e c t i v e f u n c t i o , n subsec t ion

0 . \ > > d q*tl. - :he ; r J e l + ~ i ~ i . ~ + + > : 3 2 C i S i , 3, ~i.11~ :7 jenez-il 5. : eii:if:ed,

-5

i L

provided t h a t a l l elements of - - q* a r e observed (Case I of subsec t io I I B , P a r t - *

-1 I

1/ A World Bank sponsored research p r o j e c t (RPO 672-02), "The Econ - the Supply of Perennia l Q-ops: A. Case Study of Natura l Rubber a Production i n S r i Lankal'--in which the f i r s t au thor is Pr inc ipa Inves t iga to r , with a s s i s t a n c e from R. K. Pe t e r s ( ~ e s e a r c h e r / r u b Rajendra ( ~ e s e a r c h A s s i s t a n t / t e a ) and consu l t an t s , D. E ther ingt Nerlove .

One). An identification problem, in partial equilibrium models of this

canonical form, will, however, occur when the value of the objective function,

Q ~ , is not observed for each individual i, i=1, . . . ,N (Case IIj--the typical

economic illustration being the l~ousehold's utility maximization problem,

subject to an income equality constraint. Neoclassical economic theory of

utility maximization (see, e.g., Henderson and Quandt [ 43 1 ) implies that for

any "cardinally measured" utility function, Q, with associated explicit demand

functions,

* exactly the same zi, - values will obtain under any strictly aonotonic

transformation of the original cardinal measure of Q. Hence, there will be an

identification problem with respect to certain elements of 8 for a-ny cardinal -0 -

representation of the utility function, Q(~.,~i~,~), whenever the utility . .w

level of individual i, Q is not observable or, more generally, not observed i ' -

(missing data on Q;). The particular way in which this problem manifests - itself will, of course, depend upon the functional form of Q-e.g., quadratic,

1 / Cobb-Douglas, CES, trans-log , etc .- r s

The of interest here relates to the matter of how such an

identification problem will affect the concenttated log-likelihood -

function, < ' ( ~ I x , z ) , of (1.17), and implementation '& the proposed algorithm I - -

of subsection IIC of Part One. - I 9 - < - I

1/ This is evident since, for any monotonic increasing function, say, - P(Q(~, ,3. '2)) with P'( ) > 0, t!le "first order conditions" are the same

set of implicit functions, ( 3 4 1 , in the & .

Suppose for a particular choice of Q, the vector, i, may be ~

partitioned into

8 = - (2-13j

I

where the ?41-subvectorB -1 8 ' refers to the potentially identifiable (estimable)

parameters in 2 8 is the M 2 -element subvector of clearly unidend,ifiable I

- ' -2 parameters, and M + M2 = M. Let us consider two situations:

1

(1) the investigator is able to analytically determine%(fron he explicit ti form of the demand equations) which of the elements of 2 re

-1

a potentially identifiable (8 ) and which are clearly not ( ) and

(ii) the investigator is unable to determine for the - -

Q, which elements are potentially identifiable

unidentifiable (i.e., inestimable from the

In the situation of (L), the sub-vector, 3, contains th only

parameters of potential interest, in the sense that for any feasib e pair of A- + parameter points, - 8 and 2 , say, where both are partitioned according to

+ (2.13) and where 8 = 8' but i2 # f2, the resulting optimal solut on I -1 -1' * +

values, x of (2.12) relative to either a or 5 , will b*: -i ' - Q

+ It follows that we may foranyz dc?pite the fact thapi2 #22. i

* * ' - I I *

immediately "shrink" the dimensioffality of the parameter space .t

M, and proceed as if the subvector, 8 is the parameter vector of potential -1 '

interest, whilst 8 may be assigned arbitrary feasible values and the elements -2

of 22 treated as if they were known constants. In short, the fac that Qi . --

the M elements of z2, which may be resolved by t r e a t i n g -- 2 - - - the p o t e n t i a l l y est imable parameter vec to r and the - a r b i t r a r y , but f e a s i b l e ) cons tan t s .

a l a t e n t va r i ab le c r e a t e s an . ' d e n t i f i c a t i o n problem with respec t

What happens, however, i f t h e inves t iga to r is i n s i t u a ion ( i i ) I

t o

above, and proceeds t o attempt t o e s t ima te the e n t i r e vec to r , 8, i n ignorance

of which elements a r e p o t e n t i a l l y es t imable and which a r e clear1

In the I n i t i a l i z a t i o n Step of t he a lgor i thm of subsect on IIC of i Par t One a f e a s i b l e 8 - ('1' = (O)' :e(O)'] w i l l be s e l e c t e d i n i norance of - [ i l -2 +- the "proper" p a r t i t i o n i n g . Ihe numerical approximation t o the g i a d i e n t

vec to r , - L**(O") of (1.26) i n Step 0.2, w i l l then be ca l cu la t ed nd, provided

t h e i n i t i a l guess - 0") was not " r igh t on" (i .e . , provided t O), the

vector 5 **(O'.) should exh ib i t '0.0.. .OO i n a l l pos i t i ons

c l e a r l y un iden t i f i ab le parameters wi th in , - 0, thus permi t t ing , up+ a p p r o p r i a t e

reordering of - 8 t o conform t o

a shrinkage of - 8 t o the "revealed" number of p o t e ~ t i a l l y i d e n t i f i

1 / parameters, Ml .- I

A p a r t i c u l a r example i n which Q(x z 0) is a pos i t i vb d e f i n i t e r u 7

*.'*.'- I

.. i b

I/ It is i r o n i c t h a t t o achieve " revealed p o t e n t i a l i d e n t i f i c a t i i n " of z, one -

must not make "too accurate" an i n i t i a l e s t ima te , e(O), of 91 - akin t o the need t o have "suff i c i en t l : i t e r a t i o n s , n= l , 2, . . . [I71 algori thm i n o rde r tha t ~ ( ~ 1 - l of (1.28) be a " sui tab le" the information mat r ix i n (1.33)-see Davidon [ l o ] .

This is , i n t h e DFP

es t ima to r of

footnote 'a' on page 19. Suppose

In this case, we might proceed by defining th2 "full" parameter ector as v

where

0' = 9 - 1% i 41 lx(J+Kj

partitioning of [x' : z'], we would defjne the row vector of pa ameters - 0 - r

(2.18)

I' = (yij) and, under symmetry (I' = I") and in the context of the

withtn I' as: I

particular

- . ? L .

= vec I" : vecI'i21 , 1-11 - - s

so as to "is&late" the relevant unique elements of I' 11PP21 and 122. I

respectively. Thus, the full parameter uector, 8, contains a t tal of M = 1 + 0 (J+K)(J+K+l) (J+K) + elements.

Now, since

it is imnediately obvlous that a, -2 B and r 22 will clearly be unibentifiable 1

without cbservations on %--since they do not even appear in the set of - I

implicit equations, q'(x,z,8) - - = - 0, of (2.1). 1 Now, evidently, provided r is of full-rank, J, equat on (2.1) 11 I

pcrmits the analytic solution,

where the reduced form coefftcients consist of the J-vector,

I

I - ' f ~

Consider now the concentratedLlog-likelihood function,^ I - - *

e ~ I

and the JxK matrix,

11 Indeed, with the insight provided by the analytic solution i (2.21), and, - hence, the "form" of (2.22a) and (2.22b), one classify the r .levant elements of veer , , to 82'.

I

I I

0 . z - I I I

(2.22b)

Thus, equations (2.22a) and (2.22b) represent a set of im?lied r strictions on

-1

e the reduced form coefficients, and indicate that B and r12,--on choice of

L . I

I the J(K+l) structural parameters of relevance--are "estimable up to multi-

1 plication by an arbitrary nonsingular symmetric matrix, A : - rip 11 r e s ~ e t t i v e l v .- I

I

* * N * LII ( i IX,Z) = c o n s t . - - PnlS ( j ) l , 2 I (2.23)

* where S ( - 8) is def 3 ned by ( 1.18), w i t h t h e e x p l i c i t : " so lu t ion- a l u e matrix,*

w r i t t e n both i n terms of t h e s t r u c t u r a l pa ramete rs

reduced form paramete rs (A and 0), i n which i denotes t h e u n i t v e c t o r of -N

o r d e r N. I **

The v e c t o r of f i r s t p a r t i a l s of LII(i) w i t h r e s p e c t o 0 may be f - o b t a i n e d by n o t i n g t h e fo l lowing r e s u l t s :

* a g n l ~ * ( e ) l J J a e l s * ( g l asi i,(z)

a e = C C * - j=i j*=i as a - e j , j '

agn I s * ( g I * = s*jJ* ( t h e ( j , j * ) t h e lement of ~ * ( 8 ) - ' )

as j , j '

and, in conformity with tke partitioning of 8 in (2.17), -

ax* (el iJ= a e -

where Cij is t h e JxJ matrix with arbitrary (k, 2) element,

It will be noted that equation (2.24). defines the Jacobian of 8

transformation, i .e . , I

I

(rmk) is the Jxl column vector consisting of the kth ,column 01 rll, and -1 1

the

' @' denotes the Kronecker product, i .e ., for arbitrary matr:.ces, 7 ~ x 2 * -3 - r

A@B =

mniJ .

, and

-- this time with r e spec t t o the elements of -a, - It now fol lows, from (2.25) - (2.28), t h a t

* * a L~~

a a 0

(1x1) ........................... ........ * * a L~~ -1 *-1 * - -N-rllS (X-X )'&

...................... .(+A2 ........ **

a L~~ - * * 0 - a 4 I(!!)

as2 ........ - .. m... - - - (Kxl) a~: O..........O......... ao- -

avec rll - g11 (J(Jll) x 1) ............................ ........ **

a L ~ ~ - avec r - 21 g . -2 1 .................... .(4%1.2 ....... ** a L ~ ~ - 0 airec I' K(K+l)

22 &- 2 x 11

where,

N J J * = - N o C C C ( x i j - xi j ) S *j f ' Ell vec -

is1 j=l j's1

and

, I

N -1 *-1 *

9 2 1 = - N c C = I ( r l l x z t . ) s (s*:. -zl.)* - .a C

- . - - Thus, it is evident t h a t t h e a lgor i thm of subsec t ion * L - One w i l l au tomat ica l ly s epa ra t e the fu l l- vector of parameters,

which a r e c l e a r l y unidentifiable, and those w h i c h a r e p o t e -2 '

i d e n t i f i a b l e , unidentifiable, . This f e a t u r e should be b u i l t i n t o Step 0.2 c -1

algor i thm, so t h a t t h e p o t e n t i a l l y es t imable parameter vector j

1.C Bf P a r t

I , i n t o t hose

t i a l l y

t h e

, upon

reordering, automatically "shrunk" to the MI, elements of 0 Ir remains, -1

therefore, to provide a numerical procedure to characterize the elass of I

estimable or identifiable parameters within the subvector, 8 . 4s we shall -1

see, this will involve the use of the Sore-Penrose generalized inverse in I place of the standard matrix inverse and the use of the Singular value

let us now return, for the momelt, to the gneral problbm-~ase II

of subsection 1I.B of Part One for an arbitrary (latent) objectiv (utility] I function. Suppose that we have "shrunkw thO dimensionality of thk parameter

I

vector, i, to the M1 elements of the potentially identifiable veckor, , d a I I

the numerical methods indicated for the Initialization Step. our; "estimable* > ~

I

equations may therefore-be written as: I I

* .. I

8 Eli. ' X. Li.s -1' z2) + ?i. I (2.31) I I

I

Suppose we adopt the convention that for each iteration n, we fixlthe

subvector, 0 at'gts previous iteration's value, i .e., -2'

presuming that 0") a , indeed, fea?i!le. 1 l i -2 . I

\

i ' 7 r ~ 5 t r 4 ~ 5 ~ : ~ s ~ ~ l ~ ~ ton r , ~ t30 l t t ~ s ~ ~ - ~ - < - ~ l r <. -. ,%- t; ~cl3-vecrz r 0: 3 ~ ~ e ~ i ~ d ~ 1 Y ' J I

identifiable parameters in the M1-vector, 8 In general, only cirtain of t h - * -1 '

I

elements of 8 will be -ually identifiable, but, in the absence !of an -1 w

Consfder now the arbitrary nth iteration, and note that we may

1/ I am indebted to Marcello Pagano for clarifying discussions on these - matters. See alr o Pagano and Hartley (741 for an illustration of these qetbods in the wntext of polyn~rnial distributed-lag models. ;

I explicit representation of the functional form of the solution vectors,

where J(") is the Jacobian mat r ix ,

* * Xij

i s the a r b i t r a r y ( i , j) element of X and elm is the mth -1

In terms of t he above no ta t ion , the "concentrated"

funct ion may be w r i t t e n as :

** N * * L ( 0 8 1X.Z) = cons t . - 7 h l [ ( ~ j - ~ . j ) ' ( ~ m j , - ~ m j , I (2.38) 11 -1 '-2

whence (2.25) holds with

Thus, with the use of (2.35) and (2.36) we have

i n the case where the {J I a r e f u l l (column) rank, Ml--in which ase a l l MI I rLr * e

elements of 8 a r e i d e n t i f i a b l e , &d hence, est imable. I

-1 I

In genera l , however, the mat r ix , ~ ( " 1 , w i l l be of less than f u l l I

column rank, and thus we must use the Moore-Prxose Generalized ~ n k e r s e i n

equat ion (2.39). This may be obtained by means of a Singular Valu I Decomposition (SVD) of the matri:., ~ ( " 1 , i .e.,

where u(") is an orthonormal matrix in the inner-product space d 4 fined by the weighting matrix, (S *(n)-l I), i .e . ,

where v(") is an orthonormal matrix such that

the "singular values" of J(") (see Golub [21]). It follows,

(2.43) that the interated (non-linear) SUR algorithm is given

8 (n+l) - - Iv(n) .(in)' ,,(n))-lOv(n)'] [ v ( 1 *(n~-~& x(n)l -1 *rs

1 .

-1 = v . '(n)' .(S*(n) @ 112 (n) ~ (2.44)

-- where A(~) is a matrix of the form,

. . Y."

Ah] = [i?.?] , NJxM

1

( 2.43)

and D(") is a diagonal matrix with strictly positive diagonal el ments--called

Thus, using (2 -32) .

defines the MLE--see Oberhofer and Kmenta [ 70 1 . The number of "linearly

independent" elements in fj is the rank, ml, associated with -1

Finally, the asymptotic covariance matrix

* cove -I = lim v (n)Din)2v(n)'

n+w

k t us now draw together the implications

A

of 0 is given by -1

of the above discussion.

We have argued that, except in highly specialized chcices of functional forms

which admit analytic solutions to the implied "first order cond-itions," the

situation of a latent object3ve (say, utility) functioh will result in:

(1) an identification problem with respect to estimation of the "full"

s e t ;i pclramecers in 8, and ' d i

(2) that, in general, the investigator w:Lll not be able to decide, a C - - - *

which elements of 0 are identifiable (and hence estimable . -

from data on X and Z) and which are not.

We have, however, exhibited numerical procedures which "solve" the

Identification Problem via the use of our algorithm. In particular, we have

shown that once an initial parameter vector, 0('), has been found which is -

I

f e a s i b l e ( i n the sense of subsect ion II.A), the numerical approx matior t o t h e

**(O9 ') o f ( 1 - 2 6 ) i n gradient of the (concent ra ted) log- likel ihood func t ion , - L

Step 0.1 of i t e r a t i o n 0, w i l l e x h i b i t zeroes i n a l l p o s i t i o n s a s oc i a t ed with

c l e a r l y u n i d e n t i f i a b l e parameters* Upon s u i t a b l e reorder ing of , t h e s e may 4 success ive i t e r a t i o n s .

h e remaining MI elements i n - 8, v i z ., i1, c o n s t i t u t e t h e s e t of

be l a b e l l e d a s (32, and may be t r e a t e d a s cons tants , s e t a t 8(0), -2

p o t e n t i a l l y i d e n t i f i a b l e ( o r est imable) parameters. Indeed, the numerical &

i n a l l

ax*( z , ~ ) rank, m of the Jacobian of t he J N x 5 matr ix , 1 ' , def ines t h e number

El -1 +- of func t iona l ly independent parameters wi th in e . Rote a l s o t h a t i n

c a l c u l a t i n g the numerical approximation t o the gradient vec tor , 9 in

each i t e r a t i o n , n=0,1,2,..., we a r e a l s o au tomat ica l ly

numerical approximation t o every element i n t h e requi red ~ a c o b i a n ~ ma t r ix ,

w i l l i n general %be of l e s s than f u l l column rank, i .e . , w i l l e x h i b i t a n

a x t j $1 s i n c e J(") = ( ) f o r m=1,2,. ..,MI . Hence, no a d d i t i o n a l

a em c a l c u l a t i o n s a r e requi red t o implement t h i s approach. It fol lows

unknown numerical rank of m < MI. But the Singular Value Decomp

J(") a t each i t e r a t i o n , a s L I t

automatical ly:

t h a t J(")

1 1 ) provtde a s e t of "pst, iartesl ' o<>!") at rach t t e r a t i s r i w - r

7- .. A

f o r the s e t of "implied r e s t r i z l o n s " on the " reduced-fob - parameters" i n the context of l&s-than-full rank Jacobi

- 0

and '

( i i ) def ine the numerical rank, m 1 < 5, of t h e Jacobian matrkx, and,

I hence, "inform" the i n v e s t i g a t o r a s t o the a c t u a l number of

f u n c ~ i o n a l l y independent parameters i n h i s problem*

Identification Problem--as has been customary ecocometric practi e with I

These procedures, therefore, obviate che need for an analytic

respect to the limited class of problems which admit such an a-

so:htion to this

investigation. -

Several further clarifying points seem worthy bf menti&. First,

the procedcres advocated will only tell us the number, ml, of furkctionally I

independent parameters within the M -vector, 0 It will 1 -1

determine explicitly which these elements are, in that this requ res an i arbitrary choice of so-called "normalization rules," From a pra

standpoint this does not matter. Indeed, the use of the SVD to

Moore-Penrose Generalized Inverse implicitly imposes one such no A, ualization rule--see, e.g., the discussion in Pagano and Hartley [ 74 1--vi ., unique 1 parameter estimates of 0 are obtained by invoking the extraneous1 criterion

-1

values--see, e .g ., Rao and Mitra [ 77 ] .

A A

that ^e has the "shortest" length, ( 0 ' 0 )'I2, of the class of -1 -1 -1

Second, it will be noted that we have made use of the

Jacobian matrix in resolving two different issues. In order to

admissible

feasible solutions to the optimal decision variables of each indi L

- 4

i=1, . . . , N , within rhe sample given 0, we required a non-vanishing Jacobian of

c .. , "'0. 1 n . . a , ------------ s e e s u ~ ~ e c c ~ o n l l . A , equation ( 2 . 4 ) . -4 permits use

laxim "i2nl

* of the hplicii Function Theorem to clef ine optimal xi *(z+ ,i) 8

present instance, we also make use of the Jacobian aatrix, -im

equation (2.37), to resolve the Identification Problem. In both ases, t suitable numerical algorithms are employed to convert problems th t are i tackled analytically by customary econometric practice, to problek whose

solutions may be obtained numerically. Not only does this vastly expand the I

class of problems amenable to solution, but it also eliminates the decessity

that applied economic researchers also be competent applied mathema icians/ I i - I

econometricians. Indeed, specialists in solving these types of prodless I . -

analytically will, in a manner analogous to the "mathematical econonhics ~ industry,'' be rendered "technologically obsolete."

I I I I

One way to look at the role played by the two Jacobian madrices is I

to consider the relationship between the objective function Q(x z e), the * o S - 4 e S -

* * I optimal decisions, x = *8), and the parameter vector, 2, in the context -i . x. (ziL I

of i-1,2, . . . ,h' observations on X and Z. We begin by noting that f 04 each - *

individual, i, the Jacobian matrices. {H :i=1.2, . . .,I?}, must be nor/singulars 21

i .e., recalling that w' 1 2

' 0'1, equation (2.1) implicitly dedines = 1%. - *

the {q. :i-1, . . .,NIB and for 8 ~ 0 , -

I * I * must be nonsingular in order that the (x 1 exist. Then, given the :{Ei 1, the

1-

I - * I

- I E

and suqect to normalization rules on the "revealed" 5 - ml unideatbfiable - I

elements of 28, has ra& ml, indicating the number of independent

* in 8 which will induce differential changes in the {x 1-- and, henee, in -1 -i * I

the { Q ( i , i , ) . Thus, the conditions on these Jacobians are neckssary to I

define a one-to-one correspondence between the objective functions, Q and I

the relevant elements of BrO, as revealed or mediated through minimilation of - I

* t he general ized variance, S (81X, - Z), given by

min

0 4

In s h o r t , the Jacobian condit ions a r e necessary t o " s t a t i s t i c a l l y inve r t " the

* chain r u l e corresponding t o the sequence {Qi } + h i )+is where t h e

I * nons ingular i ty of t he {M ) a r e requi red f o r d e f i n i t i o n of the f i r s t "step,"

I 21

and the rank of $ 1

n+m I

I

i .e. , ml, def ining the number of i d e n t i f i a b l e parameters wi th in is

required f o r t he second.

F ina l ly , we should note t h a t t h e SVD of J w i l l def ine q rank, I

ml C Ml, bu t , t o the ex ten t t h a t J is e s s e n t i a l l y t h e l imi t ing ( a s

n-t -) " regressor matrix" i n ' an ( i t e r a t i v e l i n e a r model of t he fo rq (2 -36) w e

w i l l be incapable of d l s t ingu i sh ing between the multicollineari it^ Problem" , I

@

(i .e. , a l i n e a r dependence i n the columns of J ) and an " I d e n t i f i c ~ a t i o n I

Ptoblera" ~ i t h rPs?ect to the elements c:f 9.. R e Forrnt-;- Cssus ' 3 d 2 + 2 -1

'3 dependent ( i .e ., depends orf the sample X and Z ma t r i ces ) , wh i l s t t he l a t t e r is - * a l o g i c a l , v m i f e s t a t i o n of *he func t iona l form of 9 f o r any X and; Z - matr ices . Zn shor t , t he re is an I d e n t i f i c a t i o n Problem with respec t t o t h e

causes of a numerical rank def ic iency , m < MI-either a ~ u l t i c o l ~ i n e a r i t ~ or 1

an I d e n t i f i c a t i o n Problem!

E. Uncertainty, Expected Values and c e r t a i n t y - ~ ~ u i v a l e b c e :

The formulation of the canonica l farm of t h e model str 4 c t u r e of Pa r t

Oneiequations (1.1) - (1.7)--involves some i m p l i c i t assumptions r e g a r d i n g t h e I

s t o c h a s t i c s p e c i f i c a t i o n of t he " error s t r u c t u r e .I1 I n par t icu la l / , we have I

i m p l i c i t l y assumed t h a t t he ind iv idua l -agent has knowledge of th$ s t r u c t u r e of

t h e optimand, Q , , , including t h e " true" parameter vecto 4, , which

is unknown t o the e x t e r n a l observer o r i n v e s t i g a t o r . This parad gm involves 4 t h e following h e u r i s t i c j u s t i f i c a t i o n :

( a ) Each agent , i, maximizes (minimizes) Q wi th r e spec t t o f o r given 4 I values of z and 8 t o determine the (unobserved) optimz(l i npu t -i I *

vector , x of (1.3). -i

( b ) A s a r e s u l t of "human f a l l i b i l i t y , " ins t ead of t h e expl

* vector , -1 x . s random " errors i n m a x i m i ~ a t i o n , " ~ ~ ,

are 'Mde agents , r e s u l t i n g i n the observed vec tor of

(1.5). Such non-systematic e r r o r s may be

* a (mul t iva r i a t e ) normal dens i ty , u - N ( 0 , C ) .

I* - ( c ) The consequence of t he u t i l i z a t i o n of t h e a c t u a l d e c i s i

vec tor , xi., r e s u i t s i n a value of the optimand, Qi,

(1.41, where the e r r o r , E may be va r ious ly i'

L

measurement e r r o r on t h e optimal value of t h e

e f f e c t of "missing vartables" fn the a p t l m n d , Q, etc .-]the l a t t e r * - inducing poss ib le covariance, a, between E and - i - -

be noted t h a t t h i s paraqgm involves a r ecu r s ive - I -

11 -

causa l 1s t ruc tu re" : I I ~ i

(2.49)

and the Jacobian

is un i ty , regard less of whether o r n o L a = 2. Thus, t h e concentraqed log-

** l i ke l ihood funct ion is properly defined by L (e l Y,X, 2) of (1.15)

I - I **

o r L (01X,Z) of (1.17), r e spec t ive ly . I1 -

Since these mat te rs appear t o have been t h e s u b j e c t of s me 0 controversy ( i n the context of c e r t a i n s p e c i a l cases t h a t have bee poss ib l e n t o t r e a t a n a l y t i c a l l y ) , it w i l l be the purpose of t h i s s e c t i o n t o bornpare and

s t r u c t u r e . In t h i s regard, it w i l l be convenient t o cons ider t h e se where I-= c o n t r a s t the present approach with a l t e r n a t i v e motivat ions f o r the

I

Q~ is observed (Case I of subsect ion 1 I . B of Pa r t One) and t o exanipne t h e

e r r o r

cases i n which the production funct ion wi th in Q(x z 0) is e i t h e r + *'-i 0' -

( i ) Cobb-Douglas, o r I ( i i ) Quadratic. I

I I

Case ( i ; has been extens ive ly inves t iga t ed by Nerlove 1671, Klein 11501, Wch

[44], Walters [88], Mundlak 1651, Ymenta [511, Elarschak and Andreds [621,

J T , denote o.itput, x , denote x vector of vririable 1 --I I

Zellner , Kmenta and Dr:ze 1931, e t c . Case ( i i ) permits us t o exa

connection between the "maximization of e x p e ~ t e d p r o f i t s" motivat

notion of "Certainty' Equivalencem--see, e .g . , Theil (841 .

A

and z denote a vector of f ixed inpu t l e v e l s . The Cobb-Douglas 1-

9 t

, The Cobb-Douglas Case:

funct ion is given by

I

I I I

- B j . n 2 Yk a- lI x B I

Yi - i k I ( 2 -50 )

j-1 i j k = l

whence ~

or , i n logarithmic u n i t s ,

f j '=1 ,2 , . . J . Note that t h i s is a l inear system of J equations in

the {tn x , : j ' = 1 , 2 , . . . , J ) , which can be wri t ten i n notat ion a s : i j

where IJ is an ident i ty matrix of order J ,

and

* Ilnx = ' -i*

This system, i n conjunct ion with the p r o f i t , o r , i n t h i s case,

I I

t h e product ion

fucc t ion i n logar i thmic u n i t s , may be w r i t t e n a s :

and estimated d i r e c t l y by FIML methods ( a s an i t e r a t e d SUR sys em). The I 1 a r b i t r a r y nth i t e r a t i o n i s given by:

Step n .l : Given a(n) B ( ~ ) , y(n) , c a l c u l a t e the Q(n) : 1.1, . . . ,N} and - 9- - --i I

t he { ~ n x * ( ~ ) : i = l , -1 - ..., N}.

Step n.2: Given {%. I and {&ti *("I 1 , c a l c u l a t e

and y - (n+?.) a s a SUR (Generalized k a s t Squares) regress ion co responding t o

I

estimates--see Oberhofer and Kmenta [70] . ~ I

(2.59).

I t e r a t i o n bet aeen Step n.1 and Step 1.2 14.11 r e s u l t

This has - not been the t y p i c a l approach t o the probl m of e s t i m a t i n g e t h e Cobb-buglac production funktion unzer the maintained- hyp+thesis of p r o f i t

i n FIML

maximization wi th p e r f e c t l y comp::.titive good and f a c t o r marke s . The * t

t r a d i t i ~ n a l r e p r i s e n t a t i o n , due t o Marechak and Andrewe ~ b i j , i n u o i v a s w r i c i i ~ g L

t h e system i n l a a r i t h m i c u n i t s , but i n t h e form:

r - Iln Y - B'2m = h a + y'llnz + E

i --I* -1 i. ~ I ~ ( 2 .6Q)

The disturbances, u , : 1 2,. . . , J in the "traditional model1l ~ I

"...are introduced to allow for rand~m, non-systematic1 errors on the part of entreprenelTrs in their attempts adjust inputs to satisfy the necessary conditions for profit maximization. On the other k..nd, the interpret& tion of E has not been as clear?^^ in the literature.)

i Marschak ar.d Andrews [62] drpcr?he the E as reflectilpg i "technical efficiency" and GcpenLing "on the techn5cal~ knowledge, the will, effort and luck of a given I

11 ~ entrepreneur- I

As noted by Zellner, Kmenta and Dr&e [93],

"...it is clear irom (2.60) that the llnx are not inde2endent of the E , since each i n p a s ' a function f 111 disturbances of {he system. Consequenrly, c1assic 1 least squares estimates of the production function par - meters wili be biased and inconsistent. This conclusi n led to the development of a number of alternative

i ~ estimatian methods based on various assumptions conced- ing the prof it maximizing conditions. In particular, the so-called "factor shares" method (Klein [50]) assumes ~ (the) abse~ce of profit maximizing restraints, and the 1 maximum-likelihood method (Kmenta [51]) requires the 1 assumption that u is distributed independently of

i*

In response to this situation, Nundlak and Hock [66] nd Zellner, a I

Kmentr and Drzze [93] have independently ~roposed an alternative to the

"traditional specjfication." They assume a I

written in logarithmic units as the f irst equation of (2,593, 1 ~ where E represents I

i I I I

i q f q - - o r s ';*::- 7 : jp-l i-hpr, * : ; I ~ , ? ~ L c ~ s L L = -;3,-:3:;3ii~ cn . . , *.-L machine or labor performance, and sc on. Whenever the^ production process is not instantaneous , the effect of 1 - the disturbance on output cannot beanown until after the

W preselected quantities of inputs ha* been employed in^ I proci~iction. Any given level of inp& will result in fm

uncertain quantity of output and, consequently, in an 1 uncertain profit. The concept of "profit maximizationl," which is unambiguous within the deterministic framework of the model postulated by economic theory, needs a moke subtle interpretation when stochastic elements are ~

l! Zellner, Kmenta and Dr$ze [93]. I

I - I I I

I

introduced: obviously no manager can maximize something which is uncertain and beyond his control .'&I

It is proposed, instead, that (a) entrepreneurs maximiz the e mathematical expectation of profit, and (b) prices (p and q) are

with certainty or statistically independent of the production fun

+ for firm i--see [ 9 1 . ZKD disturbance, E ~ , with expectations, pl and qi . 1 continue by noting that:

- -3

been proposed, the motivation for our error structure follows si ilar 1iw.s tea' 0 -

hndlak and M c h [66] and Zeuner, Kmenta and Drzze [93]. . s -

"In reality, the profit maximizing conditions are not likely to be exactly fulfilled because of managerial errors originating in inertia, ignorance, etc. These errors, resulting in deviations from the desirable position, can be assumed .to be randonly distributed over all firms in the industry. Another source of deviations from optimality ex post is the differene betwe~n anticipated and realized prices.... One important implication of the new model ip that inputs do not deperd on the disturbance in the production function... hder this assumption, the result is that simple least squares estimators are consistent; under normality, or with the stronger assumption that s and u are statistically

i -i independent, they are also unbiased ." They conclude :

"As far as estimation of the production function parameters is concerned, the simultaneous natnre of the ~

a model is releva-lt .only because it makes x stochastic.

The Quadratic Case: ~

1. As a renult, the finite sample properties of the least

We have already developed the quadratic model in the c ntext of mar 0

I

discussion of the Identification Problem with Qi being latent ut lity. Im the I

squares estimators, except for their property of unbiased;zess, remain to be established."

4 ~ It should be evident, therefore, that while the struct re of our Y

r>hh-D011~1r l~ ~ o ? P ' ( ? - 5 9 1 CLffers Eyorn t%e -r-?r?ous a i : p r n a t l ~ ~ ~ s vhlcb 52*;t -

1/ Zellner, Kmenta and Dreze [93]. -

* is a nonlinear SUR system (with zio defined by (2.63)) and FIML est~imates

I

of a, , r and L may be obtained by the algorithm of subsection 11.j~ I of Part

One.

Certainty Equivalence:

Suppose we motivate our model (1.3) - (1.5) of Part One dy the

Zellner , Kmenta , Dr&e [93] presumption that ent tepreneurs maximiz expected

-i

e normalized profits with respect to x . Thus, from (1.4), in gene a1 r

I

In the case where a quadratic production function is empioyed, ~ * Q has the f o m given bl7 (2.64a) . Clearly, i

and

- *' * - x +E$.ki. Ezi. +* +. -i.

- *' * * - y , y 4- t ; Z -1. -1.

- so that the usual result of C rtainty Equivalence (Theil [84, pp. 'i

I

holds, i.a., paximization of expected profits and maximization of

associated with the expected levels of variable inputs leads to t

decisions, since the objective functions differ by a constant.

&

414-4241) - - - the profit

ie same

E. The E-M Algorithm: I I

The so-called "E-M algorithm1'--see Dempster, Iaird an4 Rubin [ 121, I

designed both: ~ I

I

(1) as a general purpose algorithm to compute Maximum ~.ik&{ihood

Estimates (MLE's) in the context of models involving I complete Data, 4

Data problem at hand and the corresponding Complete ~ a j a model,

(2) as an instructive motivational and/or pedagogical devife to exhibit

was initially proposed by Hartley [28] in the context of a mult 4 nodal "grouped data" problem and, indeedU...covers any grouped discre e t distribution.'&/ The extension of this "score" by Dempster, Ia rd and Rubin

(121 to the gen6.ral class of incomplete data problems is, indee

the connection between the particular manifestation of

contribution, despite the fact that computational experience wi h the t

the Incomplete

Expectation-Maximization algorithm iqdicares that, in return fo guaranteed i convergence, its rate of convergence is slow. Examples of the pplication of I the E-M algorithm to econometric models may he found in Hartley [31, 33, 34,

321 and Hartley and Swanson [39]--applied to the Tobit/ProbCt, 1 e

Regression (Self-Selection),Markets in Disequilibrium and Trunc

Regres s ton models , r e s p e c t i v . l y . T t 1s the purpose of fhis &1&n ta 4??@99~TP

the connection between the E-M algorithm and the canonical form of the problem I in Neoclassical ~conometrics--see subsections II.B and j~-c of hart One.

- I

In the context of the problem at hand, "incomplete da

I 1' endogenous'' variables within the model arise in two ways:

1/ See the comments of Hartley following the paper of Dempster Laird and - Rubin [ 121 . ,

* (a) the optimal decision vectors {x :i=l,. . . ,N) may be vifwed as latent

-i I

variables in the context of a more comprehensive complkte data model,

and

(b) the value of the optinand {Qi: 1'1, . . . , N l may not be obberved (Case 11

of subsection II.B, Part One). I Case (a) always occurs 2nd case (b) occurs in instances where t e Qi are h latent, i.e., where Q( ) denotes, e .g., a utility function. 1

Since the E-M algorithm is not widely known by econo 1 etricians at I

this juncture, we shall provide some preliminary discussion. '$e following

quote from Dempster, Laird and Rubin [ 12 , p.11 is difficult to improve upon:

"The term 'incomplete data' in its general form implies the existence of two sample spaces ,l() and a many-to-one mappin The observed a realization from 83: r"k? ::?;spending x in'x is no observed directly, but only indirectly through

though, in certain examples, 5 includes what are

specifically, we assume there is a to3, and that x is known only to of 3 determined-by the equation, y = observed data. We refer to 5 as

traditionally called parameters. We postulate a family of

depending on family of sampling specification, f(. data specification, g( . . . l . . .), by

- -2 L The E-M algorithm is directed at finding a value, 2, which maximizes g(yle), given an observed -

'but it does so by making essenFial use of the I

Tamily f(x1 - - 8) . @ Each iteration of the E-M algorithm ir~volves two, -

d steps, which we call the Expectation Step (E-Step) an the Maximization Step (M-Step) ." Consider the function, I

f (TI - - 0) > 0 almost everywhere i n X f o r a l l - 0rO. W e now d e f i n e t+e E-M I

+ which is assumed t o e x i s t f o r a l l . p a i r s ( 0 - ,2). In p a r t i c u l a r ,

E-Step: Compute R(010 (.I) - -

suppose

M-Step: Choose - 0 t o be a value of 0rO which inaxi+izes

Ihe " . . .heur i s t ic idea here is t h a t w e would l i k e t o choose maximize

l og f ( x l 0 ) . - - Since w e do not know l o g f(xl?), we maximize, i n s t ad , i t s t c u r r e n t expec t a t i on , given the d a t a , 1, and the c u r r e n t f i t ,

Indeed, one may view R( - 01 - 0 ("I) a s ' the "pseudo log- likel ihood f u

0!"), a s i n Har t ley [32, 33, 3 ] . Dempsrer, r e l a t i v e t o t h e c a r r e n t i t e r a t e , _ b i r d and Rubin [12] provide a set of genera l cond i t i ons s u f f i c

A

guarantee convergence. n u s , s i n c e a t convergence - B(") = -(n+lf 0 = 2 (say), st

is i n s t r u c t i v e t o no te t h a t t he "pseudo log- likel ihood func t ion+" R(01 - - 8), and

** t h e a c t u a l log- likel ihood func t ion , L ( 8 ) , - a r e i d e n t i c a l a t a 1 iterations 1 such t h a t - 0(") = - e("+'), and, hence, w i l l produce equiva len t l ok - l i ke l i hood -

A

values a t 8.

I n t h e p re sen t contex t , i t w i l l s u f f i c e t o s p e c i f y both t h e so-

b c a l l e d E-Step and t h e M-Step a s soc i a t ed withqeilther (a ) o r both ( a ) and (.b> of

P a r t One f o r t h e canonica l form of t h e Neoclass ica l Econometric Problem, a d

LO p r o v i d e some i n c u i c l v e mociva Lon f o r t h i s approach. 4 i Suppose w e have a f e a s i b l e and "current" parameter ve t o r , B ' ~ ' , for

? I -. !m

any n=0,1, . . . . Given - 0("), we may proceed t o c a l c u l a t e the "0 t i m a l

dec i s ions , I 1 p

1/ Dempster, Laird and Rubin (12, p . 61. -

f o r every sample mesber, i, r e l a t i v e t o - € J ( ~ ) . Indeed, t h i s s t e p s equ iva l en t

t o Step n .1.0 i n our a lgor i thm of subsec t ion 1 I . C of Pa r t One. I The {x *("I} values a r e t h e r e f o r e l a t e n t v a r i a b l e s w i t h i n t h e o p t i i z a t i o n

-i t model def ined by equa t ions (1 2 1 3 ( 1.4) and 1 5 ~ u r t h e k , given

equat ion ( 5 ) , it fol lows t h a t the {si.

t he observed dependent v a r i a b l e s , x , r e l a t i v e t o 8 , i .e . , -i

* Thus, a s may e a s i l y be v e r i f i e d , c a l c u l a t i o n of t h e {x. 1 i s equ v a l e n t t o t h e

-1

E-Step of t he E-M a lgor i thm. 1 The M-Step r equ i r e s maximization of R( - 81 - 8'")) . Howeve

a l r eady noted ( i n t h e context of our d i s cus s ion of t h e I d e n t i f i c

- subsect ion I1 .C.) t h a t t h e " l i nea r i zed" vers ion of t h e model, (

by equat ions (2.35) and (2.36), and is, indeed, p r e c i s e l y the ma

M-Step of t he E-M algor i thm. This may e a s i l y be v e r i f i e d by con

func t ion S(81 - - 8(")), d i f f e r e n t i a t i n g R with r e spec t t o - 8, and s o l

f o r - 8. Thus, i n t h e contex t of t h e " ful l- rank model" i n which a B

- of 8 a r e i d e n t i f i a b l e (Case I of subsec t ion I I . B , P a r t One), t h e

f o r t h e MLE ( s u i t a b l y modified t o inc lude the a d d i t i o n a l equa t io (1.4)) w i l l b be def ined by equa t ion (2.39). In t he s i 'cuat ion of a l a t e n t s e t of {Q 1, t h e

I d e n t i f i c a t i o n Problem a r i s e s , and we nus t make use of the Sing l a r Value Y

I

Decomposition of kc om lead in^ t o t h e E-M a lgo r i t hm represenL d by equa t i on

(2.44). I It should be noted t h a t i n e i t h e r ( a ) o r (b ) t h e M- t e p f o r t h e E-M i

a lgo r i t hm does no t r e q u i r e i t e r a t i o n s , i t s e l f . This is becaus a c l o s e d form - i= I

a n a l y t i c s o l u t i o n t o t h e s e t of i m p l i c i t f u n c t i o n s de f i ned by f o r t h e

" l i nea r i z ed" r e p r e s e n t a t i o n of t h e (non l i nea r ) SUR system,

(1.4) and (1.5), e x i s t s i n t h e form of equa t ions (2 -39) and

In problems i n which such a closed- form s o l u t i o n

f o r t h e M-Step ( s ee P a r t Three below), one has a cho i ce . The k-M a l g o r i t h m

I Our proposed a lgo r i t hm of subsec t i on I I . C , P a r t One, on t h e o t

sugges t s t h a t , g iven - 8("), one should i t e r a t e w i t h i n t h e M-St=:>

o b t a i n 8 - ("+'I de f ined by

to

r of "length" A(") be made. Needless t o say , i n g e n e r a l such a

C?) > ;<::Lsf;i ( 1 . 5 5 ) , F.?., 1; d i i L nor inaxiinize i 7 ( a l a ) for it vill - - .-

simply improve R, i .e ., *.~rider (2.671,

r e q u i r e s t h a t a s i n g l e s t e p a long t h e d i r e c t i o n v e c t o r ,

a (n+l) - e(n) = n ) H(n) L**(n,-) - - 9 - ( 2 -67)

f o r any n=0 ,1 , . . . . 1n s h o r t . i f (2.65) does - not admit a closed-fdrm a n a l y t i c

s o l u t i o n , t he E-M and t h e proposed a lgo r i t hm of subsec t i on I I . C , P r t One w i l l

d i f f e r .

a F. The Aggregation Problem:

As s t a t e d by Green [ 2 3 ] : ~ " In t h e b roades t s ense , agg rega t i on i s a p roce s s whereby p a r t of t h e in format ion a v a i l a b l e f o r t h e s o l u t i o n of a prob len i s s a c r i f i c e d f o r t h e purpose of making t h e problein more e a s i l y manageable. We s h a l l be conlerned ... with t h e cgses i n which aggrega t ion takes t h e form of r ep l ac ing a s e t of numbers ( f o r i n s t a n c e , q u a n t i t i e s o r p r i c e s of commodities) by a s i n g l e number o r a sma l l e r set of numbers o r " aggregates" ( f o r i n s t a n c e , q u a n t i t y o r p r i c e i n d i c e s ) . Aggregation w i l l be judged " s a t i s f a c t o r I'

by the economist t o t he e x t e n t t h a t he b e l i e v e s t h a t t h e c o s t of handl ing infor inat ion i n g r e a t e r d e t a i l outweighs the g r e a t e r r e l i a b i l i t y of t he r e s u l t s he might ob t a in b

of course , on the purpose of t h e i n v e s t i g a t o r .

I 1 us ing more d e t a i l e d in format ion ; khe judgment must depen , d

It must be admi t ted t h a t t h e bulk of t h e l i t e r a t u r e ~ on aggrega t ion has no t been concerned with t h e compariso of t he se two k inds of cost- - the c o s t of handl ing and t h e c o s t of s a c r i f i c i n g d e t a i l e d in format ion . Most w r i t e r s appear t o have been mainly i n t e r e s t e d i n d i s cove r ing t h e

4 cond i t i ons i n which " cons i s t en t" aggrega t ion ( t o use McManus' [60] term) is p o s s i b l e . Aggregation w i l l be sa t o be c o n s i s t e n t when t h e use of in format ion more d e t a i l than t h a t con ta ined i n t h e aggrega tes would make no d i f f e r e n c e t o t he r e s u l t s of the a n a l y s i s of t h e problem a t hand.

, r

To i l l u s t r a t e t he concept of c o n s i s t e n t aggrega t ion , l e t us assume t h a t each household 's consurn~t fon is a f unc t t on of i t s income a l o n e . ht us

" . . . . d e r i n e dggregdie i i i c o i ~ ~ AS i..e su n;j,;e;.i).,2 ; . ? C 3 E l 2 ~

and agg rega t e consumption a s t h e sum - ! horsehold consumptions. In what cond i t i ons i s agg rega t e consumption, a s de f i ned , a f u n c t i o n of ( t h a t i s , unique1 determined by) aggrega te income? If t h e r e a r e no r e s t r i c t i o n s on t h e d i s t r i b u t i o n of income, t h e necessa r

Y and s u f f i c i e n t cond i t i on is t h a t a l l marginal p r o p e n s i t i t o consume a r e constant: and e q u a l . I f t he on ly t he i n v e s t i g a t o r i s t o p r e d i c t aggrega te consumption, p roper ty of t he marginal p r o p e n s i t i e s ensures t h a t t h e aggrega t ion procedure de sc r i bed is c o n s i s t e n t ; knowledge of t h e d i s t r i b u t i o n of income, f o r example, would not a f f e c t t he r e s u l t of h i s a n a l y s i s .I1

The above, a l b e i t l engthy , quote is , i n my view a l u i d e x p o s i t i o n i of the e s s e n t i a l s of the so- cal led Aggregation Problem. Indee , a s noted 1 above ( i n s e c t i o n I), I submit t h a t the f a i l u r e t o "solve" the Aggregat ion I Problem is not s u r p r i s i n g , per se, i n t h a t t h e fundamental que t i o n has b e e m

wrongly posed. The underlying presumption of most w r i t e r s on he s u b j e c t fs

behaviora l equat ions," such a s t h e Keynesian Aggregate Consump ion Functfom, i

t h a t ~ c r o e c o n o m i c s , a s c u r r e n t l y p r a c t i c e d , is a l e g i t i m a t e , J s e f u l and

Aggregate Production Functions, Aggregate Demand and Supply Fu c t i o n s f o r 4

sc i en t i f i ca l ly- sound a c t i v i t y . I.e., t he f i t t i n g of so- called "macro

output of goods and s e r v i c e s , f a c t o r s of prrjducticn (such a s 1 bor ) , etc., to .i macro t ime- series data on the r e l evan t Macro Aggregates is--ev n on cos t - 4 b e n e f i t grounds ( see Green, above)--a j u s t i f i a b l e a c t i v i t y . 14 w i l l be the

expla in the same universe would be e i t h e r o t i o s e o r ~

purpose of t h i s s e c t i o n to ques t ion t h i s premise.

Cer ta in w r i t e r s on the Aggregation Problem have been

mark ." Peston [ 7 5 ] , e .g ., argues t h a t t h i s vers ion of t he prodlem is of '*mu

wrong .I1

" c lose r t o the

g r e a t importance" : - " If macro-theory were v a l i d , any micro- theory invented

Indeed, i t is the "macro-theory" t h a t is both o t i o s e and wrong1

r)es?-?te +k-Ls. :LR n o r e r n l c n l ~ ~ e . 4 's to ? ~ s n a F r a r thl 7 7 r . > q 7 9 ~ r -

t o

- '3

t h a t "... micro- theory may%e wrong i n a way which has an imporlant b e a r i n g s m - t he problem of aggrega t ion Thus, a " s p e c i f i c a t i o n a r r r o r " i t h e % T c r e

I t four.dations may r e s u l t i n the paradoxical " r e su l t " t h a t predietkons based .a

t he aggregate equat ion a r e , i n gene ra l , more a c c u r a t e than thes based on the I 1/ Green [23, p. 41 . -

i n d i v i d u a l household equat ions- - see, e .g . , Grunf e l d and G r i l i c h e s 125) . However, i t must be acknovledgd t h a t comparison w i t h a n i s - s p e c i a i e d micro-

t h e o r y is somewhat of a "s t raw m n . " Rather , a more u s e f u l app;4ach would be I

t o s t a r t from t h e presumption t h a t t b a r e i s a " c o r r e c t l y s p e c i f i ~ b l . ? " micro- -

t h e o r y which may evolve through a s y s t e m a t i c s t u d of Model Se le t i o n ( s e e u- c h a r a c t e r i z a t i o n s of t h e mlcro-behavior of economic u n i t s . ~ s u b s e c t i o n 1I.B) t o dec ide which main ta ined hypotheses a r e u s e f u l

This is , indeed , t h e p o s i t i o n of Green 1231 i n cons ide a t i o n of t h e 1

p o s i t i v e

p r o s p e c t t h a t t h e individual - mlcro-equat ions " . . .conta in no s p e c i f i c a t i o n

e r r o r ( p . 4 ) ." However, i n su rvey ing t h e l i t e r a t u r e ( c i r c a 1964

c o n s i d e r s t h e fo l lowing c h a r a c t e r i z a t i o n of t h e Aggregat ion Prob 4 em: "Given

g one (1 ) an economic r e l a t i o n s h i p o r s e t of r e l a t i o n s h i p s e x p r e s i n

~ (2 ) an a g g r e g a t i o n procedure , more o r less wel l- def ined ,

v a r i a b l e a s an ( e s p l i c i t ) f u n c t i o n of a set of independent

( 3 ) a r e l a t i o n s h i p , more o r l e s s we l l- def ined , among t h e a g k r e g a t e s :

v a r i a b l e s ,

I What a r e t h e necessa ry and s u f f i c l e n t c o n d i t i o n s f o r c o n s i s t e n t a g g r e g a t i o n as

I d e f i n e d above?" The d i s c u s s i o n t h e n proceeds i n terms of cons id r a t i o n c a b I " t h e problem of grouping v a r i a b l e s i n a s i n g l e e q u a t i o n ," making "...ht ,vy u s e

I of t h e Sono-Leoqief [82, 581 concept of f u n c t i o n a l s e p a r a b i l i t ~ , "

- :;, >; - - - - . - - '+,&T"'J:2 ,>-"uucLA":: " :.--:-:: * -.- < , &2;2r227:2 - '-'-.. . - :,n- I - - . --- -9 6

commer'ing nri t h e s t a t e of a f f a i r s to which we have been l ed Sy pursui.12 t h i s - - - f approach, Bridge [ 7 ] observes t h a t " . . . the d e p r e a i n g c o n c l u s i o n t o be drawn - I

~ ~ r o ~ l s r n s

s p e c i f i c t o a g g r e g a t i n g over i n d i v i d u a l s t o o b t a i n market demand ' , ? c t i o n s ,

I "macro b e h a v i o r a l r e l a t i o n s h i p s ."

from t h i s a r e t h a t , even i f assuming c o n s t a n t r e t u r n s t 3 s c a l e alnd o p t i m a l

a l l o c a t i o n , f i rms have t o be very s i m i l a r f o r even n complicated

p roduc t ion f u n c t i o n t o e x i s t ( p . 352)." The same ~ a y be s a i d of

a g g r e g a t e

a l l o t h e r

As s t a t e d elsewhere (Hart ley [36]) :

research a t the f i rm (and household) l e v e l . ~ ~ ~ r $ ~ a t i o n t o t h e macro-economy i s then simply a mat te r of adding UD .'I

"It is my personal convic t ion t h a t t h e theo re t i c ihns '

Consider, f o r thc. sake of Ciscussicn, t he case i which a s i n g l e

common v~ain ta ined hypothesis is app l i cab le t o a l l f i rms an ( s e p a r a t e l y ) t o

approach t o the Aggregation Problem is a "blind ... unless we a r e i n the unfor tunate pos i t i on of only aggregate t ime- series data on which we must f i t (behavioral) r e l a t i o n s . Fu r the r , t he r e s tr5.cgive

a l l households Ll Let N* denote the s i z e of the popula t io of such u n i t s ' and

N<<N* denote the s i z e of a ( s t r a t i f i e d ) random sample of d

a:.leyl' having t r y t o

* ( Q : i = l , 2 , , . Sicce N is f i n i t e , i n c e r t a n i n s t a n c e s i t may

i

assumptions necessary f o r the p u r i s t t o f e e l comf about f i t t i n g (say) aggregate prodsc t ion funct ion never going t o be met-.-firms simply don't always e f f i c i e n t input combinations and they do d i f f e r . we should admit t h a t aggregate macro-level produc funct ions (e tc . ) make no sense and proceed with

prove t o be of importance whether o r not we "sample with r placement." Now,

l e t 1 denote the r a t i o of the population t o sample s i z e . Swppos w e have obta ined .i ML es t imates of the parameters, 5 (and poss ib ly t l s o s u b s e c t i o n I1.B)

i n the ob jec t ive funct ion, Q(zi ,zi. ,- 9.h, r ega rd le s s e r o r not Qi 1s

latent- - see subsect ion 1 I . C . Now the micro-behavioral r e l a t i o n s f o r t h e

"zepresi . i ~ A L i v e sainple u n ~ c ' are g i ven by

I - - '3 C

* = x + u -1. -i . 9

11 In a more complex s i tua t . ion involving s e v e r a l "meintained - simply t r e a t the "super maintained hypothesis" and "super instead--see subsec t ian 1 I . F

( 2 -70)

hypotheses," w e log- likel ihood"

* where it is unders tood that. t h e {x : i - 1 , 2 , ..., N } a r e s o l u t i c n v l u e s r e l a t i v e -i 4- t o 8. It then f o l l o w s t h a t t h e cor responding "aacro b e h a v i o r a l e l a t i o n s " aye -

simply def ined by -I- i n t h e c a s e where t h e "macro concept': i s t h e d i r e c t sun over i n d v i d m l I u n l t s . I n o t 5 e r i -ns tances , we may be bntereaLed i n t h e average o r mean)

macro concspt -- per u n i t , i~ vhich c a s e we would, i n s t e a d , d e f i n e 1

F i n a l l y , i t may be of i n t e r e s t t o d e f i n e "macro indexes ," which o l l a p s e t h e I r i c h e r d e t a i l p r e s e n t i n t h e e n t i r e ve-cor , I< ) t o at- index num e r (such a s -. . b an a g g r e g a t e consumption i n d e x j . Here, we may have a J- vector @ c o n s t a n t £ w e i g h t s , w, and d e f i n e t h e macro index number, - -

.I I I

w'x = w o w ' C x . = w

I n each of Lhrse i l l u s t ~ n t i v e exampies t h e expected v a l u e af t h e m c r o - c o n c e p t

i s obta'rned (under t h e assumption (1 .6 ) ) by s e t t i n g G t o 0 and e v a l u a t i n g ri -

* I

t h e r e l e v a n t e x p r e s s i o n f o r t h e {z, . } a t 0 = ^ B y t h e ~ . L . E . of 0 ~. - - -4 I n s h o r t , t h e " so lu t ion" t o t h e s o - c z l l e d Aggregation Problem 7

I * I

I r e s i d e s i n t h e use of t h e v e c t o r of s o l u t l o n v a l u e s , {x 1 , instiead of t h e - -i

I * ( u s u a l l y un;cnown) s e t of e x p l i c i t f u n c t i o n s , {z.(zi . , j ) 1. As o* 'k might - -

expec t , , t i : i s s o l u t i o n is u n r e s t r i c t i v e wi th r e s p e c t t o t h c f u n c t i o n a l form of --- I

* t h e x (3- , z ) ' s . Indeed, t h e c o n s t r c c t i o n of macro models, - -. (2.71), (2.72), (2.73), e t c ., simp1 y i n v ~ l v e s t h e i n c l u s i o n

A s I have p r e v i o u s l j noted (Ha r t l ey [36]) :

u s i n g any of

of t h e d e f i n i r f o n

of t he a p p r o p r i a t e "macro concept" a s an a d d i t i o n a l " equation"

model c o n s i s t i n g of equa t ions (1 .4) and (1.5) (Case I) o r

"It may l e g i t i m a t e l y be argued t h a t t h e "econometri approach" t o aggrega t ion of f irm- and household- lev behav io r a l r e l a t i o n s h i p s merely " f ine s se s" t h e Aggregation Problem, i . e . , w e have s imply reduced problem from one of p o s i t i n g t h e e x i s t e n c e of a ?a l e v e l behav iora l r e l a t i o n s h i p ( t o be es t imated i*

b a s i s of macro-lcvel d a t a j t o one of pcs i t inp , tile e x i s t e n c e of a n i c r o - l e v e l aggrega te r e l a t i o n s h i p " rep re sen t a t i ve" household o r f i r m withi i ! a p a r t i c sub- region o r narrowly def ined s e c t o r . In t h e 1 c a s ? , an aggrega t ion problem s t i l l e x i s t s i n d e f i " r ep re sen t a t i ve" unit--no ma t t e r how f i n e l y one "

cake ." I would respond wi th t h r e e arguments : (aj I n so fa r a s t h e t h e o r e t i c a l t r ea tment o

" aggregat ion problem" sugges t s t h a t @he aggregab be " r e l a t j v e l y homogeneous" wi th r e s p e c t t o ( s ay ) p r i o r i s e t of r e l e v a n t c h a r a c t e r i s t i c s , it seems t h a t t h s "more" one narrows t h e c l a s s , t h e "less" v i o l a t e s t h e o r e t i c a l p r o p o s i t i o n s .

(b) If t h e r e is any ho?e i n c o n s t r u c t i n g s t i ~ t i s t i c a l ,aodels of economic and s o c i a l behavio it must be t r u e t h a t t h e r e w i l l e x i s t classes of and s o c i a l u n i t s t h a t a r e r e l a t i v e l y homogeneous hence, aggregable .

( c ) Even g r a n t i n g t h e f : t h a t t h e r e a r e , i n p r i n c i p l e , t h e o r e t i c a l aggrel . . t ion prob::ems f o r bo t macro-'ieCel and micro- level behav ih r a l - r e l a t i o n s , I submit t h a t t h e r e is a world of p r a c t i c a l d i f f e r e n c between es t i rn r t ion r6f a s e t of a ic> .o- leve l r e l a t

' ~~->+.;" wi t i i n ;I c r : . , ~ . : : ~.,,nnq - - then agg rega t i ng , a s opposed t o d i r e c t e s t i m a t i o

3 s i n g l e macro-behavioral r e l a t i o n s h i p -" - * s O r c u t t is perhaps t h e c l o s e s t adheren t t o t h i s m

e v i d a n t l y , t h e Aggregation Problem was s t i l l ( c i r c a 1972)

i n t h e dcm

(1.6) (Case IP),

1/ Preface--A Personal i'-ccount bv Guy Orcu t t i n G c r o a n a l y s i s : - Research, Sys te~ is Modelling, Monte Car lo Simulat ion and

-BehavioraE Analys i s , 2972-

"Around 1955 I became increasingly aware of the large gap between microanalytic research and application of the results of research to policy problems at th? nationci level. In order to achieve estimation and testing power it is necessary to concentrate most research at less aggrcgatlve levels than that represented by national income or other national accounts data. Nevertheless, ~ while substantial gains in testing and estimation power can be achieved by use of data about components which ma be treated as members of populations rather than as national aggregates, the following of a micro analytic research strategy did lead to a serious aggregation problem. There simply was no known way of satisfactoril aggregating relations about microcomponents into

I Y I

macroeconomic relations ." ~ I

His Lntuition, however, was largely correct, once the missing lin of our "key

idea" (see subsection II.D, Part One) is provided:

"During the last two to three decades economists have gradually become aware of a variety of estimation and testing problems associated with the analysis of highly agregative time series data, Foremost among thes are problems associated with simultaneous equations,

, muiticollinearity and autocorrelated error terms, Still present, but somewhat forgotten, are problems associated I with errors of observation, Major attempts have been 1 made to fir,d ways of dealing with these problems, and 1

progress has been made in finding appropriate ways of ~ using available time series data. Nevertheless, the fact remains that the presefice of these problems seriously reduces the information available in aggregate time series data for testing and estimation. It is as though the available data has been generated by very poorly

4 designed experiments. Sophisticated methoils of analysis have been developed to take this into consideration, but no method of analysis is a substitute for evidence that h79 not hsen g n n e r ? t c d o r t h a t has \ e e ~ ??$Y. 3--.

-- -- - - - - --- -

I aggregation. Moreover, evei in the absence of statistical problems associiited with use of national accounts data, the ratio of-p~tential observations to parameters to be estimated & woefully small . oreo over,

I as will now be pointed .,ut,;there are a wrrtbzr of rather1 I general considerations lvhich favor placing a primary I

emphasis on the rnicroa-mlytic research strategy." ~ I

Orcutt, (71, Ch. 1, p. ;] I

I I

Orcutt then proceeds to list the following issues favoribg a micro- I

1. Formulation of Hypotheses:

The s o c i a l s c i e n t i s t does have a p e c u l i a r advantagel In producing

t e s t a b l e hypotheses about the behavior of i n d i v i d u a l s , f a m i l i s and f i rms no t

a v a i l a b l e a t o t h e r l e v e l s of aggrega t ion . 1 2. Spec i f i ca t ion of Causa l i ty :

We need models of s o c i a l systems which w i l l enable o l i c y makers t o

p red ic t consequences of a l t e r n a t i v e a c t i o n s , and which say mo

Successful induct ion from p a s t experience about eve t h e most li

c e r t a i n t h ings have been a s soc i a t ed i n h i s t o r i c a l per iods o r ,

of con t ro l e f f o r t s , w i l l be a s soc i a t ed i n the f u t u r e .

immediate impact of pol icy in t e rven t ions is d i f f i c u l t enough. 1 It seems most

i n t h e absence

r , zpe t i t i ons of it under varying circumstances and, p re fe rab ly , a t c l o s e l y -

a t t a i n a b l e i n cases where, t o c l e a r l y i d e n t i f y t h e a c t i o n , we

spaced t i m e i n t e r v a l s . These condi t ions a r e f r e q u e n t l y m e t i re sea rch on

dicro-components. They a r e almost .;ever m e t a t h ighly aggreg t i v e l e v e l s .

can observe many

reasonable t o e x p l o i t t e s t i n g and es t imat ion p o t e n t i a l s a t t h a

3. Estimation and Testing:

Since much of the hoped-for payoffs is a t t h e macro l e v e l , it is

g r e a t e s t op t imi s t s . ~conomic t h e o r e t i c a l foundat ions a r e ';not .uf f i c i e n t l y " 1:

However, anyone who recommends primary r e l i a n c e on e s t ima t ion ,

so'elv an the jlse of na+lor .a l accounL-- 1at.a. is s ~ : r e i . i o n p nE --

* develop$ t o permit confident s p e c i f i c a t i o n of v a r i a b l e s 8. be inc luded ,

* m = I

and t e s t i n g

t b ~ ;10713' -

func t iona l f o rq , a p r o p r i a t e l a g s and s t o c h a s t i c p r o p e r t i e s of k r r o r s . Highly

aggrega t ive time s e r i e s do not permit enough degrees of f reedo t o permit t extens ive t e s t i n g and e s t ima t ion . This is c r i t i c a l i f w e a r e o a t tempt t o f n a i l down the " correct" economic s t r u c t u r e i n t he context of m k l t i -

c o l l i n e a r i t y , a u t o c o r r e l a t i o n , e r r o r s i n v a r i a b l e s , r ap id feed acks , e t c . One 1

should not r e l y s o l e l y on macro-accounts da t a i f a l t e rna t ives- - cros s- sec t ion

micro da t a c r , b e t t e r y e t , micro panel data- - exist .

While t h e r e is l i t t l e a v a i l a b l e empi r i ca l evidence f om a t t e m p t s c o r cons t ruc t micro-based models of a n e n t i r e macro economy, t h e r e i s a ~- cons iderab le 1 ' - terature on a p p l i c a t i o n s of t h i s approach t o p r t i c u l a r

I s e c t o r s . To c i t e two examples, no t e t h e burgeoning micro- leve l i t e r a t u r e i n 1 +.he t r a n s p o r t a t i o n ( e .g ., Domencich and McFadden 1131 and ~ e s t l n and G f l l e n

[89] ) and housing market ( s e e , e .g ., b i n and Ingram [49]) f i e 1 d s . A major

empir ica l outcome of these c o n t r i b u t i o n s is t h a t r e l i a n c e on m c ro - l eve l d a t a k bases has permi t ted d e t a i l e d ana lyses of po l i cy i s s u e s such a s c h o i c e s of

t r anspo r t modes, time-of-day t r a v e l dec i s ions , shopping d e s t i n I t i o n , f requency

of t r a v e l , e t c . , i n t h e former ca se ; and t h e impact of moving o s t s and l and f p r i c e s on 1.ocation dec is ions of household and f i r m demands a n d hous ing

s u p p l i e r s , a t t r i b u t e- c h a r a c t e r i s t i c s of housing s t o c k and socib-demographic

c h a r a c t e r i s t i c s of consumers and t h e i r i s p a c t on housing p r i c e , etc., i n t h e S l a t e r . These s t u d i e s do, however, r e l y upon market- clear ing a sumptions t o

determine p r i c e s - a f u r t h e r d i f f e r e n c e from the p re sen t propos

C

It is obvious t h a t v a s t amounts of information a r e 1

aggrega t ion . The r e a l i s s u e , t he re fo re , is whether t h i s l o s s

has serioris consequences with r e spec t t o e s t ima t ion and t c s t i n i n s f t u a t t o n s - I t y p i c a l l y encountered i n r e sea rch . In s h o r t , tlie choice is bniueen: 'f -

( a ) Aggregatian pr io? t o e t i n t i o n ii I

versus I ~ ( b ) Estiu,?.tion p r i o r t o aggrega t ion .

~ I

S t a t i t i t i c a l arguments (e.g., (Goldberger [ l a ] ) t h e f a c t t h a t t4e v a r i a n c e of

2 2 t h e sample mean, q, i s equal t o .J /N when sampling from norma popu la t i ons X

'7 w i t h mean,

p*, and var iance , a") and not r e l i g i o u s f e rvo r whou d d e c i d e t h e

X !

issue in principle. Cost-benefit arguments should be made regarding the

trade-off between accuracy and costs of data collection, proqessing, and model

constraction (using the most advanced statistical methods an sophisticated d economic/social theories) to decide the issue in practice. ~ Orcu~t, himself, has pio~eered in demonstrating, via Monte Carlo

studies of relatively complex socio-economic structures with ,known paraneters, ~

incurred through aggregation carried all the way up to the ndtional level can,

indeed, be severe. Such experiments suggest that, if the "c rrect 4 specification" of a micro-baaed macro model is known, a stro g case can be

that the loss of information needed for effective estimation

made fur the virtues cf the microanalytic approach. This do s, of course, beg 1 I

the question of appropriate procedures to "search" for the "dorrect"

and testing power

specification (see subsection II.B), but, at the micro-level, one would appear I I

to have stronger guidance from economic theory as to the norCtive aspects of I

behavioral specifications. In addition, following approach (~b) does not

restrict the choice of functional forms for the micro-relatidns, in that the

model does not require the estimation of the implied macro-c~unterparts. They I

are sirupiy "derived" from the process of direct aggregation. ~ I

I P

The approach being advocated here is, in a nutsheld the 1' -

construct:ion of macro-econometric models by direct a E e atidn over individaal -g__,_ -- econcaic units--households and firms--while incorporating go ernment policy -

I P instruments explicitly in the micr~ economic-behavioral rela ionshi s. Thus, - - Ir c

the "circular flow" diagram one finds in a first c,urse A in"dcro-econosirs-- ~

exhibiting the interdependence between firms, households and government, I

through the flow of goods and services--can be explicitly cadtured at the I

national, regional and individual-unit levels. The historicail antecedents of

this proposal are Walras' 1871 theory of general equilibrium, Orcutt's [71,

72, 731 contributions towards making Walras an operational realiIty, and

Kornai's [52] skepticism regarding the plausibility of the market clearing

assumption.

It is not, to my knowledge, an approach which has bee4 given serious

attention to date:--11i part because its implementation would re d uire a sizeable team effort of subject-matter specialists and economet iciars; in t part because the statistical and econometric tools appropriate or i

Powell and Williams 1591 or Lau [55]) and disaggregated sectorat work on the

implementation are only now being developed in the context of eipirical

thecry of the firm; in part because only recent advances in corn uter I

research on the "new household economics ,'I demand systems (see,

technology--particularly storage capacity--?emit the joint estimation and

e .g ., LJ.uc;i,

investment in time and money to demonstrate feasibility via a p lot case k simulation of such models; and in part because it would require

study. I am, however, convinced that it is nQw feasible and ha e already v a sizeable

attempted to provide various types of arguments in its support.

While the construction of a complete micro/macro mode of a national 1 economy may still seen to be a visionary prospect to those endowed with a

.I stages of analyds (see Part Three, which follows). 4s such, rogress toward

. r

healthy degree of skepticism, I submit that each a project can

ron~ar~rnental.1- tntr) a scauence of manseab le , but increasingly ----- --- ----

- - -

the reality of & I w -

be logically

more com~lex --

I approach which I have sketched. First, each of the items list d above may be t

-

viewed as either a supply or demand-side household or firm dec sion. For i example, labor supply arises from the direct aggregation acros household I labor force participation decisions, while the demand for clas es of goods and

T

services- - heal th, educat ion, housing, c lo th ing , food, etc.--can be cons t ruc t ed

by d i r e c t aggregat ion across the ind iv idua l household demands Sirlce w e would 1 be dea l ing with a l a r g e sample from a presumably known popula i o n - a census a t

l e a s t a t tempts t o be exhaust ive and methods f o r e s t ima t ing t h

households - a r e a v a i l a b l e , t h i s permits the aggrega t ion of the " representa t ive 1'

household's de5and and supply func t ions wi th in each sub-region (or o t h e r

s u i t a b l e taxonomy) t o reg iona l and na t iona l l e v e l s . Closure o each such I market depends upon t h e a v a i l a b i l i t y of supplementary d a t a . I f i r m l e v e l I da ta , such a s a census of i ndus t ry , a r e a v a i l a b l e , a complemen ary ap?roach I

(M .M .G .E .) model by d i r e c t aggregat ion- - firs t t o t he reg ional l e v e l and then

may be appl ied t o e s t a b l i s h c l o s u r e . A l t e r n a t i v e l y , one may

aggrega t ive da ta from regional and/or na t iona l accounts and

p r i c e da ta t o c l o s e the model v i a macro-type behaviora l r e l a t i o n s .

t o the na t iona l l e v e l . Such markets may o r may not be c l ea red and t.his may

empi r i ca l ly be t e s t e d v i a use of e i t t . ~ . r t he Cowles Commission

u t i l i z e

co t respondins

This

Equations approach or by the more recent econometric methods p r m i t t i n g 4

would, i n p r i n c i p l e , permit cons t ruc t ion of a Micro-Macro G e n e ~ a l Equil ibr ium I

markets t o be i n disequilibrium--with e i t h e r aggrega te regiona / n a t i o n a l i . I i em and .or supply a s a binding c o n s t r a i n t a t p r e v a i l i n g price^.^

A t the micro (household) l e v e l the dependent va r i ab lds i n such a - - simulta~neous equat ions model a r e e i t l l e r ' bn t inuous o r d i s c r e t e (e .g ., t h e

permit es t imat ion of models e x h i b i t i n g these f e a t u r e s . ~ e r t a i d of t he p a r t i a l

equi l ibr ium problems can be immediately addressed, wh i l s t the +re I

comprehensive j o i n t es t imat ion problems would r e q u i r e development of s p e c i f i c

econometric methods tailor-made t o the problem a t hand. I am convinced t h a t

there are methods which can resolve the econometric issuos--see Part Three of

this paper. The issue is a matter of computational cost and no feasibility. t extending the "error structure" of the model to include "error components"--

Another option available with a micro-based model is

i .e ., error terms specific to each level of disaggregation, one may control

tkat, by

Eor the effect of unmeasured or unobserved variables that are:

(a) common to all individuals within a given household, I (b) common to all households or firms within s given subr gion/region or .i

industrial sector, I (c) common to dl1 regions or industries within a given co ntry, i (d) common to var2ous "types" of countries,

residual variation. This feature is not shared by the macro t me-series

models. I as well as statistically estimate their separate contributions

A third possibility is that since national censuses re typically li

to the total

conducted at benchmark intervals--usually on a decennial basis -the macro time

series information (as well as interim population surveys) can be used to t "statisrically interpolate" between the benchmark years--see, .g., a recent I research proposal by Hartley [30]. b sddition, if the census s are designed C . #

so that at least a subset of individual hcuseholds may be uniq ely identiffed G ~n t;uccess~ve ce~iscls years, Lr le K ~ S U L L L I L ~ i o n g l t u d i n a i (pane; j d a c a cam oe 1 employed to examine family and firm dynamic relations over tim --

I

incorporating, e .g . ,?"lif e-cycle features" of sequential decis l o on makdg at

1 / the micro level (ala Heckman [ 4 2 ] ) .-

1/ The next subsection will pursue in greater detail. the imp1 cations of the - i proposed approach to the problem of specificaticn of "lags behavior1'--see Nerlove [68] . in economic

Fourth, the household nodules should account f o r he household n o t

only a s a consumer of goods and supp l i e r of l abo r s e rv i ces , b u t a l s o as a

producer--see, e.g., Barnum and Squire [2 ] f o r t he example f a g r i c u l t u r a l

households. In my view, t h e p o t e n t i a l b e n e f i t s from the "n w household

economics" (see, e .g., Deaton and Muellbauer [ l l ] , Becker [ ] and Lehrer and I I

Nerlove ( 5 6 , 571) i n conjunct ion with the wave of recent developments i n

micro-econometric methods, have not been adequately e x p l o i t

Maintenance c o s t s between census yea r s (every t e n would only

requi re upJa t ing the reg ional and/or n a t i o n a l accounts d a t a - unless , af 1 course, w e a r e i n the f o r t u r ~ t e pos i t i on of having micro-ty$e househoId and

f i rm da ta (such as the U.S. Current Populat ion Survey) i n i t e rven ing years- i In t h a t case one would have t o weight t he marginal c o s t s an b e n e f i t s of 4 u t i l i z i n g t h e CPS type d a t a . Otherwise, r o u t i n e updati.?g o the macro d a t a 4 base is of s e v e r a l o rders of magnitude less than i n i t i a l co -- cons t ruc t ion .

While i t should be apparent t h a t t h e approach b e i g proposed i s a 1 "modem va r i an t" i n the evo lu t ion of Orcut t ' s micro- analyti approach, i n 4 I

which macro models a r e cons t ruc ted from t h e "bottom up", t hd re a r e s e v e r a l

d i f f e r ences between what I am proposing and what Orcut t a t t pted t o do: b

8 I

i d e n t i f i c a t i o n of i nd iv idua l un i t s- i n success ive survey/census I

(a ) Orcut t was p r i n c i p a l l y i n t e r e s t e d i n obta in ing hou ehold and f i r m I

yea r s . While I, too would be de l igh ted wi th such d a t a , my p r o p o s a l

survey/census data i n which ind iv idua l household ar.d

could be constructed- - this o b v i o u d y r equ i r e s the

r e s t s on the idea of aggregat ing up t o rogional lna i o n a l l e v e l s in t

f i rm time-series

~ n i q u e

o rde r t o br ing t ime- series f e a t u r e s i n t o t h e model ' whereby r e g i o n a l '~

and/or na t iona l macro accounts would be pooled w i t micro- cross- "i

sections. This is, in my view, a major difference, and emits one P to still make time-series projections, while building in a

advantages of micro structure. l1 the

(b) F r ~ m what I can gather, Orcutt's simulation models conta ned separate I

classes on a common structural form. The advantage of wrking with '?

modules for individual households and firms. I am proposing

of the "representatiq:e household and firm" within a

region/region, industry, or other suitable taxonomy--statistically

estimated by pooling data on separate individual units

the "representative unit" is that direct-aggregation to I

the use

particular sub-

w>thin such

(c) Orcutt appears to view the world from a Walrasian in

regional/regional-national or sector analogues of

relations is immediately feasible.

which all markets are cleared. l3us, all prices and

micro-'3ehavioral

observed would be viewed by him as equilibrium market po itions.

Present work (notably by Richard Quandt and his collabor tors at I Princeton) would permit testing such any hypothesis vers~us various

specifications of -- markets in disequilibrium. Methods f r estimation

of the .latter in the one and two market case are now we 1 in hand 1 (see, e.g ., Fair and Jaffe [16], Maddala and Nelson [ 6 1 ] , Hartley

- ... iji,, ~uf&&?Td and h d n d c i 2 j ; j and ~ i r e i r d S p p t 5 ; i ; p r

' i - well established (see Hartley and Mallela [38] and Amem C

) - [l] ) . Generalization to the "many-markets case" with "

now appears feasible (Ito 1451). 1 (d) Orcutt's efforts suffered from a constant problem of in uf ficient S

funding, a lack of suitable statistical methods at the lime his

attempt was made and significant constraints from preva ling computer

technology. In a sense one may say that Orcutt was bor too soon! n

Suppose it may be granted ( f o r purposes of p resen t d i s cus s ion ) t h a t

it is f e a s i b l e t o cons t ruc t macro models by d i r e c t agg rega t i n over micro I decision-making u n i t s i n the manner I am sugges t ing . I h a v e a l r e a d y a t tempted

t o argue t h a t t h i s approach avoids t h e problem of d a t a a v a i l b i l i t y I c o n s t r a i n t s , can be made t o add-up ( i n t he sense of i n t e r n a l cons i s t ency a t I sources of d a t a t o generate c o n s i s t e n t ( i n t h e s t a t i s t i c a l s nse) parameter e t h e reg iona l , n a t i o n a l and even g loba l l e v e l s ) , and u t i l i z e s

es t imates i n a sys temat ic f a sh ion . What o t h e r v i r t u e s can b claimed? The e a l l r e l i a b l e

s e v e r a l methods f o r de r iv ing the d i s t r i b u t i o n a l consequences

most obvious advantage is t h a t , s i n c e t he u n i t of I

I ( i n the p o ~ u l a t i o n census) o r t h e f i rm ( i f a I

these is a v a i l a b l e ) , a by-product of t h e

I dec i s ions , e t c . One method begins by f i t t i n g s t a t i s t i c a l d i t r i b u t i o n s t o t h e

i s sues may be d i r e c t l y addressed wi th in t h e model framework.

exogenous v a r i a b l e s con t ro l l ed f o r w i th in t he model, but exc uding those which I may be proper ly considered p o l i c y ins t ruments . Then Monte c a r l o techniques

There a r e

may be employed t o generate cond i t i ona l d i s t r i b u t i o n s (and h nce expec t a t i ons ) e of the endogenous va r i ab l e s i n t h e model. Non- stochastic silnulatEun of t he

4 est imated model w i l l y i e l d the more korhent iona l cond i t i ona l p r e ~ i c t . i o n s . A

I t h i r d a l t e r n a t i v e is the r ecen t innovat ive , bu t unpublished pproach of R.

* - Baszann, e t a l . , i n whi* the j o i n t d i s t r i b u t i o n of z11 endo enous v a r i a b l e s

L b i s e s t i z a t e d d i r e c t l y , B t h e r than j u s t modelling t h e expectLd va lues , a s i n * the convent ional "regres8ion1' approach. ~

I

Inherent i n t he t e r n , Micro-Macro General Equilibrkum <HHGE) Models,

is a non-conventional use of t he no t ion of "equilibrium." B/ equ i l i b r ium, I

lo not mean t o imply t h a t a l l markets a r e c leared-- i .e ., t h a t a l l pr2ce-

quan t i t y p a i r s observed i n the " r e a l world" r ep re sen t market

I c l e a r i n g s t a t e s ,

~ a s was assumed i n t he seminal work of Haavelmo 1261 and subsequent

developments i n t h e Simultaneous Equations Es t imat ion l i t e r a t re pioneered by Y t he Cowles Commission group. Rather , an equ i l i b r ium is simp1 used .i> r e f e r Y

properly s p e c i f i e d i n a d i s e q u i l i b r i u q form, then t h e r e is a divergence between the two d e f i n i t i o n s , and my preference is t o r e f e r t "equilibrium" a s

def ined above.

A major v i r t u e of the.World Bank's Computable Gene a 1 E q u i l i b r i u 3 T

t o t h e J o i n t s o l u t i o n of t h e endogenous v a r i a b l e s w i t h i n t h e

view t h a t markeca may o r may not be c l ea red and t h a t t h i s is,

(CGE) g loba l mod51,

pode l . It is my

presumably, a

I presen t ly being u t i l i z e d f c r the World De/velopment k , o r t

t e s t a b l e hypothes i s . I f it t u r n s ou t t h a t , a s a resalt of such tests, a l l

markets a r e s p e c i f i e d a s a Walrasian market- clear ing system t en t h e two I d e f i n i t i o n s a r e i d e n t i c a l . I f , on t h e o t h e r hand, c e r t a i n r k e t s a r e mor

problem i s s e r i o u s s i n c e , s h o r t of t r i a l- and- error search ov r t h e parameter 4

pro j ec t ions and count ry economic work, i s t h a t it i c consis tent- -

i .e . , every th ing "adds up" i n a c o n s i s t e n t f a sh ion

!dhile t h i s is a d i s t i n c t advantage over models

space va l ida ted by the " reasonablenes" of t he generated s o l u @

m o d ~ ~ l s u f f e r s from a uniform p r o t o t y p i c a l parameter va lues .

sys temat ic way t o " f i x up" these types of models i f they t u r ou t t o y i e l d

"I:-vn ~ ~ , ~ ~ ~ . h ' , = ' ' rpc: . - r - ? , -- IS - - ; r c . i ~ p ' . k t fh j? - ~ ; ~ b ? a r ? .ih LC>. ~ 7 - n : l e s

The l a t t e r

' i I US a t t he p re sen t t i m e . me matter of i n t e r n a l consistency ban, of course , - - a n o be d l r e c t l y imposed i n t h e case of each of the)conometric modelling

I I

approaches-- either the pure tin.3 s e r i e s , pure c ross- sec t ion r pooled t i n e - t s e r i e s l c r o s s - s e c t i o n approaches. Theref o r e , t h i s is not a sh f f i c i e n t

c r i t e r i o n which w i l l a i d us i n the choice between the a ove c o m p e t i e o r s , It

simp'y r u l e s out c e r t a i n models with an incomplete spec f i c a t i o n of CAOSPIH~

i d e n t i t i e s .

IS t he re a p o t e n t l a l f o r an a l t e r n a t i v e t o macro-time serges or CGE

globa l models? It would appesr t h a t , i f t he c o n s t r u c t i h of i n d i v i d n a l

country micro/macro models can be demonstrated as f e a s i l e i n t h e e e x r c of a + p i l o t ca se s tudy, t he re is, i n my view, no reason--shor of computer 1 technology c o n s t r a i r t s and t h e l a c k of a sys temat ic c o l l e c t i o n d f o a for

LDCs p r e s e n t l y without s u i t a b l e da t a t o permit tho f econometr ic me-ds-

- tha t such a v i s ion may not be r e a l i z e d by the n c f c o n n e c t h g erade-

flow equat ions between micro/macro n a t i o n a l models. I

problem t h a t admits t o quick s o l u t i o n . I have a t tempte t o p rov ide cbe best

arguments I can think of t o j u s t i f y the d i r e c t i o n t h a t uch r e s e a r c h s h d d I \

t ake . I f these arguqents a l l? persuas ive , I subrnit t h a t i t is t i m e ts fa-

t h a t re? i t y square ly . We should not be de t e r r ed by t h e l eng th of the path,

With t h i s v i s i o n i n mind, i?- is Lai r game t o r g u e t h a t , ewen if ooe "I

j . nor by the s i z e of the undertaking. Rather ve should a t tempt t o ass -b le as I

were t o commence today on a research program s i m i l a r t o

attempted t o ske tch , i t may take s e v e r a l yea r s before a -- .

econometric model is an ope ra t i ona l r e a l i t y . I concede

l a r g e a group of the "best cf the b r i g h t e s t" t h a t a v a i l a b l e r e s o u r c e s wiBE I ..

the one I ha-

micro-macro maticma3

t h a t i t is IMEC a

p e r m i t and plunge boldly ahead. Let. ,us make t l l ~ "new

economics" a macro-economic r e a l i t y .

I n e n t i v e m i c r m i c &

G. Lags i n Econ9mic Szhavior and t h e Samuelson Correspondence - Pr t n c i p i e ~

I By f a r t h e most i l l u m i n a t i n g paper on t h e s u b j e c t df "Lags i n ~

Economic Eiehavini' is Nerlove's [68] Second Henry Schul tz ~enjorial Lecture t o \ I

t h e Econametric SocieLy. Indeed, i t wocld be f a i r t o say thalt Nerlove I

"...asks a l l t h e r i g h t q u e s t i o n s , bu t f h i l s t o p r s v i d e us w i t h any aaswers." I

I n t h i s s e c t i o n w e s h a l l rpview t h e s t a t e of tk.e a r t ala ~ e r l ~ o v e and o f f e r I

some s i ~ , ~ l e s o l u ~ i o n s t o t h i s c l a s s of problems. I Narlove 1683 beg ins w i th a r o s t a p p r o r r i a t e quotati ion from Schu l t z

"Research i s no t good s imply becaose i t is mathematical o r s t a t i c v i c n l , o r because it makes u s e of ingeniousl lgeh i r ies . Research is good i f i t is s i g n i f < c a n t , i f i t is f r u i t f u l , Lf it is c o n s i s t e n t w i t h e s t a b l i s h e d p r i n c i p l e s , o r i f it h a l p s t o over throw erroneous ~ p r inc ip l - e s ." ~

I

The main purpose of Nerlove's paper i s r o make t h e p o i n t t h a t I

work Qn dynarriz ~cor lomic models i s witl lout f i r m t h e o r e t i c a l f bunda t i ons : I

"lhe e x t e n t and v a r i e t y of t o p i c s t o which d i s t r i b u t l a g a ~ a l y s i s nas been e p p l i e d i n e m p i r i c a l economics as tounding , bu t , gha t 1s more remarkable , is the l a c k of t h e o r e t i c a l j t ~ s t i f i c a t i o n f o r t h e la,:

. t superimposed on b a s i c a l l y s t a t i c models" The extra-1 r o r d i n a r y ncb lpc t c f t h e dynamic underp inn ing of t h e ~ models a c t u a l l y C I t t e d t o the d a t a Ss w h a t ~ r i l i c h e s [24, p . 42 1 has ca L Led tbe ' t hca re t i ca l ad-hoc! -rv' .:I I

I

Indeed, t h i s s i t u a t i o n i s even more remarkable i n :ha: S a t e l s o n -

I I I

[78] has examined i n d e t a i l t h e c h e o r e t i c a l v ide of t h i s i y u e i n e n u n c i a t i n g I - I

hi-;, now famous, Correspondence P r i n c i p l e : I I

L 1 I

There e x i s t s an intimate iqrrnal dependence between comparat ive s t h t i c s a t d dynamics. To my knowledge t h i s had n o t p r ev ious ly been e x p l i c i t l y e -unc la ted i n +he: economic l i t e r a t u r e , and, f o r l a c k of a b e t t e r n a n t , I s h a l l r e c e r t o i t as t h e Correspondence P t€nc ip l s_ . . .I- Ai. unders tand ing of ? h i s p r i n c i p l e 19 all. t h e more ir;yc/t.:.t;t ~t a t ime when pure ecocom!~ t h ~ o r : ~ nas andergon;. a ~ r e v o l u t i o n ok thought--from s t ~ t i c a l t o d:lnilmi~a? I

I

modes... The r e s u l t i n g change :n out look can be qonipared t o t h a t of the t r a n s i t i o n from c l a s s i c a l t o quant mechanics. And j a s t a s i n t he f i e l d of physics , w e l l t h a t the r e l a t i o n s h i p between t h e o ld and theo r i e s could be i n p a r t c l a r i f i e d , so i n our s i m i l a r i n v e s t i g a t i o n seems i n order."

The most p ~ z z l i n g aspec t of t h e e n t i r e i s s u e is, i n my ques t ion of

why it took econometricians ovar twenty-flve years- -with N e love 's ~ e m i n a l

paper on "Tagsw--to t ake up t h e Samuelson chal lenge!

Nerlove employs "...the p a t h b r e a k h g paper, 'Capi a1 Theory and

Investment Behavior' ( o f j Jorgensoo," . . . who " . . .developed what he c a l l e d t h e

neoc la s s i ca l theary of investment behavior , by which he aeah t a theory b u i l t

upon a r o d e l of or t imal a c c u m ~ ~ l a t i o n over time.. ." as a veh ic l e t o make h: r

con t r ibu t ion cons i s t i ng of:

po in t . H e cha rac t e r i ze s Jorgenson's 1471 investmerit theory

( i ) a theory of demand f o r t he s e r v i c e s of c a p i t a l bas d upon maximization of the present balue G£ the stream of n e t revenues accruing t o t he f i rm over t ime, I

a s a t r i p a r t i t e

( i i ) a theory of the replacement demand f o r c a p i t n l , ( j i l ) a theory of the way i n which two types of demand

c a p i t a l a r e t r a n s l a t e d i n t o a c t u a l investment 1 expenditures . I

Indeed, a l l t h r e e aspec ts a r e contained wi th in the dynamic p t i m i z a t i o n d problem which we have a l r eady sketched i n subsec t ion ~ I . A . , nd w i l l pursue h e x p l i c i t l y below.

-

A s Smmarized by Nerlovc, .= i " 3 e ~ g u s e r?o adjustment c o s t s o r a n t i c i p a t e d o r l a g g i n the de l ive ry of c a p i t a l goods a r e incorpora ted ~ o r g h s o n ' s o r i g i n a l folrmulation of t he net-worth i n t e g r a l , the condi t ions f o r maximization t u r n ou t e s s e n t i a l l y those of the purely s t a t i c case , a l b e i t formulat ion incorpora tes a t e r n f o r t h e i m p l i c i t value of c a p i t a l depending on cu r ren t i n t e r e s t r a t e s , p o l l c i e s and dep rec i a t ion . ll-11~8, two advantages of neoc la s s i ca l formulat ion, a s presented , a r e : ( i ) thc: e x p l i c i t dependence of the demand f o r cap i : a l

the underlying technology, a s expressed i n t h e production func t ion , r e l a t i n g ou tpu t t o capital. l abo r , and

u n c e r t a i n i n 1

t o be h i s

r e - l t a l t a x

h i s

an

and

f a c t due t o t he choice of a C~tb-Douglas product ion f u n c t i o n l a r a t h e r

( i i ) a d e r i v a t i o n of t h e app rop r i a t e "price" v a r i a b ~ l e i n t he demand f o r c a p i t a l and i ts e x p l i c i t dependknce upon i n t e r e s t r a t e s ."

r e s t r i c t i v e assumption implyieg 2 ..- u n i t a r y e l a s t i c i t y of eubsb i tu t i on ) .

Indeed, ou tput i= t r e a t e d - exog=~;=u* i n t h e long-run denand

Despite t h i s , very l i t t l e emphasis "... has been d i r e c t e d a g i n s t t he rather i.

f o r c a p i t a l (a

l

ad hoc fc rmula t ion employed, o r towards t he p l a u s i b i l i t y and r e l e v a n c e of the I

empir ica l l a g d i s t r i b u t i o n s obtained." 1 I

me b a s i c pcrpose of Nerlove's c r i t i q u e is t o suggest t h a t asfusion

models, expectat ions- formation ( i nc lud ing so- cal led l l ra t ionab expec+at%onsS*> l

adjustment over t i m e and space and should be incorpora ted ex l i c i t l y Bnto the I I

and t h e genera l c a t e g o r i z a t i o n of "adjustment costsn--search1 c o s t s ,

I ob jec t ive func t ion . I n s h o r t , I ~

t r ansac t ions c o s t s , e tc . -a l l c o n s t i t u t e t h e o r e t i c a l explana.'lions

I

"I hope t o br ing ou t t h e p o t e n t i a l f e r t i l i t y of a ~ marriage between expectat ion- formation and adjustme+t- c o s t approaches, and t o show t h a t excess ive preoccu a t i o n

, with t h e s t a t i c concept of long-run equi l ib r ium has P r e t a rded the development of econometr ical ly re levan dynamic micro-economic theory. This is not t o say comparative s t a t i c s ( a l a Samuelson) has no place i n economic theory , but merely t h a t it is not a useful^

r approach when one is expl ic i t l -1 concerned 'wfth t h e I dz-namics? of behavior .I1 I - I

f o r dyrmn3.e

I I

UQ to thfs n o f n t , N e r l o v ~ is "rlgtrt o q the ??ark." ~ ' f

"By Shul tz 's c r i t e r i a , curre* research on l a g s i n economic behavior is not "go*," because n e i t h e r is t h e t h e o r e t i c a l r e sea rch verT- s t w n g l y empi r i ca l l y o r i e n t e d . But w e a r e s t r i v i e towards t h a t dimly perceived i t may be a t times."

I

I However, i t is un fo r tuna t e t h t Nerlove's l a s t two paragraphs 1 were w r i t t e n as

I I The g r e a t e s t nroblems seem t o m e t o hr :hese: f i r 4 t ,

t h e s e r i c u a i s s u e s involved In segregating micro- I

r e l a t i o n s h i p s i n t o r e l a t i ~ n s h i p s d e s c r i b i n g t h e behdv iw I

of economic aggrega tes i n both a temporal and c ro s s- s e c t i o n a l sense ; and second, t h e d i f f i c u l t y i n fo rmula t ing models which l ead t o a

whi le I _a-s o p t i m i s t i c t h a t s t e ady progress w i l l i n t h e o r i g i n a l model. These

I am not a t a l l s u r e it w i l l dynamic behavior is s t i l l

a long t ime before w e have

hope I am wrong ." Measurement of Demand on o t h e r broad ca t ego ry of

I

I W e s h a l l b r i e l y s k e t c h an approach t o t h i s gene rd l class of problems,

I

l eav ing more s p e c i f i c i s s u e s f o r P a r t Three.

Let x denote a J- vector of observed d c i s i o n v a h i a b x e s *( i> ,n( i>Y

and z denote a K-vector of observed s t a t e v a r i d b l e s a s s o c i a t e d wit5 - t ( i > , a ( i > , - I

I

i nd iv idua l i i n year t = 1 , 2 , . . . ,T and taxonomic c l a s s 1 2 , . L. k t I

Q t ( i ) y ~ ( i ) denote the optimand ( p r o f i t , u t i l i t y , e t c . ) r e l b t i v e t o yesr L aad

I taxonomic class II f o r I nd iv idua l i--where Q may be e i t h e r bserved ar P l a t e n t . Let L denote t h e f i n i t e piannino, ho r i zon , assume common to s%l

L - -

i n d i v i d u a l s ass igned t o taxonomic c l a s s L, 1 C L C 11

d i scount r a t e i n year t . Then, i n any year t, i n d i v i d u a l gen t s a r e presmed I t o maximize t h e "present value" of t h e "expected" s t ream o f u t u r e p m f X t s .

4

(Case I ) o r u t i l . t t y (Casc. I I ) , i.e., maximize f

I - - L " 1 '3

v ~ .= = c t i , i 1 - q2,a41

t ( i ) , f - ( i ) K=o - I - # 1 *

1 - I

where I I I I I

I I

with r e spec t t o the t r a j e c t o r y of dec i s ion v a r i a b l e s , I I

{X :Il(i)=R and r=0,1, ..., t ( i ) + T , 2 ( i ) ,

L21 , -

1 / f o r a given t r a j e c t o r y of s t a t e var iables ,-

{%(i)+T, 2 ( i ) , : 2(i)=R and T=O, 1, . . . , Lgj s

v a r i a b l e s , z is ? vector paramet r ic func t ion , - t ( i ) f T , 2 ( i ) , * ' --

and under the assumption t h a t expec ta t ions of "future" s t a t e

r e l a ~ i v e t o any year t , a r e based on a parametr ic - func t ion of

* * = Z

%(i)+'t, a ( i ) , -T, 2 ( i ) , 0%) , g ( i ) , - s = t ( i ) - l , g ( i ) , * *

..., Z s 1 --t(i)-G, a ( i ) , -' - a ~

Thus, t he "T-step ahead predictor ' ' i n yea r , t ( i ) , of the vec tor

taxonomic c l a s s j of the " in fo rna t ion s e t , " c o n s i s t i n g of cu r r e t s t a t e 4--

v a r i a b l e v e c t o r s ,

t h e form

(2.78)

of s t a t e

v a r i a b l e s , and lagged va lues , i s a , - %(i)-& 2( i ) ,

g=1,2, . . . .G, where G denotes t he maxinun l a g 21 and 6 denotes i the vec tor of -2- -------+-- --

parameters a s soc i a t ed with the r- step ahead

1/ The nota t idn i s not meant t o preclude the p o s s i b i l i t y t b a t - - c e r t a i n 3 ( i ) , %L), . ' s may be contained as elements

vec to r s , z , ( ~ ) , R(i), . --as, e.g., t he e f f e c t of p r i o r f e r t l i z e r i app l i ca t ions on the cu r r en t y i e l d of rubbe r , I

2/ I: .go, t he ~ h y s i c a l d i s t r i b u t e d- l a e e f f e c t of f e r t i l i z e r on bber y i e l d - has a modal e f f e c t a t 3 years and a " lag length" of up t o f ve yea r s . t

I

Examples of t he types of expec t a t i on format ion models which might be

employed a r e given by Nerlove, Grether and Carvalho [69, cqs . V and X I ] and I

inc lude var ious types of Autoregress ive I n t e g r a t e d Moving qverage (ARIMA)

models, pure o r r a t i o n a l l i s t r i b u t e d- l a g models, adap t ive qxpec t a t i ons and so- i

c a l l e d " quas i- ra t iona l expec ta t ions" formula t ions f o r the i e c t o r f u n c t i o n ,

* z . In most of t he se formula t ions , t h e { B , ~ : r - O , l , ; . . . , Le) are -.r,,(i) , *

- func t ions of a s i n g l e parameter vec to r , say - i3,which is invak i an t w i th T.

Thus, f o r any year of obse rva t i on , t ( i ) , we employ a c t u a l z/ v a l u e s T', a(%), - when t' < t ( i ) and p r e d i c t e d - (o r expected) v a l u e s a l a (2.783 when

t* > t ( i ) . \

Depending upon t h e particular f u n c t i o n a l form ado$ted f o r Q i n \ 11

(2.75)-- implying a product ion func t i on 15 t h e Q,

a r e p o f i t s (Case I) o r a I

u t i l i t y func t i on i f they a r e l a t e n t utilities--maximizationlof V i -- t ( i ) , l ( i ) =

i n d i v i d u a l , i, i n any sgmple yea r , t ( i) , imp l i e s t h a t an opdimal t r a j e c t o r y of

t he ' ' fu ture dec i s ion v a r i a b l e s , I I

I I

may be c a l c u l a t e d r e l a t i v e t o any parameter con f igua t ion

: T-0, 1, . . . , L,) , f o r any c lass , &=I , .= v . 4 -ate +--.a,zLc {!a.>BP.r -- -- -- --- -.

pwgramming algorithm--see, e.g., Bellman [ 4 ] and Bellman and Dreyfus [ 5 1 . i

* I *- 72~1s t he c u r r e n t pe r io? . ' ~ s o l u t i o n va lues , x

* ( i ) , Ni l , * ' r e p l a c e t h e i n * I

t h e a lgor i thm .of oubsect lon 1 I . C . and i t e r a t i o n w i l l , i n genkra l , produce F W I

..timates of a l l parameters , {iL and {BkT: r = O , l , . . .,Le) . This c i rcumvents

the need--see Carvalho [8] o r Nerlove, Grether and Carvalho f69 , ch. XIV]--to

* o b t a i n closed-form a n n i y ~ i c s o l u t i o n s f o r t h e x wliich w i l l , i n

t i , 1 , genera l , be func t i ons of t h e {z

- t ( i ) , E ( i ) , 1, 3 and {JET 1. ~ i d e e d , except i n

i

I

,

highly specialized cases--Carvalho [8 ] assumes a quadratic profit function--

such analytic solutions will be impossible- without the compuiter implementat ion

of a suitable dynamic programming algorithm. In short, unde the proposed

approach FIML estimates of all structural parameters--8 wit in the production -II

or utility function and the {BET} within the model of expect tions formation-- I. I

will obtain provided that a dynamic programing algorithm is available to I

I from possible constraints relating to.the present availabiliby of dynamic

I .

* solve for the - x 's given i3%and {lllT). The availability of - such algorithms

To summariie, o: : pro~sed approach appears to apply equally well to I

depends, at this juncture, on the form of the objective funcbion Q. Apart

programming algorithms, however, the functional form of Q is

to "well-behaved" choices .

dynamic optimization models of the standard economic variety -see, e.g., t

only restricted

Henderson and Quandt [ 4 3 , ch. 01 2 Neoclassical dynamic mohels of the .Jorgenson/Nerlove type may therefore be directly estimated.

required is to incorporate the appropriate dynamic progammi

algorithm within a FIML estimation package to iterataively g

"optimal" currant decision values for given parameters. Agg egation to macro-

level equations theQ obtains by the methods proposed in sub ection I1.F.

One point regarding the type of data required at t e micro-level in n OE'ZZ i o i~plzmrnt Lho pruposed approacn is w o r t h rnenrion. f l e a r l y , t o t h e

' ? L

rurrrent and lagged values of data at the level of the micro-funit. This - I

extent that lagged values of the x - -t(i), II(j.), and +(f), z l<i), 1 are required in

11 In fact, I have just learned at an examnle 32 use of this - approach--see Wolpin [92], applied to model of

't fertility and child moibtality . e.

the ~pecification of * %(I), ~(i)

of (2.751, this implies that we must have

informat ion may e i t h e r be ob ta ined r e t r o s p e c t i v e l y A l o r f r o m :urrent and

h i s t o r i c a l records , o r f-om . - - a sequence of c ross- sec t ions on common i n d i v i d u a l I

u n i t s ( pane l d a t a ) ..%I Thus, t h e r e is a n i m p l i c i t "data cc)stv j n p o s t u l a t i n g a I

dyr.amic op t imiza t i on model as a maintained hypothes i s . I

I

11 See, e .g ., the Egyptian Li te racy Reten t ion Study of t h e !world Bank (RPO - 671-55)--Hartley and Swanson [39] , o r Har t ley , ~ e n c i v e n d a and P o i r i e r [371 * I

I 21 See Har t ley [351 i n t h e con t ex t of pane l micro d a t a a t qhe " f i e ld- leve l" -

over a sample of S r i Lankan rubber e s t a t e s from 1970 t o 1979 .

I

References : -

[iJ Amemiya, T. and G. Sen: "The Consis tency of t h e Maximum ke l ihood Es t imator i n a Disequi l ib r ium Model," Technica l Report o . 238, I n s t i t u t e f o r Mathematical S tud ies i n t h e Soc i a l Scienc s, S tanford Un ive r s i t y , 1977.

1 121 Barnum, H. N . and L.Squire: A Model of an A g r i c u l t u r a l ~ o $ s e h o l d :

Theory and Evidence, World Bank S t a f f Occas iona l papers! No. 27, Balt imore: Johns Hopkins P r e s s , 1979. I

[ 3 ] b c k e r , G .: "A Theory of t h e A l loca t i on of Tima,"Economi: 1 J o u r n a l (September 19651, 493-517. I

I I I

[ 4 ] Bellman, 8.: Dynamic Programming, Pr ince ton : P r i nce ton ~ n i i e r i s t ~ P r e s s , 1957. 1 1

I

[5 ] Bellman, R. and S .Dreyfus : Applied Dynamic P r o g r a m i n g , ~ d i n c e t o n : Pr inc ton Univers i ty P r e s s , 1962. I I

I I

[6 ] Box, G .E .Pa and G.M. J enk in s : Time S e r i e s Ana ly s i s , ~ o r e c a s l t i n ~ and '

Control., San Franc i sco : Holden-Day, 1970. I

I [ 7 ] Bridge, J.L.: &p l i ed Econometrics, Amsterdam: North-Holla d , 1971.

[8 ] Carvalho, J .: Product ion, Investment and E x p e c t a t i ~ n s : A S t dy of t h e United S t a t e s C a t t l e I ndus t ry , PhD. D i s s e r t a t i o n , Dept. f Economics, Univers i ty of Chicago, 1972. 1

I I

191 Cooley, T. and E. P r e s c o t t : "Varying Parameter Regression: A Theory and Some Appl ica t ions ," Annals of Economic and Soc i a l Measur ment, (Oct .

t 19731, 463-473. h

I

[ l o ] Davidol:, V. C.: "Variance Algorithm f o r Minimizat ion," Jou rna l , 10 (19681, 406-410.

I

I [12] Dernpater, A . P., N . M. l a i r d and D.R.Rubin: "Maximum Likel ihood from. Incomplete Data v i a t h e EM Algorithm," Journa l of _the ~ o i a l S t a t i s t i c a l Soc i e ty ,Se r i e s B, (1977), 1-22 and D i s c u s s i o ~ , 223-38.

8 I

[ l l ] Deaton,Angus and John Muellbauer, Economics and Consumer Cambridge: Cambridge ~ n i v e r s i t ~ T r e s s , 1980.

I [ I 31 Domencich, T. A. and D. McFadden: Urban Trave l ema and, ~maderdam: ~ o r t h - ~ o l l a n d , 1975. ~

Behavior,

I I

[ I41 Dubin, J . A . and D - McFadden: "An Econometric Ana lys i s of q e s i d e n t i a l E l e c t r i c Appliance Holdings and Consumption," 1980 World \Congress of t h e Econometric Soc ie ty , Aix-en-Provence- I

I [ I51 Duesenberry, J . S., G . Fromrn, L. R. Klein and E. Kuh: The B lookin=

Qua r t e r l y Econometric Model of t h e United S t a t e s , N+nd McNally, 1965. I

I 1

1161 F a i r , R. C . and D. M. J a f e e : "Methods of Es t imat ion f o r Markets i n Disequi l ibr i r lm," Econometrica, 42 (1974) , 177-190.

[17] F l e t c h e r , R. and M.J.D. Powell: "A Rapidly Convergent Descent Method f o r Mnimiza t ion ," Computer J o u r n a l , 6(1963), 163-1

Goldberl;er,A. S.: Econometric Theory, New York: Wiley 1965.

Amsterdam: North t lo l land, 1972.

1 Goldfe ld , S. M. and R.E .Quandt : Nonlinear Methods i n Econometr ics ,

Goldfeld ,S. M. an3 R. E. Quandt: "Recent Problems and Advances Xn Ect imat ing Disequi l ib r ium Models ,I' paper g iven a t Wastern Economic Assoc i a t i on Meetings, Las Vegas, June, 1979.

[21] Golub, G. " l e a s t Squares , S ingula r Values and Matr ix I

Applied Mathematics, 13(1968), 44-51.

[23] Green, H. A. J.: Aggregation i n Economic Analys i s , ~ri'pcnceton: P r i n c e t o n Un ive r s i t y Press , 1964. I I

[22] Granger, C . and M. Hatanaka: S p e c t r a l Ana lys i s of P r i nce ton : Fr ince ton Univers i ty P r e s s , 1964.

(241 G r i l i c h e s , Zvi : "Di s t r i bu t ed Lags : A Slltvey," 1967) 16-49.

Ecoqomic Time Series,

(251 Grunfeld , Y. and Z. G r i l i c h e s ; "Is Aggregat ion Necessa i l y Bad?," 1 Review of Economics and S t a t i s t i c s , 42(Feb. 1960) , 1 r 1 3 .

(261 Haavelruo, T.: "The S t a t i s t i c a l I m p l i c a t i ~ n s of a S y s t e of S imul taneous Equat ions ," Econornetrica 11(Jan. , 19431, 1-12. 1 I

(281 Har t l ey , H. 0.: "Maxi- Likelihood Es t ima t i on From Inbomplete Data," Biometr ics , 14(1958) , 174-194.

I [29] Ha r t l ey , H . 0.: "'In Dr. Baves' consult in^ Qaon," D c

1271 Haberman, S. J., Analysis of Q u a l i t a t i v e Data, Vol.. I, Topics , ?yew York: Academic P re s s , 1378.

i [30] H a r t l e ~ , ~ H . 0 . : " In t e r census I n t e r p o l a t i o n Using Pub l i Use Tapes and CPS,lvoa r e sea r ch proposa l t o t h e Bureau of t h e Censu

b 1

I n t r o d u c t o r y

I

(311 Ha r t l r y ; -Michael J.: "The Tobi t and P r o b i t Models: Maximum Like l ihood Es t imat ion by Ordinary l e a s t Squares," Discussion Pa e r N o . 374, Dept. of Economics, S.U.N.Y. a t Buf fa lo , March, 1976 t

;32] Ha~Liey , Michael .I.: "On t h e C a l c u l a t i o n of t h e Maximu Estir , lator f o r a Model of Markets i n Disequi l ib r ium," No. 409, Economics Department, S .U .N .Y. a t Buf fa lo ,

Hartley, Michael J.: "On the Est imation of a General Switch ng Regression Model v i a Maximum Likelihood Methods," Depdrtm n t of

1977.

4 Economics, Discussion Paper No. 415, S .U.N.Y. a t Buffalo, October !

Hart ley, Michael J.: Comment on Quandt, R. E. and J. Ramsey: H x t u r e s of Normal D i s t r ibu t ions and Switching Regressions

Hartley, Michael J .: "The Construct ion of Econometric Models Supply of Perennia l Crops: A Case Study of Natural Rubber S r i Lanka," Research Proposal , Development Economics Depar World Bank, January 1980.

"Estimating ( I n v i t e d

Paper)" , Journa l of t he American S t a t i s t i c a l &soc ia t ion , 730-737 and Discussion, 738-752.

Hartley, Michael J.: "Micro-Macro General Equil ibr ium Yroposal f o r Research on the Construct ion of Models," Development Econcnics Department,

73(1978),

Hartley, Michael J., V. Bencivenga and D. P o i r i e r : "A S t a t i s Methodology f o r t he Egyptian U t e r a c y / ~ u m e r a c y Retention S Development Economics Department, World Bank, August 1979.

Xartley, Michael J. and P. Mallela: "The Asymptotic P rope r t i s of a Maximum Ukel ihood Estimator f o r a Model of Markecs i n Disequilibrium," Econometrica, 45(1977), 1205-1220.

4 Hart ley, Michael J. and E. V.Swanson: "Maximum Likelihood E s

t he Truncated and Censored Normal Regression Models," Deve Economics Department, World Bank, November 1980.

[40] '~eckman, James J.: "Shadow F r i c e s , Market Wages and Labor Su ply," Econometrica 42(1974), 679-94. P

[41] Heckman, James, "Heteroge;leity, and S t a t e Dependence i n of Labor Supply," unpublished manuscript, Universi,ty , 19783

(421 Heckman, James: " S t a t i s t i c a l Models f o r Discre te Panel Data,'* ~ k ! - , . ~ ~ ? { c . ~ . 7~ p-1111 3qC) , r+-. . .- - ,- - - - . - ,i :-. ' -

- L L - - -,.--dy L > Sciences, J u l y , l W 7 ( f i r s t d r a f t ) . .$

[44] Hoch, I rving: "Jlmultaneous Equst.ion Bias i n t h e Context of t Douglas Production Function," Econometrica, 26(0ct . 1958),

- [43] 'Aenderson, James and Richard E. Quandt: ~ c r o e c o n o m i c Theory,

s McGraw H i l l , 1958. I )

I

I451 I t o , T.: "Methods of Est imation f o r Multi-&rket Disequi l ibr ium Models ,I1 RIAS Program Working Paper No. 9R, Harvard Univers:.ty, December 1977.

New York:

[ 6 1 ] S d d a l a , G . S . and F.D.Pelson: "Maximum Like l ihood Methods kor Plodels of b r k e t s i n D i s e q u i l i b r i u m , " Econometr ica , 42 ( 1974) , 1bl3-1030. ~

[62) ?larschak, J . and W ..T .Andrews : "Random S i m u l ~ ~ ~ n e u u s ~ ~ u a t i o ; l k and t h e Theory of P r o d u c t i o n ," Economet r i ca , 1 2 ( J u l y - Cct . 1944) 1 143-205.

I

[63] Yay, K.O.: "The Aggregat ion Problem f o r a One I n d u s t r y Model;" Econometric&, 14(1946) , 187-202.

[64] May, K .O .: " Techno log ica l Change and Aggrega t iun , " Economet:rica, 15 (1S47) , 51-63.

[65] Xundlak, Yair : " E s t i m a t i o n of P r o d u c t i o n and l3et:avioral ~ u n l t i o n s f r o n a combinat ion of Cross- Sect ion and Time- Series Dhta , I 1 i l l

C .F . , e t . a 1 .,Measurement i n Economics, S tzn f o r d : S t a n f o r P r e s s , 1963.

[ 6 6 ] Xundlak, Ya i r and I. Hoch: "Consequences of i n E s t i m a t i o n of Cobb-Douglas P r o d u c t i o n 3 3 ( 0 c t . 1 9 6 5 ) , 814-828.

I

[ 6 7 ] Ner love , Marc: E s t i m a t i o n and I d e n t i f i c a t i o n of , F u n c t i o n s , Amsrerdam: North-Holland, 1965.

[68] Ner love , Harc: "Lags i n Economic Behavior , I1 Econornetr ica, 4 221-252.

[69] Ner love , M., D . G r e t h e r and J . Carvalho: S e r i e s , New York: Academic P r e s s ,

[ 7 0 ] Oberhofer , W . a11d J . Kmenta: "A Genera l P rocedure f o r Likel ihood E s t i m a t e s i n G e n e r a l i z e d Regress:sn

~ 42(3) (1974) , 579-590.

( 7 1 ] O r c u t t , Guy, e t . a 1 .: N i c r o a n a l y s i s : Bekav io ra l Research, &stem

.. j Orc l l t t , Guy, A . G i a z c r , K. Harris;and R. Wertheimer: "Micr

'

Modeling and t h e A n a l y s i s of ~ d 1 i . c T r a n s f e r P o l i c i e s , " "Microeconomic S i m u l a t i o n Mode* f o r P u b l i c P o l i c y R. H . Haveman and K . H o l l e n b e c b ( e d s .), New York. 1980. m

Model l ing , Monte C a r l o S i m u l a t i o n and A n a l y s i s , - Urban 1972.

L 3 I

[ 7 2 ] O r c u t t , Guy, S . Caldwel l and R. Wertheirnsr : P o l i c y E x p l o r a t i o ~ ~ M i c r o a n a l y t i c S i m u l a t i o n , Urban I n s t i t u t e , :.976.

[ 7 4 ] Pagano, Marce l lo and Xichae l J . H a r t l e y : "On F i t t i n g ~ i s t r i b u t e d k g Models S u b j e c t t o Polynomial R e s t r i c t i o n s , " J o u r n a l of Economet r i c s , 1 6 ( J u n e , 1 9 8 1 ) , 171-198.

I n s t i t u t e ,

t h r o u g h

[ 7 6 ] P o i r i e r , kl .e: The E c o n o n e t r i c s of S t r u c t u r a l Change, Amsterdam: North Ho l l and , 1976.

[ 7 7 i Rao, C. R. and S. K . M i t r a : -- G e n e r ~ l i z e d I n v e r s e of I.t t r i c e q and i t s A p p l i c a t i o n s . New York: Wiley , 1971 .

I I

I

[ 7 8 ] Samuelson, P a u l A.: F o u n d a t i o n s of Economic A n a l y s i s , ~ a m b d i d ~ e : Harvard U n i v e r s i t y P r e s s , 1947 . I

[ 79 ] Samuelson, P a u l A.: P r i n c i p l e s of Econo~n ics , New York: ~ c ~ d a w H i l l , 1963. I

I I

[ 8 0 ] Schulez , Eenry: The Theory and Measurercent of Demand, ~ h i c a l g o : Chicago U n i v e r s i t y P r e s s , 1938.

I ~ 1811 Sims, C .: "Macroecor.ornics and R e a l i t y ," ~~~~~~~~~~~a, 48(1C(73) .

I [ 8 2 ] Sono, M.: "The Z f f e c t of P r i c e Changes on t h e Demand and Supply of

S e p a r a b l e Goods , I 1 I n t e r n a t i o n a l Econoaic Review, 2(1962) , 239-271 . ~ [83 ] Swamy, P .;.V .B.: " E f f i c i e n t I n f e r e n c e i n a Random ~ o e f f i c i e l n t

Regress ion Model , I1 - Econometr ica , (Psy , 1 9 7 0 ) , 311-323. ~ I

[ 8 4 ] n~e i l , Henr i : Economic r o r e c a s t s and P o l i c y , 2nd e d ., Amsteirdan: North Hol land, 1961. ~

[ 8 5 ] l b b i n , James : " E s t i m a t i o n of R e l a t i o n s h i p s f o r Limi ted ~ e ~ e b d e n , : V a r i a b l e s , " Economet r i ca , 26 ( Jan . l i 5 8 ) , 24-36. 1

[P6] Wald, Abraham: "A Note on t h e C o n s i s t e q c y of t!le Maximum L i E s t i m a t e ," Annals of ?lathematic? ' S t a t i s t i c s , 20(1943). 5

[ 8 8 ] W a l t e r s , A . I..: " Produc t ion and Cost F u n c t i o n s : An ~ c o n o o e k r i c Survex," g c o n o m e t r i c a , 31 (Jan .-Apr. 1 9 6 3 ) , 1-66. r I I

I I

[ 8 7 ] Walras , U o n : S lemen t s a Econornie P o l i t i q u e P u r e , L ~ u s a n n e :

[89 ] Wes t i n , R icha rd and D. V. G i l l e n : "Pa rk ing L o c a t i o n a r d ran sit Demand: A Case Study of E n d o ~ e n o u s A t t t f b t ~ t o s in Disago,reo,;lt.e Mn?k C ~ C , : ~ P

a , A ~ A d , , adL U L I ~ d t i ~ ~ ? C l i . i A C b , o ( L J / ~ ) , / >-lUl. I I I

[ g o ] ' . ? s tpha l , I a r r y and Y. W. Rhee, "A Micro, Economet r i c I n v e s t p a t t o n o f 1 Cnoice of Technology , I 1 Journal, f ( S e p t . , 1977) , 205-237. -

I t

I

L. Corbas &

I

[ 9 1 j Wiener, Norbe r t : C y b e r n 2 t i c s , Cambridge: M . 1 .T. P r e s s , 1948~. --

Co., 1874. I

1921 Wolpin, K . I.: "AT. E s t i m a b l e r3ynamic S t o c h a s t i c Model of Fe t i l i t y an-1 Chi ld M o r t a l i t y , " Yale U n i v e r s i t y , O c t o b e r , 1981 . f

I [!I31 Z e l l n e r , A . , J. Krnenta and J . Drsze: " S p e c i f i c a t i o n and ~ s t k m a t l o n o f

C ~ b b - h a g l a s P roduc t i o n P l ~ ~ i c t i o n odei is ," Econornet r ica , 3 ( O c t o b e r , - 1 3 6 6 > , 784-795. 1

I

ENC