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International Statistical Institute (ISI)is collaborating with JSTOR to digitize, preserve and extend access to Revue del'Institut International de Statistique / Review of the International Statistical Institute.

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Some Remarks about Relations between Stochastic Variables: A Discussion DocumentAuthor(s): R. C. Geary

Source: Revue de l'Institut International de Statistique / Review of the International StatisticalInstitute, Vol. 31, No. 2 (1963), pp. 163-181Published by: International Statistical Institute (ISI)Stable URL: http://www.jstor.org/stable/1401371Accessed: 01-06-2015 14:21 UTC

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REVIEW OF THE INTERNATIONAL

STATISTICAL

INSTITUTE

Volume

31:

2,

1963

163

SOME

REMARKS

ABOUT

RELATIONS

BETWEEN

STOCHASTIC

VARIABLES:

A

DISCUSSION

DOCUMENT*

by

R.

C.

Geary

The EconomicResearch

nstitute,

ublin

It

is

the

contention

f

the writer

hat

he fundamental

roblem

f

the

meaning

f

stochastic

elationship,

n

the economiccontext nd

in

general,

emains nsettled.

It

is true

hat

much

of

the work

n

thisfield s

excellent,

ut

real

progress

as been

confinedomathematics and theessence f mathematicss thecertaintyfconclu-

sions

from

tated

ypotheses.

n the

problem

f

tochastic

elationship

t

s

theformul-

ation

ofthe

hypotheses

hat

s the rouble. t is

not

surprising

hat uthors the

writer

is one

-

tend

to

return

o the

topic

at intervals f

years

to shake ts

uneasy

bones.

We

may,

r

may

not,

politely

mention ne another

n

our ists f references

ut there

is little vidence

n

our individual

writings

hat

we have

deeply

tudied

he others'

thinking;

nd

the

present aper

s

no

exception

o

this

sorry

ule.

More like

poets

than

cientists,

ach

of

us seems

o want o work his

ne

out for

himself;

he

truggle

is

in one's own soul.

The mathematicsnwhatfollows revery imple, eliberatelyo, to highlighthe

hypotheses

in

particular, ssumptions

s

to

the stochastic

haracteristics

f

the

residual error.

Also

deliberately,

he writer's

xpression

f

views

will be

forthright,

to

inspire

r to

provoke

ebate.

t

was

an Irish

tatesman

f

other

ays

who said

that

he

exaggerated

n

speech

o attainmoderation

n

ends.

Perhaps

t

s

high

ime

workers

in

thisfield

et together.

I.

WHAT IS REGRESSION?

In

the

writer's

pinion

regression

s

essentially

cause-effect

elationship,

he

n-

dependent ariablesbeingthecauses and thedependent ariablethe effect.With

Y=

-o

+

X

+

u

you

are

saying

hat

given

henumerical

alues ofa and

3,

Y is found

by

substituting

a

given

alue

for

X,

calculating

a +

3

X and

adding

random

ariable

.

Simple

regression

heory

s concerned

with

he estimation f

a

and

p

from series

of

pairs

of observations

X,

Y)

and

discussing

heconfidence

imits

f

these

stimates,

s

well

as

estimating

he

variance

f the random

lement

-

the

atter

eing

of

major

m-

portance orestimatinghe confidenceimits ftheestimate ftheaveragevalueof

Y,

given

X. Viewed

s

cause-effect,

t

becomes

lear

why

here re

in

general

wo

re-

gression

ines n

the

wo-variablease:

for ne line

X

is,

by hypothesis,

he

cause

of

Y,

forthe other

Y is the cause

of X and there

s no reason

why

hese

hould

coincide,

evenwhenthe

number

f

pairs

of observations

the

data)

is infinite.

Still

confining

neself

o

the two-variable

ase,

the basic

problem

onfronting

he

statistician

s,

given

scatter

iagram

n

X,

Y),

to

find

he

aw,

f

any,governing

he

relationships,

avingregard

o

probability,

r

stochastic,

heory.

We have

already

*

Paper presentedt theJoint uropeanConferenceftheEconometricociety,he nstitutef

Management

ciences

nd

the nstitutef

Mathematical

tatistics,

ublin

3-7

September

962.

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164

mentioned wo such

relationships,

he two

regressiontraight

ines: there ould of

course

be curvilinear

elationships,egressional

n character

i.e.

cause-effect)

o

which

random

ampling heory

an be made

apply.

We have tests

or

determining

hether

there

s

any relationshipnd,

f

there

s

a

relationship,

f

what

kind an

it

plausibly

be

regarded.

Attention ill be confined o the inear ase.

From

the arliest

tatistical

imes, owever,

tatisticiansave

recognised

hat here

were

conceivably elationships,

ther

han

regressional,

etween andomvariables.

They

aid

more

or

ess)

et

us

abandonthe

notion

f

ny

pecial

ole

e.g.

a

particular

variable

egarded

s a

cause or an

effect)

or ach

variable:

be

quite

neutral

s to the

role of the

variable,

reat hem ll as

equals,

and see what

happens.

Call

the

resulting

relationship

functional",

associative", neutral",

r

what

you

will.

shall,

n what

follows,

se the term ssociative.

What s

the aw

governing

he

oint

movement

f

pairs

of observations?

he

question

posed

in

this

way

ndicates hatthe

associative

viewpoint redominatesn the field f experimentalciencewhere o oftenwe can

believe

n

the xistence

f a

law,

f

only

we could

find

t,

our

difficulty

eing

ue

solely

to

errors

f

the

ordinary

ind

n our

observations.

An

early

favourite

s

an

associative aw

was

the

ine

or plane)

of closest

fit,

.e.

the

straight

ine whichminimises he

sum

squares

of

distances

rom

he

point

obser-

vations. he

trouble

ere

s

that,

n

general,

he

procedure

annot e

ustified

tochast-

ically, hough

t s

perfectly

ensible

n

practical rounds.

stochastic

heory

as been

developed

on

the

following

ines

[1],

[2]).

In

the

simplest

ase

of

two

variables

et

the

model

be

Yt

-

xt

(1.1) Xt = xt

+

u,

t=

1, 2,

.

. . , T,

Yt

=

Yt

+

vt

where he

Xt, Y,)

are

the

observations

ubject

o errors

f

observation

(ut,

v,)

about

which

nothing

lse

is

assumed

xcept

hat

hey

re

independent

f one another

nd

of

(xt,

yt)

the "true"measures nd

that ll

theirmoments xist.

The

problem

s

to

estimate

he coefficient

fromT

sets

of

observations.

ll

variables

re assumed

measured rom heir

means.

The

problem

s formulated

annot

be

solved

using

he

variances f

Xt

and

Yt

and/orthecovariance

(Xt,

Y,) since whenT is indefinitely

large)

theuse of these

moments

upply nly

hree

quations

o

obtain

four

nknowns

(,

x?

(E =

expectation),u2,

Ev2.

Instead,

ecoursemust e had to

higher

moments,

or

the

mathematicallyquivalent

wo-dimensional

umulants

L

for

X,

Y),

X for

(x,

y))

defined

y

the

dentity

n

(s,

t):

-

(1.2)

E

exp (sX

+

t

Y)

-

exp

{I

.E

L(i,j)

s'

t1/i

}.

1,

J

But,

from he

ndependence

ssumptions

bout the rror ariables

,

v and

using

1.1)

(1.3) E exp (sX + tY) E exp sx + ty). E expsu. E exptv

whence he

fundamental

elation

(1.4)

L(i,j)

=

(i,j)

whenboth

and are non-zero

ositive ntegers.

ut,

from

1.1),

Eexp(sx

+

ty)

=

E

exp

x

(s

+

tP3)

=

exp

C

k

S

+

t

)k/k

=

exp

Z

X

(i,j)xiyJ/i j


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165

where

Xk

s thek

thcumulant f

x.

Equating

oefficientsf

s ti

kP5j

X

=

(i,j)

Hence

Xk+1 ~j+1=

X(i,j + 1)

kk

+

1,U).

t follows

hat

(1.5)

X

i,j

+

1)-

p X(i

1,j)

=

0

or,

using

1.4)

when

,

;>

1

(1.6)

L

(i,j

+

1)-

L(i

+

1,j)

=

0

.

This

theory

an

readily

e

extended o

any

number

variableswhen

he

model s

k

:

PkXk

0

(1.7)

=1

i=

1,2,...,k

Xk=

Xk

Uk

omitting

the

cursive

subscript

t

(t

=

1, 2,...,

T).

The

equations

for

finding

the

coefficients

,i

re then

k

(1.8)

f

Pi

(c ,

c2

...,

Ci

+

Ci+

1,

...

Ck)

=

0

where he

ntegers

i

>

1.

There

are,

n

general,

n

infinity

f

relations

1.8)

which

(when

T is

indefinitely

arge)

constitutehe

necessary

nd sufficientonditions or he

acceptability

f

the

model

1.7).

It

can

easily

be shown

hat,

when

number f

setsT

of

observationss

finite,

onsistent

stimates

f

the

L

functions an

be

found

from

(1.2) (and

analogously

n

the

general

ase ofk

variables)

y

substituting

he

operation

1T

T t=1

for

E.

There is

an

asymptotic

andom

sampling heory

vailable for

the

theory

outlined

bove

[2].

It suffers

rom he

disadvantage

hat

t s

computationally

ifficult

using deskmachine xceptwhen henumber fvariabless two orthree, rperhaps

in

the

"Reiersol

ase" of

nstrumental

ariables see

vii)

below.

Also,

since

with

his

theory

we must

have

in general)

ecourse

o

cumulants

f

powergreater

han

two,

the error

ariances end to

become

arge.

That

is

why

one musthave more

than

a

sneaking

egard

or

empirical

evices ike the

straight

ine

or plane)

of

closest

fit,

which nvolves

nly

he

variances nd

covariances.

Following

re

some

remarks

n

associative

elationship:

(i)

The

theory

s not

applicable

when he observations

X1, X2,...

Xk)

are

ointly

normally

istributed,

or

hen ll

the cumulants

f

more

than

one discussion nd of

powergreaterhan2 are zeroso that heequation ystem1.8) reduces o thetrivial

0 - 0.

(ii)

The

theory

as been

very

ittle

pplied.

The writer imself sed it to estimate

formally

he

coefficient

nvolved

n

the rishman

Boyle's 17th

Century)

aw

using

Boyle's

original

5

pairs

of

observations. or constant

emperature

he

Law is

Log

P

+

P

og

V

=

Constant

The

estimate of

is

P

=

L(3,1)

/

(2,2)=

1.00404


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166

which

scarcely

equires

significance

estto establish

nsignificant

ifference

rom

unity.

n a

lecture

n

Paris,

the

writer emarked:

"Remarquons

ncidemment,

ue

la loi de

Boyle

'appelle

oi

de

Mariotte n

France,

avec la meme ogique qui faitque la loi normale, ecouverte ansdes conditions

diff6rentes

ar

de Moivre et

Laplace,

s'appelle

quelquefois

oi

de Gauss.

Sans

doute,

ous es

pays

regoivent

ventuellement

ustice

n

moyenne".

(iii)

There

s a

non-linear

wo-variable

heory

lso available

though

ere

nuisance

parameters

ntervene,

hich

under

certain dditional

hypotheses

an

be

estimated

from

he

data.

(iv)

Linear

associative

heory

an

be

regarded

s a

generalization

f

regression

theory

hrowing

ome

ight

n the

atter.

n

the

usual

notation

he

model s

k

Y=

YE

i + u,

i=1

all

variablesmeasured

rom

means.The

standard

quations

or

stimating

he

Pi

by

bi

are

1

bi

bi bk

SYXi

=

X,

Xi

+

...

+

-

X2

+

...

+

XkXi,

i

=

1,2,..,

k.

T T

T'

T

Now,

from he

viewpoint

f

arlier

heory,

here re

k

+

1

variables

ndthe

ovariances

involved

re

equal

to

the

corresponding

umulants

o

that,

or

ssociative

heory

he

covariance oefficientsreestimable ince

E

YXi

=

Eyxi

E

Xi

X

=

Exixj,

i

6

j.

But

E

X2

=

Ex

+

Eu

in

which here

ntervene

he

nuisance

arametersu?.

The

regression

quations

here-

forebecome

associative

nly

when

Eu2

=

0,

i.e.

ui =

0,

i

=

1, 2,

..

,

k.

Hence

by

a

circuitous

oute

we come

to the

basic

assumption

f

regression

heory,

amely

hat

ityields ssociative aluesofthecoefficientsnlywhen he ndependentariables re

observedwithout

rror,

he

single

error

variable

n

the

model

pertaining

o

the

dependent

ariable

Y.

(v)

The

R. A.

Fisher

tochastic

model

envisages

he

regression

ata as

a

sample,

or

realisation,

rom

universe n

which he

ndependent

ariables

re the same

for

all

samples.

J. Berkson

3]

has,

however,

solated

linear

wo-variablease

in which

regression heory

ields

he

correct

ssociative stimate

hough

both

variables re

subject

o

error.

n

the

Berkson

ase when

we think

ur measure f

the

ndependent

variable

s

X

it s

really

where

x = X

+

u,

u

being

herandom rror

ssumed

uncorrelated ith

X. The contrast

ith

ssociative

theory

will

be noted: here

the

observation

X=x+u

and u is

uncorrelated

ith

x

though

t

is,

of

course,

n

general

orrelatedwith

X,

compared

with

a

in

the

Fisher

ase. In

the

Berkson

ase the

regression

f Y

on

X

yields

consistent

stimate f

the

coefficient.he

Fisher

ignificance

heory pplies,


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167

though

he

price

paid

for he

mprecision

n measurement

fthe

ndependent

ariable

is

that

he error

ariance

V

is now

V

o2

2G2

where

a2v

and

ac

are respectivelyheerror ariances f Y (residual) nd X. A non-

linear

ignificance

heory

or

he Berkson

ase is available

4]

but there

he nuisance

parameter

a

intervenes

hich

can, however,

e estimated

rom

he observations.

(vi)

When

he

number

f

sets

of

observations

s

indefinitelyarge

n the

inear sso-

ciative

ase of

twovariables

t

s

easy

o show

hat

he

ssociative ine

must ie between

the

two

regression

ines. This

need

not

necessarily

e the case

whenthe number

f

pairs

of observations

s limited.

n fact

n the

general

ssociative

ase,

as remarked

earlier,

he

random

ampling

rror

ariance

f

P

tends

o be so

large

s

to

give

very

aberrant

esults.

(vii) Whenonehas availablemany conomic ime eries ll inter-relatednd when

one's

model

of several

behaviouristic

quations

only

few

of

these

variables

ppear

in each

equation,

for the

consistent stimation

f the

coefficients

ne can

use the

instrumental

ariable

method,

ue

essentially

o 0.

Reiersol

[5],

[6]),

the

nstruments,

in

regard

o

any

equation,

being

he

variables

which

do not

appear

n

the

equation.

This

constitutes

particular

ase

of

1.8)

above,

he

quation

ystem

or

he

stimation

of

the oefficients

i

now

containing nly

ovariances

ince

he oefficients

f he erms

in E

Xi2,

hrough

hich

ntervenehe

biassing

rror

ariances,

re assumed

o be

zero.

Suppose

the

equation

contains

only

two

variables

X and

Y

(measured

from heir

means) ndthemodel s

X=x+u

Y=

y

+Uv

Y=y+

v

y=

px,

u

and

v

being

uncorrelated

ith

one another

r with

x

and

y.

Suppose

we

have

an

additional

ariable

Z,

correlated

ith

X and Y but uncorrelated

ith

u and

v.

Then

EXZ

=

ExZ

E YZ = Eyz = PExZ

so that

P=EYX/EXZ

of

which

b

=

Z

(Y-

Y)

Z

-

Z)

/

(X-

X)

Z-

Z)

(where

he

hree

ariates

re

not

necessarily

easured

rom

heir

means)

s a

consistent

estimate.

here

s

a

theorem

hat

when

X,

Y,

Z)

are

normally

istributed

certain

function

f b

is distributed

s

the Student

Fisher

[7].

The writer

ould

wish

for

simpler

roof

f

this

heorem

han

hatwhich

he

found,

or uch

a

proof

might

ead

to a generalisation,.e. for nynumber

f

variables.

Of

course,

f

X is

non-stochastic,

.e.

if

u

is

zero,

the

nstrumental

ariable

hould

be

Z

=

X itself

when

the solution

s

the

regression

ne,

for the

reason

Markov)

that,

of all

possible

nstrumental

ariables

Z

=

X

yields

minimum

ariance

of

the

estimate

f

3.

n the

multivariate

ase the

corresponding

heorem

s that hematrix

minimizes

he

generalised

ariance

f he

stimates

f he

oefficients.

he

nstrumental

variable

procedure

orcoefficient

stimation

as

the merit f statistical

onsistency

but

at the cost

of

a measure

f

asymptotic

nefficiency.

s we know

from ampling

practice,

ometimes

t

may

be

expedient

o

sacrifice

onsistency

or

greater

fficiency

in

estimation

nd

simplicity

n

calculation.


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168

II.

A PROPERTY OF REGRESSION

COEFFICIENTS

It is curious

hat

he

following

ather undamental

roperty

s notto be found n

any

of

the

text-books hich he writer as

consulted,

hough

e is aware thatother ol-

leaguesknow t,and indeed tmight e suspected y anyone amiliar ith egression

theory.

et the

original

model,

n

matrix

orm,

e

(2.1)

y=

=pX+u

where

y

and u are

(1

x

T),

p

is

(1

x

k)

and X is

(k

x

T).

Divide the

ndependent

variables

nto

any

two

groups

of

kl

and

k2

variables o

thatk

=

k,

+

k2.

Model

(2.1)

can

thenbe written

n

the dentical orm

(2.2)

y

=

P1

X

+

P2X2

+

u,

wherenow i, s (1 x kj), Xi is (k, x T) and similarlyorthe secondterm n the

right.

et

V,

be the residual

matrix

f

the

regression

f

X1

on

X2.

The

property

s

that heestimate

,

of

p

is identical ith he

regression

oefficient

ransposed

ector

cl

of

y

on

V,*.

This is the

generalisation

f a

proposition

ue

to

R.

Frisch

and

F.

V.

Waugh [8], proved

forthe

case

of

k2

=

1. The

main nterest

f this

property

in ts

general

orm

s

computational;

s the

number f

ndependent

ariables

ncreases

(beyond

or

5)

partitioning

s an

increasingly

fficient ethod

using

desk

machine)

of

computing

he

regression

oefficients

n

terms f

number f

computational

pera-

tions nvolved.

t is

even

possible

o

determine

henumbers

kl

and

k2,given

k,

which

affords

he most

efficientartition.

Another orm

f the

property

s

that

f z

is

the residualmatrix

f

the

regression

y

on

X2

the

regression

f z on

V1

also

yields

dentically

he

coefficient

atrix

l.

This

is

why

n

the

post-war

eriod

V. Cao-Pinna

[9]

found oefficients

or

Cobb-

Douglas

function

or

taly

of

theform

(2.3)

q

=

const

x

H KY

e

t,

where

q

=

G NP

(with

ertain ndustries

xcluded)

t constant

rices,

H

=

hours

and

K

=

capital

tock t

constant

rices

had

as

Cao-Pinna

found)

oefficients

and

y

insignificantly

ifferent

rom

ero.

n

fact

,

H

and

K

were

ncreasing

lmost

inearly

with = time o thatwhen heestimatesf

p

and

y

are

nterpreted

s the

regression

of the

small)

and

probably

andom

residuals

whenthe effect

f t

is

excluded

rom

log q, log

H

and

log

K,

thenul-results understandable.

he

writers

sceptical

bout

resultsfor

many

countries or the inter-war

eriod,

where

3

was so often

found

equal

to

about

2/3

nd

y

to

1/3, ighly

ignificant

oth,

on

the

grounds

hat

i)

with

K

as

capital

stock

and

not

capital

n use or

capital

ctually

onsumed

n the

produc-

tion

process)

ormula

2.3)

could

not

possibly

e

a

good

theory

or

xplaining

ear-to-

year

variationn

q

and

(ii)

that he

"good

fits"

ound

weredue

to

spurious

orrelation

helped

by

the

pronounced

ip

of all variables

including,

xceptionally,

)

in the

de-

pressionperiod1929-35 f which the term xpat obviously ouldnot take suffi-

cient

ccount.

III.

HAVE

INDIVIDUAL

REGRESSION COEFFICIENTS

OBJECTIVE

SIGNIFICANCE?

Since

regression

s

essentially

cause-effect

elationship

he

only

valid

object

of

the

exercise

s

to be

able

to

estimate

n

average

he value

of

y corresponding

o

given

valuesof

the

ndependent,

r causal

variables. he coefficientsre

therefore

ollectively

*

The

proof

s a

pretty

xercise

n

matrix

manipulation

t student

evel. G.

Tintner

12]

has a

theorem

very

ike

this

though

he

does

not use a

matrixmethod o

prove

t.


7/23/2019 141371.pdf

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169

useful.

n most

ases,

especially

hen

conomic ime

eries re

nvolved,

he

ndividual

coefficients

re

devoid

of

nterest r

significance.

Suppose

that,

n

a

three

ndependent

ariable

ase,

one has

determinedhe

coeffi-

cients

bl,

b2

and

b3 by

east

square

procedure

nd

writes

(3.1)

Y

=

bl

xl

+

b2

X2

+

b3

x3

(all

measured

rom heir

means).

Marginal

theory

eachers

re

prone

to

interpret,

say b2

as

"a

riseof one n

x2

entails riseof

b2

n

y

when

,

and

x3

remain

onstant".

The

trouble

s

that

he

ceteris

aribus

art

does

not obtain

xcept

n

the

very pecial

and rarecase

of

xl,

x2

and

x3

being

mutually

ncorrelated,

case

which

never rises

when

one is

dealing

withmacro-economicime

eries

when

one can

find

veryhigh

correlations

ndeed;

in

fact

n

a

paper

[10]

of

manyyears

ago

the writer

ound

a

correlation f .97 between

employees'

compensation

nd consumers'

perishable

goods forU.S.A., 1921-38usingH. Barger'sdata) and, even after he removalof

terms

o

degree

in

time

he

correlation

of residuals)

emained

s

high

s .93.

It

may

evenbe of some ittle

nterest

o

consider

he

value of

y,

in

the three

nde-

pendent

ariables

ase,

corresponding

o a

value

x2

of x2

when

ccount s takenof

concomitant

ariations

n

x,

and

x3,

within he

ogic

of

regression

heory.

et

x1

and

x3

be

the

average

or

expected

values

to

be

assigned

o

x,

and

x3 respectively

consequent

o

the

value

x2

being

ssigned

o

x2.

From

simple

egression

SX2X1

-

-

X2

3

-

(3.2)

x

-

2

X2;

X3

X2

Let

y'

be the

value of

y

corresponding

o

these

values

of

xl,

x2,

x3.

Then

from

3.1),

Yl

=

bl

x

+

b2x2

+

b3

x3

(3.3)

=

x2

(bl

Z2

X1

+

b2

Z x2

+

b3

Z

x2

X3)

But,

from

he

second of the

standard

quations

for

determining

he

coefficients,

he

expression

n

brackets

quals

Z

x2y.

So

finally

e have

Z

x2y

-

(3.4)

-

X2

2

the

imple

egression

f

y

on

x2.

The

right

nswer o

the

question

f

the

verage

ffect

on

y

of a

rise of

unity

n

x2

(any independent ariable)

s

furnished

y

the

simple

regression

f

y

on

x2,

no

matter ow

many

ther

ariables

r

equations

here

re

in

the

system.

A

rational

meaning

an

therefore

n

many

ases be

attributed

o the

ingle

oefficient

in

simple

egression,

nd

perhaps

nly

n

such a case.

At

the other xtreme

ne

must

be

extremely

ceptical

n

statistical

rounds

lone about the

meaning

r usefulness

of individual oefficientsn themany-variablease when one so well knowsthat

small

changes

n

the

basic

data

(sometimes

ell within he

range

of

accuracy

f the

data)

can

result n

substantial

hanges

n

the estimates f the

coefficients.

The

writer s

aware

that the

statement

hat

values

found

for ndividual

multiple

regression

oefficients

s

meaningless

as rather

evastatingmplications

or

marginal

analysis

n

practice.

One of

the best-known

pplications

s that

of

price-income

e-

mand

analysis

based on

time

eries

n the

form

with

heusual

notation)

p

Y

(3.5)

log q =

c+

P

log +

y

log +

t + u,


7/23/2019 141371.pdf

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170

where

p

and

y

are

interpreted

s

price

nd

income

lasticity.

he

special

Providence

which watches

over virtuous

nalystsmay

ordain

that

og

p/P

and

log YIP

are

uncorrelated

ut

f

they

re

not the

usual

elasticity

nterpretation

eems nvalid.

f

we

regress

og q

on

log

p/P

and t we obtain coefficient

'

to which rational

meaning

in theelasticityontext an be attached:no matter owmany ther ausal variables

should

be

in

the

true

ormula

or

og q

and

whetherhese re nter-correlatedr

not,

the

regression

stimate

epresents

hevalue

of the oefficienthen

ll theother

aria-

bles assume

their

verage

value

consequent

n

log p/P

having

given

value.

But

n

(3.5)

P

is

not

necessarily

qual

to

P'

and

yet

P

seems o be

the

price

lasticity

or

all

values

of

log

YIP

whenthe effect

f

time

t

is eliminated. f

course,

3.5)

in its

re-

gression

orm

nd under

he usual

conditions

will

afford

perfectly

alid answer

o

the

problem

f

expected consequent

n

given

alues

of

p,

P,

Y and t. Rather

imilarly

any theory

f

marginal

ates

of return o labour and

capital

based on

partial

differ-

entials f a productionunction.g.

(3.6)

q =f

(H,

K)

are

dubious

unless ne cares

o

sponsor

he urious

ypothesis

hatH

- hoursworked

are

ndependent

fK-

capital

tock.Can hours

e worked

without

ools nd

machines

or vice versa?

The

foregoing

onsiderations

lso

lead

one to the conclusion

hat much of

the

preoccupation

ith he error ariances

f the

coefficientss

irrelevantnd

compara-

tively nimportant.

In

one

special

ase of

multiple egression

he ndividual

egression

oefficientsave

a

meaning,

amely

when he

ndependent

ariables reuncorrelated.ut n this ase,

of

course,

he coefficients

re

exactly

hose

whichwould

be found

on

regressing

he

dependent

ariable

on each

of

the

ndependent

ariables

eparately,

.e.

by simple

regression.

his

fact

ends

ome

nteresto the

problem

f

orthogonalizing

he

original

independent

ariable

ystem.

here

s

an

infinity

f inear ransformations

n which

this

may

be

affected,

n

matrix

orm s follows

(3.7)

Z B

X,

where

X

and

Z

(k

x

T)

are the

original

nd transformed

atrices

nd

B is

(k

x

k).

One transformationas themerit hat t s symmetricalntheoriginalndependents,

namely

hat

n

which

hetransformedariables

re the

principal omponents,

rtho-

gonal,

of

course,

to

one another

11].

This transformation

as

no stochastic

m-

plications

whatever:

nalysis

n Z

is

mathematically

dentical

with he

original

n X

since the estimated alue

of

y

given

X

will

be identical

with he value

of

y given

Z

when

X

is transformed

nto Z

by (3.7).

In

the context

f economic

ime

eries

he

principal

omponent

may

take

up

the

greater art

of thevariance

f

y

and so

impart

an

objective

alidity

o the

imple

egression

oefficient

fy

n

this

rincipal

omponent.

In

forecasting

hat

usually

matterss the

variance f the

forecast

hich,

nfortun-

atelyforforecasters,epends absolutely n the unit residualvariance

02

of the

seriesused

to determine he

regression,

ven f this series s

very ong

and

even

f

one makes

the most favourable

ssumptions

bout

stability

f

coefficients

nd all

therest. n

fact,

f

n

simple egression

(3.8)

Y=a + b

X,

where

X

is

given

nd

Y

is the estimate f

Y,

then,

s is

well-known,

(3.9)

VarY

=

02

-+

(X-

)2

=

0

(T-1)

.

T

(X -X)2/


7/23/2019 141371.pdf

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171

However,

his

takes

care

only

of

the

expected

r

average

value of

Y.

For

actual

Y

forecasted

hevariance s

(3.10)

VarY

=

a2

+

Var

Y,

where hefirst erm n theright redominates henthe number fobservations

is

large.

f a

is

of

the

orderof the

year

to

year

change

n Y no

valid

forecast an

be

made in

the short

erm. n

the

onger

ermwe

may

be

in

better

ase,

since

we

can

reasonably

ssume

hatwe

are

concernedwith

he

average"

or normal ituation

nd

changes

rom

ase to

reference

ear

may

be

substantial.

IV.

REMARKS

ON

SYSTEMS

OF

EQUATIONS

When

thewriter

as

actively

esearching

n

relations etween

conomic ariables

aboutfifteenr twenty ears go, itwas in thehighest egreeheretical o takethe

attitude

f

etting

he

figures

peak

for

hemselves.

o,

one

musthave

regard

o what

was

called,

and

perhaps

s still

called,

"economic

heory".Actually

he

formula

f

the

priestcraft

s enshrinedn

the

ubtitle fthe

Econometric

ociety

An

International

Society

for

the

Advancement f

Economic

Theory

n

its Relation to Statistics

nd

Mathematics",

utting

tatistics

nd

Mathematics ell nd

truly

n their ubordinate

place.

A

verypoor

view

was

taken of

the notion

that the

problem

f

establishing

relationships

etween

conomic

ime eries

hould

be

approached

n a neutral

way,

without,

owever,

ny

abdication

f

good

sense,

with he

object

of

finding

set

of

completein thesenseofthewritern [7]) linear elationsnwhich heerror ariance

of

the

ystem

s

a

whole

perhaps

he

generalised

ariance was as small

s

possible.

He well

recalls the

shock

of

disagreement

t

a

session

of

International

tatistical

Institute

many

ears

go

when

O.

Morgenstern

with

o

doubtdeliberate

xaggeration)

remarked

let's

throw ll

the

figures

nto a

computer

nd

see whatcomes

out

at the

other

nd".

He was

possibly

he

only

person

present

who had

any sympathy

ith

Morgenstern.

he

writer

oes not

assert

hat he

ack of

success

which

has

attended

efforts

o set

up

economic

quation

ystems

as

necessarily

ue to the

hackles

fthe

priestcraft:

e

does know

hat

hackles f

any

kind

re nimical o scientific

bjectivity

and

development.

While

paying

warm

ribute o the

so

very

ew

devoted

workers

whodaredto

apply

their

heory

o actual

data,

the

totalvolume

of

applied

work n

trying

o

derive

working

macro-economic

odelshas

been

puny

n the xtreme.

hat

these

efforts

ere

not on a

larger

cale

was due in a

degree

o the

scepticism,

he

inspissated

loom,

f

the

priestcraft.

s

Larochefoucauldlmost

emarked,

hey

were

not

too

unhappy

n

their

conometric

riends'misfortunes.

hose

who

genuinely

want to

know n

measured

erms ow

the

economic

ystem

works

must

ever

heir

connection

ith

very

rejudice

f

o-called

conomic

heory

nd set

up

computational

experiments

n a

vastly

arger

cale in

the

future han n

the

past.

Of

course n

practice

he

few

devoted

model-workersid not llow

themselves

o be

spancelledby economictheory.Havingmade obeisance,theyset down perfectly

sensible

utative

elationshipsapart

from

he

accounting

nd definitional

dentities)

such

as

current

onsumption

eing

related o current

and

possibly)

agged

ncome,

that

output

was

related o

manpower

nd

capital,

hat

government

xpenditure

as

related o

taxes nd all

the

rest.One

does not

need

to be an

economist

o surmise

uch

forms f

relationship.

hosewho

tried

o

verify

nything

hich

might

roperly

e

called

economics"

avenot

had

happy xperiences:

he ole f nterests

a

case

n

point.

As

already emarked,

ery

ewof the

coefficients

n

systems

f

equations

have

any

significance

n

themselves;

hose whichhave are those

occurring

n

equations

with


7/23/2019 141371.pdf

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172

only

two

variables.That the

main

object

of

model-makers

s

forecasting

s

implied

in

the

rranged quality

fthenumber fcurrent

ndogenous

ariables

o the

number

of

equations.

Such

equalitypermits

he

derivation

f

the reduced

form

after

he

determination

f he

oefficients),

.e.

of

xpressions

or ach of he urrent

ndogenous

variables n terms f predeterminedariables. t is hightime thatmodel-makers

should

shed

their

reoccupation

ith

he coefficientsnd assert

hat

predominantly

the

object

of

model-making

s

forecasting.

Any

models

withwhich hewriters familiar

re

redolent

f

cause-effect

elation-

ships.

Usually

ne

finds

hat

ach

equation

onsists

f

one current

ndogenous

ariable

on

the left nd

one

or

more

current

ndogenous

ariables

nd

predetermined

in-

cluding

agged ndogenous)

ariables

n

the

right;

t s evident

hat he

atter

ariables,

in

the

hought

f

the

model-maker,

re

regarded

s causes

and

the

variable

n the eft

as the

effect.

ince

number

f

equations quals

number f current

ndogenous,

ach

ofthe atter as a solopart n a particularquation.Now this s surely very urious

way

to

imagine

ow the

system

orks.How can a variable

e

simultaneously

cause

and

an

effect?s

one

to

imagine

hat he

causative

ariable

s to be

lagged

a

little"

in

time

as

compared

with

ostensibly

he same

variable

s

an effect?

ut,

f

so,

in

principle

here

re twovariables

nd

not one. t

is not

quite

atisfactory

o

rejoin

hat

thevalues

will

be

only

little ifferent:

ne

doesn't

know.

One

way

of

dealing

with

his

difficulty

s,

of

course,

o

insert

n

the

right

ith

ach current

ndogenous

ariable

he

same

variable

agged

one

time

unit,

with he dea that

he

wo values

weighted y

the

coefficientsre

equal

to one

lagged

value,

.e.

x,

+ P

x,_

=

X,_,Oc

p=

1.

This

devicewould

have

some

plausibility

fthevariable

was

more

or

less continuous

in

time,

which

t

rarely

s.

Considerably

moreattention

must

be

given

n the future

than

n

the

past

to

the

time

nterval hether

ne

believes

hat

ause-effect

s

a useful

approach

to

the

tudy

f economic

elationships

r not.

To

expect ood

results

when

one has

imposed

usually)

the

year

as the

time

unit,

giving

ne the choice

only

of

simultaneity

r

effectfter whole

year

s to

expect

oo

much.

Relationships,

hich

are

obviously ignificant,

fter time

ag

of

a week

or a month

when

one

has

the

statistics )

ften anishwhen the

figures

re totalled ora

year.

Of

course,

model-

makers annot

be

faulted or

not

working

with hort

ime

unitswhenthe

required

statisticsre

not

available.

From

the

forecasting

oint

of

viewwhatwe

really

want

to know

re thevalues

of

some

k

variables

n

year

T

+

7

whenwe know the

data for

years

=

1,

2,

...,

T.

We have no

direct

nterest

n

what

aused

what;

we

ust

want

o know.

The

equation

system

s

a

meansto

this

nd,

but,

s

already

emarked,

ll economic

model-makers

have

followed he

ause-effect

oute.The writer

uspects

hat

his

pproach

has some-

times

nvolved

hem n

logical

contradictiont

the

stage

whenthe

original quation

systemwith oefficientsstimated)s expressednreduced orm orforecastingur-

poses.

As

every

tatistical

eophyte

nows,

having

written

he

simple

egression

f

Y

on X in

theform

Y= a

+

bX,

one

cannot

tate hat

X=

(Y-a)/b,

in

any

very

meaningful,

s

distinct

rom

ormal,

way.

Yet this eems o be the kind

of

thing

one

does with

transformation

o reducedform. t is the writer's

rowing

conviction

hat

when everal

ariables

ppear

n an

equation

herelation etween hem


7/23/2019 141371.pdf

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173

shouldbe

associative,

ot

regressional;

t

any

rate

when he

relationship

s

associative

substitution

f variables

of the

type

ndicated s

always

permissible.

uch

sanctity

attaches

o the

full

maximum

ikelihood

method f coefficient

stimation,

hat t

is

commonly

verlooked

hat

ML does not

produce

ssociative

esults.

In somemodels t scustomaryo introduce ariables ermedpolicynstruments",

usually

those

variableswhich re

under

the direct

ontrolof

government,

evel

of

taxation

nd the

like,

the

problem

being

to

determine

he effect

n other

macro-

economicvariables

of

changes

n

the

nstruments.

ne mustbe

very

areful

here.

Suppose

the

model

consists

f a

single

quation

(4.1)

Yt

=

P

Xt-1 +

Ut,

t

=

1,

2,

...

, T,

where

y

is income nd

x is amountof taxes both measured

rom heir

means.

The

coefficient

3

(presumably egative

n

sign)

s

estimated

y regression

nd

T

is in-

definitely

arge.

There

might ppear

tobe no

difficulty

ince

xt_

,

at a timeunit

back,

can be

conceived

s the cause

of

yt,

but

n

the

ordinary

eaning

f

words,

xt,_

an-

not be

regarded

s the

"cause"

of

y,.

Yet

on

occasion

y

is

a

target

with

value

r

and

the

problem

rises

of

finding

he

value of

x,

say

?,

the

"instrument",

hich orres-

ponds

to this

arget.

From

4.1),

(4.2)

-

=

(r - u) /p,

where

u

is

a

nuisance

rror erm. o

obtain

the

right veragevalueof the?, the n-

strument

ariable

orresponding

o

given

r,

we

must

ssign

to

u

its

average

value

u

corresponding

o

r.

This value s found s

(4.3)

u

=

E

yu

/

Ey2.

Then,

on

substitution

n

(4.2),

-=

r

Ey2- Eyu)

/

Ey2

(4.4)

=

Exy

/Ey2

,

theregressionfx ony.Thismaybe obvious nthis imple ase (namely hat fone

wants

to

know the

lagged

tax

level

corresponding

o

target

ncome

and

should

regress

agged

tax on

currentncometo

find he

regression

oefficient

nd not the

other

way

about), yet

one

wonders f

this s

always

recognised

when the

issue is

complicated

y

many

variables nd

many quations.

The

writer

would

very

much ike to

provoke

a

discussion

n

this

point

at

this

Conference. e

would ike

his

colleagues

o

address hemselveso

this

ittle

roblem

in

particular.

et the

model of

a

consumption

unction

e

(4.5)

C

=

p

Y

+ u

and

suppose

that

C

personal

consumption

nd

Y

personal

ncome are so defined

that

when

Y

-

0

thenC

-

0. Can I estimate

by

(4.6)

=c

/ Y,

the

associative

ormula,

n this

ase,

or

is

my

stimate

o

be

(as

usual)

the

regression

of C

on Y?

Before

you

answer oo

hastily

et me

warn

you

that,

despite

he

nitial

hypotheses

f zero associationof Y and

C,

the

regression

ill

containa

positive

constant

erm,

ontrary

o

hypothesis.

n

fact,

n

the associative

ase,

the model

s


7/23/2019 141371.pdf

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174

C=c+u

(4.7)

Y=

y

+ v

c=

py

where and vare error ermswithmeans erouncorrelatedith ne another nd with

the

nherent utunknown ariables and

y.

When

he

number

f

sets

of

observations

is

indefinitely

arge

(4.8)

Ec

/

Ey

=

EC/EY

But

from

egression

f C on

Y

the

bsolute

erm

'

is

given y

(4.9) a'

=

EC-

P'

EY,

where

(4.10) ' = E(C - EC) (Y- EY)/E(Y-

EY)2.

From

(4.7), (4.9)

and

(4.10)

and

using

the assumed

properties

f

u and

v,

we

find

(4.11)

C'

=

P

a2

EY/E(Y-EY)2

where

2

=

Ev2.

The

constant

erm

x'

is

accordingly

ositive

when

E

Y

is

positive

(as

it will

alwaysbe)

and

a

-=

0.

The

writer's

uestion

Are the relationswe want

to be

causative

r

associative?"

is

not a rhetorical

ne;

he

really

wants

to know. There are

conceptual

difficulties

inregardingherelationships associative,nparticularnanyeconomic ariableX

being

decomposable

nto an inherent

art

x,

to which he

relationships

re

supposed

to

pertain

nd

a

purely

andom

part

u,

so

that he

observation

X

=

x

+

u,

which

many

will

have

trouble

n

accepting specially

when

the economic eries re

time

series.

From the writer's wn work

10],

associative

elationship

etween

ime

series

ends

o be

relationship

etween

ecular

rendswhereas

he results

reviously

mentionedend o show

trong elationships

etween esiduals

fter rends

reremoved.

Areassociative elationshipsuitable or onger erm orecastingutnotsuitable or

short-term?

If

one

succeeds

n

establishing

ssociative elations

and

this

will be

by

no

means

easy

in

practice,

hough

here

s

plenty

f

theory)

o

contradictions

r

paradoxes

of substitution

f

the

kinds ndicated bove

can

arise.

V.

THE ERROR

TERM IN

EQUATION

SYSTEMS

In

the

historical

evelopment

f the

mathematical

heory

f

errors,

rrorwas

conceived

s

an error

f

measurement,

ue to

human

fallibility

r the

imitationsf

themeasuringnstruments.t was easyto attributeonceptuallyhequality fran-

domness

o

errors

f

thiskind.Now

in

those

days

he

stronomical

nd other

hysical

laws

which

had

emerged

r

were

emerging

ere

simple

n character: ften ndeed

their orm

ould be inferred

heoretically,

he

ctualobservations

eing equired nly

for

verification a

simple

ituation

ndeed.

n

the ocial

sciences,

n the other

and,

the

aws,

f

ny

obtain,

re

mmensely

ore

omplicated

han n the

physical

ciences.

Theoretical nd

qualitative

conomics

ive

us but ittle

uidance

n

the

mathematical

form

f these aws. Geometrical

nd

mathematical

conomics

se

only

he

simplest

functional orms nd

these,

ne

suspects,

re

selectedmorefortheir

implicity

nd


7/23/2019 141371.pdf

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175

for classroom

purposes

han for

any

conviction

n

the

part

of their

nventors hat

they

epresent

eality.

ou willhave

noticed,

or

xample,

hat

n

time

eries

hey

lmost

invariably

ave

the

solution

y

=

Cex

,

which

no

economic ime erieshas

obeyed

except

between onsecutive

ears.

Came

the econometrician. e showed ittle

dis-

positionto work n functional elationshipsigher han n thefirst egree hough

there

may

be this o be said

for

him,

n

ustification,

hat

n

introducingagged

erms

into

his

equations,

he was

implicitly

sing

he

calculus

of

finite

ifferences,

ust

one

remove

therefore rom inear differential

quations,

which,

s

you

are

aware,

can

involve

solutions f

high

functional

omplexity.

ne

quality

hared

by

economists

and

econometricians

like

is

a

distinct

reference

or the dialectic nd for

mathe-

matical

abstractions s

against

the

brutalising iscipline

f

numerical alculation.

Inevitably

here

appeared discrepancies

etween

heory

nd

practice.

The "ex-

pected"

values

werefound

o

deviate n

greater

r

lesser

degree

rom hetrue

values,

ifone canso politelyermhe tatistics.o makeupfor hediscrepancyn error erm

was introduced.

tochastic

heory

was

then vailable for

coefficient-estimation

nd

tests f

significance,

ithR. A. Fisher's ests

f

consistency,fficiency

nd thewell-

known

properties

f

maximum ikelihood stimation.

y

far the

greater

olume

of

data were

ime eries orwhich

t

was

found

necessary

o make a

considerable

xten-

sion

of

existing

tatistical

heory,

ue

principally

o thefact f

serial

orrelationn the

statistical

ime

eries.

The

error

erm n

any

equation

s

the

measure

f

whatwe don't

know.

n

the ocial

sciences

nowledge

f

aw

of

cause and effects far

ess

than

n

the

ase

of

experimental

science. n economic nvestigatione haveto make thebestuse ofwhatwecan get

and

the

statistics

vailable tend

to

be of unsuitable

efinition,

naccurate

nd

in-

complete.

n

addition,

we don't know

n

advance the mathematical orm f the

aw

or aws

of

relationship.

t

is

really nly

n

thefield

f

sampling

ocial

surveys

hat

he

economic tatistician

s

in

anything

ike the

situation

f

the

experimental

tatistician

in

having

his measurements

nd thewhole

plan

of

his

nquiry

nder

ontrol.

It is

not

as

clearly ecognised

s it

should

be that he

ntroduction

f the random

variable

ompletely

hanged

hecharacter f

mathematicalconomics

n the

broader

sense.

Any

reasonable

ystem

f

behaviouristic

quations

n

time erieswill contain

lagged

as well as current

ndogenous

ariables nd

it

is

customary

o

arrange

hat

the number

f

endogenous

variables

quals

the

number

f

equations.

The

formal

solution

ontains

term inear

n

the random

variableswith oefficientshich re

more

or

less estimable

ut thiserror erm s

of

the

same order

of

magnitude

s the

variable

o be determined. s remarked

arlier,

n

mathematicalconomics

without

the

errors

he

solution

s

usually

n

exponential

r

Fourierform:

n

any

realistic

solution hese

ermswill

ong

sincehave vanishedwhen ccount s

takenof the

error

terms.

he

point

s

that hecharacter

f

thesolution

s

fundamentally

ltered

y

the

introduction f

error

erms:

he formal olutionfor each

endogenous

ariablefor

current ime s

an

expression

inear

n theerror

erms nd

in the

exogenous

ariables,

stretchingack intime o the start ftheseries.

The

specialproblems

f economic ime

eries

osed

theoretical

roblems

f

special

interest

o themathematiciannd

many

f

thesehave

been

ngeniously

olved.These

problems

nd

their

olution

ave

ustly

ndowed his

articular

ranch fmathematical

statistics ith

highprestige, ertainly

much

higher restige

han t deserves

or

its

practical

sefulness.

In

economic

quations, ingle

r n

sets,

eguiled y

theory,

e are

asking

hat rror

term o do

too

much.

Surely,

n

reason,

we cannot

xpect

muchof a stochastic

heory

whenwemake he

rror erm tand or ll

the

ariables

which

hould

be

in the

quations


7/23/2019 141371.pdf

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176

if

only

we

knewwhat

they

were,

forthe errors f

measurement

n

the

variableswe

have

included

nd for he nevitable

implification

f the aw of

relationship.

Since all the series xhibited he

phenomenon

f serialcorrelation

sually

n

em-

phatic

degree

nd since he

imple

models ould not

possibly epresent

he

mmensely

complicated orkingf the conomic ystem orrectly,t was inevitablehat hephe-

nomenon

fautocorrelation

f

residuals

hould

ppear

n the

results.

t was

surely

ot

surprising

hat f this

phenomenon

s admitted s

part

of the

hypothesis,

he

model

could

not

yield

atisfactory

esultsn

practice.

he

postulate

hat he

residuals re

i)

independent

f the

predetermined

ariables nd at the ame

time

nclude,

s

it

must,

(ii)

the

contributionsf variables

necessarily

erially

orrelated)

ot

explicit

n

the

equations

because

they

re not

known,

eems contradiction

n

terms,

or

he

reason

that the

unknowns

nd therefore

he

residuals

which

encompass

hem

cannot

be

postulated

s

independent

f the known

predetermined

ariables.

It is thewriter'sonvictionhatthehypothesisfauto-regressionfresidualsn

equation

systems

ased on time series

however

ttractive

t

is

mathematically)

s

inadmissable

rom

he

practical oint

f

view.

f,

fter

stimationfthe

oefficients

n

any

particular

hypothetical

quation,

the residuesexhibit

his

phenomenon

he

equation

should

be

rejected

r,

by

trial

nd

error

adding

fresh

ariables

r

taking

others

out),

the

original quation

should be

amendeduntilone

attainsnon-auto-

correlated

esidual

rrors.

Admittedly

his s a

highly

mpiricist oint

of

view. The

writer

elieves

hat,

when ll the

original

ime

eries

re so

highly

utocorrelated,

he

best

criterion f

adequacy

of

relationship

s,

that heresiduals houldbe

found

o

be

completelyandombythevon Neumann r other ests.

If

this

viewpoint

e

accepted

hen

models

ncorporating

he

hypothesis

f residual

auto-regression

re erroneous. onsider

he

model

(5.1)

Yt

=

--

t-1

+

Vt,

where

he

vt

re

autocorrelated.

he

simplest

modelof

thiswould be

(5.2)

vt

=

vt,_

ut

where

ut

are

non-autocorrelatednd

non-correlated

ith

vt-_,

t

being

uncorrelated

with

yt-i.

The

problem

s to

estimate from

series f

Ty'

s.

In

myopinion

his,

he

simplest ossible aseof hekind f hing hichsnotuncommonn more omplicated

form,

s

a

wrong

formulation. ore

correctly,

y

substitution

rom

5.1)

in

(5.2),

Yt -

Yt-1

=

(Yt-1

-

Yt-2)

+

Ut

or

(5.3)

Yt

=

(c +3-

Yt-1

P

t-2

+

Ut

The latter

urely

s

the aw we are

seeking.

We

are

interested

n

estimating

oc

P)

and

a

3

for he

purpose

f

forecasting

t.

The

oc

nd

P

n

the

original

ormulation

re

of no interest

n

themselves.*

Andhere s an example frather differentharacter iscussed ymany uthors,

though

he

present

loss

s

the

writer's

wn. The model

s

Ct

= P Yt

-+u,

(5.4) t

=

1,

2, ...,

T

Yt

=

Ct

+-

It

*

I

am ndebted

o

M. H.

Quenouille

or

he

nteresting

bservation

hat

f,

s

appears

o be the

nly

method,

he

olution f

the

problem

f

estimating

and

3,

given

y

5.1)

and

5.2)

from set

of

observations

s via

5.3),

then,

ince

ac

P

nd

ap

are

symmetrical,

he

stimatesf and

3

are

n-

distinguishable

rom

ne another.


7/23/2019 141371.pdf

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177

ut

random,

Ct

and

Yt

endogenous

nd

It exogenous.

The

object

s to

estimate

3,

presumably esigned

s

the

propensity

o

consume,

hough

we

shall see that

t

isn't.

First

and

there s a

hint

here)

ry

o set

up

this

ystem

n this

orm

iven

he

columns

of

ut

nd

It

as well as the

coefficient

.

You willfind

you

cannot.

You

can

only

do

so

byreducingheform o either

1

ut

(5.5)

Yr=

t

+

ut,

P'

=-3'

ut

-

1-

of

(5.6)

"I

+

u,

P

Actually

when

solved

by

least

squares

the

ast two

equations

are

found

o

be

con-

sistent

n

that

as

should be the

case since

S= 1.

We

may

also

remark

though

his

s

not

the

point)

that

P

estimated

y

east

squares

directly

rom he

first

f

5.4)

is

inconsistent

ith he

estimates

'

and

3".

The

point

is

that

5.4)

is a

false

formulation

hich

tates,

f t

states

nything,

hat t

given

evel

of

Y,

we

expect

subject

to

a

random

rror)

hat

Ct

will have

a

given

value,

.e.

a

consumption

unction.What

we are

really

aying

s

that

Y,

and

It

or

Ct

and

It

are

related, quite

different

conomic

heory, capital/outputheory,r,equivalently,a

capital/consumptionheory.

his

may

be

a

trivial

xample

but t raises

the

whole

question

fthe

validity

fthe

olution

f

mixture

f

behaviouristic

quations

nd

the

accounting

dentitiesnd

allowing

neself,

s

is

common,

omplete

reedom

f

action

as

regards

limination f

variables

efore

roceeding

o solution

y

maximum

ikeli-

hood or

other

methods.

When

we state

hat

n

equation

n

our

economicmodel s

k

(5.7)

Yt=

ox

I+

P

Xtt

+

ut,

t

=

1,

2,..., T,

i=1

what s ourpicture fthereality? fcourse,wehaveno illusions boutthe inearity

of the

equation:

we

usually

ross that

hurdle

which

will

trouble

s

no more

n this

section)

by

assuming,

ometimes

gainst

ll

the

evidence,

hatwe

are concerned

nly

with

he estimation

f

"small"

deviations

rom

ome

norms,

whenwe

have

Taylor's

Theorem

o sustain

us.

If

(5.7)

is

a

classical

regression

quation

the

assumption

f

linearity

ffects

nly

the

dependent

ariable

Y

since

the

X's,

being

pre-determined,

can

have

any

functional

orm

whatever,

.g.

X2

can be

X12,

r

what

we wish.

But

can

anyone

be

happy

bout

the

classical

regression

model

n

economic

pplications,

hat

Y

on

the

one

hand and

the

X's

on

the

otherhand have

so

very

different

tochastic

properties; orwhatwe are sayings thatY is envisaged s exactly qual to a linear

expression

n

the

X's

together

ith

random

cattering

f values

u uncorrelated

ith

the

inear erm?

Whatever iew

be taken

of

theassociative

oncept,

urely

ne

must

be

unhappy

bout this

s

an

image

of economic

eality

r

anything

pproaching

t?

Or do we

say

that Y

is

exactly epresentabley

a linear

expression

f which

we

know

only

k

+

1

terms?

hat,

n

fact,

has the

following

orm:

(5.8)

ut

=

*

Pk+j

Xk+j.

t,

j=1

with

no

knowledge

hatever

f

thenumber

f

additional

ariables

k',

the

coefficients


7/23/2019 141371.pdf

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178

or

the

Xk

j.

(Of

course

it

would

be more ensible n these ircumstanceso use

a

single ymbol

or each

term;

he

expression

s writtenn the

way

it is to

point

he

analogy

with

5.7)).

We assume

throughout

hatvar

(u)

is an

ordinarymagnitude:

if

it were "small"

therewould

be no

problem.

f we

knew the values of the

Xk

+jthenanyk + k' + 1setsof Y; Xi, Xk j) wouldserve o obtain he exact valuesof

the

(c;

i,

k

+j),

consistent

ith

the whole

T

>

k

+

k'

+

1

sets of values. f

the

correlation

f each of the

X1with ach of

the

Xk

j

were

exactly

ero the values

of

the

pi

found

would

be

exactly

qual

to

those

found

by regression

rom

5.7).

What

regression

as

done is

to

give

thefirst

+

1

terms

f a linear

xpression

ontaining

k

+

k'

+

1

terms.

f the orrelationsetween

heX's

in the wo

ets,

nstead f

being

all

exactly

ero,

were

imply

ot

ignificantly

ifferent

rom ero n therandom

ample

ofT

sets

of

operations

hen he

i

calculated

y

regression

ould

be

unbiased stimates

of

the

true

values

Pi.

It is difficulto believe hat heknown nd unknownndependentariableswould

divide hemselves

p

intotwo

groups

ike

this,

nless,

f

course,

he

relationship

as

associative nd

complete

n

which ase

the error

ermwould

merely ynthesize

he

random rrors

n

the

Y;

Xi).

In theknown

et,

ypically,

henumbers

re all

mutually

correlated nd

each is auto-correlated

n time s

well;

since

non-correlation

ith he

known et s

postulated

n the unknown

et,

t

s

unlikely

hatmembers

f the

atter

are

auto-correlated

nd

lack of this

property

eems

o

disqualify

hem s time

eries.

The

process

of

regressionmposes

on-correlation

etween

he

residual

u and

the

variables

Xi

in

(5.7).

If

in truth has the

form

5.8)

where

he

X

variablesexist

(thoughwe do not knowthem) nd if, n act, omeof thesevariables recorrelated

with

ome

of

theX's

in

the

known

et

hen

egression

auses

distortion

nthe stimates

of

Pi

which re

not

consistent ith heir rue alues.

f

these

rue alues

re

supposed

to

have some kind

of

economic

alidity,

o

much he

worse

or he

regressionrocess.

If,

after

stimation

f

the

coefficients

i

n

(5.7) by

regression,

ne finds

n

sub-

stitutionhat

he stimates

f

the

ndividual esiduals

re

auto-correlated

n

time,

his

result

eems

to

establish

prima acie

case

for

the

factthatu

has

in factthe

form

(5.8)

with t

least

one of

the coefficients

k

j

non-zero,

he

corresponding

ariable

values

Xk

j

having ordinarymagnitudes

nd

the variable

having

the

expected

propertyfauto-correlation.n a wordthevariable xists nd the obviouscourse s

to

go

look for

t

nstead

f

postulating

he

property

f

uto-correlation

n

the

esiduals,

of whichno

practical

ood

can

come.

VI.

INTEGRAL

SOLUTION

OR

INDIVIDUAL

LEAST

SQUARES?

On

this

famous ssue the

writer's ttitude

might

e described

s

one of

malevolent

neutrality.

s he

attaches

ittle

mportance

o

individual

oefficient-estimation,

f he

had to

choose

he

would ncline

owards

he

solution

f each

equation

n

the

system

by

east

quares,

referring

he

plane

of

closest

it o

regression,

f

one s unable

o use

the full associativesolution. The writer rankly onfesseshimself o be a rank

empiricist.

s

we

don't

know the

aws

of

economics

nd

probably

will

never

know

them,

et's

go

findwhat

has worked

estover series

f

yearsby

all kinds

141371.pdf

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