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Multiple Regression for Systems of EquationsAuthor(s): Gerhard TintnerSource: Econometrica, Vol. 14, No. 1 (Jan., 1946), pp. 5-36Published by: The Econometric SocietyStable URL: http://www.jstor.org/stable/1905702.

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MULTIPLE

REGRESSION FOR

SYSTEMS

OF

EQUATIONS*t

By

GERHARD TINTNER

1. THE PROBLEM

It has

been

apparent that the classical

method of

least-squares

fitting of

single equations in a

system of

economic equations leads fre-

quently to

unsatisfactory results.

(But it is

perfectly adequate for pre-

diction.) The reason

for this is

that the

actually observed economic

variables are really

jointly

determined by all the equations in the

sys-

tem

and

not, in

general, by a single one of

them. For

instance, the

prices and

quantities actually

established on

a market

are determined

by the intersection of the demand and supply curves. This was already

recognized early in

connection

with empirical work

dealing

with de-

mand

function, for

instance, by Frisch' and E.

Working.2

A

systematic

treatment

of

the

problem in terms of modern statistical ideas

is

due to

Haavelmo.3 Klein,4

Koopmans,

Marschak,6 Wald,7 and Smith8

made

important

contributions.

*

The author is much obliged to 0. H.

Brownlee

(Ames), W. G. Cochran

(Ames), H. Hotelling (Columbia), L.

Hurwicz (Chicago), L. R. Klein (Chicago),

J. Marschak (Chicago), T. Koopmans (Chicago), A. Wald (Columbia), and F. V.

Waugh (Washington) for advice and criticism.

t

Journal

Paper

No. J-1344

of

the

Iowa

Agricultural

Experiment

Station,

Ames, Iowa, Project No. 40-62-730.

1

R. Frisch, Pitfalls in the Statistical

Constructionof Demand and Supply Curves,

Veroffentlichungen der Frankfurter Gesellschaft fur Konjunkturforschung. Neue

Folge, Heft 5, Leipzig, 1933, 39 pp.

2

E. J. Working, What Do Statistical

Demand Curves Show, Quarterly

Journal of Economics, Vol. 41, February,

1927, pp.

212

ff.

I

T. Haavelmo, The Statistical

Implications of a System of Simultaneous

Equations,

ECONOMETRICA,

Vol. 11, January, 1943, pp. 1-12; The Probability

Approach in Economics, ibid., Vol. 12,

Supplement, July, 1944, 118 pp.

4L.

R. Klein, Pitfalls in the Statistical

Determination of the Investment

Schedule, ECONOMETRICA, Vol. 11, July-October, 1943, pp. 246-258.

5

T. Koopmans, Statistical Estimation of Simultaneous Economic Relations,

Journal of the American Statistical

Association,

Vol.

40, December, 1945, pp. 448-

466.

6

J. Marschak, Economic Interdependence

and Statistical

Analysis,

in

Stud-

ies in Mathematical Economics and

Econometrics, In Memory of Henry Schultz,

Chicago, 1942, pp. 135-150; (with W.

H.

Andrews)

Random

Simultaneous

Equations and the Theory of Production,

ECONOMETRICA,

Vol. 12, July-Octo-

ber, 1944, pp. 143-205.

7H.

B. Mann and A. Wald, On the Statistical

Treatment of Linear Stochastic

Difference Equations, ECONOMETRICA, Vol. 11, July-October, 1943, pp.

173-

220.

8

J. H. Smith, Weighted Regressions

in

the

Analysis

of

Economic

Series,

in

Studies

in

Mathematical

Economics

and

Econometrics,

n

Memory of Henry Schultz,

pp. 151-164.

ID

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8/10/2019 1905702

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6 GGERHARD TINTNER

Many of the authors following

Haavelmo suggested the introduction

of stochastic or error terms

on the right-hand side

of the

theoretical

equations of

the

contemplated system.

We

propose

here a somewhat

different approach and will also show that both lines of attack are really

special cases of a more

general method of solution

which has not yet

yielded to statistical analysis.

In the following we propose to deal

with systems of economic

equa-

tions. But it is not unlikely

that our methods may be

applicable in other

fields, especially in the case

of certain biological problems.

We

will first give an economic

theoretical justification of our

method,

then sketch the statistical procedures implied. After

this we will deal

with the troublesome question of meaningful economic relationships

(identification) and then

indicate tests of significance.

Finally we will

give

an

application to agricultural

data.

2. THEORETICAL

JUSTIFICATION

In

order to clarify our ideas we will first assume

the existence of

a

complete static Walrasian system:

(1) G(zi(Z Z2,

..

IZQ)

=0,

i

=1,

2,

,q

q variables

zj.

We can imagine that

these equations determine

completely the q economic variables

Zl, Z2, * *

Zq.

These can be

interpreted

as all the

prices,

al the

in-

terest rates, all quantities

of commodities exchanged and

produced,

wages, etc.

Apart from these q

variables we have also

Q-q parameters

ZQ+i,

- -

*, ZQ.

These

parameters

exert

an

influence

on the

system

but

are

themselves independent of

the variables

zi,

Z2,

- -

*,

Z,.

They are,

for

instance:

technological

coefficients, institutional

data like

income

distribution or property

of factors of production, periods of payment,

noneconomic variables like the weather, etc. It will

frequently depend

upon

the

exact

nature

of the system considered which

variables

will

fall

into the first and which

into the second category. One important

consideration is the

distinction

between short-term

and

long-term sys-

tems. Fixed capital for instance will belong to the

second category

in

the

short run

but to

the first in the long run.

The equations (1) are derived by some assumption of rational be-

havior, i.e., the striving for maximum profit or utility.

We will disregard

here the problem of the

existence of economically significant

solutions

for

our system which has

been discussed by von Neumann9

and Wald.10

9

J. von Neumann, Ueber

ein 6konomisches

Gleichgewichtssystem

und

eine

Verallgemeinerung

des

Brouwerschen

Fixpunktsatzes, Ergebnisse

eines

mathe-

matischen

Kolloquiums,

Heft

8, 1935-36, pp.

73

ff.


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MULTIPLE

REGRESSION

FOR SYSTEMS OF EQUATIONS

7

Stability

conditions will also

be taken

as fulfilled. It should be

men-

tioned

that Leontief

has recently attempted to

verify

such static

sys-

tems

in

a

somewhat

simplified form.

We can also consider system (1) as representing a nonstatic Wal-

rasian

system of

general equilibrium.

Anticipations

have then to be

introduced.

Such

ideas have been treated

extensively

by Hicks12and

recently by Lange.'3

The present

author

has shown that the assumption

of

specific

routines

of anticipations leads

to dynamic

terms in the

equations

(1).14 This is similar

to procedures

adopted earlier

by Evans'5

and

Roos.'6

That

is to say, some

of the

zi

in (1)

have now to

be inter-

preted

as referring to different

points in

time, as

derivatives with re-

spect to time, integrals, etc.

It

seems

to us that

such an

approach is similar

to the one

implied

in

certain

mathematical

business-cycle theories,

especially

those

of

Tin-

bergen,'7

Kalecki,'8 Davis,

and the author.20

Empirical verifications

are available

in the extensive

work of

Tinbergen2l and also

on a much

smaller

scale

in

a

publication

by the author of this

article.22

It has already been

pointed

out by Pareto2' and

emphasized

by

later

writers24

hat the labor of determining

empirically

the system

(1) would

10

A. Wald, Uebereinige Gleichungssysteme dermathematischen Oekonomie,

Zeitschrift ir

Nationalokonomie,

Vol. 7, 1936, pp.

637-670.

11

W.

Leontief,

The Structure

of American

Economy, 1919-1929,

Cambridge,

Mass., 1941,

181 pp.

12 J R. Hicks,

Value and

Capital, Oxford,

1939, 331

pp.

13

0. Lange,

Price Flexibility and Employment,

Bloomington, Ind., 1944,

114

pp.

1

G. Tintner,

A

Contribution

to the Nonstatic

Theory of Production,

in

Studies

in

MathematicalEconomics

and Econometrics, n

Memory

of Henry

Schultz,

pp.

92-109, esp.

pp. 106

ff.;

A Contribution

to the Non-Static

Theory

of

Choice, Quarterly

Journal of Economics, Vol. 56, February, 1942, pp.

274-306,

esp. pp. 302 ff.

1

G. C. Evans,

Mathematical

Introduction

to Economics,

New York,

1930,

pp.

36

ff.

18

C. F. Roos,

Dynamic

Economics,

Bloomington, Ind.,

1934,

pp.

14

ff.

17

J.

Tinbergen,

Annual

Survey: Suggestions

on Quantitative

Business

Cycle

Theory,

ECONOMETRICA,

Vol. 3, July,

1935, pp.

241-308.

18

M. Kalecki, Essays

in the

Theory of Economic

Fluctuations,

London,

1935;

Studies

in

Economic

Dynamics,

New York, 1944.

19

H. T. Davis,

The Theory

of Econometrics,

Bloomington,

Indiana,

1941,

pp.

408 ff.

20

G.

Tintner,

A

'Simple'

Theory of

Business

Fluctuations,

ECONOMETRICA,

Vol.

10,

July-October,

1942, pp.

317-320.

21

J.

Tinbergen,

Business

Cycles

in the United

States,

1919-32, Geneva,

1939.

22

G. Tintner,

The 'Simple'

Theory

of Business

Fluctuations:

A

Tentative

Verification,

Review

of

Economic Statistics,

Vol. 26,

August, 1944, pp.

148-157.

23

V. Pareto, Cours

d'gconomiepolitique,

Vol. 2, Lausanne,

1897, pp.

364 ff.

24

F. A. von Hayek,

The

Present State

of the

Debate

in Collectivist

Economic

Planning,

London,

1935,

pp. 201-243,

esp. pp.

207

ff.


8/10/2019 1905702

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8

GERHARD

TINTNER

be

prohibitive because of the great number of

variables actually enter-

ing into such a system of general equilibrium. This is true

even if

ob-

servations for all the z; were available which is

certainly

not

the

case.

Hence we are forced to investigate the possibilities of simplification.

In economic theory, such simplified models have

very often been

con-

structed. Fisher's famous equation of exchange25 can

be considered

as

a

very simplified model of the complete Walrasian

system.

This

as-

sumes of course that this equation or the

corresponding Cambridge

equation is not a mere tautology. The equations in Keynes's

earlier

Treatise26

are of a similar nature. Many other

models

have

been con-

structed, some of which are represented in

Lundberg's book.27 Keynes's

General Theory28can be thought of as a simplified model of the Wal-

rasian system. It is mathematically formulated

in the articles by

Hicks,2 Lange,30 and Modigliani,3' to mention

only a few. We want to

proceed along similar lines.

We replace a certain subset of the variables and parameters

Z1,

Z2,

.

.

*,

ZQ

by one new variable

Mh

(h=l1,

2,

* * *, p). p is now

a

much smaller number than Q. At the same time

we reduce the

num-

ber

of equations in the system.

The substitution of

Mh

for a subset of our

original variables and

parameters amounts to the following: We replace

for instance

the vari-

ous wheat prices by one representative wheat

price or by the average

of all

wheat prices. We replace the quantities of all

the producers' goods

in

the system (like iron, coal, copper, etc.) by an index of the quantity

of

producers' goods. We replace the various short-term interest rates

by

one

representative short-term rate or by the average of all

short-

term rates, etc. Time will probably also enter

explicitly into our equa-

tions.

After

replacing subsets of the original variables

and parameters

in

(1) by the new set of variables M1, M2, * * * ,

Mp we may still have

variables that are not represented. Write w for a new variable which

stands

for those variables which lack

representation.

25

I. Fisher, The

Purchasing Power of

Money,

New York,

1911.

26

J. M. Keynes,

A Treatise on

Money, New

York,

1930.

27

E.

Lundberg,

Studies

in the Theory

of

Economic

Expansion,

London,

1937.

28

J.

M. Keynes, The General Theory of Employment, Interest and Money,

London,

1936.

29

J.

R. Hicks,

Mr.

Keynes and the

'Classics'; A

Suggested

Interpretation,

ECONOMETRICA, Vol. 5,

April,

1937, pp.

147-159.

30

0.

Lange,

The

Rate of

Interest

and

the

Optimum

Propensity

to

Consume,

Economica, Vol. 5,

New

Series,

February,

1938, pp. 12-32.

31

F.

Modigliani,

Liquidity

Preference and the

Theory

of Money,

ECONO-

METRICA, Vol. 12,

January, 1944,

pp.

45-88. See

also D.

M. Fort, A

Theory

of

General

Short-Run

Equilibrium,

ECONOMETRICA,

Vol. 13, October,

1945,

pp.

293-310.


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MULTIPLE

REGRESSION FOR SYSTEMS OF EQUATIONS

9

Assume that

we have

now R equations in our simplified

system

(R

8/10/2019 1905702

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10

GERHARD

TINTNER

The

economic meaning of

the

assumption that the

Mi

are

the mathe-

matical

expectatiQns

of

the

observed variables

Xi

is

the

following:

We

assume that in the

long run

the averages(arithmetic

means) of

the dis-

turbances or deviations mentioned above tend to be zero. The conse-

quences

are

not

very serious

if this

assumption

should

not be

strictly

fulfilled. The

resulting

bias will

only influence

the constant

terms

if

we

consider linear systems, as

we propose

to do later.

The

more

impor-

tant

regression

coefficients (which

represent,

for

instance, elasticities),

are not affected.

Since we are

primarily interested in

the estimation of

these

coefficients we may

assume

that the biases will not

impair

the

significance of our

results

considerably,

even

if

they should

exist.

Write Xit for the actual observation of the theoretical variables Mi

at the

point

in

time

t. Our observations extend over the

period

t=l,

2,**, N.

We

have;

(3)

Xit

-Mit +

Yit)

i

=

1,

2, ..,

p;

t-=1, 2, ..,

N.

We

have

by

definition

Mt

=

EXi.

Mit

is the

mathematical

expecta-

tion

or systematic

part and

yit

the

random term or the

disturbance

(Frisch). These disturbances

result from

lack of

representation, fric-

tions,

and

errors

of measurement as

indicated

above. The last term

may be absent with

some of

the variables, especially

the one

represent-

ing

time. The

mathematical expectations

are not

random variables. We

will

also assume that the term

wt (t= 1, 2, *

* * , N)

which stands for the

variables

not

represented

in

the

system

is

random, with

Ewt

=0.

It

is

also

possible

that

sometimes some kinds

of frictions

may give

rise

to

this

type

of

stochastic

terms.

Now we

finally

assume the system (2)

linear in

first approximation.

p

(4)

k,o

+

Z

k,jM2t

=

w,t,

v

=

1,

2,

.,

R;

t=1, 2,

,

N.

j=1

We would

like

to mention

here that our purpose

is not

prediction

but

estimation of

the structural

coefficients

k,j

themselves

(Wald).36

Their

importance

for economic policy has

been

stressed by

Haavelmo.37

Our

system is

theoretically a

representation of the complete

Walrasian

system

but the individual

equations are

not yet meaningful in an eco-

nomic

sense. That is to say,

we cannot in

general

interpret the individ-

ual

(normalized

and

orthogonalized)

relationships (4)

as economically

meaningful

functions, e.g.,

demand functions,

supply

functions, etc.

36

A.

Wald, The

Fitting of

Straight

Lines if

Both

Variables

are

Subject

to

Error,

Annals

of

Mathematical

Statistics, Vol.

11,

September,

1940,

pp. 284-300.

37

The

Statistical

Implications of a

System of

Simultaneous

Equations,

cited in note

3.


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MULTIPLE REGRESSION

FOR

SYSTEMS

OF

EQUATIONS

11

This problem, the so-called

question of identification,

will be taken

up later.

A general theory would require the treatment of the

system (4) under

specific assumptions regarding all the various terms entering into it,

i.e., especially the

wit

in

addition to the

Mit

and

yit.

Actually, the sta-

tistical problem of estimation under these very general

conditions

is

as yet not solved. There

are, however, two different

approaches avail-

able:

A.

Assume the

yit

(disturbances)

negligible compared with the

wit.

Then we get the stochastic systems studied by

Haavelmo and his

school.

B. Assume the wit negligible. That is to say, assume that the chosen

variables

Xit

represent the

total economic system

well enough so that

the

disturbances

yit

are

actually responsible for all

or nearly all the

deviations. This is the approach which will be presented

in this paper.

We are now forced to make some additional assumptions,

some of

which are

necessary

in any case and others of which are

made only for the

sake

of convenience.

A

complete

mathematical

treatment

of the prob-

lem of estimation under

somewhat less stringent

conditions is given

elsewhere.38

First we neglect the

term

w,t

in (4) and will only try to

estimate the

k,

in the system:

p

(5)

ka

+

k,ktMit

=

O, v

=

1,2,

..

*

,R;

t

=

ly,2,

*

N.

This assumption is permissible only

if

we feel reasonably

certain that

the

most important

variables

pertinent

to

our

system

have

been

repre-

sented and that the influence of variables and parameters z; which are

not represented by the

Ma

is

negligible.

In

other

words,

the

equations

(5)

would

give

a

perfect

fit without

any

deviations

were

it

not for

the

disturbances discussed

above.

The linear form of the equations,

which has been assumed

as

a

first

approximation, may

also create

difficulties. Sometimes

we

may

assume

our

system

as linear

in

the

logarithms, especially

in

the case of

produc-

tion

functions (Douglas).3

Certainly,

the

assumption

of

linearity

will

be justified in a small region around the equilibrium. More compli-

cated

systems, would

include,

for

instance, squares, cubes, etc.,

or

or-

thogonal polynomials

as variables

in

(5).

Some of the

assumptions given

38

G.

Tintner, A

Note

on Rank,

Multicollinearity

and

Multiple

Regression,

Annals of Mathematical

Statistics,

Vol.

16, September,

1945,

pp.

304-308.

39

P. H. Douglas,

Theory

of Wages, New York,

1934, pp.

113

ff.,

and subsequent

publications.


8/10/2019 1905702

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12

GERHARD

TINTNER

below would

have

to be

slightly

modified

in

consequence,

especially the

one referring to

independence of the

disturbances.

It is clear that the

yit

(disturbances)

are

statistically

independent of

the Mit (systematic parts) and we assume that they are normally dis-

tributed.

The second assumption is

made chiefly for the

sake of

con-

venience. If, however, the

disturbances discussed above are small

and

very

numerous,

and some additional conditions regarding their

higher

moments are

fulfilled, then normality

in large samples follows from

the

Laplace-Liapounoff theorem.40

Investigations into price

dispersions, for

instance,

indicate that actually the deviations from normality

may not

be

very great

if we replace, e.g., a set of prices by a

representative

or

average price

(Mills).41

This may not be true with other data, for in-

stance

incomes where skewness is very

extensive.42 It may be necessary

in

such cases to include specifically a

measure of

skewness,

e.g.,

Pareto's

a

as one of

the variables. As far as deviations caused by errors

of meas-

urement are

concerned, there is no

reason to assume that they are not

normally

distributed. The same seems

to be true of those disturbances

which go back to frictional causes.

We

assume

further, that the

yit

have constant variance43 over the

period

considered. The empirical estimate of this variance is designated

by

Vi.

This

assumption is probably

justified

if

we

consider periods

of

time that are not too long. The

mathematical analysis would

be very

much more

complicated if the variances were changing over

time. We

assume also

that the

yit

are statistically independent of each

other.

This

assumption, which is here made

only for the sake of

convenience,

is probably not strictly justified. The

complete theory of

estimation has

been

given

elsewhere without this

restriction.44

A

very important assumption is the

following: The

systematic parts

of

our

variables

(Mit)

are smooth

functions of time.45 We have not

to make more

specific assumptions

about the nature of these functions.

This excludes stochastic

business-cycle

theories,

like, for

instance,

Frisch's.46 It

is,

on the other hand, compatible with the

theories men-

40

J.

V.

Uspensky, Introduction

to

Mathematical

Probability,

New

York, 1937,

pp. 291

fl.

41

F.

C.

Mills,

The

Behavior

of

Prices,

New

York, 1927, pp. 251 ff.

42

H. T. Davis, The

Theoryof

Econometrics,pp. 26

ff.

43

G. Tintner,

The

Variate Difference

Method,

Bloomington, Indiana, 1940,

pp. 165

ff.

44

G. Tintner, A

Note on Rank,

Multicollinearity and

Multiply

Regression,

cited

in

footnote 38.

45

G.

Tintner,

The

Variate

Difference

Method, pp. 31 ff. See

also the

review

by

Tjalling

Koopmans, Review of

Economic

Statistics, Vol. 26,

May,

1944, pp.

105-107.

4

R.

Frisch,

Propagation Problems and

Impulse

Problems in

Dynamic Eco-

nomics,

Economic Essays

in Honor

of Gustav Cassel,

London,

1933, pp. 171-205.


8/10/2019 1905702

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MULTIPLE REGRESSION

FOR

SYSTEMS

OF

EQUATIONS 13

tioned above and also with the nonmathematical

theories of the

cycle.

There is one feature in nonstatic systems that may occasionally bring

about abrupt and sudden changes which may appear as discontinuities.

This is the kink in the (real or imagined) demand curve of the firm

under oligopoly. It implies a discontinuity

in the

marginal-revenue

curve. This was recently pointed out by Lange47

on

the basis of the

earlier

work

of P.

Sweezy.48 It

does not

seem, however,

that

these dis-

continuities are actually very large and that they are widespread in the

economic

system. Continuity

and

momentum seem to us

outstanding

characteristics of our economy.

The recent book by von Neumann and Morgenstern49also stresses

discontinuities and indeterminacies arising in our economy from oligop-

olistic and

similar situations.

These are not unlike

phenomena ob-

served in games of strategy. But the actual

determination

of prices,

etc., under such conditions of indeterminacy may depend upon non-

economic factors like bargaining power, politics,

etc.

as in

the much

discussed

problem

of bilateral monopoly.50 Hence it does not seem to

us that

under normal conditions important

discontinuities must

neces-

sarily arise, as long as the total social situation

is

reasonably stable.

It

is important to note that the stochastic

features of our

system

are

introduced only through the disturbances

yit.

Economically

this is

equivalent to the following assertions:

If we knew

all

our

data

perfectly

and if

the system (1) was frictionless

and

represented perfectly

ra-

tional behavior then the

zi

would appear as smooth (and predictable)

functions

of time. This

is not

unlike

the

famous assertion of

Laplace

in classical mechanics.

Stochastic

processes

would

appear only

if

assumption

A

above

is

adopted or if we attempted to determine the complete system (4).

This

would

in

some measure correspond to

Frisch's

business-cycle theory,52

the

problem considered by Slutsky,53

the

analysis

of

Hurwicz,54

etc.

40

O.

Lange, Price Flexibility

and Employment,

pp. 40 ff.

48 P.

M. Sweezy, Demand

under Conditions of Oligopoly,

Journal

of

Politi-

cal Economy, Vol. 47, August,

1939, pp.

568-573.

49

J. von Neumann and 0. Morgenstern, Theory of

Games

and Economic Be-

havior, Princeton,

1944, pp.

43 if.

50

See, e.g., G. Tintner,

Note on

the Problem of

Bilateral

Monopoly,

Journal of Political Economy,Vol. 47, April, 1939, pp. 263-270, and the literature

quoted there,

esp. p. 270.

51

P.

S.

de Laplace,

Essai

philosophique

sur

les probabilit6s,

4th

ed.,

Paris,

1819,

p.

4.

52

R.

Frisch, Propagation

Problems

and

Impulse

Problems ...

,

cited in

note 46.

53

E. Slutzky,

The Summation

of Random Causes

as the Source

of Cyclical

Processes,

ECONOMETRICA,

Vol. 5, April,

1937, pp. 105-146.

54

L.

Hurwicz, Stochastic

Models of Economic

Fluctuations,

ECONOMETRICA,

Vol. 12, April, 1944, pp.

114-124.


8/10/2019 1905702

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14

GERHARD

TINTNER

3. THE

STATISTICAL

METHOD

A.

Estimation

We

give

now

a

short

description of our method. It

deals

essentially

with a case discussed by Frisch55and is in a sense an extension of the

work of

Koopmans,56

who considered

the

existence of only one

relation-

ship

existing between the

Mit.

This

corresponds

to the case R =1.

The

methods presented

here are

also similar

to the ones given

by the

author

in

an earlier

article.57

They are

definitely

connected with modern

work

in

multivariate

analysis.58

Under the

assumptions

stated above we

endeavor to

estimate

the

coefficients

k,

in

equation

(5). In order

to do this we have first

to

estimate R itself, i.e., the number of independent linear relationships

actually

existing

among our p variables

Mit

in

the population.

The

possibility of

estimating the

true

dimensionality

of our prob-

lem in

statistical terms (a

problem

already

envisaged by Frisch59

and

contemplated by Fisher60 and

others

in

discriminant

analysis )

dis-

tinguishes our approach

from the

one of Haavelmo

and

his

followers.

These

simply assume

R, the number of

independent linear

relationships

between the

variables

in

the

population

as

given.

Hence

it

seems

pos-

sible that they may endeavor to accomplish too much, i.e., to deter-

mine

a

greater number of

equations

than

actually

exist

in

the data.

This

would

by necessity lead to

nonsensical

results.

The reason

for

this

distinction between their

method

and

our

procedures

lies

in

the different

role of economic

theory and

the

reliance put into a priori assumed

eco-

nomic

relationships,

which seem to be

greater

with

Haavelmo

and

his

school

than with

us.

a.

Estimation

of

the

Variances

of

the Disturbances. In

order

to

esti-

mate R we have first to estimate the variances of the disturbances yit.

Since

we

have assumed above that the

Mit

are

smooth

functions of

time

we

can

use

the variate difference method for

this

purpose.

This

seems

to

us

preferable because we do not

have

to assume

that the

sys-

tematic

parts

of

our

variables

are

specific functions

of

time, e.g., poly-

55

R.

Frisch, Statistical Confluence

Analysis.

56

T. Koopmans, Linear RegressionAnalysis in Economic Time Series, Haarlem,

1937.

57G. Tintner, An Application of the Variate Difference Method to Multiple

Regression, ECONOMETRICA,

Vol.

12,

April, 1944, pp.

97-113.

58

H. Hotelling, Relations between

Two Sets of Variates, Biometrika, Vol.

28,

December, 1936, pp. 321-377.

69

Statistical Confluence Analysis

.

. .,

cited

in

note

34. See also,

R.

Stone.

T'he

Analysis of Market Demand, London, National Institute of Economic and

Social

Research, 1945.

60

R. A.

Fisher,

The

Statistical

Utilization of Multiple Measurements,

An-

nals of Eugenics, Vol. 8, 1938, pp. 376

ff.


8/10/2019 1905702

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MULTIPLE REGRESSION FOR SYSTEMS

OF EQUATIONS 15

nomials. It will be

shown below that our procedures do not necessarily

depend upon the use

of this method.

It has been shown

elsewhere6' that we can proceed as follows:

We

form the series of first, second differences, etc. of our original data Xit.

Then we compute

the variances of these difference series. The

Mit

are

by assumption smooth functions of time.

It follows that the

portion

of the variance of the difference series of

Xit

which is due to this sys-

tematic part will become smaller and smaller

the higher the

order of

the difference. Eventually we shall be left substantially with

the vari-

ance of the random terms

yit

alone.

If

this

is true in the

koth

difference

series,

it follows

that, apart from sampling

fluctuations,

the variances

of the (ko+l)th difference series, of the (ko+2)th difference series, etc.

must

be the same as the variance of the

koth

difference series

if

they

are all

correctly

computed. Test functions have been given elsewhere62

which serve to

form an

idea

about

the order of the difference series

in

which this is the case, i.e., in which we probably

are left with

the vari-

ance of

the

random element alone. Then

we

may take the

variance of

the

koth

difference series of

Xit

as an estimate of the population

vari-

ance of the corresponding random element.

It will be denoted

by

Vi.

But our procedures are by no means dependent upon the use of the

variate difference

method. There are other

ways of getting an estimate

of

the variances of the random elements

in our variables

Xit.

For in-

stance

we

may

compute orthogonal polynomials, Fourier series,

etc. to

approximate

the course of the systematic element

Mit

over time.

Then

we take the residual variance computed from

the deviations from

these

trends

as

estimates of the variances of the

random elements

or dis-

turbances

in

the

population.

Sometimes we may even be able to assume the Vi a priori. This will

especially

be the

case

if we

can interpret the disturbances

yit

substan-

tially as the results

of errors of measurement or sampling.

If modern

sampling

methods

are more

frequently

applied

in

economic

data63 we

may

often

get

an

idea of the

magnitude

of

the

sampling

error.

But

it

is not impossible to imagine that we know

in some cases

a

priori

the

relative accuracy

of our data. It is for instance

a well-known

fact

that

price data are frequently

much more reliable than

data

on quantities

consumed, national income, etc.

f,.

Estimation

of

the

Number of

Relationships. Having

found

in one

way

or another estimates of the variances

of the disturbances

yit

in

the

population

we

can now utilize

a

very ingenious

method

due

to

61

G. Tintner, The Variate

Difference

Method.

12

Ibid.,

pp. 67

ff.,

73

ff.

63

See, e.g.,

A.

J.

King

and R.

J.

Jessen,

The Master

Sample

of

Agriculture,

Journal of

the

American Statistical

Association,

Vol.

40,

March, 1945, pp.

38-56.


8/10/2019 1905702

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16

GERHARD

TINTNER

P. L. Hsu64

n order to estimate R.

This is

equivalent

to

forming

an

idea

about

the rank of the

variance-covariance

matrix

of the

Mit

in the

population. For this

and the

following

method

of estimation of the

k,1

we have

to assume

that we

are

dealing

with

large samples.

Then

our

estimates

Vi

are accurate

enough

to

enable

us to treat

them as con-

stants.

They

are assumed to be

equal

to the

corresponding

population

values. This

assumption

is

necessary

for

our

method

and

simplifies

the

procedures greatly.

It

will, however,

only rarely be strictly

fulfilled

in

practice

and

certain

errors

may appear

in

our

estimates because of this

assumption. But

these errors are

essentially

errors of weighting

and

not likely

to

be always

very important

as

Koopmans has shown.65

Hsu's method leads to the following procedure:

Consider the determinantal

equation:

al

-

XV1

a12

alp

(6)

al2

a22

-XV2

a2p

.

.

.

.

. . . .

. .

. . .

.

. .

. =

O,

alp

a2

* *

app

-

XVp

where ai is the sample covariance of Xit and

Xit.

That is to say,

ai1=(Z:'=xitxit)/(N-1)

where

xit

is

the

deviation of

Xit

from its

arithmetic

mean

Xi,

etc. The

Vi

are

our

estimates

of the

random vari-

ances.

They

may

of course be

absent

in

the

case of certain

variables

that have

no random term,

especially the

variable representing time

itself.

In

the population

equation

corresponding to

(6) there should be r

roots

X

=

1,

if there are

actually r

independent linear

relationships be-

tween the Mit in the population.

Let X1be the

smallest root of

equation (6), X2be

the next

smallest,

etc.

These

roots are

best

computed

by Hotelling's66

methods. We form

the test

function:

(7)

Ar

=

(N- 1)(Xl

+

X2

+

+

Xr)

i.e.,

the

sum

of

the

r smallest

roots of

(6)

multiplified by (N

-

1). This

quantity

(7)

is distributed like x2

with

(N-I-p

+r)

r

degrees of free-

dom. It may be used to estimate R, i.e., the true number of independent

relationships of the

type (5) which actually exist

in the

population.

The

determinantal

equation (6) is fundamental

for our

procedures.

64

P.

L. Hsu,

On the

Problem

of

Rank and

the

Limiting

Distribution of

Fisher's Test

Function, Annals

of

Eugenics, Vol. 11,

1941, pp.

39 fif.

66

T.

Koopmans, Linear

Regression

Analysis in

Economic

Time

Series,

pp.

87 ff.

6B

H.

Hotelling,

Simplified

Calculation of

Principal

Components,

Psycho-

metrika,

Vol.

1,

March, 1936,

pp. 27-35.


8/10/2019 1905702

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MULTIPLE REGRESSION FOR SYSTEMS

OF EQUATIONS

17

The number

of degrees of freedom in the distribution of

A,.

will be

inter-

preted later.

The following

hypothesis will be tested:

There are exactly

r inde-

pendent

linear relations in the population between

the

Mit.

Given a

level of significance, say 1

per cent, we will

reject the hypothesis

for a

A,

which is

larger than the

one permitted for the given

level of sig-

nificance taken

from the tables

of

X2

with the appropriate number of

degrees of freedom. Hence

we will compute

the quantity

(7)

for

r=

1,

2, 3,

*

*

-

until we find a

A,.

which

is larger than the

one permis-

sible under

the conditions

given above. The r corresponding

to this

A,

is to be taken

as an estimate

of the true number of

relationships

existing in the population between the systematic parts of our variables.

This estimate

will be denoted

by R.

ay.

Estimation

of the Weighted Regression

Coefficients.

The method

of

maximum

likelihood for the

estimation of

the

kIq

eads under our as-

sumptions

to the method of least squares.

Our estimates

can

then

be

regarded as

the best linear estimates in large

samples and

are unbiased

under the conditions stated.

This follows

from the above assumption

that

the

Vi

can be treated

as constants. Hence the

Markoff

theorem

applies.67The sum of squares to be minimized is:

N

p

(8)

Q =

E E

(xt

-M,t)2/V,

t=1

i=1

where

mit

is the deviation

of

Mit

from its

arithmetic mean. We

have

to

minimize

Q

under the R conditions (5).

These

conditions

involve actually Rp+R

coefficients

k,,

(v=1,

2,

* *

,

R;

j=O, 1, 2, * *

,

N) which have to be estimated.

The

con-

stants k,o are evidently given by the condition that the best fit has to

pass through

the means of

all the variables :68

p

(9) kao

+

k,ikXi

=

0,

v

=

ly

2,

..

I

R.

This still leaves

Rp

coefficients

k,,1

v= 1, 2, * * * R; j=

1, 2, * * *

,

p).

But it is

evident that R2 of these coefficients

are not really

necessary.

Suppose

we

suppress in (5)

the constants

k,,o

and substitute

the devia-

tions from the means

mit

for the variables Mit. Then we can, for in-

stance, express

the first p-R

variables

mit,

m2t,

.

.

.

X

mp-R

tin

terms

of the

last

R variables

mpR+l,

t,

mpR+2,

t,

.

.

.,

mpt.

Hence

we

need

only (p-R)R

independent

coefficients

k,,.

67

F. N. David and

J. Neyman, Extension of the Markoff Theorem

on

Least

Squares, Statistical

Research Memoirs,

Vol. 2, 1938, pp.

105

ff.

68

G.

Tintner,

An Application

of

the Variate Difference

Method

to

Multiple

Regression, op. cit.,

p. 105, formula

(20).


8/10/2019 1905702

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18

GERHARD

TINTNER

The

f2

conditions

imposed on the pR

coefficients

kq1

v=

1, 2, *.**,

R;

j=1, 2,

,

p) are

as follows:

p

(10) k ikwiVi =

vw,,

v, w = 1, 2, *

*,

R

i=l1

where

&vw

s

the Kronecker

delta.

It is one

for

v=

w

and zero for v

w.

(11)

E3 E

ktjx1)

(

E

kwix1t)

= O,

t=1

i==l

j=1

v

p

w;

v, w

=

1, 2,

*

,

f.

These conditions (10) and (11) normalize and orthogonalize the co-

efficients klcv v-1,

2,

.

* *,

R; j= 1,

2,

. .

*, p).

The procedure is

then

essentially the

same as

weighted regression

with

just

one

equation.69

We determine

first the

most

advantageous

values

of the

mit

(i=

1,

2, *

* *, p;

t= 1, 2, *

,

N). It can

be demon-

strated that

under

the given conditions

this

leads to a sum

of squares

in

the

form:

N

R

P

(12)

9=

E

kvj

tx)2.

t=l v=l

j=l

This

expression is very

similar

to the

one appearing

in the

earlier analy-

sis

with

just

one relation.

The complete

solution

in mathematical

form is given elsewhere.70

It

can

be shown that

the

minimization

of the expression

(12)

under

con-

ditions (10)

and (11)

leads again to

the determinantal

equation

(6).

The system

of

linear homogeneous

equations

that yields

the

solu-

tions is as follows:

(all

-

XvV,)kv,

+

al2kv2

+

+

aipkvp

=

0,

(13)

al2kv1

+

(a22

-

XvV2)kv2

+

*

+

a2,kvp

=

0,

. .

.

.

.

. .

.

.

.

.

. . .

.

. . . . . . .

.

.

aJpkvc a2pkv2

*

+

(app

-

XvVp)kvp

=

0,

v

l=

21,

,

R.

This system of equations can evidently only have a nontrivial solu-

tion if its determinant is

zero. [A

trivial solution

is excluded

by condi-

tion

(10).]

This is

achieved by inserting

for Xvone of

the roots

of the

determinantal equations

(6).

Then

the computation

of the

k,j

is possi-

ble,

if

one additional

condition

is imposed,

e.g., condition

(10) for

v

=w,

69

T. Koopmans, Linear

Regression

Analysis.

.

.

, cited in note

56.

70

G.

Tintner,

A Note on Rank, Multicollinearity

and Multiple

Regression.


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MULTIPLE

REGRESSION FOR

SYSTEMS OF EQUATIONS 19

which normalizes

the coefficients

k,j

by making their weighted

sum of

squares equal to

one.

By carrying out

this process for v= 1, 2,

,

R we get

a complete

set of estimates for the coefficients

k,i

in equations (5). The minimized

sum of squares

in

(8)

and

(12) becomes

AR(N- 1).

We can now

interpret the number of degrees of freedom

for the dis-

tribution of the sum of squares

AR.

For

each

v

we have to

minimize N

sums of

squaresDf=

1(xit-mit)2/Vi, which appear in formula

(8). They

correspond to

the deviations of the

dependent variables from the

fitted

regression

equation in ordinary regression theory. The

total num-

ber of

sums

is RN since v=

1, 2,

* *

*,

R. Each sum corresponds to

one

of the roots of equation (6), X1,

X2, * *

*, XR.

There are p-R+1 inde-

pendent constants

k,j

(j=0,

1,

2,

* * *, p) to be determined

for

each

v.

Their total number

for all v is

R(p-R)+R.

The difference

between

the number of sums of squares

and the number of

constants

is

(N-1-p+R)R as given above, the

number of degrees

of freedom

for

AR.

The

procedure

is

analogous

to the one ordinarily given

in mul-

tiple regression.

In fact, for

R

=

1

the expression becomes

N

-

p,

the

correct number

of degrees of freedom

in ordinary regression

analysis

with p variables, i.e., one dependent and p -1 independent variables.

The difference

between our new approach

and the classical method

of

fitting weighted

regression equations given by Koopmans

is

the fact

that

we determine

here all the

R

smallest

roots of the determinantal

equation (6)

instead of only the smallest. This enables us to

estimate

R

sets of

coefficients

of

the type (5) instead

of just one.

B.

Identification

We have now estimated the coefficients kq in (5) for v= 1, 2, * *

*,

R;

j=0,

1, 2,

. . .

,

p

on

the

basis

of our estimate of

R.

This is

equivalent

to

establishing

R weighted regression equations

whose

coefficients

are

orthogonalized

and

normalized

by the

conditions (10)

and

(11).

These

results represent

the system (1), i.e.,

the

Walrasian

system of general

equilibrium

with

the modifications

mentioned above.

But these equations are not yet economically

meaningful,

if

consid-

ered each

in isolation. We are for instance

not able to say

which ones

are demand equations, which one is the investment function, supply

functions,

etc.7

In

order to achieve

a meaningful system

of

equations

we

have

to make additional assumptions.

Following

a suggestion of Klein72we put these assumptions

into the

71

T. Haavelmo, The Probability

Approach in Econometrics, pp.

91 ff.,

pp.

99 ff.

72

L. R. Klein, Pitfalls in the

Statistical Determination of the Investment

Schedule.


8/10/2019 1905702

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20

GERHARD

TlNTNER

following

form: Certain

of the coefficients

kvj

(v

=l,

2,

R;

j =1,

2,

.

*,

p)

will

be

assumed

a

priori

to be

equal

to

zero and

hence

missing

from

some of

tlhe equations

(5).

A

perfect

estimation

of

eco-

nomically meaningful relationships will be possible if there is in each

equation of the

system (5)

at least

one variable

Mit

that

enters

into

this equation

alone.

Hence all the

coefficients

k,j

belonging

to

this

variable

in all other

equations

of the

system (5)

are

equal

to

zero.

We

could

say

that

the individual

economically

meaningful equations

(e.g.,

demand

functions,

supply

functions,

etc.)

are

generated

by

the

varia-

bles

MIit

that

enter

into them alone and

into

none of the

other

equa-

tions.

To give an example: We will for instance assume that certain specific

demand factors

like

income

appear

in

the

demand

functions

alone and

that cost factors

like the

prices

of

factors of

production

appear

solely

in the

supply

equations,

etc.

It

is

apparent

that

there

is a

certain

amount

of

liberty

possible

in the

choice of

the

specific variables

that

should enter into

given

economically

meaningful

equations,

like

for

in-

stance demand

functions. Statistical criteria

for

this

choice

will

be

given

below,

but

they

must not

be

taken as decisive.

We

may very

well

decide

in favor of a system that is economically meaningful and gives a

rather

poor

fit

against

one that

gives

a

more

perfect

statistical

fit

but is

difficult to

interpret

economically.

A

certain

amount of

arbitrariness enters also

into

the

selection

of

the

specific variables

that should be included

in

a

given

economically mean-

ingful equation.

If

it is a demand

function,

we

may

for

instance

take

average

income,

the

distribution

of

incomes,

taste

factors,

advertising

cost,

etc.

as

variables

that will

enter

into

this

specific

equation

apart

from the quantities bought and prices paid for this good (and possibly

also prices of

related

goods).

The demand

functions

will

differ

according

to which

particular

variables

we

include into our

equation. This

is

a

consequence

of the

fact that

we

make a

somewhat

arbitrary

selection

of

variables and

parameters

out of the

comprehensive Walrasian

sys-

tem

(1).

An

error

will

be

committed

by

neglecting

certain

variables.

We

can

perhaps

say

that

we

actually replace

the

general

Walrasian

system

by

a more restricted

system

of

a

kind

of

partial

equilibrium.

Statistical criteria

will

be

given

below

that should

enable

us

to

ap-

preciate

the

statistical

goodness

of fit

resulting

from

the

inclusion

of

certain

variables

in a

specific

equation.

A

comparison

between

different

possible

selections

is

also

feasible.

But,

as indicated

above,

such

purely

statistical

considerations

are

by

no

means decisive.

Some

of

the

initial

advantages

of

our

approach

may

be lost

or

partially

lost

in

this

fashion.

The problem

of

identification

is

important

especially

for

economic

policy.

Assume for

instance a

policy

of

government investments.

Then


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MULTIPLE

REGRESSION

FOR

SYSTEMS

OF

EQUATIONS

one of the

equations

in

(5)

which

represents

the

investment

function

will

be

replaced

by.the

new

conditions of

government

investments,

while all the

other

equations

remain

the

same.

But

without

having

solved the

problem

of identification

we cannot

say

which one of the R

normalized

and

orthogonalized

equations

(5) (or

which

of the

infinitely

many

combinations

of them that

are

equivalent)

is

actually

the

in-

vestment

function. The

system

of

equations

given

under the

previous

conditions is

useless for

purposes

of

economic

policy.

Hence,

in

order to

get

meaningful

equations

that are

useful

for eco-

nomic

policy

(and

also

more

enlightening

for

theory)

we have to

impose

the

conditions

mentioned above

and

carry

out

our

estimation under

the

new conditions

apart

from the ones

imposed

by

(10)

and

(11).

We will

later

give

statistical

methods

to

estimate

the loss

in

accuracy

suffered

in

the

process

of

obtaining

meaningful equations.

The

estimation of

the

new

coefficients

kv/

determined under the new

conditions

that

make the

equations

meaningful

is

actually

simpler

than

the

estimation

of

the

system given

before.

A number of

new conditions

is

added

to

the

orthogonalizing

and

normalizing

conditions

(10)

and

(11).

They

express

which of the coefficients

kji

are

assumed

to be zero

in any specific equation in (5).

Assume

that

s,

conditions

are

imposed

for

equation

v.

Then

p/=

p--sv

is the

number of

coefficients

kv,/

(j=1,

2,

* *

*,

pv')

to be

determined.

The

method of

estimation

is

essentially

the same as before. We

put

equal

to

zero

in

(13)

the

sv

coefficients

ckv

hat

are assumed

to

be

equal

to

zero

for the

purpose

of

identification

and

get

the new

system

(14).

Here

the

numbering

of

the

variables

has been

changed

and does not

necessarily correspond to the one previously used, e.g., in (13).

(an

-

Xv'Vi)kvl'

+

al2kv2 +

+-

aklpvkp,

v

=

0,

(14)

al2kv'

+

(a22

-

Xv'V2)kv2'

+

+.

-

a2p',kvpv,'

=

0,

(14)

alpv'kul'

+

a2pv,k2

+

+'.

+

(apvpv

-

Xv')kvpv,'

=

0,

v-1, 2, ,

R.

This is

again

a

system

of

linear

homogeneous

equations

in

the varia-

bles

kv'.

A

nontrivial

solution

is

possible only

if the

determinant

is

equal

to

zero. In

order to

accomplish

this

we have

to

modify

the de-

terminantal

equation

(6)

in such

a

fashion,

that the

rows and columns

corresponding

to

the

sv

variables that

do not

now

appear

in the

system

(14)

are left out.

Denote the smallest

latent

root of the

modified

equa-

tion

(6) by

Xv'.

Then

we

insert

this

solution

Xv'

nto

(14)

and assume

21


8/10/2019 1905702

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22

GERHARD

TINTNER

again

a

rule of

normalization (for instance

making

the

sum of

the

squares

of the

k,,'

equal to one). We get estimates for

the

k,,'

which

are valid

under our new

conditions. We repeat this process for

v= 1, 2, * - *, R with the appropriate conditions satisfied in each case.

Then

we

achieve now the solution of the

problem of estimation in such

a

fashion that each

individual equation is

identified, i.e., has

a

specific

economic meaning. Our

complete system

contains now all the original

variables

and

all the

identifying variables. This economic

advantage

may

stand against a

statistical loss because the sum of squares is

now

in general larger than

before.

The test

indicated above

under A,

a

should be used for each set

of

variables utilized for a given identified equation in order to make sure

that

there is

exactly

one

relationship between these

variables.

C.

Tests

of

Significance

We will

now give tests of

significance that should enable us to

get

an

idea about the goodness

of the fit achieved

and to estimate the

loss

of

statistical

accuracy suffered by imposing new

conditions for

the

pur-

pose

of

identification.

Let

81+82+ -

*

-

+SR=

S'

be the total number of conditions imposed

in

the

process of identification. Define a

quantity analogous to (7):

(15)

AR'

=

(N

-

1) (X1' + X2 +

- -

* + XR

)

which is now

the sum of the R smallest roots

of the modified determi-

nantal

equations (6) multiplified by N-1.

Each one of the roots

Xv'

is

computed by leaving out

in (6) the

appropriate

s,

rows and columns.

The

quantity (15) is

distributed like

x2

with

(N-1

-p+R)R+S'

de-

grees of freedom.

The

following test of significance is

analogous to a procedure given

by

Wilks.73

We form

F

=

(N-1-P

+R)R

(AR'-AR)/S'AR.

This is

ap-

proximately distributed like

Snedecor's F with

(N-1

-p+R)R

and

S'

degrees

of

freedom. We

fix

a level

of

significance and test the

following

hypothesis:

That the

difference between ARand

AR'

is zero, i.e.,

that

the

introduction of

the

S'

new

conditions

necessary

for identification

has not

materially deteriorated our fit.

There is no equivalent test for the individual equations. The reason

for

this is that in the original

system (5) the

equations are

not identified

and

we

hence do

not

know

which one corresponds to a given meaningful

equation

of

our

new

set-up.

Hence

we

must

always

determine a com-

plete

system before we can make a test of

significance. There is,

however,

a

method

by which

we

can compare

identified systems that involve

different sets of

conditions.

73S. S. Wilks,

Mathematical

Statistics, Princeton, 1943, pp. 166 ff.


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MULTIPLE

REGRESSION

FOR

SYSTEMS

OF

EQUATIONS

Assume the

number of

conditions

again

S'

in

the first

system

and

S

in

the

second. Then

we

form

a

quantity

AR

corresponding

to

(15).

The

quan-

tity

F=[(N-1-p+R)R+S]AR'/[(N-1-p+R)R+S']AR

is

again

approximately distributed like Snedecor's F with (N-1-p+R)IR+S'

and

(N-1-p+R)R+S

degrees

of freedom. The

hypothesis

to be

tested

is

again:

That

there

is

no essential difference

in

the

sum

of

squares

of

residuals

divided

by

the

appropriate

number

of

degrees

of

freedom

resulting

from

the

two

sets of

conditions

(or

in

the

two

sys-

tems).

It

is

remarkable and

entirely

in the

spirit

of

our

approach

to the

problem

that it

is

not

possible

to test

individually

the

difference

be-

tween the k,j and ki'. It shows again that it is really not individual

equations

that are

determined in

our case

but that the entire

system

is

estimated

as a

whole.

Tests of

significance

for

the

individual

kv'

can be

given

that

are

approximations

and

based

upon

an idea

indicated

by Koopmans.74

In

order

to do

this

we

have to

assume

a

different normalization rule

from

the

one

given

above.

This new

normalization

rule is

also

more

appropriate

and

useful

in

economics. It consists in making one of the variables Mit, say Mlt,

the

dependent

variable.

E.g.,

in

the

theory

of demand

we

frequently

consider

the

quantity

demanded

as

the

dependent

variable

and the

price,

income,

etc.

as

independent

variables.

It

should be

stressed,

however,

that

now this distinction is

made

only

for

the

purpose

of

normalization

and

that

it

has

definitely

nothing

to

do

with

the

distinction

between

dependent

and

independent

variables

made

in

classical

regression analysis.

There the

assumption

is

that

the

dependent

variable alone

is

subject

to disturbances

while

the

fit

is car-

red

through

for

purposes

of

prediction

assuming

fixed values

for

the

so-called

independent

variables.75

Here

our

purpose

is the

estimation

of

the

structural

coefficients

themselves

and

not

prediction.

Dis-

turbances

enter

into all

our

variables,

not

only

into

Xit

which now

as-

sumes

the

role

of

dependent

variable for

purpose

of

normalization

only.

Let

our

new

normalized

solutions

be

denoted

by kc,i

v

=

1,

2,

* *

, R;

j=2,

3,

*

,

pv').

The

system

of

equations

becomes

now:

(a22

Xv'V2)kv2 t+a23kv3 +

.

+a2pv p

=

a12,

(16)

a23k,2 +-

(a33-X'V3)kC3 -+

+a3pk ,p.'t

=a13,

(16)

a2pvkv2/

a3pkv3+

*

*

+(apvp'v

-Xv'Fpv)kpv,

=alpv.

74

T.

Koopmans,

Linear

Regression Analysis

in Economic

Time

Series,

pp.

72 ff.

75

H.

Hotelling,

The

Selection of

Variates

for Use in Prediction with Some

23


8/10/2019 1905702

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24

GERHARD

TINTNER

This

system

of linear

homogeneous equations will

again

have

a non-

trivial

solution

if we

substitute for

Xv' a solution for

the

modified de-

terminantal

equation (6). The

modification consists in

leaving

out

the

s,

rows and

columns

corresponding

to the

variables that

by

assumption

do not enter

into the

specific

vth

equation.

Using an

approximation

given

by Koopmans76

we can

now compute

the

variances

and

covariances of

the

kq '

under

the

assumption that

the

variances of the

systematic parts are

much

larger

than the ones

of

the

disturbances.

The

covariance of

ku i

and

kj'

is then

given

by the

formula:

(17)

Ekcvi kvi

Xv/Cij(

E

kh Vh

+

V1)

(N

-

pv').

Here

c

is the

element

corresponding to a

i

(i,

j-2, 3,

,

Pv') in

the

matrix inverse to the

one

used in (16) to

compute

the

k,j'.

The method

of

computation

of this matrix

is

the same

as the one given

by

Fisher77

in ordinary

classical

regression

analysis;

pv'

is the

number of

variables

actually

entering

into the

particular

equation

v.

We obtain

the

variance of

ki'

by putting

in formula

(17) i=j

and

the standard error by taking the square root. We can then use the

t-test78

to

establish

approximately

the

significance of

the

individual

kv,

'. The

quantity t'

=

kv'

/VEk,

il2

will

follow the

t-distribution with

N

-

Pv'

degrees of

freedom for

large samples.

It

should

be mentioned

that

Anderson

and Girshick

established

re-

cently

the

distributions of

the

variances and

covariances

for some spe-

cial

cases.79

These tests

of

significance

should help

somewhat

in

choosing

out of

the many possible ones a specific identifiable system that is economi-

cally

sound and

that also

has statistical

significance.

It

cannot

be

denied, however,

that the

solutions

reached are in

some degree arbi-

trary

and rest rather

heavily upon

economic

assumptions, the validity

of

which

cannot be expressed

in

terms of

probability. There does

not

seem

to be any way

in

which this

fundamental

difficulty can be

over-

Comments

on

the General

Problem of

Nuisance

Parameters,

Annals of

Mathe-

matical


September,

1940, pp.

271-283.

7 T.

Koopmans,

Linear

Regression Analysis in

Economic

Time Series, p.

80,

formula

(47).

77

R. A.

Fisher,

Statistical Methods for

Research

Workers,8th ed. New York,,

1941,

pp. 150

ff.

78

T.

Koopmans, Linear

Regression

Analysis in Economic

Time

Series, p.

95,

formula

(4).

79

T.

W.

Anderson and M. A.

Girshick,

Some Extensions

of the

Wishart

Dis-

tribution,

Annals of

Mathematical


December,

1944, pp. 345-

357.


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MULTIPLE REGRESSION

FOR SYSTEMS OF

EQUATIONS

25

come except by the

accumulation

of econometric studies which combine

sound economic theory with statistical

verification

and

may eventually

give

economics the same firm theoretical and empirically

verified basis

that, in physics, is the envy of the social scientist. It is to be hoped

that the

above methods may make a

modest contribution

towards

the

achieving

of

this goal, or at least

point

the

way

to some

more ade-

quate approach.

4.

AN

APPLICATION TO AGRICULTURAL

DATA

A. The

Data

We

will

now apply our method

to

an

endeavor to fit

a demand and a

supply curve for agricultural products in the United States. Our data

cover the

period 1920-1943 (N

=

24). The data are

annual

figures.

They are

the following:80 X1,

Prices received by farmers, 1910-14

=

100;

X2, National income (billion $);

X3, agricultural

production,

1935-39

=

100;

X4, time, origin between

1931 and 1932;

X5,

prices paid by

farmers, 1910-14=100. The means of the variables are

presented

in

Table 1.

TABLE

1

ARITHMETIC MEANS (X,)

Symbol

Variable

Mean

Xi

Prices

received by

farmers

127.417

X2

National

income

72.3975

X3

Production

100.625

X4

Time

0.000

X5

Prices

paid

by

farmers 136.460

The variance-covariance matrix of the data is given in Table 2. Since

all

matrices

are

symmetrical,

only

the elements above the

diagonal

will

be

presented.

TABLE 2

VARIANCE-COVARIANCE MATRIX

(aii)

Xi X2

X3 X4

X5

X1

1212.593

536.182

87.594

-72.696 532.000

X2

557.636

207.147 75.022

220.043

X3

106.418

54.543 30.261

X4

50.000 -38.543

X5

240.652

B. Estimation

of

the Variances

of the Disturbances

We use

the

variate difference method in order

to

get

estimates

of

80

The author is

greatly obliged

to

Dr.

James

Cavin (Bureau

of

Agricultural

Economics, Washington,

D.

C.)

for

supplying

the data and

suggesting

the

prob-

lem.


8/10/2019 1905702

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26

GERHARD TINTNER

the error variances.

We present

in Table 3 the appropriately

computed

variances of

several difference

series for our data:

TABLE

3

VARIANCES

OF DIFFERENCE

SERIES

Difference

Xi

X2 X3

X5

1

337.885

72.346

J4.156

62.863

2

130.649 19.068 10.698

25.221

3 47.901

9.319

7.961 12.472

4

28.580

7.349 7.160

9.812

5 24.349

7.056 6.827

8.987

Tests indicate

that the variances probably

become stable

for the fifth

difference series.

Hence we

will take the items in the last

line of Table 3

as

the estimates

of the error variances

Vi

of our variables.

The variable time (X4) is

evidently not

subject to error and hence we

have V4 =0.

C. Estimation of

the

Number of Independent

Linear Relationships

In order to estimate the number of independent relationships among

our variables

we form

the determinantal equation corresponding

to

formula

(6):

1212.593-24.349X 536.182

87.594

-72.696

532.000

557.636 -7.056X 207.147

75.022 220.043

(18)

106.418 -6.827X

54.543

30.261

=0.

50.000

-38.543

240.652 -8.987X

Since the variable

X4

(time) is not subject to error, we have

V4

=0 and

the term

with

X

is missing from the fourth

line.

The

following table summarizes

the latent roots

(Xr)

and the corre-

sponding

sums of squares

(Ar) resulting

from the determinantal e

Documents

1905702