A commzm smsmvm ANALYSIS. 0F * ' ‘ '
SELECTED ASSUMPTiONS IN AN ANNUAL ‘ , - ' ‘
ammo MODEL 0; THE U. s. ECONOMY: _
THE ELECTRIC mm - GOLDBERGER ‘

Thesis for the ”Degree of Ph. D.
MSCHGAN STATE UNWERSITY
ARTHUR M. HAVENNER
1973

. LIBzz ‘RY i”;
Michigan State
Univer. ity

     
   

This is to certify that the

thesis entitled

A Computerized Sensitivity Analysis of Selected Assumptions
in an Annual Econometric

Model: The Electric Klein-Goldberger
presented by

Arthur M.~ Havenner

has been accepted towards fulfillment
of the requirements for

Eh. .D. Economics

‘ degree in

 

 

22%M

Muor professor

Date February 14, 1973

 

0-7639

 

ABSTRACT
A COMPUTERIZED SENSITIVITY ANALYSIS OF
SELECTED ASSUMPTIONS IN AN ANNUAL

ECONOMETRIC MODEL OF THE U.S.ECONOMY:
THE ELECTRIC KLEIN-GOLDBERGER

By
Arthur M. Havenner

The practicing macroeconometric modeler faces a host of
problems competing for his attention. Among these are the econometric
problems of efficiency, simultaneous equations inconsistency, auto-
regressive disturbances, lag specification, and nonlinearity. Often
correcting any one of these problems precludes correcting others, and
it is therefore important to assess the relative importance of each
of these considerations. This thesis examines these five problems
in the context of a revised Klein-Goldberger model of the U.S.
economy, estimated over an extended data set encompassing the years
1929-41 and 1948-69. The results emphasize the importance of
accurate lag specification and autoregression correction, with

simultaneous equation inConsistency being relatively unimportant.

A COMPUTERIZED SENSITIVITY ANALYSIS
OF SELECTED ASSUMPTIONS IN AN ANNUAL
ECONOMETRIC MODEL OF THE U.S. ECONOMY:

THE ELECTRIC KLEIN-GOLDBERGER

By

.3”
Arthur Mf Havenner

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY

Department of Economics

1973

6 35” ACKNOWLEDGEMENTS

(Saib This thesis is truly the product of many peoples' efforts.
My wife Sandra not only provided moral support, but also spent

long hours in the library gathering data, and even longer hours over
a calculator (the disposable income by shares data is entirely her
work) and in the computing center. Without her efforts, the thesis
could never have been finished.

A debt of gratitude is also owed to the MSU computing
center staff, particularly the I/O room personnel and the computer
operators -- who bent the rules to allow me "hands on" control of
the CDC 3600 six nights a week for four months. Without this
flexibility, the computations could never have been performed.

Also important was the AES STAT group, particularly
Marylyn Donaldson and Bill Ruble, who provided me with computer
time in exchange for a large problem with which to debug their new
programs. (My own department supplies a half hour per student, even
though the CDC 3600 is idle every night.) And the Computer Institute
for Social Science Research provided me with the last hour of
computer time necessary to finish.

My chairman, Jan Kmenta,. both provided encouragement and
was commendably prompt in reading and returning material sent to him;

even while he was in Germany I got better turnaround than others with

committee members not ten miles away.

ii

Jim Pierce of the Board of Governors of the Federal Reserve
System made my life much easier by providing a perfect working
environment and time to write the thesis, after I came to a new job
with only the computations completed. Carmen Feliciano is
responsible for the excellent typing job, including the Greek letters,
while Mary Flaherty wrestled with the many tables that had to be
typed on out-sized paper and then reduced.

And, last, I should certainly mention the two late addi-
tions to my committee, Professors Officer and Rasche, who called

for massive editing changes and made the thesis what it is today.

iii

TABLE OF CONTENTS

I. INTRODUCTION

. The Gains From Full Information Estimation
Equation Simultaneity and Policy Endogeneity
The Returns From Generalized Lags
Nonlinearities

Autoregression

muom>

II. ECONOMETRIC PROBLEMS

A. Three Stage Least Squares

1. Disturbance Properties

2. Parameter Estimates, Nonautoregression Corrected Model
Simultaneous Equation Problems

The Pascal Lags

Nonlinearities

1. Parameter Nonlinearity

2. Variable Nonlinearity

E. Autoregression

COG!

III. CONCLUSION

Theoretical Changes in Equation Specification
Simultaneous Equations Problems

Variable Nonlinearities

Pascal Lags

Autoregression

Data

Three-Stage LeaSt Squares

An Interesting Implication

mommuom>

Appendix I DATA SOURCES
II DISPOSABLE INCOME BY DISTRIBUTIVE SHARES
III A GENERALIZED EXAMPLE OF INITIAL CONDITION PARAMETERIZATION
IV SIMULATIONS

iv

LIST OF TABLES

l. The Model Equations and Symbols

2. Nonautoregression Corrected Parameter Estimates

3. Nonautoregression Corrected Parameter Estimates

4. Impact Multipliers

5. Additional Instruments in the Constrained Augmented
Reduced Form Equations

6. Autoregression Corrected Parameter Estimates

7. Impact Multipliers

8. Autoregression Parameter Search Results

Tables in Appendix IV

IV-l Forecasts

IV-2 Dynamic Multipliers

IV-3 Government Expenditures Multipliers
IV-4 Policy Normalized Multiplier Effects
IV-S Policy Normalized Multiplier Effects
IV-6 Policy Normalized Multiplier Effects

LIST OF FIGURES

Figures in Appendix IV

IV-1
IV-2
IV-3
IV-4
IV-5
IV-6
IV-7
IV-8
IV-9
IV-lO
IV-ll
IV-12
IV-13
IV-14
IV-15
IV-16
IV-17
IV-18
IV-l9

Gross National Product
Consumption

Durable Investment
Residential Investment
Inventory Investment
Imports

Corporate Saving

Profits

Depreciation

Change in the Nominal Wage Rate
Nominal Wage Rate

National Income

Private Wage Bill
Disposable Income

Price Level

Manhours

Capital

Share-Weighted Consumption
Long Term Interest Rate

vi

I. Introduction

The practicing macroeconometric modeler faces a host of problems
competing for his attention. Among these are the choices of model
size, observation time period, and equation specification, as well as
the econometric problems of errors of observation, autoregressive dis-
turbances, efficiency, simultaneous equations inconsistency, lag speci-
fication, and nonlinearity. Often correcting any one of these problems
makes it more difficult or impossible to account for some, or all, of
the others, is therefore important to weigh the relative importance of
these assorted complications.

This thesis examines the importance of five problems in the con-
text of a revised Klein-Goldberger (henceforth K-G) model of the U.S.
economy, estimated over an extended data set encompassing the years
1929-41 and 1948-69. The model equations and symbol definitions are
given in Table 1.

As is apparent by inspection, the equations retain the basic
structure of the K-G model -- particularly the tripartite income
division conjoined in the consumption function -- although every
stochastic equation except corporate profits has been respecified. In
addition, the determination of investment has been disaggregated and
the equations determining household and business liquidity have been
drapped. The extended data set (two-thirds of which is post-war),
gives a more modern version of the classic K-G model.

The five econometric problems examined in the context of the K-G
model deal with (l) the gains from full information estimation proce-
dures, (2) the practical importance of equation simultaneity, (3) the
returns from generalized lag structures, (4) the importance of non-

1

lO.

2

Table 1

The Model Equations and Symbols

Consumption

rl w r +i-l
Ct = B1,0 + 51,1(14‘1) 2 < )

i=0

Investment, plant and equipment

d _ d
I: ’ p2,0 + 32,1": + p2,2(11.> + B2,3I:-1 + U2,:

Investment, residential

r .
I: ‘ p3,0 + 33,1 x: + B3,2(13): + U3,:

Investment, inventories

i i + U

_ i
It _ a4,0 + B4,1 x: +'p4,2(ISTK)t-1 4,t

Imports

(F1): = 35,0 + p5,10%): + a5,2(p15): + p5,3(FI):-1 + U5,:

Production

p ..
X: ‘ a6,0 + a6,1(hm): + B6,2K: + U6,t

Long term interest

r2 m r2+i-1 i
“1.): = B7,0 + 87,1(1'7‘2) .2 < >k2(ls)t-i + U7,:

Short term interest

(is)t = B8,0 + s23,1(ic1): + 58,2 R: + U8,:

Corporate saving
= - - - +
(3:): p9,0 419,1“): Tc)t + B9,2(Pc Tc Sc)t-l U9,:

Corporate profits

(Pc)t = B10.0 + 510.1 P: + U10,:

11.

12.

13.

14.

15.

16.

17.

18.

19.

3

Table 1 (continued)

Depreciation

' a11,0 + a11,1 K: + U11,:

Dt
Agricultural income

(APA)t = p12.0 + fl12,1(YN‘F): + a12,2(Dw): + U12,:

Private wage bill

= p p
(WI): 913,0 + 513,1(X ): + a13,2 X:-1 + U13,:

Nominal wage rate

_ . ‘1
Aw: ' a14,0 + B14,1 (AU): + 514,2 (AP): + U14,:
Identities

Gross national product

d
t

i

r
a: + -
Xt Ct +‘I + It +1t +Gt (FX)t (F1)

t

National income

Y = Xt - Dt - T

t t

Profits

Pt = Yt ' (A1): ' (A2): ' (W1): ' (W2):
Capital

Kt =I‘:+I:+I:-Dt+xt_1

Price level

wt' ht ' (NW):

P" g ("1): + (“2):

4

Table 1 (continued)

Internal definitions

20.
21.

22.

23.
24.
25.
26.
27.
28.
29.
30.

31.

i _ _ 1
xt - xt It
p - -
Xt ' xt (W2):
A
32
(Yo): = (WI): +(“12): ' aw): + (T) (A1 + A2 ' TA)

It?)

+<

(YNF t;= (w1)t + 642): ' aw): + Pt ' (Sc): - (Tp)t

) (P - sc - T )

Pt

"CD

1

o = Pt +1)t +~(A1)t +-(A2)t - (IA)t - (Tp)t
Apt = P: - P:-1

Aw: = wt - wt-l

(Yd)t = Yt - (Tw)t - (TA)t - (Tp)t

(FF): = pt/pM t

(Ap ):= [9:(A1):]/(pA):

A
(hm)t = h: (“w ' Ne):

(PC-Tc)t = (Pc)t - (Tc)t

Symbol Status*
C *
Y *

C
I(1 *
o *
1L *
Ir *
X *
is *
Ii *
xi *

1
ISTK

*
FI
*

Yd
pF *
xP *
h *

m
K *
ll:1

 

5

Table 1 (continued)

Symbols

Explanation (Value in billions of 1958 dollars
unless otherwise noted).

Consumer expenditures

Share weighted personal disposable income
Investment in plant and equipment

Disposable nonwage income

Average yield on corporate bonds (Moody's), percent
Residential construction

Gross national product

Yield on prime commercial paper, 4-6 months, percent
Change in business inventories

GNP less inventory investment

Stock of inventories from arbitrary origin(192850)
Imports of goods and services

Disposable income

Ratio of domestic to import prices, 1958 a 1.00
Private GNP

Index of manhours

End of year stock of private capital from arbitrary
origin (192850)

Average discount rate at all Federal Reserve Banks,
percent

* means the variable is endogenous to the model.

Symbol Status*
R
3 *
C
P *
C
T
C
p *
W ‘k
h *
NW *
*
w1
w2
*
A1
A2
P *
TW
TA
TP
*
YNF
D *

6

Table 1 (continued)

Explanation (Value in billions of 1958 dollars
unless otherwise noted).

Year end ratio of member banks excess to required
reserves, pure number

Corporate savings

Corporate profits

Corporate income taxes

Price index of gross national product, 195851.00
Annual earnings, thousands of current dollars

Index of hours worked per year, 19545100

Number of wage and salary earners, millions of persons
Private employee compensation

Government employee compensation

Private farm income (A1 + A = total farm income)

2
Government payments to farmers

Nonwage nonfarm income

Personal and payroll taxes less transfers associated
with wage and salary income

Taxes less transfers associated with farm income

Personal and corporate taxes less transfers asso-
ciated with nonwage nonfarm income

Nonfarm income
Capital consumption allowances

Annual change in the GNP price index

Symbols Status

Aw *
A *
pA
*
pA
NO
P -T *
C C
ZU’I
N
pM
F
x
YP *
1P *
Y'l *
C
1'1 *
S
Cc
Ci
L
T*

Table 1 (continued)

Explanation (Value in billions of 1958 dollars
unless otherwise noted).

Annual change in earnings, thousands of current
dollars

Private farm income deflated by pA

Index of agricultural prices, 195851.00

Number of government employees, millions of persons

Corporate profits less corporate taxes

Inverse of percent unemployed [N/(N-Nw)]

Number of persons in the labor force, millions of
persons

Index of import prices, 195821.00

Exports of goods and services

r w r1+i-1 i
(ml) 1 12:0 ( i )ll (Ye) :-1’ Pascal lag weighted Yc

m r +i-1 . . .
(1_X2)r2 Z ( 2 >IK; (is)t_i, Pascal lag weighted ls

1=o i

Inverted Pascal lag on share weighted income .
See Section

III.A.4.c.
Inverted Pascal lag on the short term interest
rate

Restricted Pascal lag constant term,

consumption See Section

III.A.1.,
Restricted Pascal lag constant term, Eqn. (6).
long term interest

Composite tax variable, T , T , T
w A p

each at mean share.

linearities, and (5) the Operational significance of the autoregression
correction.
A. The Gains From Full Information Estimation
Unlike information contained in overidentifying restrictions,
the information in across-equation disturbance variance-covariance
matrices does not enter -- even asymptotically -- if not specifically
allowed for in the estimation procedure. If the across-equations
covariances are not large, however, adding the additional parameters
may actually result in worse finite sample estimates due to the lost
prior information (that the covariances are approximately zero). One
of the objectives of this thesis is to examine alternative limited-
and full-information estimators and make a judgment on which is
superior under varying conditions.
B. Equation Simultaneity and Policy Endogeneity
The well-known difficulties resulting from simultaneous
equations lead to elaborate estimators that are difficult to correct
for the violation of other assumptions (e.g., disturbance autoregres-
sion and variance nonlinearity). Policy endogeneity is developed as
a special case of equation Simultaneity (as is the errors in variables
problem). One of our objectives is to examine the importance of the
resulting inconsistencies.
C. The Returns From Generalized Lags
The time-path of response of a macroeconometric model is of
critical importance and hinges directly in the specification of the
distributed lags. We shall estimate a more general lag formulation
which more closely resembles the actual underlying processes.

Generalized expectations and adjustment processes are develOped and

convoluted with each other, and then estimated by initial condition
parameterization in order to make them more robust with respect to
violation of other assumptions. The returns from this computationally
expensive process are assessed.

D. Nonlinearities

One often has at least the suspicion that the world is signi-
ficantly nonlinear. We shall work out the full conditions for general-
izing the estimators and then to examine the practical effects of non-
linearities on various structural and reduced form coefficients.

E. Autoregression

Usually the most important single feature of time-series
data is its autoregressive prOperties; and yet simultaneous macro-
economic models often are not corrected for autoregression because of
the estimation difficulty. ‘We derive an efficient, computationally
feasible method for correcting autoregression in the presence of equa-
tion simultaneity, and analyze the importance of this problem.

After considering each of the above problems in turn and deter-
mining which were important and which were not important in the con-
text of the specific model examined, we attempt to make estimation
recommendations. In addition, forecasts and dynamic analysis of the
model follow (in an appendix) to allow the reader to judge the model's

correspondence to the world -- which necessarily conditions the results.

II. Econometric Problems
A. Three Stage Least Squares
l. Disturbance PrOperties

Given values for the parameters (r1, r ll, 12) of the

2’

two Pascal lags, the model equations1 can be written in the form

(1) (N4A) Y + B Y__1 + C X 3 U*
(GxG) (GxT) (GxG) (GxT) (GxK) (KxT) (GxT)

(the subscript -1 refers to a matrix whose value at time t is equal
to the unsubscripted matrix's value at time t-l) where

G is the number of jointly dependent variables;

K is the number of predetermined variables;

T is the number of observations in the sample;

N is a normalizing matrix with n =1 if the 1th variable in Y is

11

the normalizing variable in the ith equation, and nij=0
otherwise;
N,A,B, and C are parameters;
Y is the (GxT) matrix of sample values of the jointly dependent
variables;
X is the (KxT) matrix of sample values of the predetermined
variables, including the Pascal lag pseudo-variables;
and U* is the (GxT) matrix of stochastic error terms.
Since there are only fourteen stochastic equations in the model, all
of the rows of U* after the first fourteen will be zeros. Defining
U as the residual submatrix of U* after deleting the rows of zeros
corresponding to the identities, and letting U(t) be the tsh column of
U, the following assumptions complete the model:
(1) E(U) - ¢

10

(ii)
(iii)
(iV)

(V)

(vi)

11

E[U(t)U(t)'] ==Z, t=l,2,..., T, 2 positive definite;

E[U(t)U(s)'] is diagonal, t=l,2,..., T, s=l,2,..., T, t#s;

1 1

plim '1" XU*' - plim 13" Y 1 U*' - (a;

The moment matrix of the jointly dependent and predetermined
variables is well behaved in the limit;

N+A has an inverse.

[Assumptions (ii) and (iii) state that although there may be

contemporaneous correlation of the disturbances across equations

(measured by 2), there is assumed to be no correlation of the distur-

bances of different equations at different time periods; under these

assumptions, autoregression does not result in inconsistency unless the

equation involves a lagged value of the normalizing variable.]

12 //<.

2. Parameter Estimates, Nonautoregression Corrected Model
Two sets of estimates of the parameters of

the fourteen stochastic equations will be presented: (1) classical

ZSLS and (3) 3SLS.

The format is

2SLS

_3SLS

with estimated coefficient standard errors in parentheses below the

I O I 2 I
pOint estimates where appropriate. R , the standard error of estimate,
O I I 1 D
and the Durbin-Watson statistic are reported to the extreme right

where applicable.

 

1 The Durbin-Watson statistic looses one additional observation due

to the discontinuity at the war.

17

B. Simultaneous Equation Problems
1. Causality and instrument selection
a. Definitions.
Many econometricians ignore the work of Simonl,
Koopmansz, and Fisher3 and lose some potential generality in defining
endogenous, exogenous, jointly dependent, and predetermined variables.4
The loss occurs when the system is decomposable or recursive so that
different terms are necessary to describe the statistical properties
of a variable, as opposed to its modeling relation to a complete
subsystem.

Consider the model5

(1) Y1 = Boyz +812 + szl + 83(y1)_1 + 61

and e are stochastic

where y1 and y2 are to be determined, and e 2

1

disturbances. 2, p1, and p2 are exogenous to equations (1) and (2);
that is "... they are variables which affect the economic system but
are not in turn affected by it, or at least are only affected to a

"6’7

negligible degree by it. The residual variables y1 and y2 -- which

both affect the system and are affected by it -- are endogenous.

 

[54].
2 [331.
3 [21].
4 See Goldberger [24], pp. 294-295; Christ [5], pp. 179-180; Kmenta [42],
pp. 532-534.
5 These equations are a modified version of Koopman's example [33], p. 202.
6

[52], p. 197.

7 This asymmetric relation has also be defined as "casuality"([54],
pp. 6-7).

18

Statistically, the relevant question is whether any variables
on the right of each equation are correlated with the disturbance of
that equation; correlated right hand side variables are termed "jointly
dependent."1 If the system is fully integrated, all endogenous
variables will be correlated, whether or not 2, the across equations
disturbance variance-covariance matrix, is diagonal (this is the well
known simultaneous equations problem); thus the set of jointly dependent
variables will include all current endogenous variables. Even in the
case of diagonal 2, however, although current endogeneity is sufficient
cause for correlation with the disturbance and thus joint dependence,
it is not necessary. For example, if anticipations are significant,

then y1t may depend on y1,t+1, or lagging, may depend on y1 t;

y1,t-1
then lagged endogenous variables can be jointly dependent, and not

predetermined as they are usually classified.

If we add a third equation to the system (1) and (2),
(3) Z =B7P3+€3:
then Z becomes endogenous to the system composed of (1), (2), and (3),

though still exogenous to the closed system (1) and (2): the expanded

is uncorrelated with e and 6

system is recursive. In this case, if 63 1 2,

Z may be treated as predetermined, when estimating the parameters of (l)

 

Errors in variables may be regarded as a special case of the simultaneous
equations problem where the omitted equation is the one defining the
error in the variable; not surprisingly, ordinary least squares estimates
are inconsistent even when the variable error is uncorrelated with the
equation error (2 is diagonal).

l9

and (2), although, if anticipations are important, yl,t-l may Egg.
Then the set of jointly dependent variables is not equivalent to the
set of endogenous variables, and the set of predetermined variables
is not the union of the lagged endogenous1 and exogenous variables.
Endogenous/exogenous describes the relation of a variable to a
particular complete sybsystem, while jointly dependent/predetermined
indicates presence or absence of correlation of a right hand side

variable with its equation error term.

b. Instruments

We shall consider two methods of computing the instru-
mental variables to be included in the second stage of ZSLS: (l) the
classical method of including all predetermined variables in the first
stage regression that appear with nonzero coefficients in the reduced
form equation for the particular right hand side jointly dependent
variable being considered; and (2) a tame version of Fisher's structurally

ordered instrumental variables (SOIV) procedure.

 

fifi’

Legged endogenous variables have probably been automatically included
in predetermined variables because of a confusion of temporal pre-
dence with causality; causality is the basis of exogeneity, not
temporal precedence, and it is exogeneity that guarantees no simul-
taneous equations problem. In our classification, "lagged endogenous"
is just a special case of "endogenous."

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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20

(1) Classical instrument selection
Government expenditures and exports occur only in
the GNP identity and thus can be combined into a composite variable (G+FX);

. 1 .
then 29 variables - W2, A2, pM’ NW’ (NW-NG), pA, (G+FX), R, 1d, TC, TW’

-1 d i P
TA, Tp9 T) U , Dw’ I_19 (FI)_1) W_1, (ISTK)-1! K_1) x_13 (PC TC SC)_1) p 19
(Ygl), CC’ ci , (i;1), and a constant--appear with nonzero reduced from

L

coefficients for those jointly dependent variables in the main block of
model1 (1S and iL in the recursive subblock will be computed only from the
predetermined variables occuring in that block, as discussed above). All of
the equations are overidentified and ZSLS is very close to OLS (since the
number of instruments is very close to the number of observations).

(2) Structurally ordered instrument selection

Fisher has suggested that the selection of pre-

determined variables to be used in first stage regression be based on the
casual structure of the model; since the efficiency of the estimate
depends on the correlation of the instrumental variable and the variable
being replaced, it is important to include those predetermined variables

that are most directly related to the particular jointly dependent variable

in the first stage regression for that variable.

 

1 (Y21) and (i;1) are the predetermined variables originating in the
(inverted) Pascal lags; CC and c1 are the restricted intercept
L
variables for the consumption and long-term interest rate equat ons,
respectively (see Section III.A.l. above).
2

There are 33 observations in the subsample from which the parameters
are being estimated.

21

As Fisher explains

"Consider any particular endogenous variable in the equation
to be estimated, other than the one «xplained by that equation.
The right-hand endogenous variable will he termed of zero causal
order. Consider the structural equation (either in its original
for or with all variables lagged) that explains that variable.
The variables other than the explained one appearing therein will
be called of first causal order. Next, consider the structural
equations explaining the first causal order endogenous variables.
All variables appearing in these equations will be called of second
causal order with the exception of the zero causal order variable
and those endogenous variables of first causal order the equations
for which have already been considered. Note that a given pre-
determined variable may be of more than one causal order. Take now
those structural equations explaining endogenous variables of second
casual order. All variables appearing in such equations will be
called of third casual order except for the endogenous ones of lower
causal order, and so forth. (Any predetermined variables never
reached in this procedure are dropped from the eligible set while
dealing with the given zero causal order variable).

 

"The result of this procedure is to use the g priori structural
information available to subdivide the set of predetermined variables
according to closeness of causal relation to a given endogenous
variable in the equation to be estimated."

Fisher then suggests that the predetermined variables be ordered lexi-
cographically according to the lowest causal orders at which each occurs,
and then those that are highest (causally closest) in this ranking be re-

. . . 2 . . .
tained if their contribution to the R of the first stage equation is large
enough according to some preselected criterion (the variables are deleted

. . . 2
one by one, retaining those whose deletion causes R to drop more than the

criterion allows).

 

[21], p. 266; italics his.

If one predetermined variable occurs in the first and third (1,3) casual
order, and another in the first and second (1,2), then the second
predetermined variable ranks ahead of the first. (Alphabetizing is an
instance of lexicographic ordering.)

22

This procedure has been modified in two ways in estimating the
present model: (1) the criterion for inclusion of a predetermined
variable has been that it appear by the second causal order (rather than
using an R2 test over the lexicographic ordering); and (2) all of the
right-hand jointly dependent variables of a given equation use the
same (pooled) set of predetermined variables in their first stage re-
gressions (to guarantee orthogonality of all instrumental variables and
the second stage disturbance), whereas Fisher's original1 suggestion was
to use a (possibly) different first stage regression for each right-hand
jointly dependent variable (even within an equation).

We have also recognized the endogeneity of policy in the SOIV model.
If there exist government reaction functions, then the control variables
that are exogenous to the particular complete subsystem we are examining
need not be predetermined variables. If we consider the meta-model
consisting of the subsystem and the reaction functions, then the control
variables are not uncorrelated with the equation disturbances (the
familiar simultaneous equations problem again) and are not predetermined
variables; the fact that they are potentially exogenous (the gggd Egg
be affected by the subsystem) is irrelevant statistically if they are

lg fact affected through governmental reaction to the state of the economy.

 

 

1 Fisher recognized the potential inconsistency of his original sugges-

tion and noted the above procedure as a consistent alternative [21],
p. 273.

23

Thus all taxes, the discount rate, the ratio of excess to required
reserves, and the agricultural price level have been reclassified as

jointly dependent.l

 

Government expenditures remain predetermined, an inflexible policy
instrument. The only predetermined variables Obtained from the
(otherwise unspecified) reaction functions are C and FX from the GNP
identity for the calculation if id and R (presumably some function
of GNP).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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28

C. The Pascal Lags
All variables are deflated by the implicit GNP deflator

unless otherwise noted; the U are stochastic error terms.

1,:

The simple Keynesian relation between current consump-

tion and income may be unstable for several reasons: (1) variables
other than income may be important in determining consumption; (2) the
long run marginal propensity to consume (mpc) is systematically above
its short run counterpart, suggesting a consumption "ratchet" or
permanent income effect; (3) the separate components (durables, non-
durables, and services) of consumption may depend on different
variables; (4) aggregate consumption may depend on the distribution,
as well as the level, of income; and (5) the decision variable should
be disposable rather than total income.

0f the variables other than income that are often suggested, the
most important are wealth variables, primarily liquid assets and
occasionally unrealized capital gains. It can be shown, however, that
the influence of wealth can be transformed into a distributed lag on
income. Consider the household consumption function

(1) ct = BO + E1w: + 323!:

where C = household consumption

W household wealth

Y

household income.
The change in wealth is defined to be household savings in year t,

(2) Awt e Yt - Ct ,

29

while the total household wealth is cumulated savings plus the initial

endowment, W0:

T
(3) wt = 121(Y-C)t'i +wo

th
(assuming the household is in its T—— period of existence).
Substituting (3) into the household consumption function and rearranging:

T

= a
(4) Ct+81§ Ct-i BO+B2Yt+Bl.Z_Y
i—l i-l

t-l

t-i

where B: = (50 + Blwo)°

Rewriting (4) in terms of the lag Operator L1 we Obtain

T o T s
o i _ O i
(5) (L + 8112211. )ct - B; + (321. + 315:1: )Yt .

It is then apparent that the consumption function including wealth has

. . 2 .
been transformed to one incorporating a rational lag on income:

i
(621? + B1 L)
i 1

= 1': r r -

(6) C Bo + O . T i Y .
(L +-s 2 L )

1.
L 1:1 .l

"Edie

 

 

The divergence of the short run and long run mpc has been much
3 . . . .
discussed. If consumers attempt to aVOid rapid changes in their

standard of living, they will adjust slowly to their eXpected future

 

2 [31]. Since household wealth is not infinite, this series must be

finite, the only condition necessary to transform it into a rational
lag.

3 Esp. [23], [21.

3O

consumption path. An approximation to this process will be developed
. . . l . . . .
by combining generalized adaptive expectations and partial adlust-
2

ment models.

. . . P

Consumers are assumed to adjust to permanent income (Y ),

. . . 3 .

hypothesized to be a stochastic function of past income. If, for

example, a second order rational lag describes this relation, then

(for the particular case of two roots)

p _ _ _ m h i h+i _ _ -1
(7) (y)t—(lll)(1)\2)h2:012=:0 1112 L {Yt+[(1>.1)(1>.2)l alt}

or, applying the Koyck transformation (inverting the lag),

P P P
(a) (Y )t — (l-ll(1-l2)Yt + (i1+i2)(Y ):-1 - i1i2(y ):-2 + alt,

and A,

2 are the roots of the two convoluted geometric lags that

where A 1
comprise the second order rational lag, and Yt is current income. [Note
that if A2 = 0, equation (8) reduces to the simple (geometric) adaptive

eXpectations mode1.]

Desired consumption (C*) is a function of permanent income:

(9) (2’: =5 + alop) +6

0 2t ’

 

The usual generalization allows interdependence between two first-
order adjustment processes; the generalization considered here is
the higher order adjustment of a single process.

9

“ [42], pp. 474-477.

3

This function can be very general, depending on the number of con-
volutions and/or the domain of the roots [31].

31

while actual consumption (C) depends on the (generalixcd) adjustment
mechanism:

‘ _ : 1_ -_ 3': _ J. ‘ '. __ I L! (‘
(10) Ct Ct-l (‘ )‘BMi )4)[Ct Ct-l] ‘ t3laltt-1 (t-z1 “3t

which, after recursive substitution, can be rewritten as

= _ _ m m j k j+k * - _ ‘1
(11) Ct (1 i3>(1 i4>jEO kEO i3 l4 L [Pt + [(1 i3)(1 i4)] eBt] .

Substituting (7) into (9) into (11) gives1

 

1 = _ _ jk _ _ jkj+k-.
0, 30(1 x3)(1 x4) 2, 2k l3 14 + (1 i3)(1 i4) zj 2k (3 (4 L (times)

m
l

m

(times)§51(1-li)(1'x2) 2h 21 i? l: Lh+i [Yt +[(1-ii)(1-i2)l-lelt]
+ e + (1"). NH >'1 2
2: 3 4 e3:

- _ _ _ W h b h i j k h+i+j+k
- BC + 31(1 A2)(l A3)(1-14) Zh.bi Lj bk Xi A2 A3 A4 L

_ _ -1 _ _ g j k j+k .

[capo-(4)] ’19,,

Rewriting in inverted(Koyck transformed) notation:

 

 

4 (1'1-) 0
Ct = n O 1 Yt + 4 LO 63:

i=1 (L -x,L) n (L *l.L)

i ._ i
i—l
(l-j.)
4 i -l
+ n e2t + [(1'63)(1‘€4)] €3t

i=3 (Lo-jiL)

32

€2t + [(1‘63)(1'€4)] €3t .

a fourth order rational lag on income.

The lag as specified allows positive, negative, and complex roots;

this is certainly extreme generality for estimation purposes, and perhaps
even for specification -- it seems plausible that the adjustment roots
should lie between 0 and 1, and not implausible that the expectations
roots should also lie in this interval. The orders of the expectations
and adjustment processes have not been specified (the second order
example is only for expository purposes); given this freedom in the
number of convolutions in each case, it may not be unduly restrictive

to constrain the roots of each process to be equal. Combined with the
interval restriction above, this means that the weights of each
mechanism lie along a Pascal density function.1 If we add the additional
assumption that the roots of the adaptive expectations model are equal

to the roots of the partial adjustment model,2 then it is not necessary
to determine separately the order of each process and the resulting lag

is nonlinear in only two parameters: r, the combined number of

 

The Pascal (Solow) lag is a rational lag with equal roots 1 that lie
in the interval 0 < l < 1.

In the simple geometric version of each model this implies that the
adjustment coefficient is 1 minus the decay parameter of the expec-
tations model. [This is strictly an assumption of convenience,
since the Pascal lag is (nonspectrally) estimable while the rational
lag is not (but see appendix III-A for later work to eliminate this
assumption).]

33

convolutions; and l, the value of all r roots. In addition, since
the combined result is a Pascal lag, there is a procedure available
for consistent parameter estimation.

In specifying the consumption function, some care has been taken
to allow inclusion of a partial adjustment mechanism; complete
adjustment of nondurables and services consumption may or may not be
immediate, but we would certainly not want to assume complete adjust-
ment of durables "consumption." Because durables enter the national
income accounts as they are purchased, rather than as they are consumed,
their purchase decision is an investment decision. Assuming that the
desired flow of services from durable goods depends on expected future
income, but the consumer must purchase the entire service stream at
once, in an uncertain world the prudent consumer will only partially
adjust his actual purchases to the desired level. Since a separate
equation for durables consumption has not been specified in the model,
it is important to allow this partial adjustment mechanism.

The three-way partition of income -- into wage income, farm
income, and nonwage, nonfarm (profit) income -- is a basic feature
of the K-G model, with equations being developed to determine these
shares. Since changes in the distribution of income, ceteris paribus,
result in changes in total consumption, this information should be
incorporated into the consumption function. As Klein and Goldberger

3 . . . . . . . . .
argue, this functional distribution is an approx1mation to the Size

 

1
[47].

2 . . . . .
In order to retain the income shares consumption function discussed
below.

3

[34], pp. 4-7.

34

distribution of income, and also includes the effects of: (1) the
relative stability of wage income, allowing a higher mpc at the same
level of risk, (2) business saving by individual proprietors due to
better investment outlets and more liquidity preference, and (3) "...
the influence of the rural way of life with comparatively fewer
expenditure outlets for consumer items."1 In estimating the three
mpc's of the original K-G consumption function in order to avoid multi-
collinearity the ratios of the mpc's out of wage income, farm income,
and nonwage, nonfarm income were constrained to values obtained from

(extraneous) survey data. For example, instead of estimating the

relation

(13) C = 80 + 31 (disposable wage income) + 82 (nonwage, nonfarm

income + B (farm income) + e

3 9
B B3
first —- and -— would be estimated, and then time series
B1 B1
data would be used to estimate 51 in
A
B2
(14) C = 60 + 81 [(wage income) + E—- (nonwage, nonfarm income)
1
A
E33
+ -— (farm income)] + e
l
/\ a“.
82 E53
where E" and g- are survey estimates of the ratios of nonwage,
l 1

nonfarm income and agricultural income mpc's, respectively, to the wage

income mpc. [Note that only the ratios, not the levels, of the mpc's

have been constrained.]

 

[3]. p. 6.

35

Clearly personal disposable income is the appropriate variable
for determining consumption; since disposable income by factor shares
is not available, it will be specially constructed for the model
following the method proposed by Frane and Klein.1

Putting all of these considerations together, the resulting

consumption function is

r1 w r1+ i-l i
(15) C: = 81,0 + 51,1(131) £0 ( 1 ) A1(Yc):-1 + U1,:
with
A /\
BA ER
(16) (Yc)t-: (w1 + w2 - TW)t + a; (A1+ A2- TA): + 8w (IR-SC-Tp)t

 

where r1, 11, 81,0, and 51,1 are parameters, and
W1 = private wages

W2 = government wages

Tw = net taxes and transfers on wage income

A1 = private agricultural income

A2 = government payments to farmers

TA = net taxes and transfers on agricultural income
P = nonwage, nonfarm (profit) income

SC = corporate savings

Tp = net taxes and transfers on profit income.
1

[22]. Since the National Income Accounts were revised in 1965, it
was necessary to recalculate all of the years rather than accept the
Klein and Frane estimates through 1952 (see Appendix II-B).

36

In estimating the Pascal (Solowl) lags, we have chosen a trans-
formation that eliminates the infinite string of regressors and per-
mits estimation that was originally suggested by Klein2 for the
geometric case and later generalized by Maddala3 to the Pascal case.
The basic procedure involves truncating the infinite sum and estimating
the effect outside the sample as a parameter, since it is invariant
over time. For the geometric case,

= s - 3
(2) yt 0 51(1 A) 190 x1 xt-i + 6t ’

Klein pr0posed that the equation be rewritten

(3) yt = 60 + 612 t + e z + e

1 2t t ’
r t-l i
= -1 7
where th (1 ) 130 A xt-i
t
Z = A
2t

9 3 E(yo) - 80

(8121: measures the effect on yt of the regressors inside the sample
period, while 9, the parameterized initial condition, contains the
effect of the regressors outside the sample period discounted by it).
Maddala has generalized this technique to include the Pascal
family of lag distributions, lags with multiple convolutions but

equal roots. In this case, equation (1) is written

 

1[55].
2 [41]; see also [7].

3 [47].

r
= B + +— 2 +
(4) yt 0 B1 zlt i=1 e1-1 zi+l,t et ’
Where 91-1 = E(Y1-1) ' 8O
r -r( rH-l i
and th = (1-1) igo l l Xt-l
r-l t-l t
— - i
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_ .£:l . .t. . l
ZBt 1 t-l l Zt
_ r-Z . £:l . l
Z4: :-2 i Z3:
_ 2:2 ..£:2 . l
Zot ' ' 3 t-3 l 24:

Zr+l,t - (r::1))\tnr+1

except that for t=1, Z3t=l, and the rest are zero, for t=2, Z4t-l and

the rest are zero, and so on, until for t=r-l, 2 =1, and the rest

r+l,t
are zero. A generalized example of this derivation is given in Appen-

dix III. For the particular case of three convolutions, (1) becomes

t'3 '+ ‘+1 ' t-l t-

i=0

- :(:-2)it'1[E(y1)-eO] +.£I%§ll lt'z [E(y2)-80] + at .

Since E(yk) = yl-ei, if we assume1 that the disturbances on the para-

 

1Instinct served me well; later work by Swamy and Rao ("Maximum Likeli-
hood Estimation of a Distributed Lag Model with Autocorrelated Errors,"
unpublished paper, P.A.V.B. Swamy and J.N.K. Rao, Federal Reserve System
and University of Manitoba) demonstrates that 81 is inconsistent when
the initial conditions are estimated, but consistent if the initial
conditions are set to any arbitrary constant.

38

meterized initial conditions are very small relative to yi, we can

replace E(yi) with y1 and reduce by r the number of parameters to be

estimated. The restricted equation, for the case of r-3, Obtained by

substituting yi for E(y1) and distributing the terms, is

t-3
(6) y: 3 so I: + 51(1_j)3 go .Liiglilill 11 X

+
1 2 t-i 6:

where

. _ (:-12(:-22 : _ :-1 _ : :-1 :-2
V? y: 2 A yo + t(t 2)X y1 2 A y2
I: - 1 - Ell-12933)- it + :(t:-2)>.':"1 - t :1 ,:-2 .

y: and I: are similarly defined for other values of r so that there

always remain only the two coefficients 80 and Bl to be estimated for

each trial set of r and l.

39

D. Nonlinearities

The model is nonlinear in both the parameters and the variables,
but the four parameters of the two Pascal lags (r1, A1, r2, and l2)
introduce the only parameter nonlinearity -- given these values the
model is linear in the remaining parameters and is written so that
the stochastic equations are strictly linear in both variables and
parameters and the variable nonlinearity occurs only in the
identities. h

a. farameter Nonlinearity

The model is to be estimated by two- and three-stage

least squares (ZSLS and BSLS), techniques not customarily applied
to systems nonlinear in parameters (although the generalization is
not difficult). If the model is nonlinear only in the parameters,
first stage estimation can be performed as usual, since the elements
of the reduced form coefficient matrix are stable and thus the
predicted values of the nonnormalizing jointly dependent variables

are well defined.

40

Considering ZSLS as a special case of instrumental variable
estimation that provides a particular method for obtaining the instru-
ments, it is apparent that the second stage of ZSLS is simply an
instance of the nonlinear least squares problem and can be solved in
a number of ways [including differentiating the sum of squared
errors with respect to the parameters and solving the resulting
(nonlinear) differential equations, or, if the parameter domain is
bounded (as in the present easel), by search techniques]. Models
nonlinear only in the parameters are not conceptually difficult to
estimate via ZSLS.

The extension to BSLS is equally straightforward conceptually,
although it imposes a large computational burden; in the 3SLS case
the parameters would be chosen to minimize the generalized variance
([2], the determinant of the across-equation disturbance variance-
covariance matrix). While nonlinear ZSLS estimates will be calculated
for the current model, BSLS estimates will not--they would necessitate

a four dimensional search (over r A

1, 1, r2, and A2), With a BSLS

estimate of the parameters of fourteen stochastic equations at each
combination of trial values of the nonlinear parameters. The BSLS
estimates to be calculated will be conditional on the ZSLS estimates of

the four Pascal lag parameters.

 

1 r1 and r2 are assumed to be integers between 1 and 5, and A1 and X2

are assumed to lie inside the interval from O to l.

41

b. Variable Nonlinearity
Nonlinearity in the variables is more troublesome

than parameter nonlinearity, since (1) the reduced form is often
not in closed form, and (2), even when it is, it will generally
be nonlinear in the variables such that the reduced form coefficients
are unstable (the reduced form is nonlinear in its parameters).1
To overcome these problems, Goldfeld and Quandt2 proposed approximating
the reduced form by a p£h degree polynominal in the predetermined
variables, representing the first p terms of a Taylor series expansion.
They assumed but did not prove consistency under this approximation;
however, Kelejian3 and Edgerton4 have reestablished sufficient con-
ditions for consistent ZSLS estimation using the Goldfeld-Quandt
polynomial approximation (actually, Fisher published a much clearer
discussion in an earlier article5 in a slightly different context).
Since there seems to be needless confusion on the requirements for
consistent nonlinear ZSLS estimation, and since none of the articles
summarize all of the conditions, we shall examine the problem in more
depth.

Assume that a system of G structural equations is linear in the
parameters, has additive error terms, and each equation can be noralized

on one jointly dependent variable; the first equation can be written:

(1) Y1: = ch Y1 + Flt 61 + Ult

. th . . . . .
where y1t is the t- observation on the normalizing variable in the

 

1 See especially [13], [26], [11], and [12].
2 [26], p. 116.

3 [321.

4 [12].

5

[21].

42

first equation, X is the (l x K*) vector of observations on the first

It

equation predetermined variables at time t, Y1 is the associated

(K* x 1) vector of parameters, is a [l x (GA-1)] vector of observa-

F1:
tions on (GA-1) jointly dependent functions at a time t, 81 is the
corresponding [(GA-l) x 1] vector of coefficients, and U1t is the tEh

period stochastic error term. In addition, define Xt and Yt’ respec-
tively, as the tEll observation on all predetermined and jointly dependent
variables appearing in the G equation system and note that the elements
of the vector of jointly dependent functions can be written

(2) Flt = [fl(Yt’Xt) ... fGA_1(Yt,Xt)]

To perform two stage least squares it is necessary to satisfy
what we shall call the consistency conditions and obtain instruments
that (l) in the probability limit are uncorrelated with the structural
disturbance but (2) are correlated with the structural variables they
replace. However, as Goldfeld and Quandt observe, there are two
immediate candidates:1 (1) the "hat of the function," e.g. EEYEZTKE),
and (2), the "function of the hat," e.g. fj(§t,Xt). Goldfeld and
Quandt guess that estimates based on the second method will be incon-
sistent; Kelejian proves it.2 Each jointly dependent function, then,

A
is to be considered as a random variable in order to calculate f'(Yt’Xt)’

j=l, ..., GA-l. There is an unknown, perhaps nonanalytic function

 

. .th . . .
For the instrument for the j- jOintly dependent function in the given
equation. The predetermined variables are used as instruments for
themselves.

2 [32], p. 374.

43

relating each jointly dependent function to the predetermined variables;

the nonstochastic part of this function is defined to be hj(xt)’ so that

(3) E[fj(Yt,Xt)|Xt] = hj(xt) (j=l, ..., GA-l; t=1, ..., T)

and

(4) fj(Yt’xt) = hj(xt) + vjt

(j=1. ..., GA-l. t-l. .... T)
where vjt is the actual reduced form disturbance, with E[vjtlxt] = 0.
Since hj(xt) is generally unknown (depending on the structural parameters),

and may not be in closed form, the Goldfeld-Quandt polynomial approxi-

mation is introduced: let Pj(xt) be a pgh degree polynomial in X

+...+1w p (j=l,

(w Pffi)=no.+n.x 'h“+n m’kt

j 13 1t Kj XKt

t=1. .... T).

where x x are the elements of Xt and (noj, ..., n ,) s plim

, .00,
1t Kt q_] T

(”Oj’ ..., nqj). Then (4) can be rewritten in terms of this approximation as

*
(6a) fj(Yt’xt) Pj(xt) + v,

jt

or (j=1. .... GA-l; t=1. .... T).

(61>) ffl') ia<x>

j t t j t
with Vjt defined as vjt plus the error of approximation inherent in
replacing hj(xt) by Pj(xt)° Equation (6b) forms the basis of the
instrument calculation given the following two (sufficient)1 subconditions

for meeting consistency condition (1): (a) all of the predetermined

variables that occur in the given structural equation occur as regressors

 

See Maddala, ([48], p. 2, footnote 2), for an explanation of how
recursive models can provide counterexamples to the necessity of
condition [1] for certain equations within the model.

44

in the first stage equation (i.e., X is included in Xt)’ so that

It

Xt is uncorrelated with the first stage residual; and (b) all instru-
ments in a given structural equation are calculated from the same

set of regressors (which implies that the approximating polynomials
are all of the same degree). The reasoning behind these conditions
is apparent from examination of the second stage equation after in-

strument substitution:

(7) yt = Xlt Y1 + Flt 81 + U1: + (F1: " Flt) Bl '

W

_ U~k*
‘ It

A linear combination of the first stage residual from each instrument
equation is added to the structural disturbance every time period;

thus, as Fisher remarked long before the present consistency discussions,
"... consistency requires not only zero correlation in the
probability limit between the original disturbance and all
the variables used in the final regression but also zero
correlation in the probability limit between the residuals
from the earlier-stage regression equations and all such
variables. If the same set of instruments is used when
replacing every right-hand endogenous variable, and if
that set includes the instruments explicitly in the equation,
the latter requirement presents no problem since the normal
equations of ordinary least squares imply that such corre-
lations are zero even in the sample. When different
instruments are used in the replacement of different vari-
ables, however, or when the instruments so used do not
include those explicitly in the equation, the danger of
inconsistency from this source does arise." 1

Having established uncorrelatedness of the composite disturbance
** I I o
U1t with the instruments in the second stage regression, it remains to

satisfy the second consistency condition and show correlatedness of the

 

[21], pp. 271-2.

45
instruments with the variables that they replace. Since X1t is used

as an instrument for itself the correlation is obvious, while the
correlation of F1 1 (calculated from the reduced form) with F1 is
implicit in the model assumptions: if the model is correctly

specified, then h (Xt) describes a causal2 relationship and Xt includes

J

all variables relevant to the determination of the jointly dependent

functions. Since Pj(xt) is a p term approximation to H (Xt), there

J
exists3 a p such that

7673)] 7‘ 0. 1’4

which, with the previous two subconditions, establishes the consistency

plim [fj(Y,X)' f

of ZSLS parameter estimates if they exist.

There are four nonlinear identities in the model, including the
nontrivial price identity. Two of these identities--defining agricul-
tural income and relative import prices--are sufficiently irregular as
to be poorly approximated by a first degree Taylor expansion. In order
to test model sensitivity to the linearization of these identities, re-
duced forms were calculated with (1) four Taylor expansions, (2) two

Taylor expansions and two5 identities fitted by ordinary least squares,

 

Vector symbols without t subscripts refer to matrices, each row of
which is the original vector at a given t, with each matrix consis-
ting of T rows.

The predetermined variables, by definition, affect the jointly depen-
dent variables but are not affected by them. This asymmetric relation
may be defined as causality. See the immediately following sections.

Only a pathological function would require p > 1.
Regardless of the fact that E(v§t) # 0 due to the error of approximation.

The irregular identities (mentioned above) defining Ap and PF'
A

46

and (3) four ordinary least squares approximations. Some of the
results appear in Table 4, along with sample means of the exogenous
variables and a measure of the importance of the change in the multi-
plier--the absolute value of the maximum difference between the ZSLS
multipliers times the mean of the exogenous variable.

That the linearization chosen is important in determining the p;1
and pi1 multipliers is not surprising; none of the linearizations will
provide accurate estimates of these multipliers.1 Unfortunately, however,
the government expenditures multipliers are also sensitive to the lineari-
zation chosen. Since we have used first degree polynomial approximations
to the reduced form in the first stage of ZSLS, this impact multiplier
sensitivity may be an indication that we have lost considerable first
stage information in the linearization.2 Of course we have yet to
correct for autoregression, which will lower the multipliers and perhaps

lessen the linearization sensitivity.

 

1 This disagrees with the Goldberger result ([25], pp. 136-138). The
difference undoubtedly stems from the fact that the original K-G
model was too heavily damped,while these estimates are not damped
enough--thus the effect of the nonlinearity is understated in one
case and overstated in the other.

 

E

 

 

 

 

 

 

sm.~- oH.o- no.0- -.o- so.o- so.o- om.o- am.o ~a.~- so.~ L
mmo.o- Noo.o- moo.o- mao.o- Hzc.o- «Ho.o- moo.c- mao.o omo.o- «no.0 a
mo.ma- as.o- om.c so.m- ma.a- aq.s- om.m- on.m om.ma- om.sH x
mac cap .toamma esp
mamm
E
qoh.~ sem.a omm.o Ho“.a aso.o Nao.a omm.a omo.o omm.o “so.o _ n< assaxmx_
«00.0 mmo.o cos.o soo.o mco.o muo.o mno.o Nem.o omo.o moo.o _ a< essaxmz_
mmq.mm o~a.m moo.~z moc.o~ cem.c ow~.o osm.aa HHN.~s smc.m aHm.o _ xa saeaxmz_
wocmupanH
soo.a Ham.a cao.oa som.aw czo.a moo.aa amm.0m ohm.me mao.~ ~¢H.o can: manmanas mzocmmoxm
E
am.a- Ha.o- no.0 as.o- Ho.o- No.o- sa.o- ow.o m~.~- os.~ ;
o~o.o- Hoo.o- moo.o- mao.o- ac:.c- woo.o- ooo.o- oao.o a~o.o- w~o.o a
mo.oa- Na.o- ma.a sw.s- 0N.s- m~.s- Hm.m- ~m.m ms.ma- NH.mH x
Beta;
E
mm.a- .oao.o- .o.c- no.0- om.o- ow.o- oa.o- wa.o so.~- ma.~ ;
mmo.o- Noo.o- m:c.c- Nsc.c Hac.c- Hao.o- mco.c- «Ho.o mmo.c- aao.o a
mo.oa- o~.o- mw.o N~.q mz.sa aa.¢- mu.m- wm.m aa.ma- mo.afi x
uofikmh 039 .m_c 039
E
aN.a- mm.o- who-o- “0.0, m.a- mm.o- Na.o- ms.ic N¢.H- m~.~ .i
ooc.o- “so.o- wcc.c- Hao.c- ozc.c- oao.o- sco.c- mao.o mmo.o- mac.o a
oq.~m- mm.o- o_.z n~.a- am a. am.m- oc.m- oo.s so.sa- s¢.o~ x
occuuaunum uo~hwh mach
momm
E
a-a Hma «3 as as .a9 a u an m deco:

 

 

 

 

 

 

 

 

 

 

 

ffiopoz kuoouuoo scammopwmuousmcozv

mmqumHHnDZ Ho<mZH

s «asap

48

E. Autoregression

. . . l
1. Autoregression 1n simultaneous models

The estimation of the parameters of simultaneous equation

systems with both serial correlation and lagged endogenous variables is

complicated by the inability to solve for the separate equation auto-

regression parameters piecewise even when the estimation technique is a

single equation one such as two stage least squares (ZSLS). The problem

is further complicated by tradeoffs between large and small sample

. . . 2 . . .
properties. A simple example (due to Ray Fair ) wlll make the exposition

clearer.

Assume the

(1“) ylt z ’“12
(1b) y2t z ”821
where

(2a) ”1: = r11
(2b) “2t = r22

Then the reduced

of (1a)] is

model consists of the two equations

Y2: ”b11 X1: 'b12 x2t "bla y1,t-1 + ”1: (‘21’2"°°'T)

-b

Y1: 'bzz th 23 *3: ‘b

24 y2,t-1 + ”2c (t=l’2""’l)

1t (t=l,2,...,T)

+ =
u2,c-1 V2: (t 1,2,...,T)

form for y2t [used in the first stage of ZSLS estimation

 

l

2 [16], p. 510.

This section borrows heavily from an article by Ray C. Fair, [16].

49

-1
(3) y2y ‘ (1'321 a12) [(r22‘321 a12 r11)y2,t-1’b24(Y2,t-1'r22Y2,t-2)

+

a21(r22'r11)y1,t-1 + 821 b14(y1,c-1‘r11 V1,t-2)

+a b

21 11(x1t'r11 x1,t-l) + (321 b 'b

12 22>x2t

(321 blZ r11 ‘bzz r22>X2,t-1'b23(x3t'r22 x3,c-1)

' a21 ”1: + Vzt] ’

an expression involving both autoregressive parameters (r and r22) of

11
the system. Thus the (nonlinear) constraints implied by the autoregressive
structure cannot be incorporated in the first stage unless either the
autoregressive parameters are all simultaneously estimated or at least all
but one are known. If the restrictions are forsaken, then an important
part of the structure of the model is not incorporated into the choice of
instruments (y's) for the second stage.1 The objective is to include as
much prior information as possible in the first stage without significantly

damaging the second state properties. To this end an extremely rigid

simplifying assumption--that the disturbances for each equation are not

 

The asymptotic properties of this procedure (dropping the restrictions)
are by far the easiest to prove; the small sample properties would be
expected to be less than desirable, however. In this completely unstruc-
tured case, the autoregression coefficients in the first stage are not
even constrained to lie between -1 and 1, while the number of first stage
regressors is doubled (losing much prior information about the auto-
regressive process).

50

only uncorrelated with those of other equations, but in addition the
autoregression parameter within each equation is the same for all
equations--is imposed in the first stage of estimation and later
disregarded in the second and third stages.

The model is

(A) (N+A) Y + B Y_1 + c x = U*
(ch) (GxT) (GxG)(GxT) (GxK)(KxT) (GxT)

with

(5) U* = R Ufl + V
(GxT) V(GxG)(GxT) (GxT)
and the assumptions

(i) E(V) = 0

(ii) E[v(t)v(t)'] = [i I] , t=l,2,...,T, 2 positive definite, where

v(t) = tth column of the matrix V;
(iii) E[V(t)V(t')'l = ¢. t=1.2.-.-.T, ttt';

v' = plim T'1Y_ U' = ¢;

(iv) plim T'l xv‘ = plim T'1x_ 1

l

(v) the moment matrix of the endogenous, lagged endogenous,
predetermined, and lagged predetermined variables is well
behaved in the limit;

(vi) R is a diagonal matrix of elements between zero and one;

(vii) N+A has an inverse.

 

These assumptions have been taken from the Fair article [16] with
only minor changes.

51
Solving (4) for the reduced form, we obtain
(6) Y = -(N+A)-1 BY_1 -'(N+A)-1CX + (N+A)'1 U*
which, after Koyck transformation with factor R, gives

(7) Y-RY_1 = (N+A)'IB(Y_1-RY_2)-(N+A)'IC(x-Rx_1) + (N+A)'1(R-R)U:c1 + (N+A)'1v.

At this point we introduce the assumption that R is a diagonal
matrix of identical elements, and thus can be written rI. We are now
1 . . . . .
prepared to search over r to minimize the first stage reSiduals
/\ . .
(Y-rY 1)-(Y-rY 1). Let r be the trial value for r; then the disturbance
in (8) will be at a minimum for E = r:

l

(8) Y-?Y_ = -(N+A)- B(Y -}Y_2)-(N+A)’1C(x-rx_1)+(N+A)‘1(r-?)Uff(N+A)’1v.

l -l

The transformed structural form is

-A = .. -A - -A - ..A -A * .
(9) N(Y rY_1) A(Y rY_1) E(Y-l rY_2) C(X rX_1) + (r r)U__1 + V,

if we carry out ZSLS replacing the nonnormalizing jointly dependent vari-
ables with predicted values calculated from (8) there is no guarantee
that the minimum sum of squared errors (SSE) will occur at r=r, since

the combined disturbance on (9) after the instrument substitutions will

A /\
be (r-r)U‘f1 + V + A(Y-rY_1) - A(Y-rY_1), and AY_ and Uf are not un-

l l
correlated (and thus the combined disturbance cannot be minimized by parts).
The r chosen would minimize the sum of the variances and the covariances,

and thus would differ from r. The solution is to include enough variables

in the instrument equation to guarantee the orthogonality of AY_1 and Ufl

 

1 0r iterate.

52

by taking advantage of the fact that least squares residuals are ortho-
gonal to the space spanned by the regressors.1 Since

*

(10) U_1 = (N+A)Y_1 + BY + cx_

-2 l

we must include in the instrument equations for each structural equation
lagged values of all variables that occur in that particular structural
equation.2 We call the resulting equations constrained augmented reduced

form equations and write them

Y

 

(ll) (Y-rY_1) = mg Y-: -(W:E1L:E(::l::::g)-(N+A)-1C(X-rx_1)
" M
(GxT) (0x30) x_1 (GxT) (GxT)
(BGxT)
-1 -l
+(N+A) (r-r)U__1 + (NhA) V
_———"‘\c/"‘~—-_
(GxT) (GxT)

. . . th .
where fig 15 the augmenting matrix for the g equation:

”g E [nAi nB 1 WCJg

(Gx3G) (GxG)(GxG)(GxG)

 

1 If the coefficients are restricted, then the least squares residuals
are not necessarily orthogonal; hence the latter inclusion of
separate, "extra" variables to eliminate the autoregression restric-
tions on the predetermined variables occurring in the structural
equation.

2

The inclusion of these "extra" variables does not affect the disturbance
term and may be regarded as a dropping of the nonlinear autoregression
restriction relating current and lagged values of the variables
occurring in the structural equation.

53

[UAJg = [(nA)ij] = 0 if the g,j element of A, [agj], = 0 (((g=l,2,...G),
i=l,2,...G),
1 . j=19290006)
3 [(fiA)ij] otherWise
EnBJg = [(nB)ij] = 0 if the g,j element of B,[bgj], = o (((g=1,2,...c),
i=1,2,...G),
J1 . j=1,1,...G)
— [(flB)ij otherWise
[11C]g = [(nC)ij] = 0 if the g,j element of C’Ecgj] = O (((g=l,2,...G),
1:122:00-G):
- 1 J=1,2,...G)
_ [(nC)ij] otherwise

(the fig matrix is simply a method of including lagged values of the pre-
determined variables that occur in the g£11 structural equation in the
constrained augmented reduced form used in obtaining instruments for that
equation).

Two stage least squares can now be performed on each equation
separately, both stages being calculated at some trial value ? of the auto-
regression parameter, with the instruments being obtained from.the
constrained augmented reduced form equations (11) corresponding to the
given structural equation. The second stage of this procedure, with the

2
instruments substituted in, will be (for the gEh equation)

 

1 The c fficient of the constrained augmented reduced form equation for
the g—- equation.

2 Note that (Y:;§:I) (§-§y_1).

54

-~ =- 7A- A- _ A
(12) yg,t rysvt-1 a11("1,t ryl,t-l) a12(y2,t ry2,t-l) "' alg(yg,t ryg,t-1)

L5 3

A ' .
)-b12(y2,t-1-ry2,r-2)-°°°-blg(yg,t-l-ry )

A
“b11(y1,t-1'ry1,t-2 g,t-z

"\

(x -rx )-c (x -rx )-...-c (x -ix )
l,t l,t-l 12 2,t 2,t-l 1g g,t g,t-l

"C11

+

+...+(n+a)1gu

a ll 12
(r-rflKn+a) u1,t-1 + (n+a) u g,t-l

2,t-l ]

+...+ (n+a)1gv

12
+
(n+a) V2,t g,t

(n+a)11v

+

l,t

+

-A - ...———"—-"\ ‘A
all[(yl,t r"Lt-1) (y1,t ”l,t-1)]+ a12[@2,t W2,t-1)

- mmm” al, to " >-<'y/-’r§\

g g,t'ryg.t-1 g,t g,t-1)]’

where the lower case subscripted letters refer to particular elements of
the uppercase matrix, while superscripts denote elements of an inverted
matrix.

The vij's are independent of the rest of the disturbance by assumption;
inclusion of the augmenting variables in the first stage guarantees ortho-
gonality of the disturbance terms in uij and yij' Thus the minimum SSE will
be at i=r, as desired.

The estimation is carried out for each equation, choosing the estimate
of the system autoregression parameter that minimizes SSE for the particular
equation. (The across-equations restriction that the system autoregression
parameter is assumed the same for all equations is purposely ignored
everywhere except in the first stage for each equation).

Since V is hypothesized to have a cross-equation covariance [assump-

tion (ii) above], if the model is overidentified there will be an efficiency

SS

gain from three stage least squares (BSLS) estimation. Considering
3SLS as Aitken's generalized estimation over ZSLS starting estimates
(with instruments substituted for nonnormalizing jointly dependent
variables), we can follow the usual procedure of replacing the unknown
across-equation disturbance variance-covariance matrix, x, with a
consistent estimate 2 obtained from the ZSLS residuals.1

Recalling the model

(4) (N+A)Y + BY_1 + ox = U*

*
(5) U*5 RU-1-+ v
we rewrite it after the autoregression transformation

(13) (N+A)Y - R(N+A)Y_1 + BY_1 - RBY_ +-CX - ch_ = v .

2 1

For general R (not necessarily diagonal), at time t the first equation
can be written:

(14) [a + ...+

- - r
11’1: (r11811+r12821+°"+rlgagl)y1,t-l]+[a12y2t (r11a12 lgagZ)y2,t-1]

l

+...+ - r a +r a +...+r a
[algygt ( g)y

11 lg 12 2g 1g g g,t-l

+[b b +r b +;;.+r b

11y1,t-1'(r11 11 12 21 lg g1>y1,t-2]+[b12y2,c-1'(r11b12

b +...+r

+r12 22 lgbg2)y2,t-2]

+...+[b (r b +r b +

lgyg,t-l 11 lg 12 2g °"+r1gbgg)yg,t-2]

 

1 As Ruble mentions [53], 2 must be estimated from the ZSLS residuals

calculated from the actual jointly dependent variables rather than

their instruments in order for this procedure to give 3SLS estimates.

The fact that there are as many separate estimates of the system auto-
regression parameter as there are stochastic equations in no way effects
the consistency--given the assumptions--of z estimated from the residuals
of these equations: the restriction across equations has simply not been
incorporated into the estimation process .

56

+[c11x1t-(r11c11+r12c21+. ° °+r1gcgl)xl, t-l HG12X2c' (r11°12+r12°22

+...+rlgcg2)x2,t_1]

+...+[c1gx xg,t-l] - v11.

gt-(rllclg+r12c2g+°"+rlgcgg)

The Aitken procedure does not directly include the knowledge that
R is diagonal (as does the constrained first stage of ZSLS), and is itself
consistent with a nondiagonal R (though the parameters are unidentified
without further assumptions about R); With the assumption that R is diagonal
all of the rij terms in (14) for which i#j are zero and drop out and the
equation will normally be overidentified,1 giving rise to one or more
tuxflinear (linear given rii) restrictions on the coefficients within the
equation. Two estimation possibilities present themselves: (1) estimate
the equations via BSLS ignoring the nonlinear restrictions,2 since they
will hold in the probability limit anyway; or (2) choose values for each
of the r11, and perform BSLS estimation subject to the now linear restric-

tions. Estimates from both methods will be obtained for the model, with

the rii set at their ZSLS estimates in the latter estimation procedure.

 

So long as at least one exogenous variable occurs or not all of the

jointly dependent variables occur with lags in the original specifica-
tion.

The r implied by the coefficient estimates for each equation can be
calculated and used as an index of either sampling error or non-
diagonality of R;

57
2. Structural parameters

Parameter estimates1 based on the autoregression
correction technique discussed above are presented in this section;
Figure I gives the first stage regressors that appear in each constrained
augmented reduced form: Orcutt's iterative technique has been employed
in estimating the autoregression parameters for the Pascal lag equations,
while a search method has been used for the remaining stochastic equa-
tions; the results of the search are tabled in Appendix III-B.

Three sets of estimates are given: ZSLS, BSLS restricted, and 3SLS
unrestricted. "BSLS restricted" refers to the estimation procedure
mentioned at the end of the preceding section in Which the autoregression
parameter for each equation is restricted to its ZSLS estimate--thereby
reducing the nonlinear relations between the coefficients of each current
and lagged variable to linear relations conditional on the given value
for the autoregression parameter. (The unrestricted 3SLS estimation
drops these restrictions, allowing 3SLS to affect the values of the auto-
regression parameters but loosing the information contained in the
between-coefficient restrictions.)

The format is

ZSLS

3SLS restricted
BSLS unrestricted

 

1

All estimates are conditional on the nonautoregression corrected
classical ZSLS estimates 03 the Pascal lag parameters. Five

1 .
equations-~normalized on I ,I ,F ,i , and S --were assumed to be
known nonautoregressive from their specification.

58

Table 5

Additional Instruments 13 the Constrained Augmented Reduced Form Equations

 

Additional Variables

 

First Stage Degrees
Equation YSL'I 8,-1 of Freedom
Consumption C__1,(YP)._1 (cC)_1 1
Investment durable [Id 1 0 (i ) Id 1
: -1 ’ -1’ d -1 -2
. . r .
Investment, reSidential I_1, X-1’(ls)-l l
I t ' t 1 1i x1 i
nvestmen , inven or es _1, _1 (ISTK)-2 1
Imports [(FI)_1], (Yd)_1(pF)_1 (FI)"2 1
. P
Production [X_1], (hm)_1:[K_1] 3
. . .P
Long term interest (1 ) ,(1 ) (c. ) 27
L -l -l 1L
Short term interest (13)-1 R-l’(id)-l 27(OLS)
Corporate savings (S ) 1,(P T ) 1 (PC-Tc--Sc)_2 1
Corporate profits (PC)_1,P_1 2
Depreciation D 1,[K 1] 3
Agricultural income (APA)-1,(YNF) 1 (Dw)_1 1
2
wage bill (W1)_1, [X 1] X82
Nominal wage rate (AW)_1,AP 1 (U-l)-1 l

 

Brackets indicate that the variable already occurs in the equation. There
are 33 observations and 29 predetermined variables:

. -1 d
w2.A2.q,.Nw.(Nw-NG>.PA.(0+Fx).R.1d.TC.TW.1fA.IP,T,U .DW.I_1.(FI>_1.

i -l .-1
w_1,P_1,(ISTK)_1,K_1,X.I_)1,(PC-TC-Sc)_1,(Yc ),cC,ci ,(iS ), constant.

L

59

with estimated standard errors in parentheses below the coefficients

. 2 . .
where appropriate. R , the standard error of estimates, and the Durbin-
Whtson statistic for the ZSLS estimates are reported to the extreme

right, as before.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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as oow.o one.~- +
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mouMEHumm noumEmHmm nmueoumoo mowmmoummHOumd

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64

3. Reduced form parameters
Selected reduced form parameters for all three auto-

regression corrected models are recorded in Table 7.

 

 

 

 

 

 

 

 

 

 

 

 

 

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vouuwuumwh mAmm

E
0AA0.0 Aoqo.0- 0000.0- 0A~m.0- 0500.0- 00~0.0- m0~n.0- «mmw.0 5
0500.0- ~n00.0- ~000.0- 0000.0- ~000.0- m000.0- ”000.0- mwoo.0 m

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Agape: pmuuauuou sawmmmuwoLOu=<0

mMMHAmHHADZ Ho<m2~

A «Asap

66

Table 8

1
Autoregression Parameter Search Results

Durable Investment

(1 A I‘ A . A d
It ‘ BZ,O+BZ,lOt+BZ,2(1L)t + B2,3It-1

 

 

 

a Model SSE 0 1L Constant (Id)_1
0.0 C 189 0.315 -1017 '8067 09505
0.0 F 189 0.313 '1.23 r8.37 0.509
0.1 C 185 0.343 -1.17 -8.89 0.456
001 F 186 00348 -1029 ”8.65 0.449
0.2 C 185 0.380 -1.15 “9.23 0.390
0.2 F 187 0.391 -1.36 -8.95 0.372
0.3 C 185 0.423 -1.08 -9.53 0.311
0.3 F 188 09440 -1041 -9014 00288
0.4 C 184 0.468 -0.95 '9.63 0.230
0.4 F 189 0.489 '1.46 -9.02 0.206
0.5 C 180 0.511 ~0.75 '9.38 0.154
0.5 F 189 0.533 -1.51 -8.52 0.136
0.6 C 177 0.547 -0.49 ~8.67 0.093
0.6 F 191 09573 ”1.57 -7067 0.081
0.7 C 175 0.577 '0.20. -7.46 0.049
0.7 F 195 0.610 -1.61 -6.57 0.043
0.8 C 180 0.604 0.04 -5.78 0.025
0.8 F 201 0.649 -1.48 -5.33 0.022
009 C 194 09640 0.20 “3.64 0.022
099 F 205 00687 -0988 -3069 0.020

1

C refers to the classical ZSLS model; F refers to Fisher's structurally
ordered instrumental variables model.

67
Residential Investment

r - A A A .
It " 83,O+63,1Xt+63,2(ls)t

 

 

 

OO
\00

‘ COeffICients
p Model SSE DW X is Constant

0 0 C 364 0.53 0.0573 €2.10 -0.482
0 0 F 365 0.53 0.0562 ~2.08 ~0.131
0 1 C 305 0.62 0.0578 -2.13 ~0.599
0 1 F 306 0963 090575 -2922 "09292
O 2 C 255 0974 090582 -2913 “09712
0 2 F 258 0.79 0.0592 -2.39 -0.479
0 3 C 215 0989 090583 -2909 “09807
0 3 F 220 0999 090611 -2956 “09677
0 4 C 184 1.06 0.0578 -1.97 -0.864
0 4 F 191 1.22 0.0622 -2.63 '0.823
0 5 C 161 1.23 0.0565 '1.76 '0.860
0 5 F 167 1938 09060]. -2935 “09752
0 6 F 148 1.38 0.0502 -1.39 -0.177
0.7 C 135 1.56 0.0508 -1.03 -0.641
0.7 F 158 1.43 0.0289 0.25 1.050
O 8 C 129 1976 090474 -0966 ”09519
0 8 F 215 1.73 -0.0035 1.91 2.540

C 130 1.92 0.0481 ~0.47 -0.602

F 321 1.95 '0.0618 3.19 3.950

 

68

Inventory Investment

r_‘~ " i A i
It734,0+84,1Xt:+B£+,2 (ISTK)t-1

 

 

 

Coefficients
SSE DW Xi Cons tant (IgTK1-L
O 0 254 1.55 0.118 -24.2 -0.477
0 0 265 1.27 0.142 -29.1 -0.584
0 1 238 1964 09124 “2299 ‘09504
0 1 249 1.34 0.148 -27.5 -0.613
O 2 226 1.72 0.132 -21.7 -0.538
0 2 235 1.41 09156 '2599 “09650
0 3 216 0.141 -20.4 -0.581
0 3 224 0.167 -24.2 -0.696
0 4 207 0.153 ~19.0 ~0.631
0 4 216 0.179 -22.4 -O.749
O 5 200 0.166 -17.3 -0.687
0 5 208 0.193 -20.3 -0.808
0 6 195 0.180 -15.1 -0.744
0 6 203 0.208 -17.6 -0.864
0 7 192 0.194 ~21.4 -0.800
0 7 199 0.221 -l4.2 -O.910
0 8 192 0.208 - 9.0 -0.840
0 8 197 0.233 -10.3 -0.934
0 9 196 0.225 - 5.2 -0.861
0 9 199 0.245 - 5.8 -0.926

 

_A f A -1 ’
(F1)t‘55,0+55,1(Yd)t+55,2(pf )t+BS,3(F

69

Imports

I t-l

 

 

 

Céefficients
Mbdel SSE
Yd p31 Constant ZFITI
J
0 0 C 44.0 0.0275 5.6 -9.2 0.813
0 0 F 44.5 0.0317 7.2 -11.3 0.767
0 1 C 44.5 0.0351 7.1 -10.4 0.731
0 1 F 45.2 0.0409 9.1 '12.8 0.668
0 2 C 45.5 0.0436 8.7 '11.2 0.639
0 2 F 46.6 0.0508 11.2 -13.8 0.562
0 3 C 46.7 0.0527 0.2 -11.5 0.543
0 3 F 48.2 0.0610 13.2 -14.3 0.456
0 4 C 47.7 0.0622 11.4 ~11.2 0.446
0 4 F 49.5 0.0710 14.8 -13.9 0.356
0 5 C 48.2 0.0716 12.0 -10.2 0.353
0 5 F 50.3 0.0807 15.9 -12.7 0.265
0 6 C 47.8 0.0812 11.8 - 8.5 0.268
0 6 F 50.0 0.0901 16.0 -10.8 0.187
0 7 C 4690 090911 097 - 695 09192
0 7 F 4799 090996 497 - 891 09123
0 8 C 42.4 0.0102 8.8 - 4.3 0.132
0 9 C 38.3 0.1180 7.2 - 2.5 0.0 5
0 9 F 39.9 0.1310 10.5 - 3.2 0.049

 

70

 

 

 

CO
\DW

Production
XE=B6,O+B6 , 1 (hm) t+B6 ,2Kc
SSE DW Coefficients

nm K Constant
0 0 3135 0.38 6.29 0.282 -19.4
0 1 2670 0.45 6.24 0.285 -16.0
0 1 2672 0.45 6.31 0.282 ~17.9
0 2 2269 0.55 6.25 9.285 -l4.4
0 2 2271 0.55 6.33 0.281 ~15.4
0'3 1928 0.68 6.26 0.284 -12.9
0 4 1648 0.86 6.29 0.283 -11.4
0 4 1652 0.86 6.42 0.277 -13.8
0"5 1429 1.07 6.32 0.281 - 9.9
0 5 1434 1.07 6.51 0.274 -12.5
0 6 1271 1.31 6.38 0.279 - 8.5
0 6 1277 1.31 6.61 0.269 -11.1
0‘7 1172 1.56 6.45 0.275 - 6.8
0‘7 1182 1.55 6.74 0.262 - 9.2
0 8 1134 1.77 6.52 0.271 - 4.7
0‘8 1146 1.77 6.87 0.254 - 6.3

1155 1.92 6.57 0.263 - .

1 9
1173 1.91 6.96 0.238 - 2.4

 

71

Short Term Interest Rate

9 = A A ‘ 9
(ls)t B8,0+88,1Rt+88,2(1d)t

 

 

 

b Mbdel SSE DW Coefficients
Constant R
0.0 C 5.55 1.34 -0.228 -1.43
0.0 F 8.72 1.48 -0.241 -2.60
0.1 C 5.30 1.43 *0.249 -1.35
091 F 8918 1959 -09159 ‘2953
0.2 C 5.11 1.52 ”0.264 -1.24
0.2 F 7.77 1.72 “0.077 -2.45
093 C 4.97 1961 '09272 “1.10
093 F 7946 1984 -0900]. '2934
094 C 4987 1969 -09265 “0994
094 F 7924 1996 -09070 '2.21
0.5 C 4.79 1.76 -0.241 -0.76
0.5 F 7.12 2.07 -0.125 -2.05
0.6 C 4.74 1.81 -0.197 -0.57
0.6 F 7.12 2.17 ~0.158 -1.88
097 C 4971 1985 -09138 "0939
097 F 7928 2925 “09160 “1.71
098 C 4971 1987 -09076 -0925
0.8 F 7.72 2.31 -0.128 -1.58
0.9 C 4.80 1.88 -0.025 -0.15
099 F 8966 2933 -09067 -1956

 

72

Corporate Profits

 

 

 

 

 

(P)-A +6 P
c t 510,0 10,1 t
SSE DH __§oefficients
P Constant fi_

338 0.30 0.570 -11.24
341 0.31 0.562 ~10.58
285 0.36 0.569 -10.10
288 0.37 0.561 - 9.43
239 0.43 0.569 - 9.02
242 0.45 0.559 - 8.29
C 199 0.54 0.570 - 7.93
F 203 0.57 0.557 - 7.15
C 165 0.69 0.570 - 6.87
F 171 0.73 0.554 - 6.00
C 138 0.88 0.572 - 5.85
F 145 0.94 0.550 - 4.87
C 118 1.10 0.577 - 4.88
F 128 1.20 0.545 - 3.74
103 1.30 0.588 - 4.01
F 119 1.49 0.537 - 2.62
91 1.40 0.613 - 3.24
121 1.74 0.524 - 1.53
75 1.29 0.671 - 2.39

141 1.90 0.498 - 0.53

73

Corporate Savings

(Sc)t = B9,0+B9,1(Pc-Tc)t+89,2(PC-Tc-Sc)t-1

 

COefroIEnts

 

 

CO
\D\O

p Model SSE Dw §C_f;' Constant (Pé:T¢-sc)
0 0 C 88.8 1.94 0.820 -2.61 -0.466
0 0 F 90.1 1.71 0.793 '2.79 -0.402
0 1 C 86.7 1.95 0.796 -2.68 -0.390
0 1 F 88.7 1.65 0.762 -2.86 '0.311
0 2 C 8493 1992 09769 ”2973 '09304
0 2 F 86.8 1.59 0.730 -2.86 -0.218
0 3 F 83.7 1.54 0.698 -2.76 '0;132
0 4 C 77.0 1.86 0.713 -2.59 '0.130
0 4 F 79.5 1.55 0.670 ~2.53 ~0.061
0 5 C 7390 1988 09688 I'293‘1’ -09059
0 5 F 75.1 1.61 0.647 -2.20 -0.009
0 6 C 69.8 1.94 0.667 -1.98 -0.007
0 6 F 71.4 1.73 0.627 ‘1.79 0.025
0 7 C 68.3 2.03 0.649 -1.54 0.027
0 7 F 69.7 1.86 0.609 -1.34 0.042
0.8 C 69.4 2.11 0.636 '1.09 0.046
0.8 F 70.8 1.96 0.591 ~0.90 0.048

C 73.6 2.16 0.632 -0.66 0.056

F 75.2 2.02 0.578 -0.52 0.047

 

74

Depreciation

Dt =811,O+811,1Kt

 

 

 

Coefficients

Model SSE Dw K Constant
0 0 86.5 0.25 0.0521 14.50
0 0 86.5 0.25 0.0522 14.40
0 1 71.1 0.29 0.0521 13.00
0 1 71.2 0.29 0.0522 13.00
0 2 57.8 0.34 0.0520 11.60
0 2 57.8 0.34 0.0521 11.60
0.3 46.4 0.42 0.0520 10.20
0.3 46.4 0.42 0.0521 10.20
0 4 36.9 0.54 0.0519 8.76
0 4 37.0 0.54 0.0520 8.73
0 5 29.5 0.72 0.0518 7.33
O 5 29.5 0.72 0.0520 7.30
0 6 24.0 0.96 0.0517 5.90
0 6 24.0 0.96 0.0519 5.87
0 7 20.4 1.24 0.0514 4.48
0 7 20.4 1.24 0.0517 4.44
0 8 18.8 1.51 0.0510 3.05
0 8 18.9 1.51 0.0516 3.00
O 9 19.1 1.66 0.0500 1.64
0 9 19.3 1.66 0.0515 1.55

 

75

Agricultural Income

(Ap ) = B12,0+ e12,1(YNF)t+ é12,2(Dw)t

A

 

Coefficients

 

 

OO
\OKO

SSE DW —1NF Constant DW
0.0 37.4 1.70 -0.00224 9.98 3.39
0.0 37.5 1.69 -0.00199 9.95 3.35
0.1 37.1 1.96 -0.00187 8.95 3.29
0.1 37.1 1.95 -0.00168 8.93 3.26
0.2 37.3 2.22 -0.00137 7.92 3.16
0.2 37.3 2.22 -0.00130 7.91 3.15
0.3 38.2 2.46 -0.00073 6.88 2.99
0.3 38.2 2.46 -0.00084 6.89 3.02
0.4 39.7 2.67 0.00011 5.85 2.76
0.4 39.7 2.68 0.00026 5.88 2.84
0 5 42.0 2.86 0.00131 4.81 2.43
0.5 42.0 2.87 0.00047 4.87 2.60
0.6 44.8 3.00 0.00308 3.78 1.93
0.6 44.9 3.02 0.00145 3.87 2.25
0.7 48.3 3.11 0.00308 2.74 1.08
0.7 48.5 3.13 0.00276 2.88 1.72
0.8 52.3 3.20 0.01060 1.73 -0.58
0.8 52.8 3.21 0.00438 1.92 0.82

56.9 3.28 0.02020 0.77 -5.27

58.2 3.26 0.00457 1.02 -0.93

 

76

Private Wage Bill

_ A f' ’ p
(“1): " 513,0‘L513,1(X5’1:+513,2(X )t-l

 

*—

 

 

OO
00

Coefficients

SSE DW XP Constant (XP)_1
0 0 225 1.09 0.414 -14.2 0.1510
0 0 354 0.71 0.549 -13.8 0.0092
0 1 225 1.03 0.442 -13.0 0.1220
0 1 306 0.75 0.541 -12.6 0.0187
0 2 223 0.99 0.468 -11.6 0.0957
0 2 272 0.81 0.537 -11.4 0.0234
0 3 217 1.01 0.488 ~10.3 0.0754
0 3 247 0.87 0.536 -10.2 0.0254
0 4 209 1.05 0.502 - 9.00 0.0612
0 4 229 0.95 0.537 - 8.97 0.0261
0 5 201 1.11 0.513 - 7.69 0.0517
0 5 215 1.04 0.538 - 7.77 0.0264
0 6 195 1.19 0.520 - 6.38 0.0454
0 6 206 1.13 0.540 - 6.56 0.0267
0 7 193 1.27 0.526 - 5.04 0.0414
0 7 201 1.22 0.543 - 5.37 0.0273
0 8 194 1.34 0.532 - 3.63 0.0387
0 8 203 1.30 0.549 - 4.17 0.0285

202 1939 09537 - 1998 090364

217 1.34 0.564 - 2.92 0.0307

 

77

Change in the Nominal Wage Rate

- A o '1
Awt é14,0+ e14,1Apt+ I514,24”:

 

 

 

CO
00

Coefficients
Mod 1 SSE -

e DW Apt Constant %111
0 0 C 0.0376 1.94 2.22 -0.0825 0.0298
0.0 F 0.0397 2.09 2.73 -0.0714 0.0268
0 1 C 0.0394 2.15 2.42 0.0007 0.0287
0 1 F 4 0.0410 2.24 2.80 -0.0549 0.0263
0.2 C 0.0415 2.33 2.57 -0.0007 0.0279
0.2 F 0.0423 2.36 2.77 -0.0549 0.0264
0.3 C 0.0441 2.46 2.64 -0.0516 0.0274
0.3 F 0.0445 2.47 2.74 -0.0482 0.0265
0.4 C 0.0473 2.56 2.67 -0.0433 0.0271
0.4 F 0.0475 2.56 2.73 -0.0412 0.0265
0.5 C 0.0513 2.63 2.68 -0.0355 0.0269
0.5 F 0.0514 2.64 2.72 -0.0341 0.0265
0 6 C 0.0561 2.69 2.69 -0.0278 0.0267
0.6 F 0.0562 2.69 2.71 -0.0267 0.0263
0 7 C 0.0617 2.73 2.69 -0.0203 0.0265
0.7 F 0.0617 2.73 2.71 -0.0194 0.0261
0.8 C 0.0680 2.76 2.69 -0.0129 0.0261
0.8 F 0.0681 2.76 2.71 -0.0121 0.0256

C 0.0751 2.77 2.70 -0.0058 0.0257

F 0.0752 2.78 2.71 -0.0051 0.0251

 

III. Conclusion

As is the case in classical hypothesis testing, model changes are
notoriously difficult to assess unless they are "nested," that is, the
change includes the prior specification as a special case; in this
event the researcher at least knows that in the limit the change will
not do damage, although with limited evidence information may still be
lost if the generalization is unnecessary. 0f the seven major types
of changes that have been made to the model, five include the original
K-G specifications as nested subcases, while two represent an alternate
underlying set of assumptions. The two nonnested changes are the theo-
retical changes in equation Specification and the SOIV treatment of
the simultaneous equations problem. The nested changes were the im-
proved treatment of variable nonlinearities in determining the multi-
pliers, estimation of theoretically superior lags, the autoregression
correction, use of an expanded data set, and the use of a "full-
information” estimation procedure (BSLS). We shall examine each of
the changes in turn.

A. Theoretical Changes in Equation Specification

Although we deviated from the original K-G specification in

many places, the three most important were the disaggregation of in-
vestment, the specification of the two equations of the monetary
sector, and the derivation of the consumption function. The consump-
tion function will be disucssed later, when we evaluate the Pascal
lags.

Although the disaggregation into durable, residential, and inven-
tory investment (see sections II.A.2., 3., 4. above) is an obvious and

much-needed improvement in the K-G model, the equation errors remain

78

79

relatively large. The dominant variable in the durable investment
equation, nonwage disposable income (retained from the original K-G
investment specification) is unfortunately not one that the model is
adept at tracking; the errors in Id in the single period sample data
solution correspond closely to the errors in the determination of
nonwage disposable income.1 In addition, the eXpense of estimating
the Pascal lags restricted their application to only one equation in
the main body of the model; had the consumption function not been the
logical center of the K-G model, the durable investment equation would
have been specified with the Pascal lag2 -- where the convolution of
expectations and adjustment mechanisms is likely to be very important
and the simple first-order adjustment model less satisfactory. With
the results of Appendix 111 it is feasible to estimate different
expectations and adjustment roots; in order to simplify estimation
each process may be specified a priori to have a known number of roots
(based on the desired lag shape), and further, the roots may be speci-
fied to be equal within a process but different across processes:

then the parameter search is only two dimensional. That is, the Pascal

lag with unknown convolutions but equal roots would be exchanged for a

 

See figures 13 and 24. Nonwage disposable income has not been
plotted separately; since disposable wage income is relatively
stable, the actual and solution values of nonwage disposable income
are almost identical to the corresponding values of total dispos-
able income.

The surprisinqu good estimation residuals (which did not continue
through the full model solution phase) further influenced the
decision to use the Pascal lag on the consumption function.

80

rational lag of known convolutions with subsets of roots equal.1 A
durable investment equation of this type would likely have much im-
proved the model.

Inspection of the single period residential investment solution
(Appendix III) suggests that the shape of the curve is correct but the

A A

timing is wrong -- 1: corresponds more closely to IE+1 than to 1:.
Since the initial formulation of the equation contains no lagged vari-
ables, this effect must be a result eithercflfthe autoregression
correction or of input2 model solution values of the right-hand side
endogenous variables. The Durbin-Watson statistic for the uncorrelated
equation is an autoregressive 0.53 (Appendix III-B), and the sum of
squared errors (SSE) drOps from 364 to 129 when 6 is set to 0.8. Thus
we conclude that the autoregression correction is necessary and the
minimum is meaningful. Examining the input variables we note the
slight lag in the solution value of GNP (Figure 11); this is evidently
the source of the residential investment lag, and the equation itself
performs adequately.

The inventory investment equation at least performs the function
of confining the errors resulting from excessive time aggregation to

the equation in which they originate; the poor tracking of this equa-

tion is primarily a result of the use of annual data.

 

Estimation algorithms are currently being develOped for this case.

We assume that the primary determinants of the endogenous variables
are the variables in the equations normalized in the endogenous
variables in question; in a fully integrated system there are
clearly feedback effects from the other equations, but these are
assumed small relative to the other effects.

81

B. Simultaneous Equations Problems
An annual model provides an Opportunity to assess the impor-

tance of inconsistency resulting from simultaneous equations problems,
since the longer the observation time period the more nearly simultan-
eous the equations become. First, however, a digression is in order.

There has been considerable confusion regarding the calculation
of ZSLS parameter estimates when the number of first stage regressors
equals or exceeds the number of observations; specifically, ZSLS is
often said to be undefined when this condition prevails.1 As Ruble
demonstratesz, however, the ZSLS estimates are perfectly well-defined
in this case and are in fact equal to ordinary least squares (OLS)
estimates. Two-stage least squares is an estimator in which the right-
hand side jointly dependent variables (Y) are replaced by their esti-
mated values Y, the part of Y in the space spanned their first stage
regressors (denoted XI). If the number of observations equals the
number of regressors, we can obtain coefficients relating (perfectly)

A

Y and XI: that is, Y E Y, or the entire Y vector is contained in the
space spanned by XI. If the number of first stage regressors exceeds
the number of observations, unique coefficients cannot be obtained --
but the coefficients are not required, only the part of Y in the space
of XI’ and all of Y is in this space again, so that Y E Y. But when
2 E Y, ZSLS E OLS. Asymptotically, of course, the two estimates will

diverge, since the number of first-stage regressors will not increase

as the number of observations does, but for the given sample the

 

1 See, for example, [37], p. 175.

2 [53].

82

estimates are equivalent. Thus classical ZSLS is not a good way to
take equations Simultaneity into account.

In this particular model (estimated from 33 observations), there
are 29 predetermined variables in the first-stage regressions for
those right-hand jointly dependent variables in the main body of the
model before the autoregression correction; after the required auto-
regression augmentations (see II.C.1 above), the median first-stage
degree of freedom value is 1 (Table 5 above). Thus the Y's, although
never quite equivalent to the Y's, are very close indeedl, and our
classical model is one that assigns little importance to simultaneous
equations problems and results in estimates very close to ordinary
least squares estimates.

The structurally ordered instrumental variable (SOIV) model is the
opposite extreme,2 placing heavy emphasis on equation Simultaneity,
since it is based not only on causal structure within the model, but
also recognizes the existence of policy variable reaction functions
outside the model. How different, then, were the parameter estimates
resulting from these two models?

The unambiguous conclusion is that the estimates are not very
different. The parameter estimates from the two models at each trial

value of the autoregression parameter are tabled in Appendix III-B; in

 

1 Subsequent calculations showed that the correlation between Y and Y
was never less than 0.99 for any right-hand side jointly dependent

variable.

The identification of the equations provides an index to the polarity
of the two models: all of the SOIV model equations have between 0
and 3 overidentifying restrictions (Table 2 above), while the clas-
sical model equations have between 30 and 32 overidentifying res-
trictions.

83

only three cases are the differences worthy of comment, in the equations

determining the short term interest rate, private wage bill, and im-
ports.

In the short term interest rate equation the generalization to
endogenous policy is likely to be important, since there may be strong
feedbacks from the short term rate to the discount rate. The SOIV
model tends to increase the coefficient on the reserve ratio and de-
crease the coefficient on the discount rate, although at the means
(with 6 = 0) the net change in the short rate is only 0.2%.

In the private wage bill equation the classical and SOIV estimates
are in approximate agreement at 8 = 0.7 (the minimum SSE), but diverge
as drops; at 6 = 0 there is considerable difference between the two
models, with the SOIV model estimates remaining much closer to the
previous estimates while the classical model begins to put more
weight on the lagged private output term. It seems that the SOIV
model is less likely to overstate the importance of lagged terms in
the presence of autoregression, principally because many of the lagged
endogenous first stage regressors were not included in calculation of
the instruments -- thus lowering the correlation between the instru-
ments and the second stage error and decreasing the inconsistency.
Similarly, in the import equation the SOIV model consistently puts
less weight on the lagged term, although in neither of these equations
are the parameter differences important; when the SOIV model estimates
are different at all, it is primarily due to better estimates in the
presence of autoregression.

Thus the conclusion must be that the inconsistency introduced by

equation Simultaneity, whether derived from unstated policy reaction

84

functions or from supposedly inapprOpriate estimation procedures, is
negligably small; in fact, in large econometric models ZSLS is "sample
equivalent" to OLS since (1) in the absence of significant expectations
processes all of the lagged endogenous variables become predetermined
(increasing the number of first stage regressors), (2) if the auto-
regression correction does not constrain the augmented reduced form
equations as above the number of first stage regressors is vastly
increased, and (3) if the model is nonlinear the reduced form Taylor
expansion need not be arbitrarily stOpped at first—order, and any
number of first-stage regressors may be included. Any one of these,
and certainly all together, virtually assumes that the number of first-
stage regressors will exceed the number of observations, guaranteeing
the sample equivalency of OLS and classical ZSLS. Further, if our
results generalize,enmn1the polar SOIV estimates will differ little
from the OLS estimates, and estimation might just as well be carried
out by the less expensive OLS method as far as consistency is concerned.

C. Variable Nonlinearities

Goldberger calculated the original K-G multipliers from a

simplified, linearized reduced form;1 we compared impact multipliers
obtained from an unsimplified reduced form linearized in three separate
ways and found them somewhat different (Table 4), although the com-
parison was distorted by overlarge multipliers calculated from the
model as yet uncorrected for autoregression.

The full generalization of the Goldberger calculations is to
numerically differentiate the model at the apprOpriate point; this is

the technique underlying the multipliers in tables in Appendix IV.

 

1 [25], pp. 71-77.

85

Comparison with Table 7 (the autoregression corrected linearized impact
multipliers) leads to the conclusion that the linearization loses
significant information, even when the multipliers are smaller: the
government expenditures-GNP multiplier differs by 14 per cent, the
-manhours multiplier by 13 per cent, and the -price multiplier changes
sign. In this particular model, at least, the multipliers are somewhat
sensitive to variable nonlinearities. However, since the preferred
technique (numerical differentiation) is usually the least expensive,
all modern multipliers are calculated by this method (in 1959 Goldberger
did not have available the nonlinear solution programs of today, and
thus had to use the less-accurate labor intensive linearization); the
linearization inaccuracy is of consequence only in its effects on the
first-stage instrument calculation. Expansion of the approximating
polynomial to a higher degree reduces ZSLS to OLS in most econometric
models in the course of obtaining better first-stage estimates, and
even then serious inefficiency may result if all of the variables

(not just those suspected a priori of being important nonlinearly)

are included with higher order terms, and inconsistency may result if
they are not (see above). Thus the conclusion is that for most macro-
economic samples ZSLS has no advantages over OLS in the presence of

variable nonlinearities.

D. Pascal Lags
Two equations, determining consumption and the long-term
interest rate, have been estimated incorporating Pascal lags. The
long term rate equation was discussed above (V.A.) with the conclusion

that although the root is perhaps a little too large (due to sampling

86

error) the equation still tracks very well. The consumption lag remains
to be examined.

The original K-G model included, among other regressors, lagged
consumption; if consumption decisions are based on an adaptive expecta-
tions/permanent income mechanism, the lag inversion will introduce
serial correlation into the (transformed) disturbance. Except in the
special case in which the autoregression parameter of the untransformed
disturbance is equal to the root of the lag the resulting parameter
estimates will be inconsistent, with the seriousness on the inconsis-
tency depending on the difference vetween the lag root and the auto-
regression parameter. (This incidentally is Egg the case with the
parameterized initial condition estimation we have undertaken since
the lagged dependent variable does not appear). The K-G estimate1 of
the lagged consumption coefficient is 0.26, while our estimate of the
untransformed disturbance autoregression parameter is 0.54; the dif-
ference (0.28) is large, resulting in an upward biased lagged consump-
tion coefficient and too small an income coefficient. These effects
are offset by the depression-dominated sample, which tends to raise
the short run marginal prOpensities to consume (mpc's) -- although they
remain low (see [14], p. 66). Our estimates of the short run mpc's
from wage, agricultural, and profit income are 0.60, 0.37, and 0.45
respectively, compared to the K-G estimates2 of 0.55, 0.34, and 0.41.
While our short run estimates are considerably better than the K-G

estimates, this is not entirely due to the Pascal lag; without the

 

fifi

1 [34]. p. 90.

2 [34], p. 90.

87

autoregression correction the estimates would be too high (the usual

result in annual data including the depression).

In calculating the long run mpc's we have followed Evans'

1
example and assumed an equilibrium growing at 2 per cent per year,

giving long run mpc's of 0.94, 0.58, and 0.70. The aggregate long run

mpc depends on the relative income shares and ranges from a sample

 

l

2

frf' V—

’[14], pp. 70~72.

-1 .
Implying Yt 1°02Yt-1’ or Yt-l (1.02 Yt' The inverted Pascal lag
equation
P ’ _ P a
(1) Y: - 0.24Y1;_1+0.0216r:_2 -0.000864 Y1;_3+0.00001296 YH‘ 0.780%)t

becomes

..0-24 9.0216 ‘_0.000864 0.00001296 3
(2) Y: 1.02 Y:'*(1.02)2 Y: (1102)3 Y:'*‘ (1.02)h Y: 0'78(Yc)t

 

or

P
(3) Yt = 0.88(Yc)t.

The estimated consumption function

(4) 0C a -2.21 + 0.76 Y: = 0.45 Y:_1 + 0.71 ct_1

Similarly discounted becomes

(5) (1 1«02) 0C 2.21 + (0.76 1.02) it.

Substitution of (3) for YP gives

1 -"o 9 + ' 9 + - +.
(6) Ct 2 21 + 0 94 [W1 W Tw + 0 62 (A1 A TA) 0 7S

2 2

(P-SC ' TC)],

the discounted consumption function from which the long run mpc's

'are obtained.

88

high of 0.886 in 1969 to a low of 0.841 in 1933 (the corresponding

short run aggregate mpc's were 0.565 and 0.537). These values are
slightly higher than expected a priori [Evans' comparisons ([14], p.65),
suggest 0.834 as the concensus on the aggregate long run marginal
prOpensity to consume nondurables and services; the inclusion of
durables would raise this estimate].

Altogether, the conclusion is that in the particular case of
annual consumption data there are offsetting misspecifications in the
simple inverted geometric lag framework that result in approximately
the same estimares as are obtained from the more general Pascal formu-
lation estimated by the statistically superior initial condition
parameterization technique. This results from the offsetting biases
discussed above combined with (l) the lessened importance of inverted
V-lags in annual data, and (2) the absence of a partial adjustment
mechanism, or the presence of a partial adjustment mechanism with a
root radically different from the expectations root. Quarterly data
will provide a better test of the Pascal lags, and investment data --
with the strong expectations/adjustment convolution -- will provide a
better test of the generalized lags of Appendix III.

E. Autoregression

Serial correlation of the model disturbances may occur for
several reasons; besides the economist's usual explanation of long
cycles in unmeasurable omitted variables included in the disturbance,

autoregression may also result from algebraic transformations (e.g.,

89

stock-flow translations1 or lag inversionz), or from aggregation over
time when there are lagged normalizing variables.3

Two of the equations, determining durable investment and imports,
have lagged normalizing variables on the right; fortunately, aggrega-
tion is unlikely to be serious in these cases, although long cycles in
the disturbance generation may be. As is well known, the Durbin-
Watson test is not applicable in the presence of lagged normalizing
variables; Durbin, however, has prOposed an alternative test that also
has the desirable characteristic of not requiring estimation of the
autoregression parameter.4 The test was originally designed for the
ordinary least squares model; we shall temporarily reverse our assump-
tions and assume the simultaneous equation problem is negligible in our
model and thus use the test as an approximation. The assumption that
the disturbances are normally distributed is also added.

Consider the model

(1a) Yt so + 51X: + 523.1 + at (t-l,...,T)

+ u ut ~2N(0,02)

(1b) et Det-l t

and let 5 denote the set of parameters (80,81,82,02); the logarithm of

the likelihood function [L(o,B)] is

 

T
2 1 2
L<o,e>2-llog<2m)-— '2 (e -oe ).
2 2 _ t t-l
o t-l
1 See [40] for an application of this often neglected fact.
2 See section II.A.1 above, for example.
3
See for example [49].
4

[9], p. 416 and p. 420.

90

If we set the derivatives of (2) with respect to o and 3 equal to zero
and solve the resulting nonlinear differential system, the estimates
obtained will be asymptotically normal with known variances and co-
variznces. One way to solve such a system is to choose arbitrary
starting estimates and use the Newton-Raphsen iterative technique.

But we do not really need an estimate of 0, we just want to know if it
is significantly different from zero; we need a test on the change in

5 from its starting estimate of zero. If we expand the likelihood
function (2) abOut p=0 and B = 0, where 0 is the ordinary least squares

. 1
estimate of B, the result can be rearranged as

2

T
1 ‘ C15o dBIXt-dBZYt-Z)

T
— 2 _l__
(3) No.8) = - 2 1080M) - 2 E (et ‘do et_

where the e's are the ordinary least squares residuals from (la).
Setting the derivatives with respect to do and d8 -- not 0 and B --
equal to zero and solving the differential system obtained is equiva-

lent to carrying out ordinary least squares on the equation

(4) = (d0)et_1 - (d81)Xt - (dSZ)Yt_1 + vt.

et:

Since the Taylor expansion implies multiplying each change in the para-
meters, dp and d5, by the partial with respect to that parameter of

the likelihood function (evaluated at 0 = 0 & B = 8), this is also
equivalent to taking the first step in a Newton-Raphson iterative
estimation of 0 and 8. Thus we are testing whether the change in 6
resulting from the first Newton-Raphson iteration is significantly

different from zero.

 

1 [9]) p0 4209

91

Durbin shows that the estimate of dp is asymptotically normal and
the test of p = 0 can then be "carried out by testing the significance
of the coefficient of [et_1] by ordinary least-squares procedures."1

0n the other hand, if the original structural disturbance (before
lag inversion) was autoregressive, the models are reversed, with
fi=0.0 the AE model and §=0.9 the PA model. These models are hard to
distinguish on the basis of the coefficient estimates alone. Theo-
retically we are included toward the PA hypothesis (see Section II.A.3.
above), but important serial correlation would decrease our faith in

this hypothesis. Applying the Durbin test gives the following results:

Imports

w = 0.218 e
t

-1
t + 0.529 + 0.126 (Yd )‘-8.833(pF )t-+0.197 (F1)t-l.

t
(0.148) (8.320) (0.023) (6.680) (0.291)

-1

The apprOpriate t ratio is 1.470 so that we fail to reject the null
hypothesis of no autoregression, choosing the estimates of fi=0.0 based
on a partial adjustment hypothesis. In this case the autoregression
decision is very important.

Autoregression decisions on the remaining stochastic equations
were based<>n the Durbin-Watson (D.W.) statisticu‘ Although all of the
differenced equations (e.g., Ir, Ii, Aw) were candidates for negative

2
autoregression , all of the D.W.'s were 2.0 or below, precluding

 

l
[9], p. 420.
2 If, for example, the disturbance originates on the stock of houses
r r r r r
= + E - = ..
(K ), so that Kt Ext st, then It Kt K _1 E(Xt Xt-l)
+ - d - - = -. ' ' '
st €t_1 an COV (et et_1, €t_1 et_2) 0 even if at is serially

independent.

92

negative serial correlation. 0f the twelve equations, seven --
normalized on C, Ir, Xp, iL, PC D, and W1 -- had D.W.'s below the
lower bound; these were searched or iterated over 6, and the results
obtained were reasonable (see Appendix III-B). Two more equations,

determining Aw and AP , had D.W. statistics above the positive auto-

A
regression upper bound, allowing us to reject the hypothesis of auto-
regressive disturbances. These equations were searched anyway and the
minimum SSE was at 6=0.0, where the coefficients were plausible.
Three other equations-normalized on Ii, is, and SC -- resulted in D.W.
statistics either above the upper bound for positive autoregression
(giving strong evidence of no autoregression) or else in the inconclu-
sive region. These equations were also searched over the positive 6
domain for completeness, although in the model B was constrained to
zero in these cases (even though the minimum SSE was elsewhere) since
the inventory investment and short—term interest rate equations were
specifically hypothesized to be short run processes, while the corporate
savings equation resulted in a negative adjustment coefficient at the
minimum SSE (with B=0.8).

Casual inspection of Appendix III-B shows that, as expected, the
autoregression correction is of critical importance in almost all
cases. In the short-term interest rate equation, for instance, the
coefficient of R, the ratio of excess to required reserves, changes
from -l.43 at B=0.0 to -0.398 at 6=0.7, the minimum SSE over positive
values of 6; the policy implications are enormous. Even with this
particular 5 constrained to zen), the ZSLS monetary policy impact
multipliers on GNP are reduced to less half of this previous values by

the full model autoregression correction (line 1, Tables 4 and 7),

93

primarily through changes in the residential investment coefficients.
The fiscal policy multipliers, on the other hand, all increase slightly.
Both of these results carry through the 3SLS estimation (also Tables 4
and 7). If these results are indicative the autoregression parameter
is the single most important parameter of an equation.
F. Data

The original K-G model was estimated from data for the 18
years 1929-1941, 1946-1950; less than a third of this data lies in the
postwar period. The present revision has been estimated from 33
observations of which 21 lie in the postwar period, with the unrepre-
sentative immediately postwar years drOpped from the estimation and
used as a forecasting test; it is hOped that this gives the model a
much more current flavor, in addition to almost doubling the data set.
The expanded data set also allowed us to perform the autoregression
correction (autoregression was a serious problem in the original
model)1 and discontinue the arbitrary instrument set truncation (Klein
and Goldberger used 15 predetermined variables in place of the 21 that
actually occurred in the system in their calculationsz), although this
latter point is of small importance (see Section V.B. above).

C. Three-Stage Least Squares

Three-stage least squares can be proven to be asymptotically
efficient relative to ZSLS. In the small sample, however, there is no
guarantee that 3SLS is superior; indeed, if the number of stochastic

equations exceeds the number of observations, or if the number of

 

1 [34], ppo 51-539

2 [34], pp. 50-51.

94

coefficients to be estimated is greater than the product of equations
and observations, then conventional 3SLS is undefined, since -- the
across-equations disturbance variance-covariance matrix -- is singular.
In our particular case this is not a problem (since there are 14 sto-
chastic equations and 33 observations and the number of coefficients
to be estimated is 59, against an equation-observation product of 462),
but it will be interesting to examine the finite sample gain from 3SLS
estimation.

Since at least three separate effects -- due to 3SLS estimation,
autoregression correction, and lag specification -- are intermixed in
the estimates, it is difficult to accurately assess the importance of
3SLS estimation alone. There are two comparisons that potentially
might remove the effects of the autoregression corrections and allow
us to concentrate on the changes due to 3SLS: (l) autoregression
corrected ZSLS versus autoregression corrected but constrained2 3SLS
(Table 6); and (2) nonautoregression corrected ZSLS versus uncorrected
3SLS (Table 3). In both cases we immediately observe that the standard
errors of the coefficients are all much smaller than the 2818 estimates;
3SLS is efficient relative to ZSLS, as expected. But how much did the
point estimates of the coefficients themselves change?

In the first (autoregression corrected) case, the coefficients of

r

d o
twelve of the equations -- determining I , I , 11, F , Xp, i , i , D,

I s L

Pc’ Sc’ W1, and 2 -- do not change much, usually less than one standard

deviation. There is a larger change in the coefficients of the equa-

tion normalized on AP , and an even larger change in the coefficients
A

1 See [37], p. 175, for example.

 

See Section III.c.2. above.

95

of the consumption function [the marginal prOpensity to consume wage
income increases 14%, which results in impact multipliers that are far
too large (see Table 7)]. The change in the agricultural income equa-
tion is probably due to misspecification, since the nonfarm income
variable is not significant; the consumption function changes probably
result from a bad ZSLS estimate of 0, obtained by premature cessation
of the Orcutt iterative technique (the later unrestricted 3SLS estimate
of p is quite different). Thus over the equations that are comparable
the three stage coefficients did not change much, even with noncompar-
able equations elsewhere in the system.

In the second and far more clear cut case, ZSLS and 3SLS are com-
pared before any autoregression correction. Here the decrease in coef-
ficient errors is even more striking, and almost1 all of the 3SLS
coefficients are withon one (3SLS) standard deviation of the ZSLS esti-
mates. Not surprisingly, we find in Table 4 that the impact multipliers
are also very similar, with the BSLS monetary policy multipliers slightly
lower while the fiscal policy multipliers are slightly higher.

The conclusion, then, is that 3SLS is very important for hypothe-
sis testing, but has only marginal effects on the actual coefficient

estimates obtained -- at least for this particular model.

 

Only the coefficients of the long-term interest rate and corporate
savings equations were not within one standard deviation; the maxi-
mum iL coefficient difference is three standard deviations, while
the SC coefficients are all within two.

96

H. An Interesting Implication
Much of the previous discussion has been leading inexorably
toward the suggested use of an alternative estimator. Let us briefly
review some of the major points.

First, variable nonlinearities are relatively important and at
the same time most difficult to properly treat in the first stage
regression, and sometimes (if nonlinear within an equation) very
hard to handle in the second and third stages of estimation. At the
same time, a crucially important nonlinearity relating GNP and the
price level occurs in all meaningful macroeconomic models.

Second, some of the most important parameters of a model enter
nonlinearly. This is true of the lag parameters--in this case causing
limited use of the Pascal lag as a result—-and also of the auto-
regression parameters, which are extremely important in model estima-
tion. Not only does this nonlinearity cause first-stage difficulties
resulting in grossly inefficient estimates except under certain
relatively restrictive sets of assumptions, but even more important
it renders BSLS inefficient in small samples-- either p is
arbitrarily fixed at some value for each equation or else the non-
linear restrictions between coefficients are dropped (assuming that the
search over all possible, combinations of the 3's is computationally
impossible, even with modern electronic equipment).

Third, we have found that in this case at least the inconsistency
that results from simultaneous equations is negligibly small. Two-

stage least squares is a technique with the main purpose of eliminating

97

this inconsistency, even at the price of efficiency. Three-stage
least squares adds information from the disturbance variance-
covariance matrix, which in our sample primarily affects the coeffi-
cient standard errors and only marginally changes the coefficients
themselves; thus 3SLS coefficients are very similar to the ZSLS
results. What is the price of using ZSLS with its emphasis on
consistency above all?

Asymptotically we lose nothing, as the standard textbooks show;
in the finite sample the cost is higher, however. Comparing uncon~
strained reduced forms to those derived from the estimated structure,
it is apparent that they are quite different, as a result of the
overidentifying restrictions not being included in the first stage.
Thus the calculated instruments inputted into the second stage are
very inefficient estimates, which in turn condition the second stage
parameter estimates. Asymptotically the unconstrained and desired
reduced forms converge, since each parameter estimate goes to the
true parameter in the limit, and the restrictions hold between the
true parameters. In a finite sample, however, these restrictions do
not necessarily hold; to the extent that they do not, we are losing
efficiency. In actual practice, the information included in these
restrictions is likely to be far more important in the determination
of coefficients than any minor inconsistency1 as a result of

Simultaneity. It is critically important that simultaneous estimators

 

Wold's Proximity Theorem reassures us that statistically we are not
in a Lipsey-Lancaster second best world where any deviation from

the optimum results in completely lost properties; statistically,
minor inconsistencies can be proven to be better than major inconsis-
tencies so that the coefficients will be relatively unaffected if

the inconsistency is small. See [21].

98

not sacrifice small sample efficiency in the quest for consistency.
Maddala has done some interesting work in attempting to
incorporate these restrictions into first-stage estimation.
Essentially he proposes iterating until the estimated "instruments"
obtained from the derived reduced form (derived RF) when substituted
into the second stage calculation result in the same derived reduced
form. (This is equivalent to stating that the structural §'s, when
substituted back into the second stage, result in the same structural
y's.) Without reviewing the entire Maddala paper, we note that in this
case ZSLS is not equivalent to instrumental variable estimation (IVE),
and also that there are two different iterative techniques that can
be followed. The first iterative technique Maddala terms the solved
reduced form (SRF) method; it involves starting with the ZSLS
derived RF and substituting the implied y's back into the second stage,
reestimating the structural parameters, recalculating the derived RF,
etc., until convergence is achieved. The second iterative technique
is called the method of successive substitution (SS); it involves
starting with an initial set of jointly dependent variable values
and performing the second stage calculation over them to obtain
structural coefficients, which are then solved for the implied
(structural values of the jointly dependent variables, which are
then substituted back into the second stage calculation, etc., until

convergence is achieved. Maddala examined via Monte Carlo methods the

 

1 [as]

99

properties of both of these iterative techniques for each of the two
estimators, ZSLS and IVE, and concluded that the IVE-SRF method was

by far the most likely to converge to unique values. The SS iterative
method often failed to converge; although the inconsistency of the non-
iterative procedure may be small (as the SOIV model demonstrated
above), continued iteration causes it to accumulate. The ZSLS
estimates often were not unique.

Maddala then reconciled the IVE-SRF estimator with full information
maximum likelihood (FIML) estimation,1 demonstrating that whereas
IVE-SRF uses the estimated values from an unchanging derived RF as
second stage instruments, FIML is equivalent to an instrumental
variable estimator that adds a weighted structural disturbance to the
above instruments. Thus FIML enforces a stochastic version of the
overidentifying restrictions.

Two- and three-stage least squares have been touted for their
computational simplicity relative to FIML; this simplicity entirely
disappears in the face of parameter and variable nonlinearity however,
both of which our results indicate are important. Further, ZSLS and
3SLS are inefficient to the extent that the derived reduced form coeffi-
cients differ from the unconstrained first-stage estimates; that these
restrictions are important has been borne out by modeling experience --
a major problem in all large models is that each single equation tracks
well given the actual values of the other endogenous variables, but
the model as a whole performs poorly. It can be argued that loss of

the across-equation information in the overidentifying restrictions

 

Assuming normality of the disturbances.

100

is the cause. Any attempt to include these restrictions §_l__Maddala
is computationally impossible in the presence of even minor parameter
nonlinearity, and ZSLS is not at all well adapted to handle variable
nonlinearity. Thus for actual models FIML would seem to be the best
estimator;1 although computationally more difficult in the simple

cases, it generalizes readily and makes better use of prior information

in finite samples.

 

The "conventional wisdom" holds that FIML is undesirable because
it is more sensitive to specification error. What is not
considered is that the researcher wants to find the specification
error; if the error does not appear in the estimated coefficients,
it is likely that the model solution will inherit it. The present
procedure of estimating and then solving for multipliers,
elasticities, and long run properties in order to detect error is
far more computationally demanding than would be direct FIML.

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APPENDIX I

Data Sources

There are 33 basic series from which the rest of the data are
calculated. The primary 1929-65 data source (for 26 of the 33 series)
is The National Income and Product Accounts of the United States,
1929-1965;1 while the primary 1966-69 data source is the Survey of
Current Business,2 Julyyp1970; table and line numbers refer to these
two sources (and are the same) unless noted.

The first eight entries are in constant (1958) dollars; an
asterisk denotes a series based on current dollar sources but deflated
for use in the model. All dollar magnitudes except the wage rate have

been transformed to billions of dollars. The sources are:

Personal consumption expenditures; Table 1.2, line 2.

I Gross private domestic investment, nonresidential,
Table 1.2, line 8.

r . . . . .
I Gross private domestic investment, re51dent1al structures;
Table 1.2, line 11
i . . . . .
I Gross private domestic investment, change in bu81ness

inventories; Table 1.2, line 14.
G Government purchases of goods and services; Table 1.2, line 20.
F Net exports of goods and services, exports; Table 1.2, line 18.

F Net exports of goods and services, imports; Table 1.2, line 19.

 

 

 

1 The National Income and Product Accounts of the United States,y1929-l965:
A Supplement to the Surveypof Current Business, U.S. Department of
Commerce, Office of Business Economics (August, 1966).

2

Survey of Current Business: July 1970, U.S. Department of Commerce,
Office of Business Economics, Washington, D.C.

X Gross national product; Table 1.2, line 1.

N Labor force; 1929-38 from [37], p. 191; 1939-1968 from
1969 Business Statistics, p. 67; 1969 from The Statistical
Abstract of the U.S., 19702, p. 213, Table 316.

Nw Average number of full-time and part-time employees, all
industries, total; Table 6.3, line 1.

N Average number of full-time and part-time employees,
government and government enterprises; Table 6.3, line 73.

t Time, l929=0, continues through the war.
w Annual earnings, thousands of dollars; 1929-63 from SCB,

7-70, Table 6.5, line 1; 1961-69 from assorted later
SCBs.

 

p Implicit price deflator gross national product; Table 8.1,
line 1 (All times 10‘2).

pM Implicit price deflator for gross national product, imports;
Table 8.1, line 17 (All times 10-2).

pA Implicit price deflators for gross farm product, total
value of farm output; Table 8.5, line 1 (All times 10'2).

id Average discount rate at all Federal Reserve Banks; 1929-65

from [37], p. 191; 1966-69 from The Spatistical Abstract
of the U.S., 1970, weighted by days in effect.

iL Bond and stock yields, corporate (Moodys) total; 1929-61 from
[37], p. 191; 1962-69 from The Federal Reserve Bulletin,3
January, 1970 (henceforth FRB, 1-70) p. A34.

4R Year-end ratio of member banks' excess to required reserves;
1929-65 from [37], p. 191; 1966-69 from FRB, 1-70, p. A6.

1 Yield on prime commercial paper, 4-6 months; 1929-61 from
[37], p. 191; 1962-69 from FRB, 1-70, p. A33.

 

1969 Business Statistics: A Supplement to the Survey of Current
Business, U.S. Department of Commerce, Office of Business Economics,
Washington, D.C.

The Statistical Abstract of the United States: 1970. (9lst edition)
U.S. Bureau of the Census, Washington, D.C. 1970.

FederalyReserve Bulletin: January 1970, Board of Governors of the
Federal Reserve System, Washington, D.C.

D*

T*

Capital consumption allowances; Table 1.9, line 2.

Undistributed corporate profits after tax plus corporate
inventory valuation adjustment; Table 1.10, lines 23 plus 24.

Corporate profits and inventory valuation adjustment less
corporate profits before tax, agriculture, forestries,
and fisheries; Table 1.10, line 18 less Table 6.13, line 2.

Private employee compensation; Table 1.10, line 4 plus line 7
less Table 6.7, line 14.

Government employee compensation; Table 1.10, line 5 plus
line 6 less Table 6.7, line 14.

Private farm income; Table 6.8, line 2 plus Table 6.13, line 2
less Table 1.17, line 13.

Government payments to farmers; Table 1.17, line 13.

Nonwage, nonfarm income; Table 1.10 line 1 less (WT +-W§
+ A1 + Ag).

Indirect business tax and nontax liability, business transfer
payments, statistical discrepancy, and subsidies less
current surplus; Table 1.9, lines (4+5+6-7).

Federal and State corporate profits tax liability, all industries

less agriculture, forestry, and fisheries; Table 6.14, line 1,
less line 2.

See Appendix II

APPENDIX II
Disposable Income by Distributive Shares

I. The Reconciliation

The U.S. Department of Commerce publishes data on National Income
disaggregated into five distributive shares: (1) Compensation of
employees, (2) Proprietors' income, (3) Rental income of persons,
(4) Corporate profits and inventory valuation adjustment, and (5) Net
Interest. The model supposes three distributive shares -- compensation
of employees (W), agricultural income (A), and the residual (P) -- and,
in the consumption function, requires data on personal disposable
income by these three shares. To obtain these data, we must build the
desired three shares from the National Income shares, and then distribute
to our three shares the reconciling items added and subtracted to obtain

personal disposable income. The reconciliation is as follows:

 

VARIABLE USDC TABLES
pX Gross National Product
pD Less: Capital consumption allowances

Equals: Net National Product
Less: Indirect business tax and nontax liability
Business transfer payments
Statistical Discrepancy
pT
Plus: Subsidies less current surplus of

government enterprises

Equals: National Income

 

VARIABLE

USDC TABLES

 

pY

X-

pT

22

X-

pT

S

a.

-pT

2‘.

-pT

-pT

>X—‘UX-‘UX-

..pT

..pT

 

National Income
Employees Compensation
Private Employee Compensation
Government Employee Compensation
Agricultural Income
Private Agricultural income
Government payments to farmers

Residual

Less: Corporate profits before tax, agricultural
corporations

Corporate profits before tax, nonagricultural
corporations

Inventory valuation adjustment

Contributions for social insurance

Wage accruals less disbursements

Plus: Government transfer payments to persons

Interest paid (net) by government & consumers

Dividends, nonagricultural corporations

Dividends, agricultural corporations

Business transfer payments
Equals: Personal Income
Less: Federal personal tax

Federal personal income taxes less refunds

 

‘k
* Add all pr*,

a)" 7’: * .
PTA , pTw to obtain total pr, pTA, pTw.

VARIABLE USDC TABLES

*
pTW Share from employees' compensation

* See part II
pTA Share from agricultural income of this

* Appendix
pTP Residual

*
pTP Federal estate and gift taxes

State and local personal tax payments

*
pT Share from employees' compensation

W

* See part II
pTA Share from agricultural income of this

* Appendix
pTP Residual
pT: State and local death and gift taxes
pT; State and local property taxes

Nontax payments, licenses, permits, etc.
(all other Federal, state and local payments)
Equals: Personal Disposable Income

pW-pTw Disposable employees compensation
pA-pTA Disposable agricultural income
pP-pTP Disposable residual income

 

 

II. Allocation of Federal, State, and Local Income Tax to Distributive
Shares

A simplifying assumption has been made in the allocation of federal,
state, and local income taxes: we assume that the relative tax shares
vary directly with the relative incomes. Specifically, we assume the

equations

 

 

hold identically over the sample, providing us with definitions of XW’ XA’
and xP from which we calculate their values (where x is the total federal

income tax bill, XW is the portion from pW, x

A is the portion from pA,

and xP is the residual; the zero subscript refers to an arbitrary base

x o ‘ w1+w2 0 Y t ' Xt
_ X_A) ( Y ) (Al+ A2)

(xp)t

year):

III
III

(xw)t kth

xt - (xw)t - (XA)t '

We require XW’ xA, and XP for some (representative) base year,
here 1959. The U.S. Internal Revenue Service provides data on sources
of income and loss by twenty-six adjusted income classes, and the
amount of income tax after credits paid by each adjusted income class.
Ten sources of income are tabled; they have been distributed to the

three income shares (W, A, P) of the model as follows:

(1) Salaries and Wages (net) W
(2) Dividends (after exclusions) P
(3) Interest received P

 

1 Statistics of Income,pIndividual Income Tax Returns for 1959, U.S.
Treasury Dept., Internal Revenue Service. Publication No. 79
(9-61), pp. 24-26.

(4) Business or professional profit: allocated to

 

 

A,P on the basis of

Agricultural net profit1 = 2,913,642
Total net profit 21,516,876

= 0.1354

(5) Partnership profit: allocated to A,P on the
basis of

2
Agricultural Partnerships with net profit
Total Partnerships with net profit

750 842
=-—————L——— =
9,720,805 O°O772

(6) Sales of capital assets: allocated to A,P on
the basis of

Sales of agricultural assets
Total sales of assets

1,579,384

= 12,331,867 = 0°1281

(7) Sales of property other than capital assets
(8) Rents and royalties
(9) Estates and trusts

(10) Other income

 

2

3

Statistics of Income, U.S. Business Tax Returns 1959-1960, I.R.S.
p. 16.

Ibid., p. 44.

P,A

P,A

Statistics of Income,»1959,_Supplemental Report: Sales of Capital

Assets, I.R.S., p. 10.

From these data the 1959 relative shares were found to be:

1‘2 2 2| I
kA I x I x1 I'll-m2 JAa+A2 P2 Y2

W] $510 0311l1. 047zlo. 2339l38.645299l274.69l 11.69I 107.33l393.58

 

 

1:ng

The state and local income tax bill was distributed using the ki's

calculated above:

(I:
B>
ll
“K
>1
/"\
>
H
"<+
>1
N
v
n
m
n

(where s is the total state and local income tax bill, and s

w,
SA’ and sP are the portions from W, A, and P income respectively).
This state and local allocation is somewhat suspect -- wage income

undoubtedly bears a relatively heavier burden at this level than at the
federal level -- but the figures themselves are small so the allocation

error is assumed to be negligible.

 

1 In billions of current dollars.

From the model data, not from IRS estimates.

APPENDIX III

A Generalized Example of Initial Condition Parameterization

A model based on a normalized rational lag of two convolutions

can be written

(1) =e(1-A1-2)(1x) E Z xixj Lk+3x +6 ,
i=0 j=o 6t

which, upon expansion, is

= _ _ 2 3 4
(13) 8(1 >‘1)(l x2) {[ Xt + >‘ZXt-l-ip )‘ZXt-2+ )‘th-3+ Ath-4 + "
+ [ l X +A A Xt_ WA XZX +A X3X +
1 t-l 1 2 1 2x t- -3 1 2X t- -4 °
2 2 2 2x
+ [ klxt-zfilkzxcafilxzx t_ _4 + ..]
‘ i i 2X1 j
+ [xlxt-ifi1A2xt-i-1R1A2Xt-i-j2 "‘fllxzxt-i-f

+.
}€t

Regrouping coefficients by time period and partitioning the infinite sum

into the part t-r (here t-Z) periods back and the residual, we obtain

t 2 1 . . . 00 i
= _ .. J 1.] j 1" j
(2) yt B“ )\l)(1 A2) (i=0 j=o A1A2 Xt-i + 1?:- 1 E992 X 11"1) + at .

The objective is to write the second term -- the effect of the values that
variable X assumed more than t-r periods back in time -- as a finite number
of parameters (stable as t changes) so that they can be estimated. For

notational ease, we name the second term of (2) Zt:

5(1- x 12)(1->.) 2 2x51" jx

(3) Z 12 -i .
t- l j=o

Rebasing the summation over i at zero,
t-1+i
(Ba) Zt =_.-B(1-)\1)(1->\2) Z Z (I; 1) Ail; jxl

i=0 j=o 1

and partitioning the inner summation such that the upper limit of one

of the terms is i, we obtain

m t-1+i
t-l —\ j i-j
(4) 2 Ex 3(1-x )(1-x )L E x A _
c 2 1 21=oj=i+l 12 11

w i
t-l _ _ j i-j

t-l
The second term of (4) is simply X2 E(yl), as is apparent by ignoring
the partition in (2) and subsituting 1 for t. Adding and subtracting

t times the last term of (4) and rebasing the inner summation of the
first term at j=o, we obtain
t-l m i t'2 j+1 -j-1 t-l
E - - +
(5) 2t x2 5(1 x1)(1 A2)23A. Z) x1 A x txz E(yl)

. 1 ._ 2 l-i
1=o J-O

 

 

t-l m i j i-j
- (t-1)x2 3(1-x1)(1-x2) 52; jg; Alxz X1-1 .
-——‘ —‘\/’—————_ c_.
E(yl)

Combining the first and last terms of (5) and expanding

 

 

 

 

 

2 t-l
._ t-l t'l { 0 -1 0 X1 x1 )‘1
(6) 2t a txz E(yl) (t 1n2 8(1-l1)(1 x2) [x1 (t-l) x1 (;e+;§+,,,+—E:i)]x1
2 2 A2
2 t-l
+ ( o °>X + [Al-(t-1>'1 1051+§l +§l——)]x
x2 x1)? 0 1 X1 )x 2+... t’]. O
2 X2 A2
A x2 lt-l
o 1 l o 2 _1 2 1
+x2(x1x2+xlx2) X-1 + [kl-(t-l) x1(;f+;% ...-+—%:I)]X_1
2 2 x
2
+...}.‘
V?

 

 

1
But the terms enclosed in bold lines, when combined with the factor in

front of the brace, can be written as -(t-l) 1: E(yo):

_ t-l t
(7) Zt = t x2 E(yl) - (t-ln2 E(yo)

j+l
t-l E i tiz x1
+ x 8(1-x1)(1-x2)i=0 A1 j=o 'X;31I - (t-l) X1_i

Note that the quantity inside the brackets depends on t but not on i,

and can be brought outside the summation over i:

(8) 2t 2 t x§‘1E<y1>-<t-1> x:E(yo)

)x ' oo
t-l t-2 1 J+1_ z 1
+ )‘2 on()\2) (t-l) B(l-kl) (1 x2)i=o )‘lxl-i'

Substituting (8) for Zt in (2) gives

t-2 i

= _ - z: j i'j t']. - _ t
(9) yt B (1 11)(1 12)i;5 j=0 xlxz xt_i + t x2 E(yl) (t 1)x2E(yo)
W M W
data data data

C’)

x o o
t']. t-Z 1 +1 “5

j=o A2 i=0 1

m
data

i
where B, E(yl), E(yo), and 6(1-xl)(l-A2)i§o A1X1_i are stable over t
and therefore parameters, while their factors all vary with t and there-
fore can be treated as data; the rational lag is estimable in a manner
similar to the Pascal lag except for an additional correction term.

Note that if XI = l the fourth term in (9) becomes zero since

2’
t-2 .
g (l)3+1 = t-l, and (9) can be written
1=o
t-2
2 . i t-l t
(10) yt - 5(1-11) iEO (1+1)x xt_i + t 1 E(y1)-<t-1>1 E(yo) + at:

the Maddala result.

APPENDIX IV

Simulations

Unless otherwise noted, all parameter references are to the

autoregression corrected three-stage least squares estimates.

A. Forecasts

The theoretical underpinnings of a model cannot and should
not be independent of the data; consequently, subsampling and fore-
casting are necessary to validate the model. In our case the years
1945-1948 have not been used in the estimation process and have been
saved for use as a forecasting test, a relatively stringent test since
these immediately post war years are somewhat abnormal.

Three forecasts have been made, each using the actual values
of the exogenous variables. Two of the forecasts using observed1 values

of the lagged endogenous variables. Over the twenty tabled variables,

 

l Reliable (but war-distorted) data are available for Id and XP in 1945.
Data for the Pascal lag variable (YP) in the consumption function was
obtained by noting that since

t=4
P __ 4 1+31
(1) (Y )t — (1-x) i2 ( 1 )A (Yc)t_i ,
i=0
by inversion
P 3'. 4 P 2 P 3 P 4 P
(2) (Y )t - (l-A) (Yc)t + 6AYt_l - 10x Yt_2 + 6x Yt-3 1x Yt-4 .

(The relation would hold exactly if the limit on the summation in (1)
was m rather than t-4. For t=l947, t-4=14 which is close enough to m
when we remember that A was estimated to be 0.06: (0.06)14 f:0, and
the approximation is good enough.)

The war values of YP were estimated by the simple autoregressive
scheme

P P
(3) Yt - 0.87 + 1.03 Yt_1

where the coefficients were obtained by ordinary least squares over
the sample data; since we have the actual 1945 and 1946 values of YP
and since A is so small, the error is not excessive.

war1 (the initial conditions of the 1947 forecast include data from the
last war year, 1945, with government expenditures of $156.4 billion
compared to $48.4 billion in 1946; the distortion continued well past
the war, however, with, for example, government wage bill figures of
$61.6 billion, $34.1 billion, and $25.1 billion in 1945, 1946, and 1947,
respectively). If we were forecasting Ex 3353, extraneous information
about specific residuals for the most affected equations would be included
as add factors to those equations; £§.E2§£: the information is perfect
and thus invalidates the comparison. Hence our forecast of two rather
exceptional years must be made without utilizing gay extraneous information
about the residuals.

Ex ag£g_forecasts of the Klein-Goldberger-Suits (henceforth
K-G-S) model have been tabulated for the years 1953-1960.2 A com-
parison of the forecast errors may be interesting even if not completely
legitimate.3 The GNP forecast errors listed in the Suits article
range from a low of 0.1% in 1958 to a high of 4.0% in 1959. Our GNP
forecast errors for 1947 and '48 were 6.6% and 4.8% respectively.
The maximum Suits consumption error was 4.9%, while our '47 and '48
consumption errors were 4.1% and 2.7%. The Suits private wage bill
errors ran from 0.1% to 8.1%; our 1947 error was 8.5%, reduced to 4.1%
in 1948. Given the distortion of the initial conditions and the years
to be forecast (combined with the inability to use add factors), our

forecasts must be considered something of a success.

 

1 Leaving these particular Years for the forecast conserved data for
estimation purposes, as previously explained.

2 [58], p. 602.

3

Problems (1) and (2) complicated the Suits forecasts, while (3)
and (4) were not particularly important -- the opposite of our case.

B. Sample Data Simulations

While forecasts provide the only truly independent test of the
model, there is still considerable information in the sample data con-
cerning the model's performance. Various summary statistics based on
the sample residuals have been presented with the parameter estimates.
In addition, two other tests of the model fit to the sample data have
been performed: (1) the sample data have been simulated by using
actual exogenous and lagged endogenous values to solve the model for
each single period, and (2) the postwar data have been simulated fully
endogenously by using the model's own past solution values for the
lagged endogenous variables. The plots1 of selected variables are

included as figures 11 through 29.

 

These plots can be misleading: the same percentage error on each
variable is not represented by the same acutal distance on each
plot.

6L3.

II}. '75 b”- m 339-

US

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r11r11

 

1W 1",» 1‘38 191.2 lei-ob L950 1% .606 1a.: 1“».

Gross National Product

Figure 11

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.1 _
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1930 1‘13. 1‘38 10“.? I‘LL: 102.0 1% 191% 1% 1%

Consumption

Figure 12

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ill?

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Durable Investment

Figure 13

at ".8 0.0

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Figure 14

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Figure 15

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Figure 16

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1‘30 .15th 19.35 1%? lq-oh J‘TJO A'Du 1:135 1:13 .L‘ldr

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Figure 17

 

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Figure 18

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Figure 19

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Figure 20

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Figure 21

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Figure 22

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Figure 23

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Figure 24

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Figure 25

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Figure 27

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Share-Weighted Consumption

Figure 28

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3J0 320 3 to “10 1m 1” b-&1 h.” 1.4.” ton

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111141LillijllllillllLLlJJllllllJLllLJl}

1‘1“ .1 “.314 1 “35‘ 1‘1»? 1‘Lb 1‘13) 1 ‘Tb'q 19$ 1‘13? 1‘55

liltiilllifufiEMIISH

 

11111111111111lllliJlllllllll11L1111111%

1% 1533. 1:158 1‘14? LWb 1W7 1‘64 1","?! Lu 1‘“-

Long Term Interest Rate

Figure 29

1. Single Period Simulations
Examining the single period prewar simulation it is

evident that the model malfunctioned in 1933 at the depth of the
depression. The model was extremely difficult to solve1 at this
point; the solution that was finally obtained is economic nonsense,
with wages, prices, and profits being far too severely depressed, and
total income, manhours, and consumption not declining enough. This
error can be traced to the particular nonlinearity incorporated into
the Phillips curve, which, although satisfactory in the other sample
years, is out of its range of adequate approximation in 1933. (Since
the sample data include the depression they provide an excellent
opportunity to allow the data to partially determine the nonlinearity
by using a Box-Cox transformation.2 A priori, the potential return
did not appear to justify the cost of estimating an additional
parameter nonlinearly; a posteriori, it seems the effort would have
been well spent.)

The postwar performance is considerably better, although there
is evidence that the model overstates the lags somewhat. There are
four postwar recessions in the period covered by the sample, with

troughs in 1949, 1954, 1958, and 1960;3 in all four cases the model

 

1 Both Newton-Raphson and short-step Newton-Raphson algorithms were
unsuccessful in solving the model for this year (although the model
normally solved in four N-R iterations); the 1933 solution was
ultimately obtained after 30 relaxation iterations.

2 [4].

3

Due to the timing of the '49 and '60 recessions, there is no actual
decline in annual real GNP figures for these years.

solution was low in the immediately following year. The 1949 solution
is surprisingly accurate; the 1954 solution is higher than the actual
data, but the downturn from '53 is generally evident. It is in 1958
that the model does most poorly -- the variables continue steadily
upward, only to crash in 1959. By the 1960 recession the model is back
on course, and minor errors continue through the period of steady
growth in the sixties.

What caused the model to miss the '58 recession and still
track the others, including the relatively severe 1954 downturn?
The answer lies in the causes of the 1958 recession, which appears
to have been the result of restrictive monetary policy from late
1956 until mid 1958. While it is true that both of the model's monetary
variables (R and id) fell during this period, these do not adequately
reflect the much larger change in the growth rate of M1. In addition,
it is likely that the importance of money is underestimated while the
impact of the sizeable (5%) change in government expenditures in 1958
is overstated. Similarly, the model missed the large 1959 residential
investment figure-~primarily a result of much looser monetary policy in
late '58 and early '59, after the extended through in residential invest-
ment due to the preceeding tight monetary policy. Thus the model, with
its two equation monetary sector, did not track the 1958 recession,
even though its performance in the other three post-war recessions was

adequate.

 

1 It should perhaps be reemphasized that the plots are of full model

solution values; the individual equation (estimation) residuals are
of course much smaller and less systematic.

2. Fully Endogenous Simulations

The fully endogenous simulation is a very severe test;
surprisingly, the turning point behavior of the broad aggregates--GNP,
national income, and consumption--is better here than in the single
period simulation. The model begins to run high in the '54 recession,
which it projects as a major slowdown but not an absolute decline; the
'58 recession increases the upward error, which diminishes to a reasonable
size by the early '60's. The unexpected result is that, with 21 years
of compunded errors, the general shape of the actual and simulated
curves is not dissimilar (figures 11, 22, and 12), and the ending
GNP error is only 0.7%. The turning points of the volatile components

i

of GNP--Id, Ir, I , and FI-- and the income shares--P, A SC--are'not

1,

nearly as accurate, however.

C. Dynamic Multipliers
Since the model is nonlinear in the variables the dynamic multi-
pliers depend on the levels of the exogenous and lagged endogenous
variables and are difficult to obtain analytically; consequently, these
multipliers have been calculated by numerical differentiation at the

1 . . . .
sample means. Dynamic multipliers for ten exogenous variables are

 

The method used is simple ans well known: first a control solution is
calculated with the exogenous variables set at their sample means over
the entire simulation, and then a disturbed solution is calculated with
one exogenous variable set at its mean plus one for the first time
period only. The differences between the second and first solutions
are the dynamic multipliers for the sidturbed exogenous variable.

presented in Tables 9 through 18. The ten period sum is given as an
approximation to the long run multiplier.

The multipliers of the original K-G model were far too low,
a fact not unrecognized by the authors.1 Two important demand leakages--
endogenous taxes and exportsZ--were not included in the original K-G
model nor in the current revision, implying that the calculated multi-
pliers should be higher than those obtained from models including these
functions. The original multipliers were considerably lower than those
computed from models with some or all of these additional endogenous
functions3; the multipliers from the current revision are higher, as

expected.

 

See discussion in [25], p. 68-71.

R.)

Endogenous due to price level effects.

See [15], p. 355, and [141, pp. 567-568 for multiplier comparisons.

Variable

“1.804
“0.984
“1.259
-O.«52
“0.072

0.248

1.251
-0.972

“0.118
“0.286
“2.243

0.010

0.005
“4.425
“2.577
“0.609
“4.310

Table IV—Z

0 Y N A H I C

t+1
-2 0139

“1.300
“0.430
“0.218
“0.140
0.186
0.0
“0.655
“0.939
“1.435
“0.202
“0.138
“2.178
0.005
0.002
“3.944
“6.323
“0.490
“3.745

DISCOUNT RATE

“0.109
“0.20?
0.148
0.0
“0.556
“0.641
“1.320
“0.280
“0.050
“2.106
0.002
0.001
“3.754
“5.896
“0.415
“3.~77

t+3
'10803

“1.309
“0.296
“0.010

“0.253

0.0
-O.411
“0.602
-1.091
-U.345

0.066
“1.861
“0.002
“0.001
“3.223
-7.106
“0.291
“2.881

“2.683

M U L T 1 P L 1 E R S

t+4

~1.1«3
-0.233
0.068
-0.291
0.10.
0.0
-0.255
-0.450
-O.600
-0.395
0.164
-1.470
~0.000
~0.003
-2.49«
-a.079
-o.1~4
—2.102

c+5
-1.021

“0.926
“0.175
0.123
-0.316
0.093
0.0
“0.105
-0.234
“0.492
-0.428
0.202
-1.031
“0.009
“0.004
“1.681
“6.629
0.008
“1.256

Ten Period

Sum

“10.117
“8.660
“2.973

0.040
“2.538
1.186
1.251
“2.275
“3.561
“6.334
“3.511
1.695

“10.900
“0.061
“0.027

“18.973

“70.583
“0.689

“15.668

Variable

t
1.777

0.966
1.241
0.444
0.072
-0.245
-1.234
0.959
1.191
1.701
0.116
0.209
2.204
“0.010
“0.004
4.358
2.536
0.600
4.240

4.230

0 Y N A H 1 C

Table IV-2 (cont.)

H U L T I P L I E R S

RATIO OF EXCESS TO REQUIRED RESERVES

t+1
2.105

1.276
0.423
0.213
0.139
“0.184
0.0
0.643
0.922
1.410
0.198
0.129
2.135
“0.005
“0.002
3.880
4.250
0.482

3.680

3.677

t+2
2.072

1.357
0.355
0.107
0.200
“0.146
0.0
0.545
0.825
1.301
0.275
0.047
2.062
“0.002
“0.001
3.692
5.794
0.407
3.415

3.412

t+3
1.834

1.286
0.291
0.010
0.250
-0.121
0.0
0.403
0.649
1.0I0
0.339
“0.056
1.811
0.002
0.001
3.172
7.041
0.286
2.830

2.828

t+4
1.457

1.123
0.229
“0.067
0.287

“0.104

0.250
0.441
0.784
0.368
“0.160
1.438
0.006
0.003
2.458
7.938
0.142
2.067

2.065

t+5
1.009

0.911
0.171
“0.121
0.309

’00091

0.104
0.231
0.485
0.421
“0.258
1.008
0.009
0.004
1.663
8.478
“0.007
1.240

1.237

Ten Period

Sum

9.997
8.341
2.924
“0.038
2.492
“1.171
“1.234
2.260
3.532
6.276
3.452
“1.724
10.670
0.060
0.027
18.750
69.382
0.667
15.273

15.250

Table lV-2 (cont.)

0 Y I! A 11 1 C M L} L 1 1 F’ L 1 IE R S

GUVfRNHtNT EXPLNUITUEES

Ten Period
Variable t n+1 t+2 t+5 t+0 t+5 Sun
0 1.305 0.958 0.593 0.320 0.112 —0.047 2.204
1d 0.400 0.413 0.314 0.200 0.100 0.027 1.107
1' 0.119 0.003 0.044 0.020 0.013 0.002 0.209
1' 0.323 0.007 -0.052 -0.074 -0.070 -0.009 -0.090
r1 0.049 0.070 0.000 0.001 0.077 0.070 0.017
'1 0.0 0.0 0.0 0.0 0.0 0.0 0.0
is 0.0 0.0 0.0 0.0 0.0 0.0 0.0
5C 0.707 0.155 0.001 -0.000 —0.050 -o.o77 0.415
PC 0.070 0.204 0.144 0.030 -0.039 -0.090 0.001
p 1.290 0.454 0.257 0.102 -0.010 -0.088 1.344
0 0.040 0.001 0.073 0.079 0.079 0.015 0.021
A1 0.221 0.033 -0.013 -0.049 -0.073 -0.007 -0.333
01 1.002 0.017 0.497 0.202 0.070 ~0.002 2.257
w -0.000 ~0.001 0.000 0.002 0.003 0.003 0.012
p —o.004 ~0.001 0.000 0.001 0.001 0.001 0.005
x 3.168 1.372 0.321 0.401 0.002 -0.155 3.900
x 0.070 1.292 1.520 1.007 1.574 1.400 12.279
hm 0.454 0.173 0.081 0.013 —o.030 —0.074 0.179
Y 3.120 1.309 0.740 0.320 0.000 —0.z32 3.319
Y 3.123 1.300 0.741 0.310 -0.002 -0.235 3.297

Variable

“1.338
“0.269
“0.008
“0.164

“0.502

“0.742

0.005

0.002

“0.498

“0.256

“1.781

‘22 a 7113

Ind L3 I\"2 (C(Hh

D Y N A H I C

t+l
“0.853
-0.290
-0.051
“0.045

“0.424

“0.040

'000198

0.002

0.001

“0.851

HAGE TAXES

r+3
“0.409
-0.237
“0.034
0.020

”00072

“0.060

0.001

”0.903

0.000

0.000

“0.613

“0.615

t4}

-0015“

0.000

“0.034

$01111“

“0.211

“0.001

“0.311

“1.120

“0.661

M U L T 1 P L I t R S

t+.
“0.087
“0.064
“0.011
0.056
“0.064
0.0
0.0
0.036
0.026
0.005
“0.055
0.049
“0.067
“0.002
“0.001
“0.059
“1.104
0.025
“0.007

-1).ch

t+5
0.035
“0.022
“0.002
0.051
“0.057

0.0

0.057

0.066

0.066

0.060

0.039

0.122

-1¢UZ‘O

fixtkerfw

“2.256
“0.809
“0.146

0.069

“0.274
“0.452
“0.097
“0.4;2

0.215
“1.502
“0.007
“0.003
“2.590
“6.382
“0.110
“2.193

Table IV—Z (cont.)

[3 Y 11 A 11 I L. M 11 L 1' I P l. I E 51 5

AGRICULTURAL TAXES

Ten Period
Variable c t+1 t+2 t+3 t+4 t+5 Sum
c -1.195 -0.977 —0.001 -0.413 -0.100 0.012 -2.300
Id -0.001 -0.520 -0.410 -0.205 -0.107 -0.000 “1.698
11’ -0.o70 -0.007 ~0.001 -0.034 ~0.019 -0.005 “0.188
11 —0.200 -0.070 0.009 0.052 0.009 0.072 0.109
FI -0.001 -0.000 -0.091 -0.094 -0.092 -0.005 -0.743
1L 0.0 0.0 0.0 0.0 0.0 0.0 0.0
18 0.0 0.0 0.0 0.0 0.0 0.0 0.0
3C “0.448 -0.251 -0.120 -0.031 0.031 0.072 “0.315
Pc -0.557 -0.370 -0.211 —0.007 0.000 0.070 -0.037
p -0.023 -0.509 -0.340 -o.171 -0.032 0.073 -1.090
0 -o.039 ~0.007 -0.000 “ —0.090 —0.099 «0.090 -0.779
A1 —0.130 -0.059 0.000 0.045 0.078 0.099 0.470
w1 -1.023 -0.070 -0.013 -0.309 -0.157 0.010 -2.000
w 0.005 0.002 0.000 -0.002 -0.003 -0.004 -0.017
p 0.002 0.001 0.000 -0.001 -0.001 -0.002 -0.007
x -2.015 -1.504 -1.040 —0.504 -0.203 0.099 -1.322
x -0.005 -1.447 -1.a13 —1.904 -2.001 -1.905 “15.478
hm -0.204 -0.200 -0.109 -0.032 0.030 0.077 0.001
Y -1.979 -1.499 -o.950 -0.«90 ~0.1o0 0.193 -2.500
Y —2.902 -1.502 -0.950 -0.«92 —0.109 0.191 -3.591

“1.370
“0.637
“0.065
“0.231

“0.502
“0.624
“0.922
“0.042
“0.155
“1.145

0.006

0.002
~2.256
“0.911
“0.318
“2.217

Table IV—Z (cont.3

0 Y N A H 1 c

t+1
“1.092
“0.568
“0.079
“0.082
“0.087
0.0
0.0
“0.270
—o.408
“0.628
“0.073
“0.000
—O.969
0.002
0.001
-1.728
“1.502
“0.222
“1.658

PROFIT [AXES

t+2
“0.751
“0.444
“0.056
0.012
“0.099
0.0
0.0
“0.130
“0.231
-0.301
“0.093
“0.000
“0.672
0.000
0.000
“1.138
“1.957
“0.120
—1.047

t+3
“0.453
“0.308
“0.038
0.058
-0.102
0.0
0.0
“0.033
“0.094
“0.166
“0.10%
0.049
“0.402
“0.002
“0.001
“0.636
“2.140
“0.035
-0.536

“0.537

M U L T 1 P L 1 E R S

0+4
-0.197
“0.181
—o.021
0.077
-0.100
0.0
0.0
0.034
0.010
-0.o34
-o.107
0.004
“0.171
—0.003
“0.001
-0.221
“2.136
0.032
-0.110

t+5
0.011
“0.071
“0.006
0.079
“0.092
0.0
0.0
0.079
0.085
0.079
“0.104
0.107
0.017
“0.004
“0.002
0.107
“2.052
0.083
0.209

0.206

Ten Period

Sum
“2.637
“1.821
“0.209

0.118
“0.803
0.0
0.0
“0.359
“0.609
“1.229
“0.839
0.500
“2.235
“0.018
r0.008
“3.727

-16.681

“0.020
‘2091‘9

“3.936

Table IV-2 (cont.)

0 7"! A 11 1 C M LII. T I F’ L 1 IE R S

INDIRECT TAXtS

Ten Period
Variable t t+1 t+2 t+3 t+4 t+S Sum
c -0.011 —0.795 -o.012 -0.401 -o.202 -0.032 -1.900
Id -0.522 -0.454 -0.302 -0.200 -0.102 -0.075 -1.s73
1’ -0.050 -0.055 —0.045 —0.032 ~0.019 -0.007 -0.100
I1 —0.152 —0.071 —o.005 0.035 0.054 0.059 0.007
PI -0.053 -0.000 ~0.070 -0.002 -0.001 -0.070 -0.000
1L 0.0 0.0 0.0 0.0 0.0 0.0 0.0
13 0.0 0.0 0.0 0.0 0.0 0.0 0.0
sc -o.a75 -0.140 —0.004 -0.022 0.027 0.003 -0.050
P0 -1.007 —0.290 -0.103 “0.080 0.003 0.005 -1.039
p -1.007 -0.400 -0.323 -0.170 —0.054 0.040 -1.971
0- -0.032 ~0.057 -0.074 —0.004 “0.008 —0.000 —0.092
11 -0.100 -o.051 -0.005 0.033 0.003 0.003 0.013
w1 —0.755 ~0.724 ~0.5«9 -0.355 —0.174 ~0.022 -1.779
V 0.004 0.002 0.000 -0.001 -0.002 -0.003 -0.015
p 0.002 0.001 0.000 -0.001 —0.001 -0.001 -0.000
x -1.407 -1.305 —0.943 -0.57« -0.2«0 0.023 -2.953
x -0.099 —1.222 -1.559 —1.733 -1.772 ~1.709 -13.79o
hm -0.200 -0.107 -0.101 “0.038 0.010 0.050 -0.003
Y -2.457 -1.251 -o.072 -0.«93 —0.101 0.107 -3.200
Y -2.459 -1.253 ~0.a74 —0.q95 —0.103 0.104 -3.300

Table IV‘2 (Cont.)

D Y'll A I! I C H ll L 1'1! P l. 1 E El S

GOVERNMENT RAGE BILL

Ten Period
Variable t t+1 t+2 t+3 t+4 t+5 Sum
c 0.050 —0.104 -0.212 “0.188 -0.139 -0.003 -0.013
Id —0.101 -0.139 —o.124 -o.100 “0.082 -0.057 £0.090
1’ -0.000 -0.015 —0.017 —0.015 —0.012 -0.000 -o.070
I1 -o.022 -0.o31 —0.019 -0.004 0.000 0.014 0.013
F1 0.057 0.035 0.018 0.005 -o.004 -0.010 0.042
1L 0.0 0.0 0.0 0.0 0.0 0.0 0.0
18 0.0 0.0 0.0 0.0 0.0 0.0 0.0
8C -0.139 -0.040 -0.057 -0.039 -0.019 -0.003 —0.210
Pc -0.172 -0.077 -0.000 -0.004 -0.037 -0.013 “0.338
P -0.255 -o.120 -0.137 -0.107 -0.o7o -0.034 -0.012
D -0.009 -0.017 -o.024 -0.029 —0.032 -o.033 “0.266
A1 -0.337 -0.007 —0.007 0.004 0.014 0.023 -o.112
"1 -0.014 -0.240 -0.220 -0.191 -0.139 -0.005 -1.425
w 0.012 0.000 0.000 ~o.ooo -0.000 -0.001 0.000
p 0.005 0.000 0.000 -0.000 -0.000 -0.000 0.003
x -0.200 -0.303 -0.303 -0.315 -0.219 -0.122 -1.399
K -o.zo1 -o.309 -0.505 —o.001 -0.050 -0.073 -s.345
hm -0.170 -0.040 —0.045 -0.030 -0.013 0.003 ~0.108
Y -0.201 “0.368 -0.307 _ —0.209 -0.190 ~0.091 -1.150
Y —0.203 —0.370 -0.309 —0.291 -0.192 —0.09« —1.100

Table IV-2 (cont.)

0 Y N A M 1 C M U L T 1 P L 1 E R S

GOVERNMENT PAYMENTS T0 FARMERS

Ten Period

Variable t t+l t+2 t+3 t+4 t+5 Sum
0.385 0.181 0.068 0.011 “0.022 “0.044 0.329
0.077 0.071 0.046 0.022 0.003 “0.010 ‘0.110
0.020 0.011 0.006 0.002 “0.001 “0.002 0.017
0.053 0.004 “0.014 “0.017 “0.016 “0.012 “0.024
0.008 0.012 0.013 0.012 0.011 0.009 0.084

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0
“0.429 0.102 0.035 0.008 “0.004 “0.010 “0.341
“0.533 0.072 0.027 0.005 “0.007 “0.013 “0.512
“0.788 0.085 0.023 “0.008 “0.024 “0.034 “0.897
0.007 0.010 0.012 0.012 0.011 0.010 0.081
0.037 0.007 “0.006 “0.012 “0.015 “0.016 “0.062
0.265 0.148 0.058 0.008 “0.024 “0.044 0.170
“0.001 “0.000 0.000 0.000 0.001 0.001 0.002
“0.001 “0.000 0.000 0.000 0.000 0.000 0.001
0.528 0.257 0.094 0.008 “0.045 “0.077 0.366
0.143 0.219 0.245 0.240 0.216 0.181 1.566
0.076 0.033 0.007 “0.006 “0.014 “0.019 “0.002
0.519 0.245 0.080 “0.007 “0.058 “0.089 0.261
0.517 0.242 0.078 “0.009 “0.060 “0.091 0.238

Table IV-2 (cont.)

1) Y hi A P! I C. H 11 L.'I 1 P’l. 1 E II S

INVERSE OF THE IMPORT PRICE LEVEL

Ten Period

Variable t 0+1 t+2 t+3 t+4 t+5 Sum
0.344 0.522 0.500 0.520 0.445 0.337 3.020
0.122 0.204 0.243 0.240 0.222 0.102 1.470
0.031 0.041 0.044 0.042 0.030 0.029 0.200
0.004 0.000 0.040 0.012 -0.010 -0.020 0.017
-0.250 -0.170 —0.119 -0.072 -0.030 -0.010 -0.579

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.104 0.100 0.103 0.120 0.000 0.040 0.703
0.229 0.255 0.230 0.197 0.145 0.090 1.002
0.330 0.300 0.371 0.319 0.240 0.172 1.003
0.010 0.024 0.030 0.050 0.000 0.007 0.525
0.050 0.054 0.039 0.010 -0.005 -0.020 -0.117
0.420 0.544 0.557 0.500 0.419 0.313 3.007
-0.002 -0.002 -0.001 -0.001 0.000 0.001 0.004
-0.001 -0.001 -0.001 -0.000 0.000 0.000 0.002
0.034 1.015 1.013 0.902 0.730 0.533 5.374
0.220 0.517 0.000 1.055 1.242 1.359 10.500
0.120 0.139 0.131 0.100 0.073 0.030 0.470
0.021 0.909 0.973 0.049 0.007 0.404 4.024
0.019 0.907 0.970 0.040 0.005 0.402 4.002

Multiplier comparisons are difficult since superficially
similar models often have quite different sets of endogenous and
exogenous variables; thus even though we have the original K-G model
and the Suits revision to compare to the present model (denoted K-G-H),
the only1 multipliers that are available and approximately comparable
are the government expenditures multipliers on GNP, consumption, and
durable investment (reproduced in table 19).

Examination of table 19 shows that there is considerable
disagreement between the models about the timing of the impact of
government expenditure changes; while the long run multipliers of the
K-G-H and K-G models are in approximate agreement, the K-G-H model
concentrates the effects more heavily in the beginning years. We
take this to be a marked improvement over the original K-G model which,
as Goldberger states3 ". . . must be judged deficient on the basis
of its sample period record. It is reasonable to infer that the
structural lags specified in the model were, in general, too long." In
table 19, all of the K-G-H multipliers peak in the first year, while
the K-G multipliers peak in the second year (with large third year

effects), and the K-G-S model exhibits a peak consumption multiplier

 

1 The three models use completely different monetary instruments for
instance.

2 Even these are not directly comparable: The Suits multipliers include
the effects of endogenous tax functions, while the K-G investment
multipliers apply to total--not just durable--investment.

3

[25], p. 68.

in the fourth year--a clearly unreasonable result. Both the K-G and

K-G-S models constrain all impact multipliers on durable and residential
investment1 to zero--a questionable practice in an annual model-~which
further lowers the GNP impact multiplier. The K-G-S investment multi-
pliers are questionably low, while the long run GNP multiplier (admittedly
only a five year approximation) seems excessive. Thus, of the three

sets of multipliers tabled, the timing of the K-G-H appears to be most
reasonable.

Since a one unit change in each of the instruments brings about
vastly differing changes in the target variables, to examine the policy
implications of the estimated multipliers we shall first normalize the
exogenous variable changes to those magnitudes necessary to effect a
10% increase in mean GNP (or a 5.34% increase in 1969 GNP) in the long
run.2 Three instruments will be normalized: (1) the discount rate,

(2) government expenditures, and (3) a composite personal tax variable (T*)
that maintains wage, farm, and profit taxes at their mean proportions

while the combined changes result in the required 10% increase in GNP.3

 

l . . . . . .
The K-G model also constrains the impact multiplier on inventory invest-
ment to zero, since investment is not disaggregated.

2 . . .
The sum of the first ten dynamic multipliers lS taken as the long run
multiplier.

3

Let tw, tA, tP be the solution tax changes. Then
tw/(tw+tA+tP) = the mean wage tax proportion = 0.366

and
tP/(tw+tA+tP) = the mean profit tax proportion = 0.609,

while from the long run multipliers

-2.590tw - 3.322tA - 3.727tP = X = 38.853.

Solving the three equations gives the values of t
make up the composite personal tax change.

W’ tA, and tP that

Table 21 gives the policy effects on the income side of
the accounts. The expected relative stability of private wages and
volatility of profits appears; less expected, perhaps, is the long
run negative effect on private agricultural income brought about
through price level changes-~all of the policies exhibit important

percentage changes in farm income in the long run.

 

The result should be tempered with misgivings about the lag structure
of price changes and the specification of the farm income equation;
farm income is almost certain to be price level sensitive, however,
even if the exact timing and magnitude of the changes are unknown.

Table IV—3

GOVERNMENT EXPENDITURES MULTIPLIERS1

 

 

 

Variable Model Y E A R 2
t t+l t+2 t+3 t+4 Long run
K-G-H 3.168 1.372 0.821 0.401 0.082 3.966
X K-G 1.386 1.421 1.078 0.680 0.323 4.7453
K-G-S 1.304 1.619 1.582 1.545 1.335 7.385
K-G-H 1.305 0.958 0.593 0.320 0.112 2.264
C K-G 0.398 0.619 0.574 0.418 0.242 2.302
K-G-S 0.295 0.426 0.460 0.500 0.465 2.146
K-G-H 0.468 0.413 0.314 0.206 0.108 1.187
Id K-G 0.0 0.826 0.533 0.294 0.112 1.624
K-G-S 0.0 0.186 0.173 0.133 0.082 0.574

 

 

 

 

 

 

 

 

The Klein-Goldberger multipliers are from [25], p. 87; the Suits
multipliers are from [58], p. 610 (the aggregated consumption multi-
plier is the sum of the four separate Suits consumption multipliers).

The long run multipliers are ten, seven, and five period sums for the
K-G-H, K-G, and K-G-S models, respectively.

The long run K-G GNP multiplier is elsewhere estimated at 2.93
([25], p. 87 ff.).

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Table 22 shows the effects of the normalized policies on
prices, wages, and manhours; both prices and manhours move in the
opposite direction from the expected, and remain negative for three
years (although the long run effects are plausible). This unlikely
result becomes even more doubtful when we remember that the impact
multipliers calculated from the Taylor expanded identities were
positive for both price and manhours (table 7 above). Thus we
conclude that these multipliers are very sensitive to the nonlinearities
while the other multipliers are only moderately sensitive. Since the
price level and manhours are of such critical importance to policy
makers they should be determined as normalizing variables in stochastic
equations if possible in order to avoid the accumulated multiplier
errors that result from determining them residually.

To be honest, we must accept all of the conclusions based
on the dynamic multpliers with reservation. Examination of figures 11-
29 shows that although the fully endogenous solution tracks better than
expected §_priori, there remain large inaccuracies in the model solution,
especially in the less aggregative variables. In addition, the dynamic
multipliers are only point estimates: they may have very large
variances, or variances that increase rapidly as the time from the
impact multiplier increases.2 Consequently, the results should be
interpreted as suggestive, pending an elaborate sensitivity analysis

and variance estimation.

 

1 Unfortunately, solving the production function for manhours (trans-
forming it to a labor demand function) results in multi-collinearity
among the regressors (capital and output). See [14], pp. 253-255.

2

Only the acute shortage of computer time kept these variances from
being calculated (see [28] for the method).