View original document

The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.

Working Paper 74-2

ALTERNATIVE

OPTIMAL
OPEN MARKET STRATEGIES

A CLASSICAL
OPTIMIZATION
CERTAINTY
EQUIVALENCE
APPROACH

Joseph M. Crews

Federal Reserve Bank of Richmond
April 29, 1974

Presented at the 49th Annual Conference
of the Western Economic Association,
Las Vegas, Nevada, June 9-12, 1974.

The views expressed here are those of
the author and do not necessarily reflect
the views of the Federal Reserve Bank of
Richmond.

ALTERNATIVE OPTIMAL OPEN MARKET STRATEGIES
A CLASSICAL OPTIMIZATION,CERTAINTY EQUIVALENCE APPROACH

The Federal Reserve System has historically been criticized for
placing too much emphasis on insuring the short-run stability of financial
markets and too little emphasis on real sector goals.

Critics generally

contend that the Federal Reserve is often misled by concentrating on money
market indicators, that it formulates strategy in terms of short-run targets,
and therefore reacts improp e rly to changing economic conditions.

Guttentag,

for example, finds that, in the period before 1966, the strategy of monetary
policy was incomplete, fail ing to include "specific quantitative

target

values . . . for the money Supply or [some] other strategic variable that
could serve as a connecting link between open market operations and System
objectives."'
Implicit in these criticisms is a set of requirements for specifying
the nature of the policy process and for evaluating current policy.
requirements

These

include a theory of policy formulation, a theory of how the

effects of policy actions are transmitted through the economy, and precisely
quantified measures of the goals of policy.

These requirements may be met

by specifying a policy regime and strategy.

A policy regime is defined as a

policy model of the economy, consisting of an instrument proxy and a theory
of monetary policy transmission.
determination,

A policy strategy is a framework for policy

consisting of a set of target variables and a theory of policy

formulation.2

'Jack M. Guttenta
"The Strate y of Open Market Operations," Quarterly
Journal of Economics, 80 9 February 1966 4 , l-30.

‘Joseph M. Crews, "Alternative Optimal Open Market Strategies: A Simulation Approach," Unpublished doctoral dissertation, University of North Carolina,
Chapel Hill, 1972.

-2-

The purpose of this paper is to present an optimal control framework for the analysis of monetary policy problems, using the target/indicator
problem for illustration.

This involves accepting as given two of the ele-

ments listed above--a theory of policy formulation and a theory of policy
transmission.

The remaining elements, the instrument proxy and target

variables, are altered under controlled conditions to simulate the effects
of alternative

policy actions on the economy.

OPTIMAL CONTROL

The theory of policy formulation adopted is the "theory of quantitative economic policy" developed by Theil."

According to this approach,

the policy maker maximizes a social preference function, in terms of targets
and instruments, subject to the constraints of the economic structure, represented by an econometric model.

This constrained optimum problem is transformed

into a system of simultaneous equations by the Lagrangean multiplier technique.
This approach is usually stated in terms of a linear model and quadratic
preferences.

Since most large-scale econometric models in use today are

non-linear, this approach should be extended to cover this case.

This paper

presents a computational algorithm based on Theil's non-linear case.
Central to extension of this approach to non-linear models is the
problem of uncertainty.

Theil's theory of policy formulation recognizes

that the constraining model may not be a true representation of the economic
structure and that the preference function may not truly represent the
preferences of the decision maker.

Uncertainty may exist in the multiplica-

tive structure of the model, in its predetermined structure, or in specification
of the preference

(or loss) function.

These difficulties may be overcome by

3Henri Theil, Optimal Decision Rules for Government and Industry,
Amsterdam:
North-Holland, 1964.

-3-

assuming that the decision maker seeks to minimize the expected value of
the loss function.
theorem:

In this case Theil develops a certainty equivalence

If the loss function is quadratic and the constraining model is

linear and stochastic only by additive random disturbances

that are inde-

pendent of the instruments and whose expected values are zero, then the
optimal values of the instruments are the same as if there were no uncertainty.4
A number of extensions of the certainty equivalence
required if it is to apply generally.

First, Theil demonstrates

uncertainty exists in the multiplicative
equivalence"

theorem are
that when

structure, a "first order certainty

results in the linear static and multi-period

cases.5

He also

extends these results to the non-linear static case.b
Concerning

the non-linear dynamic case the picture remains unclear.

Malinvaud demonstrates

that the first order certainty equivalence

is general

and applies "with exceptions for singular cases, as long as the various
functions involved are twice differentiable.l17

In a recent dissertation,

however, Porter shows that Malinvaud's results are more limited, applying
only in cases where the degree of uncertainty is very small.'

4Henri Theil
Weltwirtschlaftliche;

"Econometric Models and Welfare Maximization,"
Archiv, 72 (1954), 19-70.

5Henri Theil, Optimal Decision Rules, p. 59 and pp. 72-74.
'Henri Theil, Economic Forecast and Policy, Second Edition, Amsterdam:
North-Holland, 1965, pp. 405-424, 510-513.
7E. Malinvaud, "First Order Certainty Equivalence, Econometrica, 37
(October 1969), 706L718.
8Richard D. Porter, "Strategies for Discrete-time Decision Models,"
Unpublished doctoral dissertation, University of Wisconsin, Madison, 1971,
pp. 133-151.

-49

Given the state of these generalizations;
are based on the non-linear

static case.

the results reported here

The policy maker is viewed as

planning policy in a given period with future goals in mind, but revising
his plans at the beginning of each new period as additional

information

becomes available.
The solution to the non-linear constrained
derived in the following manner.

optimum problem may be

Assume that the economic structure is

represented by the model:'
= fj(yl 9** *

Y

*'j-l,

j+l

,**

l

, ‘,,,’

j=l

xjz)

,-**,N

(1)

'j

where Yj are N noncontrolled
of instruments,

endogenous

variables,

X is an ?I element vector

and Z is a p element vector of predetermined

variables.

Assume also that the preferences of the policy maker can be represented
by the general quadratic

+ 2

C

loss function (2).

Wij

(Yi-Yz) (Yj-Y;) + 2

i#j

+2 ~ Z

Wik(Yi’Y*i)

i=l

(Xk-xIz)]

C Wkh(Xk’X;) (Xh-X~)
k+h

’

k=l

where there are n target variables and m instruments entering the preference
function.
To determine the optimal instrument.vector
transformation,

X0 using the Lagrangean

proceed as follows:

'This development

parallels Crews, QQ. C&.,

PP. 37-40.

-5-

(1) Establish the Lagrangean

L = w(X,YlZ)

+

function:

ii x g (X#YJZ)
j=l j j

where the gj(X,YlZ) are the constraints

(1) expressed

'Jj(X,YIZ) = Yj - fj(Y1,...,Y.,-l'yj+lt".t

(3)
in the form:

YN' X12) = 0
(4)

(2) Differentiating

(3) partially with respect to each endogenous

variable and setting the result to zero:

N

aL
aYi

-

aw
ay

+
i

Since each -

asj
aYj

Ehagj&)

= 1, each equation

A.=-aw
3

_
zj

j-l
c

p .i-

i=l

(3) Differentiating

i=l I-**,N

j=i jaYi

aYj

(5)

(5) can be normalized on a unique X.J:

N
asi
Aic
ay
i=j+l
j

j-1 I*-*,

N

(6)

(3) partially with respect to each instrument

and setting the result to zero:

k=l ,...,m

Each of these equations may be normalized on a unique instrument.
necessitates

(7)

This

solving for the partial derivative of the loss function (2)

with respect to each instrument:

-6-

aw

n

3 - wk(xk-xi)

-

%-k

C Wa(Xh-<)
h#k

-

C Wik(Yi'Yz)
id
(8)

k=l , . . . ,m

Substituting

(8) into (7):

aL
%k

= - wk (xk-xc)

-

c w& (xh-x;;)
h#k

n
- c wik(Yi-Y;)
i=l

N
+ c
j=l

(9)
hj

agi
axk

= 0

k=l

,...,A

I"lormalizing(9) on Xk:
Xk = x;

- -

1

n
(C
h~k wfi

(Xh’Xt)

+ iflwik(yi-y~)

Wk
(10)
k=l ,...,m

(4) Finally, differentiating

(3) partially with respect to X and

setting the result equal to zero determines the original equation system (4).

A&
=

gj(x,Yp)

= 0

j=l

aAj

The resulting sets of equations
of simultaneous equations,

,a-*,

(6), (lo), and (4) constitute a sys'tem

the solution of which establishes

the first-order

conditions for the optimal policy vector XO, the corresponding
and the vector of Lagrangean multipliers

N

A.

real vector, Y"

The second order conditions are

-7-

presented by Theil."
The Guass-Seidel

algorithm,

used in solving many large-scale econo-

metric models, may be extended slightly for use in solving optimizing
This solIution technique is discussed by Evans
paper.

11

and in the Appendix

models.

to this

Optimal simulation-- in which the policy instrument is determined

by

the equation system rather than being read in as an exogenous variable--is
therefore feasible with relatively

straightforward

extensions of current

methodology.
The theory of policy transmission utilized in this study is contained
in the most recently published version of the FRB-MIT econometric
This model was specifically

designed to capture econometrically

of monetary policy actions on the real sectors of the economy.
sector is highly developed and its financial-real
separate channels--the
ability.

cost-of-capital,

interest rates.

Its financial

linkage includes three

interest sensitivity of

expenditure flows related to appropriate

Its portfolio adjustment mechanism

While particular

the effects

the wealth effect, and credit avail-

The model is based on a neo-Keynesian

investment theory, with particular

model.12

is broadly inclusive.

theories of policy formulation and policy transmission

are accepted as given for punposes of this study, alternative
proxies and policy targets are assumed in the experimental
sections develop the target/indicator

instrument

design.

The next

problem and establish particular

vari-

ables for evaluation.

"Henri

Theil, Optimal DeNcision Rules, pp. 37-40.

"Michael
Evans, Computer Simulation of Non-Linear Econometric Models,
Discussion Paper Number 97. Philadelohia:
The Wharton School of Finance and
Comnerce, 196&, pp. 5-7.
12
Frank de Leeuw and Edward M. Gramlich, "The Channels of Monetary
Policy: A Further Report on the Federal Reserve-MIT Model ," Journal of
Finance, 24 (May 1969), 265-290.

-8-

THE TARGET PROBLEM
Monetary policy is conducted in an atmosphere of uncertainty.
Knowledge of the economic structure is incomplete, the chain of causation
from policy action to ultimate goals is long, the speed of monetary impulses
is slow and variable, and information regarding current policy and economic
conditions is available only after a time lapse.

In view of these uncer-

tainties, the policy maker finds it useful to direct his actions toward
intermediate variables, closer in time and under more positive control than
ultimate goals.

The function of intermediate targets is to facilitate

control over a sequence of successively

longer-term targets so that ultimate

This suggests several criteria by which intermediate

goals may be achieved.

targets may be evaluated.

The target should (1) be readily observable with

minimal lag, (2) bear some relation to the transmission of policy, as reflected in a stated structural hypothesis,

(3) be sensitive to, but not

necessarily dominated by policy actions, and (4) be strongly correlated
with longer-term goals.
suggested targets.

13

Economic literature abounds with evaluations.of

Six alternative quarterly target candidates, which

meet the above criteria, were chosen for this study.

First, for comparative

purposes two money market targets, free reserves RF and the Treasury bill
rate RTB are included to reflect "incomplete" strategies.
targets include:

(1) total reserves RT, which constitutes

Longer-term
the base upon

which the banking system generates money and credit, (2) the money supply
MS, a strategic variable in both the neo-Keynesian and Monetarist

views of

the transmission process, (3) bank credit BC, the commercial bank asset

l3For example, see Thomas R. Saving "Monetary Targets and Indicators,"
Journal of Political Economy," 75 (August 1967), 446-456.

-9counterpart of money supply creation, and (4) long-term interest rates,
specifically the corporate bond rate RC, especially critical in a neoKeynesian cost-of-capital

transmission channel.

These six alternative

intermediate targets, together with a non-optimal control solution, provide
seven strategies

to be evaluated.

THE INDICATOR
PROBLEM
Indicators are variables used by market participants

to separate

the impact of current policy actions from concurrent forces operating in
financial markets.
are conceptually

Within the context of econometric models, indicators

equivalent to instrument proxies. l4

gests the following criteria for indicators.
observable,

The literature sug-

They should (1) be readily

(2) be important links in the transmission process, (3) reflect

the impact of policy action apart from all other forces affecting the target,
and (4) provide reliable information regarding current and future movements
in economic activity.
centered on criterion

Recent controversy on the indicator question has
(4), the exogeneity problem.

15

This controversy nar-

rows the question to whether the monetary base or one of its components is
more nearly exogenous.
struments:

Accordingly,

this study utilizes the following in-

(1) the monetary base MB, defined as to its uses as total reserves

plus currency,

(2) the adjusted base--nonborrowed

(3) total reserves, and (4) free reserves.

reserves plus currency BA,

These instrument proxies and the

14Thomas J. Sargent, "Framework of the Economic System--Discussion,"
American Economic Review, 60 (May 1970), 57-58.
15

Frank de Leeuw and John Kalchbrenner, "Monetary and Fiscal Actions:
A Test of Their Relative Importance in Economic Stabilization--Comment," Review,
Federal Reserve Bank of St. Louis, 51 (April 1969), 6-11.

- 10 FRB-MIT model, modified slightly as required, constitute

the four policy

regimes to be evaluated.
THE EXPERIMENTS
The experimental

design that incorporates

strategy framework of this analysis
regimes are specified as rows.
solutions

representing

the alternative

is presented in Table 1.

The four

The columns include non-optimal

actual economic developments,

regime/

control

two incomplete

strategies,

and four complete strategies.

experiments

the FRB-MIT model was solved in optimizing mode over a 16

quarter period from 1959 to 1962.

In each of these 28 simulation

This period was chosen because the

version of the model used is not capable of handling the rapid inflation
of later years.16

TABLE 1
AVERAGE WELFARE LOSS RESULTING
FZOM ALTERNATIVE REGIMES AND STRATEGIES
-------

--------

------------------------------------------------------

----------Nonoptimal

Strategy
-------------- --------------------------Incomplete

r

Complete
Average

Regime
--$--I
----------- t-------------- i4:28
';;:yg'
.
4:'41
40:65 $I,
48:83 t--------------------------36:91 is34
32:35
33:22
3:60
4:94
37:61
MB
I
43.98
I 41.25
51.43 41.05
34.84
33.99
41.67
BA
40.35
40.66
48.75 42.19
32.32
34.83
32.22

41.01
38.45
41.17
39.48
m-------

Average I

40.04
--------

41.38

1 41.24

50.70 140.11

32.71

33.91

40.36

16Robert H. Rasche and Harold T. Shapiro, "The FRB-MIT Econometric
Model:
Its Special Features," American Economic Review, 58 (May 1968),
123-149.

- 11 -

Optimal policy levels were obtained by constraining
policy-determining

loss functions by the model.

alternative

Each policy-determining

loss function includes as arguments the assumed instrument, the intermediate
target of the specified strategy, a financial market stability target and
Typical is the total reserves/money

ultimate goals.

w

ij

=-4

supply case.

[(RT-RT*)~ + (M~-Ms*)~ + (RCP-RCP*)~

t (GNP-GNP*)2 t (P-P*)2 t 2(RT-RT*)(MS-MS*)
+ 2(RT-RT*)(RCP-RCP*)

t 2(MS-MS*)(RCP-RcP*)]

Where * indicates a target desired level, RCP is the commercial paper rate,
serving as a financial stability proxy that is not altered experimentally,
GNP is gross national product, and P is the price level.

Since GNP* is

"potential GNP," which implies an unemployment target, no separate employment goal is specified.
The policy-determining

loss function changes for each. regime/strategy

case, and the values produced are not comparable with any other.

This problem

is solved by assuming that the policy maker determines policy with regard to
the particular loss function specified for the strategy being studied, but
policy performance

is evaluated in terms of all intermediate and ultimate

objectives together. l7
regime/strategies

The ultimate loss function used to evaluate all

in this study is of the form:

u.. = - 4 [(GNP-GNP*)~ + (P-P*)~ + (RCP-RcP*)~
1J

+ (RT-RT*)2 t (MS-MS*)~ + (BC-BC*)~ + (RC-RC*)2]

17Gary Frown and Paul Taubman, Policy Simulations With an Econometric
Model, Washington:
The Brookings Institution, 1968, pp. 106-123.

- 12 -

EVALUATING THE RESULTS
This study focuses on two economic questions:
instrument proxy candidates

(1) Which of four

best measures the thrust of monetary policy?

(2) Which of six monetary targets provides for maximum effectiveness

of

open market policy?
These questions, together with the 4 x 7 experimental

design, en-

ables us to formulate the following hypotheses:

I.
II.
III.

There is no difference among instrument regimes (Row effect).
There is no difference among policy strategies

(Column effect).

The performance of the Federal Reserve under alternative strategies
is independent of the instrument regime (Interaction
effect).

In addition, the inclusion of a non-optimal control solution,
representing

the actual time path of the economy in the period of study,

allows the testing of the general hypothesis:

IV.

The Federal Reserve responds in a systematic manner to intermediate
targets.
For each of the 28 regime/strategy

function is evaluated.

combinations

the ultimate loss

The results are summarized in Table 1, which reports

the average welfare loss (cell means) for each case.
order from lowest (best) to highest as:
strategies are ranked in the order:

The regimes rank in

RF, BA, RT, MB.

Similarly, the

MS, BC, RT, RF, CONTROL, and RTB.

The

data was subjected to analysis of variance using a randomized bloc, two
variables of classification,

with replication model.

There are sixteen

blocks (time periods), four replications of rows (regimes) and seven columns
(strategies) in the problem.

Individual differences among regimes and

strategies were further tested using a least significant difference test.

18Ya-lun Chou, Statistical Analysis, New York:
Winston, Inc., 1969, pp. 407-409, 417-423.

18

Holt, Rinehart and

- 13 -

The results may be briefly summarized.
no differences among regimes.

Hypothesis

I:

There are

The regimes produce significantly

different

welfare levels, and fall into two sets, [RF, BA] and CRT, MB], whose members
are not statistically

distinguishable

generates smaller welfare losses.

from each other.

The former set

These results are consistent with those

of de Leeuw and Kalchbrenner, who argue that the monetary base is made up
of endogenous components and is, therefore, also endogenous. lg

Since the

present analysis is in terms of an open market proxy only, leaving the discount rate and reserve requirements aspects of monetary policy as given, RF
and BA are expected to perform best.

The results are consistent with these

expectations.
Hypothesis

II:

There are no significant differences among strategies.

This test concerns the choice of the optimal intermediate target.

Preliminary

analysis of the target candidates indicated that incomplete strategies provide
closer control over short-term targets, while complete strategies provide
closer control over intermediate and ultimate targets.
confirmed by the statistical tests,

These results were

The strategies were found to separate

into three sets whose members are indistinguishable

from each other.

One set

[CONTROL, RF, RT, RC] shows a consistency between the Federal Reserve's actual
behavior and the other members of the set.
general Monetarist

position that a money supply target should be adopted by

the Federal Reserve.
advocacy.

But this is not the "money supply rule" of Monetarist

It is, rather, an anti-cyclical

reflecting the use of Hendershott's

19

The set [MS, BC] represents the

use of the money supply as a target

neutralized money stock to establish target

de Leeuw and Kalchbrenner, o&.

- 14 -

values. 2o

The RTB strategy is, as expected, the least effective strategy.

It is included in the experimental design as a typical incomplete strategy
for comparative purposes only.

We conclude that the Federal Reserve's per-

formance is improved by adopting a money supply target.
Hypothesis

III:

The performance of the Federal Reserve under alter-

native strategies is independent of the instrument regime.

This hypothesis

of an interaction effect is rejected, implying independence of the target
and indicator concepts.
The general hypothesis of this study, Hypothesis

IV, is that the

Federal Reserve responds in a systematic manner to intermediate targets.
The evidence is consistent with this hypothesis in the sense that several
strategies are not significantly

different from the CONTROL solution.

The

RF strategy is the subject of Guttentag's criticism that free reserves is
too short-run a variable to be a proper policy guide.

Guttentag and Brunner

and Meltzer agree, however, that if the,control period is three months or
longer free reserves may perform better as a target.21

Our results show that

if quarterly targets are specified, there is no significant difference between the RF and RT strategies; and, in fact, the Federal Reserve has been
behaving in a manner consistent with these strategies.

The CONTROL solution

is also not significantly different from the RC strategy, implying that the
Federal Reserve holds a neo-Keynesian

view of the monetary process, as distinct

20Patric H. Hendershott, The Neutralized Money Stock, Homewood, Illinois;
Richard D. Irwin, Inc.,
1968.
21Karl Brunner and Allan Meltzer, "Genesis and Development of the Free
Reserves Conception," Readings in Money, National Income, and Stabilization
Policy, ed. Smith & Teigen, 1965, pp. 197-210.

- 15 -

from a Monetarist view.

This evidence is consistent with the view that

the Federal Reserve responds systematically
view, are important in transmitting

to the targets which, in its

the impact of policy to ultimate ob-

jectives.
Finally, aside from the results of this particular set of experiments,
the optimal simulation approach presented offers a method for studying the
problems of economic policy using non-linear models.

While general proofs

are as yet available only for the non-linear static case, the implicit assumption that the policy maker revises his plan when new information becomes
available is not unrealistic.

APPENDIX

SOLUTION
METHODS
Of the three sets of equations derived above, only the set (6) is

generally linear.

Sets (10) and (4) are generally non-linear.

Two alter-

native solution methods are discussed.

MATRIXINVERSION
System (6) is linear in the partial derivatives as coefficients
can be solved either by generalized
discussed

Gauss-Seidel

interactive

in the next section, or by the following matrix

technique,

technique.

and

as
Express

.(5) as:

."I

jil

agj

ayi

h~.

l

=

J

_

i=l ,...,N

x

(11)

aYi

or, in matrix notation:

Ah’ =B
where A is an NxN partial derivative matrix, X' is an N element Lagrangean
multiplier

aw

vector.
vector , and B is the negative of the ayi

be solved with ease by any one of several matrix manipulation
a0ae in computer libraries.'

System (11) can
programs avail-

However, these programs involve some type of

matrix inversion and are extremely time-consuming.
next section shows that the Gauss-Seidel

On the other hand, the

solution method requires initial

'The Present research uses PROGRAM SIMQ, described in System 1360
Scientific Subroutine Package (360A-C+03X)
Version III Programmers Manual
H20-0205-3.
(International Business Machines Corporation, 1966). p. 120.

-2estimates of the endogenous
bility and efficiency,

variables.

therefore,

The matrix inversion procedure

As a matter of computational

the two procedures are used in combination.

is used on the first iteration to obtain

initial estimates of the x vector.

On all subsequent

iterations the entire

equation system (4), (6), and (10) is solved by the Gauss-Seidel

THE GUASS-SEIDEL

feasi-

procedure.

ITERATIVEPROCEDURE

Computer simulation

of large-scale econometric models requires a

numerical solution technique for systems of simultaneous equations.

If

the model is linear, a matrix method such as that discussed above may be
applied.

In addition to the time-costliness

of these matrix methods,

the

non-linear nature of many current models of the U. S. economy requires
rapid non-linear

solution methods.

Until recently these models were solved

by some variant of the Newton iterative method, which essentially

linearizes

each equation of the system by a Taylor series expansion about the trial
solution vector and solves the resultant linear system by matrix

inversion. 2

The several variants of the Newton method have been found to be extremely
costly in terms of computer time because inversion of large matrices and
iterative solution-seeking

are both involved.

To overcome these difficulties,
are currently solved by the Gauss-Seidel
iterative technique.

most large-scale econometric models
method.3

This is a straightforward

After a first trial solution is assumed, each successive

iteration adopts the previous trial solution as a starting estimate.

Iteration

2

G. Hadley, Nonlinear and Dynamic Proqramminq, pp. 56-57. C. C. Holt
et aJ., Proqram Simulate II (Madison, Wisconsin:
University of Wisconsin, 1967),
Zztions
9.3-9.4.
31bid., Section 9.5. See also, Michael K. Evans, Computer Simulations
of Non-Linear Econometric Models, pp. 5-7; Jorge J. More, A Class of Nonlinear
Functions and the Convergence of Gauss-Seidel and Newton-Guass-Seidel Iterations.

-30
continues until successive solutions agree to the preassigned degree of
precision.
Algebraically, the method may be expressed as follows:

Let the

jth equation of the system be represented as:4

=

fj

j=l ,.**,N

,* -0 ,yNIx,z)

(Yl'Y2"'*'Yj_lyyj+

yj

where Y is endogenous,

Assume an

rector and Z is exogenous.

X is a policy

initial trial value for each endogenous variable, denoted Yj

(0) .

Evaluate

the equation system:

=

y 2b)

(YJ (01,

(1)
fj

,...,

j=b4
Yj-l(O),

Using the values of Yj obtained

Yj+l("),"'Y

=

fj(y,

(r-l),

y2(r-l),

Z)

in the first trial solution, solve the system

again to obtain a second trial solution.

(4

Ytj(o)IX,

.

.

.

,yj-,

Iterate in this manner such that:

(r-l),

'jc,(r-lJ,-.

.yY:,(r-')

1x9

Z)

Yj

j=l , .. .,."I

until:

‘j

(4 _ y (r-1)
j

c tolerance

yj (r-11

While the above presentation
several additional

indicates the basic Gauss-Seidel

features may be used to improve its.solution

First, convergence may be enhanced by using the values yi

(4,

routine,

characteristics.

i c j, already

4This exposition derives from that of Evans, Computer Simulations of
Non-Linear Econometric Models, pp. 6-7.

-4calculated for the rth iteration to determine yj

While this method may speed convergence,
equations are not properly ordered.

0").

That is:

it may also lead to divergence

if the

5

Secondly, as an alternative

to the straightforward

substitution

of

current iteration values as initial estimates of the succeeding solution, more
control over the solution may be obtained if the following up-date routine is
used:

‘ii

b+l) = y (r) + d*S
j

j

(Y

(4 _ y (r-1 1)
j

j

where a is a dampening factor and sj is a Sign factor.

6

The dampening

factor

a allows the starting estimate of yj for a particular iteration to be approximated as the weighted average of the previous and current estimates.
a may speed convergence,

but at the risk of inducing divergence.

less than 1.0 will enhance convergence,

Increasing

A value of 3

but at a slow pace.

The sign factor is needed to indicate the direction of change necessary
to reduce the residual error; that is, to move toward convergence.

It is cal-

culated for the initial trial solution, and is positive or negative, depending
on the sign of the partial derivative of the function fj with respect to the
endogenous variable y..

J

That is:

'Evans, Computer Simulations of Non-Linear Econometric Models, p. 7.
See Also, Michael J. Hartley, “Instructions
for the Use of the Econometric
Model Solution Program," (Durham, North Carolina: Duke University, 1969,
mimeographed).

6Holt et al,, Program Simulate II,

Section 9.5b.

-5-

+l

2

11

= -1

afj

if

aYj

(y

(‘))

yb)

[I
2

where y(O) is the vector of initial trial solutions for the rth iteration. 7
Changing the sign factor for one or more errant variables may turn a divergent
system into a convergent one.
In addition to these factors, several other changes in a system of
equations may enhance convergence.
the equations.
is automatic.

These include the possibility

of reordering

In some cases a recursive ordering may be found, and convergence
In more complex systems some simultaneity

is present, and the

correct ordering of the equations may be a question of trial and error.

Secondly,

the equations themselves must be normalized on their "dominant" variable. 8
Finally, if all else fails, the tolerance may be relaxed to allow a less precise
solution.

Reasoned and diligent use of these various controls, together with

familiarity with the logic of the equation system being solved, should result in
convergence

of any soluable system.

71dem.
81bid., p. 9.5c.


Federal Reserve Bank of St. Louis, One Federal Reserve Bank Plaza, St. Louis, MO 63102