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Working Paper 74-2
ALTERNATIVE
OPTIMALOPEN MARKET STRATEGIES
A CLASSICALOPTIMIZATION
CERTAINTYEQUIVALENCE
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‘Joseph M. Crews, "Alternative Optimal Open Market Strategies: A Simulation Approach," Unpublished doctoral dissertation, University of North Carolina,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 INDICATORPROBLEM
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
--$--I
Regime
----------';;:yg'
.
t-------------4:'41
40:65 $I,
48:83 i4:28
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
SOLUTIONMETHODS
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.
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