SHOULD MACROECONOMIC POLICY MAKERS CONSIDER PARAMETER
COVARIANCES?
HANS M. AMMAN AND DAVID A. KENDRICK
Abstract. Many macroeconomic policy exercises consider the mean values of parameter
estimates but do not use the variances and covariances. One can argue that the uncertainty
of these parameter estimates is sufficiently small that it can safely be ignored. Or one
can take the position that this kind of uncertainty cannot be avoided no matter what one
does. Thus it is just as well to ignore it while making policy decisions.
In this paper we address both of these positions in the presence of learning and find
that they are lacking. To the contrary, we find evidence that the potential damage from
ignoring the variances and covariances of the parameter estimates is substantial and that
taking them into account can improve matters.
1. Introduction
Many macroeconomic policy exercises consider the mean values of parameter estimates
but do not use the variances and covariances. One can argue that the uncertainty of these
parameter estimates is sufficiently small that it can safely be ignored. Or one can take the
position that this kind of uncertainty cannot be avoided no matter what one does. Thus
it is just as well to ignore it while making policy decisions.
In this paper we address both of these positions in the presence of learning and find
that they are lacking. To the contrary, we find evidence that the potential damage from
ignoring the variances and covariances of the parameter estimates is substantial and that
taking them into account can improve matters.
We use the simplest possible model, namely the Chow-Abel macroeconometric model
of the U.S. economy, and find that in this case (1) the uncertainty of parameter values in
the U.S. economy is sufficiently great that it cannot be safely ignored and (2) a method is
available for considering parameter uncertainty which can significantly improve on policy
outcomes. The method is the Passive Learning method from control theory. We compare
this method with a certainty equivalence method which ignores the parameter uncertainty.
In these experiments we find with Monte Carlo methods that the Passive Learning method
has substantially lower cost than the certainty equivalence method. Thus we find an
indication that it is prudent to consider parameter uncertainty in macroeconomic policy
analysis.
2. The Model
We have chosen to use the simplest possible model for this study. It is a model
based on the work of Chow (1967) and Abel (1975) which has only two state variables
Date: October 26, 1997.
Key words and phrases. Macroeconomics, learning, stochastic optimization, numerical experiments.
JEL Classification: C63, E61. We would like to thank Gregory Chow, Henk Don, Ray Fair and Peter
Tinsley for useful comments. Corresponding author: Hans M. Amman, Department of Economics, University of
Amsterdam, Roetersstraat 11, Room E1-913, 1018 WB Amsterdam, the Netherlands, Email [email protected].
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H.M. AMMAN AND D.A. KENDRICK
(consumption and investment) and two control variables (government expenditures and the
money supply). It was important to us to have a model with at least two control variables
because we wanted the decision maker to have a choice between policy channels which
included variables whose coefficient were more and less uncertain. Thus the decision
maker can choose between policies which offer different degrees of uncertainty.
To be more specific, the Abel (1975) model has the following structure
(1)
&W
(2)
,W
&Wb b ,Wb *Wb 0Wb b &Wb ,Wb b *Wb 0Wb b where &W is consumption, ,W investment, *W government expenditures and 0W the
money supply at time W. The parameters in these equations are uncertain. Moreover,
the diagonal variance elements in the covariance matrix for these parameters indicate, for
example, that we can be more certain about the impact on consumption and investment of
changes in government expenditures than of changes in the money supply. Which policy
mix should be used?
In order to study the question of whether or not it is wise to consider the covariance
matrix of the parameters while making policy decisions, we wanted the smallest model
which was large enough to capture the essence of the problem. We did not want a model
with made-up coefficients, but rather one with coefficients estimated from the U.S. data
because we wanted to know whether or not the uncertainty in the U.S. economy was large
enough that accounting for the uncertainty was important. Also, we did not seek a model
with elaborate lag structures, with forward variables, or with time-varying parameters
- not because these things are unimportant but because we wanted to begin with the
simplest possible model which focused on the question at hand.
3. The Method
The method we used was drawn from control theory. In particular we used 10000
Monte Carlo runs and compared two methods:
s Certainty Equivalence (CE)
s Passive Learning
An early and influential treatment of uncertainty in macroeconomic parameters was
done by Brainard (1967). The Passive Learning method used in our study dates back to
the works of Aoki (1967), Baum (1977), (Shupp (1972) (1976), Chow (1973) and (1975),
Tinsley, Craine and Havenner (1974), Turnovsky (1975) and (1976), Craine (1979), Pohjola (1981) and Don (1983). Two recent approaches, applied to monetary policy, are
presented by Caplin and Leahy (1996) and Fair and Howrey (1996). Caplin and Leahy
apply a game theoretical approach while Fair and Howrey follow a control framework.
This paper builds upon the work of Fair and Howrey in the sense that we use a linearquadratic control approach extended for a Passive Learning information structure in which
parameter uncertainty is taken into consideration.
The comparison of CE and Passive Learning methods as used in this study is described
in Kendrick (1981), (1982) and Amman and Kendrick (1994), (1995). Both of these
methods are passive learning in the sense that at each time period the new observations
on the state of the economy are used to update parameter estimates. The two methods
differ in that CE uses only the mean values of the parameter estimates to compute the
MACROECONOMIC POLICY
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feedback rule while the Passive Learning method uses both the means and the covariances
of the parameter estimates.
There are four sources of uncertainty in our approach
s
s
s
s
additive noise terms in equations and uncertain parameters in equations and measurement errors on the consumption and investment variables
uncertain initial state of consumption, & and investment ,
All these random variables are generated by Monte Carlo routines using the appropriate
covariance matrices. For example, the additive noise term generation uses the covariance
of the model noise terms. Parameter uncertainty is modeled by generating, for each
Monte Carlo run, estimates of the parameters using the variance-covariance matrix from
the model estimation. Thus the true values of the parameters are treated as fixed but
unknown. So the decision maker begins each run with parameter estimates generated by
the Monte Carlo routines which are not the same as the true values. These estimates
are used to compute the feedback rules which in turn yield the government expenditures
and the money supply. These instruments are then used along with the true values of
the parameters and with the noise terms to calculate consumption and investment for
the next time period. Measurement error is added to the model and then the resulting
observations are used to update the parameter estimates and the estimates of consumption
and investment. Since there is measurement error, the true values are not known to the
decision maker but rather he or she only has estimates of the states.
Also, the covariance for the additive noise terms in the measurement error equations
is used to generate initial estimates & and , . Thus, as in the real world, the decision
maker is faced at each time period with parameters and &W and ,W which he knows are
not the true values. However, these values are created using the appropriate probability
distributions.
4. The Results
We ran 10000 Monte Carlo runs on the Abel model. For each run we computed the
values of the quadratic tracking function as described in Kendrick (1982). The means
and standard errors for these criterion values were
mean standard error
Certainty Equivalence 930.60
29.79
Passive Learning
883.44
28.01
Thus the Passive Learning method performed substantially better by having a lower
criterion value than the certainty equivalence method. What does this mean? Simply that
if we ignore the fact that we have more accurate estimates of some parameters than of
others, we do so at our peril.
Also we compared the number of runs on which CE or Passive Learning has the lowest
criterion value. Here we found
Certainty Equivalence
Passive Learning
number of runs with lowest criterion value
5124
4876
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H.M. AMMAN AND D.A. KENDRICK
From these two results one can see that CE has the lowest criterion value roughly
51% of the time so it might seem wise to simply ignore the uncertainty in the parameters.
However, in a substantial number of cases faulty parameter estimates lead the CE solution
astray in a serious way, see Figure 1. The points in Figure 1 lie on the 45 degree line
when the CE and Passive Learning criterion values are equal on a Monte Carlo run. So
the greater number of points above the 45 degree line is one way to illustrate the higher
average cost associated with the CE solutions as well as the existence of a number of
outliers high on the CE side of the line.
Figure 1.
Scatter Diagram CE versus Passive Learning
12000
10000
8000
CE
6000
4000
2000
0
0
2000
4000
6000
8000
Passive Learning
10000
12000
Thus if a policy maker takes the risky approach of using the CE method and ignoring
the parameter uncertainty he or she will do well if the estimates are close to the true
parameter values. On the other hand, if the parameter estimates are distant from the true
values, the policy maker may do really badly. In contrast, with the Passive Learning
method the policy maker is cautious about using policy variables which have associated
parameters along the channel of effect with high variances. Thus when the parameter
estimates are far from the true values, he or she does better than they would have done
with the CE approach. So, on average, the more conservative Passive Learning strategy
does a better job.
Thus when policy makers use the parameter estimates to make simulations and to
compute feedback rules while ignoring the uncertainty of these estimates, they run the
risk that the estimates they use will be off from the true values by large enough amounts
to cause serious problems. Therefore, the results from this case suggest that it would
be prudent to factor the degree of uncertainty about parameter estimates into their policy
calculations.
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Department of Economics, Roetersstraat 11, 1018 WB Amsterdam, the Netherlands.
E-mail address: [email protected]
Department of Economics, University of Texas, Austin, Texas 78712, USA.
E-mail address: [email protected]