Should macroeconomic policy makers consider parameter covariances?

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/5144411

Should Macroeconomic Policy Makers Consider Parameter Covariances?, Working Paper 9701 Article in Computational Economics · February 1999 DOI: 10.1023/A:1008724519121 · Source: RePEc

CITATIONS

READS

24

18

2 authors, including: Hans M Amman University of Amsterdam 114 PUBLICATIONS 594 CITATIONS SEE PROFILE

Some of the authors of this publication are also working on these related projects:

Approximating the Value Function for Optimal Experimentation View project

All content following this page was uploaded by Hans M Amman on 08 December 2016. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately.

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 Date: September 2, 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]. 1

2

H.M. AMMAN AND D.A. KENDRICK

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 (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 &Wb b ,Wb  *Wb  0Wb b  (2) ,W &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: q Certainty Equivalence (CE) q 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), Shupp (1972) (1976), Chow (1973) and

MACROECONOMIC POLICY

3

(1975), Tinsley, Craine and Havenner (1974), 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 linear-quadratic 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 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

q q q q

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

H.M. AMMAN AND D.A. KENDRICK

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

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.

MACROECONOMIC POLICY

5

6FDWWHU 'LDJUDP &( YHUVXV 3DVVLYH /HDUQLQJ

   &(

   







  3DVVLYH /HDUQLQJ





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. References [1] Abel, A.B., 1975, A Comparison of Three Control Algorithms to the Monetarist-Fiscalist Debate, Annals of Economic and Social Measurement 4, 239-252. [2] Amman, H.M. and D.A. Kendrick, 1994, Active Learning: Monte Carlo Results, Journal of Economic Dynamics and Control 18, 119-124. [3] Amman, H.M. and D.A. Kendrick, 1995, Nonconvexities in Stochastic Control Models, International Economic Review 36, 455-475. [4] Aoki, M., 1967, Optimization of Stochastic Systems, Academic Press, New York. [5] Brainard, W.C., 1967, Uncertainty and the Effectiveness of Policy, American Economic Review 57, 411-425. [6] Caplin, A. and J. Leahy, 1996, Monetary Policy as a Process of Search, American Economic Review 86, 689-702.

6

H.M. AMMAN AND D.A. KENDRICK

[7] Chow, G., 1967, Multiplier, Accelerator and Liquidity Preference in the Determination of National Income in the United States, Review of Economics and Statistics 49, 1-15. [8] Chow, G., 1973, Effect of Uncertainty on Optimal Control Policies, International Economic Review 14, 632-645. [9] Chow, G., 1975, Analysis and Control of Dynamic Systems, John Wiley, New York. [10] Craine, R., 1979, Optimal Monetary Policy with Uncertainty, Journal of Economic Dynamics and Control 1, 59-83. [11] Craine, R., A. Havenner and P. Tinsley, 1976, Optimal Macroeconomic Control Policies, Annals of Economic and Social Measurement 5, 191-203. [12] Don, F. J. H., 1983, Uncertainty and the Vigor of Policy, Journal of Economic Dynamics and Control 6, 187-191. [13] Fair, R.C. and E.P. Howrey, 1996, Evaluating alternative monetary policy rules, Journal of Monetary Economics 38, 173-193. [14] Kendrick, D.A., 1981, Stochastic Control for Economic Models, McGraw-Hill, New York. [15] Kendrick, D.A., 1982, Caution and Probing in a Macroeconomic Model, Journal of Economic Dynamics and Control 4, 149-170. [16] Pohjola, M.T., 1981, Uncertainty and the Vigor of Policy: Some Implications of Quadratic Preferences, Journal of Economic Dynamics and Control 3, 299-305. [17] Shupp, Franklin, 1972, Uncertainty and Stabilization Policies for a Nonlinear Macroeconomic Model, Quarterly Journal of Economics 80, 94-110. [18] Shupp, F., 1976, Uncertainty and Optimal Stabilization Policy, Journal of Public Economics 6, 243-253. 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]