یادگیری در مورد قواعد سیاست پولی

We study macroeconomic systems with forward-looking private sector agents and a monetary authority that is trying to control the economy through the use of a linear policy feedback rule. We use stability under recursive learning a la Evans and Honkapohja (Learning and Expectations in Macroeconomics, Princeton University Press, Princeton, New Jersey, 2001) as a criterion for evaluating monetary policy rules in this context. We find that considering learning can alter the evaluation of alternative policy rules.

1.1. Overview Monetary policy rules have been the subject of a good deal of recent research in the literature on monetary economics and monetary policy.1 While some of this work has focussed on systems which abstract from or suppress private sector expectations, many of the more recent papers analyze systems where private sector expectations enter the model explicitly. Most of these models involve small, forward-looking representations of the macroeconomy, such as those found in Clarida et al. (1999), McCallum and Nelson (1999), and Woodford (1999). In many cases the small model is a log-linearized and simplified version of a larger model derived from optimizing behavior in a dynamic stochastic general equilibrium context. When private sector expectations enter such models explicitly, recent research has emphasized the possibility that certain policy rules may be associated with indeterminacy of rational expectations equilibrium, and therefore might be viewed as undesirable. Some of the authors who discuss this issue include Bernanke and Woodford (1997), Carlstrom and Fuerst 2000 and Carlstrom and Fuerst 2001, Christiano and Gust (1999), Clarida et al. (2000), Rotemberg and Woodford 1998 and Rotemberg and Woodford 1999, and Woodford (1999). In a typical analysis, the authors compute the rational expectations solutions of the system with a given monetary policy rule, and if the rule induces indeterminacy then it is viewed as undesirable. The idea is that if the monetary authorities actually followed such a rule, the system might be unexpectedly volatile as agents are unable to coordinate on a particular equilibrium among the many that exist.2 In contrast, when equilibrium is determinate, it is normally assumed that the agents can coordinate on that equilibrium. It is far from clear, however, exactly how or whether such coordination would arise. In order to complete such an argument, one needs to show the potential for agents to learn the equilibrium of the model being analyzed. In this paper, we take on this task. We assume the agents of the model do not initially have rational expectations, and that they instead form forecasts by using recursive learning algorithms—such as recursive least squares—based on the data produced by the economy itself. Our methodology is that of Evans and Honkapohja 1999 and Evans and Honkapohja 2001. We ask whether the agents in such a world can learn the fundamental or MSV equilibrium of the system under a range of possible Taylor-type monetary policy feedback rules. We use the criterion of expectational stability (a.k.a. E-stability) to calculate whether rational expectations equilibria are stable under real time recursive learning dynamics or not. The research of Marcet and Sargent (1989) and Evans and Honkapohja 1999 and Evans and Honkapohja 2001 has shown that the expectational stability of rational expectations equilibrium governs local convergence of real time recursive learning algorithms in a wide variety of macroeconomic models. 3 We think of learnability as a necessary additional criterion for evaluating alternative monetary policy feedback rules. In particular, in our view economists should only advocate policy rules which induce learnable rational expectations equilibria. Central banks adopting monetary policy rules that are not associated with learnable rational expectations equilibria, under the assumption that private sector agents will coordinate on the equilibrium they are targeting, are making an important mistake. Our analysis suggests that such policymakers will encounter difficulties, as the private sector agents instead fail to coordinate, and the macroeconomic system diverges away from the targeted equilibrium. Learnable equilibria, on the other hand, do not have such problems. This is because the agents can indeed coordinate on the equilibrium the policymakers are targeting, so that the learning dynamics tend toward, and eventually coincide with, the rational expectations dynamics. Learnable equilibria are therefore to be recommended. 1.2. Model environment We consider monetary policy rules which have been suggested by various authors. All of these rules envision the central bank adjusting a short-term nominal interest rate in linear response to deviations of inflation from some target level and to deviations of real output from some target level. We take up four variants of such rules which we believe are representative of the literature: rules where the nominal interest rate set by the central bank responds to deviations of current values of inflation and output (we call this the contemporaneous data specification); rules where the interest rate reacts to lagged values of output and inflation deviations (lagged data specification); rules where the interest rate responds to future forecasts of inflation and output deviations (forward looking rules); and finally, rules which respond to current expectations of inflation and output deviations (contemporaneous expectations). The novel contribution of this paper is to evaluate these policy rules based on the learnability criterion in a standard, small, forward-looking macroeconomic model which is currently the workhorse for the study of such rules. We analyze the stability of equilibria under learning dynamics, and we also provide conditions for unique equilibria. Conditions for unique equilibria may be found sporadically for some of these policy rules in the existing literature, and we put these results into a unifying framework. Thus, we are able to evaluate monetary policy rules based not only on whether they induce determinacy but also based on whether they induce learnability. 1.3. Main results We find that monetary policy rules which react to current values of inflation and output deviations can easily induce determinate equilibria. Moreover, when equilibrium is determinate it is also learnable under this specification. However, contemporaneous data rules have often been criticized because they place unrealistic informational demands on the central bank, since precise information on current quarter values of inflation and output is usually not available to policymakers. One of our important findings is that rules which react to contemporaneous expectations of inflation and output deviations lead to exactly the same regions of determinate and learnable equilibria. Consequently, our results suggest that rules where the central bank responds to current expectations of inflation and output deviations are the most desirable in terms of generating both determinacy and learnability. Our reading of the policy rules literature is that such rules have not been given adequate attention and our results suggest more emphasis on them may prove fruitful. We find that rules which respond to lagged values or to future forecasts of inflation and output deviations do not have the same desirable properties. Determinate rational expectations equilibria are not necessarily learnable under the lagged data specification. In addition, rules which respond to lagged data can easily fail to generate determinacy. Forward-looking rules can easily induce equilibrium indeterminacy (see also Bernanke and Woodford, 1997). We find that determinate equilibria are always learnable for forward-looking rules, but when equilibrium is indeterminate, those equilibria which correspond to the minimum state variable (MSV) solutions may also be learnable. We do not examine the learnability of sunspot equilibria, which may exist when the equilibrium is indeterminate, in this paper. Taylor (1999a) recommends a “leaning against the wind” policy rule which calls for nominal interest rates which are adjusted positively, and more than one-for-one, in response to inflation above target, and positively to levels of production above target. We call this the Taylor principle following Woodford 2001 and Woodford 2002. Taylor's intuition is that under such a rule, a rise in inflation brings about an increase in the real interest rate which reduces demand and inflationary pressures, bringing the economy back towards the targeted equilibrium. On the other hand, a policy rule which does not obey the Taylor principle brings about a decrease in the real interest rate which adds to inflationary pressures, pushing the economy away from the targeted equilibrium. In this paper we support Taylor's intuition based on the criterion of learnability. In fact, we find that the Taylor principle completely characterizes learnability. If agents do not have rational expectations of inflation and output and instead start with some subjective expectations of these variables, learning recursively using some version of least squares, then a “leaning against the wind” policy on the part of the central bank does indeed push the economy towards the rational expectations equilibrium (REE) across all the specifications of policy rules we consider. Finally, we find that across all the information structures we consider, policy rules which respond relatively aggressively to inflation with little or no reaction to the output gap (or output gap forecasts) generally induce both determinate and learnable rational expectations equilibria. To the extent that both determinacy and learnability are desirable criteria, central banks may want to consider adopting such rules. 1.4. Recent related literature In an analysis complimentary to ours, Evans and Honkapohja (2000) consider the learnability of equilibria induced by optimal monetary policy rules in a structural model like the one used in this paper. By optimal, Evans and Honkapohja mean a policy rule derived from minimization of a loss function for the monetary authority, given the structure of the economy. A rule derived in such a way may or may not generate either determinacy or learnability of equilibrium, and Evans and Honkapohja investigate both of these properties. One important finding is that if the central bank assumes rational expectations on the part of private agents, then the equilibria induced are always unstable in the learning dynamics. On the other hand, if optimal policy is conditioned directly on the observed (subjective) private sector expectations, then the REE becomes stable under learning dynamics. In contrast to Evans and Honkapohja (2000), we study simple policy rules which are of the form recommended in the widely-cited work of Taylor (1993). We locate the set of rules in this class which are associated with both determinacy and learnability. A practitioner wishing to find an optimal policy rule in this set could then postulate an objective criterion for the central bank and use it to locate the best rule. This is essentially the same process that Rotemberg and Woodford 1998 and Rotemberg and Woodford 1999 and other authors have used to analyze these types of rules. The advantage of remaining in the class of Taylor-type rules hinges on the alleged robustness of these rules across models, as discussed at length in the Taylor (1999c) volume. For this reason we think it is interesting to consider learnability under either optimal rules or Taylor-type rules. McCallum (1999) and Taylor (1999b) have argued that it is important to check the robustness of policy rules in different monetary models since in general there is little agreement among economists about the appropriateness of any particular model. In this respect, Carlstrom and Fuerst 2000 and Carlstrom and Fuerst 2001 show that the equilibrium determinacy properties in models like the one we analyze are sensitive to certain key assumptions. These assumptions include which money balances, in terms of timing, enter the utility function, and also the nature of the sticky price assumption along off-equilibrium paths. They conclude that under their alternative assumptions, in setting the nominal interest rate, central banks should react aggressively to lagged inflation in order to preserve determinacy of equilibrium. Our model also provides support for policy rules which react aggressively to lagged inflation (and mildly to output). We also advocate policy rules based on contemporaneous forecasts. Since these forecasts are in effect based on past data in our systems under learning, we support the intuition that a central bank should look backward, as in the models analyzed by Carlstrom and Fuerst. We also think it would be interesting to carry out our learning analysis in the classes of models analyzed by Carlstrom and Fuerst. 1.5. Organization In the next section we present the model we will analyze throughout the paper. We also discuss the types of linear policy feedback rules we will use to organize our analysis, and a calibrated case which we will employ. In the subsequent section we present results on determinacy of equilibrium, and then on learnability of equilibrium, for each of four different classes of policy rules. We conclude with a summary of our findings

We have studied the stability of a simple macroeconomic system under learning for various monetary policy rules using methods developed by Evans and Honkapohja 1999 and Evans and Honkapohja 2001. We found that the Taylor principle—that nominal interest rates should be adjusted more than one-for-one with changes in inflation—is closely linked with stability in the learning dynamics across all of our specifications. We also found that, in general, determinacy alone is insufficient to induce learnability of a rational expectations equilibrium. We conclude that it may be unwise to simply assume that coordination on a unique equilibrium can occur under a reasonable description of agent learning. We stress that the methodology we employ to analyze the effects of learning imparts a lot of information to the agents in our model. By endowing the agents with a perceived law of motion that coincides with the MSV solution of the system, we are in effect giving the agents the correct specification of the vector autoregression they need to estimate in order to learn the rational expectations equilibrium. The local nature of the analysis further imparts initial expectations which are in the immediate neighborhood of the equilibrium. If, under these circumstances, the system is nevertheless driven away from the rational expectations equilibrium, then we do not hold out too much hope that the system can be rendered stable under some other plausible learning mechanism (although of course that remains an open question).15 For this reason we think of the learnability criterion as a minimal requirement for a policy rule to meet. In this paper, we have only considered “simple” policy rules, in which policymakers do not respond to the lagged interest rate. In part this was because this is the type of policy rule studied by Taylor (1993) which fueled the current wave of interest in monetary policy rules. However, estimated policy rules usually include a lagged interest rate in order to better capture the interest rate smoothing observed in actual central bank behavior. In a companion paper, Bullard and Mitra (2000), we are considering the case when the central bank also reacts to a lagged interest rate.