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Recent models of monetary policy can have indeterminacy of equilibria, which is often viewed as a difficulty of these models. We consider the significance of indeterminacy using the learning approach to expectations formation. We employ expectational stability as a selection criterion for different equilibria and derive the expectational stability and instability conditions for forward-looking multivariate models, both without and with lags. The results are applied to several monetary policies.

In recent years, there has been a large amount of research studying the performance of alternative monetary policies in dynamic macroeconomic settings; for example, see the survey (Clarida et al., 1999) and the papers in the 1999 Special Issue of the Journal of Monetary Economics and in the volume ( Taylor, 1999). A difficulty has emerged in this literature: many recent models of monetary policy are plagued by the problem of indeterminacy, i.e. there are multiple, even continua of rational expectations equilibria (REE). 1 The issue of indeterminacy can be important from a practical point of view. Pursuit of optimal monetary policy on the part of the central bank, or flexible inflation targeting in the sense used by Svensson (1999), implies that the instrument of monetary policy, the short-term nominal interest rate, should respond to inflation forecasts; see Clarida et al. (1999). Clarida et al. (1998) provide evidence that monetary policy in a number of industrialized countries (like Germany, Japan, and the U.S.) has been forward-looking since 1979.2 A number of theoretical studies have considered the issue of indeterminacy, with interest rate rules and different views been taken on this problem. Bernanke and Woodford (1997) have argued against inflation forecast targeting on the basis of indeterminacy—these rules may lead to too much volatility in inflation and output which any central bank ought to avoid; see also Woodford, 2003b and Woodford, 2003a. At the other end of the spectrum, indeterminacy is viewed as an unimportant curiosum. For instance, in McCallum, 2001a, McCallum, 2001b and McCallum, 2003, the use of the minimal state variable (MSV) solution is advocated for applied analysis. To fix terminology, we will use the terms fundamental vs. non-fundamental equilibria to distinguish between the MSV and other REE. A recent paper taking indeterminacy as an empirically relevant possibility is Clarida et al. (2000). They estimate a forward-looking policy reaction function for the postwar U.S. economy, both before and after the appointment of Paul Volcker as Fed Chairman in 1979. They conclude that monetary policy in the pre-Volcker era was compatible with the possibility of bursts of inflation and output that resulted from self-fulfilling changes in expectations of the private sector. In this way, monetary policy of the Federal Reserve contributed to the high and volatile inflation of the 1960s and 1970s. Analytically, the pre-Volcker period is modelled as a non-fundamental REE. In contrast, monetary policy in the Volcker–Greenspan era is compatible with the existence of a unique fundamental equilibrium delivering low and stable inflation. In this paper, we take a new perspective on the problem of indeterminacy induced by monetary policy by introducing a selection criterion among the REE to narrow down the set of plausible equilibria. We use the adaptive learning approach to expectation formation that has recently gained some popularity.3 In general terms, the learning approach suggests that expectations might not always be fully rational, and the REE of interest should satisfy a natural stability criterion in expectations formation. If economic agents make forecast errors and adjust their forecast functions over time, the economy will reach an adaptively stable or learnable REE asymptotically, where these forecast errors eventually disappear. In contrast, adaptively unstable REE will not emerge as an outcome from such adjustment processes. Even though our motivation is primarily an analysis of recent models of monetary policy, we in fact do more as we provide general results that are applicable to a wide variety of multivariate linear models. Our applications to monetary policies illustrate how to use the results in an economic framework. We first consider models without lags of endogenous variables for which general theoretical results can be obtained. We then develop stability conditions for models with lags. The latter can be used in numerically calibrated models even though general analytical results are not obtainable. As our application, we develop stability/instability conditions of non-fundamental REE under learning for versions of the New Keynesian model of monetary policy.4 Learnable non-fundamental REE can be ruled out if the structural and policy parameters of the model satisfy certain specific conditions that have an economic interpretation and yield important insights about specific interest rate rules. We also assess the plausibility of the Clarida et al. (2000) explanation of the Pre-Volcker and Volcker era from the learnability view point. We remark that expectational errors can naturally arise in practice. The economy might be subject to changes in its basic structure or in the practices of policy makers. The assumption that agents somehow have rational expectations (RE) immediately after such changes is clearly strong and may not be correct empirically. The policy maker would naturally like to adopt policy that is conducive to coordination by the private sector on a desirable equilibrium entailing low inflation and output volatility. The key general message of our paper is that the monetary policy rule used by the central bank plays a pivotal role in determining the equilibrium selection. Good policy design should ensure that (i) the fundamental equilibrium is stable under adaptive learning and (ii) that possible non-fundamental REE are not stable under learning. The paper is organized as follows. Section 2 develops a general linear bivariate model without lags, the different types of REE for such models and the conditions for stability under least-squares learning for these REE. Section 3 applies the results to the standard New Keynesian model when monetary policy is conducted either through a forward-looking Taylor rule or an optimal discretionary rule proposed by Clarida et al. (1999). Section 4 incorporates lagged endogenous variables to the general model of Section 2. Section 5 applies the generalized framework to the issues analyzed by Clarida et al. (2000). Conclusions and appendices follow.

We have carried out a general analysis of learnability of non-fundamental equilibria for multivariate forward-looking linear models with and without lags. Our results apply to models of monetary policy that are being used to give advice to policy makers. Learnability of the fundamental REE and unlearnability of non-fundamental REE are an important constraint that good monetary policy design should aim to meet. Otherwise undesirable fluctuations may result. While it is clearly desirable to achieve learnability of the MSV REE by appropriate choice of the policy rule, the assessment of the fluctuations arising as non-fundamental REE is less clear-cut. While these endogenous fluctuations are usually inferior to MSV REE, their normative comparison is ambiguous with respect to cases where the economy fluctuates as a result of there being no learnable REE. Nevertheless, it would seem possible to conduct a positive analysis between stable SSEs and non-rational fluctuations due to non-learnable equilibria. The scenario in Clarida et al. (2000) illustrates the possibilities for a positive analysis. Clarida et al. suggest that the high and volatile U.S. inflation in the 1960s/70s may have been indeterminate equilibria caused by the policy. Our analysis has shown that neither the fundamental nor the non-fundamental REE were learnable during this period. The volatile period was perhaps a situation of agents trying unsuccessfully to find some equilibrium and not necessarily a sunspot equilibrium. A further analysis of this issue would seem worth while. The Clarida et al. (2000) explanation is based on RE and it would be consistent with agents not making systematic errors in their forecasts of inflation and output gap. Our explanation has agents making forecast errors that do not disappear over time. Agents might believe in PLMs corresponding to the fundamental REE but, since errors do not disappear over time, they might also entertain the possibility of PLM matching the form of some non-fundamental REE. However, even in the latter case, the forecast errors would not disappear over time as all REE are unstable under learning. Thus, one way to test the competing hypotheses is to study the behavior of forecast errors in inflation and output gaps in the pre-Volcker era. We have also found policy rules and domains for policy parameters which satisfy the Taylor principle but are nevertheless associated with indeterminacy and existence of learnable non-fundamental REE. Both fundamental and some non-fundamental REE are potentially learnable for some domains of policy parameters under the rules considered in Sections 3.1 and 3.2. This result shows that the Taylor principle does not always guarantee determinacy and it is important to avoid indeterminacies when forward-looking policy rules conform to the Taylor principle. Indeterminacy can be avoided with moderate aggression to inflation and/or output gap forecasts. Furthermore, even when indeterminacies exist, a modest response to output leads to instability of all types of sunspots (as in Section 3.1) and to learnability of the MSV solution. A further way to reduce indeterminacy is to make the interest rule react directly to its own past values, which makes it easier to satisfy the Taylor principle. These inertial rules have been found to have desirable properties: they can lead to the existence of a unique learnable fundamental equilibrium and also have the potential to implement optimal policy of the central bank; see Bullard and Mitra (2004) and Rotemberg and Woodford (1999). The U.S. interest rule estimated since the 1980s by Clarida et al. (2000) has such an inertial component that, in conjunction with an appropriate response to the inflation forecast and output gap, leads to a unique learnable fundamental REE. In summary, we do not advocate policies that violate the Taylor principle. Policies satisfying the Taylor principle are recommended as long as they do not lead to indeterminacy. In addition, our results in Section 3.2 suggest that inflation-targeting central banks should adopt a policy of flexible inflation targeting instead of strict inflation targeting, since the latter can lead to the existence of learnable, indeterminate equilibria. This is what most inflation-targeting central banks seem to do in practice.