سیاست های پولی، عدم قطعیت و یادگیری

Forward-looking monetary models with Taylor-type interest rate rules are known to generate indeterminacies, with a potential dependence on extraneous ‘sunspots,’ for some structural and policy parameters. We investigate the stability of these solutions under adaptive learning, focusing on ‘common factor’ or ‘resonance frequency’ representations in which the observed sunspot has a suitable time-series structure. We consider specifications incorporating both lagged and expected inflation in the Phillips Curve, and both expected and inertial elements in the policy rule. We find that some policy rules can indeed lead to learnable sunspot solutions and we investigate the conditions under which this phenomenon arises.

The development of tractable forward looking models of monetary policy, together with the influential work of Taylor (1993), has lead to an explosion of research on the implications of adopting Taylor-type interest rate rules. These rules take the nominal interest rate as the policy instrument and direct the central bank to set this rate according to some simple (typically linear) dependence on current, lagged, and/or expected inflation and output gap, and possibly on an inertial term to encourage interest rate smoothing. While these simple policy rules for many reasons are advantageous to both researchers and policy makers, it has been noted by some authors, e.g. Bernanke and Woodford (1997), Woodford (1999) and Svensson and Woodford (1999), that the corresponding models exhibit indeterminate steady states for large regions of the reasonable parameter space. This presence of indeterminacy is thought undesirable because associated with each indeterminate steady state is a continuum of sunspot equilibria, and the particular equilibrium on which agents ultimately coordinate may not exhibit wanted properties. Though having their informal origins in Keynes’ notion of animal spirits, analysis of sunspots has, in the past, been couched principally in the theoretical literature. However, applied macroeconomists began to take notice when, in the mid nineties, Farmer and Guo (1994) showed that calibrated real business cycle models, modified to include externalities or other non-convexities, exhibited sunspots; and furthermore, these sunspots could be used to explain fluctuations at business cycle frequencies. This applied interest has spread to the literature on monetary policy, and, in an empirical sense, has culminated with the argument of Clarida et al. (2000) that the volatile inflation and output of the seventies may have been due to sunspot phenomena. In particular, they combine a standard forward-looking ‘New Keynesian’ IS–AS model1 with a simple estimated forward-looking Taylor rule, using data from the 1960s and 1970s, and find that the corresponding steady state is indeterminate; they conclude that the fluctuations in output gap and inflation may be well explained by agents coordinating on a volatile sunspot equilibrium. The existence of sunspot equilibria raises the question of whether it is plausible that agents will actually coordinate on them. One natural criterion for this is that the sunspot equilibria should be stable under adaptive learning.2 Although it has been shown by Woodford (1990) that stable sunspots can exist in simple overlapping generations models,3 the sunspots in many calibrated applied models are lacking this necessary stability. For example Evans and Honkapohja (2001) show that sunspots in the Farmer–Guo model are unstable, and Evans and McGough (2002a) describe a stability puzzle surrounding the lack of stable indeterminacies in a host of non-convex RBC-type models.4 The existence of indeterminacies in monetary models, together with the instability of indeterminacies in RBC-type models, raises a natural question: Are sunspot equilibria in the New Keynesian models stable under learning? This specific question has been addressed by Honkapohja and Mitra (2004), who consider a purely forward looking AS equation (‘Phillips’ curve) and analyze a variety of interest rules including those dependent on current, lagged, and expected inflation and output gap, and those also dependent on an interest rate smoothing term. They find that if the interest rate depends only on expected inflation and expected output gap then there can exist stable equilibria that depend on finite state sunspots; otherwise, the sunspot equilibria they consider are not learnable.5 Independently of the monetary policy literature, recent research has emphasized that stability under learning of sunspots can depend upon the way in which a particular equilibrium is viewed, or represented. Evans and Honkapohja (2003c) found that finite state sunspots in a simple forward looking model can be stable even though previous research had suggested that no stable sunspots exist in these models. The apparent paradox is resolved as follows: all sunspot equilibria in these models can be represented as a linear dependence on lagged endogenous variables and a sunspot variable taking the form of a martingale difference sequence. These representations are always unstable under learning. However, when the sunspot is a finite state Markov process, the associated equilibrium is also finite state and thus has an alternate representation depending solely on the sunspot. When represented in this manner, the associated learning dynamics indicate stability for some (but not all) regions of the parameter space. In Evans and McGough (2005), we studied sunspot equilibria in a univariate stochastic linear forward looking model that incorporates a lag. We found that any given equilibrium may be viewed, or represented, in two fundamentally different ways: in the usual way, as a linear dependence on once and twice lagged endogenous variables and on a sunspot having zero conditional mean; and in a new way, on once lagged endogenous variables and on a sunspot exhibiting serial correlation. We referred to the usual way of viewing sunspots as the ‘general form’ representation of the equilibrium, and to the new way of viewing sunspots as the ‘common factor’ representation of the equilibrium. 6 We found that the stability of the equilibrium in question depended on the chosen representation. In particular, for the model we considered, stable common factor sunspots were found to exist in abundance, even though, as was already well known, there exist no stable general form sunspots. This new line of research indicates the need for careful analysis of sunspot stability in applied models. Every sunspot equilibrium has a common factor representation, and the stability properties of common factor representations are different from their general form counterparts. Thus, stability analysis must incorporate both general form and common factor representations. In this paper, we generalize common factor analysis to apply to standard models of monetary policy, and carefully investigate the stability of the resulting representations. We follow Bullard and Mitra (2002) and Honkapohja and Mitra (2004) by specifying a simple New Keynesian IS–AS model, except that, for added generality, as in Gali and Gertler (1999) and much applied work, we allow for some dependence on lagged inflation in the Phillips curve. We close the model with a variety of interest rate rules: like Bullard and Mitra, we consider rules depending on current, lagged, and expected inflation; and like Honkapohja and Mitra, we also consider rules depending on lagged nominal interest rates. For each model we consider three calibrations of the IS–AS structure, as well as some alternative parameter values. Analytic results are, in general, unavailable, and so we test stability numerically by considering, for each calibration, a lattice over the space of policy parameters. At each point in the lattice, indeterminacy and stability of the corresponding equilibria are examined. Our main result supports the findings and advice of Honkapohja and Mitra, and indeed it makes their cautionary note more urgent: All models in which the policy rule depends on some form of expectations of future variables exhibit stable common factor sunspots for some parameter values. To be sure, these parameter values are not always reasonable, but, in some cases, they closely match calibrations. Furthermore, these stable sunspots exist even when the policy rule also depends on other aggregates, such as current inflation or output, and lagged interest rate. We also find that no general form sunspots are stable, thus emphasizing the importance of analyzing common factor representations. This paper is organized as follows. Section 2 presents the various monetary models under consideration, as well as the associated learning theory and the extension of common factor analysis to monetary models. To conserve space and facilitate comprehension, we include explicit computations of equilibrium representations in the appendix and for only one policy rule, and simply note that the remaining policy rules can be analyzed in a similar fashion. Section 3 contains the results of our investigations. The policy rules are classified into four types and discussed in separate subsections. In each case we consider numerous permutations of calibration, Phillips curve structure, indeterminacy nature, and representation type, and thus a careful catalog of all possible results would be tedious if not infeasible. Therefore, we provide a summary of the main features followed by a more careful discussion of the particularly interesting results. Section 4 concludes.

This paper has examined the question of whether macroeconomic fluctuations, taking the form of coordination on extraneous exogenous variables, are likely to emerge under adaptive learning when the economy is characterized by New Keynesian IS–AS equations and monetary policy follows a form of Taylor rule. Both purely forward-looking and hybrid, partly backward-looking inflation equations were examined. We have emphasized that the possibility of ‘sunspot equilibria’ that are stable under adaptive learning depends critically on the representation of the solution, i.e. on the econometric specification used by agents when they estimate and update their forecasting model. In many cases stationary sunspot equilibria can be represented either as ‘general form’ VARs, driven by serially uncorrelated sunspots, or as ‘common factor’ sunspot solutions, in which the extraneous sunspot variables are autoregressive processes with resonance frequency coefficients. Common factor sunspots generalize finite state Markov sunspots, which were an early focus in the sunspot literature and which have recently been shown to yield the possibility of stable sunspots in purely forward looking linear models. In the New Keynesian model, we find that common factor sunspots can indeed be stable under learning, in many cases, even though the general form solutions with serially uncorrelated sunspots are not. In particular, Taylor-type interest rate rules that depend on forecasts of future inflation can generate stable common factor sunspot solutions, and this risk is particularly high when there are strong IS and AS effects. This possibility arises even if the AS equation includes backward looking components and the interest rate rule includes inertia. This result is deeply troubling since monetary policy is often viewed as forward looking. If the structural model and its key parameters are known, or can be estimated fairly precisely, then an appropriately designed forward looking policy can deliver a stable determinate equilibrium (indeed an optimal stable equilibrium) and the sunspot problem will not arise. However, for some structural parameters the margin of error is small and the impact of an error is great. In contrast, policy rules depending on forecasts of current output and inflation do not appear to be subject to these difficulties.