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Description
1. MONETARY POLICY ADVICE
1.1 Determinacy
A fundamental issue in the evaluation of alternative monetary policy rules is the question of whether a proposed policy rule is associated with a determinate equilibrium or not. Starting with the work of Sargent and Wallace (1975), it has been shown that certain types of policy rules may be associated with large sets of rational expectations equilibria (REE) and that some of these equilibria may involve fluctuations in variables like inflation and real output due solely to self-fulfilling expectations. (1) Such rules and the associated equilibria arguably ought to be avoided if one wishes to stabilize these variables. Perhaps disconcertingly, this problem appears to be particularly acute for policy rules that may otherwise seem to be fairly realistic in terms of actual central bank behavior. For example, Clarida, Gali, and Gertler (1998) have provided evidence that suggests that monetary policy for the major industrialized countries since 1979 has been forward looking: nominal interest rates are adjusted in response to anticipated inflation. This empirical finding is somewhat puzzling in light of the fact that such forward-looking rules are associated with equilibrium indeterminacy in many models (see, in particular, Bernanke and Woodford 1997). Similarly, in many models policy rules that call for the monetary authority to respond aggressively to past values of endogenous variables (such as the previous quarter's deviations of inflation from a target level, or the output gap) can be associated with explosive instability of rational expectations equilibrium. Yet at the same time, such policy rules might also be viewed as fairly realistic in terms of actual central bank behavior in some contexts. Thus, at least two empirically relevant and seemingly ordinary-looking classes of policy rules seem to be associated with important theoretical problems.
These theoretical concerns impinge on the design of stabilization policy. Even aside from broad modeling uncertainty, there is considerable sampling variability about the estimated parameters of a given model of the macroeconomy. When a candidate class of policy rules may or may not generate indeterminacy, or explosive instability, depending on the particular parameter values of the structural model and of the policy rule, it creates something of a minefield for policy design. (2) One might, for instance, recommend a particular rule on the basis that it would generate a determinate rational expectations equilibrium and that the targeted equilibrium would have desirable properties based on other criteria, such as utility of the representative household in the model. And yet, in reality, important parameters may lie (because of sampling variability alone) in a region associated with indeterminacy of equilibrium, or with explosive instability. Actually implementing the proposed rule could then lead to disastrous consequences. Thus, from the perspective of the design of stabilization policy, one would greatly prefer to recommend policy rules such that, even if the structural parameters actually take on values somewhat different from those that might be estimated, a determinate rational expectations equilibrium is produced.
1.2 Learnability
Even when a determinate equilibrium exists, coordination on that equilibrium cannot be assured if agents do not possess rational expectations at every point in time. It therefore seems important to analyze these systems when agents must form expectations concerning economic events using the actual data produced by the economy. In general terms, the learning approach admits the possibility that expectations might not initially be fully rational, and that, if economic agents make forecast errors and try to correct them over time, the economy may or may not reach the REE asymptotically. Thus, beyond showing that a particular policy rule reliably induces a determinate REE, one needs to show the potential for agents to learn that equilibrium. (3) In this article, 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. We ask whether the agents in such a world can learn the equilibria of the system induced by different classes of monetary policy feedback rules. We use the criterion of expectational stability (aka. E-stability) to calculate whether rational expectations equilibria are stable under real time recursive learning dynamics or not. The research of Evans and Honkapohja (2001) and Marcet and Sargent (1989) 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.
1.3 The Benefits of Monetary Policy Inertia
We conclude that it is important to recommend to central banks those policy rules which have desirable determinacy and learnability properties, taking into consideration possible imprecision in our knowledge of structural parameters. Our main finding is that a wide variety of monetary policy rules are desirable in this sense provided the monetary authorities move cautiously in response to unfolding events. This is true from the point of view both of determinacy and of learnability of equilibrium. We model this caution, or inertia, on the part of the central bank by allowing the contemporaneous interest rate to respond to the lagged interest rate in the policy rule.
Inertia is a well-documented feature of central bank behavior in industrialized countries: policymakers show a clear tendency to smooth out changes in nominal interest rates in response to changes in economic conditions. Rudebusch (1995) has provided one statistical analysis of this fact. More casually, actual policy moves are discussed among central bankers and in the business press in industrialized countries as occurring as sequences of adjustments in nominal interest rates in the same direction. This is so much the case, in fact, that policy inertia has been the source of criticism of the efforts of central bankers, as suggestions are sometimes made that policymakers have been unwilling to move far enough or fast enough to respond effectively to incoming information about the economy.
Our study provides analytical support for the idea that monetary policy inertia enhances the prospects for equilibrium determinacy and learnability in the context of a standard, small, forward-looking model, which is currently the workhorse for the study of monetary policy rules. More specifically, we consider two variants of monetary policy feedback rules made famous by the seminal work of Taylor (1993, 1999a, 1999b). In one case, the central bank is viewed as adjusting a short-term nominal interest rate in response to deviations of past values of inflation and output from some target levels and, in order to capture interest rate smoothing, we also include a response to the deviation of the lagged interest rate from some target level. We call this the lagged data specification. Our second specification calls for the policymakers to react to forecasts of inflation deviations and the output gap, in addition to the lagged interest rate, and we call this the forward-looking specification. (4)
In previous studies it has been observed that there are important determinacy problems with both of these rules in the absence of inertia (see Bernanke and Woodford 1997, Rotemberg and Woodford 1999, Bullard and Mitra 2002, Woodford 2003). We find that by placing a sufficiently large weight on lagged interest rate deviations in each of these classes of policy rules, the policy authorities can mitigate the threats of indeterminacy or explosive instability, and that this is one of the primary benefits of monetary policy inertia. We also argue that policy inertia actually promotes learnability of rational expectations equilibrium. Our contribution is to provide analytical results to this effect and to highlight some of the intuition behind them.
1.4 Recent Related Literature
Our results suggest why other, non-inertial types of policies might leave the economy vulnerable to unexpected dynamics, and hence why central banks might willingly adopt inertial behavior. Recently, several very different theories have been proposed as to why policy inertia might be observed (e.g., Caplin and Leahy 1996, Sack 1998, Woodford 1999). Our results are probably best viewed as complementary to these theories.
Bullard and Mitra (2002) study the determinacy and learnability of simple monetary policy rules, that is, of policy rules that only respond to inflation and output deviations but not to lagged interest rate deviations, and so do not comment on the question of monetary policy inertia. Evans and Honkapohja (2003a) analyze learnability in a similar model and consider different ways of implementing optimal monetary policy under discretion, which leads to non-inertial rules. (5)
The finding that interest rate inertia is conducive to the existence of determinate REE has been noted by Rotemberg and Woodford (1999), Woodford (2003), Benhabib, Schmitt-Grohe, and Uribe (2003), and Carlstrom and Fuerst (2000). Our contribution with regard to determinacy is to elaborate in greater detail the reasons for the numerical findings in Rotemberg and Woodford (1999) and to show that the beneficial effects of inertia are true for a wider class of policy rules than considered in Woodford (2003). In addition, our results on determinacy are useful in understanding the effects of inertia on learning dynamics. Benhabib, Schmitt-Grohe, and Uribe (2003) find support for super-inertial interest rate policies in a somewhat different class of models where a supply-side channel of monetary policy transmission is emphasized. Carlstrom and Fuerst (2000) consider models where the timing of money balances entering the utility function and the nature of sticky price assumption along off-equilibrium paths is important. They find that inertial forward-looking policies are subject to indeterminacy problems whereas backward policies that react aggressively to past inflation can be associated with a determinate equilibrium independently of the degree of inertia.
1.5 Organization
In the next section we present the model analyzed throughout the article. We also discuss the types of linear policy feedback rules we will use to organize our analysis, and a calibrated case that we will occasionally employ. In the subsequent sections, we present conditions for determinacy of equilibrium for the lagged and forward-looking policy rules. We then turn to the question of learnability of REE under our various specifications. Section 5 considers the robustness of our results in Preston's (2005a) model. We conclude with a summary of our findings.
2. ENVIRONMENT
2.1 The Model
We study a model developed by Woodford (2003), which we write as
[x.sub.t] = [[??].sub.t][x.sub.t+1] - [sigma]([r.sub.t] - [r.sup.n.sub.t] - [[??].sub.t][[pi].sub.t+1]), (1)
[[pi].sub.t] = [kappa][x.sub.t] + [beta][[??].sub.t][[pi].sub.t+1], (2)
where [x.sub.t] is the output gap, [[pi].sub.t] is the period t inflation rate defined as the percentage change in the price level from t - 1 to t, and [r.sub.t] is the nominal interest rate; each of the two latter variables is expressed as a deviation from the long-run level. Since we will also analyze learning we use the notation [[??].sub.t][[pi].sub.t+1] and [[??].sub.t][x.sub.t+1] to denote the possibly non-rational private sector expectations of inflation and output gap next period, respectively, whereas the same notation without the hat symbol will denote rational expectations (RE) values. (6) The parameters [sigma], [kappa], and [beta] [member of] (0, 1) are structural and assumed positive on economic grounds--see Woodford (1999, 2003) for an interpretation of these constants. The "natural rate of interest" [r.sup.n.sub.t] is an exogenous stochastic term that follows the process
[r.sup.n.sub.t] = [rho][r.sup.n.sub.t-1] + [[epsilon].sub.t],
where [[epsilon].sub.t] is i.i.d, noise with variance [[sigma].sup.2.sub.[epsilon]], and [less than or equal to] [rho] < 1 is a serial correlation parameter.
2.2 Alternative Policy Rules
We close the system by supplementing equations (1) through (3), which represent the behavior of the private sector, with a policy rule for setting the nominal interest rate representing the behavior of the monetary authority. We stress that we view identification of classes of rules that reliably produce determinacy and learnability as a prior exercise to locating an optimal rule according to some objective function assigned to the central bank. Once we isolate the characteristics of rules that reliably produce both determinacy and learnability, then one could go about finding an optimal or best-performing rule from among the ones in this set.
Taylor (1993, 1999a) popularized the use of interest rate feedback rules that react to information on output and inflation. Our first specification considers a case in which interest rates are adjusted in response to the last quarter's observations on inflation and the output gap. This is our lagged data specification for our interest rate equation
[r.sub.t] = [[phi].sub.[pi]][[pi].sub.t-1] + [[phi].sub.x][x.sub.t-1] + [[phi].sub.r][r.sub.t-1]. (4)
This specification is considered operational by McCallum (1999) since it does not call for the central bank to react to contemporaneous data on output and inflation deviations.
Our second specification assumes that the authorities set their interest rate instrument in response to their forecasts of output gap and inflation, so that the policy rule itself is forward looking. Forward-looking rules have been found to describe well the actual behavior of monetary policymakers in countries like Germany, Japan, and the United States since 1979, as documented by Clarida, Gali, and Gertler (1998). We consider a simple version of this rule, namely, (7)
[r.sub.t]... |
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