Robust Monetary Policy Rules with Unknown Natural Rates

Robust Monetary Policy Rules with Unknown Natural Rates Athanasios Orphanides Board of Governors of the Federal Reserve System and (cid:3) John C. Williams Federal Reserve Bank of San Francisco December 2002 Abstract We examine theperformanceandrobustnesspropertiesof alternative monetarypolicyrules inthepresenceofstructuralchangethatrendersthenaturalrates ofinterestandunemployment uncertain. Using a forward-looking quarterly model of the U.S. economy, estimated over the 1969-2002 period, we show that the cost of underestimating the extent of misperceptions regarding the natural rates signi(cid:12)cantly exceeds the costs of overestimating such errors. Naive adoption of policy rules optimized under the false presumption that misperceptions regarding the natural rates are likely to be small proves particularly costly. Our resultssuggestthatasimpleande(cid:11)ective approachfordealingwithignoranceaboutthedegreeof uncertainty inestimates ofthenaturalrates is toadoptdi(cid:11)erencerulesformonetary policy, in which the short-term nominal interest rate is raised or lowered from its existing level in response to inflation and changes in economic activity. These rules do not require knowledge of thenaturalrates of interest or unemploymentfor setting policy andare consequently immune to the likely misperceptions in these concepts. To illustrate the di(cid:11)erences in outcomes that could be attributed to the alternative policies we also examine the role of misperceptions for the stagflationary experience of the 1970s and the disinflationary boom of the 1990s. Keywords: Inflation targeting, policy rules, natural rate of unemployment, natural rate of interest, misperceptions. JEL Classi(cid:12)cation System: E52 Correspondence: Orphanides: FederalReserveBoard,Washington,D.C.20551,Tel.: (202)452-2654, e-mail: Athanasios.Orphanides@frb.gov. Williams: Federal Reserve Bank of San Francisco, 101 Market Street, San Francisco, CA 94105,Tel.: (415) 974-2240,e-mail: John.C.Williams@sf.frb.org. (cid:3) Prepared for the September 2002 Brookings Panel on Economy Activity. We bene(cid:12)ted from presentationsofearlierdrafts atthe EuropeanCentralBank,the Deutsche Bundesbank,The Johns HopkinsUniversity,andtheUniversityofCalifornia,SantaCruz. Thisresearchprojecthasbene(cid:12)ted fromdiscussionswith Bill Brainard,Flint Brayton,RichardDennis, ThomasLaubach,Andy Levin, DavidLindsey,JonathanParker,MikePrell,GeorgePerry,DaveReifschneider,JohnRoberts,Glenn Rudebusch, Bob Tetlow, Bharat Trehan, Simon van Norden, Volker Wieland, and Janet Yellen. We thank Mark Watson, Bob Gordon, and Robert Shimer for kindly providing us with updated estimates. KirkMoore providedexcellentresearchassistance. Any remaining errorsareare the sole responsibilityoftheauthors. Theopinionsexpressedarethoseoftheauthorsanddonotnecessarily reflect views of the Board of Governors of the Federal Reserve System or the management of the Federal Reserve Bank of San Francisco.

\The natural rate is an abstraction; like faith, it is seen by its works. One can only say that if the bank policy succeeds in stabilizing prices, the bank rate must have been brought in line with the natural rate, but if it does not, it must not have been." (Williams, 1931, p. 578) 1 Introduction The conventional paradigm for the conduct of monetary policy calls for the monetary authority to attain its objectives of a low and stable rate of inflation and full employment by adjusting its short-term interest rate instrument|in the United States, the federal funds rate|in response to economic developments. In principle, when aggregate demand and employment fall short of the economy’s natural levels of output and employment, or when other deflationary concerns appear on the horizon, the central bank should ease monetary policy by bringing real interest rates below the natural rate of interest for some time. Conversely, the central bank should respond to inflationary concerns by adjusting interest rates upward so as to bring real interest rates above the natural rate. In this setting, the natural rateofunemploymentistheunemploymentrateconsistent withstableinflation; thenatural rate of interest is the real interest rate consistent with the unemployment rate being at its 1 natural rate and, therefore with stable inflation. In carrying out this strategy in practice, thepolicymaker wouldideallyhaveaccurate, quantitative, contemporaneousreadingsofthe natural rate of interest and the natural rate of unemployment. Under those circumstances, economic stabilization policy would be relatively straightforward. However, an important di(cid:14)culty that complicates policymaking in practice and may limit the scope for stabilization policy, however, is that policymakers do not know the values of these natural rates in real time, that is, when they make policy decisions. Indeed, even in hindsight there is considerable uncertainty regarding the natural rates of unemployment and interest, and ambiguity about how best to model and estimate natural rates. Milton Friedman, arguing against natural rate-based policies in his AEA presidential address, posited that \One problem is that [the policymaker] cannot know what the ‘natural’ rate is. Unfortunately, we have as yet devised no method to estimate accurately and read- 1This de(cid:12)nition leaves open the question of the length of the horizon over which one de(cid:12)nes stable inflation. Rotemberg and Woodford (1999), Woodford(2002), and Neiss and Nelson (2001), among others, consider de(cid:12)nitions of the natural rates whereby inflation is constant in every period while many other authors (cited later in thispaper) examine estimates of a lower frequency or \trend" naturalrates. 1

ily the natural rate of either interest or unemployment. And the ‘natural’ rate will itself change from time to time." (Friedman, 1968, p. 10). Friedman’s comments echo those made decades earlier by Williams (1931, quoted above) and Cassel (1928), who wrote of the natural rate of interest: \The bank cannot know at a certain moment what is the equilibrium rate of interest of the capital market." Even earlier, Wicksell (1898) who stressed that \the natural rate is not (cid:12)xed or unalterable in magnitude" (p. 106). Recent research using modern statistical techniques to estimate the natural rates of unemployment, output, and interest indicate that this problem is no less relevant today than it was 35, 75, or 105 years ago. These measurement problems appear to be particularly acute in the presence of structural change when natural rates may vary unpredictably, making estimates of the natural rates subject to increased uncertainty. Staiger, Stock, and Watson (1997a) document that estimates of a time-varying natural rate of unemployment are very imprecise (see also Staiger, Stock, and Watson 1997b and Laubach 2001). Orphanides and van Norden (2002) show that estimates of the related concept of the natural rate of (or potential) output are likewise plagued by imprecision (see also Lansing 2002). Similarly, Laubach and Williams (2002) document the great degree of uncertainty regarding estimates of the natural rate of interest. These di(cid:14)culties have led some observers to discount the usefulness of natural rate estimates for policymaking. Brainard and Perry (2000) conclude \that conventional estimates from a NAIRU [nonaccelerating-inflation rate of unemployment] model do not identify the full employment range with a degree of accuracy that is useful to policymaking." (p. 69). Staiger, Stock, and Watson suggest a reorientation of monetary policy away from reliance on the natural rate of unemployment, noting that a rule in which monetary policy responds not to the level of the unemployment rate but to recent changes in unemployment without reference to the NAIRU (and perhaps to a measure of the deviation of inflation from a target rate of inflation) is immune to the imprecision of measurement that is highlighted in this paper. An interesting question is the construction of formal policy rules that account for the imprecision of estimation of the NAIRU. (Staiger, Stock, and Watson, 1997a, p. 249) 2

This question, coupled with the related issue of mismeasurement of the natural rate of interest, is the focus of this paper. We employ a forward-looking quarterly model of the U.S. economy to examine the performance and robustness properties of simple interest rate policy rules in the presence of real-time mismeasurement of the natural rates of interest and unemployment. Our work builds on an active literature that has explored the implications of mismeasurement for monetary policy, including Orphanides (1998, 2001, 2002a), Smets (1998), Wieland (1998), Orphanides et al (2000), McCallum (2001), Rudebusch (2001, 2002), Ehrmann and Smets (2002), and Nelson and Nikolov (2002). A key aspect of our investigation is the recognition that policymakers may be uncertain as to the true data-generating processes describing the natural rates of unemployment and interest and the extent of the mismeasurement problem that they face. As a result, standard applications of certainty equivalence based on the 2 classical linear-quadratic-Gaussian control problem do not apply. To get a handle on this di(cid:14)culty, we compare the properties of policies optimized to provide good stabilization performance across a large range of alternative estimates of natural rate mismeasurement. We then examine the costs of basing policy decisions on rules that are optimized with incorrect baseline estimates of mismeasurement, that is, rules that attempt to properly account for the presence of uncertainty regarding the natural rates but inadvertently overestimate or underestimate the magnitude of the problem. These robustness exercises point to a potentially important asymmetry with regard to possible errors in the design of policy rules attempting to account for natural rate uncertainty. We(cid:12)ndthatthecostsofunderestimatingtheextentofnaturalratemismeasurement signi(cid:12)cantly exceeds the costs of overestimating it. Adoption of policy rules optimized under the false presumption that misperceptions regarding the natural rates are likely to be small proves particularly costly in terms of stabilizing inflation and unemployment. By comparison, the ine(cid:14)ciency associated with policies incorrectly based on the presumption that misperceptions regarding the natural rates are likely to be large tends to be relatively modest. As a result, when policymakers do not possess a precise estimate of the magnitude of misperceptions regarding the natural rates, a robust strategy is to act as if the 2See Swanson (2000) and Svensson and Woodford (2002) for recent expositions of certainty equivalence in the absence of any model uncertainty. Hansen and Sargent (2002) o(cid:11)er a modern treatment of robust control in thepresence of possible model misspeci(cid:12)cation. 3

uncertainty they face is greater than their baseline estimates suggest it may be. We show that overlooking these considerations can easily result in policies with considerably worse stabilization performance than anticipated. Our results point towards an e(cid:11)ective, simple strategy that is a robust solution to the di(cid:14)culties associated with natural rate misperceptions. This is to adopt, as guidelines for monetary policy, di(cid:11)erence rules in which the short-term nominal interest rate is raised or lowered from its existing level in response to inflation and changes in economic activity. These rules, which do not require knowledge of the natural rates of interest and unemployment and are consequently immune to likely misperceptions in these concepts, emerge as the solution to a robust control exercise from a wider family of policy rule speci(cid:12)cations. Although these rules are not \optimal" in the senseof delivering (cid:12)rst-beststabilization performance under the assumption that policymakers have precise knowledge of the form and magnitude of uncertainty they face, they are robust in that they e(cid:11)ectively ensure against major mistakes when such knowledge is not held with great con(cid:12)dence. Finally, ourresultssuggestthatsomeimportanthistoricaldi(cid:11)erencesinmonetarypolicy and macroeconomic outcomes over the past forty or so years can be traced to di(cid:11)erences to the formulation of monetary policy that closely relate to the treatment of the natural rates. As we illustrate, misperceptions regarding the natural rates, importantly due to the steady increase in the natural rate of unemployment, could have contributed to the stagflationary outcomes of the 1970s. Paradoxically, a policy that would be optimal at stabilizing inflation and unemployment if the natural rates of unemployment and interest were known can account for such dismal outcomes in a period when natural rates were rising. In contrast, our analysis suggests that had policy followed a robust rule that ignores information about the levels of natural rates during the 1970s, economic outcomes could have been considerably better. Conversely, outcomes during the disinflationary boom of the 1990s appear consistent with a policy closer to our robust rule. The natural rate of unemployment apparently drifted downward signi(cid:12)cantly during the second half of the decade, which might have resulted in deflation had policymakers pursued the policy that real-time assessments of the natural rates might have dictated. In the event, policymakers during the mid- and late 1990s avoided this pitfall. 4

2 Policy in the Presence of Uncertain Natural Rates As a starting point, we look at the nature of the problem in the context of a generalization of the simple policy rule proposed by John Taylor (1993) ten years ago. Let f t denote the federal funds rate, (cid:25) t the rate of inflation, and u t the rate of unemployment, all measured in quarter t. The Taylor rule can then be expressed by f t = r^ t (cid:3) +(cid:25) t+(cid:18) (cid:25)((cid:25) t −(cid:25)(cid:3) )+(cid:18) u(u t −u^ (cid:3) t ); (1) where(cid:25)(cid:3) is thepolicymaker’s inflation target andr^ (cid:3) andu^ (cid:3) are thepolicymaker’s estimates t t of the natural rates of interest and unemployment, respectively. Note that here we consider a variant of the Taylor rule that responds to the unemployment gap instead of the output 3 gap for our analysis, recognizing that the two are related by Okun’s (1962) law. As is well known, rules of this type have been found to perform quite well in terms of stabilizing economic fluctuations, at least when the natural rates of interest and unemployment are accurately measured. Inhis1993 exposition, Taylor examined responseparameters equalto 1/2forinflationgapandtheoutputgap,which,usinganOkun’scoe(cid:14)cientof2,corresponds to setting (cid:18) (cid:25) = 0:5 and (cid:18) u = −1:0. We also consider a revised version of this rule with double the responsiveness of policy to the output gap ((cid:18) u = −2:0 in our case), which Taylor (1999b) found to yield improved stabilization performance relative to his original rule. The promising properties of rules of this type were (cid:12)rst reported in the Brookings volume edited by Bryant, Hooper and Mann (1993) which o(cid:11)ered detailed comparisons of the stabilization performance of various interest rate-based policy rules in several macroeconometric models. The contributions in Taylor (1999a), as reviewed in Taylor (1999b), provided additional support for this (cid:12)nding. However, historical experience suggests that policy guidance from this family of rules may be rather sensitive to misperceptions regarding the natural rates of interest and unemployment. The experience of the 1970s, discussed in Orphanides (2000a, 2000b, 2002a), o(cid:11)ers a particularly stark illustration of policy errors that may result. We explore two dimensions along which the Taylor rule has been generalized that in combination o(cid:11)er the potential to mitigate the problem of natural rate mismeasurement. 3Inwhatfollows,weassumethatanOkun’slawcoe(cid:14)cientof2isappropriateformappingtheoutputgap to the unemployment gap. This is signi(cid:12)cantly lower that Okun’s original suggestion of about 3.3. Recent views, as reflected in thework by various authors place this coe(cid:14)cient in the2 to 3 range. 5

The(cid:12)rstaimstomitigatethee(cid:11)ectsofmismeasurementofthenaturalrateofunemployment by partially (or even fully) replacing the response to the unemployment gap with one to the change in the unemployment rate. This modi(cid:12)cation parallels that made by McCallum (2001), Orphanides (2000b), Orphanides et al. (2000), Leitemo and Lonning (2002), and others, who have argued in favor of policy rules that respond to the growth rate of output rather than the outputgap whenreal-time estimates of the naturalrate of outputare prone to measurement error. Although in general it is not a perfect substitute for responding to the unemployment gap directly, responding to the change in the unemployment rate is likely to be reasonably e(cid:11)ective because it calls for a easing of policy when unemployment 4 is rising andtightening when unemploymentis falling. Thesecond dimension we explore is incorporation of policyinertia, represented bythepresenceofthelagged short-term interest rate in the policy rule. As shown by Williams (1999), Levin et al. (1999, 2002), Rotemberg and Woodford (1999) and others, rules that exhibit a substantial degree of inertia can signi(cid:12)cantly improve the stabilization performance of the Taylor rule in forward-looking models. The presence of inertia in the policy rule also reduces the influence of the estimate of the natural rate of interest on the current setting of monetary policy and, therefore, the extent to which misperceptions regarding the natural rate of interest a(cid:11)ect policy decisions. To see this, consider the generalized Taylor rule of the form f t = (cid:18) f f t−1+(1−(cid:18) f)(r^ t (cid:3) +(cid:25) t)+(cid:18) (cid:25)((cid:25) t −(cid:25)(cid:3) )+(cid:18) u(u t −u^ (cid:3) t )+(cid:18) (cid:1)u(u t −u t−1): (2) The degree of policy inertia is measured by (cid:18) f (cid:21) 0; cases where 0 < (cid:18) f < 1 are frequently referred to as \partial adjustment"; the case of (cid:18) f = 1 is termed a \di(cid:11)erence rule" or \derivative control" (Phillips 1954), whereas (cid:18) f > 1 represents superinertial behavior (Rotemberg and Woodford 1999). These rules nest the Taylor rule as the special case when (cid:18) f = (cid:18) (cid:1)u = 0. 5 To illustrate more precisely the di(cid:14)culty associated with the presence of misperceptions regarding the natural rates of unemployment and interest it is useful to distinguish the real-time estimates of the natural rates, u^ (cid:3) and r^ (cid:3) , available to policymakers when policy t t 4Interestingly, as Woodford (1999) has shown, the optimal policy from a \timeless perspective" in the purelyforward-looking\newsynthesis"modelrespondstothechangeintheoutputgap,butnottoitslevel. 5Policy rulesthatallow foraresponsetothelagged instrumentandthechangeintheoutputgaporunemploymentrateasinequation(2)havebeenfoundtoo(cid:11)erasimplecharacterizationofhistoricalmonetary policy in the United States for the past few decades in earlier studies (Orphanides 2002b, Orphanides and Wieland 1998, McCallum and Nelson 1999, and Levin et al 1999, 2002). 6

decisions are made, from their \true" values u(cid:3) and r(cid:3) . If policy follows the generalized rule given by equation (2), then the \policy error" introduced in period t by misperceptions in period t is given by (1−(cid:18) f)(r^ t (cid:3)−r(cid:3) )+(cid:18) u(u^ (cid:3) t −u(cid:3) t ): Although unintentional, these errors could subsequently induce undesirable fluctuations in the economy, worsening stabilization performance. The extent to which misperceptions regardingthenaturalratestranslateintopolicyinducedfluctuationsdependsontheparameters of the policy rule. As is evident from the expression above, policies that are relatively unresponsivetoreal-timeassessments oftheunemploymentgap, thatis, thosewithsmall(cid:18) u minimize the impact of misperceptions regarding the natural rate of unemployment. Similarly, inertial policies with (cid:18) f near unityreducethe directe(cid:11)ect of misperceptions regarding the natural rate of interest. That said, inertial policies also carry forward the e(cid:11)ects of past misperceptions of the natural rates of interest and unemployment on policy, and one must takeaccountofthisinteraction indesigningpoliciesrobusttonaturalratemismeasurement. One policy rule that is immune to natural rate mismeasurement of the kind considered here is a \di(cid:11)erence" rule, in which (cid:18) f = 1 and (cid:18) u = 0: 6 f t = f t−1+(cid:18) (cid:25)((cid:25) t −(cid:25)(cid:3) )+(cid:18) (cid:1)u(u t −u t−1): (3) We note thatthis policy ruleis as simple, in terms of thenumberof parameters, as theoriginal formulation of the Taylor rule. In addition, this rule is certainly simpler to implement in practice than the Taylor rule, because it does not require knowledge of the natural rates of interest or unemployment. However, because this type of rule ignores potentially useful information about the natural rates of interest and unemployment, its performance relative to the Taylor rule and the generalized rule will depend on the degree of mismeasurement and the structure of the model economy, as we explore below. It is also useful to note that this rule is closely related to price-level and nominal income targeting rules, stated in (cid:12)rst-di(cid:11)erence form. 6This speci(cid:12)cation is similar to those examined by Judd and Motley (1992) and Fuhrer and Moore (1995b),inwhichthechangeintheshort-termraterespondstothegrowthofnominalincomeortoinflation, respectively. 7

3 Historical Estimates of Natural Rates Considerable evidence suggests that the natural rates of unemployment and interest vary signi(cid:12)cantlyover time. Inthecaseoftheunemploymentrate, anumberoffactors havebeen put forward as underlying time variation, including changing demographics, changes in the e(cid:14)ciency of job matching, changes in productivity, e(cid:11)ects of greater openness to trade, and the changing rates of disability and incarceration (Shimer 1998, Katz and Krueger 1999, Ball and Mankiw 2002). However, a great deal of uncertainty surrounds the magnitude and timing of these e(cid:11)ects on the natural rate of unemployment. Similarly, the natural rate of interest is likely influenced by variables that appear to change over time, including the rate of trend income growth, (cid:12)scal policy, and household preferences, as discussed in Laubach and Williams (2002). But the factors determining the natural rate of interest are not directly observed, and the quantitative relationship between them and the natural rate remains poorly understood. Even with the bene(cid:12)t of hindsight and \best practice" techniques, our knowledge about the natural rates remains cloudy, and this situation is unlikely to improve in the foreseeable future. Staiger, Stock, and Watson (1997a) highlight three types of uncertainty regarding natural rate estimates. For estimated models with deterministic natural rates, sampling uncertainty related to the imprecision of estimates of model parameters is one source of uncertainty. Sampling uncertainty alone yields 95 percent con(cid:12)dence intervals with widths between 2 and 4 percentage points for the natural rate of unemployment (Staiger, Stock, and Watson 1997a), and between 3 and 4 percentage points for the natural rate of interest (Rudebusch 2001, Laubach and Williams 2002). Allowing the natural rate to change unpredictably over time adds an another source of uncertainty; for example, the 95 percent con(cid:12)dence intervals for a stochastically time-varying natural rate of interest is over 7 percentage points, twice that associated with a constant natural rate. Finally, there is considerable uncertainty and disagreement about the most appropriate approach of modeling and estimating natural rates, and this model uncertainty implies that the con(cid:12)dence intervals based on any one particular model may understate the true degree of uncertainty that policymakers face. Importantly for the analysis in this paper, policymakers cannot be con(cid:12)dent that their natural rate estimates are e(cid:14)cient or consistent, but most realistically 8

must make due with imperfect modeling and estimating methods. Of course, in practice, policymakers are at an even greater disadvantage than theeconometrician who attempts to estimate natural rates retrospectively, because policymakers must act on \one-sided," or real-time natural rate estimates, which are based only on the data available at the time the decision is made. As documented below, such estimates typically are much noisier than the smooth retrospective, or \two-sided," estimates generally reported in the literature. For a given model, the di(cid:11)erence between the one-sided and two-sided estimates provides an estimate of natural rate misperceptions resulting from the real-time nature of the policymaker’s problem. To illustrate the extent of these measurement di(cid:14)culties, we provide comparisons of retrospective and real-time estimates of the natural rates of unemployment and interest. The various measures correspond to alternative implementations of two basic statistical methodologies that have been employed in the literature: univariate (cid:12)lters and multivariate unobserved-componentsmodels. Theunivariate(cid:12)ltersseparatethecyclical componentof a seriesfromitsseculartrendandusethelatterasaproxyofthenaturallevelofthedetrended series. Univariate (cid:12)lters possess the advantages that they impose very little structure on the problem and are relatively simple to implement. Because multivariate methods bring additional information to bear on the decomposition of trend and cycle, they can provide more accurate estimates of natural rates assuming that the underlying model is correctly speci(cid:12)ed. However, there is a great degree of uncertainty about model misspeci(cid:12)cation, especially regarding the proper modeling of low-frequency behavior, and as a result the theoretical bene(cid:12)ts from multivariate methods may be illusory in practice. We examine two versions each of two popular univariate (cid:12)lters, the Hodrick-Prescott (1997) (cid:12)lter (HP) and the Band-Pass (cid:12)lter (BP) described by Baxter and King (1999). For the HP (cid:12)lter, we consider two alternative implementations, one with the smoothness parameter (cid:21) = 1;600, the value most commonly used in analyzing quarterly data, and one with (cid:21) = 25;600 which smoothes the data more and is also closer to the approach advocated by Rotemberg (1999). Application of the BP (cid:12)lter requires a choice of the range of frequencies identi(cid:12)ed as associated with the business cycle, which are to be (cid:12)ltered from the underlying series. We examine two popular alternatives, an 8-year window favored by Baxter and King (1999) and Christiano and Fitzgerald (2002) and a 15-year window 9

employed byStaiger, Stock andWatson (2002) to estimate a \trend" for the unemployment rate. We apply these four univariate (cid:12)lters to obtain both one-sided (real time) and twosided (retrospective) estimates of the natural rates of unemployment and interest. We also obtain estimates of the natural rates based on two multivariate unobserved components models, and we o(cid:11)er comparisons with models similar to those proposed by other authors. These models suppose that the \true" processes for the natural rates of interest and unemployment can be reasonably modeled as random walks: u(cid:3) t =u(cid:3) t−1 +(cid:17) u;t (cid:17) u (cid:24) N(0;(cid:27) (cid:17) 2 u ); (4) r t (cid:3) = r t (cid:3) −1 +(cid:17) r;t (cid:17) r (cid:24) N(0;(cid:27) (cid:17) 2 r ): (5) Forthenaturalrateofunemployment,weimplementaKalman(cid:12)ltermodel,similartothose inStaiger, Stock,andWatson (1997a, 2002)andGordon(1998), toestimates atime-varying 7 NAIRU rate from an estimated Phillips curve. (In what follows, we treat the NAIRU and thenaturalrateofunemploymentassynonymous.) Wealsoexamineestimates followingthe proceduredetailed byBallandMankiw(2002). Theseauthorspositasimpleaccelerationist Phillips curve relating the annual change in inflation to the annual unemployment rate. They estimate the natural rate of unemployment be applying the HP (cid:12)lter to the residuals from this relationship. For the natural rate of interest, we apply the Kalman (cid:12)lter to an equation relating the unemployment gap and the real interest rate gap (the di(cid:11)erence between the real federal funds rate and the natural rate of interest). The basic speci(cid:12)cation and methodology are close to that used by Laubach and Williams (2002), but we assume that the natural rate of interest follows a random walk, whereas they allow for an explicit relationship between the natural rate and the estimated trend growth rate of GDP. Thebasic identifying assumption is that the unemployment gap converges to zero if the real rate gap is zero. Thus, stable inflation in this model is consistent with both the real interest rate and the unemployment 8 rate equaling their respective natural rates. 7Inthemeasurementequation,theinflationratedependsonlagsofinflationwiththeunitysumrestriction onthecoe(cid:14)cients,relativeoilandnon-oilimportpriceinflation,andtheunemploymentrategap. Weapply StockandWatson’s(1998)medianunbiasedestimatorforthesignal-to-noiseratioandestimatetheremaining parameters bymaximum likelihood overthesample period 1969:1-2002:2. 8In two papers, Bom(cid:12)m uses other approaches to estimate the natural rate of interest. Bom(cid:12)m (2001) usesyieldson indexedbondstoestimate investors’viewof thenaturalrateof interest;unfortunately,these 10

As notedabove, thesemultivariate approaches to estimating naturalrates aresubjectto speci(cid:12)cation error and therefore the resulting estimates may be ine(cid:14)cient or inconsistent. For example, the models used for estimating the natural rate of unemployment impose the accelerationist restriction that the sum of the coe(cid:14)cients on lagged inflation in the inflation equation equals unity. But as Sargent (1971) demonstrated, reduced-form characterizations of the Phillips curve consistent with the natural rate hypothesis do not necessarily imply this restriction andimposingitis invalid. Averydi(cid:11)erentview, whichlikewise comes to the conclusion that these models are misspeci(cid:12)ed, is provided by Modigliani and Papademos (1975), whoview the Phillips curve as a structural relationship but, instead of imposingthe natural rate hypothesis, propose the concept of a \noninflationary rate of unemployment, or NIRU" (p. 145) Following this approach, Brainard and Perry (2000) report estimates of the natural rate of unemployment when the assumption of constant parameters and the accelerationist restriction are relaxed. Retrospective estimates of the natural rate of unemployment exhibit variation over time and across methods at given points in time. Table 1 reports estimates for the natural rate using the methods described above as well as the most recent Congressional Budget O(cid:14)ce (2001, 2002) NAIRU estimates, the Kalman (cid:12)lter-based NAIRU estimates in Staiger, Stock, and Watson (2002) and Gordon (2002), and Shimer’s (1988) estimates based on demographic factors. All of these estimates are two-sided in the sense that they use data over the whole sample period to arrive at an estimate for the natural rate at any given past quarter. Figure 1 plots a representative set of these estimates over 1969-2002; for comparison, the average rate of unemployment over that period was nearly 6 percent. Theretrospectiveestimatesshareacommonpattern: generallytheyarerelativelylowat the end of the 1960s, rise during the late 1960s and 1970s, and trend downward thereafter, reaching levels in the late 1990s similar to those in the late 1960s. However, these estimates alsoexhibitsubstantialdispersionatmostpointsintime, indicating that, even inhindsight, precisely identifying the natural rate of unemployment is quite di(cid:14)cult. For example, the estimates for both 1970 and 1980 cover a 2-percentage point range. As stressed above, the estimates of the natural rate of unemployment that are relevant securities haveonly been inexistencefor arelatively short timesowe havescant timeseries evidenceusing this approach. In earlier work, Bom(cid:12)m (1997) estimated a time-varying natural rate of interest using the Federal ReserveBoards’s MPS model. 11

for setting policy are not those shown in Table 1 and Figure 1 butrather the one-sided estimates that incorporate only information available at the time. Figure 2 shows the one-side estimates for a range of the methods described above. In the case of the univariate (cid:12)lters, the reported series are constructed from the estimates of the trend at the last available observation at each point in time. In the case of the multivariate (cid:12)lters, the natural rate estimates are likewise based only on observed data, but the estimates of the model parameters are from data for the full sample. Given the relative imprecision of the estimates of many of the latter estimates, the true real-time estimates in which all model parameters are estimated using only data available at the time are likely to be considerably worse than the one-sided estimates reported here. A striking feature of univariate (cid:12)lter real-time estimates is how much more closely they track the actual data than do the smooth, retrospective estimates reported in Figure 1. This excess sensitivity of univariate (cid:12)lters to (cid:12)nal observations is a well known problem (see e.g. St. Amant and van Norden (1998), Christiano and Fitzgerald (2001), Orphanides and van Norden (2002), and van Norden (2002)). Evidently, these (cid:12)lters have di(cid:14)culty distinguishing between cyclical and secular fluctuations in the underlying series until the subsequent evolution of the data becomes known. This problem is less evident in the multivariate (cid:12)lters where the natural rate estimate is updated based on inflation surprises as opposed to movements in the unemployment rate itself. Figures 3 and 4 plot a set of two-sided and one-sided estimates, respectively, of the natural rate of interest. Throughout this paper, the real interest rate is constructed as the di(cid:11)erence between the federal funds rate and ex post rate of inflation (based on the GDP price index). Each (cid:12)gure shows two multivariate estimates (our Kalman (cid:12)lter estimate 9 described above as well as that from Laubach and Williams (2002) ) and estimates from the same univariate (cid:12)lters used for the natural rate of unemployment. As in the case of the natural rate of unemployment, the various techniques yield a broad range of possible retrospective and real-time estimates of the natural rate of interest over time. Given the wide dispersion in these natural rate estimates, especially the more policyrelevant one-sided estimates, a natural question is whether one can discriminate between 9Laubach and Williams (2002) construct the real interest rate using the inflation rate of personal consumption expenditure prices; we have adjusted their natural rate estimates to place them on the basis of GDP price inflation. 12

the methods based on their empirical usefulness in predicting inflation and unemployment. To test the forecasting performance of methods using the natural rate of unemployment, we compare inflation forecast errors using a simple Phillips curve model in which inflation dependson four lags of inflation, the lagged change in the unemployment rate, and two lags of the unemployment gap based on the various one-sided estimates of the natural rate of unemployment. We also consider the performance of a simple fourth-order autoregressive, or AR(4), inflation forecasting equation without any unemployment rate terms. For this exercise, we use the revised data current as of this writing. As seen in the upper part of Table2, theequations thatincludetheunemploymentgap outperform(thatis, havealower forecast standard error than) the AR(4) speci(cid:12)cation, but inflation forecasting accuracy is 10 virtually identical across the speci(cid:12)cations that include the unemployment gaps. To test the forecasting performance of methods using the natural rate of interest, we apply the same basic procedure to a simple unemployment equation, where the unemployment rate depends on two lags of itself and the lagged real rate gap. This yields the parallel result, shown in the lower panel of the table. Evidently, one cannot easily discriminate across speci(cid:12)cations of the natural rates based on forecasting performance. We now use the di(cid:11)erent natural rate estimates presented above to gauge the likely magnitude and persistence of natural rate misperceptions. We start by computing natural rate misperceptions solely due to the limitation that only observed data can be used in real time, assuming that the correct model for the natural rate is known. Given the problems of samplingandmodeluncertainty, weviewtheseestimates aslowerboundsonthetruedegree of uncertainty of natural rate estimates. The (cid:12)rst column of the upper portion of Table 3 reportsthesamplestandarddeviationsofthedi(cid:11)erencebetweenthetwo-sidedandone-sided estimates of thenaturalrateof unemployment(u(cid:3)−u^ (cid:3) )for thevariousestimation methods. This standard deviation ranges from about 0.5 to 0.8 percentage point, with the Kalman (cid:12)lter lying in the center at 0.66 percentage point. The lower parnel of the table reports the corresponding results for estimates of the natural rate of interest. The standard deviations inthiscaserangefrom0.9to1.7percentagepoint,withtheKalman(cid:12)lterat1.44percentage point. In our subsequent analysis, we use the estimates from our multivariate Kalman (cid:12)lter 10However,thesuggestedforecastimprovementfromincludingtheunemploymentgapisbasedonwithinsample performance. The usefulness of unemployment or output gap estimates for out-of-sample forecasts of inflation is muchless clear (Stock and Watson, 1999; Orphanidesand van Norden, 2001.) 13

method as a baseline measure of the uncertainty regarding real-time perceptions of the natural rates of interest and unemployment in the historical data. Natural rate misperceptions are highly persistent. The persistence of these series can be characterized with the (cid:12)rst order autoregressive processes: (u(cid:3) t −u^ (cid:3) t ) = (cid:26) u(u(cid:3) t−1 −u^ (cid:3) t−1 )+(cid:23) u;t ; (6) (r t (cid:3)−r^ t (cid:3) )= (cid:26) r(r t (cid:3) −1 −r^ t (cid:3) −1 )+(cid:23) r;t ; (7) where the errors (cid:23) u;t and (cid:23) r;t are independent over time but may be correlated with each otherandwithothershocksrealized duringperiodt,including,importantly, theunobserved errors of the underlying processes for the natural rates, (cid:17) u;t and (cid:17) r;t. Table 3 also presents least squares estimates of (cid:26) and (cid:27) (cid:23) for the various misperceptions measures. In all cases, misperceptions arehighlypersistent, with theKalman (cid:12)lter lyingin themiddleof the range on this dimension also. Note that the persistence in misperceptions does not necessarily imply any sort of ine(cid:14)ciency in the real-time estimates, but merely reflects the nature of these (cid:12)ltering problems. We now extend our analysis of the mismeasurement problem to include model uncertainty. For this purpose we compare the one-sided estimate using each method to each of the two-sided estimates. For our set of six methods, this yields thirty-six measures of misperceptions for the natural rates of unemployment and interest. Table 4 summarizes the frequency distribution of the standard deviations and persistence from these alternative estimates of misperceptions. Both the standard deviations and the persistence measure of our baseline (Kalman (cid:12)lter) estimates for both natural rates, from Table 3, are close to the 25th percentile as shown in Table 4. Table 4 indicates generally larger and much more persistent misperceptions than those based on comparing the one- and two-sided estimates from a single model; indeed, the magnitude of misperceptions can be as much as twice that implied by the Kalman (cid:12)lter model. Moreover, these calculations do not reflect sampling uncertainty. In summary, combining the three forms of natural rate uncertainty suggests that conventional estimates of misperceptions based on comparing one-sided and two-sided estimates using a single estimation method are overly optimistic about the magnitude and persistence of the problem faced by policymakers. 14

4 A Simple Estimated Model of the U.S. Economy We evaluate monetary policy rules using a simple rational expectations model, the core of which consists of the following two equations: (cid:25) t = (cid:30) (cid:25) (cid:25) t e +1 +(1−(cid:30) (cid:25))(cid:25) t−1+(cid:11) (cid:25) u~ e t +e (cid:25);t ; e (cid:25) (cid:24) iid(0;(cid:27) e 2 (cid:25) ); (8) u~t = (cid:30) u u~ e t+1 +(cid:31) 1 u~t−1+(cid:31) 2 u~t−2+(cid:11) u r~ t a −1 +e u;t ; e u (cid:24) iid(0;(cid:27) e 2 u ): (9) Here we use u~ to denote the unemployment gap and r~ a to denote the real interest rate gap based on a one-year bill. This model combines forward-looking elements of the New Synthesis model studied by Goodfriend and King (1997), Rotemberg and Woodford (1999), Clarida, Gali and Gertler (1999), and McCallum and Nelson (1999), with intrinsic inflation and unemployment inertia as in Fuhrer and Moore (1995a), Batini and Haldane (1999), and Smets (2000). Given, the uncertainty regarding the proper speci(cid:12)cation of inflation and unemployment dynamics, later in the paper we also consider alternative speci(cid:12)cations, including one with no intrinsic inflation and one with adaptive expectations. The \Phillips curve" in this model (equation 8) relates inflation (measured as the annualized percentchangein theGDP priceindex)duringquartert tolagged inflation, expected future inflation, and expectations of the unemployment gap during the quarter, using retrospective estimates of the natural rate discussed below. The estimated parameter (cid:30) (cid:25) measures the importance of expected inflation on the determination of inflation. The unemployment equation (equation 9) relates the unemployment gap during quarter t to the expected future unemployment gap, two lags of the unemployment gap, and the lagged real interest rate gap. Here two elements importantly reflect forward-looking behavior. The (cid:12)rst element is the estimated parameter (cid:30) u, which measures the importance of expected unemployment, and the second is the duration of the real interest rate, which serves as a summary of the influence of interest rates of various maturities on economic activity. Because data on long-run inflation expectations are lacking, we limit the duration of the real rate to one year. In estimating this model we are confronted with the di(cid:14)culty that expected inflation and unemployment are not directly observed. Instrumental variable and full-information maximum likelihood methods impose the restriction that the behavior of monetary policy and the formation of expectations be constant over time, neither of which appears tenable 15

over the sample period that we consider (1969{2002). Instead, we follow the approach of Roberts (1997, 2001) and Rudebusch (2002) and use the median values of the Survey of Professional Forecasters as proxies for expectations. We use the forecast from the previous quarter; that is, we assume expectations are based on information available at time t−1. To match the inflation and unemployment data as best as possible with the forecasts, we 11 use (cid:12)rst announced estimates of these series. Our primary sources for these data are the Real-Time Dataset for Macroeconomists and the Survey of Professional Forecasters, both currently maintained by the Federal Reserve Bank of Philadelphia (Zarnowitz and Braun (1993), Croushore(1993) andCroushoreandStark(2001)). Usingtheleastsquaresmethod, we obtain the following estimates over the sample 1969:1 to 2002:2 (this choice of sample reflects the availability of the Survey of Professional Forecasters data): (cid:25) t = 0:540 (cid:25) t e +1 +0:460 (cid:25) t−1 −0:341 u~ e t +e (cid:25);t ; (10) (0:086) (−−) (0:099) SER =1:38, DW = 2:09, u~t = 0:257 u~ e t+1 +1:170 u~t−1 −0:459 u~t−2+ +0:043 r~ t a −1 +e u;t ; (11) (0:084) (0:107) (0:071) (0:013) SER =0:30, DW = 2:08; In these results the numbers in parentheses are the estimated standard errors of the corresponding regression coe(cid:14)cients. The estimated unemployment equation also includes a constant term (not reported) that captures the average premium of the one-year Treasury bill rate we use for estimation over the average of the federal funds rate, which corresponds to the natural rate of interest estimates we employ in the model. In the model simulations we impose the expectations theory of the term structure whereby the one-year rate equals the expected average of the federal funds rate over four quarters. In addition to the equations for inflation and the unemployment rate, we need to model the processes that generate both the true values for the natural rate of unemployment and interest and policymakers’ real-time estimates of these rates. For this purpose we use our Kalman (cid:12)lter estimates as a baseline for the speci(cid:12)cation of the natural rate processes. Throughouttheremainderofthepaper,weassumethatthetruevaluesforthenaturalrates 11RomerandRomer(2000)followasimilarprocedurewhencomparingFederalReserveBoardGreenbook forecasts tothe data. 16

are given by the two-sided retrospective Kalman (cid:12)lter estimates. Speci(cid:12)cally, we append the basic macroeconomic model to include equations (4) and (5) for u(cid:3) and r(cid:3) , respectively, and compute the equation residuals|the \shocks" to the true natural rates|using the two-sided Kalman (cid:12)lter estimates. For the policymakers’ estimates of natural rates, we assume the di(cid:11)erence between the true and estimated values follows an AR(1) process described by equations (6) and (7), with the AR(1) set equal to that based on the regression using the di(cid:11)erence between the one- and two-sided Kalman (cid:12)lter estimates reported in Table 3. As seen in that table, this speci(cid:12)cation approximates several common (cid:12)ltering methods. The residuals from these equations represent the shocks to mismeasurement under the assumption that the policymaker possesses the correctly speci(cid:12)ed Kalman (cid:12)lter models. Because we are interested in the possibility that the policymakers’ natural rate estimates result from a misspeci(cid:12)ed model, we allow for a range of estimates of the magnitude of natural rate mismeasurement, indexed by s, in our policy experiments. The case of s = 0 corresponds to the \best case" benchmark (a standard assumption in the policy rule literature), in which the policymaker is assumed to observe the true value of both natural rates in real time. For this case, we set the residuals of the two mismeasurement equations to zero. The case of s = 1 corresponds to the assumption that the policymaker possesses the correctly-speci(cid:12)ed Kalman (cid:12)lter models (including knowledge of all model parameters). In this case, the residuals from the mismeasurement equation are set to their historical values. As discussed above, owing to the possibility of modelmisspeci(cid:12)cation, this calculation most likely yields a conservative (cid:12)gure for the magnitude of real-world natural rate misperceptions. To approximate the policymakers’ use of a misspeci(cid:12)ed model of natural rates, we examine simulations where we amplify the magnitude of misperceptions by multiplying the residuals to the mismeasurement equations by s. As indicated by the results in Table 4, incorporating model misspeci(cid:12)cation can yield di(cid:11)erences between one- and two-sided on average twice as large as those implied by comparing the one- and two-sided Kalman (cid:12)lter estimates, implyinga valueof s of upto 2. 12 In addition, these calculations ignore sampling 12Forexample,s=2approximatelycorrespondstothecaseofapolicymakerwhomayincorrectlyrelyon the HP (cid:12)lter (with (cid:21) = 1600) for real-time estimates of the natural rates when the true process continues to be described by our two-sided Kalman (cid:12)lter. In terms of the policy evaluations we report later on, we con(cid:12)rmed that using s = 2 with the Kalman (cid:12)lter errors are also very similar to those based on these mispeci(cid:12)ed errors. This suggests that our approach of summarizing the magnitude of misperceptions by 17

uncertainty associated with estimated models; inconsideration of thissourceof uncertainty, we also consider the case of s = 3. For a given value of s, we estimate the variance-covariance of the six model equation innovations (correspondingtoequations 4{7, 10, and11)usingthehistorical equation residuals, where the misperception residuals are multiplied by s, as described above. Note that, byestimatingthevariance-covariancematrixinthisway,wepreservethecorrelationsamong shocks to inflation, the unemployment rate, changes in the natural rates, and natural rate misperceptions present in the data. For example, shocks to misperceptions of r(cid:3) are positively correlated with shocks to the unemployment rate and to u(cid:3) misperceptions, and shocks to u(cid:3) misperceptions are negatively correlated with shocks to inflation. For a given monetary policy rule of the form of equation (1), we solve for the unique stable rational expectations solution, if one exists, using the Anderson and Moore (1985) 13 implementation oftheBlanchardandKahn(1980) method. Given themodelsolution and the variance-covariance matrix of equation innovations, we then numerically compute the unconditional moments of the model. This method of computing unconditional moments is equivalent to, but computationally more e(cid:14)cient than, computing them from stochastic simulations of extremely long length (see Levin, Wieland, and Williams 1999 for a detailed discussion). 5 Policy Rule Evaluation We now examinehow uncertainty regardingthenaturalrates ofinterest and unemployment influences the design and performance of policy rules. We assume that the policymaker is interested inminimizingtheloss, L,equaltotheweighted sumoftheunconditionalsquared deviations of inflation from its target, those of the the unemployment rate from its true natural rate, and the change in the short-term interest rate: L = !Var((cid:25)−(cid:25)(cid:3) )+(1−!)Var(u−u(cid:3) )+ Var((cid:1)f): (12) a single parameter, s, captures the key implications of policymakers’ misspeci(cid:12)cation of the natural rate process. 13We abstract from the complications arising from imperfections in the formation of expectations (OrphanidesandWilliams, 2002). Forsimplicity,wealsoabstractfrom errorsinwithin-quarterobservationsof therates of inflation and unemployment. 18

As a benchmark for our analysis and for comparability with earlier policy evaluation work, we consider preferences equivalent to placing equal weights on the variability of inflation and the output gap. Assuming an Okun’s law coe(cid:14)cient of 2, this weighting implies setting ! = 0:2. We include a relatively modest concern for interest rate stability, setting = 0:05 Later in the paper, we show that the main qualitative results are not sensitive to changes in ! and . In all our experiments, we assume the policymaker has a (cid:12)xed and known inflation target, (cid:25)(cid:3) . 14 We start our analysis of the e(cid:11)ects of natural rate mismeasurement by examining macroeconomic performanceundertheclassicandrevisedformsoftheoriginal Taylor rules: f t = r^ t (cid:3) +(cid:25) t+0:5((cid:25) t −(cid:25)(cid:3) )−1:0(u t −u^ (cid:3) t ) (the classic rule); f t = r^ t (cid:3) +(cid:25) t+0:5((cid:25) t −(cid:25)(cid:3) )−2:0(u t −u^ (cid:3) t ) (the revised rule): The direct e(cid:11)ects of natural rate mismeasurement on the setting of policy are transparent under these rules: a 1-percentage-point error in r(cid:3) translates into a one percentage point error in the interest rate, while a 1-percentage-point error in u(cid:3) translates into a {1-percentage-point error in the classic Taylor rule and a {2-percentage-point error for the revised rule. The (cid:12)rst panel of Table 5 reports the standard deviations of the inflation rate, the unemployment rate gap, and the change in the federal funds rate, as well as the associated loss under the classic Taylor rule in our model, for values of s between 0 and 3. The next panel does the same for the revised Taylor rule. Figure 5 illustrates some of these results graphically, tracing out the unconditional standard deviations of inflation (top panel) and the unemployment gap (bottom panel) for our model economy when policy is based on the classic Taylor rule or the revised Taylor rule for di(cid:11)erent values of s. Starting with the case of no misperceptions, s = 0, we see that both the classic and revised Taylor rules are e(cid:11)ective at stabilizing inflation and the unemployment rate gap. The revised variant of the rule is more responsive to the perceived degree of slack in labor markets and thereby achieves lower variability of both inflation and the unemployment gap, at the cost of modestly higher variability of the change in the interest rate. This result is consistent with the (cid:12)ndings reported in the studies collected in Taylor (1999a) and 14Weassumethattheinflationtargetissu(cid:14)cientlyabovezerotominimizeissuesrelatedtothezerobound on interest rates and other nonlinearities associated with very low inflation or deflation (Akerlof, Dickens and Perry, 1996; Orphanidesand Wieland, 1998; Reifschneider and Williams, 2000). 19

elsewhere. However, policy outcomes for both rules deteriorate markedly and increasingly soas thedegreeofmisperceptionsregardingthenaturalrates increases. For example, under the classic Taylor rule, thestandard deviation of inflation is 2.14 when s is assumed to be0, but increases to 3.67 under the assumption that s = 1. In addition, and of greater interest from a policy design perspective, Figure 5 illustrates that the performance deterioration owing to natural rate uncertainty is worse for the revised Taylor rule, because it places greater emphasis on the unemployment gap. Indeed, for even modest levels of natural rate misperceptions, the classic Taylor rule performs better than the revised version, a result consistent with (cid:12)ndings based on output gap mismeasurement in Orphanides (2000b). We now examine the e(cid:14)cient choices for the two parameters, (cid:18) (cid:25) and (cid:18) u, that measure the responses to the inflation and unemployment gaps, respectively, in a policy rule of the same functional form as the Taylor rule with natural rate uncertainty. In this exercise, we assume that the policymaker is interested in identifying a simple (cid:12)xed policy rule that can provide guidance for minimizing the weighted variances in the loss function (12) with the weights described above. Figure 6 presents the optimal choices of the two parameters for various values of s. As the left-hand panel shows, the optimal responsiveness to inflation increases with uncertainty in this case. From theright-hand panelit is also evident that the optimal response to the unemployment gap drops (in absolute value) and approaches zero as the degree of mismeasurement increases to values of s beyond 2. This (cid:12)nding con(cid:12)rms the parallel result, reported by Orphanides (1998), Smets (1998), Rudebusch (2001, 2002), McCallum (2001), and Ehrmann and Smets (2002), of attenuated responses to the output gap as an e(cid:14)cient response to uncertainty regarding the measurement of the output gap in level rules. Thisattenuation resultcontrasts with standardapplications of the principleof certainty equivalence whereby, under certain conditions, the policymaker could compute the optimal policy abstracting from uncertainty and apply the resulting optimal rule by substituting into it, for the unobserved values, estimates of the natural rates based on an optimal (cid:12)lter (Swanson (2000) and Svensson and Woodford (2002) o(cid:11)er recent expositions on this issue.) Rather, our result is similar to Brainard’s (1967) conservatism principle, where attenuation is shown to be optimal when policy e(cid:11)ectiveness is uncertain. Two key conditions that are necessary for the standard application of certainty equiv- 20

alence are violated in our analysis. First, we focus on \simple" policy rules that respond to only a subset of the relevant state variables of the system, and certainty equivalence only applies to fully optimal rules. The distinction is especially important in the presence of concern about model misspeci(cid:12)cation. As discussed by Levin, Wieland, and Williams (1999, 2002), simple rules appear to be more robust to general forms of model uncertainty than rules optimized to a speci(cid:12)c model, arguing that in the broader context of the types of uncertainty that policymakers face, an exclusive focus on fully optimal rules may be misguided. Second, and especially relevant for our analysis, the traditional applications of certaintyequivalencerelyontheexistenceofamodelthatispresumedtobetrueandknown with certainty, and which policymakers can apply to obtain \optimally" (cid:12)ltered estimates of the natural rates. In light of the uncertainty about how to best model and estimate the 15 natural rate processes discussed earlier, we (cid:12)nd this assumption untenable. We now assess the implications of ignorance regarding the precise degree of uncertainty policymakers may face about the natural rates. We start by examining the costs of basing policy decisions on rules that are optimized with incorrect baseline estimates of this uncertainty. We examine the performance of rules optimized for natural rate mismeasurement of degree s = 0 and s = 1 when the true extent of mismeasurement may be di(cid:11)erent. The economic outcomes associated with this experiment are shown in Figure 7 and the third panel of Table 5, for true values of s ranging from 0 to 3. As seen in the (cid:12)gure, the rule optimized on the assumption of no misperceptions performs poorly even at the baseline value of s = 1, whereas the rule optimized assuming s = 1 is much more robust to natural rate mismeasurement. 15Togain some insight intothebreakdown ofthetraditional certainty equivalenceresultsin thepresence of(cid:12)lteruncertainty,considerthesimplestaticproblemofminimizingtheexpectedsquaredvalueofvariable y = x−c, where x is a random variable and c is the policy control. If x is observed, then the solution is trivial: set c=x. Suppose, instead, thatx is not directly observable but instead must beinferred from the variable z = (cid:24)x+(cid:17). Let x and (cid:17) be zero mean independently and normally distributed random variables with constant and known variances (cid:27) x 2 and (cid:27) (cid:17) 2 = (cid:27)(cid:22)(cid:17) 2, respectively, and without loss of generality let (cid:24) = 1. Then,ifalltheseparametersareknown,certaintyequivalenceappliesandtheoptimalcontrolisc=x^=(cid:20)z, where(cid:20)= (cid:27)x 2 istheoptimal(cid:12)lterappliedtoz. Next,toillustrate(cid:12)lteruncertainty,supposethatinstead (cid:27)x 2+(cid:27)(cid:22)(cid:17) 2 ofbeing(cid:12)xedandknown,(cid:27) (cid:17) and(cid:24)areindependentlydrawnwithequalprobabilitiesfromf(cid:27)(cid:22)(cid:17) −s (cid:17) ;(cid:27)(cid:22)(cid:17)+s (cid:17) g, and f1−s (cid:24) ;1+s (cid:24) g, respectively. In this case, if we consider the optimal linear policy c=(cid:18)z, the optimal choice of (cid:18) is given by: (cid:27)2 (cid:18)= x : (1+s2 (cid:24) )(cid:27) x 2+((cid:27)(cid:22)(cid:17) 2+s2 (cid:17)) Notethat(cid:18)=(cid:20)fors (cid:24) =s (cid:17) =0butisstrictlydecreasinginboths (cid:24) ands (cid:17). Thus,theoptimallinearpolicy attenuates theresponse relative to that implied assuming certain and known(cid:27) (cid:17) and (cid:24). 21

These experiments point to an asymmetry in the costs associated with natural rate mismeasurement: the cost of underestimating the extent of misperceptions signi(cid:12)cantly exceeding the cost of overestimating it. Policy rules optimized under the false presumption that misperceptions regarding the natural rates are likely to be small are characterized by large responsestotheunemploymentgap. Thiscan prove extremely costly. Bycomparison, policies incorrectly based on the presumption that misperceptions regarding natural rates are likely to be large are more timid in their response to the unemployment gap, but this is associated with relatively little ine(cid:14)ciency. In the case where there are in fact no misperceptions, the policy optimized under the assumption of s = 1 delivers modestly worseresultsthanthepolicyoptimizedundertheassumptionofnomisperceptions;however, in the presence of even a modest degree of misperception, the performance of the policy designed on the assumption of no misperceptions deteriorates dramatically as the degree of mismeasurement increases. Giventhepotentialdi(cid:14)cultiesassociatedwiththeoptimizedTaylorrulesinthepresence of natural rate mismeasurement, it is of interest to compare the performance of these rules to our alternative family of \robust" di(cid:11)erence rules of the form given by equation (3). In the present context, this class of rules is robust to natural rate mismeasurement because natural rate estimates do not enter into the implied policy setting decision. The (cid:12)nal row of Table 5 presents the e(cid:14)cient choice of the parameters (cid:18) (cid:25) and (cid:18) (cid:1)u corresponding to this robustrulechosentominimizethesamelossastheoptimizedTaylorrules. Thestabilization performanceof this ruleis also shown in Figure 7. In this model this ruleperformsabout as well as the Taylor rules (1) when the natural rates are assumed known, and, consequently, dominates these rules in the presence of uncertainty, since with greater uncertainty about misperceptionsregardingthenaturalrates,theperformanceoftheTaylor rulesdeteriorates, whereas the performance of the robust rule remains unchanged. The key reason that the robust di(cid:11)erence rule performs so well relative to the Taylor-type rules even absent natural rate uncertainty is that it naturally incorporates a great deal of policy inertia. As noted above, thisisanimportantingredientofsuccessfulpoliciesinforward-lookingmacromodels when policymakers are concerned about interest rate variability. Given these results, we now consider a more flexible form of policy rule that combines leveland(cid:12)rst-di(cid:11)erencefeatures. Figure8presentstheoptimizedparameterscorresponding 22

tothegeneralizedpolicyrulesgiveninequation(2)fordi(cid:11)erentvaluesofs,whichisassumed to be known by the policymaker. If the natural rates of interest and unemployment are assumedto beknown, thenthe e(cid:14)cient policy ruleexhibits partial adjustmentanda strong responseto theunemployment gap, along with a responseto inflation and thechange in the unemployment rate. We now examine how the optimal policy responses are altered when the degree of mismeasurement is increased and this is known by the policymaker. First, the response to the unemployment gap diminishes sharply and approaches zero as the degree of uncertaintyincreases. Second,compensatingforthereducedresponsetotheunemployment gap,inthefaceofincreaseduncertaintythee(cid:14)cientrulescallforlargerresponsestochanges in the rate of unemployment. Third, the degree of inertia in the e(cid:14)cient rules increases as the degree of uncertainty rises, approaching the limiting value (cid:18) f = 1. In the limit as the degree of uncertainty increases, the generalized rule collapses to the robust di(cid:11)erence rule. The performance of optimized generalized rules is shown in Figure 9, which repeats the experiments reported in Figure 7 but using optimized generalized policy rules. As in the case of Taylor rules, the performance of the generalized rule optimized assuming no natural rate misperceptions deteriorates dramatically if natural rates are in fact mismeasured. In contrast, the rule optimized assuming s = 1 is quite robust to natural rate mismeasurement. As noted, this rule features very modest responses to estimates of r(cid:3) and u(cid:3) . The performance of the robust di(cid:11)erence rule is invariant to the degree of mismeasurement and exceeds that of the generalized rule optimized assuming s = 1 for all values of s > 1:5. The asymmetry in outcomes due to incorrect assessments, shown in Figure 9, suggests that, when policymakers do not possess a precise estimate of the magnitude of misperceptions regarding the natural rates, it may be advisable to act as if the uncertainty they face is greater than their baseline estimates. We examine this issue in greater detail with an example shown in Figure 10. To facilitate comparisons, the (cid:12)gure plots pairs of the policy responses, (cid:18) u and (cid:18) f, corresponding to di(cid:11)erent values of a known degree of uncertainty (from Figure 8). Note in particular the location of the e(cid:14)cient policies corresponding to s = 0, 1, and 2 and the limiting case of di(cid:11)erence rules (\Robust policy" in the (cid:12)gure). Consider the following problem of Bayesian uncertainty regarding s. Suppose that the policymaker has a di(cid:11)use prior with support [0,2] regarding the likely value of s. By construction, the baseline estimate of uncertainty is thus s= 1. As the (cid:12)gure shows, however, 23

the e(cid:14)cient choice based on the optimization with the di(cid:11)use prior over s, corresponds to a choice of (cid:18) u and (cid:18) f that is closer to the certain e(cid:14)cient choice with value s = 2, the worse outcome for this distribution. In this sense a policymaker with a Bayesian prior over the likely degree of uncertainty he may face about the natural rates should act as if he were con(cid:12)dent that the degree of uncertainty he faces is greater than his baseline estimates. Of course, complete ignorance regarding the distribution of s leads to the robust control solution, which here corresponds to the limiting case of the robust di(cid:11)erence rule given by equation (3). The precise parameterization of the robust di(cid:11)erence rule for our model depends on the loss function parameters, ! and . As noted earlier, in our analysis thus far we set ! = :2, and = 0:05 which can be interpreted as a \balanced" preference for output and inflationstabilitybutexhibitsrelativelylowconcernforinterestvariability. Forcomparison, in Table 6, we present alternative robust rules corresponding to di(cid:11)erent values of the loss function parameters: 0.1, 0.2, and 0.5 for ! and 0.05, 0.5 and 5.0 for . Given , higher values for ! correspond to a larger inflation response coe(cid:14)cient, (cid:18) (cid:25), with a relatively small e(cid:11)ect on (cid:18) (cid:1)u. Given !, a greater concern for interest rate smoothing reduces both response coe(cid:14)cients, (cid:18) (cid:25) and (cid:18) (cid:1)u. This leads to a noticeable reduction in the standard deviation of interest rate changes, but at the cost of higher variability in both inflation and the unemployment gap. 6 Robustness in Alternative Models Thus far our analysis has been conditioned on the assumption that the baseline model we estimated in section 4 o(cid:11)ers a reasonable characterization of the workings of the economy in our sample, including, importantly, the role of expectations. This assumption may be critical for interpreting our policy evaluation analysis and (cid:12)ndingthat the simple di(cid:11)erence policy rule we identify o(cid:11)ers a useful and robust benchmark for policy analysis. Given that researchers and policymakers may hold di(cid:11)erent views about the most appropriate model for characterizing the role of expectations, and given the uncertainty associated with any estimated model, it is of interest to examine whether the basic insight regarding the robustness of di(cid:11)erence rules in the face of unknown natural rates holds in alternative models. To that end we also examined two alternative models based on the same historical data as 24

our baseline model butreflecting quite di(cid:11)erent views regarding the role for expectations: a new synthesis model in which economic outcomes depend much more critically on expectations than in our baseline model, and an accelerationist model in which the role of rational expectations is largely assumed away. 6.1 A New Synthesis Model In the new synthesis model we examine, no lagged terms of inflation and unemployment appear in equations (8) and (9), the short-term interest gap enters the unemployment equation, and there is no lag in the information structure regarding expectations (that is, we assume time t expectations): (cid:25) t = (cid:25) t e +1jt +(cid:11) (cid:25) u~ e tjt +e (cid:25);t ; (13) u~t = u~ e t+1jt +(cid:11) u r~t+e u;t : (14) We calibrated this model to the 1969-2002 sample so that the characteristics of the underlying data are the same as in our baseline model. As is well known, this speci(cid:12)cation does not capture the dynamic behavior of the inflation and unemployment (or output gap) data very well when the shocks to the inflation and unemployment equations, e (cid:25) and e u are serially uncorrelated (Estrella and Fuhrer, 2002). Following Rotemberg and Woodford (1999), McCallum (2001) and others, we therefore allow the errors e (cid:25) and e u to be serially correlated and estimated the model with this modi(cid:12)cation using the same data as in our baseline model, with the changes noted above. Because our unrestricted least squares estimate of (cid:11) u was essentially zero, and therefore inconsistent with the theoretical foundations of this model, we imposed a value for that parameter. We set (cid:11) u = 0:05, following with the theoretically motivated calibration presented in McCallum (2001) based on a model of the output gap (see Nelson and Nikolov (2002) for further discussion). The resulting estimated form of this model is (cid:25) t = (cid:25) t e +1jt + −0:408 u~ e tjt +e (cid:25);t (15) (0:103) (cid:26) e;(cid:25) = 0:26, SER = 1:33, DW = 2:04 u~t = u~ e t+1jt +0:05r~t +e u;t ; (16) (cid:26) e;u = 0:72, SER =0:21, DW = 2:23. 25

Usingtheseestimates andtheassociated covariance structureoftheerrorsinthismodel,we computed e(cid:14)cient policy responses for the generalized rule given by equation (2) without andwith uncertainty regardingthenaturalrates as with ourbaseline model. An interesting feature of the new synthesis model that di(cid:11)ers from our baseline model is that, in the absence of uncertainty about the natural rates, the e(cid:14)cient policies are super-inertial, that is (cid:18) f > 1. (This is explored in detail by Rotemberg and Woodford (1999).) In the presence of uncertainty, of course, such policies also introduce policy errors from misperceptions about the natural rate of interest similar to policies with (cid:18) f < 1. The only di(cid:11)erence is that the sign of the error is reversed. Figure 11, which repeats for this model the experiments shown in Figure 8 for our baseline model, con(cid:12)rms that, in the presence of increasingly higher uncertainty regarding the real-time estimates of the natural rate, the e(cid:14)cient policy again converges towards (cid:18) f ! 1 and (cid:18) u ! 0. Evidently, the di(cid:11)erence rule of the form given by equation (3) represents the robust policy for dealing with natural rate uncertainty in this model as well as in the baseline model. This can also be con(cid:12)rmed in Table 7, which compares the values of the loss function corresponding to the robust rule given by equation (3) andthegeneralized rulegiven byequation (2) optimized fors = 0. From thesecond row of the table it is evident that the cost of adopting the robust rule relative to the optimized one is modest when s = 0, and the bene(cid:12)ts considerable if the true level of uncertainty is s = 1 or higher. This is similar to the result indicated earlier for our baseline model, as shown in the (cid:12)rst row of the table. 6.2 An Accelerationist Model A key feature of the baseline and new synthesis models is the assumption of rational expectations. As noted above,di(cid:11)erence rules perform reasonably well in those models even in the absence of natural rate misperceptions. In \backward-looking" models with adaptive expectations, however, di(cid:11)erence rules generally perform very poorly and may be destabilizing because of the instrument instability problem. Moreover, in such models the costs associated with responding to the change in the output gap or the unemployment rate, as opposedto the levels of thegaps, tendto much greater than in forward-lookingmodels with rational expectations. To explore the sensitivity of policy to a di(cid:11)erent speci(cid:12)cation of expectations, we estimate a backward-looking model that imposes an accelerationist Phillips 26

curve and assumes that rational expectations are unimportant for determining aggregate demand, with the exception of the determination of the real interest rate, where we retain the ex ante real rate of interest from our baseline model: (cid:1)(cid:25) t = +0:477 (cid:25) t−1 +0:099 (cid:25) t−2 +0:255 (cid:25) t−3 +0:123 (cid:25) t−4 (0:089) (0:094) (0:093) (0:088) −0:278 u~t−1 −1:189 (u t−1 −u t−2)+e (cid:25);t (17) (0:096) (0:323) SER = 1:36, DW = 1:96 u~t = 1:415 u~t−1 −0:485 u~t−2+ +0:049 r~ t a −1 +e u;t (18) (0:074) (0:072) (0:014) SER = 0:31, DW = 2:14 Figure 12, which parallels Figures 8 and 11 for our baseline and new-synthesis models, respectively, presents the simulated e(cid:14)cient response coe(cid:14)cients of the generalized rule in equation (2) for this model. Two (cid:12)ndings are apparent. As in the baseline and new synthesis models, uncertainty regarding the natural rates raises the e(cid:14)cient degree of inertia in the policy rule and leads to a signi(cid:12)cant attenuation of the policy response to the unemployment gap. However, the e(cid:14)cient policy for this model does not converge to the robust di(cid:11)erence rule given by equation (3) as quickly as in the other two models. Evidently, in a backward-lookingworld, therearecostsfromcompletelyignoringtheestimated levels ofthe unemployment gap and the natural rate of interest, even when the uncertainty regarding natural rates is signi(cid:12)cant. The last row of Table 7 con(cid:12)rms this result. However, even in this model our experiments suggest that policies should exhibit signi(cid:12)cant smoothing and attenuated responses to the unemployment gap. As the last row in also Table 7 indicates, even in this case the robust rule for this model performs better than the rule optimized under the assumption of no misperceptions when the true degree of misperceptions is as high as s = 3. However, this is a much higher threshold than that for our baseline and new synthesis models. 6.3 Robustness to Both Model and Natural Rate Uncertainty McCallum (1988) and Taylor (1999b) argue that monetary policy should be designed to performacrossawiderangeofreasonablemodels. Inthissection, wefollow Levin,Wieland, 27

and Williams (2002) and compute the optimized policy rule given priors over the three models discussed above. For this experiment we assign equal weights to the three models and compute theoptimal choice of parameters for the robustpolicy rule. Theresults of this exercise are reported in Table 8, which follows a format similar to that of Table 6, which was based on the baseline model alone. The third and fourth columns show the optimal rule parameters for the objective of minimizing the sum of the losses in the three models. Thelastthree columnsshow the correspondinglosses. Comparison of thetwo tables reveals that the optimal rule allowing for model uncertainty features slightly larger responses to the change in the unemployment rate, but the response to the inflation rate is from 3 to 5 times larger than in the baseline model. Although not shown in the table, the parameters of the generalized rule that accounts for model uncertainty lie between those of the baseline and accelerationist models. 7 Misperceptions and Historical Policy Outcomes Ourpolicyevaluationexperimentshighlightthatovercon(cid:12)denceregardingthepolicymaker’s ability to detect changes in the natural rates|that is, the pursuit of policies that are \optimal" under the false assumption that misperceptions regarding real-time assessments of the natural rates are smaller than they actually are|can have potentially disastrous consequences for economic stability. The sensitivity of economic outcomes to policy design is potentially informative for understandingthe historical performanceof monetary policy, especially during episodes when natural rates changed signi(cid:12)cantly and real-time assessments of these rates were likely subject to substantial misperceptions. As an illustration, we perform two experiments comparing outcomes from the Taylor, optimized, and robust rules, designed to highlight some elements we (cid:12)nd important for understandingthe stagflationary experience of the 1970s and the disinflationary boom of the 1990s. 7.1 The 1970s The stagflationary experience of the 1970s has proven a rich laboratory for understanding potential pitfalls in policy design. A number of plausible explanations that boil down to inherently \bad" policy have already been put forward for the dismal outcomes of that period: possible confusion of real and nominal interest rates, insu(cid:14)cient responsiveness 28

of policy to inflation, attempted exploitation of a Phillips curve that was misspeci(cid:12)ed to include a with a stable long-run tradeo(cid:11) between inflation and unemployment, and so forth. In our illustration we instead highlight the more subtle complication arising from comparing policies that, as already pointed out, would appear to be \good" under certain circumstances, but have di(cid:11)erent degrees of sensitivity to the presence of misperceptions regarding the natural rates. To set the stage, consider (cid:12)rst the evolution of perceptions regarding the natural rates of interest and unemployment following unanticipated increases in the natural rates such as appear to have been an integral part of the 1970s experience. (We review some direct evidence from the historical record on the evolution of beliefs below.) To illustrate the misperceptions that we wish to consider for this experiment, Figure 13 traces an example that assumes that both naturalrates increase over a periodof 2-1/2 years by1.5 percentage points. Weassumethat,atthebeginningofthesimulation,beforetheunexpectedincreases, policymakers know the correct levels of the natural rates. Despite starting with correct estimates, their gradual learning of the evolution of the naturalrates when theyunexpectedly riseresultsintemporarybutnonethelesspersistentmisperceptions. Giventheaveragespeed of learning implied by our baseline estimates of historical misperceptions in our sample, the 1.5 percentage increase shown by the solid lines in Figure 13 results in real-time estimates shown by the dashed lines. For both natural rates, errors in real-time estimates|the di(cid:11)erence between the true natural rate and the real-time estimates|gradually increase at (cid:12)rst, to about 1 percentage point, and then dissipate slowly over a period of many years. The e(cid:11)ect of these misperceptions on economic outcomes for the classic and revised Taylor rules are compared in Figure 14. The upper panel shows that, when policy follows the classic Taylor rule, natural rate misperceptions lead to a persistent rise in inflation, whcih peaks at 3 percentage points above the policymaker’s objective. The bulk of this unfavorable outcome is due to the strong response of this policy rule to an incorrectly estimated unemployment gap, which can be seen in the lower panel. As the policymaker’s perceptions of the natural rate lag behind reality, the policymaker incorrectly and strongly attempts to stabilize the rate of unemployment at a level that is persistently too low. Throughout the simulation, the policymaker believes that the actual unemployment rate is above the natural rate, and policy actions impede the movement of the economy towards 29

the true natural rate. The outcome is the modest stagflationary experience shown in the (cid:12)gure. The magnitude of the increase in inflation is greater for the Revised Taylor Rule because this rule is more responsive to the size of the perceived unemployment rate gap. The magnitude of the peak inflationary e(cid:11)ect depends on the parameters of the policy rule,butaslongaspolicyrespondstonaturalrates, thee(cid:11)ects arequitepersistent. Thetop two panels of Figure 15 show the responses from the generalized rule optimized under the assumption of no misperceptions. The peak rise in the inflation rate is nearly 7 percentage points and even after seven years inflation is nearly 3 percentage points above target. The robustpolicycannot avoid the initial increase inunemploymentand inflation either, as seen in the bottom two panels of the (cid:12)gure. However, because the robust policy is not guided by perceptions of the unemployment gap, but only by the evolution of inflation and changes in the unemployment rate, policy does not impede the movement of the economy towards the true natural rate in the way the optimized policy does. Consequently, the increase in the natural rates leads to a much less persistent deviation of inflation from its target in this case (bottom left-hand panel). The relevance of this comparison for explaining the events of the 1970s rests on two elements. The (cid:12)rst is that the misperceptions regarding the natural rate of unemployment, and to a lesser degree the natural rate of interest, signi(cid:12)cantly influenced policy. Second, and perhaps more controversial element is that policymakers at the time actually operated in a way resembling the Taylor rule or our \optimal" policy approach, instead of a more robust policy. Bearing on this are the fascinating intellectual debates regarding \activist" countercyclical stabilization policies and the observation that proponents of such policies appeared to have won the day at the turn of the 1970s. (See Orphanides, 2000a,b for a historical review.) The perceived triumph of activist stabilization policy is reflected in many writings, including those of Heller (1966) and Okun (1970), and appeared to capture the hopes of both academic economists and policymakers across a wide spectrum of ideologies and backgrounds. One succinct accounting of the policy errors committed using this lens was o(cid:11)ered by Stein (1984) who reflected on policymakers’ attempts to guide the economy to its \optimum feasible path" (p. 171) at the turn of the 1970s by targeting \ ‘the natural rate of unemployment’ which we thought to be4 percent" (p. 19). In contrast, our baseline 30

estimates, as well as those by the Congressional Budget O(cid:14)ce, suggest that the natural rate of unemployment at the beginning of the 1970s was nearly 6 percent. Stein’s account is corroborated by the recent retrospective on Paul McCracken’s service at the Council of Economic Advisers (Jones 2000). The view from the Federal Reserve suggests a similar picture. Shortly after he left the Federal Reserve Board, Arthur Burns (1979), who had served as Chairman from 1970 to 1978, expressed his anguish over the the deleterious effects of underestimating the natural rate of unemployment; like Stein, he noted that the initial estimate of 4 percent proved, retrospectively, to have been too low. As Orphanides (2000a,b) documents, the related estimates of potential output and the output gap during the early 1970s proved, retrospectively, to have been exceedingly high. Many issues complicated the measurement of the natural rate of unemployment in the early1970s,includingdisagreementsregardingmodellinginflationdynamicsandthePhillips curve,themeaningof\fullemployment," theproperaccounting ofdemographics, modelling expectations and so forth. Starting with the (cid:12)rst volume in 1970, the (cid:12)rst few years of the Brookings Papers on Economic Activity provide a valuable source documenting the debate andevolution ofviews regardingthenaturalrateof unemployment. Indeed,inthevery(cid:12)rst meeting of the Brookings panel, Okun and Teeters (1970) presented an analysis of the \full employment" surplus assuming that the appropriate de(cid:12)nition for \full employment" was the widely accepted during the previous decade 4 percent rate. Hall (1970) identi(cid:12)ed the \equilibrium level of unemployment" or \full employment unemployment" as the level that, \...if maintained permanently, would producea steady rate of inflation of 3 or 4 percent per year," (p. 370) and noted that \[m]ost economists agree that this is somewhere between 4 and 5 percent unemployment." (p. 370). Perry (1970) presented estimates of the shifting Inflation-Unemployment tradeo(cid:11) adjusting for changes in the demographic composition of the employment force (what later became known as \Perry weighting"), and the dispersion of unemployment among the age-sex groups of the labor force. According to his estimates (Figure 2, p. 432), whereas an unemployment rate of about 4 percent was consistent with a 3 percent annual increase in the consumer price index during the mid-1950s, by 1970 the unemployment rate would have had to be around (cid:12)ve percent to be consistent with the same 3 percent rate of inflation. Finally, in one of the earliest exercises of policy design based on an estimated econometric model at the Federal Reserve (and,as far as we are 31

aware, the earliest such exercise using a model consistent with the natural rate hypothesis), Poole(1971) presentedexperimentsusingtheFederalReserve’seconometricmodelwithtwo versions ofaPhillipscurve, the\standardmodel"(with asloping\long-run"Phillipscurve) and an \accelerationist model." Poole’s simulations using the standard model showed that inflation could be stabilized below 3 percent with a 4 percent rate of unemployment. In simulations of the accelerationist model the implicit \natural" rate of unemployment was 4.5 percent. Already from this work from 1970 and 1971 it is clear that estimates of the natural rate were beginning to rise from the 4 percent view that had prevailed during the 1960s. Nonetheless, the evidence is compelling that misperceptions regarding the natural rate of unemployment were sizable at the turn of the 1970s. Whereas such real-time estimates of the natural rate of unemployment are well documented, real-time estimates of the natural rate of interest are hard to come by. One source is the report prepared each year by the trustees of the Social Security system; for several decades this report has included projections of long-term interest rates. The forecast longrun real interest rate reported by the trustees rose from 2-1/2 percent in 1972 to 3-1/4 percent in 1975. Before 1972 only nominal rates were projected, and estimates of this rate rose by a full percentage point between 1969 and 1972. Given the relatively modest rise in inflation during that period, this rise in nominal rates can be interpreted as a signi(cid:12)cant increaseinlong-runrealrates. Overall, thisevidenceprovidessomesupportforasigni(cid:12)cant increase in the perceived natural rate of interest over this period. 7.2 The 1990s What Blinder and Yellen (2001) have called the \fabulous decade" arguably constitutes, in some respects, an exact opposite of the dismal experience of the 1970s. During the 1990s the natural rate of unemployment apparently drifted downward, and signi(cid:12)cantly so. This lower level of the natural rate of unemployment went hand in hand with somewhat lower inflation; however, inflation more or less remained in line with policymaker descriptions of their price stability objectives. One possible di(cid:11)erence from the experience of the 1970s is that natural rate misperceptions may have been smaller and less persistent in the more recent episode. Ball and Tchaidze (2002), for example, argue that Federal Reserve’s implicit NAIRU estimates may 32

have fallen rapidly in the second half of the 1990s. Even so, the record indicates the possibility of signi(cid:12)cant misperceptions. The FOMC transcripts for 1994 and 1995, for example, indicate that some members of the Committee as well as Federal Reserve Board sta(cid:11) held the view that the natural rate of unemployment was around 6 percent at the time. By 2000, then Governor Meyer, indicated that a range of 5 to 5-1/4 percent was a better estimate (Meyer, 2000). This points towards a nontirivial misperception, perhaps as high as 16 1 percentage point, for the middle of the decade. Table 9 suggests similar revisions in responses from the Survey of Professional Forecasters as well as the estimates published by the he Congressional Budget O(cid:14)ce and the Council of Economic Advisers. An alternative possibility is that, despite signi(cid:12)cant misperceptions regarding the natural rate of unemployment, economic outcomes were better because monetary policy was morerobusttosucherrorsthanthepolicyframeworkinplaceduringthe1970s. Tohighlight this possibility, Figure 16 presents two alternative illustrations for this period, tracing the evolution of the economy following a reduction in the natural rate of unemployment under our optimized and robustpolicies. Here we assumethat the natural rate of interest remains unchanged and that the change in the natural rate of unemployment has the the same size andtimingasthatshownintheright-handpanelsofFigure13,butoppositesign. Assuming the 1.5 percent reduction in the natural rate of unemployment underlying the simulation, policy under the optimized rule would have led to deflation over this period|with infation falling byalmost 6 percentage points duringthesimulation andstaying wellbelow its initial value for many years. By contrast, our robust policy appears more successful in replicating the \Goldilocks"-like of economic outcomes of this period. 8 Concluding Remarks This paper has critically reexamined the usefulness of the natural rates of interest and unemployment in the setting of monetary policy. Our results suggest that underestimating theunreliabilityofreal-time estimates ofthenaturalrates maylead topolicies thatarevery verycostly in termsof thestabilization performanceof theeconomy. Itis importanttonote thatourcritiquedoesnotnecessarilyimplyanydisagreement withthevalidity orusefulness 16Transcripts and other documents relating to FOMC meetings are released with a (cid:12)ve-year lag and are therefore not yet available for years after 1996. 33

of these concepts for understanding and describing historical macroeconomic relationships. Indeed, our analysis and conclusions are based entirely on models where deviations from natural rates are the primary drivers of inflation and unemployment. Instead, we argue that uncertainty about natural rates in real time recommends against excessively relying on these intrinsically noisy indicators for monetary policy decisions. In that respect, our critiqueechoessimilarconcernsvoiceddecadesagoabouttheoperationalusefulnessofpolicy based on natural rates|concerns also reflected, at least in part, in more recent discussions 17 of monetary policy. A key aspect of natural rate measurement is the profound uncertainty regarding the degree of mismeasurement. Because the losses from underestimating measurement error exceed those from exaggerating it, Bayesian and robust control strategies indicate that the policy rule should incorporate a biased protection against measurement error and respond only modestly to estimates of the natural rates of interest and unemployment. Indeed, in forward-looking models, a \di(cid:11)erence" policy rule in which the change in the interest rate responds to the inflation rate and the change in the unemployment rate, and not to levels of the natural rates, performs nearly as well as more complicated rules that incorporate both level and di(cid:11)erence features. Only in a backward-looking model do we (cid:12)nd a strong argument for maintaining a nontrivial response to natural rates, but even in this model the basic conclusion of our analysis holds: natural rate uncertainty implies very muted responses to both the natural rates of interest and unemployment relative to policy rules 18 designed in the context of no measurement error. Thehistoricalexperiencesofthe1970sandthelate1990sprovideinsightsintothedesign of monetary policy in light of natural rate uncertainty. In the former episode, arguably, policymakers mistakenly held to the belief that the natural rate of unemployment was lower than we now (with hindsight) believe it was, and they actively sought to stabilize 17For example, As Chairman Greenspan (2000) recently pointed out that \However one views the operational relevance of a Phillips curve or the associated NAIRU (the nonaccelerating inflation rate of unemployment)|and I am personally decidedly doubtful about it|there has to be a limit to how far the poolof availablelaborcan bedrawndown withoutpressing wagelevelsbeyondproductivity. Theexistence or nonexistence of an empirically identi(cid:12)able NAIRU has no bearing on the existence of the venerable law of supply and demand." 18Interestingly, Walsh (2002) reaches similar conclusions in a recent paper that assumes no measurement problem but in which policymakers cannot commit to a policy rule. He shows that in a forward-looking model it is optimal to assign an objective of stabilizing inflation and the change in the output gap to a policymakerwhoactswithdiscretion, whenthetruesocial welfare objective istostabilize inflationandthe level of theoutput gap. 34

unemployment at that level. The result was rising inflation and eventually stagflation. In the 1990s, the reverse shock took place, but inflation remained relatively stable. 35

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Table 1. Retrospective Estimates of the Natural Rate of Unemployment Source or method 1960 1970 1980 1990 2000 Congressional Budget O(cid:14)ce (2002) 1 5:5 5:9 6:2 5:9 5:2 Gordon (2002) 1 5:6 6:3 6:3 6:2 5:0 Ball and Mankiw method 2 5:0 6:0 6:9 6:2 4:5 Staiger, Stock, and Watson (2002) 1 5:8 4:7 7:7 6:3 4:5 Kalman (cid:12)lter 2 | 5:7 6:4 5:8 5:0 Brainard and Perry (2000) 1 3:8 4:7 9:8 5:8 3:8 3 Shimer (1998) 1 5:3 6:5 7:1 5:9 5:9 BP (cid:12)lter (8-year window) 2 6:0 4:2 7:3 5:9 4:9 BP (cid:12)lter (15-year window) 2 5:6 4:4 7:9 6:3 5:0 HP (cid:12)lter ((cid:21)= 1600) 2 5:9 4:6 7:5 6:1 4:5 HP (cid:12)lter ((cid:21)= 25600) 2 5:3 5:0 7:4 6:4 4:6 Memoranda: Median of estimates 5:6 5:0 7:3 6:1 4:9 Range of extimates 3.8{5.9 4.2{6.5 6.2{9.8 5.8{6.4 3.8{5.9 Actual unemployment rate 5:5 5:0 7:2 5:6 4:0 Notes: 1. Estimates are taken from the indicated source, in some cases updated by source author. 2. Estimates are authors’ calculations, based on methods described in sources cited in the text. 3. Estimate is for 1998. 42

Table 2. Forecast Errors of Alternative Natural Rate-Based and Autoregressive Methods 1 Standard error of the regression 1-quarter 4-quarter 8-quarter Method horizon horizon horizon 2 Forecasting inflation Constant natural rate of unemployment 1:11 1:12 1:74 Kalman (cid:12)lter 1:10 1:14 1:80 Ball and Mankiw method 1:14 1:11 1:73 BP (cid:12)lter (8-year window) 1:10 1:13 1:78 BP (cid:12)lter(15-year window) 1:11 1:16 1:74 HP (cid:12)lter((cid:21) = 1600) 1:13 1:13 1:79 HP (cid:12)lter((cid:21) = 25600) 1:14 1:16 1:80 AR(4) (no unemployment rate term) 1:18 1:24 1:92 3 Unemployment rate Constant natural rate of interest 0:26 0:55 1:10 Kalman (cid:12)lter 0:25 0:52 1:07 Laubach and Williams method 0:26 0:54 1:11 BP (cid:12)lter (8-year window) 0:26 0:53 1:09 BP (cid:12)lter (15-year window) 0:25 0:52 1:06 HP (cid:12)lter ((cid:21) = 1600) 0:26 0:54 1:07 HP (cid:12)lter ((cid:21) = 25600) 0:25 0:51 1:03 AR(2) (no real rate term) 0:26 0:55 1:12 Notes: 1. The sample period is 1970:1{2002:2. For the one-quarter forecast horizon the forecast rate is that in the next quarter; for the four-quarter forecast horizon it is the average of the next four quarters; for the eight-quarter horizon it is the average of the subsequent four quarters. 2. All except the AR(4) equation include four lags of inflation, one lag of the change in the unemployment rate, and two lags of the unemployment gap. 3. All except the AR(2) equation include two lags of the unemployment rate gap and one lag of the four-quarter moving average of the real rate gap. 43

1 Table 3. Natural Rate Misperceptions Assuming the Model is Known Standard deviation of di(cid:11)erence between Persistence measures real-time and Persistence Standard error retrospective coe(cid:14)cient of regression Method or source estimates (cid:26) (cid:27) (cid:23) Natural rate of unemployment Kalman (cid:12)lter 0:66 0:95 0:21 Ball-Mankiw method 0:58 0:97 0:14 BP (cid:12)lter (8-year window) 0:52 0:89 0:23 BP (cid:12)lter(15-year window) 0:61 0:92 0:23 HP (cid:12)lter((cid:21) =1600) 0:75 0:97 0:18 HP (cid:12)lter((cid:21) =25600) 0:78 0:98 0:12 Natural rate of interest Kalman (cid:12)lter 1:44 0:93 0:55 Laubach-Williams 0:90 0:91 0:38 BP (cid:12)lter(8-year window) 1:04 0:92 0:42 BP (cid:12)lter(15-year window) 1:34 0:96 0:41 HP (cid:12)lter((cid:21) =1600) 1:26 0:96 0:37 HP (cid:12)lter((cid:21) =25600) 1:70 0:99 0:25 Note: 1. For each method, the real-time misperception is de(cid:12)ned as the di(cid:11)erence between the real-time and the retrospective estimate of the natural rate. The sample period for these statistics is 1969:1{1998:2. 44

Table 4. Misperceptions of the Natural Rates Allowing for Model Uncertainty Frequency distribution based on alternative 1 measures of natural rate misperceptions 25th 75th Minimum percentile Median percentile Maximum Natural rate of unemployment Standard deviation 0:48 0:63 0:75 1:04 1:34 Persistence coe(cid:14)cient ((cid:26)) 0:89 0:95 0:96 0:97 0:99 Natural rate of interest Standard deviation 0:90 1:44 1:96 2:84 3:24 Persistence coe(cid:14)cient ((cid:26)) 0:91 0:96 0:98 0:98 0:99 Note: 1. The sample is the thirty-six alternative measures of natural rate misperceptions corresponding to all possible pairwise combinations of the six methods listed in each panel of Table 3. Each of the two statistics is computed separately. 45

Table 5. Macroeocnomic Performance under Alternative Policy Rules 2 3 Rule parameter Standard deviation Loss Rule and (! = 0:2, misperception index 1 (cid:18) f (cid:18) (cid:25) (cid:18) u (cid:18) (cid:1)u u−u(cid:3) (cid:25) (cid:1)f ( = 0:05) Classic Taylor rule s= 0 0.0 0:5 −1:0 0:0 0:81 2:14 2:83 1:84 s= 1 0.0 0:5 −1:0 0:0 0:88 3:67 2:88 3:73 s= 2 0.0 0:5 −1:0 0:0 1:01 6:11 3:38 8:85 s= 3 0.0 0:5 −1:0 0:0 1:18 8:72 4:15 17:18 Revised Taylor rule s= 0 0.0 0:5 −2:0 0:0 0:71 2:03 2:89 1:64 s= 1 0.0 0:5 −2:0 0:0 0:77 4:13 2:91 4:32 s= 2 0.0 0:5 −2:0 0:0 0:91 7:28 3:56 11:89 s= 3 0.0 0:5 −2:0 0:0 1:09 10:57 4:59 24:36 Taylor rule optimized for s= 0 s= 0 0.0 0:31 −3:81 0:0 0:61 2:05 2:83 1:54 s= 1 0.0 0:31 −3:81 0:0 0:71 7:15 3:09 11:11 s= 2 0.0 0:31 −3:81 0:0 0:94 13:64 4:54 38:94 s= 3 0.0 0:31 −3:81 0:0 1:22 20:22 6:41 85:05 Taylor rule optimized for s= 1 s= 0 0.0 1:37 −1:23 0:0 0:73 1:86 4:25 2:02 s= 1 0.0 1:37 −1:23 0:0 0:79 2:07 4:90 2:56 s= 2 0.0 1:37 −1:23 0:0 0:82 2:50 4:94 3:01 s= 3 0.0 1:37 −1:23 0:0 0:86 3:05 5:11 3:76 Generalized rule optimized for s = 0 s= 0 0.72 0:26 −1:83 −2:39 0:62 1:82 2:23 1:23 s= 1 0.72 0:26 −1:83 −2:39 0:70 4:49 2:32 4:71 s= 2 0.72 0:26 −1:83 −2:39 0:95 8:36 3:01 15:16 s= 3 0.72 0:26 −1:83 −2:39 1:27 12:35 4:00 32:58 Generalized rule optimized for s = 1 s= 0 0.97 0:39 −0:23 −5:39 0:66 1:94 2:45 1:40 s= 1 0.97 0:39 −0:23 −5:39 0:66 1:95 2:42 1:40 s= 2 0.97 0:39 −0:23 −5:39 0:66 2:08 2:40 1:50 s= 3 0.97 0:39 −0:23 −5:39 0:66 2:32 2:40 1:71 Robust di(cid:11)erence rule s= 1 1.0 0:35 0:0 −5:96 0:66 2:01 2:49 1:46 Notes: 1. s indexes the magnitude of policymakers’ misperception of the true natural rates. 2. Parameters measurepolicymakers’responsetothelagged federalfundsrate, theinflation gap, the unemployment gap, and the change in the unemployment rate, respectively. 3. Unconditional standard deviation of the unemployment gap, the inflation rate, and the change in the federal funds rate, respectively. 46

1 Table 6. Robust Policy Rule Parameters under Alternative Policymaker Preferences 2 Loss parameters Rule parameter Standard deviation ! (cid:18) (cid:25) (cid:18) (cid:1)u u−u(cid:3) (cid:25) (cid:1)f 0.5 0.05 0:57 −6:29 0:67 1:94 2:78 0.5 0.50 0:25 −3:56 0:82 2:22 1:77 0.5 5.00 0:13 −2:43 1:05 2:67 1:48 0.2 0.05 0:35 −5:96 0:66 2:01 2:49 0.2 0.50 0:17 −3:34 0:85 2:32 1:66 0.2 5.00 0:12 −2:34 1:09 2:76 1:46 0.1 0.05 0:24 −5:79 0:65 2:08 2:36 0.1 0.50 0:14 −3:25 0:87 2:38 1:62 0.1 5.00 0:11 −2:30 1:11 2:80 1:46 Notes: 1. See Table 5 for de(cid:12)nitions of parameters and performance measures. 2. Parameters of the robust rule in equation (3) of the text. 47

Table 7. Performance under Optimized and under Robust Rules for Alternative Models 1 Loss when policy follows Robust Generalized Taylor rule optimized for s =0 Model rule True s = 0 True s = 1 True s = 2 True s = 3 Baseline 1:46 1:23 4:71 15:16 32:58 New-Synthesis 0:63 0:56 0:69 1:02 1:56 Accelerationist 5:13 2:19 2:53 3:54 5:24 Note: 1. Loss as calculated by equation (12) in the text assuming ! = 0:2; = 0:05. 48

1 Table 8. Robust Policy Rules across Alternative Models Rule Loss when true model is: 2 Loss parameters parameter Baseline Accelerationist New synthesis ! (cid:18) (cid:25) (cid:18) (cid:1)u model model model 0.5 0.05 1:56 −7:13 2:89 5:45 1:12 0.5 0.50 0:84 −4:23 5:84 10:19 2:20 0.5 5.00 0:56 −3:21 24:21 32:06 9:61 0.2 0.05 1:28 −7:85 1:88 5:27 0:74 0.2 0.50 0:76 −4:41 4:60 9:73 1:84 0.2 5.00 0:54 −3:26 22:55 30:72 9:32 0.1 0.05 1:15 −8:19 1:53 5:14 0:60 0.1 0.50 0:72 −4:49 4:17 9:51 1:72 0.1 5.00 0:53 −3:28 21:98 30:22 9:23 Notes: 1. See Table 5 for de(cid:12)nitions of parameters and performance measures. 2. Parameters of the robust rule (equation (3) in the text) chosen to minimize the expected loss for the indicated values of the loss parameters, when the model is unknown and each of the models is assigned equal likelihood of being the true model. Loss is calculated by equation (12) in the text. 49

Table 9: Estimates of the Natural Rate of Unemployment, 1995{2002 Council of Survey of Professional Congressional Economic 1 Forecasters (real-time) Budget O(cid:14)ce Advisers 2 3 4 Year Low Median High Real-Time Retrospective (real-time) 1995 | | | 6:0 5:3 5.5{5.8 1996 5:00 5:65 6:00 5:8 5:2 5:7 1997 4:50 5:25 5:88 5:8 5:2 5:5 1998 4:50 5:30 5:80 5:8 5:2 5:4 1999 4:13 5:00 5:60 5:6 5:2 5:3 2000 4:00 4:50 5:00 5:2 5:2 5:2 2001 3:50 4:88 5:50 5:2 5:2 5:1 2002 3:80 5:10 5:50 5:2 5:2 4:9 Notes: 1. Responses are those from the third-quarter in the indicated year. 2. Estimates are from the The Budget and Economic Outlook published in the indicated year (usually in January). 3. Estimates are from Congressional Budget O(cid:14)ce (2002). 4. Estimates are from the Economic Report of the President published in the year shown (usually in February) and reflect either explicit references to a NAIRU estimate, or, when no explicit reference appears, the unemployment rate at the end of the long-term economic forecast presented in the report. 50

Figure 1 Retrospective Estimates of the Natural Rate of Unemployment. 1969{2002 9 8 7 6 5 4 1970 1975 1980 1985 1990 1995 2000 tnecreP HP filter (l = 1,600) Band−pass filter (15−year) Staiger−Stock−Watson KF Kalman filter (this paper) CBO 51

Figure 2 Real-Time Estimates of the Natural Rate of Unemployment, 1969{2002 11 10 9 8 7 6 5 4 3 1970 1975 1980 1985 1990 1995 2000 tnecreP HP filter (l = 1,600) Band−pass filter (15−year) Kalman filter (this paper) Actual unemployment 52

Figure 3 Retrospective Estimates of the Natural Rate of Interest, 1969{2002 8 7 6 5 4 3 2 1 0 −1 1970 1975 1980 1985 1990 1995 2000 tnecreP HP filter (l = 1,600) Band−pass filter (15−year) Kalman filter (this paper) Laubach−Williams KF 53

Figure 4 Real-Time Estimates of the Natural Rate of Interest, 1969{2002 14 12 10 8 6 4 2 0 −2 −4 1970 1975 1980 1985 1990 1995 2000 tnecreP HP filter (l = 1,600) Band−pass filter (15−year) Kalman filter (this paper) Laubach−Williams KF Actual real rate 54

Figure 5 Macroeconomic Performance of Taylor Rules for Given Degrees of Natural Rate Misperceptions 8 4 2 1 0 0.5 1 1.5 2 2.5 3 )elacs gol( DS Inflation Standard Taylor rule Revised Taylor rule S 1.25 1 0.75 0.5 0 0.5 1 1.5 2 2.5 3 )elacs gol( DS Unemployment Gap S Notes: The two panels show the asymptotic standard deviation of inflation and the unemployment gap (vertical axis, in percent) corresponding to the degree of misperceptions regarding the natural rates, s (horizontal axis) when policy follows the original and revised Taylor rules, reflected by the solid and dashed lines, respectively. 55

Figure 6 Optimal Policy Response Parameters under Taylor Rules for Given Degrees of Natural Rate Misperceptions q q 0 p u 2 −1 1.5 −2 1 −3 0.5 0 −4 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 S S Notes: Thelines indicate the optimal choices of theparameters (cid:18) (cid:25) and (cid:18) u in thepolicy rule: f t = r t (cid:3) +(cid:25) t+(cid:18) (cid:25)((cid:25) t −(cid:25)(cid:3) )+(cid:18) u(u t −u(cid:3) t ) for di(cid:11)erent degrees of misperceptions regarding the natural rates, s. 56

Figure 7 Performance with Optimized and Robust Taylor Rules for Given Degrees of Natural Rate Misperceptions Inflation 8 4 2 1 0 0.5 1 1.5 2 2.5 3 S )elacs gol( DS Unemployment Gap Optimized for s=0 Optimized for s=1 1.25 Robust rule 1.00 .75 .50 0 0.5 1 1.5 2 2.5 3 S )elacs gol( DS Change in Funds Rate 6 5 4 3 2 0 0.5 1 1.5 2 2.5 3 S )elacs gol( DS Loss 8 4 2 1 0 0.5 1 1.5 2 2.5 3 elacs gol S Notes: The three lines in each panel show the asymptotic standard deviations/loss (vertical axes) corresponding to the degree of misperceptions regarding the natural rates, s (horizontal axis) for three alternative policy rules: the Taylor rule (1) optimized with the assumption that s = 0 (solid lines); the Taylor rule (1) optimized with the assumption that s = 1 (dashed lines); and the robust rule (3) (dash-dot lines). 57

Figure 8 Optimal Policy Response Parameters under Generalized Policy Rules for Given Degrees of Natural Rate Misperceptions q q 1 p 0.45 f 0.95 0.4 0.9 0.85 0.35 0.8 0.3 0.75 0.7 0.25 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 S S q 0 q −2 u D (u) −0.5 −3 −1 −4 −1.5 −5 −2 −6 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 S S Notes: The lines indicate the optimal choices of the parameters (cid:18) f, (cid:18) (cid:25), (cid:18) u and (cid:18) (cid:1)u in the policy rule: f t = (cid:18) f f t−1+(1−(cid:18) f)(r t (cid:3) +(cid:25) t)+(cid:18) (cid:25)((cid:25) t −(cid:25)(cid:3) )+(cid:18) u(u t −u(cid:3) t )+(cid:18) (cid:1)u(u t −u t−1) for di(cid:11)erent degrees of misperceptions regarding the natural rates, s. 58

Figure 9 Performance with Robust and Generalized Taylor Rules for Given Degrees of Natural Rate Misperceptions Inflation 8 4 2 1 0 0.5 1 1.5 2 2.5 3 S )elacs gol( DS Unemployment Gap 1.25 1.0 Optimized for s=0 Optimized for s=1 Robust rule .75 .5 0 0.5 1 1.5 2 2.5 3 S )elacs gol( DS 6 5 4 3 2 0 0.5 1 1.5 2 2.5 3 S )elacs gol( DS Change in Funds Rate Loss 8 4 2 1 0 0.5 1 1.5 2 2.5 3 S elacs gol Notes: Thethree lines in each panel show the asymptotic standard deviations/loss (vertical axes) corresponding to the degree of misperceptions regarding the natural rates, s (horizontal axis) for three alternative policy rules: the generalized rule (2) optimized with the assumption that s= 0, (solid lines); the generalized rule (2) optimized with the assumption that s = 1 (dashed lines); and the robust rule (3) (dash-dot lines). 59

Figure 10 E(cid:14)cient Policy Response Parameters under Generalized Taylor Rules for Given Degrees of Natural Rate Misperceptions q u 0 Robust policy S=2 S=1 −0.5 Bayesian over [0,2] −1 −1.5 S=0 −2 0.7 0.8 0.9 1 q f Notes: The solid line traces the pairs of optimal choices of the parameters (cid:18) f (horizontal axis) and (cid:18) u (vertical axis) for di(cid:11)erent known degrees of misperceptions shown in Figure 10. Movements along the line in the northeast direction correspond to higher values of s and the pairs corresponding to s = 0;1;2 are marked with an x. \Bayesian" indicates the optimal choices when the policymakers has a uniform prior about s on the [0,2] range. \Robust" indicates our simple di(cid:11)erence rule. 60

Figure 11 Optimal Policy Response Parameters in New Synthesis Model for Given Degrees of Natural Rate Misperceptions q 1.2 q 0.9 f p 1.15 0.8 1.1 0.7 1.05 0.6 1 0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 S S q 0 q −1 u D (u) −0.25 −1.2 −1.4 −0.5 −1.6 −0.75 −1.8 −1 −2 −1.25 −2.2 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 S S Notes: The lines indicate the optimal choices of the parameters (cid:18) f, (cid:18) (cid:25), (cid:18) u and (cid:18) (cid:1)u in the New-Synthesis model when policy follows the rule: f t = (cid:18) f f t−1+(1−(cid:18) f)(r t (cid:3) +(cid:25) t)+(cid:18) (cid:25)((cid:25) t − (cid:25)(cid:3) ) + (cid:18) u(u t − u(cid:3) t ) + (cid:18) (cid:1)u(u t − u t−1) for di(cid:11)erent degrees of misperceptions regarding the natural rates, s. 61

Figure 12 Optimal Policy Response Parameters in Accelerationist Model for Given Degrees of Natural Rate Misperceptions q 1 q 1.3 f p 1.2 0.9 1.1 1 0.8 0.9 0.8 0.7 0.7 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 S S q 0 q −2 u D (u) −0.5 −3 −1 −4 −1.5 −5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 S S Notes: The lines indicate the optimal choices of the parameters (cid:18) f, (cid:18) (cid:25), (cid:18) u and (cid:18) (cid:1)u in the accelerationist model when policy follows the rule: f t = (cid:18) f f t−1+(1−(cid:18) f)(r t (cid:3) +(cid:25) t)+(cid:18) (cid:25)((cid:25) t − (cid:25)(cid:3) ) + (cid:18) u(u t − u(cid:3) t ) + (cid:18) (cid:1)u(u t − u t−1) for di(cid:11)erent degrees of misperceptions regarding the natural rates, s. 62

Figure 13 Misperceptions of Natural Rates Following an Unexpected Increase R* U* 1.5 1.5 1 1 0.5 0.5 True value Real−time est. 0 0 0 2 4 6 0 2 4 6 Real−time R* Errors Real−time U* Errors 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 2 4 6 0 2 4 6 Years Years Notes: Thetop panels indicate the evolution of the trueand perceived natural rate of interest (left) and unemployment (right) over time, following a series of unanticipated increases in the natural rates which cumulate to 1.5 percentage points over a period of 10 quarters (2 1/2 years). The bottom panels trace the resulting evolution of misperceptions about the natural rates over time. In all panels, we plot deviations from steady state values, in percent. 63

Figure 14 Performance under Taylor Rules Following a Misperceived Increase in Natural Rates Inflation 7 Unemployment Rate Classic Taylor Rule Revised Taylor Rule 1.6 6 1.4 5 1.2 4 1 3 0.8 2 0.6 0.4 1 True U* Real−time U* Est. 0.2 Classic Taylor Rule 0 Revised Taylor Rule 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Years Years Notes: The two panels trace the evolution of inflation and unemployment (deviations from steady state values, in percentage points) in an economy subjected to the unexpected increases in the natural rates of interest and unemployment shown in Figure 13 for the classic and revised versions of the Taylor rule. 64

Figure 15 Performance under Optimized Generalized and Robust Rules Following a Misperceived Increase in Natural Rates Optimized generalized rule Inflation 7 Unemployment Rate 1.6 6 1.4 5 1.2 4 1 3 0.8 True U* 2 0.6 R U e n a e l m −t p im lo e ym U e * nt rate 0.4 1 0.2 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Years Years Robust Rule Inflation 7 Unemployment Rate 1.6 6 1.4 5 1.2 4 1 3 0.8 True U* 2 0.6 R U e n a e l m −t p im lo e ym U e * nt rate 0.4 1 0.2 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Years Years Notes: The top and bottom panels trace the evolution of inflation and unemployment in an economy subjected to the unexpected increases in the natural rates of interest and unemployment shown in Figure 13 for two alternative policies: The optimized generalized rule for s = 0 (row 17 in Table 2) shown in the top panels, and the robust rule (row 25 in Table 2) in the bottom panels. In all panels, we plot deviations from steady state values, in percent. 65

Figure 16 Performance under Optimized Generalized and Robust Rules Following a Misperceived Decrease in the Natural Rate of Unemployment Optimized generalized rule Inflation Unemployment Rate 0 0 T R r e u a e l − U ti * me U* −0.2 Unemployment rate −1 −0.4 −2 −0.6 −0.8 −3 −1 −4 −1.2 −5 −1.4 −6 −1.6 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Years Years Robust Rule Inflation Unemployment Rate 0 0 True U* Real−time U* −0.2 Unemployment rate −1 −0.4 −2 −0.6 −3 −0.8 −1 −4 −1.2 −5 −1.4 −6 −1.6 0 1 2 3 4 5 6 7 Years 0 1 2 3 Years 4 5 6 7 Notes: The top and bottom panels trace the evolution of inflation and unemployment in an economy subjected to an unexpected decrease in the natural rate of unemployment for two alternative policies: The optimized generalized rule for s = 0 (row 17 in Table 2) shown in the top panels, and the robust rule (row 25 in Table 2) in the bottom panels. For this experiment, we assume that the natural rate of interest remains unchanged and that the change in the natural rate of unemployment has the the same size and timing but reverse sign of that shown in the right panels of Figure 13. In all panels, we plot deviations from steady state values, in percent. 66