Board of Governors of the Federal Reserve System and European Central Bank

A Review of Recent Empirical Evidence

Brian Sack

and

Volker Wieland

Board of Governors of the Federal Reserve System

and European Central Bank

August 1999

We are grateful for helpful comments by James Clouse, Bill English, Mark Hooker, Donald Kohn, David

Lindsey, Brian Madigan, Athanasios Orphanides, Richard Porter, David Reifscheider, Vincent Reinhart, David Small,and John Williams. The opinions expressed are those of the authors and do not necessarily reflect the views of theBoard of Governors of the Federal Reserve System or the European Central Bank.

Sack, Division of Monetary Affairs, Federal Reserve Board, Washington, DC 20551, Tel: (202) 736 5671,bsack@frb.gov.

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Figure 1Intended Federal Funds Rate Percent9March 1990 to December 1998876543199019911992199319941995199619971998Percent12March 1984 to March 19901110 9 8 7 6198419851986198719881919903

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While for each argument we

refer to several supporting recent studies, we limit the more detailed review of empiricalmethodology and results to one study per argument. In particular, this paper reviews the resultsof Levin, Wieland, and Williams (1998) regarding efficient simple interest-rate rules underforward-looking expectations, Orphanides (1998b) regarding the impact of data uncertainty onoptimal Taylor rules, and Sack (1998a) regarding optimal policy under parameter uncertainty. Thethree studies have in common that they apply to the U.S. economy and were conducted at theFederal Reserve Board. While we have not extended our review to cover the experience of othercountries and central banks, we again note that choosing a smooth path for short-term interestrates is a strategy common to many central banks.

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The latter paper

is perhaps of special interest since the authors use the Federal Reserve’s own FRB/USmacroeconomic model with forward-looking expectations. Levin, Wieland and Williams (LWW)conduct two different experiments to assess the benefits from interest-rate smoothing in terms ofoutput and inflation stabilization

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Figure 2Forward-Looking Expectations and Interest Rate SmoothingLevin, Wieland, Williams (1998) Evaluating Policy RulesCompute output, inflation, and interest rate variability for the estimated rule (2).Compute inflation and output variability for policy rules such asrt = ρrt-1+ (1-ρ)(rr*+πt) + α ( πt - π* ) + β ytconstraining interest volatility to the value observed under the estimated rule.Policy Frontiers in the FRB/US Model4.0Standard Deviation of the Output GapSimple Rules with and without Partial Adjustment3.53.02.5XEstimated Rule (2)Without Partial Adjustment ( ρ = 0 )With Partial Adjustment ( ρ > 0 )2.01.51.01.31.51.7 1.92.12.32.5Standard Deviation of Inflation11

As to the partial adjustment coefficient , relaxing the constraint on interest rate volatility

results in only a slight reduction in its optimal value. For example, doubling the standarddeviation of the change of the federal funds rate reduces the optimal value of only by about

0.07, on average, from a range of [.96, 1.02] to [.93, .96]. LWW obtain similar results in thethree other models they consider, Taylor’s multi-country model, the model by Orphanides andWieland, and the Fuhrer-Moore model. They conclude that even with relatively high interest ratevolatility, a high degree of partial adjustment is preferred in these models. While all the resultsreported by LWW still incorporate some constraint on interest rate volatility, optimal ruleswithout such a constraint have been reported by Fuhrer (1997) for a version of the last modelconsidered by LWW. He finds optimal values of the partial adjustment coefficient between 0.76and 1 depending on the weight on output versus inflation volatility. These results indicate thatthe smooth interest rate changes induced by a high degree of partial adjustment are optimal evenwhen no constraint on interest rate volatility is imposed.

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Figure 3Forward-Looking Expectations and Interest Rate SmoothingLevin, Wieland, Williams (1998)Policy Frontiers under Alternative Constraints on Interest VolatilityStandard Deviation of the Output GapSimple Rules with Partial Adjustment4.03.53.0Standard Deviation of theChange of the Funds Rateσ( ∆ r ) = 1.2σ( ∆ r ) = 2.4σ( ∆ r ) = 3.72.52.01.51.01.31.51.7 1.92.12.32.5Standard Deviation of Inflation13

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Figure 4Data Uncertainty and Interest Rate SmoothingOrphanides (1998b)Policy Frontiers in a Small Backward-Looking Model4.0Standard Deviation of the Output Gap3.5Without Measurement ErrorWith Measurement Error (Naive Control)3.02.52.0Actual Outcome(1980-1992)1.51.01.01.52.02.53.03.54.0Standard Deviation of Inflation 17

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Figure 5Data Uncertainty and Interest Rate SmoothingOrphanides (1998b)Simple Rules and Optimal Control with Data Uncertainty4.0Standard Deviation of the Output Gap3.5Naive ControlEfficient Simple RulesOptimal Control3.02.52.0Actual Outcome(1980-1992)1.51.01.01.52.02.53.03.54.0Standard Deviation of Inflation 19

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Up to this point, the models discussed assume that policymakers have the luxury of

knowing the structure of the economy with certainty. In fact, parameters from economic modelsare estimated imprecisely and typically vary unpredictably through time. Moreover, policymakersmay not even know the correct structure of the model. As a result, monetary policy must beimplemented in the presence of a considerable amount of uncertainty, without full confidenceabout the effect of interest-rate choices on macroeconomic variables such as the output gap orinflation. Although the exact impact of parameter uncertainty on the optimal policy choice candepend on specific modeling assumptions, a general result tends to emerge: Parameter uncertaintyadds caution to the optimal policy rule. By responding more tentatively to macroeconomicdevelopments, policymakers can limit the undesired impact that their policy may have on outputand inflation due to incorrect estimates of the model’s parameters.Some Background

Theoretical analysis of optimal policy under parameter uncertainty dates back to Brainard

(1967). More recent papers include Wieland (1998), who analyzes optimal policy underuncertainty about the natural rate of unemployment and the slope of the short-run Phillips curve,and Sack (1998b), who investigates the impact of uncertainty about the policy multiplier. Bothtypes of uncertainty lead to interest-rate movements that often are not large enough to correct theentirety of the expected output gap or the deviation of inflation from its target. In the words ofAlan Blinder (1995), optimal policy involves “a little stodginess” in which “the central bankshould calculate the change in policy required to get it right and then do less.” The reason is thataggressive interest-rate changes typically increase the uncertainty in the response of output and

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inflation when parameters are unknown.

To see this, consider the optimal policy response in a situation where output is currentlyabove the level consistent with the policymaker’s objectives. If the negative response of outputto the interest rate were known with certainty, as shown in the top panel of Figure 6, the optimalpolicy decision would be trivial: increase the interest rate by the amount required to reach thedesired level of output, shown in the figure by the interest rate i1.

The presence of uncertainty need not materially complicate this decision. If uncertaintyenters additively--say through unpredictable variation in the level of household spending--theaggregate demand relationship would shift about in a parallel fashion, as in the middle panel ofthe figure. The expected level of output is indicated by the solid line, but the actual output levelmay be above or below the expected level, as shown by the dashed lines. If the policymakerimplements the same interest rate change to i1, output will reach the desired level on average, butthe actual outcome could lie anywhere in the range indicated by the thick line segment at i1.This uncertainty leaves the policymaker worse off than in the case with no uncertainty, as therealized level of output will most likely differ from the level consistent with the policymaker’sobjectives. However, the policymaker cannot reduce the magnitude of the potential outputdeviation by moving less, for example to the interest rate i2. Additive uncertainty does not affectthe optimal policy decision because the uncertainty about the response of output is independentof the policy choice. This is the certainty equivalence result that was discussed in the context ofdata uncertainty.

Unfortunately, the uncertainty confronting policymakers is more pervasive than assumedin the case of additive uncertainty, considerably complicating the policy choice. In particular, the policymaker may also be uncertain about the slope of the aggregate demand relationship, asshown in the bottom panel of Figure 6. The expected shape of the curve given the policymaker’sestimates of the parameters is again shown by the solid line, but because of the imprecision inthose parameter estimates, the level of output observed at the current interest rate could haveinstead resulted from an alternative set of parameter values, such as those generating the twoschedules shown by the dashed lines in the figure. The important difference from the case of

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Figure 6Parameter Uncertainty and Interest Rate Smoothing InterestNo UncertaintyRatei1Current iDesired YCurrent YOutputInterestAdditive UncertaintyRatei1i2Current iDesired YCurrent YOutputInterestParameter UncertaintyRatei1CAi2DBCurrent iDesired YCurrent YOutput23

Policy decisionis trivial: increasethe interest rate enough to reach thedesired level ofoutput.Uncertainty aboutthe response of output is independent of theinterest rate change.Uncertainty aboutthe response of output increases with the interest rate change.additive uncertainty is that the uncertainty in the response of output now increases with themagnitude of the interest rate change. An aggressive interest rate movement, for example to i1,could leave output far away from the desired level if the sensitivity of output to interest ratechanges turned out to be much different than believed. Reducing the size of the interest rateresponse to i2 significantly reduces the size of the output deviation if output turns out to be moresensitive to the interest rate than believed (point B instead of point A) and only leaves a slightlylarger output deviation if output is instead less sensitive to the interest rate (point D instead ofpoint C). As a result, implementing a more moderate interest rate change reduces the amount ofuncertainty in the response of output, as indicated by the narrower band of potential outcomes.Parameter uncertainty therefore creates a trade-off to be considered in the optimal policydecision: More aggressive interest rate changes will move output closer to the desired level onaverage but will generate more uncertainty in those outcomes. This is analogous to the problemfacing an investor choosing between a safe and a risky asset. The investor may be willing to holdsome of the safe asset even though it has a lower expected return in order to reduce theuncertainty of the payout. Similarly, the optimal policy may damp interest rate changes, decidingnot to move the full extent required to reach the desired level of output in expected terms so asto reduce the range of potential deviations in output.

Of course, the degree of uncertainty that policymakers face will likely change over time.Figure 7 extends the static results to a dynamic setting, showing that parameter uncertainty mayresult in gradual and persistent movements in the interest rate. The initial response of the interestrate to macroeconomic developments will be damped for the reasons discussed above, reachingonly i2 for example, but the degree of uncertainty may subsequently decline as the policymakerlearns about the impact of interest rate changes. That is, the fan-shaped region of uncertainty

shifts up and narrows around the current interest rate as information about the transmission ofpolicy begins to accumulate, as shown in the top panel of the figure. Having observed thereaction to the initial policy move to i2, the policymaker faces a narrower range of potentialoutcomes around the desired level of output from reaching i1 in a second policy step, comparedto implementing a single, large policy move as shown in the bottom panel. Even if the

In addition, the uncertainty may also be reduced if the initial policy change pushes the economy

into a region where the effect of additional policy moves has been estimated more precisely.

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Figure 7Parameter Uncertainty in a Dynamic Context Two Policy StepsInterestRatei1i2Current iDesired YCurrent YOutputSingle Policy StepInterestRatei1Current iDesired YCurrent YOutput25

uncertainty does not decline much after the initial policy move, the optimal interest rate responsemay still be very smooth and persistent. In particular, the policymaker may optimally choose notto push the interest rate as high as i1 because of the uncertainty in the response of output at thatinterest rate level, but may instead implement a more limited but persistent increase in theinterest rate, one that moves inflation more slowly towards its desired level by sustaining theoutput gap for a longer period of time. Some Results

Recent empirical evidence regarding the importance of parameter uncertainty as arationale for interest-rate smoothing has been provided by Estrella and Mishkin (1998),Rudebusch (1998), and Sack (1998a). By taking as given a specific model of the economy andspecifying a set of goals for the central bank, these papers calculate the optimal path for thefederal funds rate. In the following we present results taken from Sack (1998a), who calculatesthe optimal funds rate policy that minimizes a weighted sum of deviations of the unemploymentrate from an equilibrium level and inflation from a target, given the dynamic behavior of thesevariables as summarized in a vector autoregression (VAR). The optimal policy that is calculatedis not a simple policy rule as equation (1), but instead determines the interest rate in responseto all current and lagged variables in the VAR. This policy will depend on lagged values of theinterest rate because previous policy choices continue to affect the targeted variables due to thesubstantial lags in the impact of monetary policy.

This optimal policy is compared to the actual setting of the federal funds rate, which isestimated by one of the equations in the VAR. The monthly interest rate changes implementedunder this estimated policy rule over the period 1984 to 1998 are shown in the top panel ofFigure 8. Many of the characteristics of interest-rate smoothing described in the introduction areevident: Interest rate changes tend to be limited in size, and successive changes often have thesame sign. As indicated to the right of the graph, the standard deviation of monthly interest ratechanges is 15 basis points, and 79 percent of these changes were continuations in the directionof policy.

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The estimated policy can be compared to the optimal policy to determine whether theobserved degree of smoothing is appropriate. To do so, the analysis calculates the interest ratechanges that would have been realized had the optimal policy rule been implemented over thisperiod, assuming that the economy was subjected to the same set of shocks. The second panelshows the results from this exercise under the assumption of additive uncertainty, in which thepolicymaker takes the estimated VAR coefficients to represent the true values of the parametersand assumes that uncertainty enters only through an additive disturbance--the error terms in theVAR equations. In this case, the optimal policy adjusts the interest rate sharply in response tomacroeconomic developments, which more than doubles the volatility of interest rate changes andlimits the tendency for the interest rate to move in a sequential steps in a particular direction.These results indicate that when the relationship between the interest rate and the targetedvariables is fixed and perfectly known, the optimal policy can more effectively stabilize thetargeted variables by implementing more aggressive interest rate changes than observedhistorically.

Once the policymaker considers the full scope of the uncertainty surrounding the

estimated dynamics of the economy, the policy choice must be adjusted for the fact that thecoefficients of the VAR are estimated imprecisely. In particular, the uncertainty surrounding theresponse of unemployment and inflation increases the further that the actual interest rate deviatesfrom the level predicted by the historical sample. Intuitively, the policymaker has the mostprecise estimates about the effect of the interest rate on the economy when the contemplatedpolicy action is similar to those in the sample over which all the coefficients were estimated. Anoutsized policy move by that standard implies that many of the observations used to estimate thecoefficients are not relevant, so that the response of the targeted variables is more uncertain.Thus, the high degree of activism found to be optimal under additive uncertainty has to betempered in the presence of parameter uncertainty.

To account for parameter uncertainty, Sack calculates the policy that is optimal once thepolicymaker considers the imprecision of the parameter estimates as measured by the variance-covariance matrix of the VAR coefficients. The optimal policy under parameter uncertainty

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implements the interest rate changes shown in the third panel. As evident in the figure, parameteruncertainty dampens the responsiveness of the interest rate, substantially reducing the volatilityof interest rate changes and leading to more continuations in the direction of policy movements.The optimal policy under parameter uncertainty therefore captures several of the prominentcharacteristics of interest-rate smoothing.

Recent empirical work by Rudebusch (1998) and Estrella and Mishkin (1998) differssomewhat from the results reported here. These authors conclude that the extent of parameteruncertainty that they have identified would not account for the observed degree of timidity in theFederal Reserve’s response to output and inflation, finding a smaller effect of parameteruncertainty and less inertia in the interest rate under the optimal policy. Several factors mayaccount for the discrepancy from the above results. First, these papers focus on quarterly andannual movements in the interest rate and are estimated across much longer samples. Second,these papers evaluate the effect of parameter uncertainty on simple policy rules in models withmore limited dynamics, whereas the above results consider optimal policy rules that account forthe rich dynamics of the VAR. Finally, these studies consider uncertainty about fewer parameters,which limits some of the covariance between parameters that may be driving the interest-ratesmoothing.Extensions

Empirical papers using estimated models such as Sack (1998a), Rudebusch (1998), andEstrella and Mishkin (1998) have measured uncertainty by the estimated variance of modelparameters for a given sample period; that is, they have treated the degree of uncertainty as fixed.In practice, however, policymakers’ beliefs about relevant structural parameters as well as theprecision of available estimates will vary over time, particularly if the unknown parametersthemselves are changing. This has been recognized in recent work using theoretical and calibratedmodels that incorporate learning by the policymaker (see Sack (1998b) and Wieland (1996) and(1998)). Although the impact of learning and time-varying parameters on the optimal degree ofinterest-rate smoothing has not yet been investigated empirically, several important considerations

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emerge from these models. First, because the degree of uncertainty about relevant parametersvaries over time, the incentive for cautious policymaking is not constant. Furthermore, if the trueunderlying parameters are in fact time-varying, past observations should be discounted relativeto more recent observations, which may increase the importance of the lagged interest rate in asimple policy rule. In particular, uncertainty about the effects of policy will tend to be minimizednear the lagged interest rate, since the most relevant information is provided by recent outcomesnear this level. More generally, even though the structure of the parameter uncertainty may berather complicated, the lagged interest rate may capture much of its effect in a simple policy rule,because it represents the optimal decision implemented under a similar degree of uncertainty inthe previous period.

One should also note that there are circumstances when parameter uncertainty may justifysome activism. For example, policymakers may wish to implement larger interest-rate movementsto more effectively learn about relevant parameters, an effect referred to as experimentation.Hence, in models that incorporate learning, the policymaker typically faces a tradeoff betweencaution and experimentation. Yet even with experimentation, the optimal policy is still found tobe less aggressive in most situations than a policy that disregards parameter uncertainty.

Finally, it would also be interesting to consider the impact of uncertainty over thestructure of the model itself, as opposed to the parameters in a given model. Policymakers mayface greater uncertainty than captured in the above empirical studies because of uncertainty aboutthe specification of structural relationships and the possibility of structural changes in thoserelationships. To address this issue, Levin, Wieland, and Williams (1998) have analyzed theperformance of simple policy rules across four different models. In general, they find that a rulewith interest-rate smoothing performs well in all four models considered, so that such a rule maybe robust to model uncertainty in addition to parameter uncertainty.

An alternative approach to studying efficient policy under model uncertainty is pursuedby Sargent (1998), Onatski and Stock (1998), and Tetlow and von zur Muehlen (1999). Thesepapers apply robust control techniques to design interest rate policies that not only perform wellif the reference model turns out to be close to the true economy, but that are also reasonablyrobust to model misspecification. One particularly interesting finding from this research is thata more aggressive policy stance may be called for in the presence of model uncertainty whenpolicymakers are guarding against worst-case alternatives.

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Summary

Parameter uncertainty may provide an incentive to implement gradual interest-ratemovements in order to limit any undesired impact on the economy arising from incorrectparameter estimates. Recent empirical evidence shows mixed results about the importance of thiseffect, but at least in some cases parameter uncertainty has been shown to have a significanteffect on the optimal policy. In particular, Sack’s analysis based on a VAR model shows that asubstantial degree of interest-rate smoothing at a monthly frequency can be attributed to anoptimal policy response in the presence of parameter uncertainty. Moreover, models that involvestochastic parameters and learning by the policymaker are likely to attribute a more importantrole to partial adjustment of the interest rate under simple policy rules than has been suggestedby existing empirical studies on the impact of parameter uncertainty.

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