Keywords: Technology shocks, monetary policy rules, natural rate of output, natural rate of interest.
Abstract:
JEL Code: E5
This paper examines welfaremaximizing monetary policy in an estimated dynamic stochastic general equilibrium model of the U.S. economy where the central bank faces uncertainty about the values of model parameters. In this framework, parameter uncertainty implies uncertainty not only about the dynamics of the economy, but also about the central bank loss function and the natural rates of interest and output. Household welfare is maximized when output equals its "natural rate," i.e. the value that would obtain absent nominal rigidities, at which point the real interest rate equals its corresponding natural rate. These natural rates change over time in response to shocks and their dynamic behavior depends on the model parameters describing preferences and technology. Owing to the presence of sticky prices and wages, the firstbest outcome is not attainable and the central bank faces a trade off between minimizing deviations of output from its natural rate, the "output gap," and minimizing fluctuations in price and wage inflation, where the relative weights on the three objectives depend on the model parameters. Thus, the model parameters jointly determine the dynamics of the economy, the dynamics of the natural rates, and the weights in the welfaremaximizing central bank's objective function.
We analyze the implications of parameter uncertainty on the design of implementable optimal monetary policies and, in particular, how it affects the usefulness of measures of the natural rates of output and interest in determining policy decisions. This paper contributes to the large literature, going back at least to Brainard (1967), that examines the design and performance of monetary policies when model parameters (including natural rates) are uncertain. Most of the past research has been conducted with an ad hoc policy objective and has treated movements in natural rates as exogenous and not directly related to parameter uncertainty (see, for example, Rudebusch 2001, Orphanides and Williams, 2002, and references therein). In such a setting, natural rate uncertainty is a form of additive uncertainty and, under certain stringent conditions, has no implications for the design of optimal monetary policy. In particular, if the data generating process for the natural rates is known, then both the separation principle and certainty equivalence hold. In that case, the optimal estimates of the natural rates are inserted into the optimal policy rule and the parameters of the optimal policy are unaffected by natural rate uncertainty. More generally, this literature has found that in the context of simple, that is, not optimal, monetary policy rules, monetary policy should respond less to variables related to natural rates like the output gap and the natural rate of interest and more to indirect measures, such as the rate of inflation and the rate of output growth (see, for example, Orphanides and Williams, 2002).
In contrast to the assumptions underlying much of the previous research on this topic, in microfounded models, parameter uncertainty regarding the "slopes" of macroeconomic relationships also affects the dynamic responses of natural rates to shocks. Thus, natural rate uncertainty is intrinsically connected to parameter uncertainty and certainty equivalence does not apply. In recent papers, Giannoni (2002), Levin and Williams (2005), and Levin, Onatski, Williams and Williams (2005; henceforth LOWW) have explored aspects of monetary policy under parameter uncertainty in microfounded models where the central bank aims to maximize household welfare. However, these papers do not explicitly analyze the usefulness of natural rates as guides for policy. For example, LOWW analyzes monetary policy rules that do not respond to natural rates at all.
We use a small estimated microfounded model as a laboratory to explore how parameter uncertainty and the associated uncertainty about natural rates affects the design of optimal monetary policy. We use the estimated covariance of model parameters as a measure of parameter uncertainty. We analyze the implications of parameter uncertainty on policy design and outcomes, from the perspective of a Bayesian policymaker who aims to maximizes expected household welfare. We first show that parameter uncertainty implies a nontrivial degree of uncertainty about the natural rates of output and interest and that natural rate misperceptions on the part of the central bank are likely to be persistent. We then show that under parameter uncertainty, optimal Bayesian policies rely less on estimates of the output gap, and more on prices and wages than would be optimal if natural rates were known. We also find that in the presence of parameter uncertainty, policies that respond to the level of labor hours do as well or better than policies that respond to the gaps between hours or output and their respective natural rates. Despite the very different analytical framework used in this paper compared to much of past research, our main qualitative results are similar to those in the previous literature; that is, natural rates may be unreliable guides for policy and a more robust strategy is to respond to other, potentially better measured, variables.
The remainder of the paper is organized as follows. Section 2 describes the microfounded model that we use for our analysis. Section 3 describes the model estimation and reports the results. Section 4 examines optimal monetary policy assuming model parameters are known. Section 5 considers optimal policy under parameter uncertainty. Section 6 concludes.
In this section, we describe a small closedeconomy dynamic stochastic general equilibrium (DSGE) model that we use for monetary policy evaluation. In the model, households choose consumption and set wages for their differentiated types of labor services, and firms produce using a CES aggregate of households' labor services as input and set prices for their differentiated products. The dynamics of nominal and real variables are determined by the resulting firstorder conditions of optimizing agents. We allow for various frictions such as habit formation and adjustment costs that interfere with instantaneous full adjustment of quantities and prices in response to shocks. We analyze two sources of aggregate disturbances: shocks to monetary policy and aggregate technology. We begin by presenting preferences and technology and then describe firms' and households' optimization problems. We loglinearize the equations describing the dynamic behavior of the economy, as described in Appendix A; throughout the following, we denote the log of variables by lower case letters.
In order to make the analysis as tractable as possible, we have chosen to specify a relatively simple model of the economy that abstracts from many features present in recentlydeveloped larger DSGE models, such as investment, partial wage and price indexation, fiscal policy, and international trade (see, for example, Christiano, Eichenbaum, and Evans (2005), Smets and Wouters (2003), and Lubik and Schorfheide (2005)). The primary purpose of the modelbased monetary policy evaluation in this paper is to illustrate the connections between parameter and natural rate uncertainty and monetary policy design. We leave for future work the development and analysis of a more comprehensive model of the U.S. economy that is better suited to provide concrete quantitative guidance regarding the effects and implications of parameter uncertainty.
The economy's final good, , is produced according to the DixitStiglitz technology,
Final goods producers obtain their differentiated production inputs used in production from the economy's differentiated intermediate goods producers who supply an output . Not all of the differentiated output produced by the intermediate goods producers is realized as inputs into final goods production; some is absorbed in price formulation, following the adjustment cost model of Rotemberg (1982). Specifically, the relationship between and is given by,
Our choice of quadratic adjustment costs for modeling nominal rigidities contrasts with that of many other recent studies, which rely instead on staggered priceand wagesetting in the spirit of Calvo (1983) and Taylor (1980). We prefer the quadratic adjustment cost approach over staggered price and wagesetting because the latter imply heterogeneity among agents. Partly for this reason, models utilizing staggered price and wage setting typically assume that utility is separable between consumption and leisure, in which case perfect insurance among households against labor income risk eliminates heterogeneity of their spending decisions. By contrast, if wages are staggered and household utility is nonseparable, differences across households in labor supply (which will result due to differences in wages set) lead to differences across household in the marginal utility of consumption (and hence consumption), even if perfect insurance is able to equalize wealth across households. The quadratic adjustment cost model allows us to avoid heterogeneity across agents. In any case, the resulting price and wage inflation equations are very similar to those derived from Calvobased setups as in Erceg, Henderson, and Levin (2000).
The differentiated intermediate goods, for , are produced by combining each variety of the economy's differentiated labor inputs that are supplied to market activities (that is, for ). The composite bundle of labor, denoted , that obtains from this aggregation implies, given the current level of technology , the output of the differentiated goods, . Specifically, production is given by,
Households derive utility from their purchases of the consumption good and from their use of leisure time, equal to what remains of their time endowment after hours of labor are supplied to nongratifying activities. We assume the household members live forever and there is no population growth. Its preferences exhibit an endogenous additive habit (assumed to equal a fraction of its consumption last period) and are nonseparable between consumption and leisure.^{1}Specifically, preferences of household are given by
Nongratifying activities include supplying hours to the labor market and devoting time to setting wages. Consequently, we define as
The final goods producing firm, taking as given the prices set by each intermediategood producer for their differentiated output, , chooses intermediate inputs, , so as to minimize the cost of producing its final output , subject its production technology, given by equation (1). Specifically, the competitive firm in each sector solves
Each intermediate firm chooses the quantities of labor that it employ use for production and the price that it will set for its output. It is convenient to consider these two decisions as separate problems. In the first step of the problem firm , taking as given the wages set by each household for its variety of labor, chooses to minimize the cost of attaining the aggregate labor bundle that it will ultimately need for production. Specifically, the materials firm solves:
In setting its price, , the intermediate good producing firm takes into account the demand schedule for its output that it faces from the final goods sectors and the factas summarized in equation (2)that by resetting its price it reduces the amount of its output that it can sell to final goods producers. The intermediategood producing firm , taking as given the marginal cost for producing , the aggregate price level , and aggregate finalgoods demand , chooses its price to maximize the present discounted value of its profits subject to the cost of resetting its price and the demand curve it faces for its differentiated output. Specifically, the firm solves,
subject to  
and  (9) 
The household taking as given the expected path of the gross nominal interest rate , the price level , the aggregate wage rate , its profits income, and its initial bond stock , chooses its consumption and its wage to maximize its utility subject to its budget constraint, the cost of resetting its wage, and the demand curve it faces for its differentiated labor. Specifically, the household solves:
subject to  
(10) 
The nonstochastic steady state is summarized by the steadystate levels of the real interest rate and hours. The steadystate oneperiod real interest rate is given by:
(11) 
(12) 
The model has a counterpart in which all nominal rigidities are absent, that is, prices and wages are fully flexible. In this model the cost minimization problems faced by the final goods producing firm and the intermediate goods producing firms continue to be given by equations (7) and (8). The intermediate goods producing firms' profit maximization problem is similar to equation (9) but with the price adjustment cost parameter set to zero. Likewise the households' utility maximization problem is given by equation (10) but with the wage adjustment cost parameter set to zero. We refer to the level of output and real oneperiod interest rate in this equilibrium as the natural rate of output, , and interest, . We also define log deviations of these variables from their steadystate values, and . These natural rates are functions of our model's structural shocks and are derived in Appendix A.
Our complete model consists of the firstorder conditions (derived in Appendix A) describing firms' optimal choice of prices and households' optimal choices of consumption and wages, the production technology (3), the monetary policy rule, the market clearing conditions and , and the law of motion for aggregate technology (4). We now turn to the parametrization of our model.
In order to analyze optimal Bayesian monetary policy under parameter uncertainty, we need a posterior distribution of the model parameters. One approach to obtaining a posterior distribution, consistent with the Bayesian approach to decisionmaking assumed for the policymaker, is to estimate the model using Bayesian methods, as is done in LOWW. This approach necessitates making specific assumptions regarding the prior joint distribution of the model parameters. Because we want to avoid having the choice of the prior distribution overly influence our results, we instead follow a limitedinformation approach to estimating the posterior distribution of the model parameters.^{3} In particular, we estimate several of the structural parameters of our model using a minimum distance estimator based on impulse responses to monetary policy and technology shocks.
Specifically, we estimate a VAR on quarterly U.S. data using empirical counterparts to the theoretical variables in our model, and identify two of the model's structural shocks using identifying assumptions that are motivated by our theoretical model. We then choose model parameters to match as closely as possible the impulse responses to these two shocks implied by the model to those implied by an structural VAR.^{4} In this section we first describe the VAR and the identification of the two shocks, and then discuss our parameter estimates.
The specification of our VAR is determined by the model developed in the previous section and our identification strategy for the structural shocks. Concerning the latter, we follow Galí (1999) and assume that the technology shock is the only shock that has a permanent effect on the level of output per hour. The monetary shock is identified by a standard restriction on contemporaneous responses. Our model and identifying assumptions combined suggest the inclusion of five variables in the VAR: the first difference of log output per hour, price inflation (the first difference of the log of the GDP deflator), the log labor share, the first difference of log hours per person, and the nominal funds rate. Output per hour, the labor share, and hours are the Bureau of Labor Statistics' (BLS) measures for the nonfarm business sector, where the labor share is computed as output per hour times the deflator for nonfarm business output divided by compensation per hour.^{5} Population is the civilian population age 16 and over. Letting denote the vector of variables in the VAR, we view the data in the VAR as corresponding, up to constants, to the model variables
The dashed lines in the panels of Figure 1 show the impulse responses to a permanent one percent increase in the level of technology. The dasheddotted lines present onestandard deviation bands around the impulse responses, computed by bootstrap methods.^{7} Upon impact, output immediately rises about halfway to its new steadystate level, whereas hours worked decline by about 1/4 percent. Over the following eight quarters, output completes its adjustment while hours worked return to their original level. Interestingly, the response of inflation to a technology shock suggests only a limited role for price stickiness, with inflation declining upon impact by almost a percentage point. Wage rigidity, by contrast, seems to be more important, as the initial response of the real wage is driven by the initial price response, not nominal wage adjustment. The real wage completes its adjustment over the following two quarters. The estimated response of monetary policy is to accommodate the increase in output by keeping the real funds rate on balance unchanged.
Figure 2 shows the impulse responses of the variables to a one percentage point positive funds rate shock. The estimated responses of output, the real wage, and inflation to a funds rate shock are consistent with many studies on the effects of monetary policy. Output falls within three quarters by about 3/4 percent in response to one percentage point increase in the funds rate that takes eight quarters to die out. Hours decline closely in line with output, and the real wage falls. The response of inflation exhibits a price puzzle that lasts for two quarters; thereafter, inflation declines for six quarters, to about 0.3 percent below its original level. The responses to a funds rate shock are more precisely estimated than the responses to the technology shock.
With the VAR impulse responses to a funds rate shock and a technology shock in hand, we proceed to estimate the structural and monetary policy parameters of our model. First, we calibrate four model parameters that have little effect on the dynamic responses to shocks. We set the discount factor, , which corresponds to discounting the future at a 3 percent annual rate. We normalize the time endowment to unity. We set the steadystate rates of price and wage inflation to zero. Finally, we set both aggregation parameters and to 6, following LOWW (2005).
The remaining parameters are estimated by minimizing the squared deviations of the responses of the five variables implied by our model from their VAR counterparts. The IRFs of these five variables in quarters 0 through 8 following a technology shock in quarter 0, and in quarters 1 through 8 following a funds rate shock (the response in the impact quarter being constrained by the identifying assumption) provide a total of 85 moments to match. These moments are weighted inversely proportional to the standard error around the VAR responses, as in Christiano et al. (2005). This has the effect of placing more weight on matching the impulse responses to the monetary shock, which, as noted before, are estimated with greater precision than the impulse responses to the technology shock.
For purposes of model estimation, we assume that monetary policy is set according to a simply policy rule in which the interest rate depends on the lagged interest rate and the current inflation rate only
where is an i.i.d. monetary policy shock.^{8} Note that we have suppressed the constant that incorporates the steadystate levels of the interest and inflation rate.In addition, because the parameters and appear only as a ratio in the linearized version of the model (see Appendix A), they are not separately identified; the same is the case for the parameters and . We therefore estimate the ratios and . Note that and are equal to the coefficients on the driving process in the wage and price Phillips curves, respectively. In the end, we estimated seven free parameters: .
The estimated parameters and associated standard errors are shown in the first two columns of Table 1. The correlation coefficients of the structural parameter estimates are shown in the final five columns of the table. The covariance matrix of the estimates is computed using the Jacobian matrix from the numerical optimization routine and the empirical estimate of the covariance matrix of the impulse responses from the bootstrap. The estimates of the structural parameters are all statistically significant, with the preference parameters, especially and , relatively imprecisely estimated, while those associated with wage and price adjustment costs are estimated with a great deal of precision.
Model Parameter 
Point Estimate 
Standard Error 
Corr. 
Corr 
Corr. 
Corr.

Corr.


5.365  2.372  1.000  0.996  0.995  0.960  0.495  
0.389  0.094  1.000  0.990  0.947  0.491  
1.186  0.386  1.00  0.971  0.478  
0.004  0.001  1.000  0.436  
0.041  0.001  1.000  
0.809  0.001  
1.004  0.027 
As discussed before, one feature of both sets of impulse responses is that real output and hours adjust gradually in response to the shocks. In the case of a permanent technology shock, Rotemberg and Woodford (1996) demonstrated that DSGE models without intrinsic inertia will not display such humpshaped patterns; instead, these variables jump on impact and adjust monotonically to their new steadystate values. We therefore find a significant role for habit persistence. Our estimate of the habit parameter is somewhat smaller than those estimated by Fuhrer (2000), Smets and Wouters (2003) and Christiano et al (2005), but slightly larger than that estimated by LOWW (2005). The estimate of is higher than typical estimates based on macroeconomic data, but this estimate is very imprecise.
As noted before, the VAR responses of real wages and inflation differ substantially depending on the source of the shock: rapid responses to technology shocks, and sluggish ones to funds rate shocks. This is a feature that our price and wage specification cannot deliver. Our estimates of and imply that wages are very slow to adjust, but prices adjust relatively rapidly to fundamentals. The evidence for relatively flexible prices comes from the IRFs to the technology shock; indeed, the IRFs to monetary policy shocks alone suggest very gradual price adjustment, consistent with the findings of Christiano et al (2005). Despite the greater weight placed on matching the more tightly estimated responses of inflation and real wage to the funds rate shock, our model does better at matching the responses to a technology shock, as shown by the solid lines in figures 1 and 2. Our estimates of the parameters of the monetary policy rule, and , are broadly consistent with the findings of many other studies that estimate monetary policy reaction functions, such as that of Clarida, Galí, and Gertler (2000).
In this section we compute the optimal policy response to a technology shock assuming all model parameters are known. We assume that the central bank objective is to maximize the unconditional expectation of the welfare of the representative household. We further assume that the central bank has the ability to commit to future policy actions; that is, we examine optimal policy under commitment, as opposed to discretion. We consider only policies that yield a unique rational expectations equilibrium.
By focusing only on technology shocks, we are arguably examining only a relatively small source of aggregate fluctuations in output and wage and price inflation and hence welfare losses. For example, LOWW (2005), using a mediumscale DSGE model, find that other shocks, especially those to price and wage markups, have much larger effects on welfare than technology shocks. In order to conduct welfarebased monetary policy analysis incorporating other sources of fluctuations, we would need to take a stand on the precise source and nature (i.e., distortionary vs. fundamental) of the other shocks to the economy, as discussed in LOWW (2005). This issue remains controversial and would take us afield of the primary purpose of the paper, and we therefore leave it to further research. Nonetheless, we recognize that by abstracting from other shocks, our quantitative results regarding welfare costs under alternative policies likely dramatically understate those that would obtain if we included a full specification of all shocks that impact the economy.
As is now standard in the literature, we approximate household utility with a secondorder Taylor expansion around the deterministic steady state. We denote steadystate values with an asterisk subscript. As shown in Appendix D, the secondorder approximation of the period utility function depends on the squared output gap (the log difference between output and its natural rate, that is ), the squared quasidifference of the output gap, the crossproduct of the output gap and its quasidifference, and the squared price and wage inflation rates. As shown in the appendix, in the linearized model, the natural rate of output, , is a function of leads and lags of the technology shock.
After numerous manipulations, the secondorder approximation to period utility can be written as
and where T.I.P refers to terms that are independent of monetary policy, which we ignore in our welfare calculations. The terms , , and defined asTable 2 reports the implied relative weights on the terms related to the output gap, wage inflation, and price inflation.^{10} The first row reports the sum of the weights on the three terms in the loss associated with the output gap and its quasidifference.^{11} For this table, we have normalized the values of the weights by the weight on price inflation evaluated at the parameter point estimates. The first column reports the weights based on the parameter point estimates. The second column reports the mean values of the weights based on the estimated distribution of the parameter values, approximated using 1000 draws from the normal distribution with the estimated covariance for the parameter estimates, where we truncate the parameter values at the lower ends of their distributions as follows: at 0.5, at 0.1, and at 0. The third column reports the corresponding standard deviations of the weights. The final three columns report the crosscorrelation of the weights.
Weight in Loss 
Point Estimate 
Mean Value 
Standard Deviation 
Corr.

Corr.

Corr.


0.14  0.38  0.49  1.000  0.996  0.996  
1.00  2.31  2.86  1.000  0.989  
10.56  29.09  38.11  1.000 
Based on the point estimates, the variance in wage inflation gets a weight of over 10 times that of price inflation in the welfare loss owing to the estimated value of being one tenth as large as that for . The weights on the variances of the output gap and the quasidifference of the output gap are somewhat smaller than that of inflation, but are somewhat higher than typically seen in the literature owing to our relatively high estimate of . The mean values of the weights exhibit the same pattern, but are between two and three times larger than those based on the point estimates, reflecting the fact that the weights depend in part on the inverse of some parameter values.
The relative weights are highly positively correlated, reflecting the fact that the structural parameter estimates are highly correlated with one another and that each component of the welfare loss depends on the steadystate level of utility. As seen in Table 1, the estimates of and are highly negatively correlated, which is not surprising given that these two parameters enter multiplicatively in the utility of leisure. More interestingly, the estimate of is negatively correlated with the estimated values of and , implying that a large weight on output gap terms is associated with large weights on price and wage inflation terms and vice versa.
To compute the optimal certainty equivalent policy for a given set of parameter values, we maximize the quadratic approximation of welfare subject to the constraints implied by the linearized model. Throughout, in computing the welfare loss we assume a discount rate arbitrarily close to zero, so that we are maximizing the unconditional measure of welfare. We compute the fully optimal policy using Lagrangian methods as described in Finan and Tetlow (1999). We assume that the technology shock is the only stochastic element in the model and calibrate the standard deviation of its innovations to equal 0.64 percentage point. This value is slightly larger than the corresponding estimate in LOWW (2005).
The results under the fully optimal policy are shown in the first column of Table 3. The middle portion of the table shows the welfare loss and the breakdown into its component parts (the components add to the total welfare loss, subject to rounding).^{12} Note that we do not normalize the welfare loss in this table or in those that follow. The lower part of the table reports the resulting unconditional standard deviations of output gap, price and wage inflation rates, and the nominal interest rate. The remaining entries in the table are discussed below.
Optimal Policy  Policy Rule Coefficients (1)  Policy Rule Coefficients (2)  Policy Rule Coefficients (3)  Policy Rule Coefficients (4)  

1.00  1.00  
1000.00  
1.93  
0.10  0.00  94.94  0.00  
2.50  1000.00  
Welfare Losses:  1.758  1.760  1.758  1.760  1.760 
Welfare Losses:  0.002  0.000  0.001  0.003  0.001 
Welfare Losses:  1.618  1.608  1.622  1.612  1.605 
Welfare Losses:  0.138  0.152  0.135  0.145  0.154 
Standard Deviations:  .02  .00  .02  .02  .01 
Standard Deviations:  .19  .19  .19  .19  .19 
Standard Deviations:  .02  .02  .02  .02  .02 
Standard Deviations:  .87  .88  .88  .80  .82 
Under the fully optimal monetary policy, output gap and wage inflation variability are reduced to nearly zero, while some price inflation variability remains. In terms of the annualized rate, the standard deviation of price inflation is 0.8 percentage points, about 10 times greater than that of wage inflation. Note that this policy induces considerable interest rate variability in response to a single source of shocks, with the standard deviation of the nominal interest rate 3.6 percentage points on an annualized basis.^{13} Under the optimal policy, variability in price inflation accounts for most of the welfare loss.
The fully optimal monetary policy can be implemented in a number of equivalent ways if all model parameters are known with certainty. In the presence of parameter uncertainty, we need to restrict ourselves to representations of monetary policy that are constrained by the information set that the policymaker possesses. For this purpose, we choose to study monetary policies in terms of feedback or "instrument" rules where the shortterm interest rate is determined by a small number of observable variables and the coefficients of the policy rule are chosen to minimize the welfare loss. We consider four different specifications of monetary policy, each of which yields welfare very close to the fully optimal policy when all parameters are known.
The general specification is given by a Taylortype monetary policy rule where the nominal interest rate is determined by the central bank estimate of the natural rate of interest, , the central bank estimate of the output gap, , the level of hours, , and the rates of price and wage inflation:
(14) 
We start by considering a textbook Taylor rule with a unit response to the natural rate of interest and a free coefficients on the rates of price inflation and the output gap. We optimized the coefficients of this rule to maximize unconditional welfare of the representative household using a numerical hillclimber routine, as described in Levin, Wieland, and Williams (1999). Throughout the following, we restrict policy rule coefficients to be nonnegative and to not exceed an upper bound of 1000.^{14} The results for the optimized Taylor rule are given in the second column in the table. The optimized Taylor rule has a small coefficient on the price inflation rate and the maximal allowable coefficient on the output gap. This rule strives to keep the output gap at zero. The resulting outcome yields a welfare loss nearly identical to the fully optimal policy, with very slightly too much wage inflation variability.
We next consider a variant of the Taylor rule where policy responds to the rate of wage inflation instead of the output gap. This rule features a zero optimal response to price inflation and a moderate response to wage inflation. The resulting outcomes and corresponding welfare are virtually identical to those under the fully optimal policy.
Finally, we consider two alternative specifications of the monetary policy rule that do not depend on estimates of the natural rates of output or interest. Interestingly, in both cases, the optimized versions of these rules nearly match the outcomes under the fully optimal policy. First, we consider a rule that does not respond to the natural rate of interest (except for its longrun mean) or any measure of economic activity, but instead responds only to price and wage inflation. The optimized parameterization of this rule features huge responses to price and wage inflation with the response to wage inflation about 10 times larger than that to price inflation. Second, we consider a rule that responds to price inflation and the loglevel of hours. For this rule, the optimized response to price inflation is zero and that to hours is about 2.
In this section, we analyze the performance and robustness of various monetary policies under parameter uncertainty. We assume that the central bank knows the true model and that the model is estimated using a consistent estimator and that the central bank is certain that the model and the estimation methodology are correct.^{15} The only form of uncertainty facing the policymaker is uncertainty regarding model parameters owing to sample variation. We abstract from learning and assume that the policymaker's uncertainty does not change over time. We assume that private agents know everything, including the central bank's parameter estimates. For a given policy rule, expected welfare is computed by numerically integrating over the distribution of the five estimated structural parameters as measured by the estimated covariance matrix. Note that in these calculations, we fully take into account the effects of parameter values on the parameters of the loss function as in Levin and Williams (2005).
Before proceeding with the analysis of policy rules, we first provide some summary measures of the degree of uncertainty regarding the natural rates of hours, output, and interest owing to parameter uncertainty. In this model, the responses of the natural rates to a technology shock depend on three parameters describing household preferences: , , and . Throughout the following, we assume that the distribution of model parameters is jointly normal distributed with mean zero and covariance given by the estimated covariance matrix. In the following, we approximate this distribution with a large number of draws from the estimated covariance matrix, truncated as described in section 3.
In general, parameter uncertainty implies uncertainty both about the steadystate values of natural rates as well as their movements over time. In the stylized model that we study here, however, the steadystate natural rate of interest depends only on the household's discount rate, which is assumed to be known by the policymaker. Therefore, uncertainty about the natural rate of interest is limited to its deviations from steadystate. The steadystate level of hours depends on estimated structural parameters and the value of the time endowment. Our estimation methodology does not use information on levels of variables, so we do not have an empirical measure of uncertainty regarding the time endowment. For simplicity, we assume that the policymaker, by observing a long time series on hours, is able to estimate the mean level of hours precisely. We assume that policymaker has no independent knowledge of the time endowment, so perfect knowledge of the mean level of hours has no implications for uncertainty about other preference parameters. We note that under less restrictive assumptions, there exist tight links between estimated structural parameters and steadystate values, which affect both model estimation and the analysis of parameter uncertainty.^{16}
Parameter uncertainty implies considerable uncertainty regarding the responses of natural rates to technology shocks. The thin solid line in the upper panel of Figure 3 plots the impulse response of the log of the natural rate of hours to a one percentage point positive permanent technology shock based on the point estimates of the model parameters. (Note that the log of the natural rate of hours equals the log of the natural rate of output minus the log of TFP.) The thick solid line shows the median response calculated from impulse responses corresponding to 10,000 draws from the estimated parameter distribution. The dashed and dasheddotted lines show the boundaries of the 70 and 90 percent confidence bands of the impulse responses, respectively. The lower panel of the figure shows the corresponding outcomes for the natural rate of interest measured at an annualized rate. Note that the model implies that there is no uncertainty about the longrun effects of technology shocks on the natural rates of hours and interest, both of which eventually return to their respective steady state values.
Natural rate misperceptions owing to parameter uncertainty are sizable and persistent. We measure natural rate misperceptions as the difference between the level of the natural rate implied by the model's true parameter values and the level implied by the point estimates of the model parameters.^{17} To compute unconditional moments in our model, we calibrate the standard deviation of innovations to technology to equal 0.64 percentage point, equal to the sample average of the identified shocks from our VAR. The resulting unconditional standard deviation of the difference between the true natural rate of output and the central bank's estimate (based on the model with the parameter point estimates) is 0.32 percentage point. The firstorder autocorrelation of this difference is 0.86. The unconditional standard deviation of the difference between the natural rate of interest and the central bank's estimate is 3.15 percentage points (measured at an annual rate), with a firstorder autocorrelation of this difference equal to 0.50.
In order to provide a benchmark for policies under uncertainty, we first compute the optimal outcome if the policymaker knew all the parameter values and followed the fully optimal policy in each case. We average the outcomes and losses over 1000 draws of the parameters and report the results in the first column of Table 4. Of course, given that the parameters are uncertain, this outcome is not obtainable in practice, but provides a benchmark against which we can measure the costs associated with parameter uncertainty. As can be seen from comparing the first columns of Tables 3 and 4, the mean welfare loss under the firstbest optimal policy is considerably larger than that computed at the parameter point estimates. This reflects the fact that the mean weights in the welfare loss are higher than the weights evaluated at the point estimates. Indeed, under the firstbest optimal policy, the variability of the objective variables is about the same as for the case of the parameter point estimates.
Optimal Policy  Policy Rule Coefficients (1)  Policy Rule Coefficients (2)  Policy Rule Coefficients (3)  Policy Rule Coefficients (4)  Policy Rule Coefficients (5)  

1.00  1.00  0.83  
0.83  1.80  
3.69  2.99  
0.00  75.78  82.58  0.00  14.40  
1000.00  1000.00  210.00  
Welfare Losses:  4.056  4.112  4.082  4.094  4.089  4.065 
Welfare Losses:  0.003  0.050  0.028  0.036  0.026  0.012 
Welfare Losses:  3.775  3.734  3.783  3.760  3.748  3.786 
Welfare Losses:  0.278  0.328  0.272  0.298  0.315  0.267 
Standard Deviations:  .02  .08  .07  .07  .08  .04 
Standard Deviations:  .19  .19  .19  .19  .19  .19 
Standard Deviations:  .02  .02  .01  .02  .02  .02 
Standard Deviations:  .88  .87  .82  .76  1.08  .99 
We now examine the characteristics and performance of the implementable monetary policy rules introduced in the previous section, but now we reoptimize the coefficients to minimize the expected welfare loss under parameter uncertainty. In implementing these rules, we assume that the central bank's estimates of the natural rates of output and interest are computed using the point estimates of the model parameters, but that the actual model parameters and therefore natural rates differ from the values assumed by the policymaker. The central bank is assumed to observe the technology shocks without error since these do not depend on model parameters.
Relative to the case of no parameter uncertainty, the optimized standard Taylor rule under parameter uncertainty responds far less aggressively to the estimate of the output gap. Recall that in the case of known parameters the optimized response to the output gap is 1000 (the imposed upper bound). In contrast, under parameter uncertainty, the optimal response coefficient is less than unity. This reduction in the response to the output gap is a consequence of the mismeasurement of the natural rate of output. Indeed, if the central bank faced parameter uncertainty but somehow knew the true values of the natural rates, the optimized Taylor rule would have the same huge response to the output gap that obtains absent parameter uncertainty and the mean welfare loss would be nearly the same as the benchmark firstbest outcome.^{18} Thus, if natural rates were known, parameter uncertainty would be of little consequence for monetary policy or the mean welfare loss. But, in the presence of natural rate uncertainty, a large response to the output gap generates correspondingly large policy errors. In order to minimize this source of undesired fluctuations, the optimized rule responds much more modestly to the perceived output gap. As a result, this policy does not stabilize the output gap as well as the firstbest policy.
The alternative policy that responds to the natural rate of interest and the rates of wage and price inflation yields a loss nearly identical to the firstbest. As in the case of no parameter uncertainty, this policy rule responds extremely aggressively to the wage inflation rate and less so to the price inflation rate. The ratio of the response to wages to prices is slightly larger under parameter uncertainty than with no uncertainty. Interestingly, this policy does a better job on average of stabilizing the output gap than the optimized Taylor rule that responds directly to he output gap.
Optimized policy rules that respond to wage and price inflation and the natural rate of interest are very effective at minimizing the welfare loss under parameter uncertainty. Evidently, responding to aggressively to the wage inflation rate substitutes for responding to the output gap in this model, at least when evaluated using responses to technology shocks. Eliminating the response to the natural rate of interest causes a small deterioration in performance, as seen by the third alternative policy rule.
The fourth alternative optimized policy rule, that responds to only the level of hours (with the optimized response to price inflation of zero), performs nearly as well as any of the other rules, including those that respond to estimates of the natural rates. Given the emphasis on these natural rate concepts in the literature, this result that natural rates are nearly superfluous for optimal policy under parameter uncertainty may appear somewhat surprising. But, in an environment where natural rates are uncertain, their usefulness as indicators for monetary policy is reduced. Indeed, "imperfect" indicators such as the level of hours can perform as well or better than natural ratebased indicators like the output gap.
Finally, we examine optimized monetary policy rules that combined features of the simple policy rules considered above by responding to all five variables: the natural rate of interest, the output gap, the level of hours, and the rates of price and wage inflation. The results are reported in the final column of Table 4. The optimized version of this rule yields a welfare loss only slightly larger than the artificial firstbest benchmark. The policy features a very aggressive response to wage inflation and a moderate response to the level of hours and relatively muted responses to the natural rate of interest and the output gap.
The optimal policy taking account of parameter uncertainty yields a larger response of the interest rate to a technology shock than the optimal policy under known parameters (the certainty equivalent policy). Figure 4 shows the impulse responses to a one percentage point positive technology shock under the optimal policy assuming known parameters (the solid lines) and the optimal "Bayesian" policy that takes account of parameter uncertainty by minimizing the mean welfare loss. In both cases, the impulse responses are computed using the parameter point estimates. Under the optimal Bayesian policy, the lower initial interest rate causes the output gap to rise above zero, which then gap swings back below zero where it remains for several years. Compared to the certaintyequivalent policy, the Bayesian policy generates greater variability in the output gap, but smaller variability in the wage inflation rate. The variability of the rate of price inflation is nearly identical under the two policies.
Three crucial factors determine the design of optimal monetary policy: the dynamics of the economy, the natural rates of output and interest, and the weights in the central bank objective function. Traditional analysis of monetary policy under uncertainty has treated these three factors as being independent and studied them separately. But, modern microfounded models imply that the structural parameters describing preferences and technology jointly determine all three factors and that parameter uncertainty affects all three factors together. This paper has shown in an estimated microfounded macroeconomic model that uncertainty about the natural rates of interest and output owing to parameter uncertainty has significant implication for the design of optimal monetary policy in the face of parameter uncertainty. In particular, we find that parameter uncertainty implies that monetary policy should be reoriented away from directly responding to measures of the output gap, which are based on naturalrate estimates that are likely measured with error, and toward responding to variables that are not contaminated by such mismeasurement.
This paper has taken a first step at analyzing the implications of uncertainty about natural rates for monetary policy aimed at maximizing household welfare. In order to make the analysis as tractable as possible, we have used a smallscale stylized model and limited our analysis to parameter and natural rate uncertainty that results from sampling variation. As a result, the analysis has likely understated the true degree of uncertainty that central banks face regarding the economy and natural rates. The analysis can be extended to include additional features of the economic landscape and associated uncertainty, including a richer description of the macroeconomy, a wider set of sources of aggregate fluctuations, and uncertainty regarding the structure of the economy. One potentially important direction for future research is the incorporation of imperfect information on the part of private agents as well. As shown by Orphanides and Williams (2006), the combination of imperfect knowledge by both the central bank and the public can amplify the effects of uncertainty.
Notes: The dashed lines show the impulse responses implied by the VAR following an identified technology shock that raises output per hour permanently by 1 percent. The solid lines show the impulse responses implied by the model to a permanent shock to technology that has the same longrun effect on productivity as the technology shock in the VAR. The dasheddotted lines are one standard error confidence intervals around the VAR responses. 

Notes: The dashed lines show the impulse responses implied by the VAR following a one percent funds rate shock. The solid lines show the impulse responses implied by the model to the same shock under the assumption that the contemporaneous response of all variables other than the funds rate is zero. The dasheddotted lines are one standard error confidence intervals around the VAR responses. 

Notes: The thin solid lines show the model impulse responses of the natural rates to a one percentage point positive shock to technology based on the parameter point estimates. The thick solid lines show the median responses computed from the distribution of the parameter estimates. The dashed and dasheddotted lines indicate the corresponding 70 percent and 90 percent confidence intervals, respectively. 

Notes: The solid lines show the model impulse responses to a one percentage point positive shock to technology, where the model parameters equal the point estimates and monetary policy follows the certainty equivalent optimal policy. The dashed lines show the corresponding responses when monetary policy follows the optimal Bayesian policy that takes account of parameter uncertainty. 

The first subsection of this appendix reports the model's nonlinear equations while the second subsection reports the model's loglinear equations. We limit ourselves throughout in reporting just the equations from the symmetric model. The third subsection reports the model's natural rate of output.
The firstorder conditions from the intermediate goods producing firms' costminimization problem (equation 8), the labor demand curve and marginal cost function, are:
The firstorder conditions implied by the intermediate goods producing firm's cost minimization problem, given by equations (15) and (16), loglinearize to
Before concluding this section we note the following about the steadystate solution to the model. We know from equations (16), (17), (19), (20), and (21) that in the steady state:
Since we can rewrite this as:The natural rate of output, i.e. the level of output in the equilibrium with perfectly flexible prices and wages, is determined by the condition that the marginal rate of substitution between consumption and leisure be (up to constants) equal to the marginal product of labor at all dates . In log terms, is determined implicitly by the equation
and  
The natural rate of interest, denoted , is the real rate prevailing in the equilibrium with perfectly flexible prices and wages. Letting denote the expression (37) with substituted for , the Euler equation (28) in this equilibrium can be expressed as
(41) 
As discussed in section 3, we are interested in matching the VAR's impulse responses to two of the structural shocks of our model, a permanent shock to the level of technology, and a transitory shock to the funds rate. To identify these shocks, we use one longrun and one shortrun identifying restriction. The shortrun identifying restriction is the usual one, that the last variable in the VAR (the funds rate) is Woldcausal for the preceding variables. The structural form of the VAR is given by
(43) 
In order to estimate the VAR in structural form, we need a further set of assumptions to justidentify the elements of . We follow Altig et al. (2002) by assuming that the submatrix consisting of columns 24 and rows 24 of is lower triangular. This assumption is without loss of generality as we do not attach any structural interpretation to elements 2 through 4 of . With these assumptions, we estimate the first equation of the structural VAR imposing the longrun restrictions in the manner of Shapiro and Watson (1988) by including contemporaneous and lagged variables of elements 2 through 4 of in firstdifferenced form. To control for simultaneity, we estimate the equation by 2SLS, using a constant and as firststage regressors for elements 2 through 4 of . We then sequentially estimate equations 2 through 4 by IV, using the residuals from the previous regressions as instruments for contemporaneous variables. Equation 5 can be estimated by OLS by virtue of our shortrun identifying assumption.
We modify this identification strategy in one respect. Because, in contrast to Altig et al., our VAR includes hours per capita in first differences, we would like to assure that the longrun response of hours to a technology shock is zero, consistent with the observation that hours worked have remained broadly unchanged despite the secular trend in real wages. When this second longrun restriction is not imposed, the IRF of usually does not integrate to zero. We therefore reorder our vector of endogenous variables to include as the second variable, and apply the ShapiroWatson method to the first two equations. This leaves the interpretation of the first element of unchanged, but the second element is now the only shock that permanently affects hours per capita. Contrary to the findings reported by Francis and Ramey (2005), Laubach and Williams (2006) find that imposing this second longrun restriction can have a substantial effect on the response of hours to a technology shock.
To derive the welfare criterion we first take a secondorder approximation to the withinperiod utility function