Evaluating Interest Rate Rules in an Estimated DSGE Model∗ Vasco Cúrdia

Andrea Ferrero

Andrea Tambalotti

Federal Reserve Bank of New York†

May 1, 2009

Abstract The empirical DSGE literature pays surprisingly little attention to the behavior of the monetary authority. Alternative policy rule specifications abound but their relative merit is rarely discussed. We provide some of this discussion by comparing the fit of a large set of interest rate rules (44 in total), which we estimate as part of a simple New Keynesian model. We find that specifications in which monetary policy responds to inflation and to deviations of output from its efficient level — which would prevail in the absence of distortions — have the worst fit within the set we consider. Policies that respond to measures of the output gap based on statistical filters perform better, but the best fitting rules are those that also track the evolution of the model-consistent efficient real interest rate.

∗

Preliminary version. Please do not quote without permission. The views expressed in this paper do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal Reserve System. † E-mail: [email protected]; [email protected]; [email protected].

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Introduction

Most central banks around the world have a dual mandate: Stabilize inflation and real activity. This dual mandate is explicit and symmetric in the United States, where the Federal Reserve Act (section 2A) instructs the Federal Reserve to “...promote effectively the goals of maximum employment,” and “stable prices.” Even in inflation targeting countries, whose formal mandates tend to focus on inflation, the implementation of monetary policy usually involves balancing this objective with the stabilization of a real criterion. In fact, according to Svensson (2007), “...inflation targeting is never ‘strict’...but always ‘flexible’...”. But what does it mean, in practice, to “promote maximum employment”? While the interpretation of the price stability mandate has become increasingly transparent around the world during the last decade, the definition of the real stability objective remains everywhere vague. In part, this lack of clarity reflects the absence of a consensus in the academic literature and among policymakers. Most economists agree that inflation should be low and stable. Conversely, there is significant divergence on the definition and measurement of the appropriate real target for monetary policy. In applied contexts, full employment (potential) output is usually defined as a smooth trend for GDP, often measured through some filtering or detrending procedure.1 From a more theoretical perspective, the New Keynesian literature suggests that output should be stabilized around its efficient level, which would prevail in the absence of nominal frictions (Woodford, 2003).2 The problem is that potential and efficient output can differ significantly, since the latter incorporates the efficient response of the economy to shocks, and hence might be far from smooth.3 The heterogenity in the proposed definitions of the real objective of monetary policy manifests itself in the normative — which output gap should central banks stabilize? — but also in the positive sphere — which output gap do central banks pay the most attention to? In 1 See, for instance, Orphanides and Van Norden (2002) for a review of several of these statistical procedures. Growth accounting represents another popular approach to the measurement of potential output as a smooth trend (see CBO, 2001). 2 A third approach to the measurement of potential output, which is intermediate between the two described above, involves positing a statistical relationship between inflation and the output gap (a Phillips curve). This relationship then forms the basis for a multivariate Kalman filter to extract potential output (see for example Kuttner, 1994, and Laubach and Williams, 2003). Mishkin (2007) provides an excellent survey of various statistical and model-based methods for the estimation of potential output and discusses their policy implications. 3 For example, in an estimated DSGE model with several frictions, Edge, Kiley and Laforte (2007) find that the time series of efficient output deviates significantly from the more traditional potential output derived within FRB/US. Justiniano and Primiceri (2009) reach opposite conclusions in their simpler model comparing efficient output with an HP trend.

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this latter respect, the lack of a consensus is particularly evident within the dynamic stochastic general equilibrium (DSGE) literature. In the last few years, DSGE models have incorporated ever more detailed and empirically accurate descriptions of private sector behavior and of the monetary transmission mechanism, following the seminal work of Christiano, Eichembaum and Evans (2005) and Smets and Wouters (2007).4 When it comes to modeling the behavior of the monetary authority, though, most studies simply posit an interest rate feedback rule broadly inspired by Taylor (1993), usually with no discussion of its details and of potential alternatives. As a result, the literature has witnessed a proliferation of estimated policy rules, especially with respect to the specification of the real gap the central bank reacts to, but with very little guidance on their relative merit, either positive or normative.5 This paper attempts to put some order in this wilderness, by comparing the fit of a large set of interest rate rules within a small scale estimated DSGE model of the U.S. economy. Most of the rules we consider previously appeared in the literature. Others, including the best fitting ones, did not. Comparing rules on the basis of their empirical performance is an important step for the evaluation and further development of DSGE models, since it allows us to abstract from specifications whose fit is obviously inferior and to focus on a small set of rules with better empirical properties.6 Our analysis proceeds in three steps. First, we propose a method to integrate statistical measures of the output gap, in particular those obtained through filtering, into a general equilibrium model. The idea is to use the DSGE model as a forecasting device to construct a real-time, two-sided version of the filter, in which the model’s forecasts substitute the realized forward values of the variable of interest (in our application, GDP).7 The application of this filter produces a real-time statistical measure of the output gap as one of the endogenous variables in the DSGE model. We can then allow the central bank to respond to this variable in setting the interest rate. In the second step, we embed this statistical output gap, as well as its model-based 4

See, for instance, Christiano, Motto and Rostagno (2007) for the introduction of financial frictions, Adolfson, Laséen, Lindé and Villani (2007) for the extension to an open economy setting and Edge, Kiley and Laforte (2007) for an application to monetary policy. 5 The inflation targeting approach (see Svensson (2003) and the references therein) models central banks as optimizing agents that maximize an objective function, as it is customary for the private sector, rather than as automatons committed to an interest rate feedback rule. The formulation of the optimal targeting rule obtained in this framework then crucially depends on the arguments of the loss function the policymakers seek to minimize. See Adolfson, Laséen, Lindé and Svensson (2008) for a state-of-the-art implementation of this recommendation within a DSGE model for Sweden. 6 A useful complement to this approach is to compare the alternative rules in terms of their welfare implications. This normative side of the study is the subject of ongoing research. 7 This idea is adapted from Christiano and Fitzgerald (2003) and is similar to a procedure proposed by Watson (2007) in the context of a purely statistical model.

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counterpart (the deviation of actual output from its efficient level), into a set of candidate interest rate rules of the general form  = −1 + (1 − ) [ +  (  −  ∗ ) +   ] + 

(1)

where  is the Federal Funds Rate,  is a potentially time-varying intercept,   −  ∗ is the deviation of inflation from its target,  is the output gap,  is an i.i.d. monetary policy shock and  is a smoothing parameter. As a baseline, we adopt a simple specification with a constant intercept and inflation target and the model-consistent notion of the efficient output gap. We then compare this baseline to three main classes of policies, with several variants within them, for a total of 44 estimated rules.8 The first alternative class of policies replaces the efficient output gap with a statistical one, measured using several types of filters, such as the HP filter with various smoothing parameters and an exponential filter. The second class of policies includes a time-varying intercept. In particular, we study specifications in which the monetary authority tracks the evolution of a “neutral” real interest rate. This variable is the real interest rate that would prevail in the model if output were always equal to its potential. Within this class, we allow potential output to be measured either by statistical procedures or in deviations from its efficient level. A rule of this kind is consistent with some normative analyses of the New Keynesian model and has recently attracted some interest in policy circles (Amato, 2005). However, to our knowledge, it has never been considered as a possible empirical specification in an estimated DSGE model. Finally, the third class of policies has a time-varying inflation target, which we model as an exogenous latent variable. In the third step of the analysis, we estimate alternative specifications of the model, each closed by one of the interest rate rules whose empirical performance we wish to compare. As for the private sector, we assume that the tastes and technology describing its behavior remain the same across policy specifications. Within the Bayesian framework we adopt for the estimation, the models’ posterior data densities are the ideal tool to compare their relative fit. This analysis therefore allows us to isolate the set of policy rules that best account for the behavior of the data and to discard rules whose fit is clearly inferior. We can summarize the main results as follows. First, and to our surprise, the baseline rule with the model-based output gap is one of the worst fitting one among the 44 rules we have estimated. The main reason is that the model-consistent output gap implied by this 8

A complete catalog of the rules we have estimated and of their relative empirical performance is in Appendix . Details on each specification are available upon request.

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specification displays almost no fluctuations at business cycle frequencies. As a result, it provides little explanatory power for interest rates as one of the arguments of the policy rule. Second, the fit of the model improves significantly when we substitute the efficient output gap with a statistical one. In this context, the traditional quarterly HP filter, with a smoothing parameter equal to 1600, performs particularly well. Once again, the reason is that the HP filtered output gap computed within the model, unlike the efficient gap, effectively tracks the business cycle and thus helps account for the observed movements in interest rates. Third, the fit improves even further when we let the intercept track the model’s efficient real interest rate. Moreover, the best fitting rule within this class features the efficient output gap as its measure of real slack. However, rules that replace the efficient gap with the statistical gap perform only marginally worse. The intuition for these results is that the efficient real rate appears to provide a better description of the “real” reference point followed by interest rates in our sample than any of the measures of potential otuput we have considered. Conditional on the signal provided by this rate, most of those measures do not add much in terms of fit. Fourth, the best-fitting rule among all the specifications we estimated is  = −1 + (1 − ) [ +  ∗ +  (  −  ∗ ) +   ] +   where  is the efficient output gap,  is the efficient real rate and the inflation target  ∗ is an exogenous persistent AR(1) process. The improvement in fit for this model over the comparable one with a constant inflation target is significant. The main reason is that inflation and nominal interest rates hover around 4% and 6% respectively in the first few years of our sample (1987q3:2007q4) until the recession of the early nineties, and then step down to their more recent averages. A persistent inflation target helps explain these low frequency movements in the two series. We refrain from overemphasizing the good empirical performance of this rule, though, since an exogenous inflation target, in the absence of restrictions that tie it more closely to the rest of the model, is bound to improve the fit of the model. Providing a more satisfactory account of the low frequency movements in nominal variables remains on the agenda for future research. Finally, we show that our main results and the relative ranking of alternative policy rules survive in forward-looking specifications in which current inflation and output are substituted with their expectations, as for example in Clarida, Galí and Gertler (2000). The rest of the paper proceeds as follows. The next section presents the model under the baseline interest rate rule. Section 3 discusses the data and the estimation method with particular focus on our choice of the prior. We introduce the alternative interest rate rules 5

and compare their empirical performance in section 4. Section 5 concludes.

2

The Model

We augment the purely forward-looking textbook New Keynesian framework (Woodford, 2003) with two sources of inertia to improve its ability to fit the data. On the demand side, we include habits in consumption in the utility specification. On the supply side, we allow for partial indexation to past inflation of the subset of prices that are not reoptimized in each period. The resulting model is smaller than the workhorse empirical DSGE model of Smets and Wouters (2007). It abstracts from capital accumulation and the attending frictions, such as endogenous utilization and investment adjustment costs, and from non-competitive features in the labor market, such as monopolistic competition and sticky wages. The reason for this modeling choice is that a small scale model allows us to estimate and compare the fit of a very large set of policy rules (44 in the current version) with a relatively low computational cost.9 The remainder of this section presents the linearized equilibrium conditions that constitute the basis for the estimation. Appendix  illustrates the details of the model. We defer the description of the parameters to section 3.

2.1

Aggregate Demand and Supply

An Euler equation summarizes the demand side of the model  ˜ = −−1 ( −   +1 −  ) +   ˜+1 

(2)

where  is a combination of structural parameters. The measure of real activity  ˜ in the   Euler equation (2) is a distributed lag of the efficient output gap  ≡  −  ¡ ¢¡ ¢  ˜ ≡ 1 −    1 −     

(3)

where  measures the log deviation of detrended output from steady state,  is the lag operator ( = −1 for a generic variable  ) and  is the forward operator (  =  +1 ). Our definition of the efficient level of output in the current period is conditional on last period 9

We are currently assessing the robustness of our results within the Smets and Wouters (2007) model.

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actual output10 ©£ ¡ ¢¤ £ ¡ ¢ ¤ ª  +  1 −  2 +    =  −1 +    1 −   (1 −  )   − (1 −  )   

The expected growth rate of the measure of real activity  ˜ is a function of the gap between the real interest rate (equal to the nominal interest rate  minus expected inflation   +1 ) and the efficient real interest rate  , defined as ¡ ¢   ≡   +1 −   +1 −  − (  +1 −   ) 

A Phillips curve summarizes the supply side of the model

 ˜  =  ( + ˜  ) +   ˜ +1 +  

(4)

where  is a cost-push shock and the slope parameter  is a combination of structural parameters. The cost-push shock is the only source of tradeoff between inflation and real activity. Cost-push shocks arise in our model because of exogenous fluctuations in the markup.11 The indexation scheme implies that the measure of inflation in the Phillips curve  ˜  is a function of deviations of current and lagged inflation from the long run inflation target  ∗  ˜  =   −  −1  where   measures the log deviation of inflation from its long run target. We can also rewrite the Phillips curve as  ˜ = 

¢ ¡ ¢¤ £¡   − −1 −   +1 −  +   ˜ +1 +  

(5)

where  is a linear function of . This alternative representation features the welfare-relevant ˜  as measure of real activity ( − −1 ) and expresses the quasi-difference inflation rate  the present discounted value of current and expected future distributed lags of the efficient output gap. 10

Our results are robust to using the unconditional definition of the efficient level of output. Examples of fundamental source of fluctuations in the markup are exogenous changes in sales taxes or in the elasticity of substitution among varieties. 11

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2.2

Monetary and Fiscal Policy

We assume that the central bank sets the short term nominal interest rate  . In our baseline specification, the nominal interest rate responds, with a certain degree of inertia, to the contemporaneous inflation rate and output gap12  = −1 + (1 − ) (   +   ) +  

(6)

Expression (6) represents a natural starting point for our analysis as it brings the basic ingredients of the empirical literature on interest rate rules into the context of our DSGE framework. Inflation and real activity are standard arguments of monetary policy rules since Taylor (1993). The empirical fit of single equation estimates typically improves in presence of interest rate inertia (Clarida, Galí and Gertler, 2000). We choose the efficient output gap as the baseline measure of discrepancy between actual and potential real activity for internal consistency with the rest of our theoretical apparatus. Specifically, two intertwined reasons justify our choice. First, the efficient output gap represents a sufficient statistic to summarize the equilibrium real activity in the model. Second, the efficient output gap corresponds to the measure of slack in real activity which is relevant for welfare analysis (see Giannoni and Woodford, 2003). The fiscal authority chooses in each period lump-sum taxes (or transfers) to satisfy the government budget constraint so that fiscal decisions are irrelevant for the equilibrium outcome.13

3

Data and Estimation

We perform a Bayesian estimation of the model (see An and Schorfheide (2007) for a survey) using seasonally adjusted quarterly observations at annual rates for the United States on the growth rate of real GDP, the inflation rate of core PCE and the effective Federal Funds rate averaged over the quarter. All series are in percentage points. The Bureau of Economic Analysis publishes the output and inflation data. The effective FFR is from the Board of Governors of the Federal Reserve. The sample period is 1987q3:2007q4. Bayesian estimation combines the prior information for the parameters with the likelihood 12 Del Negro, Schorfheide, Smets and Wouters (2007) also adopt this specification. Augmenting the baseline rule with the growth rate of the efficient output gap (as in Smets and Wouters, 2007) does not significantly affect our baseline results. 13 Note that we have abstracted from government spending. If we assume that the government does not issue debt, then, lump-sum transfers rebate to households the revenues from sales taxes. More generally, lump-sum transfers can be negative if the government needs to finance an exogenous stream of debt.

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of the model to form the posterior probability density function. We obtain the likelihood using the Kalman filter based on the state space solution of the rational expectations equilibrium of the model under each specification considered. The observation equations are ∆ =   + 400 ( − −1 +   )   =  ∗ + 400 

 = ( +  ∗ ) + 400  where ∆ is the growth rate of real GDP,   is the inflation rate,  is the Federal Funds target rate,   is the constant productivity growth rate along the balanced growth path,  ∗ is the long run inflation target, and  is the steady state real interest rate. We assume no measurement errors. We initialize the Kalman filter in steady state and set the initial covariance matrix equal to the long run unconditional convariance of the states. Table 1 reports our choice of the priors.14 We impose fairly loose priors between zero and one on the habit coefficient  and the degree of indexation , only slightly favoring higher values. We impose a degenerate prior on the subjective discount factor and set  equal to 099. The steady state restriction arising from the Euler equation would imply a value for  larger than one given the sample average of productivity growth  and real interest rate . Instead, we independently estimate  and  without imposing the steady state restriction.15 The slope of the Phillips curve is a convolution of deep parameters ≡

(1 − ) (1 − )   (1 + )

where  is the probability that a firm keeps its price fixed in any given period,  is the elasticity of substitution among differentiated goods and  is the inverse Frisch elasticity of labor supply. We choose to impose a prior directly on  centered at 01, which is slightly higher than most of the existing evidence (Galí and Gertler, 1999, Sbordone, 2002). We calibrate  (otherwise not identified) equal to 8, which implies a steady state markup of about 14%. Finally, our prior for  is a Gamma distribution with mean equal to one and standard deviation of 02. Our priors for the interest rate rule parameters conform well with the existing evidence. 14

Unless explicitly stated, we keep the priors unchanged throughout the evaluation of alternative interest rate rules. 15 rate of the output gap to the real interest rate gap in (2) is −1 ≡  The elasticity of expected growth 1 −   1 −   , where   ≡ − .

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The prior for the interest rate smoothing parameter  takes value in the interval [0 1] and is centered at 07. The priors for the feedback coefficients on inflation and real activity are normally distributed with mean 15 and 05 respectively, in line with the original estimates in Taylor (1993). The prior for the inflation target is also normally distributed with mean equal to 2% and a standard deviation of 1%. We use Beta priors centered at 05 for the persistence of the exogenous shocks and Inverse Gamma priors for the standard deviations. When we allow for a time-varying inflation target, we impose a tight prior on its persistence around 095. We obtain the joint mode and inverse Hessian of the parameters by minimizing the negative of the log posterior probability density function. We then use Markov Chain Monte Carlo methods (random walk Metropolis algorithm) to build a representative sample of the posterior joint distribution of the parameters. We monitor the convergence of the chains of draws in each step using a variety of tests.16 Finally, upon convergence, we combine the chains in the last step, after discarding the initial 25% of the draws in each chain, to form a full sample of the posterior distribution which represents the source of our inference information.17

4

Interest Rate Rules

This section compares the results of the model estimation under the baseline rule (6) with several alternative specifications. We find that the inclusion of a time-varying inflation target and, crucially, the efficient real interest rate substantially improves the empirical fit of the model relative to the baseline case. The presence of time-varying intercept and inflation target also makes the measure of output gap that we include in the interest rate rule irrelevant. In this case, the feedback coefficient on the output gap  is not well identified. In fact, the omission of a real activity measure does not significantly worsen the fit of the model. The inclusion of the efficient real interest rate in the monetary policy rule is the distinctive feature of our results. The fit of the model partly deteriorates if we exclude the time-varying inflation target but the improvement relative to the baseline rule is still substantial. Most importantly, we can give the efficient real interest rate a straightforward economic interpretation in light of the model. To the contrary, the inclusion of a time-varying inflation target alone only marginally improves upon the baseline specification. In this case, we find that the feedback coefficient 16 For most specifications, convergence occurred with 200,000 draws in each chain. In few cases, we had to increase the number of draws up to 1,000,000. 17 The detailed convergence and inference analysis for each specification is available upon request.

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on the output gap is again not identified. Overall, we see the time-varying inflation target mostly as a useful latent variable to improve the fit of the model at low frequencies. Our main conclusion, therefore, is that the presence of the efficient real interest rate in the monetary policy rule is a necessary condition for our model to fit the data well. A partial exception to this finding is a rule that responds to the deviations of output from a trend constructed with a statistical filter of the data. The rest of this section introduces the elements of the best fitting rule starting from the baseline specification.

4.1

Baseline

Table 2 compares the estimated posterior distributions of the parameters under the baseline interest rate rule (6) with the priors discussed in the previous section. Most of the posterior estimates fall in a reasonable range. Nonetheless, we note two anomalies. First, the parameter  is not well identified. This result is robust across most of the alternative specifications considered below. Second, the posterior mode and median of the slope of the Phillips curve gives very different implications for the degree of price rigidity. The average duration of price contracts is six quarters when we evaluate the slope parameter  at the posterior mode. This estimate is well above the evidence in Bils and Klenow (2004) and Nakamura and Steinsson (2008). On the contrary, the posterior median implies an average duration of price contracts of about three quarters, roughly in the middle of the direct estimates. The posterior median estimate of the efficient output gap displays little variability during the sample period, in spite of a substantial degree of uncertainty. Figure 1 compares the posterior estimates of the efficient output gap with an HP filter of GDP. The posterior median efficient output gap clearly fails to capture the two recession episodes in our sample, which correspond to the dips of the HP filter. In this sense, the efficient output gap may not be the most relevant measure of real activity gap for describing the conduct of U.S. monetary policy in the last two decades. We can appreciate this result also from a slightly different angle. The only tradeoff between inflation and output gap in the model arises from the existence of markup shocks. As the variance of markup shocks goes to zero, the tradeoff disappears. Indeed, we do find evidence of fairly small and transitory markup shocks (see figure 2). Therefore, adding the efficient output gap to an interest rate rule which already responds to inflation provides little extra useful information for monetary policy.18 18

The magnitude of the markup shocks affects the tradeoff between the efficient output gap  and the rate

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We can further investigate the relevance of including the efficient output gap in the monetary policy rule by estimating the model under the restriction  = 0. We compare the two models using a standard log-marginal likelihood criterion (An and Schorfheide, 2007). Excluding the efficient output gap from rule (6) worsens our performance metric by less than three points, a relatively small deterioration of fit. This result squares well with the recent literature on optimal monetary policy in medium scale DSGE models. For example, Schmitt-Grohé and Uribe (2007) find that the optimal Taylor rule for an economy with capital accumulation and fiscal policy decisions features a muted coefficient on output.

4.2

Statistical Filter

We concluded the previous section showing that the exclusion of the efficient output gap from the baseline interest rate rule does not substantially deteriorate the fit of the model. This result does not necessarily mean that in practice the Fed did not respond to departures of output from some reference target. Rather, the target might have just been an object other than the efficient level of output. Abandoning the model-based definition of output gap, however, has a drawback. The choice of the appropriate measure of output gap to include in the interest rate rule becomes very much judgemental. In this paper, we evaluate the possibility that the monetary authority had historically responded to a purely statistical filter of raw output.19 In particular, we choose to focus on the deviations of output from a low frequency trend constructed with the HP filter (Hodrick and Prescott, 1997). One reason behind our choice is the popularity of this concept in the literature on statistical measures of real activity gaps (e.g. Orphanides and van Norden, 2002). A second rationale is the reliance on the HP filter to gauge departures of output from potential in the policy debate (Gramlich, 2004). More generally, we see the HP filter as a reasonable device to extract a “traditional” output gap measure from real time GDP data. We embed the HP filter in our model adapting the methodology proposed by Christiano and Fitzgerald (2003) for finite sample approximations of band pass filters.20 The trend of inflation adjusted for indexation  ˜  . Therefore, the indexation parameter adjusts to explain the degree of inflation persistence present in the data. 19 We emphasize the empirical fit of the model while remaining agnostic on the reasons why the Fed might be willing to target a statistical measure. 20 Christiano and Fitzgerald (2003) suggest to expand a time series  with its optimal projections under the assumption that  follows a random walk. Watson (2007) recommends a very similar approach to real time band pass filters. Our twist is to use the optimal projections of the endogenous variables from the model which coincide with their rational expectation forecasts.

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component of a time series implied by the HP filter results from the application of the lag polynomial   ( ) (Baxter and King, 1999)   ( ) =

1  1 + (1 − )2 (1 −  )2

where  is a smoothing parameter which depends on the frequency of the data. We follow Hodrick and Prescott (1997) and most of the business cycle literature in setting  = 1600 is just the difference between the output for quarterly data. The HP filter output gap   ¡ ¢   series  and the trend component   ( ≡ 1 −    ). is an infinite moving average of the data. We approxiThe problem, however, is that   mate this process by substituting for the infinite lags and leads using the rational expectation forecasts implied by the model. The expression that defines the HP gap is then i h = (1 − )2 (1 −  )2   1 + (1 − )2 (1 −  )2  

(7)

The use of the rational expectation forecasts is what makes the model-based HP filter output gap differ from its pure statistical counterpart. We can rewrite the right-hand side of equation (7) as a lagged polynomial in the growth rate of output ¡ ¢ (1 − )2 (1 −  )2  =  3 −  − 3 +  2 ∆ 

(8)

Therefore, the non-stationarity of GDP data does not affect our rendition of the HP filter output gap. We estimate the model replacing the efficient output gap  with the model-based HP in the interest rate rule (6). Overall, the marginal likelihood improves by about filter   20 log-points relative to the baseline case. The posterior estimates of the HP filter output gap exactly identify the two recessions in our sample. In fact, the model-based HP filter output gap captures the recession episodes even more precisely than the statistical construct (figure 3). in the interest rate rule also contributes to generate a more regular The inclusion of   shape for the posterior distribution of the slope of the Phillips curve. The posterior estimates imply an average frequency of price changes between 35 and 4 quarters, independently of whether we evaluate the slope at the median or at the mode. Interestingly, replacing the efficient output gap with the HP filter measure significantly changes the estimates of feedback parameters of inflation and output relative to the baseline case. In particular, the posterior median of  decreases from 175 to 131 while the posterior 13

median of  increases from 065 to 108. The HP filter seems to convey important information for describing U.S. monetary policy in recent years. The aggressive response to departures of output from the HP trend is consistent with the Fed believing the Phillips curve to be an incomplete description of the joint determination of inflation and the output gap. We also check the robustness of our findings along two dimensions. First, we choose  to minimize the sum of squared residuals between the HP filter output gap and some empirical measures often used in policymaking (the CBO output gap and the output gap implied by the FRB/US model). This strategy points in the direction of a higher smoothing parameter (between one and two orders of magnitudes). The overall performance of the model, however, worsens by about 10 log-points. Second, we directly estimate  together with the rest of the parameters of the model. We choose a normal prior for the log of  with mean equal to 9 and standard deviation equal to 3. Our prior is quite diffused and upward biased relative to the standard business cycle assumption of 1600 to accommodate high values consistent with the measures discussed above. The estimation returns a posterior median equal to 2372. Not surprisingly, then, the performance of the model is substantially unchanged (1 log-point worse) relative to the case with  = 1600. Our approach to include the HP filter in the model raises one question. Why would the central bank use the HP filter when it is sophisticated enough to compute the rational expectation forecasts based on the true model? We argued that our objective here is simply to investigate whether the inclusion of a statistical measure of the gap improves the fit of the model. In this sense, our approach incorporates some degree of judgement in the context of our model. Nevertheless, we also try to solve the tension between the naïve measure of gap and the sophistication of the forecasts by allowing for an even simpler statistical filter. In a different specification of the interest rate rule, we assume that the monetary authority of the observed output data responds to a simple exponential filter exp  exp = 

˜ (1 − )   ˜ − )  1 + (1

˜ = 615 to equalize the gain of the exponential filter and of the HP filter (with We choose   = 1600) at a frequency that corresponds to an eight year cycle. The fit of the model is about 10 log-points worse than in the case of the HP filter. As for the baseline case, the implied average frequency of price adjustment is about one and a half year.

14

4.3

Time-Varying Intercept

The previous two sections make two important points. First, the estimated tradeoff between inflation and model-consistent output gap is small. Therefore, including the efficient output gap in an interest rate rule which responds to inflation does not add much power in fitting the data. Second, we nevertheless find evidence that the Fed reacted to deviations of output from some measures of potential. A pure statistical construct, like the HP filter output gap, seems to be a good proxy for the measure of real activity that the monetary authority has targeted. This section asks if in fact the HP filter output gap just captures the information from some variable of the model that we have failed to include in the interest rate rule so far. In particular, we augment the baseline rule (6) with a time-varying intercept which corresponds to the efficient real interest rate   = −1 + (1 − ) ( +    +   ) +  

(9)

Part of the motivation for including the efficient real interest rate comes from normative results. The interest rate rule that implements the optimal policy in the baseline New Keynesian model features the efficient real interest rate as a time-varying intercept (see Clarida, Galí and Gertler, 1999, and Woodford, 2003). In presence of financial frictions, changes in credit spreads also contribute to variations of the efficient real interest rate (Cúrdia and Woodford, 2008). From a positive perspective, policymakers consider the “equilibrium” real interest rate (the real interest rate compatible with price stability and full employment) a useful concept to assess the stance of monetary policy (Meyer, 2002). Laubach and Williams (2003) provide empirical evidence in support of the hypothesis that the equilibrium real interest rate varies substantially over the cycle. In our model, the efficient real interest rate  represents the theoretical counterpart of the object discussed in the policy debate. The introduction of the efficient real interest rate in the monetary policy rule improves the log-marginal likelihood by 30 points relative to the baseline. We note three differences relative to the previous cases concerning the posterior estimates. First, the posterior median and mode of the slope of the Phillips curve imply a shorter average duration of price changes of about two quarters. Second, the posterior medians of the feedback parameters on inflation and the output gap fall in a range consistent with most of the literature (14 for  and 06 for  ). The posterior modes are very similar. Third, markup shocks are highly persistent. The posterior median of the autoregressive coefficient of

15

 is 091, compared to 043 in the baseline case and 023 in the case of a statistical filter. More persistent markup shocks lessen the degree of indexation necessary to fit observed inflation. The estimated posterior median of  decreases from 022 with the baseline interest rate rule to 013. The introduction of  in the interest rate rule makes  more volatile and persistent (see figure 4). Yet, the efficient output gap does not clearly identify the recession episodes in our sample. The negative level of the gap during the 1990 − 1991 recession seems to be mostly the result of a persistent pattern since the beginning of the sample. The efficient output gap also turns negative in the 2001 recession but barely so. Moreover, such a level is consistent with other non-recessionary periods, such as 1996. It is the efficient real rate, instead, that seems to provide the most precise information on the state of the business cycle (see figure 5). With the benefit of hindsight, we go back and check the correlation between the HP filter output gap and the efficient real rate, when we include the former in the interest rate rule. We find it to be quite high (equal to 07). This observation (see figure 6) adds statistical support to the economic motivations for including  to the monetary policy rule. In a nutshell, the efficient real interest rate turns out to be an appropriate summary of the relevant information about real developments that the Fed responded to during the last twenty years or so. The inclusion of the efficient real interest rate in the monetary policy reaction function improves upon a statistical filter and has the added benefit of being fully consistent with the structure of the model. The decisive improvement in the model’s fit when the interest rate rule includes  is the main result of this paper. We believe that this finding represents a novel contribution to the literature on interest rate rules in DSGE models. The result might not be fully surprising from the perspective of fit alone. In a sense, we are allowing for a more flexible shock structure in the interest rate rule. Nevertheless, we are also imposing a fairly tight cross-equation restriction between the interest rate rule and the aggregate demand equation. Therefore, the additional flexibility does not come completely unconstrained. More importantly, the mapping between the definition of the efficient real interest rate in theory and the “equilibrium” real interest rate used in the policy debate strengthens the economic interpretation of our result beyond the specific convolution of shocks arising from the model. Finally, we also find interesting the normative connection to the implementation of optimal monetary policy in small scale DSGE models. Based on the existing literature, we suspect that a rule like (9) would provide a better approximation of optimal policy than the other

16

interest rate rules that we have estimated. Like for the baseline case, the results in this section are fairly robust to the exclusion of the efficient output gap from the interest rate rule. The log-likelihood worsens by about 5 points if we set  = 0 in (9). Importantly, however, substituting the efficient output gap with the HP filter output gap does not improve the fit. In fact, the likelihood decreases by 6 log-points. This result provides further evidence that the efficient real interest rate conveys much of the same information contained in the statistical filter, with the advantage of being a well-defined theoretical object.

4.4

Time-Varying Inflation Target

We noted in the previous section that the efficient output gap still has troubles in capturing the recession episodes in our sample. This failure occurs in spite of a specification of the interest rate rule which helps the model to fit the data substantially better than the baseline case. The objective of this section is to investigate to which extent the poor performance of the output gap as an indicator of recessions depends on the properties of inflation in the sample. The actual inflation process contains an important low frequency component, mostly evident in the first half of the 1990s. Inflation persistently declines from peaks above 5% at the beginning of the decade to less than 2% within five years (figure 7). This section introduces a time-varying inflation target in the interest rate rule to evaluate if this shock helps the model to improve the fit of the inflation process at low frequency. We modify rule (9) in this vein  = −1 + (1 − ) [ +  ∗ +  (  −  ∗ ) +   ] +  

(10)

where  ∗ is a stationary autoregressive process in deviations from the long run target  ∗ with persistence given by |∗ |  1 and normal i.i.d. innovations.21 We impose a fairly tight prior on ∗ so that innovations to the inflation target have nearly permanent effects. From a normative perspective, a stationary but persistent  ∗ captures the monetary authority willingness to delay reaching its ultimate target, presumably to avoid excessive output losses. If the monetary authority follows rule (10), the interest rate does not need to move much at the beginning of the sample, even though inflation is well above its long run target. Rather, the inflation target  ∗ might just be temporarily (although persistently) above average. Therefore, the introduction of the time-varying inflation target also contributes to 21

Smets and Wouters (2003) adopt a similar formulation.

17

better fit the process of the Fed Funds rate. The log-marginal likelihood improves by 40 points relative to the baseline case. This specification of the interest rate rule provides the best fit of the model among the 44 rules that we considered. This result is perhaps not very surprising since we are adding a purely latent variable to the interest rate rule. From a purely statistical perspective, the inflation target is just an additional shock to the interest rate rule. In economic terms, the model does not impose any cross-equation restriction on the inflation target, which is then free to explain different frequencies compared to the i.i.d. monetary innovation  .22 The introduction of a time-varying inflation target also leads the model to generate better posterior estimates of the efficient output gap as predictor of recessions. Figure 8 shows that the efficient output gap becomes significantly negative in conjunction with the two recessions in the sample. Nonetheless, some noise remains. The efficient output gap is persistently negative during the second half of the 1990s, although at lower levels than in recessions. One issue with the introduction of a time-varying inflation target is that we lose identification of the feedback parameter for the output gap  in the interest rate rule. Estimation of a restricted version of (10) with the degenerate prior  = 0, returns a virtually unchanged (05 log-points lower) marginal likelihood. The marginal likelihood is also approximately the same (1 log-point lower) if we replace the efficient output gap with the HP filter output gap discussed in section 42. In this case, however, the feedback coefficient  is identified, with a posterior median of about 04. The presence of the efficient real interest rate in (10) remains nevertheless central for the fit of the model. The introduction of a time-varying inflation target alone in the interest rate rule leads to a log-marginal likelihood 13 points higher than in the baseline case. This improvement is significant, yet much less so than for the rules considered in section 42 and, especially, 43. Modifying the baseline rule to allow for a time-varying inflation target only is also interesting because the results confirm that the lack of identification of  is related to the presence of  ∗ in the interest rate rule. The introduction of a time-varying inflation target is a partial success. On the one hand, the fit of the model improves and the efficient output gap becomes a better indicator of recessions. On the other hand, the gains appear to be mostly the consequence of statistical flexibility rather than economic structure. An additional drawback of this specification is the loss of identification of the feedback coefficient on the efficient output gap. 22 Shortening the sample does not fully eliminate the relevance of a time-varying inflation target. The inflation and interest rate data feature low frequency components beyond the downward drift of the late 1980s and early 1990s.

18

4.5

Forecast-Based Interest Rate Rules

In this section, we ask if our findings are robust to replacing the contemporaneous values of inflation and output gap in the interest rate rule with their forecasts. The modified baseline rule becomes ¡ ¢ (11)  = −1 + (1 − )    +1 +   +1 +  

Single-equation estimation of interest rate rules often features forecast variables to formalize the idea that central banks consider all available information to form expectations (see, for example, Clarida, Galí and Gertler, 2000). This problem should not be particularly relevant in the context of DSGE model estimation since the forecast of future inflation and output gap is based on the same information set that determines their contemporaneous values. In our case, the fit of the model under rule (11) worsens by about 20 log-points relative to our original formulation (6). The introduction of forecast inflation and output gap does not alter any of our main results in terms of the ranking of alternative rules. The fit of the model is virtually unchanged if we exclude the efficient output gap from (11). The replacement of the efficient output gap with the HP output gap increases the log-marginal likelihood by about 20 log-points. An additional improvement of about the same magnitude occurs if we allow the efficient real interest rate to become the time-varying intercept of rule (11). Finally, the introduction of a time-varying inflation target contributes to a further 20 log-points amelioration of the marginal likelihood. Overall, the fit of the model is very similar whether we use current or expected inflation and output gap once we introduce the efficient real interest rate and the time-varying inflation target.23

5

Conclusions

TO BE WRITTEN

23

This specification did not achieve full convergence with 200,000 draws. We are in the process of augmenting the number of draws to check the robustness of the results.

19

References [1] Adolfson, M., Laséen, S., Lindé, J., and M. Villani (2007), “Bayesian Estimation of an Open Economy DSGE Model with Incomplete Pass-Through,” Journal of International Economics 72, pp. 481-511. [2] Adolfson, M., Laséen, S., Lindé, J., and L. Svensson (2008), “Optimal Monetary Policy in an Operational Medium-Sized DSGE Model”, NBER Working Paper No. 14092. [3] Amato, J. (2005), “The Role of the Natural Rate of Interest in Monetary Policy,” BIS Working Paper No. 171. [4] An, S. and F. Schorfheide (2007), “Bayesian Analysis of DSGE Models,” Econometric Reviews 26, pp. 113-172. [5] Baxter, M. and R. King (1999), “Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time Series,” Review of Economics and Statistics 81, pp. 575-593. [6] Bils, M. and P. Klenow (2004), “Some Evidence on the Importance of Sticky Prices,” Journal of Political Economy 112, pp. 947-985. [7] CBO (2001), “CBO’s Method for Estimating Potential Output: An Update,” available at www.cbo.gov. [8] Christiano, L., Eichembaum, M. and C. Evans (2005), “Nominal Rigidities and the Dynamic Effect of a Shock to Monetary Policy,” Journal of Political Economy 113, pp. 1-45. [9] Christiano, L. and T. Fitzgerald (2003), “The Band Pass Filter,” International Economic Review 44, pp. 435-465. [10] Christiano, L., Motto, R. and M. Rostagno (2007), “Financial Factors in Business Cycles,” Unpublished. [11] Clarida, R., Galí, J. and M. Gertler (1999), “The Science of Monetary Policy: A New Keynesian Perspective,” Journal of Economic Literature 37, pp. 1661-1707. [12] Clarida, R., Galí, J. and M. Gertler (2000), “Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory,” Quarterly Journal of Economics 115, 147-180. [13] Cúrdia, V. and M. Woodford (2008), “Credit Frictions and Optimal Monetary Policy," Unpublished. 20

[14] Del Negro, M., Schorfehide, F., Smets, F. and R. Wouters (2007), “On the Fit and Forecasting Performance of New Keynesian Models,” Journal of Business and Economic Statistics 25, pp. 123-143. [15] Edge, R., Kiley, M. and J.-P. Laforte (2007), “Natural rate measures in an estimated DSGE model of the U.S. economy,” Finance and Economics Discussion Series 2007-08. [16] Giannoni, M. and M. Woodford (2005), “Optimal Inflation Targeting Rules,” in B. Bernanke and M. Woodford (eds.) Inflation Targeting, University of Chicago Press — Chicago, IL. [17] Gramlich, E. (2004), “Oil Shocks and Monetary Policy,” Remarks at the Annual Economic Luncheon, Federal Reserve Bank of Kansas City, Kansas City, Missouri. [18] Hodrick, R. and E. Prescott (1997), “Post-War U.S. Business Cycles: An Empirical Investigation,” Journal of Money Credit and Banking 29, pp. 1-16. [19] Justiniano, A. and G. Primiceri (2009), “Potential and Natural Output,” Unpublished. [20] King, R. and S. Rebelo (1993), “Low Frequency Filtering and Real Business Cycles,” Journal of Economic Dynamics and Control 17, 201-231. [21] Kuttner, K. (1994), “Estimating Potential Output as a Latent Variable,” Journal of Business and Economic Statistics 12, 361-368. [22] Laubach, T. and J. Williams (2003), “Measuring the Natural Rate of Interest,” The Review of Economics and Statistics 85, 1063-1070. [23] Meyer, L. (2002), “Rules and Discretion,” Remarks at the Owen Graduate School of Management, Vanderbilt University, Nashville, Tennessee. [24] Mishkin, F. (2007), “Estimating Potential Output,” Remarks at the Conference on Price Measurement for Monetary Policy, Federal Reserve Bank of Dallas, Dallas, Texas. [25] Nakamura, E. and J. Steinsson (2008), “Five Facts About Prices: A Reevaluation of Menu Cost Models,” Quarterly Journal of Economics 123, pp. 1415-1464. [26] Orphanides, A. and S. Van Norden (2002), “The Unreliability of Output-Gap Estimates in Real Time,” The Review of Economics and Statistics 84, 569-583. [27] Schmitt-Grohé, S. and M. Uribe (2007), “Optimal Simple And Implementable Monetary and Fiscal Rules,” Journal of Monetary Economics 54, pp. 1702-1725. 21

[28] Smets, F. and R. Wouters (2003), “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area,” Journal of the European Economic Association 1, pp. 1123-1175. [29] Smets, F. and R. Wouters (2007), “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach,” American Economic Review 97, pp. 586-606. [30] Svensson, L. (2003), “What Is Wrong with Taylor Rules? Using Judgment in Monetary Policy through Targeting Rules,” Journal of Economic Literature 41, pp. 426-477. [31] Svensson, L. (2007), “Inflation Targeting,” in The New Palgrave Dictionary of Economics (2 edition), L. Blum and S. Durlauf (eds.), Palgrave Macmillan — New York, NY. [32] Taylor, J. (1993), “Discretion versus Policy Rules in Practice,” Carnegie-Rochester Conference Series on Public Policy 39, pp. 195-214. [33] Watson, M. (2007), “How Accurate Are Real-Time Estimates of Output Trends and Gaps?” Federal Reserve Bank of Richmond Economic Quarterly 93, pp. 143-161. [34] Woodford, M. (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press, Princeton, NJ.

22

A

The Model

This appendix presents the microfoundations of the model.

A.1

Households

A continuum of households of measure one populates the economy. All households, indexed by  ∈ (0 1), discount the future at rate  ∈ (0 1) and have the same istantaneous utility function, additively separable over consumption and labor 0 ≡ 0

(∞ X

  

=0

"

( )1+  )−  log( − −1 1+

#)



The aggregate preference shock   shifts the intertemporal allocation of consumption without affecting the intratemporal margin between labor and leisure.24 We assume that   follows a stationary process with mean zero of the form   =   −1 +   The consumption index  is a constant elasticity of substitution (equal to   1) aggregator over differentiated goods indexed by  ∈ (0 1) 

≡

∙Z

0

1



()

−1 

 ¸ −1  

(12)

Households supply their specialized labor input for the production of a specific final good. As a consequence of labor market segmentation, the wage  differs across households. However, household  can fully insure against idiosyncratic wage risk by buying at time  state at price +1 . Besides labor income, households earn after-tax contingent securities +1  Γ from ownership of the firm. The flow budget constraint for household  is Z

0

1

  ()  ()  +  (+1 +1 ) =   +  + Γ 

where  () is the dollar price of the  variety. 24

We could have also introduced a purely intratemporal shock affecting labor supply decisions only. However, in our empirical implementation of the model, hours and wages are not included among the observables. Therefore, such a shock would only affect the flexible price level of output, making it indistinguishable from a technology shock.

23

A.2

Firms

Firm  produces the differentiated consumption good  () with a linear technology in labor  () =   () 

(13)

We assume that productivity ( ) grows at rate   and that growth rate shocks display some persistence ¡ ¢ (14)   = 1 −   +   −1 +  

Firms take wages as given and sell their products in monopolistic competitive goods markets, setting prices on a staggered basis (Calvo, 1983). Every period, independently of previous adjustments, each firm faces a probability (1 − ) of optimally choosing its price. The fraction  of firms that do not fully optimize in a given period adjust their price according to the indexation scheme  () = −1 ()

µ

−1 −2

¶

∗

(1−) 

where  is the aggregate price level consistent with the consumption aggregator (12). We allow for partial indexation to the long run central bank’s inflation target  ∗ . In the short run, the target can vary exogenously according to the process ∗

 ∗ = (1 − ∗ )  ∗ + ∗  ∗−1 +   with ∗  1.25 In the event of a price change at time , firm  chooses  () to maximize the present discounted value of profits net of sales taxes   Υ () = 

(∞ X =

 −



"

 (1 −   )  ()

µ

−1 −1

¶ µ

∗ ∗

¶1−

#)

 () −  ()  ()

 (15)

subject to its technology (13) and the demand for its own good conditional on no further price change after period 26 ∙ ¸  () −  () =   (16)  25 Obviouslty, this formulation is general enough to encompass cases in which the inflation target is constant ∗ also in the short run ( = 0 ∀). 26 The form of demand (16) follows from the households’ expenditure minimization, after imposing market 1 clearing ( () = 0  () ) and assuming that the aggregator for GDP (denoted by  ) corresponds to (12).

24

A.3

Monetary Policy

The central bank sets the nominal interest rate  with a certain degree of inertia in response to departures of detrended output  and inflation Π from their target. The non-linear formulation of the baseline interest rate rule is µ

1 +  1+

¶

=

µ

1 + −1 1+

⎡ ! ⎤1− ¶ µ ¶ Ã  Π    ⎣ ⎦   ∗    

(17)

where the nominal interest rate  is defined as (1 +  )−1 ≡  +1  ∗

The steady state Fisher equation (1 +  = (1 + )  ) defines the real interest rate . The baseline specification of the interest rate rule (17) assumes that the target for inflation is constant and the target for real activity is its detrended level in the absence of nominal  ). rigidities and markup shocks (the efficient level 

B

Statistical Filters in a DSGE Model

This appendix illustrates how to embed a statistical filter in a rational expectation model. We begin with a brief general description of filtering problems. We then focus on the HodrickPrescott (HP) filter (Hodrick and Prescott, 1997). The general idea builds upon Christiano and Fitzgerald (2003). In theory, implementation of an ideal band pass filter requires an infinite data set. In practice, we rely on some approximation which nevertheless need knowledge of the spectral density of the process to be filtered. In the case of macro variables, Christiano and Fitzgerald (2003) suggest to apply the approximation that would be optimal for a random walk process, even if the series under study is not exactly a random walk. This approximation consists of augmenting the available dataset with the least squares optimal guesses of the missing data based on the available data. For a random walk, such guesses are simply 1 for all observations from  = 0  −∞ and  for all observations from  =  + 1  +∞ Our proposal is to use the rational expectations forecasts of those same values from the model.27 27

Watson (2007) recommends a very similar approach to real-time filtering with band pass filters.

25

B.1

Filters

The objective of “filtering” is to decompose the stochastic process  into two orthogonal components ˜   =  +  where the process  has power only in some frequency interval {( ) ∪ (− −)} ∈ (− ). Then, we can represent  as  =  ()   where  () — the ideal band-pass filter — is of the form ∞ X

 () =

  

=−∞

Therefore, implementation of the ideal filter requires an infinite dataset. We can think about approximating the ideal filter as a projection problem. Given a sample  = [1    ], the estimate of  = [1   ] is ˆ =  [|], which is of the form ˆ =

 X

ˆ  −   

=−

where  =  −  and  =  − 1. The main problem of this estimates is that the  coefficients require knowledge of  (), the spectral density of  Christiano and Fitzgerald (2003) show that, for most macro variables, the coefficients obtained by assuming that  is a random walk work quite well. One approach to the calculation of these coefficients is then to “expand” the available sample with the least squares optimal guesses of the missing data at the beginning and end of the sample. For the random walk, these data are just 1 and  . Our proposal is to adopt the same philosophy (i.e. to expand the available dataset) in the context of our framework, using the rational expectations forecasts of the missing data obtained from the model.

B.2

Application to the HP Filter

In this section, we discuss the application of our methodology to the HP filter. We focus on the HP filter because its wide use in macroeconomics as a flexible device (through the choice of ) to draw a smooth trend through the data. For our purposes, the HP filter provides a reasonable measure of a “traditional” output gap, that is, a statistical filter extracted in real time from data on GDP. The expression for 26

the ideal filter is a relatively simple function of lag polynomials. The result is a parsimonious (i.e. two leads and lags) recursive representation, with only a minor impact on the size of the model’s state space. The ideal HP filter is of the form (Baxter and King, 1999)   =   =

(1 − )2 (1 −  )2 1 + (1 − )2 (1 −  )2 1 1 + (1 − )2 (1 −  )2

where   denotes the filter whose application results in the “gap”, while   denotes the filter whose application produces the trend.28 To reiterate, practical applications of this filter require an approximation because of the infinite moving average representation. In the context of a rational expectations model, the approximation consists of substituting the infinite lags and leads implicit in   and   with the rational expectations forecasts. In our case, given observations on log  =  , we define the HP gap with parameter  as i h  () = (1 − )2 (1 −  )2   1 + (1 − )2 (1 −  )2  which implies  ()

 +2

 ()

− 4 +1

 ()

+ (1 + 6) 

 ()

− 4−1

 ()

+ −2

=  {3∆ − ∆−1 − 3 ∆+1 +  ∆+2 }  (18) Equation (18) constitutes the basis for estimating the model under the assumption that the central bank responds to a statistical output gap measure. The presence of first differences only in the right-hand side of the filtering equation is consistent with our assumption of non-stationary output data.

C

List of Estimated Interest Rate Rules

This section presents the list of the interest rate rules that we have estimated and orders their performance in terms of log-marginal likelihood. 28

King and Rebelo (1993) originally derived these expressions as the solution of a “smoothing” problem. However, they also showed that this filter, with  = 1600, approximates very well a high pass filter with cutoff frequency 16 or 32 quarters.

27

The statistical output gaps ( and exp   ) depend on a smoothing parameter (). We consider three possibilities. First, we fix  at 1600 for the HP filter in line with real business cycle literature. For the exponential filter, we set  = 615 to equalize the gain at a frequency that corresponds to an eight year cycle to that of the HP filter with  = 1600. Second, we explore the case  = 160 000 for the  filter to capture a very smooth trend. Finally, we ˆ for both filters. allow for direct estimation of the smoothing parameter  We also consider a number of specifications in which the time-varying intercept is consistent with the statistical filters. In these cases,  and exp are the real interest rates in the Euler equation when the output gap is respectively  and exp   . ¤ £ 1.  = −1 + (1 − )  +  ∗ +  (  +1 −  ∗ ) +   +1 +  ; LML = -305.0744. 2.  = −1 + (1 − ) [ +  ∗ +  (  −  ∗ ) +   ] +  ; LML = -306.4379. h i ˆ +  ; LML = -306.9033. ( ) 3.  = −1 + (1 − )  +  ∗ +  (  −  ∗ ) +     4.  = −1 + (1 − ) [ +  ∗ +  (  −  ∗ )] +  ; LML = -307.0159.

£ ¤ () +  ; LML = -307.5586. 5.  = −1 + (1 − )  +  ∗ +  (  −  ∗ ) +    h i ˆ +  ; LML = -308.5413. ( ) 6.  = −1 + (1 − )  +  ∗ +  (  −  ∗ ) +  exp  

7.  = −1 + (1 − ) [ +  ∗ +  (  −  ∗ ) +  ( − −1 )] +  ; LML = -309.8001. ()] +  ; LML = -310.1968. 8.  = −1 + (1 − ) [ +  ∗ +  (  −  ∗ ) +  exp 

£ ¡  ¢¤  +  ; LML = -310.3796. 9.  = −1 + (1 − )  +  ∗ +  (  −  ∗ ) +   

10.  = −1 + (1 − ) ( +    +   ) +  ; LML = -316.0236.

()] +  ; LML = -317.1820. 11.  = −1 + (1 − ) [ +    +  exp 

£ ¡  ¢¤ 12.  = −1 + (1 − )  +    +    +  ; LML = -317.8773.  h i ˆ +  ; LML = -317.9220. ( ) 13.  = −1 + (1 − )  +    +  exp   h i  ˆ 14.  = −1 + (1 − )  +    +    () +  ; LML = -319.2683.

15.  = −1 + (1 − ) ( +    ) +  ; LML = -320.8629.

()] +  ; LML = -321.8055. 16.  = −1 + (1 − ) [ (  −  ∗ ) +  exp 

£ ¤   17.  = −1 + (1 − )  (  −  ∗ ) +    ( ) +  ; LML = -321.9160. 28

£ ¤ 18.  = −1 + (1 − )  +    +   () +  ; LML = -321.9655.  h i  ˆ 19.  = −1 + (1 − )  (  −  ∗ ) +  exp  () +  ; LML = -322.2679.

h i ˆ +  ; LML = -322.3903. ( ) 20.  = −1 + (1 − )  (  −  ∗ ) +    

21.  = −1 + (1 − ) [ +    +  ( − −1 )] +  ; LML = -322.8532.

£ ¤ () +  ; LML = -322.9607. 22.  = −1 + (1 − )  (  −  ∗ ) +   

23.  = −1 + (1 − ) [exp +  ∗ +  (  −  ∗ ) +  exp ()] +  ; LML = -323.1635. 

¡ ¢ 24.  = −1 + (1 − )  +    +1 +   +1 +  ; LML = -323.3717.

£ ¤ 25.  = −1 + (1 − )  +  ∗ +  (  −  ∗ ) +   () +  ; LML = -323.7173. 

£ ¡  ¢¤  +  ; LML = -324.4024. 26.  = −1 + (1 − )  +  ∗ +  (  −  ∗ ) +   

£ ¤ () +  ; LML = -325.1442. 27.  = −1 + (1 − )    +    h i ˆ +  ; LML = -326.1282. 28.  = −1 + (1 − )    +   ( )  

()] +  ; LML = -332.0006. 29.  = −1 + (1 − ) [exp +    +  exp 

30.  = −1 + (1 − ) [ (  −  ∗ ) +  ( − −1 )] +  ; LML = -333.0426. 31.  = −1 + (1 − ) [ (  −  ∗ ) +   ] +  ; LML = -333.3899. h i ˆ +  ; LML = -333.5812. ( ) 32.  = −1 + (1 − )    +  exp  

()] +  ; LML = 336.3676. 33.  = −1 + (1 − ) [   +  exp 

34.  = −1 + (1 − ) [   +  ( − −1 )] +  ; LML = -336.5676. 35.  = −1 + (1 − )  (  −  ∗ ) +  ; LML = -337.1657.

£ ¤   36.  = −1 + (1 − )    +    ( ) +  ; LML = -337.1807.

£ ¡  ¢¤  +  ; LML = -338.5821. 37.  = −1 + (1 − )  +    +   

£ ¡ ¢¤ 38.  = −1 + (1 − )    +   + ∆  − −1 +  ; LML = -346.1662.

39.  = −1 + (1 − ) (   +   ) +  ; LML = -346.9183.

£ ¤  40.  = −1 + (1 − )    +1 +    +1 () +  ; LML = -347.0577. 29

£ ¤ 41.  = −1 + (1 − )  +    +   () +  ; LML = -347.7626.  42.  = −1 + (1 − )    +  ; LML = -348.8098.

43.  = −1 + (1 − )    +1 +  ; LML = EPi: -368.4967.

¡ ¢ 44.  = −1 + (1 − )    +1 +   +1 +  ; LML = -368.8848.

We have also estimated rules 2, 10, 31 and 39 on a shorter sample (1990q1-2007q4). The relative ranking of those four rules resulted unchanged, with an improvement of about 10 log-points at each step in moving from the worse to the best performing.

30

D

Tables

       ∗      ∗     ∗

Distribution Gamma (1 2) Gamma (01 005) Beta (06 02) Beta (06 02) Beta (07 015) Normal (15 025) Normal (05 02) Normal (2 1) Normal (2 1) Normal (3 035) Beta (05 02) Beta (05 02) Beta (05 02) Beta (095 004) Inverse Gamma 1 (05 2) Inverse Gamma 1 (05 2) Inverse Gamma 1 (05 2) Inverse Gamma 1 (05 2) Inverse Gamma 1 (02 1)

5% 0.6953 0.0342 0.2486 0.2486 0.4266 1.0888 0.1710 0.3551 0.3551 2.4243 0.1718 0.1718 0.1718 0.8710 0.1663 0.1663 0.1663 0.1663 0.0661

Prior Median 0.9867 0.0918 0.6143 0.6143 0.7166 1.5000 0.5000 2.0000 2.0000 3.0000 0.5000 0.5000 0.5000 0.9601 0.3433 0.3433 0.3433 0.3433 0.1377

95% 1.3501 0.1938 0.9024 0.9024 0.9162 1.9112 0.8290 3.6449 3.6449 3.5757 0.8282 0.8282 0.8282 0.9943 1.2367 1.2367 1.2367 1.2367 0.4959

Table 1: Prior distributions (mean and standard deviation are in parenthesis).

31

       ∗         

Dist G G B B B N N N N N B B B IG1 IG1 IG1 IG1

5% 0.6953 0.0342 0.2486 0.2486 0.4266 1.0888 0.1710 0.3551 0.3551 2.4243 0.1718 0.1718 0.1718 0.1663 0.1663 0.1663 0.1663

Prior Median 0.9867 0.0918 0.6143 0.6143 0.7166 1.5000 0.5000 2.0000 2.0000 3.0000 0.5000 0.5000 0.5000 0.3433 0.3433 0.3433 0.3433

95% 1.3501 0.1938 0.9024 0.9024 0.9162 1.9112 0.8290 3.6449 3.6449 3.5757 0.8282 0.8282 0.8282 1.2367 1.2367 1.2367 1.2367

Mode 1.0187 0.0037 0.2772 0.1366 0.7831 1.2732 1.2963 2.4477 2.3744 2.9577 0.9181 0.8901 0.6583 0.6549 0.3270 0.2333 0.3896

Posterior 5% Median 0.6830 0.9729 0.0025 0.0148 0.2828 0.5405 0.0684 0.2306 0.7769 0.8443 1.1970 1.7445 0.2713 0.6593 1.7061 2.4569 1.6776 2.3362 2.5266 2.9639 0.4147 0.9109 0.1300 0.6370 0.0822 0.4533 0.2347 0.5446 0.2650 0.7936 0.1729 0.3331 0.4049 0.5137

Table 2: Baseline estimation results

32

95% 1.3346 0.0563 0.7095 0.7616 0.8895 2.2500 1.3108 3.1443 2.9588 3.3999 0.9654 0.9272 0.9762 1.2918 4.5182 0.6115 0.6126

E

Figures

Model efficient Empirical HP 0.02

0.01

0

−0.01

−0.02

−0.03

−0.04 1990

1995

2000

2005

Figure 1: Baseline interest rate rule — Efficient output gap (posterior median and 90% uncertainty bands) vs. HP filter of output data (point estimates).

33

−3

x 10 6

5

4

3

2

1

0

−1

−2

−3

−4 1990

1995

2000

2005

Figure 2: Baseline interest rate rule — Estimated markup shocks (posterior median and 90% uncertainty bands).

34

0.03

Model HP Empirical HP

0.02

0.01

0

−0.01

−0.02

−0.03

−0.04 1990

1995

2000

2005

Figure 3: Interest rate rule with statistical output gap — Model-based HP filter (Posterior median and 90% uncertainty bands) vs. Emprical HP filter (Point estimates).

35

Model Efficient Empirical HP 0.02

0.01

0

−0.01

−0.02

−0.03 1990

1995

2000

2005

Figure 4: Interest rate rule with time-varying intercept — Efficient output gap (posterior median and 90% uncertainty bands) vs. Empirical HP filter (Point estimates).

36

Model Efficient Real Interest Rate Empirical HP Output Gap

0.02

0.015

0.01

0.005

0

−0.005

−0.01

−0.015

−0.02 1990

1995

2000

2005

Figure 5: Interest rate rule with time-varying intercept — Efficient real interest rate (posterior median and 90% uncertainty bands) vs. Empirical HP filter (Point estimates).

37

0.03

Model HP Output Gap Model Efficient Interest Rate

0.02

0.01

0

−0.01

−0.02

−0.03

−0.04 1990

1995

2000

2005

Figure 6: Interest rate rule with statistical output gap — Efficient real interest rate vs. Modelbased HP filter (both variables: posterior median and 90% uncertainty bands).

38

Core PCE Inflation FFR

9

8

7

%

6

5

4

3

2

1

0

1990

1995

2000

2005

Figure 7: Core PCE inflation and Fed Funds Rate (both variables: seasonally adjusted quarterly observations at annual rates).

39

Model Efficient Empirical HP

0.02

0.015

0.01

0.005

0

−0.005

−0.01

−0.015

1990

1995

2000

2005

Figure 8: Interest rate rule with time-varying intercept and inflation target — Efficient output gap (posterior median and 90% uncertainty bands) vs. Empirical HP filter (point estimates).

40