Simple Rules Versus Optimal Policy: What Fits?
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Bache, Ida Wolden; Brubakk, Leif; Maih, Junior Working Paper Simple Rules Versus Optimal Policy: What Fits? Working Paper, No. 2010/03 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Bache, Ida Wolden; Brubakk, Leif; Maih, Junior (2010) : Simple Rules Versus Optimal Policy: What Fits?, Working Paper, No. 2010/03, ISBN 978-82-7553-546-5, Norges Bank, Oslo, https://hdl.handle.net/11250/2497461 This Version is available at: https://hdl.handle.net/10419/209948 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/deed.no
2010 | 03 Simple rules versus optimal policy: what fits? By Ida Wolden Bache, Leif Brubakk and Junior Maih Working Paper Monetary Policy Department
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Simple rules versus optimal policy: what …ts? Ida Wolden Bachey , Leif Brubakk and Junior Maih Monetary Policy Wing, Norges Bank (Central Bank of Norway) This version: 7 April 2010 First version: 4 June 2008 Abstract We estimate a small open-economy DSGE model for Norway with two speci…cations of monetary policy: a simple instrument rule and optimal policy based on an intertemporal loss function. The empirical …t of the model with optimal policy is as good as the model with a simple rule. This result is robust to allowing for misspeci…cation following the DSGE-VAR approach proposed by Del Negro and Schorfheide (2004). The interest rate forecasts from the DSGE-VARs are close to Norges Bank’s o¢ cial forecasts since 2005. One interpretation is that the DSGE-VAR approximates the judgment imposed by the policymakers in the forecasting process. Keywords: DSGE models, forecasting, optimal monetary policy JEL classi…cation: C53, E52 We are grateful for comments from Jesper Lindé, Marco Del Negro, Øistein Røisland, Tommy Sveen, Shaun Vahey, Lutz Weinke and participants at the CEF conference in Paris 2008, the ESEM 2008 meeting in Milan, the Third annual Dynare conference in Boston 2008, the Norges Bank Workshop on Optimal Monetary Policy in November 2008 and the National Bank of Poland Workshop on Experiences and Challenges of Forecasting at Central Banks November 2009. The views in this paper are our own and should not be interpreted as re‡ecting the views of Norges Bank. yCorresponding author. Address: Monetary Policy Department, Norges Bank (Central Bank of Norway). E-mail: [email protected] 1
1 Introduction The purpose of this paper is to compare the empirical merits of di¤erent approaches to modelling monetary policy within the context of a dynamic stochastic general equilibrium (DSGE) model. To this end we evaluate a New Keynesian small open economy model estimated on Norwegian data under alternative speci…cations of monetary policy. We believe the case of Norway to be of general interest. First, to our knowledge, Norges Bank is the only central bank that has stated publicly that it uses ‘optimal’policy as a normative benchmark for monetary policy (see e.g., Holmsen et al. (2007)). Second, since 2005 Norges Bank has published its own interest rate projections along with forecasts of other key macrovariables. In most DSGE models, the central bank is assumed to set the interest rate according to a simple instrument rule (e.g., a Taylor rule). In addition to computational simplicity, one reason behind the popularity of this approach is that simple instrument rules have been shown to give a reasonable empirical description of actual monetary policy in many countries. Moreover, simple rules are perceived to be more robust in that they perform reasonably well in terms of welfare across models. An alternative approach is to assume that monetary policy is conducted optimally. By optimally, we mean that the central bank chooses the interest path that minimizes an intertemporal loss function. The optimal policy approach gives a more symmetric treatment of central bank and private sector behaviour and, moreover, allows the central bank to make e¢ cient use of all relevant information. As pointed out by Svensson (2003), it seems somewhat odd to assume a priori that the central bank has a less sophisticated approach to optimization than the private agents. Finally, the optimizing framework appears to be more in line with the way monetary policy is actually conducted in most developed countries. From an empirical point of view it is not obvious which of the two policy assumptions provides the most plausible account of the data. There are two opposing mechanisms at play. On the one hand, the optimal policy framework is more ‡exible than the simple instrument rule in the sense that the implied interest rate rule contains a larger set of variables than the simple instrument rule. However, this ‡exibility comes at the cost of introducing a new set of restrictions on the reduced form solution of the model, restrictions that could potentially be at odds with the data. The estimated model is similar in size and structure to NEMO, the model that is used as the core model in the policy process in Norges Bank1, and thus constitutes a real world example of empirical interest. The model is estimated using Bayesian techniques on data for the Norwegian mainland economy over the period 1987Q1–2007Q4. We consider two di¤erent speci…cations of monetary policy: a simple instrument rule and optimal policy based on a loss function that is consistent with the monetary policy remit. The di¤erent speci…cations of the model are compared using both in-sample and out-of-sample measures of …t, where the latter exercise is based on recursive forecasts from 1998Q1 to 2007Q4. We 1See Monetary Policy Report 3/07 (available at www.norges-bank.no). 2
stress the out-of-sample forecasting properties of the models for two reasons. First, model comparisons based on Bayesian measures of in-sample …t can be problematic (see e.g., Sims (2003)). Second, and more importantly, forecasting is a key activity of an in‡ation targeting central bank. Hence, in practical policy work, models are ultimately judged by their forecasting properties. In order to shed some light on the accuracy of the pure model projections, we also compare the model forecasts of the interest rate and in‡ation to the o¢ cial forecasts actually published by Norges Bank from 2005Q4 onwards.2 There exists a small, but increasing literature estimating New Keynesian models with optimal monetary policy. Dennis (2004) jointly estimates the parameters in the central bank’s objective function and the parameters in the optimizing constraints in a New Keynesian model of the US economy, under the assumption that monetary policy is conducted optimally under discretion. In two recent papers Ilbas (2008a) and Ilbas (2008b) use a Bayesian approach to estimate the monetary policy preferences in New Keynesian closed-economy models for the euro area and the US assuming that the central bank minimises an intertemporal loss function under commitment. Adolfson et al. (2009) estimate an operational medium-scale, small open economy DSGE model for the Swedish economy and compare the in-sample …t of models with alternative assumptions about monetary policy. We supplement and add to their results by also considering the out-of-sample forecasting performance of the models. Our …ndings can be summarised as follows. First, the in-sample …t of the model with optimal policy is superior to the model with a simple instrument rule. However, in terms of forecasting accuracy, which is our favoured measure of model …t, the models perform about equally well. Turning to the absolute performance, the estimated models signi…cantly overshoots both the actual outcomes and the o¢ cial Norges Bank forecasts for in‡ation and the interest rate. This overshooting is more pronounced for the model with optimal policy than in the model with a simple instrument rule, re‡ecting in part the fact that optimal policy is solved under the assumption of timeless commitment. Interestingly, the parameter estimates appear to be quite robust to the choice of monetary policy. Thus, it would be tempting to conclude that the model parameters in this sense are structural. However, this would ignore the issue of misspeci…cation. In the above exercise, we implicitly assume that, under each of the two approaches to modelling monetary policy, the resulting theoretical model provides an accurate probabilistic description of our data. This is obviously a strong assumption. Despite the recent progress in getting DSGE models to …t the data (see e.g., Smets & Wouters (2004), Edge et al. (2010), Adolfson et al. (2007b) and Adolfson et al. (2007c)), potential model misspeci…cation remains a key concern. As discussed in Del Negro & Schorfheide (2009), the issue of model misspeci…cation in DSGE models can be approached in a number of ways. The more practical approach, favoured by most central banks, is to account for model misspeci…cation by adding signi…cant amounts of judgment to the forecasts from their core models. In order to investigate the importance of model misspeci…cation, we employ the DSGE-VAR approach proposed by Del Negro & Schorfheide (2004). In their framework, 2This coincides with the quarter where the Norges Bank …rst started publishing its interest rate paths. 3
the DSGE model is used as a prior to inform the parameters of an unrestricted vector autoregression (VAR). The idea is to impose some of the structure from the theoretical model on the less dogmatic data representation provided by the VAR. The DSGE-VAR approach still produces estimates of the parameters in the DSGE model that can be compared to those obtained using the traditional full-information approach. In some sense, the DSGE-VAR captures the dichotomy between model and judgement in practical policy work. One of the questions we ask in this paper is to what extent accounting for misspeci…cation a¤ects the parameter estimates and the forecasts from the models. In a related paper, Adjemian et al. (2008) use the DSGE-VAR framework to compare the in-sample …t of a closed economy DSGE model for the US economy when monetary policy is conducted optimally under commitment and when the central bank follows a Taylor-type rule. Recognising that in-sample model comparison within a Bayesian framework can be problematic, we extend their results by considering the out-of-sample forecasting performance of the two models. Our recursive estimation procedure has the added advantage that it allows us to investigate the stability of the parameters over time. Our paper also di¤ers from Adjemian et al. (2008) in that we assume the same set of stochastic disturbances in the two models, which makes the model comparison more transparent. Based on the marginal data densities, we …nd that the data clearly favour the DSGEVAR model with optimal policy. This runs contrary to the …ndings in Adjemian et al. (2008). However, as in our benchmark case, the forecasting performance of the two DSGEVAR models are almost identical. Interestingly, allowing for misspeci…cation brings the projected interest and in‡ation paths from the DSGE-VARs much closer to both the actual outcomes and Norges Bank’s o¢ cial interest rate forecasts. In this sense, the DSGE-VAR models can be said to better capture the judgment imposed by the policymakers. The remainder of the paper is organized as follows. In section 2 we give a brief description of the DSGE model used in the empirical exercise. In section 3 we present the estimation strategy and the empirical results for the two speci…cations of the DSGE model. In section 4 we discuss the results obtained when using the DSGE-VAR approach. Section 5 concludes the paper. 2 The DSGE model The benchmark DSGE model used in the forecasting exercise is a medium-scale New Keynesian open economy model. The theoretical framework builds on the New Open Economy Macroeconomics (NOEM) literature (see e.g., Lane (2001) for a survey) as well as the closed economy models in e.g., Christiano et al. (2005) and Smets & Wouters (2003), and is similar in structure to existing open-economy models such as the Global Economy Model (GEM) model at the International Monetary Fund and the model developed in Adolfson et al. (2007a).3 The economy has two production sectors. Firms in the intermediate goods sector produce di¤erentiated goods for sale in monopolistically competitive markets at home and 3We refer to Brubakk et al. (2006) for a more thorough discussion of the model and literature references. 4
abroad, using labour and capital as inputs. Firms in the perfectly competitive …nal goods sector combine domestically produced and imported intermediate goods into an aggregate good that can be used for private consumption, private investment and government spending. The household sector consists of a continuum of in…nitely-lived households that consume the …nal good, work and save in domestic and foreign bonds. The model incorporates real rigidities in the form of habit persistence in consumption, variable capacity utilisation of capital and investment adjustment costs, and nominal rigidities in the form of local currency price stickiness and nominal wage stickiness. The model is closed by assuming that domestic households pay a debt-elastic premium on the foreign interest rate when investing in foreign bonds. The model evolves around a balanced growth path as determined by a permanent technology shock. The …scal authority runs a balanced budget each period, and we consider two alternative speci…cations of monetary policy. The exogenous foreign variables are assumed to follow autoregressive processes. Final goods sector The perfectly competitive …nal goods sector consists of a continuum of …nal good producers indexed by x2[0;1] that aggregates composite domestic intermediate goods, Q, and imports, M, using a constant elasticity of substitution (CES) technology: At(x) = h 1 Qt(x)11 + (1 ) 1 Mt(x)11 i 1;(1) The degree of substitutability between the composite domestic and imported goods is determined by the parameter > 0, whereas (01) measures the steady-state share of domestic intermediates in the …nal good for the case where relative prices are equal to 1. The composite good Q(x)is an index of di¤erentiated domestic intermediate goods, produced by a continuum of …rms h2[0;1]: Qt(x) = 2 4 1 Z0 Qt(h; x)11 tdh3 5 t t1 ;(2) where the time-varying elasticity of substitution between domestic intermediates is captured by tand evolves according to: ln t =ln t1 +" t;0<1; " tiid 0; 2 (3) where > 1is the steady-state value. Similarly, the composite imported good is a CES aggregate of di¤erentiated import goods indexed by f2[0;1]: Mt(x) = 2 4 1 Z0 Mt(f; x)11 fdf3 5 f f1 ;(4) 5
where f>1is the steady-state elasticity of substitution between imported goods. Intermediate goods sector Each intermediate goods …rm his assumed to produce a di¤erentiated good Tt(h)for sale in domestic and foreign markets using the following CES production function: Tt(h) = (1 ) 1 ZtzL tlt(h)11 + 1 Kt(h)11 1 ;(5) where 2[0;1] is the capital share and denotes the elasticity of substitution between labour and capital. The variables lt(h)and Kt(h)denote, respectively, hours used and e¤ective capital of …rm hin period t. There are two exogenous shocks to productivity in the model: Ztrefers to an exogenous permanent (level) technology process, which grows at the gross rate z t, whereas zL tdenotes a temporary (stationary) shock to productivity (or labour utilization). The technology processes are modelled as ln(Zt) = ln(Zt1) + ln(z) + ln z t z;(6) where ln z t z=zln z t1 z+"z t;0z<1; "z tiid 0; 2 z;(7) and ln zL t zL=Lln zL t1 zL!+"L t;0L<1; "L tiid 0; 2 L:(8) The variable Kt(h)is de…ned as …rm h’s capital stock that is chosen in period tand becomes productive in period t+ 1. Firm h’s e¤ective capital in period tis related to the capital stock that was chosen in period t1by Kt(h) = ut(h)Kt1(h);(9) where ut(h)is the endogenous rate of capital utilization. When adjusting the utilization rate the …rm incurs a cost of u t(h)units of …nal goods per unit of capital. The cost function is u t(h) = u 1eu 2(ut(h)1) 1;(10) where u 1and u 2are parameters determining the cost of deviating from the steady state utilization rate. The steady state utilization rate is normalized to one.4 Firm h’s law of motion for physical capital reads: Kt(h) = (1 )Kt1(h) + t(h)Kt1(h);(11) where 2[0;1] is the rate of depreciation and t(h)denotes capital adjustment costs. The adjustment costs take the following form: 4Note that u 1is not a free parameter. It is set to ensure that the marginal cost of utilisation is equal to the rental rate of capital in steady-state. 6
2.1.1 Data and estimation method The model is estimated on quarterly, seasonally adjusted data for the Norwegian economy covering the period from 1987Q1 to 2007Q4. The sample period available for presample estimation is 1981Q4-1986Q4. The estimation is based on the following eleven variables: GDP, private consumption, business investment, exports, the real wage, the real exchange rate, overall in‡ation, imported in‡ation, the 3-month nominal money market rate, the overnight deposit rate (the policy rate) and hours worked. Since the model predicts that domestic GDP, consumption, investment, exports and the real wage are non-stationary, these variables are included in …rst di¤erences. We take the log of the real exchange rate and hours worked. The data series relate to the mainland economy, that is, the total economy excluding the petroleum sector. The series for GDP, exports, consumption, business investment and hours worked are measured relative to the size of the working age population (16-74 yrs.). The real wage is measured as total wage income per hour divided by the private consumption de‡ator. The quarterly series for growth in wage income per hour is obtained by taking a linear interpolation of the annual series from the national accounts. The nominal exchange rate is an e¤ective import-weighted exchange rate based on the bilateral exchange rates of the Norwegian krone versus 44 countries. Consumer price in‡ation is measured as the total CPI adjusted for taxes and energy (CPI-ATE), and imported in‡ation is measured as the in‡ation rate for imported goods in the CPI-ATE. The money market rate is the 3 months e¤ective nominal money market rate (NIBOR). All the series are demeaned prior to estimation. The choice of information set is based on data availability and on the perceived quality of the data series as well as a desire to obtain good estimates of the structural parameters in the DSGE model.11 In general, the issue of parameter identi…cation points to including a large number of variables in the information set.12 Within the context of a DSGE-VAR, however, the price of working with a large set of variables is that the size of the VAR becomes large relative to the sample size, resulting in imprecise estimates of the VAR parameters and wide forecast error bands. In particular, the VAR becomes much larger than what is typically used in standard forecasting applications.13 We estimate the DSGE models from a Bayesian perspective. The estimation of the DSGE model is based on the state-space representation (40). The likelihood function is evaluated using the Kalman …lter and we use a Metropolis-Hastings (MH) algorithm to draw from the posterior distribution of the structural parameters starting from the posterior mode of the parameters computed in a …rst step. The full-sample results reported below are based on 3million draws from the posterior distribution. In the forecasting experiment, the number of draws in each recursion is 100000.14 11 E.g., due to perceived poor quality of the national accounts data, imports are not used as an observable variable. 12 See e.g., the discussion in Adolfson et al. (2007a). 13 For example, a typical VAR for a small open economy contains a measure of real activity, in‡ation, the exchange rate and the interest rate in addition to foreign variables. 14 The results are obtained using Dynare (see http://www.cepremap.cnrs.fr/dynare/) and our own Matlab 13
The shape, the mean and the standard deviation of the prior distributions for the estimated parameters are given in tables 3 and 4. Priors for the means are partly taken directly from other studies and partly chosen in order to provide shock responses that are consistent with our prior beliefs on the transmission mechanism of the Norwegian economy. Note that we apply the same priors independent of the choice of monetary policy. This is meant to re‡ect the somewhat heroic assumption that these parameters are truly structural. Another way to choose the priors, would be to follow the approach of Del Negro & Schorfheide (2008a). In our case, their approach would imply having di¤erent sets of priors for the structural parameters depending on the choice of monetary policy. However, since we deal explicitly with the issue of misspeci…cation in the DSGE-VAR setup, it makes sense to assume that the priors on the structural parameters are independent of the policy assumptions. Some of the parameters were …xed at the outset. This can be interpreted as a very strict prior, where all the probability mass is concentrated on a single value. The steadystate per capita growth rate zis calibrated to equal 2:25 per cent on an annualised basis. Based on current estimates,15 we assume a long-run annual real interest rate of 2:5per cent. Consistent with this, we set the discount factor to 0:9994. The quarterly depreciation rate of capital is set to 1:8per cent, which is in line with the recent estimates from the national accounts. The steady-state elasticity of substitution between di¤erentiated intermediate goods, and is set to 6corresponding to a price mark-up on marginal cost of 20 per cent. The home bias parameter,16 , is set close to 0:65 to ensure a steady state import share of roughly 30 per cent, and the elasticity of substitution between capital and labour, , is set to 0:7, which yields a steady state wage income share of 0:6. The utilization cost parameter, u 2;is set to 0:38. Some parameters, such as the parameters related to investment costs, I 1and the adjustment cost parameter in export prices Mfturned out to be di¢ cult to identify. Furthermore, it is not possible to identify both intermediation cost parameters B 1and B 2, using a …rst order approximation of the model. We therefore set B 1=I 1=Mf= 1. 2.1.2 Full-sample estimation results Table 1 reports measures of the in-sample …t of the DSGE model for alternative assumptions about the conduct of monetary policy. The marginal data density is measured using the modi…ed harmonic mean estimator proposed by Geweke (1999). A key result is that the model with a simple instrument rule is clearly dominated by the model with optimal policy in terms of in-sample …t. Hence, the implicit rule following from the assumption of optimal monetary policy appears to give a more accurate description of the way monetary policy was conducted over the sample period than does a simple instrument rule. However, this result depends to a large extent on the symmetric treatment of the shock processes in the codes for estimating of DSGE models with optimal policy under commitment and forecasting with a DSGEVAR. 15 See Norges Bank’s In‡ation Report 2/06. 16 This parameter represents the share of domestic intermediates in the …nal goods aggregate that would prevail in the hypothetical case where the prices on domestic and imported goods were equal. 14
two models. Including a policy shock in the simple rule brings the marginal data density much closer to the model with optimal policy. Adolfson et al. (2009) also make the point that the ranking of the models in terms of in-sample …t will depend on whether one or both of the models include a monetary policy shock. E.g., they …nd that when the instrument rule includes a monetary policy shock, but the model with optimal policy does not, the model with a simple instrument rule gives a better …t. Note, however, that the instrument rule considered in Adolfson et al. (2009) is somewhat more ‡exible than the instrument rule considered in this paper; in addition to the level variables, it includes both the change in in‡ation and in the growth rate of GDP. Turning to the parameter estimates, table 2 reports the estimates of the monetary policy preferences from the DSGE model. The estimates imply a high relative weight on interest rate changes in the loss function. The posterior estimates of the remaining parameters are reported in tables 3 and 4. Comparing the DSGE models, the parameter estimates do not seem to be signi…cantly in‡uenced by the choice of monetary policy, consistent with the …nding in Adolfson et al. (2009). This conclusion is supported by the impulse responses of the estimated shocks, which appear fairly similar. However, there is one notable exception to this conclusion. The stickiness of domestic good prices is estimated to be signi…cantly higher in the model employing a simple instrument rule. As we shall see in the next section, this could potentially explain the di¤erences in the forecasting properties of the two models, in particular with respect to in‡ation and the interest rate. 2.1.3 Forecast comparison The forecast experiment is constructed as follows. We estimate each model on a sample period ending 1998Q4 and compute forecasts for horizons of one up to twelve quarters. We then extend the sample by one quarter, demean the data, re-estimate the models and compute new forecasts. The implicit steady-states of the variables are allowed to vary over time; we demean the data prior to estimation in each recursion. This exercise is repeated until the end of the sample. All the parameters in the DSGE model are re-estimated in each recursion. The forecasts are based on 100000 MH draws starting from the posterior mean of the previous recursion. We measure forecasting accuracy by univariate root mean squared forecast error (RMSE). The point forecasts used to calculate the RMSEs are the posterior means of the forecast draws. Following Adolfson et al. (2007c) we also compute a measure of multivariate forecast accuracy, namely the trace of the mean squared forecast error (MSE) matrix for horizon h. The MSE matrix is denoted M(h)and is de…ned as M(h) = 1 Nh T+Nh1 X t=TYt+hb Yt+hjtM1Yt+hb Yt+hjt0;(42) where Nhis the number of forecasts and Mis a diagonal matrix with the sample variances of the variables as diagonal elements. For the variables that enter the model in growth rates, we follow Del Negro et al. (2007a) and report the RMSE for the cumulative changes 15
in the variables. Figure 1 plots the univariate RMSEs from the DSGE model under the di¤erent assumptions about monetary policy. The ranking of the models is less clear than was the case when using measures of in-sample …t based on the full sample. In terms of forecasting accuracy the models perform about equally well. The model with optimal policy produces more accurate forecasts of the growth rates of GDP, consumption and investment, whereas the model with a simple instrument rule produces more accurate forecasts of the in‡ation and interest rates. We conjecture that one reason why the model with a simple instrument rule produces more accurate forecasts of in‡ation and interest rates is that price stickiness parameter is estimated to be higher in this version of the model, giving rise to weaker equilibrium-correction, which is an inherent feature of both interest rates and in‡ation over the out-of-sample period. As a next step we compare the model projections of in‡ation and the interest rate with the o¢ cial Norges Bank projections. The exercise is somewhat restricted by the fact that o¢ cial forecasts are only available from 2005 onwards, and the fact that forecasts are published only three times per year, however, we still believe that it provides some interesting insights. Figure 4 shows the DSGE forecasts and the o¢ cial forecasts for each quarter in the period 2005q3-2008q2.17 As is evident from the …gures, both versions of the DSGE model consistently predict a sharper increase in interest rates than the o¢ cial forecasts. This is especially true for the model assuming optimal policy. Furthermore, we observe that Norges Banks o¢ cial forecasts are more in line with the actual interest path. However, in contrast to the DSGE models, there seems to be a slight tendency for the Norges Bank forecast to under-predict the actual interest path. The di¤erences between the model forecasts and the o¢ cial Norges Bank forecasts re‡ect to some extent the use of judgment and o¤-model considerations in arriving at the …nal projections. This can be interpreted as an attempt to correct for model misspeci…cation. 3 Acknowledging model misspeci…cation In the above exercise, we implicitly assume that, under each of the two approaches to modelling monetary policy, the resulting theoretical model provides an accurate probabilistic description of our data. In this section, we assess the robustness of our results to model misspeci…cation using the DSGE-VAR approach proposed by Del Negro & Schorfheide (2004). The DSGE-VAR approach allows us to relax the tight cross-equation restrictions implied by the DSGE model for the parameters in a VAR. The DSGE-VAR approach also produces estimates of the parameters in the DSGE model that can be compared to those obtained using the traditional full-information approach. 17 Norges Bank publishes forecasts three times a year. To compare these forecasts to the forecasts from our quarterly model we have added a "synthetic" forecast round with forecasts equal to the previously published path. In general, the forecasts made by Norges Bank are made later in time and in that sense incorporate more information than the model forecasts. 16
3.1 The DSGE-VAR approach As alluded to in the introduction, the basic idea of the DSGE-VAR approach is to use the DSGE model to construct prior distributions for the VAR. The starting point for the estimation is an unrestricted VAR of order p Yt=0+1Yt1+2Yt2+ +pYtp+ut;(43) where Ytis an n1vector of observables, 0is an n1vector of constant terms, iare nnmatrices of autoregressive parameters i= 1; : : : ; p and utN(0;u):If we let the vector of regressors in the VAR be denoted xt= [1; yt1; yt2; : : : ; ytp], the VAR can be written compactly as Y=X + U; (44) where Yis Tnwith rows y0 t,Xis T(1 + np)with rows x0 t,Uis Tnwith rows u0 t and = 0 0; 0 1; : : : ; 0 p. The likelihood function for the VAR is given by p(Yj;u)/ jujT=2(45) exp (1 2tr "1 u Y0Y0X0Y Y0X+0X0X!#) The prior distribution for the VAR parameters proposed by Del Negro & Schorfheide (2004) is based on the VAR approximation to the DSGE model. Let xx; yy; xy and yx be the theoretical second-order moments of the variables in Yand Ximplied by the DSGE model. Then ()=1 xx () xy()(46) u()= yy() yx()1 xx () xy() can be interpreted as the probability limits of the coe¢ cients in a VAR estimated on arti- …cial observations generated by the DSGE model. Conditional on the vector of structural parameters in the DSGE model , the prior distribution for the VAR parameters p(;uj); is of the Inverted-Wishart (IW) - Normal (N) form uj=IW (T u(); T k; n)(47) ju; =N();u(T xx)1 where k= 1+np. The tightness of the prior distribution is governed by the hyperparameter 2[0;1]. This hyperparameter can be loosely interpreted as the size of the sample of arti…cial or dummy observations generated by the DSGE model relative to the size of the actual sample in the estimation. The posterior distribution of the VAR parameters is also of the Inverted-Wishart - 17
Normal form (see Del Negro & Schorfheide (2004)) ujY; =IW (+ 1) Te u();(1 + )Tk; n(48) jY; u; =Ne ();uT xx +X0X1 The matrices e ()and e u()have the interpretation of maximum likelihood estimates of the VAR parameters based on the combined sample of actual observations and arti…cial observations generated by the DSGE model, that is e () = T xx +X0X1T xy() + X0Y(49) e u() = 1 (+ 1)TT yy() + Y0Y(50) 1 (+ 1)TT yx() + Y0XT1 xx () + X0X1T xy() + X0Y From the above expressions we see that if is small, the prior on the DSGE model restrictions is di¤use. In particular, setting = 0 we would retrieve the unrestricted OLS estimates. Notice, however, that in order for the prior distribution (47) to be proper, has to take a value larger than min = (k+n)=T (see e.g., Adolfson et al. (2007b)). The higher is ; the more the VAR estimates will be tilted towards the parameters in the VAR approximation of the DSGE model (()and u()). Del Negro et al. (2007a) choose by maximising the marginal data density p(Y)over a pre-speci…ed grid for . In this paper we specify a uniform distribution for over the interval [min;1). The speci…cation of the VAR prior is completed with the speci…cation of prior distributions for the DSGE model parameters : The DSGE-VAR approach allows us to draw posterior inferences about the DSGE model parameters . As explained by Del Negro & Schorfheide (2004), the posterior estimate of has the interpretation of a minimumdistance estimator, where the minimand or distance function is given by the discrepancy between the unrestricted OLS estimates of the VAR parameters and the coe¢ cients in the VAR approximation to the DSGE model, the latter being functions of . Obviously, then, the posterior estimates of will depend on the hyperparameter . In the limit !0; there will not be any information about in p(Yj);and hence, the posterior estimates of will be equal to the prior estimates. 3.2 Empirical results The estimation of the DSGE-VAR is based on the MH algorithm to draw from the joint posterior distribution of ; u; described in Del Negro & Schorfheide (2004). An important modelling choice for the DSGE-VAR is the choice of lag length. As argued by Del Negro et al. (2007b) there are essentially two dimensions to the choice of lag length for a DSGE-VAR. The …rst dimension is related to the accuracy of the VAR approximation to the DSGE model. This suggests we choose the lag-length to minimise the approximation error, that is, to minimise the discrepancy between the dynamics of the DSGE-VAR(1) 18
and the dynamics of the DSGE model. Since, in general, the accuracy of the approximation increases with lag length, this criterion points to having a fairly large number of lags. In the previous literature (see e.g., Adolfson et al. (2007b) and Del Negro et al. (2007a)), the lag-length has commonly been set to four based on this criterion. The second dimension to the choice of lag length is the empirical …t of the DSGE-VAR with the optimal value of ;that is the DSGE-VAR(b ). This suggests that we choose the lag length to maximise the marginal data density associated with the DSGE-VAR(b ). As emphasized by Del Negro et al. (2007a), there is no requirement that the auxiliary model (the DSGE-VAR) nests the underlying theoretical model (the VAR approximation to the DSGE model) for the exercise to be meaningful. For our model(s), we …nd that the marginal data density is maximised for the model with two lags. The optimal value of the hyperparameter, , is smaller in the model with two lags compared with the model with four lags, however. This re‡ects that the gains from shrinking towards the theoretical model are smaller in the former case, since there are fewer free parameters in the VAR. Similar …ndings were reported by Del Negro & Schorfheide (2008b). 3.2.1 Full-sample estimation results Table 1 reports the posterior mean of the hyperparameter in the DSGE-VAR and the marginal data densities for the two speci…cations of monetary policy. The estimated weight on the DSGE model in the DSGE-VAR is higher in the case of optimal policy than in the model with a simple rule (the posterior mean of the hyperparameter is 1:14 in the case of optimal policy and 0:89 in the model with a simple rule). We also see that the …t of the model is improved if we shrink the VAR parameters towards the restrictions implied by the DSGE model, or, alternatively, if we relax the DSGE model restrictions in the direction of the unrestricted VAR estimates. That is, the marginal data density is higher for the DSGE-VAR than for the DSGE model. This is true under both assumptions about monetary policy. In the next subsection we examine whether this holds true also in terms of out-of-sample forecasting performance. Table 2 reports the estimates of the monetary policy parameters obtained using the DSGE-VAR approach. The parameters in the loss-function do not appear to be much a¤ected by allowing for model misspeci…cation. This does not hold true for the parameters in the simple instrument rule: the weight on in‡ation increases signi…cantly once we allow for misspeci…cation. In this sense, the optimal policy framework appear to be more robust to misspeci…cation than a model with a simple Taylor-type rule. As evidenced in tables 3 and 4, the estimates of the other parameters in the model di¤er even less than for the DSGE models. This indicates that part of the di¤erences in the estimated DSGE parameters are due to misspeci…cation. One way to think about this is that misspeci…cation adds an extra source of variation to the estimated parameters. Another robust …nding is that the degree of external persistence as measured by the …rst-order autocorrelation of the exogenous shock processes is reduced signi…cantly once misspeci…cation is taken into account. It is clear from table 4 that both the autocorrelation coe¢ cient and the standard deviation of the shock processes are in general lower in the 19
DSGE-VAR models than the DSGE models. Hence, taking into account misspeci…cation reduces the need for exogenous persistence.18 3.2.2 Forecast comparison In addition to the DSGE-VAR forecasts we compute forecasts from a Bayesian VAR (BVAR) with a Minnesota-type prior. The prior in the BVAR will tilt the VAR towards univariate random walks of the variables in levels. The lag-length in the BVAR is set to two. All the parameters in the BVAR and DSGE-VARs, including the hyperparameter ; are re-estimated in each recursion. The forecasts are based on 100000 MH draws starting from the posterior mean of the previous recursion. Figure 2 compares the univariate RMSEs from the DSGE model with optimal policy to those obtained using a DSGE-VAR approach and the BVAR. For most variables, relaxing the cross-equation restrictions in the DSGE model towards an unrestricted VAR improves the forecasting performance. However, in terms of forecasting performance the DSGE-VAR models are inferior to the BVAR with a Minnesota prior. This …ndings is con…rmed in …gure 3 which reports a multivariate measure of forecast accuracy. The fact that the BVAR outperforms the DSGEs is perhaps not surprising. The BVAR prior tilts the unrestricted VAR towards univariate random walks. Given that in‡ation and interest rates are only borderline stationary in our sample, this seems like a very reasonable prior. From …gure 4, we note that the interest rate forecasts from the DSGE-VAR are quite close to the o¢ cial forecasts. Hence, accounting for misspeci…cation brings the model interest rate projections much more in line with the published forecasts. The same holds for in‡ation (see …gure 5): the DSGE-VAR forecasts are closer to both actual in‡ation and the o¢ cial projections than the DSGE model forecasts. One tentative conclusion one could draw from this exercise is that the DSGE-VAR model mimics the combination of pure model forecasts and judgment inherent in the o¢ cial Norges Bank forecasts. As noted above, the DSGE model employed in this paper is broadly similar to the core model used for policy projections at Norges Bank. However, arriving at the …nal o¢ cial projections is a complex process, involving input from other forecasting models, add factors and o¤-model considerations. Our results indicate that the iterative forecasting process used by the Norges Bank can be well represented by a DSGE-VAR model, where the restrictions from the core DSGE model can be interpreted as a prior on the VAR parameters. A notable feature of the interest rate and in‡ation projections from the DSGE model with optimal policy is that they ‘overshoot’the long-run level in the medium run. This feature of optimal policy under commitment is less pronounced in Norges Bank’s projections since 2005, and is not a feature of the DSGE-VAR forecasts. This is a sign that the model with optimal monetary policy is misspeci…ed. One interpretation is that the Norges Bank does not fully exploit the expectations channel when setting policy, or alternatively, that it perceives the gains from commitment in the current speci…cation of the DSGE model to 18 Similar …ndings are again reported by Del Negro & Schorfheide (2008b). 20
be too large (e.g., that the price-setters in the model are in a sense too forward-looking). 4 Concluding remarks The results in this paper suggest that the empirical merits of the DSGE model estimated with optimal monetary policy is comparable to the performance of a model with a simple Taylor-type rule –both with regards to in-sample and out-of-sample measures of …t. This conclusion also holds when taking account of misspeci…cation. One way of interpreting the DSGE-VAR results, is that introducing optimal monetary policy reduces the degree of misspeci…cation. Interestingly, in contrast to the model using a simple rule, the policy parameters in the optimal policy model appear to be quite robust to misspeci…cation. Hence, based on our empirical …ndings and given the superior theoretical (and intuitive) appeal of the optimizing approach to monetary policy, we argue that the optimal policy framework should be a natural ingredient in any DSGE model describing central bank behaviour. However, as is evidenced both by the optimal value of the DSGE-VAR hyperparameter and the forecasting performance of the di¤erent models, model misspeci…cation remains a serious concern for the use of DSGE models in practical policy analysis. Hence, one tentative conclusion that could be drawn from our analysis is that the empirical gains from assuming optimal monetary policy is of second order relative to improving the modelling of the transmission mechanism itself. In this respect, it is interesting to note that o¤-model considerations appear to bring the o¢ cial projections closer to the DSGE-VAR model, and, as a result, also reduce the forecast errors. 21
References Adjemian, S., Pariès, M. D., & Moyen, S. (2008). Towards a Monetary Policy Evaluation Framework. Working Paper 942, ECB. Adolfson, M., Laséen, S., Lindé, J., & Svensson, L. E. (2009). Optimal monetary policy in an operational medium-sized DSGE model. Mimeo June 2009, available at Lars E. O. Svensson’s research page at www.iies.su.se. A previous version was published as NBER Working Paper 14092 in 2008. Adolfson, M., Laseén, S., Lindé, J., & Villani, M. (2007a). Bayesian estimation of an open economy DSGE model with incomplete pass-through. Journal of International Economics, 72, 481–511. Adolfson, M., Laséen, S., Lindé, J., & Villani, M. (2007b). Evaluating an estimated New Keynesian small open economy model. Journal of Economic Dynamics and Control. forthcoming. Adolfson, M., Lindé, J., & Villani, M. (2007c). Forecasting performance of an open economy DSGE model. Econometric Reviews, 26, 289–328. Brubakk, L., Husebø, T. A., Maih, J., Olsen, K., & Østnor, M. (2006). Finding NEMO: Documentation of the Norwegian Economy Model. Norges Bank Sta¤ Memo 2006/6. Christiano, L. J., Eichenbaum, M., & Evans, C. L. (2005). Nominal rigidities and the dynamic e¤ects of a shock to monetary policy. Journal of Political Economy, 113, 1–45. Del Negro, M. & Schorfheide, F. (2004). Priors from general equilibrium models for VARs. International Economic Review, 45, 643–673. Del Negro, M. & Schorfheide, F. (2008a). Forming priors for DSGE models (and how it a¤ects the assessment of nominal rigidities). Journal of Monetary Economics, 55, 1191–1208. Del Negro, M. & Schorfheide, F. (2008b). In‡ation Dynamics in a Small Open-Economy Model under In‡ation Targeting: Some Evidence for Chile. Sta¤ Reports 329, Federal Reserve Bank of New York. Del Negro, M. & Schorfheide, F. (2009). Monetary policy analysis with potentially misspeci…ed models. American Economic Review, 99, 1415–1450. Del Negro, M., Schorfheide, F., Smets, F., & Wouters, R. (2007a). On the …t of New Keynesian models. Journal of Business Economic Statistics, 25, 123–143. Del Negro, M., Schorfheide, F., Smets, F., & Wouters, R. (2007b). Rejoinder. Journal of Business and Economic Statistics, 25, 159–162. Dennis, R. (2004). Inferring policy objectives from economic outcomes. Oxford Bulletin of Economics and Statistics, 66 (Supplement), 735–764. 22
0 2 4 6 8 10 12 0 10 20 30 40 50 60 70 DSGE (SIMPLE) DSGE (OPTIMAL) DSGE−VAR (SIMPLE) DSGE−VAR (OPTIMAL) BVAR Figure 3: Multivariate trace statistic for DSGE model with optimal policy, DSGE model with simple instrument rule, DSGE-VAR with optimal policy, DSGE-VAR with simple instrument rule and BVAR 29
2001Q2 2004Q1 2006Q4 2009Q3 1 2 3 4 5 6 7 8 Actual DSGE (OPTIMAL) NORGES BANK 2001Q2 2004Q1 2006Q4 2009Q3 1 2 3 4 5 6 7 8 Actual DSGE (SIMPLE) NORGES BANK 2001Q2 2004Q1 2006Q4 2009Q3 1 2 3 4 5 6 7 8 Actual DSGE−VAR (OPTIMAL) NORGES BANK 2001Q2 2004Q1 2006Q4 2009Q3 1 2 3 4 5 6 7 8 Actual DSGE−VAR (SIMPLE) NORGES BANK Figure 4: Actual policy rate, Norges Bank’s o¢ cial forecasts and model forecasts 30
2001Q2 2004Q1 2006Q4 2009Q3 0 0.5 1 1.5 2 2.5 3 3.5 Actual DSGE (OPTIMAL) NORGES BANK 2001Q2 2004Q1 2006Q4 2009Q3 0 0.5 1 1.5 2 2.5 3 3.5 Actual DSGE (SIMPLE) NORGES BANK 2001Q2 2004Q1 2006Q4 2009Q3 0 0.5 1 1.5 2 2.5 3 3.5 Actual DSGE−VAR (OPTIMAL) NORGES BANK 2001Q2 2004Q1 2006Q4 2009Q3 0 0.5 1 1.5 2 2.5 3 3.5 Actual DSGE−VAR (SIMPLE) NORGES BANK Figure 5: Actual four quarter in‡ation, Norges Bank’s o¢ cial forecasts and model forecasts 31