Practical Tools for Policy Analysis in DSGE Models with Missing Channels

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Practical Tools for Policy Analysis in DSGE Models with Missing Channels Dario Caldara, Richard Harrison, and Anna Lipinska 2012-72 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Practical Tools for Policy Analysis in DSGE Models with Missing Channels∗ Dario Caldara† Richard Harrison‡ Anna Lipi«ska§ September 13, 2012 Abstract Inthispaperweanalyzethepropagationofshocksoriginatinginsectorsthatarenot present in a baseline dynamic stochastic general equilibrium (DSGE) model. Specifically, we proxy the missing sector through a small set of factors, that feed into the structural shocks of the DSGE model to create correlated disturbances. We estimate the factor structure by matching impulse responses of the augmented DSGE model to those generated by an auxiliary model. We apply this methodology to track the e(cid:27)ects ofoilshocksandhousingdemandshocksinmodelswithoutenergyandhousingsectors. JEL Classi(cid:28)cation: C51, C53. Keywords: Misspeci(cid:28)cation, DSGE Models, Policy Analysis. ∗The views in this paper are those of the authors and do not necessarily re(cid:29)ect those of the Bank of England’s Monetary Policy Committee and of the Federal Reserve System. We would like to thank Giancarlo Corsetti, Lars Lundqvist, Elmar Mertens, John Roberts, Ulf S(cid:246)derstr(cid:246)m, James Warren, seminar participants at the Bank of England, the Federal Reserve Board, the Sveriges Riksbank, the European Central Bank, and the 2012 CEF annual meeting for valuable comments. †Board of Governors of the Federal Reserve System, Washington DC, United States. Email: dario.caldara@frb.gov. ‡Bank of England, London, UK. Email: richard.harrison@bankofengland.co.uk. §Board of Governors of the Federal Reserve System, Washington DC, United States. Email: anna.lipinska@frb.gov. 1

1 Introduction Over the past decade, there has been a marked increase in the use of dynamic stochastic general equilibrium (DSGE) models in policy institutions. The seminal work of Smets and Wouters (2003, 2007) is regarded by many as a proof of concept that medium-scale DSGE models can be useful tools for policy analysis (Sims, 2008). Smets and Wouters (2007), SW henceforth, showed that models of this type could deliver reasonable forecast performance as well as the story-telling capabilities that (cid:29)ow from explicit assumptions about the optimization decisions of economic agents. Indeed, a number of central banks have recently 1 developed operational policy models based on this blueprint. Though DSGE models in use at central banks follow the approach pioneered by SW, they dwarf them in scale. While the SW model is estimated on seven data series, operational models are designed to explain the behavior of two to three times as many data series. One reason why operational central bank models are larger than their academic counterparts surely stems from policymakers’ desire to have detailed and comprehensive discussions about a large number of shocks and transmission channels. But all models, regardless of size, are misspeci(cid:28)ed. For example, DSGE models in use at central banks typically contain only basic modelling of (cid:28)nancial frictions, banking, and the labor market. This is not to say that models with such features do not exist. Indeed, research on these issues is currently a very fertile area, and one response to the observation thatoperationalmodelsexcludesomechannelsandmechanismsofinterestistoexpandthem accordingly. Naturally, there are some di(cid:30)culties associated with this approach: if the model is to be estimated, then computational considerations place a (practical) upper bound on the number of observable variables; and larger models are inherently harder to understand and explain to busy policymakers. But even if this strategy is a desirable long-term objective, in 1ProminentexamplesincludetheFederalReserveBoard’sEDOmodel(Chungetal.,2010),theRAMSES model developed at Sveriges Riksbank (Adolfson et al., 2007), the NAWM model of the European Central Bank (Christo(cid:27)el et al., 2008), the Norges Bank’s NEMO (Brubakk et al., 2006), and the Bank of Spain’s MEDEA (Burriel et al., 2010). 2

the short run it is possible that the economic issues relevant for policy discussions develop more quickly than the operational models used to support those discussions. For instance, during the (cid:28)nancial crisis policy-makers were interested in the e(cid:27)ects of (cid:28)nancial shocks and the interaction between the (cid:28)nancial sector, the macro-economy, and the conduct of monetary and (cid:28)scal policy. Re-designing the policy models from scratch to inform policymakers would have been too costly, and more importantly, too slow. In this paper, we describe a practical approach for modelling the propagation of shocks originating in sectors that are not included in a baseline estimated DSGE model (henceforth the policy model) used for forecasting and policy analysis. As an example, suppose that policy-makers want to know how an increase in house prices due to unexpectedly strong demand for housing might a(cid:27)ect GDP growth and in(cid:29)ation. Unfortunately, their policy 2 model does not contain a housing sector. Our procedure works as follows. First, we identify a shock to the housing sector and the associated impulse response functions(IRFs)usinganauxiliarymodel. Inourexample,weidentifyashockinhouseprices using a structural vector autoregression (SVAR) as in Iacoviello (2005). More generally, we select a suitable auxiliary model that is able to capture the dynamic response of a (sub)set of variables that have a clear counterpart in the policy model. Auxiliary models could be smaller DSGE models, SVARs, or forecasting models already in use within the central bank. Impulse responses do not necessarily have to come from a formal model. For instance, they might re(cid:29)ect institutional knowledge, including the policy-makers’ views, or (cid:28)ndings documented in memos and policy reports written within the organization. Second, we introduce the missing housing sector in the policy model through some observable variables and, if necessary, a small set of unobserved factors, which we model as a reduced-form VAR. We allow both the observable variables and the factors to feed into the exogenous processes of the policy model to create correlated disturbances. The unobserved factors capture propagation mechanisms that are speci(cid:28)c to the housing sector and are not 2We implement this exercise in Section 5. 3

present in the policy model. Third, we estimate the additional parameters in the augmented policy model by matching its impulse responses to those generated by the auxiliary model. We do not re-estimate the deep parameters of the policy model because re-estimation of the entire model could create identi(cid:28)cation problems and would make the method inapplicable in a short time frame. Impulse responses from the auxiliary model capture the likely propagation of the shock of interest, summarizing moments in the data that are a(cid:27)ected by the additional parameters. We use our methodology to study the propagation of two missing shocks in a threeequation New Keynesian (NK) model (Clarida et al., 1999). We consider two examples in which data used in the estimation exercise are simulated from larger DSGE models. In the (cid:28)rst example, the data generating process (DGP) is the oil model of Nakov and Pescatori (2010b). In the second example, the DGP is the housing model of Iacoviello (2005). We (cid:28)nd that oil shocks are propagated as correlated disturbances to technology and price mark-up, without relying on unobserved factors. Instead, the NK model does a poor job at propagating house demand shocks without relying on unobserved factors. The reason is that this simple model misses the (cid:28)nancial accelerator mechanism present in Iacoviello (2005), which provides hump-shaped and persistent dynamic responses. We are able to capture such mechanisms through two unobserved factors, loading either on the technology or the mark-up process. We show that the policy implications derived from the augmented policy model are similar to those derived from the DGP. We also provide an empirical application, estimating the e(cid:27)ects of housing shocks in the SW model using U.S. data. We (cid:28)nd impulse responses to housing shocks in line with the existing theoretical and empirical literature. Furthermore, our results suggest that the richer the structure of the policy model, the less reliant the augmented policy model may be on unobserved factors to propagate the missing shock. Since operational DSGE models in use at central banks are large-scale models, policy experiments conducted in augmented versions of such models might only moderately rely on unobserved factors, which reduces the reliance 4

on nonstructural dynamics. We think that augmenting the policy model to study the e(cid:27)ects of missing shocks, as opposed to directly using di(cid:27)erent models, is sensible for at least three reasons. First, the policy model provides a careful micro-foundation of the main transmission channels of monetary policy, which we would like to preserve in the analysis of the alternative scenarios involvingmissingsectors. Second, theaugmentedpolicymodelcanbeusedtoconductpolicy experiments, assuming that the modelling of the missing sector is policy invariant. Third, communications between sta(cid:27) and policy-makers often rely on the policy model. We emphasize that the approach presented in this paper is not an ideal approach to deal with misspeci(cid:28)cation. The only correct approach is to develop models with a careful articulation of the sector of interest, and the interaction of that sector with the rest of the economy. Our methodology provides a compromise between the use of reduced-form models and fully structural models, preserving the structural dimensions of the policy model that best (cid:28)t the data. This argument is also the corner stone of the DSGE-VAR approach discussed in Del Negro and Schorfheide (2009). Although we design our modelling and estimation approaches to be of practical use in policy institutions, these tools can be of broader applicability, for instance by macroeconomists who want to evaluate the ability of a DSGE model to propagate shocks of interest. OnepaperthatisrelatedtooursisCœrdiaandReis(2010), whichestimateDSGEmodels withcorrelateddisturbances, andusethesemodelstoaccountforempiricalregularitiesinthe US business cycle. We di(cid:27)er from their study because we generate correlated disturbances through a speci(cid:28)c channel, which we identify using information from auxiliary models. Theremainderofthepaperisstructuredasfollows. InSection2, weprovideadescription of the methodology. In Sections 3 and 4 we illustrate the approach using simple examples with a known data generating process. In Section 5, we turn to an empirical example using the DSGE model of Smets and Wouters (2007) to track the e(cid:27)ects of house price shocks. Section 6 concludes the paper. 5

2 Methodology 3 The baseline model (cid:21) the policy model (cid:21) has the following form: X = A E X +A s (1) t 1 t t+1 2 t where X is a vector of endogenous variables, E is the expectation operator and s is a vector t t t of exogenous processes. All variables are measured as log-deviations from steady state. The matrices A and A are functions of the DSGE parameter vector ΘP, though we suppress 1 2 this notation for convenience. The exogenous processes are modeled as a VAR: s = Bs +Cε , (2) t t−1 t where matrices B and C are again functions of the DSGE parameter vector ΘP, and ε is a t vector of orthogonal structural shocks. TheendogenousvariablesX canbepartitionedintopredeterminedendogenousvariables, t Z , and non-predetermined endogenous variables, z : t t   z t X ≡   t   Z t so that the state space representation of the rational expectations equilibrium can be written as: z = DS , (3) t t S = GS +Hε , (4) t t−1 t 3This formulation is not restrictive. Lags and expectations of variables in periods beyond t+1 can be included by de(cid:28)ning them as additional variables to be included in X. 6

where   Z t S ≡  , (5) t   s t Here, S is the state vector collecting together the n ×1 vector of exogenous processes t s s , and the predetermined endogenous variables, Z . 4 t t In the VAR model for the exogenous processes, (2), B and C are usually assumed to be diagonal. The assumptions that B is diagonal and that the shocks ε are orthogonal have t two key advantages. First, these assumptions reduce the number of parameters in the model. Second, they add to the ability of the model to tell coherent stories. Because the structural shocks are given an economic interpretation, it is important that innovations to them are orthogonal. Orthogonality makes it easier to trace through the e(cid:27)ects of an exogenous impulse through the structural shock processes and onto the endogenous variables in the model. We model the variables that proxy the e(cid:27)ects of the missing channel, assuming that the model is now driven by a new vector of exogenous processes s : (cid:101)t X = A E X +A s (6) t 1 t t+1 2(cid:101)t The process s is de(cid:28)ned as follows: (cid:101)t s = s +sF, (7) (cid:101)t t t s = Bs +C(cid:15) , (8) t t−1 t sF = Λ F +Λ m , (9) t 1 t 2 t F = Φ F +Ξ u , (10) t 1 t−1 1 t m = Φ m +Ξ u . (11) t 2 t−1 2 t 4Again, matrices D, G and H are functions of the DSGE parameter vector ΘP, but we suppress this dependence for notational convenience. 7

This means that the state vector S(cid:101) of the model becomes: t   Z t      s  t S(cid:101) ≡  , (12) t    F   t    m t and that the rational expectations solution is given by: z = D(cid:101)S(cid:101), (13) t t S(cid:101) = G(cid:101)S(cid:101) +H(cid:101)ν , (14) t t−1 t where ν ≡ [ε(cid:48) u(cid:48)](cid:48) , and E [ε(cid:48) u ] = 0. t t t t t t We refer to the model described by equations (7)-(11) and (12)-(14) as the augmented 5 policy model. The vector s that enters in the augmented policy model is the sum of two components. (cid:101)t The (cid:28)rst component s is the vector of traditional DSGE exogenous processes. Innovations t to this component can be traced through the model and the story of how that shock a(cid:27)ects the endogenous variables can be constructed as usual. The second component sF is a n ×1 t s (cid:101) vector of exogenous processes, which consists of weighted-averages of unobserved factors F , t and observable variables m . The factors are driven by an exogenous disturbance u , which t t captures the missing shock. m is an n ×1 vector of observable variables that summarizes t m the evolution of the missing sector. 6 For instance, in the housing model m contains data on t house prices. Φ and Φ are coe(cid:30)cient matrices that capture the dynamics of the factors and 1 2 the proxy for the missing sector. 7 The matrices Ξ and Ξ control how the shock u a(cid:27)ects 1 2 5Ireland (2004) addresses model misspeci(cid:28)cation generalizing the measurement equation to include measurement errors modelled as VAR , while Boivin and Giannoni (2006) introduce additional observable variables in the estimation through a dynamic factor structure. 6In Section 5.3 we generalize equation (11), allowing for proxy variables m to depend on variables in the t policy model. 7We assume, without loss of generality, that the equations for the factors F and missing channel proxy 8

the factors and missing channel proxy variables. Intheexercisespresentedinthepaper, weconsidertwodi(cid:27)erentspeci(cid:28)cationsofequation (9). In the (cid:28)rst speci(cid:28)cation , we set all elements of Λ to zero, i.e. we drop the unobserved 1 factors F . The missing shock u propagates in the augmented policy model through the t t transmission mechanisms already embedded in the model, and through the law of motion 8 (11). This assumption implies that the structure of the policy model is su(cid:30)ciently rich to propagate the missing shock. In the second speci(cid:28)cation, we set all elements of Λ to zero. 2 The missing shock u propagates in the augmented policy model through the transmission t mechanismsalreadyembeddedinthemodel,andthroughthelawofmotionoftheunobserved factors (10). We assume that there are two unobserved factors, F and F , which follow 1,t 2,t a VAR(1) process. We (cid:28)nd that this parsimonious speci(cid:28)cation is su(cid:30)cient to generate impulse responses with hump shapes and persistence similar to those produced by largescale DSGE models. To identify the factors, we restrict the top 2×2 block of Λ to be an 2 identity matrix. This normalization solves the rotational indeterminacy problem ruling out linear combinations that lead to observationally equivalent models (Bernanke et al., 2005 and Baumeister et al., 2010). It would be possible to consider speci(cid:28)cations where both Λ 1 and Λ are di(cid:27)erent from zero, but we leave that for future research. 2 We assume that the unobserved factors can either load on all exogenous processes or on a subset, imposing zero restrictions in one or more rows of matrices Λ or Λ . If and where 1 2 to impose zero restrictions depends on two considerations. First, in operational models with 20 or more exogenous processes, estimating a full matrix Λ or Λ could create identi(cid:28)ca- 1 2 9 tion problems and would require a non-trivial amount of time. Second, the choice of zero restrictions depends on the type of propagation (and story) we believe is plausible. For invariablesmarewrittenincompanionformsothatΦ andΦ maycontainthecoe(cid:30)cientsoflagpolynomials 1 2 of any order. 8The missing shock u propagates through the existing transmission mechanism in the model because it t a(cid:27)ects m via equation (11) and hence sF via equation (9) and thus s via equation (7). The e(cid:27)ect on s is t t (cid:101)t (cid:101)t propagated through the model’s existing transmission channels because the A matrix in (6) is the same as 2 the A matrix in the policy model (1). 2 9Whethertherewouldbeidenti(cid:28)cationproblemswoulddependonhowmanyimpulseresponseswetarget, and the number of target periods in the estimation exercise described in the next subsection. 9

stance, if we judge that the missing shock mostly a(cid:27)ects aggregate demand, we could impose zero restrictions on the coe(cid:30)cients loading on processes that predominantly a(cid:27)ect the model through their e(cid:27)ect on potential supply (eg the TFP process). We would narrow down the channels through which the shock propagates (and rule out potentially counterfactual behavior of variables that are not targeted in the estimation), and simplify the interpretation of the results. For these considerations, we think that the judgment of the economist is a central part of the process, and cannot be replaced by purely statistical tools. One feature of our approach is that the modelling of the missing sector does not a(cid:27)ect the transmission of shocks already present in the policy model. While this dimension might be of interest, allowing for feedback between existing shocks and the missing sector would complicate substantially the estimation exercise, making the procedure less practical to use in short periods of time. In particular, to use the impulse-response matching procedure we would need to rely on auxiliary models to estimate impulse responses to more shocks in the presence of the missing sector. In a companion paper (Caldara et al., 2012), we use likelihood-based Bayesian techniques to estimate the e(cid:27)ects of missing shocks, allowing for feedback between the missing sector and the remaining model structure. A key advantage of a full information approach is that it does not require the use of an auxiliary model, which can be useful when the economist has little knowledge about the e(cid:27)ects of the missing 10 shocks. 2.1 Estimation Our methodology involves the estimation of three di(cid:27)erent models. First we estimate the parametersofthepolicymodel, ΘP. Incentralbanks, theseestimatesarealreadyavailableto theeconomist. Inourexposition,weestimateΘP usinglikelihood-basedBayesianestimation, asitiscommonpracticeinmanycentralbanks. 11 WedenotebyΘ ˆP themeanoftheposterior 10One possibility to bridge the two approaches is to elicit prior distributions on the deep parameters of the DSGE model from impulse response functions as proposed by Lombardi and Nicoletti (2011). 11For an overview of the estimation of DSGE models, see FernÆndez-Villaverde (2010) and Schorfheide (2011). 10

distributions. Second, we estimate the auxiliary model. We summarize inference from auxiliary models by impulse responses of selected variables to the shock of interest, which we denote by Ψ ˆ . 12 Third, we estimate the augmented policy model. We partition the parameters in two groups. The (cid:28)rst group is composed of all the parameters of the baseline policy model ΘP. The second group includes all parameters of the augmented block: ΘAP ≡ {Λ ,Λ ,Φ ,Φ }. 1 2 1 2 We estimate the parameters ΘAP to minimize a measure of the distance between the impulse responses generated by the augmented policy model, denoted by Ψ(ΘAP,ΘP), and those from the auxiliary model Ψ ˆ . We estimate ΘAP, (cid:28)xing the parameters of the baseline policy model to their posterior mean Θ ˆP. This choice is dictated by practical considerations, as the analysis needs to be conducted in a short period of time. Furthermore, impulse responses to the missing shock might have little information on the deep parameters of the baseline policy model. Our estimator of ΘAP is the solution to: J = min[Ψ ˆ −Ψ(ΘAP|Θ ˆP)](cid:48)V−1[Ψ ˆ −Ψ(ΘAP|Θ ˆP)], ΘAP where V is a matrix of weights. 13 We include in the objective function J the (cid:28)rst 20 periods of each impulse response. To perform the minimization of the loss function, we use the version of the CMA-ES evolutionary algorithm developed by Andreasen (2010). This algorithm performs well in (cid:28)nding global optima of ill-behaved objective functions such as the likelihood functions of DSGE models. In our experience, this algorithm is more reliable and robust than gradient 12Here we use Ψ to represent NK×1 vectors formed by stacking the impulse responses of N variables for K periods. 13We typically choose V to be the identity matrix. However, when Ψˆ are impulse responses from a SVAR identi(cid:28)ed using a Cholesky decomposition, we give smaller weight to the responses that are zero by assumption, as in Christiano et al. (2005). For the responses that are assumed to be zero by the Cholesky identi(cid:28)cation assumption, we set the corresponding entry in V to 10. 11

based search methods. 2.2 Evaluation We validate the augmented policy model by studying the IRFs of variables that are not 14 targets in the estimation. The idea is to check whether the dynamics of such variables behave reasonably. For instance, in Section 5 we validate the SW model augmented with a housing sector studying the (unrestricted) IRFs of private consumption, investment, real wages, and hours worked to a housing demand shock. If we think the behavior of some of these variables is in contrast with the likely e(cid:27)ects of a housing demand shock, we either re-estimate the model modifying the target variables, or we re-estimate the model imposing additional restrictions in the augmented (housing) block. This procedure is well suited to validate the model, given that our main objective is to havean augmentedpolicymodel capable ofexplainingthe propagationof missing shocksand that is useful for policy analysis. We do not use root mean square errors or likelihood-based criteria because a good forecasting model might not necessarily be a good model for policy analysis. 3 The E(cid:27)ects of Oil Price Shocks In this section, we investigate how to augment a policy model to track the e(cid:27)ects of oil price shocks. We (cid:28)rst describe a DSGE model with a micro-founded oil sector. We use this model as the data generating process in the estimation exercise. We then describe and estimate a policy model that does not contain the oil sector. Finally, we augment the policy model as described in Section 2 to track the e(cid:27)ects of oil shocks. 14In addition we also study the behavior of variables that are targets in the estimation, but at horizons that are not targeted, i.e. in our estimation exercise at horizons larger than 20. 12

3.1 The Data Generating Process We take the oil model described by Nakov and Pescatori (2010a,b) as the data generating process (DGP). In this model, the oil sector has two players: a dominant producer, representing the OPEC cartel, which has monopoly power, and a set of atomistic producers, who act under perfect competition and restrain the market power of the cartel. These assumptions imply that the oil price and the oil supply are endogenous variables, that react to all shocks in the economy and to the conduct of monetary policy. For convenience, we report the log-linear equilibrium conditions of the model in the Appendix. We describe the calibration of the model in Table 1, which relies on the estimates for the 15 great moderation period documented in Nakov and Pescatori (2010a). We use the DGP to produce 500 observations for output growth, in(cid:29)ation, interest rate, and the growth rate of oil prices. [ADD TABLE 1 HERE] 3.2 The Policy Model The policymaker has access to a smaller model, which does not contain the oil sector. The log-linear equilibrium conditions are: y = E y −(i −E π −rre), t t t+1 t t t+1 t rre = −(1−ρ )a , t a t π = βE π +λ(y +ν ), t t t+1 t t i = φ i +(1−φ )(φ π +φ y )+r . t i t−1 i π t y t t There are three exogenous processes: a technology shock a , a mark-up shock ν and t t 15Compared to Nakov and Pescatori (2010a), we increase the standard deviation of the technology shock. The reason is that, compared to the original paper, we drop the shock to the time discount factor because the presence of four shocks while using only three observed variables created identi(cid:28)cation problems. 13

monetary policy shock r . The exogenous processes evolve as: t a = ρ a +(cid:15)a, t a t−1 t ν = ρ ν +(cid:15)ν, t ν t−1 t where r is a an iid innovation with mean zero and standard deviation σ . t r [ADD TABLE 2. HERE] We estimate the policy model using Bayesian maximum likelihood on data simulated 16 from the model in section 3.1, which includes the oil sector. Thus, relative to the true DGP, the estimated policy model is misspeci(cid:28)ed. The observed variables are output growth, in(cid:29)ation, and the nominal interest rate. Estimation results are reported in Table 2. The mean estimate for nearly all parameters is close to the true value. The only coe(cid:30)cient for which the true value does not lie within one standard deviation is φ . This bias is largely y 17 due to the misspeci(cid:28)cation of the policy model. Yet, misspeci(cid:28)cation seems to be mostly captured by the exogenous processes. In particular, the correlation between the smoothed series for the technology and the mark-up processes is 0.51. Hence the assumption that the processes are uncorrelated is clearly violated. 3.3 The Augmented Policy Model We now assume that an economist wants to estimate the e(cid:27)ects of a 10% increase in oil prices on output, in(cid:29)ation, and the interest rate, without having access to the data generating process described in Section 3.1. [ADD TABLE 3 HERE] 16Since we estimate the model using 500 observations, the prior distributions receive a very small weight in the estimation. With such a long sample, we could have as well estimated the model using maximum likelihood. 17The use of 500 observations should rule out small sample bias. We re-estimated the model using 1000 observations and the results were largely unchanged. 14

The policy model described in the previous section is similar to the DGP. The main di(cid:27)erenceisthatinthepolicymodel,themarkupprocessisexogenous. Thismisspeci(cid:28)cation meansthattheeconomistcannotidentifycorrectlythesourcesof(cid:29)uctuationsinthemark-up and in the e(cid:30)cient real interest rate. Yet, the transmission mechanisms embedded in the two models are nearly identical. For this reason, we augment the policy model without relying on unobserved factors (that is, assuming that Λ = 0 in the context of equation 9): 1 s = s +Λ po, (cid:101)t t 2 t s = Bs +C(cid:15) , t t−1 t po = ρ po +uo. t o t−1 t Weassumethatoilpricespo followanAR(1)process, andtheyonlyreacttoanexogenous t oil price shock uo. We allow the oil price process to a(cid:27)ect all of the existing exogenous t processes, i.e. we do not restrict any element of Λ to zero. 2 We (cid:28)x all parameters of the policy model to the posterior means reported in Table 2. We estimate the loading factors Λ and the persistence parameter for the oil process ρ 2 o matching impulse responses to an oil shock produced by an auxiliary model, which in this exercise is the DGP described in Section 3.1. The reason is that in this controlled experiment we want to test whether our method is able to match the true IRFs. In real life applications, 18 when the DGP is unknown, the auxiliary model can be a SVAR. We report in Table 3 the parameter estimates. We plot in Figure 1 the target impulse responses (blue line), and the impulse responses from the augmented model (red dashed). The estimation targets the (cid:28)rst 20 periods only (in the (cid:28)gure, periods 21 to 40 are not targeted). [ADD FIGURE 1 HERE] We target the impulse responses of in(cid:29)ation, interest rate, oil prices, and output growth. 18Using our data set, the identi(cid:28)cation of an oil shock using a SVAR produces impulse responses that are very close to those of the data generating process. Hence results based on matching impulse responses from the SVAR are almost identical to those presented here. 15

The augmented model is able to match these responses almost perfectly, loading the process for oil prices on the technology and mark-up processes. These loadings are consistent with how the oil technology shock enters in the true model, where it a(cid:27)ects both the mark up and the real e(cid:30)cient rate. The (cid:28)gure also shows the response of the output level which, not surprisingly, perfectly resembles the response in the DGP. However, the response of the output gap between the two models is very di(cid:27)erent. In the augmented model, the oil shock loads mostly on the mark-up shock, which does not a(cid:27)ect potential output. As a result, the output gap in the augmented model closely follows the dynamics of the output level. [ADD FIGURE 2 HERE] Figure 2 plots impulse responses from the DGP (left column) and from the augmented policy model (right column) to a 10% increase in oil prices for di(cid:27)erent values of the Taylor rule coe(cid:30)cient on in(cid:29)ation. The augmented policy model is able to correctly identify changes intheresponsesofoutput,in(cid:29)ation,andtheinterestrategeneratedbyalessaggressivestance of monetary policy on in(cid:29)ation. The small quantitative errors are due to the fact that in the augmented policy model, we neglect the feedback e(cid:27)ect of the change in monetary policy on oil prices, which in the DGP happens via the output gap in the oil mark-up determination. The augmented policy model does a poor job in tracking the e(cid:27)ects of the policy change on the output gap (for the same reasons explained in the previous paragraph). 4 The E(cid:27)ects of House Price Shocks In this section, we consider a model where the missing channel is more deeply embedded within the endogenous structure of the economy. Speci(cid:28)cally, we assume that the data generatingprocessisamodelwherethereisanimportantroleforhousepricesindetermining consumption. We follow the same steps as in Section 3. First, we specify the data generating process; then we specify and estimate the policymaker’s (misspeci(cid:28)ed) model; we (cid:28)nally augment the policymaker’s model to try to account for the missing channel. 16

4.1 The Data Generating Process We use the model of Iacoviello (2005). Here, we provide a general description of the model, the structural equations of the model can be found on page 745 in Iacoviello (2005). The model is a variant of the Bernanke et al. (1999) New Keynesian model where endogenous changes in the balance sheets of (cid:28)rms create a (cid:28)nancial accelerator e(cid:27)ect. The model also includes collateral constraints tied to the value of housing property for (cid:28)rms which is used as one of the factors of production. These features create a (cid:28)nancial accelerator where demand shocks are ampli(cid:28)ed. When demand rises, asset prices rise, which in turn increases the borrowingcapacityofdebtors(i.e(cid:28)rms). Thisboostsconsumptionspendingandinvestment. As consumer prices rise, the real value of debtors’ outstanding obligations falls and real net worth rises. Because borrowers have a higher propensity to spend than lenders, there are further increases in demand. [ADD TABLE 4 HERE] As noted, the main innovations of the model are related to the behavior of demand. The remainder of the model is standard. Calvo price setting leads to a conventional New Keynesian Phillips curve relating in(cid:29)ation to marginal costs. The monetary policymaker is assumed to operate a reaction function for the nominal interest rate, which has a Taylor (1993) formulation adjusted to include interest rate smoothing. The model is driven by four shocks: to technology (a ), to the Phillips curve (u ), to monetary policy (r ), and to housing t t t preferences (j ). The housing preference shock is a stochastic variation in the relative weight t on housing in consumers’ utility functions. We refer to this shock as a house price shock (following Iacoviello, 2005) in what follows. Iacoviello (2005) sets the parameters of the model using a minimum distance estimator that matches the impulse responses of the model to those in an identi(cid:28)ed VAR estimated on US data. For our data generating process, we largely rely on Iacoviello’s reported parameter estimates, and the calibration is reported in Table 4. We use the DGP to produce 500 observations for output growth, in(cid:29)ation, interest rate, and house prices. 17

4.2 The Policy Model We assume that the policy model is the same three-equation New Keynesian model described in Section 3.2. The only exception is that, following Iacoviello (2005), we write the model in terms of output growth instead of the output gap and the Taylor rule is assumed to respond to the output level rather than the output gap. We estimate the parameters of the policy model using Bayesian Maximum likelihood. The central bank observes data on output growth, in(cid:29)ation, and the nominal interest rate. [ADD TABLE 5 HERE] Estimation results are reported in Table 5. The policy model lacks mechanisms capable of generating persistence in the e(cid:27)ects of exogenous shocks embedded in the true model. For this reason, the degree of price stickiness and the autocorrelation coe(cid:30)cient for the technology process display a marked upward bias. Furthermore, part of the volatility in the data generated by the house preference shock, is accounted by the estimated volatility of the mark-up shock, which is also substantially larger than the true volatility of this shock in the data generating process. 4.3 The Augmented Policy Model To incorporate the e(cid:27)ect of house prices in the policy model, we consider two alternative augmentedpolicymodels. Inbothexerciseshouseprices, theproxyvariable, followanAR(1) process: ph = ρ ph +uh. (15) t h t−1 t For the (cid:28)rst exercise, we check how a shock to house prices uh propagates in the augt mented policy model without relying on unobserved factors (Λ = 0). In the estimation 1 exercise, we keep all parameters of the policy model (cid:28)xed to the posterior means reported in Table 5. [ADD TABLE 6 HERE] 18

We estimate the loading factors Λ and the persistence parameter for the housing process 2 ρ matching the responses for output, in(cid:29)ation, the interest rate, and house prices to a h housing shock obtained from the DGP. We report in Table 6 the estimated parameters. House prices load mostly on the technology process a . t [ADD FIGURE 3 HERE] We plot in Figure 3 the target impulse responses (blue line), and the impulse responses from the augmented model (red dashed). The estimation targets the (cid:28)rst 20 periods (solid line), while we leave periods 21 to 40 unrestricted. The augmented model captures well the dynamics of house prices, but does a fairly poor job at mimicking the dynamics of the remaining variables. The reason is simple: the propagation mechanisms embedded in the policy model are not capable of generating the hump-shaped and persistent response due to the (cid:28)nancial accelerator in the DGP. To mimic such dynamics, without altering the propagation of other shocks, we augment the policy model introducing two unobserved factors as described in equations (7)-(11). The loading factor matrix Λ is: 1   1 0     Λ =  0 1 , 1     λ λ r,1 r,2 and we set Λ = 0. We estimate the coe(cid:30)cients Λ , Φ , Ξ and ρ targeting the same impulse 2 1 1 1 h responses as in the previous exercise. [ADD FIGURE 4 HERE] Figure 4 plots the target responses (blue line) and the responses generated by the augmented policy model (red dashed line). The augmented policy model does a very good job at tracking the true impulse responses, although it slightly underestimates the persistence at longer (untargeted) horizons. Instead of reporting the estimates for the loading coe(cid:30)cients Λ , which are hard to interpret, Figure 5 plots sF, the process-speci(cid:28)c factors. 1 t [ADD FIGURE 5 HERE] 19

The mark-up factor sF is 1 order of magnitude larger than the factors loading on the u,t technology and monetary processes. In fact, setting λ , λ , and λ to zero, the impulse 1,a 1,r 2,r responses from the augmented policy model are nearly unchanged. These results are not surprising, given that the housing shock in the Iacoviello (2005) model induces negative correlation between output and in(cid:29)ation, which can be also generated by a mark-up shock. Since the Taylor rules in the policy model and the DGP are similar, the policy response in both models is also very close. The technology process is also capable of generating negative correlation between output and in(cid:29)ation. In a separate exercise (not reported), we re-estimate the augmented policy model restricting all elements of Λ to zero, except for the 1 loading factors on the technology process. This restricted version of the model is capable of generating impulse responses that are nearly identical to the responses reported in Figure 4. [ADD FIGURE 6 HERE] Figure 6 plots impulse responses from the data generating process (left column) and from the augmented policy model (right column) to an increase in house prices for di(cid:27)erent values of the Taylor rule coe(cid:30)cient on in(cid:29)ation. The augmented policy model is able to correctly identify the qualitative changes in the responses of output, in(cid:29)ation, and the interest rate generated by a less aggressive response of monetary policy to in(cid:29)ation. The augmented policy model also does a good job capturing the quantitative response of output, while it underestimates the stronger response of in(cid:29)ation and the interest rate when monetary policy responds less aggressively to in(cid:29)ation. The reason is that the augmented policy model lacks su(cid:30)cient endogenous persistence to generate changes in the in(cid:29)ation response, generating such persistence through the (policy invariant) unobserved factors. Notice that di(cid:27)erences in quantitative results are not due to the lack of endogenous response in house prices, which 19 in the data generating process and in the augmented policy model are nearly identical. 19Itisworthnotingthathousepriceresponsesinthedatageneratingprocessarelittlea(cid:27)ectedbychanges in the coe(cid:30)cients of the monetary policy rule. 20

5 Housing in the Smets and Wouters (2007) Model In Sections 3 and 4 we estimated the policy and augmented policy models on simulated data. Results suggest that the speci(cid:28)cation of the augmented policy model is (cid:29)exible enough to generate responses in line with the data generating process and that are easy to interpret. In this section, we apply our methodology in a more realistic environment. In particular, we use the Smets and Wouters (2007) (henceforth SW) model as the policy model. We augment the SW model to study the implications of alternative assumptions about the future path of house prices for the variables in the model. Although the SW model includes a wide range of frictions and transmission channels, it does not include the housing market. Therefore, we use a small VAR to help us adjust the baseline DSGE model projections in the light of alternative house price scenarios. We conduct estimation exercises using US data. We proceed as follows. In Section 5.1, we brie(cid:29)y describe the SW model and the US data setthatweassumetobeavailablefortheforecaster. InSection5.2, wedescribetheVARthat is used to identify the e(cid:27)ects of house price shocks on a small number of key macroeconomic variables. In Section 5.3 we incorporate shocks to house prices into the DSGE model. This is done along the lines discussed in Section 4 for the Iacoviello (2005) model. 5.1 The Smets and Wouters (2007) Model We use the medium-scale DSGE model of Smets and Wouters (2007). As noted in the Introduction, this model has been used as a blueprint for the operational DSGE models developed at a number of central banks. It is also an important benchmark model in the literature. Given that the model is very well known, we only provide a sketch of its structure. The model includes a wide variety of nominal and real frictions. Households maximize utility subject to habit formation in their consumption choices. They accumulate capital (whichtheyrentto(cid:28)rms)subjecttocostsofadjustingtheratesofinvestmentandutilization. Households (via unions) also supply di(cid:27)erentiated labor to (cid:28)rms and set the nominal wage 21

according to a Calvo scheme. Wages that are not re-optimized are increased in line with a weighted average of trend nominal wage growth and lagged in(cid:29)ation. Firms rent capital services and labor from households which are used to produce output. Output is used for consumption, government purchases, and investment. Retailers set prices accordingtoaCalvomechanism, withapartialindexationofpricesthatarenotre-optimized that is analogous to the scheme for nominal wages described above. Monetary policy is conducted through a reaction function for the nominal interest rate. The reaction function speci(cid:28)es that nominal interest rates respond to deviations of in(cid:29)ation from the target, the outputgap,andthechangeoftheoutputgap. Theoutputgapisde(cid:28)nedusinga(cid:29)exible-price speci(cid:28)cation of the model. The model is driven by seven shocks: to the level of TFP; to the investment adjustment cost function; to household preferences; to government spending; to price and wage markups; and to the monetary policy reaction function. Government spending and TFP shocks are assumed to be correlated with each other. These shocks are designed to explain the movements of seven data series: GDP growth; consumption growth; investment growth; in(cid:29)ation (GDP de(cid:29)ator); the Fed funds rate; real wage growth; and hours worked. [ADD TABLES 7 AND 8 HERE] As in Smets and Wouters (2007), we estimate the parameters of the model using Bayesian techniques. Weestimatethemodelfortheperiod1984-2004usingthesamedatasetandprior distributions as Smets and Wouters (2007). Estimation results are reported in Tables 7 and 8. 5.2 The VAR Model We construct a small VAR along the lines of that estimated by Iacoviello (2005). We use the output, in(cid:29)ation and interest rate data from the Smets and Wouters (2007) data set. For house prices, we use the OFHEO house price index (all transactions). We apply the X12 seasonal adjustment process to seasonally adjust the data. We measure the house price 22

relative to the GDP de(cid:29)ator, which is the price series used to de(cid:28)ne the in(cid:29)ation measure in Smets and Wouters (2007). The house price series starts in 1984Q1 and the Smets-Wouters data set ends in 2004Q4. So this de(cid:28)nes our sample. We estimate a VAR(2) and identify a house price shock using a Cholesky decomposition with the following ordering: nominal interest rate, in(cid:29)ation, house prices and output. This orderingfollowsIacoviello(2005). Wereporttheimpulseresponsestoashocktohouseprices in Figure 7 (blue lines). 5.3 Incorporating House Price E(cid:27)ects into the DSGE Model We explore two options to introduce shocks to house prices in the SW model. Both options rely on two unobserved factors F , as the SW lacks a (cid:28)nancial accelerator mechanism. t Furthermore, we assume that house prices follow an exogenous AR(2) process: ph = φ ph +φ ph +s ε , (16) t 1 t−1 2 t−2 h t where ε is an iid Gaussian disturbance with unit variance. We choose an AR(2) process for t house prices because it generates an impulse response for ph following an housing shock in t line with the response generated by the auxiliary model. We (cid:28)rst assume that the unobserved factors F load on all seven exogenous processes. t Results are presented in Figure 7. This approach does a very good job at matching the target impulse responses (for output, in(cid:29)ation, house prices and the nominal interest rate). This is of little surprise, since loading the unobserved factors on seven exogenous processes grants much (cid:29)exibility. The model does also a good job at capturing the persistent response of output, in(cid:29)ation, and the interest rate from quarter 21 to 40 (untargeted). The four bottom panels of Figure 7 plot the response of variables that are not targets, namely consumption, investment, the real wage, and hours worked. These variables are key in providing a coherent story for policy-makers. The response of consumption, investment, and hours worked is 23

negative, while the response the real wage is very strongly positive. These responses are not in line with most evidence on the e(cid:27)ects of housing demand shocks. For instance, Iacoviello and Neri (2010) document an increase in both private consumption and investment. Figure 8 provides an explanation for this (cid:28)nding. The dynamic responses are predominantly driven by three shocks: investment-speci(cid:28)c technology, government spending, and price mark-up shocks. The decline in private consumption is due to the increase in government spending, which increases output (target variable), but in the SW model crowds out private demand. The decline in investment is due to the negative response in investment speci(cid:28)c technology. Finally, the decline in hours worked and the sharp rise in real wages is due to a large but short-lived increase in the wage mark-up. Guided by the counter-factual responses of private demand and labor market variables, in the second experiment, we load the unobserved factors only on three processes: general technology, risk premium, and price mark-up. We load on general technology because we want to generate an expansion in real activity. We load on the risk premium because we want to induce an increase in the interest rate faced by households and introduce a wedge between such interest rate and the one controlled by the central bank. Finally, we load on the price mark-up process because it helps to (cid:28)ne-tune the response of in(cid:29)ation. Figure 9 shows that, despite loading on only three processes instead of seven, our procedure still does well in terms of matching the target responses, both in the short-medium run (targeted) and in the long run (not targeted). In addition, the short-run responses of private demand and labor market variables are all positive. The negative response at long horizons mimics the 20 negative response of output to the housing demand shock. [ADD FIGURE 11 HERE] Figure 11 plots impulse responses to an increase in house prices for di(cid:27)erent values of the Taylor rule coe(cid:30)cient on in(cid:29)ation. As we can see, in contrast to the augmented policy model in Section 4, the augmented SW model is able to generate di(cid:27)erences in the response 20Instead of imposing zero restrictions on the loading matrix, we could have included a subset of these variables in the auxiliary model and used the associated IRFs as targets in the estimation exercise. 24

of in(cid:29)ation associated with di(cid:27)erent parameterizations of the monetary policy rule. One possible explanation for this result may be the richer structure of the SW model, which may allow the augmented policy model not to rely much on the (policy invariant) unobserved factors. Finally, we consider an alternative speci(cid:28)cation of the house price equation (16). In particular, we use in the augmented policy model the house price equation estimated in the auxiliary VAR model. The advantage of this speci(cid:28)cation is that we allow house prices to depend on variables included in the policy model, so that we can produce a model-consistent baseline forecast for ph. We use the equation from the auxiliary model because it provides a t reasonable reduced-form description of the evolution of house prices. The speci(cid:28)cation of the augmented policy model with the VAR equation for house prices generates IRFs and policy 21 implications in line with those reported for the benchmark model. The main reason is that the evolution of output, in(cid:29)ation, and the interest rate described by the policy model does not explain much volatility in house prices. Yet, we believe that this generalization can be useful when the proxy variable is endogenous to the variables modelled in the policy model. 6 Conclusions In this paper, we consider the problem of how to analyze the e(cid:27)ects of shocks that do not appearexplicitlywithinaDSGEmodelthatisusedtoinformpolicyandforecastdiscussions. To this end, we augmented a baseline DSGE model with an exogenous block that is intended to capture the e(cid:27)ects of shocks in the un-modelled sector. We estimated the parameters of the additional block by matching impulse responses to the shock of interest from an auxiliary model. We believe that our approach has broad applicability, and provides a practical way to address an important problem. We used our method to study the e(cid:27)ects of oil price shocks and house price shocks in 21Notsurprisingly,onlythehousepriceresponseshowsabetter(cid:28)ttothetarget,althoughthisimprovement does not a(cid:27)ect the overall estimation. Results available on request. 25

a three-equation New Keynesian model. We showed that the impulse response functions produced by the augmented DSGE models are similar to those produced by richer models with micro-founded oil and housing sectors. Furthermore, policy experiments conducted in with the augmented DSGE models and the micro-founded models delivered very similar conclusions. We then discussed an empirical application, studying the e(cid:27)ects of house price shocks in the United States using the Smets and Wouters (2007) model. 26

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A The Nakov-Pescatori (2010a,b) Model. We provide a parsimonious description of the model. We refer to the original papers for details. The IS equation is: y = E y −(i −E π −rre), t t t+1 t t t+1 t where y is the output gap, i is the nominal interest rate, π is in(cid:29)ation, and rre is the t t t t e(cid:30)cient real interest rate, described by: (1−ρ ) s (1−ρ ) rre = − a a + o z z , t (1−s ) t (1−s ) t o o where s is the oil elasticity of gross output, a is the exogenous technology process, and z 0 t t is the exogenous oil technology process. Both a and z follow an AR(1) process: t t a = ρ a +(cid:15)a, t a t−1 t z = ρ z +(cid:15)z, t z t−1 t where ρ and ρ are persistence parameters and (cid:15)a and (cid:15)z are iid innovations with mean zero a z t t and standard deviations σ and σ respectively. a z The Phillips curve is: π = λ((1−s )y +s ν )+βE π , t o t o t t t+1 where (1+ψ)(µ−s )(1−θ)(1−βθ) o λ = (µs +(µ−1)(1+ψ)s )θ l o is the slope of the Phillips curve. ν is the optimal oil price mark up, which in Nakov and t Pescatori (2010b) is a non-linear function of many state variables, including Lagrangian mul- 30

22 tipliersassociatedwiththemaximizationproblemoftheOPECproducer. Weapproximate 23 the optimal oil mark-up assuming that in(cid:29)ation in the oil importing country is always zero. As a result, we obtain: ν = ν a +ν z +ν y . t a t z t y t The mark up is a function of the technology process in the oil importing economy a , the t oil technology shock z , and the output gap in the oil importing economy y . The coe(cid:30)cients t t ν , ν , and ν are functions of various steady state ratios. a z y Finally, we close the model with the following Taylor rule: i = φ i +(1−φ )(φ π +φ y )+r , t i t−1 i π t y t t where r is a an iid innovation with mean zero and standard deviation σ . t r 22The OPEC producer chooses its price in orderto maximize welfare of its owners. Italso internalizes the impact it has on global output and oil demand. 23Ourassumptiongreatlysimpli(cid:28)estheoilmark-upfunction. Thenumericalsimulationofthemodelwith this approximated oil mark-up matches closely the original Nakov and Pescatori (2010b) model. Results are available upon request. 31

Name Value Policy Rule φ 3.00 π φ 0.54 y φ 0.69 i Other Deep Parameters θ 0.48 ψ 1.00 β 0.9926 s 0.64 l s 0.33 k s 0.03 o µ 15% π∗ 0 Shock Autocorrelations ρ 0.98 a ρ 0.96 x ρ 0.88 z Shock Standard Deviations σ *100 0.82 a σ *100 0.21 r σ *100 0.00 x σ *100 15.18 ν Table 1: Calibration of structural parameters for the Nakov and Pescatori (2010b) oil model used as data generating process and described in Section 3.1. 32

Name Prior (1) Posterior (2) Policy Rule φ Normal 2.00 0.50 2.84 (0.33) π φ Normal 0.40 0.10 0.36 (0.10) y φ Beta 0.50 0.20 0.70 (0.06) i Other Deep Parameters θ Beta 0.50 0.20 0.59 (0.04) ψ Gamma 1.00 0.25 1.00 (0.24) Shock Autocorrelations ρ Beta 0.50 0.20 0.98 (0.00) a ρ Beta 0.50 0.20 0.92 (0.02) ν Shock Standard Deviations σ *100 InvGamma 0.50 4.00 0.86 (0.03) a σ *100 InvGamma 0.50 4.00 0.21 (0.03) r σ *100 InvGamma 0.50 4.00 0.33 (0.07) ν Table 2: Prior and posterior moments - Policy model used in the oil price example described in Section 3.2. Column (1) reports the parameters of the prior distributions. Column (2) reports posterior means and standard deviations (in parenthesis). See Section 3.3 for details. λ λ λ ρ a ν r o 0.0495 0.0756 0.0100 0.8814 Table 3: Estimates of parameters in the augmented policy model, where the targets in the estimation are impulse response from the DGP. See Section 3.3 for details. 33

Name Value Policy Rule φ 3.00 π φ 0.50 y φ 0.60 i Other Deep Parameters θ 0.75 η 1.01 β 0.99 γ 0.98 j 0.1 ν 0.03 m 0.89 X 1.05 Shock Autocorrelations ρ 0.50 a ρ 0.85 j ρ 0.59 u Shock Standard Deviations σ *100 2.740 a σ *100 24.89 j σ *100 0.150 u σ *100 0.290 r Table 4: Calibration of structural parameters for the Iacoviello (2005) housing model used as data generating process and described in Section 4.1. 34

Name Prior (1) Posterior (2) Policy Rule φ Normal 3.00 0.20 2.93 (0.10) π φ Normal 0.40 0.20 0.46 (0.06) y φ Beta 0.50 0.10 0.63 (0.02) i Other Deep Parameters θ Beta 0.50 0.20 0.90 (0.03) Shock Autocorrelations ρ Beta 0.50 0.20 0.80 (0.02) a ρ Beta 0.50 0.20 0.44 (0.01) u Shock Standard Deviations σ *100 InvGamma 0.50 2.00 2.92 (0.10) a σ *100 InvGamma 0.50 2.00 0.25 (0.01) u σ *100 InvGamma 0.50 2.00 0.28 (0.01) r Table 5: Prior and posterior moments - Policy model used in the house price example described in Section 4.2. Column (1) reports the parameters of the prior distributions. Column (2) reports posterior means and standard deviations (in parenthesis). λ λ λ ρ a u r h 0.6447 0.00351 0.00015 0.8142 Table 6: Estimates of parameters in the augmented policy model, where the targets in the estimation are impulse response from a SVAR. 35

Name Prior (1) Posterior (2) φ Normal 4.00 1.50 5.80 (1.11) σ Normal 1.50 0.37 1.03 (0.12) c h Beta 0.70 0.10 0.56 (0.04) ξ Beta 0.50 0.10 0.78 (0.08) w ξ Beta 0.50 0.10 0.80 (0.04) p ι Beta 0.50 0.15 0.45 (0.16) w ι Beta 0.50 0.15 0.30 (0.09) p ψ Beta 0.50 0.15 0.62 (0.11) Φ Normal 1.25 0.12 1.48 (0.09) r Normal 1.50 0.25 1.68 (0.22) π ρ Beta 0.75 0.10 0.86 (0.02) r Normal 0.12 0.05 0.14 (0.04) y r Normal 0.12 0.05 0.17 (0.03) (cid:52)y π Gamma 0.63 0.10 0.61 (0.07) 100(β−1 −1) Gamma 0.25 0.10 0.18 (0.04) l Normal 0.00 2.00 0.60 (0.58) γ Normal 0.40 0.10 0.50 (0.02) α Normal 0.30 0.05 0.18 (0.02) Table 7: Prior and posterior moments - Smets and Wouters (2007) model used in Section 5.3.Column (1) reports the parameters of the prior distributions. Column (2) reports posterior means and standard deviations (in parenthesis). 36

Name Prior (1) Posterior (2) σ InvGamma 0.10 2.00 0.38 (0.03) a σ InvGamma 0.10 2.00 0.09 (0.02) b σ InvGamma 0.10 2.00 0.40 (0.03) g σ InvGamma 0.10 2.00 0.40 (0.05) I σ InvGamma 0.10 2.00 0.12 (0.01) r σ InvGamma 0.10 2.00 0.13 (0.02) p σ InvGamma 0.10 2.00 0.23 (0.03) w ρ Beta 0.50 0.20 0.92 (0.03) a ρ Beta 0.50 0.20 0.82 (0.08) b ρ Beta 0.50 0.20 0.97 (0.01) g ρ Beta 0.50 0.20 0.61 (0.07) I ρ Beta 0.50 0.20 0.28 (0.06) r ρ Beta 0.50 0.20 0.43 (0.16) p ρ Beta 0.50 0.20 0.67 (0.15) w µ Beta 0.50 0.20 0.34 (0.20) p µ Beta 0.50 0.20 0.44 (0.20) w ρ Beta 0.50 0.20 0.42 (0.11) ga Table 8: Prior and posterior moments - Smets and Wouters (2007) model used in Section 5.3.Column (1) reports the parameters of the prior distributions. Column (2) reports posterior means and standard deviations (in parenthesis). 37

Output gap Output level 0.2 0 -0.05 0 -0.1 -0.2 -0.15 -0.4 -0.2 -0.25 -0.6 -0.3 -0.8 -0.35 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Inflation Nominal interest rate 0 0.07 0.06 -0.01 0.05 -0.02 0.04 -0.03 0.03 0.02 -0.04 0.01 -0.05 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Oil price Output growth 10 0.2 0 8 -0.2 6 -0.4 4 -0.6 -0.8 2 -1 0 -1.2 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Figure 1: Impulse responses to an oil price shock in the DGP (blue solid line) and in the augmented policy model (red dashed). The DGP is the Nakov and Pescatori (2010b) model. See Section 3.3 for additional details. 38

Output Gap (DGP) Output Gap 0.2 0 0.15 -0.2 0.1 -0.4 0.05 -0.6 0 -0.8 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Output (DGP) Output 0 0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Inflation (DGP) Inflation 0 0 -0.05 -0.05 -0.1 -0.1 -0.15 -0.15 -0.2 -0.2 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Interest Rate (DGP) Interest Rate 0.15 0.08 0.1 0.06 0.05 0.04 0 0.02 -0.05 0 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Figure 2: Impulse responses to an oil price shock in the DGP (left column) and in the augmented policy model (right column) when the Taylor rule coe(cid:30)cient on in(cid:29)ation equals 3 (blue solid line), 2.25 (green dashed line), and 1.5 (red dotted line). The DGP is the Nakov and Pescatori (2010b) model. See Section 3.3 for additional details. 39

Output Inflation 0.15 0.2 0.1 0.1 0.05 0 0 -0.1 -0.05 -0.1 -0.2 0 10 20 30 40 0 10 20 30 40 Nominal interest rate House price 0.2 2 1.5 0.1 1 0 0.5 -0.1 0 -0.2 -0.5 0 10 20 30 40 0 10 20 30 40 Figure 3: Impulse responses to a house price shock in the DGP (blue solid line) and in the augmented policy model without unobserved factors (red dashed). The DGP is the Iacoviello (2005) model. See Section 4.3 for additional details. 40

Output Inflation 0.15 0.2 0.1 0.1 0.05 0 0 -0.1 -0.05 -0.1 -0.2 0 10 20 30 40 0 10 20 30 40 Nominal interest rate House price 0.2 2 1.5 0.1 1 0 0.5 -0.1 0 -0.2 -0.5 0 10 20 30 40 0 10 20 30 40 Figure 4: Impulse responses to a house price shock in the DGP (blue solid line) and in the augmented policy model with unobserved factors (red dashed). The DGP is the Iacoviello (2005) model. See Section 4.3 for additional details. 41

-3 uhatfactor -3 Ahatfactor x 10 x 10 5 2 0 1 -5 0 -10 -1 -15 -20 -2 0 10 20 30 40 0 10 20 30 40 -3 Rhatfactor x 10 1 0 -1 -2 -3 -4 0 10 20 30 40 Figure 5: Impulse responses of the process-speci(cid:28)c factors in the augmented policy model with unobserved factors. The DGP is the Iacoviello (2005) model. See Section 4.3 for additional details. 42

Output (DGP) Output 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Inflation (DGP) Inflation 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 -0.2 -0.2 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Interest Rate (DGP) Interest Rate 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 -0.2 -0.2 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 House Prices (DGP) House Prices 2 2 1.5 1 1 0 0.5 -1 0 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Figure 6: Impulse responses to a house price shock in the DGP (left column) and in the augmented policy model with unobserved factors (right column) when the Taylor rule coef- (cid:28)cient on in(cid:29)ation equals 3 (blue solid line), 2.25 (green dashed line), and 1.5 (red dotted line). The DGP is the Iacoviello (2005) model. See Section 4.3 for additional details. 43

Output Inflation 0.5 0.5 0 0 -0.5 -0.5 0 10 20 30 40 0 10 20 30 40 Nominal interest rate House price 0.5 1 0 0 -0.5 -1 0 10 20 30 40 0 10 20 30 40 Consumption Investment 0 5 -1 0 -2 -5 0 10 20 30 40 0 10 20 30 40 Real Wage Hours Worked 20 1 10 0 0 -1 0 10 20 30 40 0 10 20 30 40 Figure 7: Impulse responses to a house price shock in the augmented Smets and Wouters (2007) model with unobserved factors. See Section 5.3 for additional details. 44

afactor bfactor gfactor 0.5 1 0.4 0 0 0.2 -1 -0.5 -2 0 0 20 40 0 20 40 0 20 40 msfactor qsfactor spinffactor 0 0 0 -0.1 -0.5 -0.1 -0.2 -1 -0.2 0 20 40 0 20 40 0 20 40 swfactor 20 10 0 -10 0 20 40 Figure 8: Impulse responses of the process-speci(cid:28)c factors in the augmented Smets and Wouters (2007) model with unobserved factors. See Section 5.3 for additional details. 45

Output Inflation 0.5 0.5 0 0 -0.5 -0.5 0 10 20 30 40 0 10 20 30 40 Nominal interest rate House price 0.4 1 0.2 0 0 -1 0 10 20 30 40 0 10 20 30 40 Consumption Investment 0.5 1 0 0 -0.5 -1 0 10 20 30 40 0 10 20 30 40 Real Wage Hours Worked 0.5 0.2 0 0 -0.5 -0.2 0 10 20 30 40 0 10 20 30 40 Figure 9: Impulse responses to a house price shock in the augmented Smets and Wouters (2007) model with unobserved factors. See Section 5.3 for additional details. 46

afactor bfactor 0.15 0.04 0.1 0.03 0.05 0.02 0 0.01 -0.05 0 0 10 20 30 40 0 10 20 30 40 -3 spinffactor x 10 5 0 -5 -10 0 10 20 30 40 Figure 10: Impulse responses of the process-speci(cid:28)c factors in the augmented Smets and Wouters (2007) model with unobserved factors. See Section 5.3 for additional details. 47

Output Inflation 0.4 0.3 0.2 0.2 0 0.1 -0.2 0 -0.4 -0.1 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Nominal Interest Rate House Price 0.4 1 0.3 0.2 0.5 0.1 0 0 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Consumption Investment 0.6 1 0.4 0.5 0.2 0 0 -0.5 -0.2 -1 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Real Wage Hours Worked 0.4 0.2 0.2 0.1 0 0 -0.2 -0.1 -0.4 -0.2 4 8 12 16 20 24 28 32 36 40 4 8 12 16 20 24 28 32 36 40 Figure 11: Impulse responses to a house price shock in the Smets and Wouters (2007) augmented with house price shocks when the Taylor rule coe(cid:30)cient on in(cid:29)ation equals 3 (blue solid line), 2.25 (green dashed line), and 1.5 (red dotted line). See Section 5.1 for details. 48