Identifying the Sour es of Model Misspe i� ation

Atsushi Inoue

∗Chun-Hung Kuo

†Barbara Rossi

‡§

Southern Methodist Univ. International Univ. of Japan ICREA-Univ. Pompeu Fabra

Tohoku University Bar elona GSE and CREI

February 2015.

Abstra t

The Great Re ession has hallenged the adequa y of existing models to explain key ma roe o-

nomi data, and raised the on ern that the models might be misspe i�ed. This paper investigates

the importan e of misspe i� ation in stru tural models using a novel approa h to dete t and identify

the sour e of the misspe i� ation, thus guiding resear hers in their quest for improving e onomi

models. Our approa h formalizes the ommon pra ti e of adding �sho ks� in the model and iden-

ti�es potential mis-spe i� ation via fore ast error varian e de omposition and marginal likelihood

analyses. Simulation results based on a small-s ale DSGE model demonstrate that the method an

orre tly identify the sour e of mis-spe i� ation. Our empiri al results show that state-of-the-art

medium-s ale New Keynesian DSGE models remain mis-spe i�ed, pointing to asset and labor mar-

kets as the sour es of the mis-spe i� ation.

∗Department of E onomi s, Southern Methodist University, PO Box 750496, Dallas, Texas 75275-0235, USA. Email:

atsushi.inoue�vanderbilt.edu.

†Graduate S hool of International Relations, International University of Japan, 777 Kokusai- ho, Minami Uonuma-shi,

Niigata 949-7277, Japan. Email: hkuo�iuj.a .jp

‡ICREA-Univ. Pompeu Fabra, Bar elona GSE and CREI, C/ Ramon Trias Fargas, 25-27 08005 Bar elona, Spain.

Email: barbara.rossi�upf.edu.

§The �rst and third authors a knowledge �nan ial support from the National S ien e Foundation through grants 102159

and 1022125, respe tively. The authors thank the onferen e and seminar parti ipants at the Applied Time Series E ono-

metri s Workshop at the Federal Reserve Bank of St. Louis, Chung-Hua Institution for E onomi Resear h, International

University of Japan, National Taipei University, the 2013 EC(2) Conferen e and the Triangle E onometri s Conferen e for

helpful omments on earlier versions of the paper.

1

1 Introdu tion

The re ent �nan ial risis (or Great Re ession) has hallenged the adequa y of existing models in

explaining key ma roe onomi aggregate data. While the fa t that stru tural (and, in parti ular,

dynami sto hasti general equilibrium, or DSGE) models may fore ast poorly is well-known (see

Edge and Gurkaynak, 2010, among others), the severity and the prolonged duration of the Great

Re ession has brought to the forefront of the ma roe onomi debate the possibility that the predi tors

typi ally used might be inadequate and that existing models might be misspe i�ed. In fa t, Ng and

Wright (2013), among others, have argued that the features of the Great Re ession are di�erent from

those of "typi al" re essions, driven primarily by supply or monetary poli y sho ks. Thus, the need

to identify alternative models and predi tors, and the ne essity to identify whi h parts of the models

are misspe i�ed. For example, did the Great Re ession result from hanges in the transmission

me hanism or from unexpe ted sho ks? This is the obje t of an ongoing debate. On the one hand,

Sto k and Watson (2012) have argued that the transmission me hanism has not hanged during the

Great Re ession, and that the latter has resulted from larger sho ks that have hit the e onomy. If

indeed the transmission me hanism has not hanged, there is little e onomists an do to improve

their models' �t. However, if it did, then the models are misspe i�ed and should be improved by

in luding the missing features or hannels. But whi h features should be added? Resear hers are

often left wondering whether indeed their models are misspe i�ed, and, if they are, whi h parts of

their models are misspe i�ed. Several re ent ontributions have indeed worried that the transmission

me hanism might have hanged and, as a onsequen e, have proposed alternative models that in lude

either �nan ial fri tions or measures of interest rate spreads. While in some ases resear hers may

use their expertise in designing new features that better �t the data, in others they may have several

potential alternative andidates in mind, or none at all. If the resear her has no idea whi h aspe t

of the model is responsible for the poor �t, how an he/she identify whi h one is the best?

In this paper, we ask the question: are DSGE models misspe i�ed, and, if so, whi h features are

they missing? This is an important question, as DSGE models are routinely used for poli y analysis

in Central banks to guide poli ymakers' de isions. To examine model misspe i� ation in stru tural

models, we propose to introdu e two types of exogenous pro esses into a model. The �rst type of

exogenous sho ks are stru tural and are interpreted in the onventional way; the se ond type of

exogenous sho ks are not stru tural and are labeled as wedges. In fa t, by estimating the wedges, we

are able to assess: (i) where misspe i� ation might be lo ated (that is, whi h parts of the model are

most a�e ted by the misspe i� ation); and (ii) how qualitatively important it is. While wedges ould

be interpreted as additional stru tural sho ks, we in lude wedges only after several "traditional"

stru tural sho ks are modeled. The spe i� way we in orporate the time-varying wedges in the

model is by in luding them into the agents' optimization problem (su h as the households' budget

onstraints): this way, we allow for deviations in the relative pri es of relevant goods be ause the

model misspe i� ation will eventually lead to distorted relative pri es. These wedges an be used

to measure both the nature and importan e of misspe i� ation. While it is ommon in pra ti e to

2

add sho ks to a given model to get a better �t, our ontribution is to formalize this pro edure and

evaluate the performan e of su h approa h.

While the te hnique that we propose is very general, in this paper we fo us on DSGE models given

their widespread use in a ademia and entral banks as standard tools for analyzing ma roe onomi

poli ies. However, DSGE models are often highly parameterized to better �t the data, whi h raises

the on ern that they may be misspe i�ed. To provide more intuition and better illustrate our

framework, we onsider a medium-s ale DSGE model, embedded with most New Keynesian features.

If we treat the stru tural sho ks as an integral part of the model, our model is mildly misspe i�ed

in the sense that: (i) the ross equation and equilibrium restri tions imposed by the model do not

exa tly hold in every time period; and (ii) the deviations from equilibrium are zero on average. We

assume that the e onomi agents in the model (both �rms and households) take into a ount the

exogenous sto hasti pro esses of the deviations from equilibrium when solving their optimization

problems. We interpret the varian es of the deviations as a measure of the degree of misspe i� ation

of the model. To examine where and how large the misspe i� ation is, we ondu t fore ast error

varian e de ompositions (FEVDs) and marginal likelihood omparison. Note that our goal is not to

ompare models but rather is to evaluate the spe i� ation of a given model. While wedges may have

a stru tural interpretation, our goal is to start with a model of interest without a wedge and evaluate

its spe i� ation by measuring the ontribution of wedges without ne essarily giving the wedges a

stru tural interpretation. Finally, note that testing for over-identi� ation would not be useful in

our ontext: the over-identi� ation J-test would allow resear hers to determine whether a model's

moment onditions are orre tly spe i�ed, but would not shed light on the sour es of misspe i� ation

if the model fails the test.

Our empiri al appli ation to a state-of-the-art DSGE model highlights two interesting �ndings.

First, our te hnique points to misspe i� ation in the labor demand omponent of the model. Se ond,

the bond market also shows eviden e of misspe i� ation, whi h is persistent in its nature. Our

�ndings on�rms the existing view that the asset and labor markets in the New Keynesian DSGE

model are misspe i�ed and suggest that further work in these areas would be bene� ial.

Our method is related to several re ent ontributions. First, this paper is related to the literature

on the hallenges in the estimation of stru tural ma roe onomi models. The hallenge we fo us on

is misspe i� ation, and has been onsidered by Corradi and Swanson (2007), Curdia and Reis (2010)

and Canova and Ferroni (2011). Corradi and Swanson (2007) provide new tools for omparing the

empiri al joint distribution of histori al time series with the empiri al distribution of simulated time

series based on stru tural ma roe onomi models. Their fo us is on dete ting whether the whole

distribution of a ma roe onomi model is orre tly spe i�ed, whereas our fo us is on the �rst moments

of the model and on providing guidan e on the sour es of the models' misspe i� ation. Curdia and

Reis (2010) relax the restri tion that exogenous disturban es in stru tural models are independent

pro esses. They argue that estimating models with orrelated disturban es provides a useful he k

for model misspe i� ation. We onsider instead a di�erent way to in orporate misspe i� ation in

3

the models by in luding independent disturban es in the equilibrium onditions of the model. We

also suggest that the analysis of the estimated disturban es provides a way to identify the sour e of

the misspe i� ation and its importan e over time. Canova and Ferroni (2011) show that in orre t

�ltering distorts both y li al and non- y li al omponents and results in model misspe i� ation.

1

Our paper is also related to Watson (1993) and Del Negro and S horfheide (2009). The former

proposed a measure of �t for alibrated models; we instead estimate the stru tural models and

provide a di�erent methodology to identify the sour es of misspe i� ation. The latter develop a

framework for Bayesian estimation of possibly misspe i�ed DSGE models by using DSGE-implied

parameters as priors for ve tor autoregressive (VAR) models;

2

our framework omplements theirs in

that we an identify whi h parts of the model are misspe i�ed.

More generally, other hallenges that resear hers fa e in the estimation of stru tural ma roe o-

nomi models are related to identi� ation. Note that this paper does not deal with potential la k

of identi� ation or weak identi� ation of the models' parameters. The latter is a di�erent problem,

and has been analyzed by Dufour, Khalaf and Ki hian (2013), Canova and Sala (2009), and Iskrev

(2010), among others. If the models' parameters are not identi�ed, then adding wedges will not help.

Indeed, it is possible that adding wedges to an unidenti�ed or weakly identi�ed model may make

the estimation more di� ult. Our paper is related to Chari, Kehoe and M Grattan (2007) who in-

trodu e time-varying �wedges� into a ma roe onomi model to apture deviations from equilibrium

onditions. There are two main di�eren es between our work and theirs. One di�eren e is that

they onsider a neo lassi al sto hasti growth ben hmark model, while we onsider a New Keynesian

DSGE ben hmark model whi h in orporates several fri tions. Thus, our wedges (i.e. distortions

due to misspe i� ation) re�e t model misspe i� ation that is not already a ounted for by fri tions

built into the model. The other di�eren e is that their analysis is based on alibrated parameter

values, while ours is based on an estimated model, where the estimation takes into a ount possible

misspe i� ation.

The rest of the paper is organized as follows: Se tion 2 reports results based on simulation

exer ises whi h demonstrate that the proposed method is apable of identifying the sour es of model

misspe i� ation in a small New Keynesian model. Se tion 3 des ribes the model we use in our main

empiri al analysis, whi h is inspired by state-of-the-art New Keynesian models. Se tion 4 dis usses

the estimation results and the sour es of model misspe i� ation and Se tion 5 on ludes.

2 The Proposed Methodology and Simulation Analysis

In this se tion, we provide a detailed des ription of our proposed methodology. We also provide

Monte Carlo simulation eviden e to illustrate how to use our methodology to determine whether a

model is misspe i�ed and, in ase it is misspe i�ed, how to dete t the sour e of the misspe i� ation.

1

Ferroni (2011) proposes alternative detrending pro edures to avoid the mis-spe i� ation.

2

Note that they assume that DSGE models have a VAR representation in their implementation. Many DSGE models

have VARMA representations and we do not need to assume a VAR representation although the error due to VAR

approximations may be small if the lag is su� iently large.

4

Chari, Kehoe and M Grattan (2007) show that many models an be written as a mu h simpler

model with time-varying wedges. Based on this insight, we introdu e wedges into a model and

ompare models with wedges via their fore ast error varian e de ompositions (FEVDs) and marginal

likelihoods. The des ription of the methodology is based on a simple New Keynesian model, whi h

we use as the data generating pro ess (DGP) for a series of ma roe onomi variables. We then use

these variables as observables to estimate three misspe i�ed models as well as the orre tly spe i�ed

one. For ea h model, we report FEVD analyses and marginal likelihoods and illustrate how these

two measures an be used to dete t the sour es of model misspe i� ation.

2.1 The Data-Generating Pro ess (DGP)

We des ribe our methodology in the ontext of a simple New Keynesian model without the api-

tal a umulation de ision.

3

The model in ludes a representative �nal good �rm, a ontinuum of

intermediate good �rms, a representative household and a monetary and a �s al authority.

In period t, a representative �nal good �rm transforms a ontinuum of intermediate goods indexed

by j ∈ [0, 1] , Yj,t, into a homogeneous �nal good Yt by the following te hnology:

Yt =

[� 1

0

(Yj,t)1

λf dj

]λf

. (1)

Taking the pri es of intermediate goods (Pj,t) and that of the �nal good (Pt) as given, the �nal good

�rm solves the pro�t maximization problem:

max{Yj,t}j∈[0,1]

PtYt −

� 1

0

Pj,tYj,tdj,

subje t to equation (1).

The intermediate goods �rms are monopolisti providers of di�erentiated goods. They are, hen e,

able to set their own produ t pri es. Following Calvo (1983), we assume that in ea h period of time

only a fra tion 1 − ξ of intermediate goods �rms are able to set their desired optimal pri e. As a

onsequen e, when an intermediate good �rm is able to re-optimize its pri e in period t, its pri ing

problem an be expressed as follows:

4

maxPt

Et

∞∑

l=0

(βξ)l vt+l

{

Pt − Pt+lst+l

}

Yt+l (2)

s.t. Yt+l =

(

Pt

Pt

)−λf

λf−1

Yt+l

The onstraint is the intermediate good demand fun tion in period (t+ l) with the pri e set in period

t; st+l is the real marginal ost in period (t+ l) ; vt+l is the marginal value of per unit of money for

households at time (t+ l), be ause households hold equities of intermediate good �rms. Lastly, the

3

See for example An and S horfheide (2007).

4

Note that there is no need to spe ify the �rm subs ripts in this pri ing problem sin e �rms, whi h are able to re-optimize,

fa e the same problem and make the same de ision.

5

presen e of ξl in the obje tive fun tion indi ates the probability of not being able to re-optimize the

pri e after l periods.

Intermediate goods �rms only use labor to produ e their produ ts, and their produ tion fun tions

are assumed linear: Yj,t = ztLj,t. In terms of deviations from the steady state, the sto hasti

te hnology sho k zt follows an AR(1) pro ess:

5

zt+1 = ρz zt + σzεz,t+1, εz,t+1 ∼ iid N(0, 1).

Taking the nominal interest rate Rt and nominal wage rate Wt as given, ea h intermediate good �rm

j ∈ [0, 1] determines its labor input by solving the following ost minimization problem:

minLjt

WtLjt

subje t to its produ tion fun tion.

A representative household re eives utility from onsuming �nal goods and disutility from pro-

viding its labor servi es to intermediate goods �rms. The representative household maximizes the

expe ted dis ounted lifetime utility fun tion:

max{Ct,Lt}

∞

t=0

Et

∞∑

t=0

βt

{

C1−γt

1− γ− χt

L1+ϕt

1 + ϕ

}

,

subje t to the budget onstraint:

PtCt +Bt+1 = Rt−1Bt +WtLt + Dt + Tt.

The household has several sour es of in ome in ea h time period. First, the household olle ts returns

from holding government bonds (Bt) with gross nominal interest rate Rt. Se ond, by providing labor

servi es (Lt), the household obtains labor in ome at the overall nominal wage rate Wt. Third, the

household re eives dividends (Dt) from holding equities of intermediate goods �rms. Finally, there

exists a lump-sum transfer or taxation (Tt) from the government se tor. The household uses its

in ome to buy Ct units of the �nal good under the ompetitive pri e Pt and to pur hase the gov-

ernment bond Bt+1, whi h fa ilitates onsumption smoothing a ross time. In terms of log deviation

from its steady state value, the labor supply sho k (χt) follows:

χt+1 = ρχχt + σχεχ,t+1, εχ,t+1 ∼ iid N(0, 1)

The government is the sole provider of bond assets, and it onsumes �nal goods for non-produ tive

purposes. We assume that the government spending Gt is sto hasti , and governed by a sto hasti

pro ess ζt su h that Gt = ζtYt . To simplify the equilibrium onditions of the model, we re-

5

Following the onvention, variables with hats denote log deviations from the respe tive steady-state values.

6

parametrize ζt as gt ≡1

1−ζt, and assume that in terms of log deviation, gt follows an AR(1) pro ess:

gt+1 = ρg gt + σgεg,t+1, εg,t+1 ∼ iid N(0, 1).

The Central Bank determines the nominal interest rate (in log deviation from its steady state

value) by a Taylor rule:

Rt = ρrRt−1 + (1− ρr){

γππt + γY Yt

}

+ νt.

The lag of the nominal interest rate aptures the persisten e in the short-term nominal rate. The

random variable νt aptures the non-systemati part of the nominal rate de ision and follows:

νt+1 = ρν νt + σνεν,t+1, εν,t+1 ∼ iid N(0, 1).

The linearized equilibrium onditions of the above model are listed in Appendix A.

6

2.2 The Proposed Methodology

We estimate four di�erent models that re�e t possible misspe i� ation. We refer to them as M0,

Ml, Mm, and Mg, respe tively:

• M0 : This denotes the orre tly spe i�ed model (i.e., the DGP).

• Ml : This model is misspe i�ed be ause the labor supply sho k is ex luded from the set of

stru tural sho ks. In other words, we assume that there are only three stru tural sho ks in the

model (χt is not in luded in the model).

• Mm : This model is misspe i�ed be ause the non-systemati omponent of the nominal rate

in the monetary poli y de ision, νt, is assumed to be an iid random variable rather than the

AR(1) pro ess des ribed in the DGP.

• Mg : This model is misspe i�ed be ause the government spending sho k gt is ex luded from

the set of the stru tural sho ks. Thus, there are only three stru tural sho ks left in the model.

It is worth noting that we know exa tly whether or not ea h of the above models are misspe i�ed

and the sour es of their misspe i� ation. In ontrast, empiri al studies do not have su h valuable

information. In order to mimi su h realisti ir umstan es, we assume that we do not know whi h

models are misspe i�ed and introdu e some wedges into the above four models. Spe i� ally, we

introdu e a time-varying labor wedge τl,t into the ost minimization problem of ea h intermediate

good �rms su h that the problem is modi�ed as:

minLj,t

τl,tWtLj,t,

6

For the derivation of these onditions, we refer to the te hni al appendix of the paper, whi h is available form the

authors upon request.

7

subje t to its produ tion fun tion. Moreover, we introdu e a �nal good wedge τc,t and a bond market

wedge τb,t into the household's budget onstraint, whi h is modi�ed as follows:

τc,tPtCt +Bt+1 = τb,tRt−1Bt +WtLt + Dt + Tt.

We assume that these wedges all follow independent AR(1) pro esses. That is, for x ∈ {l, c, b}

τx,t+1 = ρxτx,t + σxεx,t+1, εx,t+1 ∼ iid N(0, 1).

Note that there is no unique way to in lude the wedges in the model. One might onsider introdu ing

wedges in �rst-order onditions. We prefer introdu ing wedges into agents' optimization problems

rather than into �rst order onditions for two reasons. One reason is that there are many ways

to write �rst-order onditions. Another reason is that introdu ing wedges in agents' optimization

problems allows us to interpret the wedges more learly. Our similation results show that our

approa h su essfully identify the ause of misspe i� ation.

We propose to ompare the models estimated with wedges using two tools. The �rst tool for

investigating the sour es of model misspe i� ation is the FEVD. Traditionally, FEVDs are used to

evaluate the ontribution of various stru tural sho ks onto the observables. See, for example, Ireland

(2001). Usually, resear hers assume that the model is orre tly spe i�ed and all the variation in the

observables are fully explained by the stru tural sho ks. In ontrast, we entertain the possibility that

all the models are potentially misspe i�ed. Thus, we propose to in lude the wedges into the model

and to examine their ontribution in terms of FEVDs. We expe t that the wedge that ontributes

the most to the FEVD identi�es the sour e of the misspe i� ation. The se ond tool is the marginal

likelihood. We expe t that removing the wedge that is most related to the type of misspe i� ation

has the largest impa t on the marginal likelihood. Thus, we propose to remove one wedge at a

time, and identifying the sour e of the misspe i� ation with the wedge whi h redu es the marginal

likelihood the most. The next sub-se tion examines the performan e of the methods we propose in

a Monte Carlo simulation exer ise.

2.3 Simulation Results

Using the data simulated from the DGP des ribed in sub-se tion 2.1, we estimate the four models

des ribed in sub-se tion 2.2. The observables we simulate are hours worked, Lt; real output, Yt; the

in�ation rate, πt; and the nominal interest rate, Rt.7

The sample size of the simulated data is 100,

whi h is more or less the typi al sample size in business y le studies based on quarterly data. The

parameter values used to simulate the data are summarized in the olumn labeled �DGP� in Table

1, and are set within widely a epted ranges.

INSERT TABLE 1 HERE

7

As one an see from the notation of the variables, the pseudo-true data are expressed in terms of log deviation from

the orresponding steady-state values.

8

The model is estimated by Bayesian methods using the random walk Metropolis-Hastings algo-

rithm (An and S horfheide, 2007). The priors used for estimation are listed in the last three olumns

of Table 1 and are the same for ea h model. However, it is worth noting that depending on a par-

ti ular model at hand, some parameters are not ne essarily estimated. Spe i� ally, in model Ml,

whi h ex ludes labor supply sho ks, ρχ and σχ are set to 0; in model Mg, whi h ex ludes govern-

ment spending sho ks, ρg and σg are set to 0; in model Mm, where the non-systemati part of the

Taylor rule is misspe i�ed, ρν is set to 0. Moreover, some parameters are not estimated in any of the

models: the subje tive dis ount fa tor β, the risk aversion oe� ient γ, and the inverse elasti ity of

labor supply ϕ are all �xed at their true parameter values. In our estimation exer ises, the number

of MCMC draws are 120,000, and the �rst 60,000 draws are dis arded.

The FEVDs of the onsidered models are summarized in Table 2.

8

INSERT TABLE 2 HERE

Panel (a) shows the FEVD analysis of model M0. Note that the stru tural sho ks, and the labor

supply sho k in parti ular, explain most of the FEVD. In parti ular, note that the ontribution of

the wedges to the FEDV are indeed negligible for any of the four variables, whi h indi ates that the

model is orre tly spe i�ed. This is what we expe ted: sin e model M0 is orre tly spe i�ed, the

three wedges are essentially redundant.

Panel (b) shows the FEVD results for model Ml. Note that the ontribution of the labor wedge

(τl) is substantial for all the observables. Comparing these results with those in panel (a), learly the

role of the labor supply sho k is almost repla ed by the labor wedge. This is be ause the omission of

the labor supply sho k distorts the wage in the labor market in model Ml. Thus, the labor wedge

orre tly identi�es the sour e of the misspe i� ation, whi h is the omission of the labor supply sho k.

Panel ( ) reports the FEVD of model Mm. In this ase, the bond market wedge explains the

majority of the fore ast error de ompositions of all variables, while the labor supply and �nal good

wedges explain almost nothing. Again, the bond market wedge orre tly identi�es the sour e of the

misspe i� ation whi h is related to the bond pri e, Rt, and therefore points to misspe i� ation in

the monetary poli y equation.

Finally, omparing the results of panel (d) with those in panel (a), it is lear that the �nal good

wedge somehow repla es the role of the government spending sho k in panel (d), even though the

absolute magnitude of the ontribution is not very large ompared to the other two ases (Ml and

Mm). The latter may be due to the fa t that, in our setting, the role of government spending is

relatively small. The la k of the government spending in model Mg distorts the resour e onstraint

whi h in turn auses distortion in the �nal goods market.

Table 3 reports the marginal likelihood of the models with the three wedges as well as the

likelihood of the models with one wedge removed at a time. The �rst olumn shows that when the

orre tly spe i�ed model (M0) is �tted to the data, the marginal likelihood does not hange mu h

8

We only report the varian e de omposition of 12 period-ahead fore ast errors. The results for longer and shorter

fore ast periods are similar. Unreported results are available upon request from the authors.

9

regardless of whi h wedge is dropped. Thus, the marginal likelihood orre tly signals that wedges

are negligible in the orre tly spe i�ed model. However, when the labor supply sho k is negle ted

(Ml), the model without the labor wedge has the lowest marginal likelihood. Similarly, when the

pro ess for the monetary poli y sho k (model Mm) is misspe i�ed, the marginal likelihood is lowest

when the bond market wedge is removed. When government spending is omitted from the model

(Mg), removing the �nal good wedge has some e�e t.

INSERT TABLE 3 HERE

To provide intuition why our methodology works, let us examine the linearized equilibrium on-

ditions in Appendix A. In model Ml, the only equation a�e ted by the misspe i� ation is the New

Keynesian Phillips urve (NKPC, eq. 37), where the labor supply sho k is missing relative to the

orre tly spe i�ed NKPC (eq. 23), and where two of the wedges (τc,t, τl,t) show up. Note that the

dynami IS urve is not a�e ted by the misspe i� ation, and that two wedges (τc,t, τb,t) show up in

the IS urve, eq. 33. Thus, the misspe i� ation in the Phillips urve must be aptured by the τl,t

wedge, whi h is the only wedge that appears in the NKPC but not in the IS urve. This is exa tly

what we �nd.

Conversely, in model Mm, the only equation a�e ted by the misspe i� ation is the Taylor rule,

hen e the nominal interest rate. Sin e the nominal interest rate appears in the IS urve but not in

the NKPC, the misspe i� ation will be aptured by the wedge that appears in the IS but not in the

NKPC, that is τb,t. In model Mg, the misspe i� ation a�e ts both the NKPC and the IS equations;

the only wedge that enters in both is (τc,t), whi h will thus apture the misspe i� ation.

The lesson we learn from the simulation exer ise is that introdu ing wedges in the model has

the potential to apture the missing hannels. For example, the wedges orre tly reveal if there are

important stru tural sho ks missing from the model, and whi h those sho ks are. To summarize,

FEVD and marginal likelihood analyses provide useful tools for dete ting and identifying model mis-

spe i� ation. When a model is misspe i�ed, the ontribution of a wedge to the FEVD is substantial

in at least one of the variables; the type of the wedge tends to be related to that of the misspe i-

� ation, thus signaling the possible ause of the misspe i� ation. Also, removing the wedge that is

most related to the type of misspe i� ation that truly a�e ts the model will have the largest impa t

on the marginal likelihood.

2.4 Suggestions for Pra titioners

In pra ti e, one does not know whi h parts of a model are misspe i�ed and may wonder in whi h

parts of the model should in lude the wedges. In theory, one an introdu e a wedge in every market in

the model but that might yield an over-parameterized model even for Bayesian estimation methods.

We suggest two approa hes. One is to introdu e wedges in markets whi h the resear her suspe ts

are misspe i�ed. Another is to introdu e wedges everywhere, but impose tight priors on the AR(1)

parameters and innovation varian es of the wedge if misspe i� ation is unlikely. The �rst approa h

orresponds to the ase in whi h the resear her has a strong view on whi h parts of the model

10

are orre tly spe i�ed and whi h parts are potentially misspe i�ed. The se ond approa h is more

agnosti about the nature of the misspe i� ation.

3 The Medium-S ale New Keynesian Model

In this se tion, we dis uss a medium-s ale New Keynesian DSGE model for identifying the sour es of

model misspe i� ation. Essentially, the model is a sto hasti neo lassi al growth model with various

real and nominal fri tions whi h is routinely used by resear hers and poli ymakers. The fri tions

that we in orporate in lude imperfe t ompetition in the intermediate goods and labor markets,

sti ky pri es and wages, habit formation in onsumption, investment adjustment ost and variable

apital utilization. The model is based on Justiniano et al. (2010). We explore whether this model

may be still misspe i�ed by in luding several time-varying wedges and evaluating their importan e.

Note that we annot simply list the equilibrium onditions of the model by Justiniano et al. (2010).

The reason is that we treat the wedges as integral parts of agents' (in luding households and �rms)

de ision problems. Thus, we have to spe ify the environment expli itly to show how these wedges

a�e t agents' de isions. The implied linearized equilibrium onditions are listed in Appendix B.

3.1 Final Good Firms

In the model, �nal goods are produ ed by a ontinuum of perfe tly ompetitive �nal good �rms.

The produ tion fun tion of �nal good �rms is

Yt =

[� 1

0

Q1

1+λp,t

i,t di

]1+λp,t

, (3)

where Qi,t is the intermediate good pur hased from the i-th intermediate good �rm, and λp,t is an

exogenous stru tural sho k governing the elasti ity of substitution among intermediate goods. In

line with Smets and Wouters (2007), λp,t follows the ARMA(1,1) pro ess:

log (1 + λp,t) = (1− ρp) log (1 + λp) + ρp log (1 + λp,t−1) + εp,t − θp,tεp,t−1, εp,t ∼ N(

0, σ2p

)

. (4)

In the literature, this sho k is often alled as the pri e mark-up sho k, as it determines the time-

varying desired mark-up of intermediate good �rms.

Taking the �nal good pri e Pt and intermediate good pri es Pj,t's as given, the �nal good �rms

de ide their intermediate good inputs (Qi,t) by solving the pro�t maximization problem:

max{Qi,t}

i∈[0,1]

PtYt −

� 1

0

τq,tPi,tQi,tdi,

subje t to the �nal good produ tion fun tion (3). Noti e that we introdu e a time-varying interme-

diate good wedge τq,t into this problem, and it evolves a ording to:

log τq,t = (1− ρq) log τq + ρq log τq,t−1 + εq,t, εq,t ∼ N(

0, σ2q

)

, (5)

11

with τq = 1 in the steady state. The value of τq,t is assumed to be known by the �nal good �rms

when they make their de isions. The reason of introdu ing τq,t is to apture model misspe i� ation

in the demand of intermediate goods. To see this, onsider the demand fun tion of intermediate

goods:

Qi,t =

(

τq,tPi,t

Pt

)

−(1+λp,t)λp,t

Yt.

The intermediate goods demand Qi,t is a�e ted not only by the relative pri e

Pi,t

Ptbut also by the

intermediate good wedge, τq,t. Evidently, introdu ing τq,t helps apture the ontribution of fa tors

a�e ting �nal good �rms' de isions other than

Pi,t

Pt. The wedge τq,t an be interpreted as a proxy

that aptures the fa t that the model builder ignores that, in reality, �rms are likely to onsider more

aspe ts than merely the relative pri es of intermediate goods when they make their input de isions.

Moreover, τq,t helps relaxing the tight restri tion between intermediate good pri es and the pri e of

the �nal good. To see this, onsider the following equilibrium ondition:

Pt = τq,t

[� 1

0

(Pi,t)− 1

λp,t di

]−λp,t

, (6)

whi h is obtained by substituting the intermediate good demand fun tion into (3). Equation (6)

implies that if the intermediate good pri es (Pi,t) annot hara terize well the the dynami s of the

overall pri e level Pt, the deviation is aptured by τq,t. Besides, our assumption of τq = 1 in the

steady-state indi ates that we take the stand that the relationship between pri es of intermediate

goods and the �nal good is, on average, orre tly spe i�ed.

3.2 Intermediate Goods Firms

We assume that there exists a ontinuum of intermediate goods �rms, whi h are operating in imper-

fe tly ompetitive markets. Thus, they have the ability to set their desired pri es. Below, we adopt

the typi al two-stage pro edure to des ribe the optimal behavior of intermediate goods �rms. We

�rst lay out the ost minimization problem and then the pro�t maximization one.

Cost Minimization In period t, the i-th intermediate goods �rm rents apital servi es Ki,t and

homogeneous labor servi es Li,t from perfe tly ompetitive fa tor markets to produ e di�erentiated

intermediate goods, Qi,t. Its produ tion fun tion fun tion is de�ned as:

Qi,t = max{

A1−αt Kα

i,tL1−αi,t −AtF ; 0

}

, (7)

where At denotes the time t te hnology level, whi h is ommon a ross all intermediate goods �rms,

9

α is the apital elasti ity of produ ing the intermediate good, and AtF is the �xed produ tion

ost in period t. Following Justiniano et al. (2010), we assume that the te hnology growth rate

9

Following Christiano et al. (2005), we assume that there is neither exit nor entry de ision for intermediate goods �rms,

while theoreti ally some �rms might not be able to produ e anything due to high �xed osts.

12

(zt = ∆ logAt) follows a stationary AR(1) pro ess:

zt = (1− ρz) γ + ρzzt−1 + εz,t, εz,t ∼ N(

0, σ2z

)

. (8)

This assumption implies that the e onomy grows along the sto hasti trend in At.

The ost minimization problem of the intermediate goods �rms is

minKi,t,Li,t

τk,tRk,tKi,t + τl,tWtLi,t

subje t to the produ tion fun tion (7). Here, Rk,t and Wt are the nominal rental rate of apital

and the nominal wage rate, respe tively. Note that two time-varying wedges are introdu ed in this

problem to apture the model misspe i� ation in the ost minimization problem. The �rst one is the

apital market wedge τk,t, and the se ond one is the homogeneous labor market wedge τl,t. As in

the ase of the intermediate good wedge (τq,t), the two wedges are assumed to be known when the

intermediate goods �rm makes the input de ision in period t. To be parsimonious, the two exogenous

wedge pro esses both follow AR(1) pro esses:

log τk,t = (1− ρk) log τk + ρk log τk,t−1 + εk,t, εk,t ∼ N (0, σ2k), (9)

log τl,t = (1− ρl) log τl + ρl log τl,t−1 + εl,t, εl,t ∼ N (0, σ2l ). (10)

The interpretation of the two wedges is similar to that of the intermediate good wedge τy,t.

That is, they apture the model misspe i� ation in hara terizing demand with respe t to apital

and labor inputs. For example, they an represent some missing fa tors a�e ting apital and labor

demand. As in the ase of the intermediate good wedge τq,t, the steady state values of these two

wedges are assumed to be one. Moreover, the ost minimization problem implies that the nominal

marginal ost of the intermediate good �rm is

St =

(

1

αα (1− α)1−α

)

(τk,tRk,t)α

(

τl,tWt

At

)1−α

,

whi h reveals how model misspe i� ation in fa tor demands propagates to the marginal ost fun -

tion.

10

Pro�t Maximization Sin e intermediate goods �rms produ e di�erentiated goods and have

monopoly power on their own produ ts, they seek to maximize the present value of the expe ted

future pro�t �ows by setting optimal produ t pri es. Following Calvo (1983), for all intermediate

goods �rms, the probability of being able to re-optimize their pri es is (1− ξp), whi h is independent

a ross �rms and onstant a ross time. Thus, we an view this un ertainty as idiosyn rati sho ks

fa ing intermediate goods �rms. Any intermediate goods �rm unable to re-optimize its desired pri e

10

Be ause the ost minimization problem of intermediate good �rms is essentially stati , all �rms make the same input

renting de ision, implying that the marginal ost fun tion is the same a ross all �rms. Thus, there is no need to atta h the

�rm subs ript i on the marginal ost fun tion.

13

in period t resets its produ t pri e by a partial indexation s heme:

Pi,t = πιpt−1π

1−ιpPi,t−1,

where πt ≡ Pt

Pt−1denotes the period t gross in�ation rate, and π is the steady-state in�ation rate.

Re ently, this partial indexation s heme has be ome popular in New Keynesian models be ause it

allows the dynami s of in�ation to be both forward- and ba kward-looking. Thus, it is a onvenient

way of apturing the in�ation rate persisten e observed in the data.

Any intermediate goods �rm able to set its optimal pri e in period t, Pt, hooses it a ording to

the pro�t maximization problem:

11

maxPt

Et

{

∞∑

l=0

(βξp)l Λt+s

Λt

[

PtFt,l − Pt+lst+l

]

Qt+l

}

subje t to:

Qt+l =

(

τq,t+lFt,lPt

Pt+l

)

−(1+λp,t+l)λp,t+l

Yt+l, for l = 0, 1, 2, . . . .

Here, st+l is the real marginal ost of the �rm in period (t+ l) and Λt is households' marginal value

of nominal in ome in period t, sin e we assume that households hold equities of intermediate goods

�rms.

12

The onstraints of the problem are the intermediate good demands in periods t + l, when

the optimal pri e of the intermediate good is set in period t. Ft,l is de�ned as

Ft,l =

∏l

k=1 πιpt+k−1π

1−ιp , if l ≥ 1;

1 if l = 0,

whi h aptures the ompound e�e t of partial indexation between periods t and (t+ l). Note that

the intermediate good wedge (τq,t) in the �nal good �rms a�e ts the pri ing de ision of intermediate

goods �rms as well.

3.3 Households

There exists a ontinuum of in�nitely lived households denoted by j ∈ [0, 1]. Following Er eg, Hen-

derson and Levin (2000) and Christiano, Ei henbaum and Evans (2005), we assume that households

are monopolisti providers of di�erentiated labor. That is, they are able to set their own wage rates,

Wj,t. The j-th household's budget onstraint is

τc,tPtCt+PtIt+Tt+Bt+Pta(ut)Kt−1 ≤ τr,tRt−1Bt−1+Wj,thj,t+Rk,tutKt−1+Dt+Ft+Aj,t. (11)

The right hand side of (11) re�e ts various sour es of the household's in ome in period t; Wj,thj,t is

the household's wage in ome. The household re eives the dividend Dt from intermediate goods �rms

11

Sin e �rms fa e exa tly the same problem when they re-optimize their produ t pri es, there is no need to distinguish

them expli itly. We thus drop the �rm index to simplify notation.

12

One an obtain Λt by the Lagrange fun tion asso iated with the household se tor problem. See the next subse tion.

14

and the lump-sum transfer Ft from the government. We assume that there exists a omplete set of

Arrow se urities, and Aj,t is the return on holding su h se urities. The omplete market assumption

implies the all households make identi al de isions on onsumption Ct, investment It, apital Kt,

and bond holding Bt, while their labor in ome is heterogeneous.

13

We assume that households an determine the utilization rate of apital, ut. Hen e, the e�e tive

apital servi e used in produ ing intermediate goods is Kt = utKt−1, and the nominal return of

providing apital to intermediate goods �rms is Rk,tutKt−1. By buying the government bond Bt−1

in period t− 1, the household re eives (gross) interest in ome τb,tRt−1Bt−1 in period t, where Rt−1

is the nominal interest rate determined in period t− 1. Note that we introdu e a time-varying bond

market wedge τr,t, whi h aptures any potentially missing fa tors that might a�e t the demand for

the bond. Similar to other wedges in the model, it follows an AR(1) pro ess (in logs):

log τr,t = (1− ρr) log τr + ρr log τr,t−1 + εr,t, εr,t ∼ N (0, σ2r). (12)

It is worth noting that τr,t an also be interpreted as a stru tural sho k: for instan e, Smets and

Wouters (2007) refer to τr,t as the risk premium sho k. Be ause we are evaluating the spe i� ation

of the medium-s ale DSGE model without a risk premium sho k, we interprete it as a wedge.

The left hand side of (11) re�e ts the allo ation of the household in ome for various purposes.

First of all, Bt is the bonding holding to be arried over to period t + 1. Se ond, the household

spends PtIt to a umulate physi al apital. The evolution of physi al apital an be expressed as:

Kt = (1− δ) Kt−1 + µt

[

1− S

(

ItIt−1

)]

It, (13)

where S(·) re�e ts the investment adjustment ost and is restri ted su h that S(1) = S ′(1) = 0 in

the steady state.

14

Following Justiniano and Primi eri (2008), we introdu e the investment sho k

µt, su h that:

log µt = (1− ρµ) log µ+ ρµ log µt−1 + εµ,t, εµ,t ∼ N(

0, σ2µ

)

. (14)

Moreover, the household fa es the apital utilization ost Pta(ut)Kt, whi h varies with the level of

the apital utilization rate ut. Finally, the household uses its in ome to onsume the �nal good Ct

with pri e Pt. When the household makes its onsumption de ision, it also onsiders the e�e ts of

a time-varying onsumption wedge τc,t, whi h is designed to apture model misspe i� ation in the

onsumption de ision. While we treat τc,t as a wedge, it an also be viewed as a stru tural sho k. For

example, Leeper at al. (2010) interpret τc,t as the taxation sho k on onsumption. The onsumption

wedge τc,t follows

log τc,t = (1− ρc) log τc + ρc log τc,t−1 + εc,t, εc,t ∼ N(

0, σ2c

)

. (15)

13

Thus, there is no need to add the �rm index subs ript i with respe t to onsumption, investment, apital, and the

bond holding in the budget onstraint.

14

See S hmitt-Grohe and Uribe (2006) for a justi� ation of the restri tion.

15

As before, we assume τc = 1.

The j-th household re eives utility from onsuming the �nal good Ct and disutility from pro-

viding di�erentiated labor hj,t. Its behavior an be hara terized by the following lifetime utility

maximization problem:

maxEt

∞∑

s=0

βt+sbt+s

{

log (Ct+s − ~Ct+s−1)− ϕ(hj,t+s)

1+υ

1 + υ

}

subje t to its budget onstraint (11) and the apital a umulation equation (13). Essentially, the

household makes a sequen e of de isions on onsumption, apital a umulation, apital utilization,

the bond holding and the desired wage. In this problem, β is the subje tive dis ount fa tor, ν is

the inverse of Fris h elasti ity of labor supply, and ϕ ontrols the weight on labor disutility. The

parameter ~ governs the extent of onsumption habit formation.

15

In line with Justiniano et al.

(2010), we assume that there exists an intertemporal preferen e sho k bt following

log bt = (1− ρb) log b+ ρb log bt−1 + εb,t, εb,t ∼ N(

0, σ2b

)

. (16)

Labor-Pa ker Firms A notable feature of New Keynesian models is that households provide

di�erentiated labor servi e hj,t, but intermediate good �rms rent homogenous labor servi es. Fol-

lowing Smets and Wouters (2007), we introdu e perfe tly ompetitive labor-pa ker �rms whi h are

onvenient to re on ile the above distin tion. The representative labor-pa ker �rm hires household

labor servi es hj,t's and transforms them to the homogeneous labor servi e Lt by the following

te hnology:

Lt =

[� 1

o

(hj,t)1

1+λw,t dj

]1+λw,t

, (17)

where λw,t governs the time-varying elasti ity of substitution among heterogenous labor, hj,t, and

follows an ARMA(1,1) sto hasti pro ess:

log (1 + λw,t) = (1− ρw) log (1 + λw) + ρw log (1 + λw,t−1) + εw,t − θw,tεw,t−1, εw,t ∼ N(

0, σ2w

)

.

(18)

Some e onomists interpret λw,t as the wage mark-up sho k, for it re�e ts households' desired wage

mark-up over the marginal rate of substitution between onsumption and leisure.

Taking the market wage rate Wt and idiosyn rati wage rates Wj,t of households as given, the

labor pa ker �rm solves the following problem:

max{hj,t}j∈[0,1]

WtLt −

� 1

0

τh,tWj,thj,tdj,

subje t to (17). Here, we introdu e a time-varying wedge τh,t as follows

log τh,t = (1− ρh) log τh + ρl log τh,t−1 + εh,t, εh,t ∼ N (0, σ2h). (19)

15

Christiano et al. (2005) show that introdu ing habit formation helps to explain the hump-shaped response of onsump-

tion to a monetary poli y sho k.

16

The �rst order ondition of the above problem implies that the demand for the j-th household's

labor servi e is:

hj,t =

(

τh,tWj,t

Wt

)−1+λw,tλw,t

Lt.

Besides, it is easy to show that

Wt = τh,t

[� 1

0

(Wj,t)− 1

λw dj

]−λw,t

,

whi h reveals how the model misspe i� ation with respe t to demand of hj,t a�e ts the determination

of the overall wage rate Wt.

Wage Setting Following Er eg, Henderson and Levin (2000), we assume that not all households

are able to set the desired optimal wage rate at every period of time. Spe i� ally, the probability

of being able to set the optimal wage rate in ea h period is 1 − ξw, whi h is independent a ross

households and onstant over time. On the other hand, a household unable to set its optimal wage

rate in period t resets its wage rate a ording to the following partial indexation s heme:

16

Wj,t = Wj,t−1 (πt−1ezt−1)ιw (πeγ)1−ιw .

The households' wage setting de ision an be represented by

maxWt

Et

∞∑

s=0

(βξw)s

−bt+sϕ

(

ht+s

)1+ν

1 + ν+ Λt+s

(

WtGt,s

)

ht+s

,

subje t to the labor demand fun tion:

ht+s =

(

τh,t+sWtGt,s

Wt+s

)−1+λw,t+sλw,t+s

Lt+s.

Here, ht+s is the household labor demand in period t + s, and the optimal wage is determined in

period t. In period t+ s, the e�e tive wage rate is WtGt,s, where

Gt,s =

∏s

k=1 [πt+k−1ezt+k−1 ]ιw [πeγ ]1−ιw , if s ≥ 1;

1 if s = 0,

aptures the ompound e�e t of partial indexation from period t to period t + s. Note as well how

τh,t's a�e t the labor demand fun tions. Moreover, Λt+s is urrent value of the marginal utility in

period t+ s, whi h is the Lagrange multiplier asso iated with the household budget onstraint.

16

As in the ase of intermediate good pri es, this partial indexation s heme makes the determination of the wage rate

not only forward- but also ba kward-looking. Thus, it is onvenient for apturing the persisten e of wage rates.

17

3.4 The Government

In this model e onomy, the government de ides both �s al and monetary poli ies. For the �s al side,

we assume that the government onsumes a variable proportion of �nal output for non-produ tive

purposes:

Gt =

(

1−1

gt

)

Yt,

where gt follows

log gt = (1− ρg) log g + ρg log gt−1 + εg,t, εg,t ∼ N(

0, σ2g

)

. (20)

We assume that the government fully �nan es its spending by lump-sum taxes and issuing government

bonds. Thus, there is no need to expli itly spe ify the taxation rea tion rule of the government.

17

As for the monetary side, we assume that the government implements an interest rate rule á

la Taylor (1993). Spe i� ally, we assume the government determines the short-run nominal (gross)

interest rate by:

Rt

R=

(

Rt−1

R

)ρR[

(πt

π

)φπ

(

Xt

X∗t

)φx

]1−ρR [

Xt/Xt−1

X∗t /X

∗t−1

]φdX

ηmp,t.

Here, ρR is a smoothing parameter apturing the nominal interest rate persisten e. We de�ne

Xt ≡ Ct + It + Gt as the measure of gross domesti produ t (GDP) of the e onomy. On the other

hand, we denote X∗t ≡ C∗

t +I∗t +G∗t as the output level, in whi h pri es (and wages) are fully �exible

and with no pri e and wage mark-up sho ks. Thus, Xt/X∗t an be interpreted as the GDP gap. The

last square bra ket aptures the hange of the GDP gap. We assume that the monetary poli y sho k

ηmp,t follows

log ηmp,t = ρmp log ηmp,t−1 + εmp,t, εmp,t ∼ N(

0, σ2mp

)

. (21)

3.5 Aggregation

Be ause of the omplete markets assumption, the aggregation of the model e onomy is straightfor-

ward, even though the households and intermediate goods �rms fa e idiosyn rati sho ks due to the

Calvo-type setting. By integrating the produ tion fun tion of the intermediate goods �rms, we have

the relationship between aggregate output and the aggregate fa tor inputs:

Yt = A1−αt Kα

t L1−αt −AtF.

Sin e our model is a losed e onomy, the total output of the e onomy an only be distributed to

four outlets: household and government onsumption, investment, and the apital utilization ost.

Hen e, the aggregate resour e onstraint is:

Ct + It +Gt + a(ut)Kt = Yt.

17

See Christiano et al. (2005) for similar arguments.

18

We assume all innovations of stru tural and wedge pro ess are mutually independent over time.

4 Estimation

We estimate the stru tural parameters of the model as well as the time-varying wedge pro esses by

Bayesian methods. We �rst transform the original equilibrium onditions, whi h are non-stationary

due to the growing te hnology level At, into stationary ones. Then, we linearize the transformed

equilibrium onditions around the steady state values and estimate them.

18

Following Justiniano, Primi eri and Tambalotti (2010), we utilize seven quarterly aggregate time

series of the U.S. from 1954:III to 2004:IV to estimate the model. Spe i� ally, we use real output,

real onsumption, real investment, hours worked, in�ation rate, and the Federal funds rate as our

observables. Real output, real onsumption, real investment, and hours worked are all in per apita

terms. Real output and onsumption per apita are obtained by dividing nominal GDP and nominal

onsumption by population and the pri e index. Nominal onsumption is de�ned as the sum of

non-durable goods and servi es expenditures. Nominal investment is de�ned as the sum of private

domesti investment and personal durable goods expenditure. Real investment per apita is nominal

investment divided by both the population and the pri e index. Hours worked per apita are de�ned

as total hours worked in the non-farm business se tor divided by the population. Real wages are

de�ned as non-farm business se tor hourly ompensation divided by the pri e index. We use the

ivilian non-institutional population as our population measure. The pri e index is the GDP de�ator,

and the quarterly in�ation rate is its growth rate. Estimation results are based on 120,000 Markov

Chain Monte Carlo (MCMC) draws, and the �rst 60,000 draws are dis arded.

4.1 Priors

It is well-known that some stru tural parameters in DSGE models are not well-identi�ed, and they

should be alibrated before estimating the model. Spe i� ally, we alibrate the apital depre iation

rate δ to 0.025, whi h is a value ommonly used in the literature.

19

The steady-state ratio of

government spending to output is set to 0.22, whi h is the average value in our sample. Moreover,

we impose the zero pro�t ondition on the intermediate goods �rms' problem, and hen e the �xed

ost (F ) is endogenously determined.

The remaining parameters an be lassi�ed into three groups. The �rst group ontains the

fundamental stru tural parameters of the model, su h as the inverse Fris h elasti ity of labor supply ν,

the onsumption habit parameter ~, et . The se ond group ontains the stru tural sho ks parameters.

Note that if we remove all wedges from the model, it will be the same as Justiniano et al. (2010).

Therefore, we use exa tly their prior distributions with respe t to parameters in the �rst and the

se ond groups, whi h are broadly line with the literature. The third group ontains the parameters

of the exogenous wedge pro esses. We do not have information on these parameters, and thus we

18

See Appendix B for the linearized equilibrium onditions.

19

See Christiano et al. (2010) and Smets and Wouters (2007) for instan e.

19

set quite disperse prior distributions. For example, the prior distribution of ρl has mean 0.5 and

standard deviation 0.2. As usual, the prior distributions are assumed mutually independent from

ea h other. The prior distributions are summarized in Table 4.

20

INSERT TABLE 4 HERE

4.2 Empiri al Results

The estimation results are summarized in the last four olumns of Table 4, whi h displays median,

standard deviation, 5th and 95th per entiles of the posterior distribution for ea h of the estimated

parameters. To simplify the dis ussion, we treat the posterior median as the point estimate. Overall,

our results are similar to the existing literature with a few ex eptions, whi h we dis uss in turn.

First of all, the point estimate of the onsumption habit parameter ~ is 0.83, whi h is larger than in

Christiano et al. (2005) and Smets and Wouters (2007). The point estimate of ν is 3.84, implying

that the Fris h elasti ity of labor supply is around 0.26. Comparing to Levin et al. (2006) and Smets

and Wouters (2007), our Fris h labor supply elasti ity value is a bit smaller. However, it is in line

with empiri al eviden e from mi ro data.

21

For example, MaCurdy (1981) showed that the labor

supply elasti ity of white, married, prime-aged men is between 0.1 and 0.5.

Our point estimate of the Calvo pri e parameter is ξp = 0.85, implying that intermediate goods

�rms, on average, reset their optimal pri es every 6.6 quarters. Our estimate of ξp is similar to

that reported by Levin et al. (2006), a ording to whom it is 0.83.

22

Our estimate of the Calvo

wage parameter is ξw = 0.72, whi h implies that average duration of a wage ontra t is around 4.5

quarters. Moreover, our results reveal that the extent of partial indexation on previous in�ation and

the wage rate is moderate: the point estimates are ιp = 0.19 and ιw = 0.10, respe tively. The results

of low partial indexation e ho Del Negro and S horfheide (2008), who laim that the model with

indexation s hemes on pri es and wages are observationally similar to those with auto orrelated

mark-up sho ks: sin e our model in ludes the moving average terms for pri e and wage mark-up

sho ks, the ontributions of ιp and ιw be ome smaller.

The point estimate of ρz, the persisten e of te hnology sho ks, is merely 0.24, whi h is mu h

lower than the existing estimates. For example, the parameter estimates in Smets and Wouters

(2007) and Levin et al. (2006) are 0.95 and 0.96. Moreover, the point estimate of σz is 0.87 per ent,

implying that the un onditional standard deviation of the te hnology sho k is 0.92 per ent. The

implied un onditional standard deviation of te hnology sho ks is about 60 and 40 per ent of the

values in Smets and Wouters (2007) and Levin et al. (2006), respe tively. See also Ireland (2004)

for further dis ussion about the role of te hnology sho ks on several key ma roe onomi variables

in a New Keynesian DSGE model. It is worth to note that Del Negro and S horfheide (2009) �nd

20

In Table 4, "SS" denotes "steady state".

21

The point estimates of inverse Fris h elasti ity of labor supply are 1.49 and 1.92 in Levin et al. (2006) and Smets and

Wouters (2007), respe tively.

22

Altig et al. (2011) explain that the implied duration of pri e sti kiness shortens on e �rm-spe i� apital is in luded

in the model.

20

that the persisten e oe� ient of the te hnology sho k be omes smaller on e model misspe i� ation

is taken into a ount.

The pri e and wedge mark-up sho ks are both quite persistent, and the point estimates of ρp

and ρw are 0.94 and 0.99, respe tively. Moreover, the moving average oe� ients (θp and θw) both

play non-trivial roles in shaping the dynami s of pri e and wage mark-up sho ks. The government

spending sho k is highly persistent and behaves almost like a unit root pro ess. Among the stru tural

sho ks and wedge pro esses, the investment sho k has the biggest innovation standard deviation: the

point estimate of σµ is 5.24 per ent, whi h is about 6 and 52 times the innovation standard deviations

of te hnology and preferen e sho ks, respe tively. Sin e the investment sho k innovation has su h a

high standard deviation, we expe t that investment sho ks play a dominant role in determining the

�u tuations of the endogenous variables of the model.

Our estimates also suggest that the wedges are serially orrelated. This suggests that the model

misspe i� ation annot be aptured by iid measurement errors in the measurement equations.

Finally, in order to understand whether in luding wedges has substantial onsequen es on the

parameter estimates, we report parameter estimates for the model without wedges in Table 5. By

omparing Tables 4 and 5, we on lude that estimating the model with the wedges does not sub-

stantially modify the parameter estimates relative to those of a model without wedges.

23

INSERT TABLE 5 HERE

Fore ast Error Varian e De omposition As shown in the Se tion 2, FEVDs an help

identify the sour es of model misspe i� ation. For onvenien e, Table 6 reports a summary of the

notation orresponding to all the wedges and sho ks in the model.

INSERT TABLE 6 HERE

Table 7 reports the FEVD ontribution of the sho ks and wedges (listed in the olumns) to the

overall varian e of the observable variables (i.e., output growth, onsumption growth, investment

growth, et .) at di�erent fore ast horizons (H = 1, 4, 20).

INSERT TABLE 7 HERE

Table 7 shows that the apital, the homogenous labor market, and onsumption good wedges are

extremely small for all the variables of interest. In ontrast, the bond market, the intermediate goods

demand, and the household labor wedges ontribute to explain the variability of several variables,

sometimes substantially. For example, the household labor wedge explains more than 60% of one

quarter-ahead fore ast error varian e of wage growth. The e�e ts of the household labor wedge are

persistent: it explains more than 40% of the wage growth variation after twenty quarters. This result

indi ates that the model without household labor wedge ould be misspe i�ed in the wage growth

dynami s. Our results di�er from Justiniano et al. (2010), whose wage mark-up sho ks explain

about 56% of the wage growth variation at the business y le frequen y: a ording to our results,

23

The stru tural impulse responses do not hange substantially after introdu ing wedges into the model.

21

the wage mark-up sho k does not explain more than 11% between 1 and 20 quarters. That is, the

model without household labor wedge might attribute the wage �u tuation to the wage mark-up

sho k.

At the one-quarter ahead fore ast horizon, the intermediate good wedge (τq) explains around

20% of the variation in in�ation. A tually, its ontribution is even larger than that of te hnology

sho ks, whi h only explain about 14% of in�ation �u tuations at the same fore ast horizon. Note also

that the importan e of the intermediate goods wedge on in�ation �u tuations de ay as the fore ast

horizon in reases. Sin e the intermediate good wedge mainly a�e ts short run �u tuations, our

results warn against using the model for fore asting short run in�ation if the model misspe i� ation

is not properly addressed. It is worth noting that while the intermediate good (τq) and household

labor wedges (τh) have ru ial e�e ts on nominal variables (that is, wage growth, in�ation rates, and

interest rates), they have hardly any e�e ts on other variables.

The bond market wedge τr mainly a�e ts the �u tuations of nominal interest rates, and its

ontribution in reases as the fore ast horizon in reases: at the twenty quarters horizon (H = 20),

more than 40% of the interest rate variability an be attributed to �u tuations in the bond market

wedge. The dominant e�e t of the bond market wedge omes at the expense of the wage mark-up and

investment sho ks, whose total ontribution is merely around 30%. This result is sharply di�erent

from that of Justiniano et al. (2010), where about 67% of the interest rate variability is explained

by wage mark-up and investment sho ks. In addition, the bond market wedge also ontributes to

the �u tuations of output, investment, and onsumption growth as well as hours worked at various

fore ast horizons, although its e�e ts are smaller in magnitude.

Based on the above FEVD analysis, we on lude wedges mainly a�e t wage growth, the in�ation

rate, and the interest rate. Our results with respe t to other observables are similar to those in

Justiniano et al. (2010). That is, the investment sho k is the main driving for e behind output

growth, investment growth, and hours worked, and the inter-temporal preferen e sho k explains

most of the variability in onsumption growth.

24

INSERT TABLE 8 HERE

For omparison, we report the FEVD of the ben hmark model without wedges in Table 8. To

simplify the dis ussion, we fo us on the twenty quarter-ahead fore ast horizon. Wedges mainly a�e t

the variability of wage growth, in�ation, and the interest rate, and we dis uss them in turn. Regarding

the wage growth, the model without wedges explains its �u tuations mainly by the te hnology sho k

(37.58%) and the wage mark-up sho k (37.64%). However, on e wedges are taken into a ount,

the ontribution of the wage mark-up sho k redu es to 6.8% only. In ontrast, the household labor

wedge explains more than 41% of wage growth in the model with wedges. Regarding the in�ation

rate, investment sho ks are the key di�eren e: in the model without wedges, investment sho ks

explain around 10% of in�ation variability, while their e�e ts are negligible (1%) in the model with

all wedges. Finally, regarding the interest rate, the role played by investment sho ks is very di�erent

24

See Table 3 of the te hni al appendix in Justiniano et al. (2010).

22

depending on whether the model does or does not in lude wedges: in the model without wedges,

investment sho ks explain more than 60% of interest rate �u tuations; however, on e wedges are

in luded, investment sho ks only explain around 20%, and the bond market wedge largely repla es

the investment sho k.

Overall, our FEVD results indi ate that misspe i� ation is mostly aptured by the household

labor, the bond market and the intermediate goods demand wedges, whi h on�rms the existing

view that the asset and labor markets in the standard New Keynesian DSGE model are misspe i�ed

(see Krause, Lopez-Salido and Lubik, 2008, and Christiano, Motto and Rostagno, 2008, for example).

Marginal Likelihoods As previously dis ussed, we also propose to use the marginal likelihood

to un over the sour es of model misspe i� ation. To do so, we onsider (i) the model with all wedges;

(ii) the model with no wedges; (iii) the models that remove one wedge at a time; and (iv) the model

that removes the three wedges that, a ording to the FEVD, are the sour e of the misspe i� ation.

The marginal likelihood is al ulated as the modi�ed harmoni mean. The logi of this exer ise is

to investigate the relative importan e of wedges: if removing a parti ular wedge obviously redu es

the marginal likelihood value, it is a signal that the wedge is ru ial in explaining the dynami s of

the observed data. Table 9 summarizes the marginal likelihood values of the various versions of the

ben hmark model. First of all, the �rst two rows of Table 9 display the results for models with and

without wedges. Surprisingly, while the FEVDs of these two models are di�erent, their marginal

likelihood values are not very di�erent. Interestingly, however, the model that removes the three

wedges the FEVD exer ise identi�es as the sour e of misspe i� ation is the one with the lowest

likelihood. This �nding on�rms that the latter are the sour es of misspe i� ation, while the small

di�eren es in the marginal likelihood suggest that the degree of misspe i� ation may be mild.

25

INSERT TABLE 9 HERE

Figures 1 and 2 report time series estimates of the wedges and their innovations, respe tively.

By examining the estimate of the bond market wedge over time (τr,t), it is lear that the wedge was

espe ially large in the early 1980s, possibly due to the hanges in monetary poli y around that time.

Serial Correlation in the Sho ks DSGE models rely on a stru tural interpretation of the

exogenous sho ks. The exogenous sho ks should be invariant to any poli y hanges (to be robust to

the Lu as ritique) as well as un orrelated among ea h other. If a sho k in luded in a DSGE model is

in reality a ombination of other sho ks, that sho k should be interpreted as a redu ed-form, rather

than stru tural, sho k. Here, we assess the orre t spe i� ation of the ARMA stru ture of the wage

mark-up sho k, as in Chari et al. (2009).

We onsider a restri ted model whi h is the same as the ben hmark model ex ept that the moving

average oe� ient θw of the wage mark-up sho k is set to zero. We estimate several variants of the

25

We did not investigate all possible ombinations of wedges, as that would result in too many models to onsider. The

s ope of the marginal likelihood analysis was mostly to on�rm the results of the FEVD previously reported.

23

ben hmark model, depending on whi h wedges are in luded or removed, and ompare their marginal

likelihoods.

The last row of Table 10 displays the marginal likelihood of the model with all wedges ex ept

the household labor wedge. Clearly, the marginal likelihood value is lower than that of the other

models. One an interpret this �nding in two ways. A �rst interpretation is that the wage mark-up

sho k indeed follows an ARMA pro ess. Thus, by estimating the model with the wage mark-up

sho k following an AR pro ess, we are a tually estimating a misspe i�ed model. By in luding the

household labor wedge into the model, the potential gap due to the labor market misspe i� ation is

�lled by the labor market wedge. This point an be seen learly by omparing marginal likelihood

values of the models with and without the household labor market wedge. Given that the true model

is that with an ARMA-type wage mark-up sho k, our results indi ate that the marginal likelihood

values indeed help us dete t the sour e of model misspe i� ation. A se ond interpretation is that

the wage mark-up sho k instead follows an AR pro ess. Sin e in luding the household labor wedge

improves the overall �tting of the model, then some fa tors related to the labor market must be

missing. In other words, the models without the household labor wedge are misspe i�ed in the labor

market. Either way, our exer ise asts doubt on the stru tural interpretation of the wage mark-up

sho k.

INSERT TABLE 10 HERE

Our results are related to Del Negro and S horfheide (2009), who suggest to estimate models

with so- alled generalized stru tural sho ks. Our methodology, whi h identi�es the sour es of model

misspe i� ation by examining the FEVD and log marginal likelihood values of models with wedges,

interprets generalized sho ks as a way to apture potential model misspe i� ation.

5 Con lusion

This paper investigates whether existing state-of-the-art stru tural models are misspe i�ed using em-

piri al methods for dete ting and onstru tively identifying the sour e of the misspe i� ation. Our

approa h is based on analyzing FEVD and marginal likelihoods of DSGE models augmented with

wedges, where the wedges are introdu ed to apture potential misspe i� ation. Monte Carlo simu-

lation results demonstrate that our method an orre tly identify the sour e of the misspe i� ation.

Our empiri al results show that a medium-s ale New Keynesian DSGE model that in orporates fea-

tures in the re ent empiri al ma ro literature is still severely misspe i�ed. We �nd that �nan ial and

labour markets were the sour e of the misspe i� ation: they were not orre tly modeled in traditional

representative ma roe onomi models and, onsequently, their poor performan e showed up during

the �nan ial risis. Our results may help shed light on Ng and Wright's (2013) results that the study

of the Great Re ession might require alternative models and predi tors, and suggest that additional

work to in lude more labour and asset market fri tions in the models would be espe ially useful. In

addition, our empiri al �ndings on�rm the onventional wisdom that model misspe i� ation annot

24

be ignored in poli y analyses.

Finally, we should note that there are two potential issues with implementing our method: ex-

ogeneity of the wedges and over-parametrization. First, be ause the wedge pro esses are assumed

to be exogenous, our method might not orre tly identify the lo ation of misspe i� ation if it was

endogenous. One solution may be to let the wedges depend on state variables. In our simulation

results, however, we are able to su essfully identify the misspe i� ation due to serially orrelated

omitted fri tions in the monetary poli y rea tion fun tion, whi h suggests that our approa h an be

useful for �nding omitted fri tions in the model. Se ond, when many markets are in luded in the

model, there may be too many hoi es on where to introdu e the wedges and for forming priors on

them. We suggest to either use fewer wedges when the prior on the lo ation of the misspe i� ation is

strong (e.g. the resear her is on�dent that there is no misspe i� ation in some parts of the model,

but unsure whether there might be misspe i� ation in others, and the resear her has strong opinions

about where the misspe i� ation is potentially lo ated), or introdu e many wedges and impose prior

information when the misspe i� ation lo ation is more un ertain (i.e. every part of the model an

potentially be misspe i�ed). When neither is possible, another approa h is to use a Bayesian model

averaging approa h to take into a ount many wedges, and let the Bayesian model average pro edure

provide information on the lo ation of the wedges. We leave these extensions to future resear h.

26

26

Also, note that our method of omparing models via their marginal likelihood is learly appropriate if one of the

models with wedges is the true model. Otherwise, one ould onsider alternative ways to perform the omparison, su h as

ross-validation and out-of-sample predi tions (see Bernardo and Smith, 2000, p.403 and Geweke, 2010).

25

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28

A Supplementary Material for Simulation Exer ises

In this se tion, we provide supplementary materials of our simulation exer ises.

A.1 The Data-Generating Pro ess

The DGP of our simulation exer ises an be summarized by the following linearized equilibrium

onditions:

Yt = Et

{

Yt+1

}

−1

γ

(

Rt − Etπt+1

)

+ (gt − Etgt+1) , (22)

πt = κ{

(γ + ϕ) Yt − (ϕ+ 1) zt − γgt + χt

}

+ βEt {πt+1} , (23)

Rt = ρr Rt−1 + (1− ρr){

γππt + γyYt

}

+ νt, (24)

Yt = zt + Lt, (25)

where (22) is the dynami IS urve, (23) is the New Keynesian Phillips urve, (24) is the monetary

poli y rule, and (25) is the linearized aggregate produ tion fun tion. Note that in the New Keynesian

Phillips urve, κ = (1−βξ)(1−ξ)ξ

.

For the sake of onvenien e, we list the stru tural sho ks pro esses below:

zt+1 = ρz zt + σzεz,t+1, εz,t+1i.i.d.∼ N(0, 1), (26)

νt+1 = ρν νt + σνεν,t+1, εν,t+1i.i.d.∼ N(0, 1), (27)

gt+1 = ρg gt + σgεg,t+1, εg,t+1i.i.d.∼ N(0, 1), (28)

χt+1 = ρχχt + σχεχ,t+1, εχ,t+1i.i.d.∼ N(0, 1). (29)

A.2 Models Containing Wedges

The wedge pro esses of the four estimated models are:

τl,t+1 = ρlτl,t + σlεl,t+1, εl,t+1i.i.d.∼ N(0, 1). (30)

τc,t+1 = ρcτc,t + σcεc,t+1, εc,t+1i.i.d.∼ N(0, 1). (31)

τb,t+1 = ρbτb,t + σbεb,t+1, εb,t+1i.i.d.∼ N(0, 1). (32)

Equilibrium Conditions of the M0 model Sin e the M0 model is basi ally orre tly spe -

i�ed, the equilibrium onditions of the model M0 are highly similar to those of the DGP. The

only di�eren e is that the dynami IS urve and the New Keynesian Phillips urve ontain three

time-varying wedges. The orresponding equilibrium onditions are listed below:

29

Yt = Et

{

Yt+1

}

−1

γ

(

Rt − Etπt+1

)

+ (gt − Etgt+1)−1

γ(τc,t − Etτc,t+1)−

1

γEt {τb,t+1} ,(33)

πt = κ{

(γ + ϕ) Yt − (ϕ+ 1) zt − γgt + χt

}

+ βEt {πt+1}+ κ (τc,t + τl,t) , (34)

Rt = ρrRt−1 (1− ρr){

γππt + γyYt

}

+ νt, (35)

Yt = zt + Lt, (36)

In sum the M0 model is hara terized by

• four equilibrium onditions: (33), (34), (35), and (36);

• four exogenous stru tural sho k pro esses: (26), (27), (28), and (29)

• three exogenous wedge pro esses: (30), (31), and (32).

Equilibrium Conditions of the Ml model Compared to model M0, model Ml does not

ontain labor supply sho ks. Thus, the New Keynesian Phillips urve has to be modi�ed as

πt = κ{

(γ + ϕ) Yt − (ϕ+ 1) zt − γgt

}

+ βEt {πt+1}+ κ (τc,t + τl,t) . (37)

In sum, Ml is hara terized by:

• four equilibrium onditions: (33), (37), (35), and (36);

• three exogenous stru tural sho k pro esses: (26), (27), and (28);

• three exogenous wedge pro esses: (30), (31), and (32).

Equilibrium Conditions of the Mm Model In this model, the only sour e of model mis-

spe i� ation is due to the spe i� ation of the monetary sho k. That is, the monetary sho ks are

treated as identi ally and independently distributed rather than serial orrelated:

νt+1 = σνεν,t+1, εν,t+1i.i.d.∼ N(0, 1). (38)

In sum, Mm is hara terized by

• four equilibrium onditions: (33), (34), (35), and (36);

• four exogenous stru tural sho k pro esses: (26), (38), (28), and (29);

• three exogenous wedge pro esses: (30), (31), and (32).

Equilibrium Conditions of the Mg Model In this model, sin e the government spending

sho ks are ex luded, this ex lusion a�e ts both the dynami IS urve and the New Keynesian Phillips

urve. They are repla ed by

Yt = Et

{

Yt+1

}

−1

γ

(

Rt − Etπt+1

)

−1

γ(τc,t − Etτc,t+1)−

1

γEt {τb,t+1} , (39)

πt = κ{

(γ + ϕ) Yt − (ϕ+ 1) zt + χt

}

+ βEt {πt+1}+ κ (τc,t + τl,t) . (40)

30

In sum, the Mg are hara terized by

• four equilibrium onditions: (39), (40), (35), and (36);

• three exogenous stru tural sho k pro esses: (26), (27), and (29);

• three exogenous wedge pro esses: (30), (31), and (32).

B Linearized Equilibrium Conditions

To linearize the equilibrium onditions, we �rst de�ne the following stationary variables

yt = Yt/At, xt = Xt/At, kt = Kt/At, kt = Kt/At, ct = Ct/At, it = It/At, qt = Qt/At

pt = Pt/Pt, wt = Wt/AtPt, st = St/Pt, πt = Pt/Pt−1, λt = ΛtAtPt, φt = ΦtAt wt = Wt/AtPt

After that, we express the above stationary variables in terms of log deviation forms. Basi ally, for

any variable ζt, we de�ne its log deviation as

ζt = log ζt − log ζ

where ζ denotes the steady-state value of ζt. However, there are three ex eptions: We de�ne zt =

zt − γ, λp,t = log (1 + λp,t) − log (1 + λp) and λw,t = log (1 + λw,t) − log (1 + λw) . We lassify the

linearized equilibrium onditions into three ategories: the endogenous part, the exogenous stru tural

sho ks, and the exogenous wedge pro esses.

The Endogenous Part This ategory onsists of 16 equilibrium onditions, derived from the

optimal de isions of agents in the e onomy. They are

1. The produ tion fun tion:

yt =y + F

y

(

αkt + (1− α)Lt

)

2. The apital-labor ratio:

kt − Lt = (wt − rk,t) + (τl,t − τk,t)

3. The real marginal ost:

st = α (τk,t + rk,t) + (1− α) (τl,t + wt)

4. The New Keynesian Phillips urve (NKPC):

27

πt =ιp

1 + ιpβπt−1 +

β

1 + ιpβEt {πt+1}+ κpst + λn

p,t

−

[(

1

1 + ιpβ

)

ϑ−1q

]

τnq,t−1 + τn

q,t −

[(

β

1 + ιpβ

)

ϑ−1q

]

Et

{

τnq,t+1

}

27

Following Justiniano et al. (2010), we normalize the pri e mark-up sho k su h that it enters the NKPC with a oe� ient

equal to one. We perform a similar transformation on τq,t.

31

where

λnp,t = ϑpλp,t

τnq,t = ϑq τq,t

ϑp = κp =(1− βξp) (1− ξp)

(1 + ιpβ) ξp

ϑq =

(

1 + βξ2pξp (1 + ιpβ)

)

5. Capital used in produ tion:

kt = ut +ˆkt−1 − zt

6. The apital evolution equation:

kt = (1− δ) e−γ(

kt−1 − zt)

−(

1− (1− δ) e−γ)

(

µt + it)

7. The marginal utility of onsumption:

τc,t + λt =

(

eγ

eγ − ~β

)(

zt + bt −eγ

eγ − ~(zt + ct) +

~

eγ − ~ct−1

)

−

(

~β

eγ − ~β

)(

bt+1 −eγ

eγ − ~(zt+1 + ct+1) +

~

eγ − ~ct

)

.

8. The Euler equation:

λt = Et

{

λt+1 + τr,t+1 + Rt − zt+1 − πt+1

}

9. The apital utilization rate:

rk,t = χut

where χ = a′′

/a′ = a′′

/rk

10. The apital supply:

φt =[

1− β(1− δ)e−γ]

(

λt+1 − zt+1 + rk,t+1

)

+ β(1− δ)e−γ(

φt+1 − zt+1

)

11. The investment:

λt = φt + µt − (e2γS′′)(

it − it−1 + zt)

+(

βe2γS′′)Et

(

it+1 − it + zt+1

)

12. The wage equation:

wt =1

1 + β(wt−1) +

β

1 + βEt {wt+1} − κwgw,t

+ιw

1 + β(πt−1 + zt−1)−

(

1 + βιw1 + β

)

(πt + zt) +β

1 + βEt {πt+1 + zt+1}

+λnw,t −

[

1

1 + βϑ−1h

]

τnh,t−1 + τn

h,t −

[(

β

1 + β

)

ϑ−1h

]

τnh,t+1

32

where

λnw,t = ϑwλw,t

τnh,t = ϑhλh,t

ϑw = κw =

(

1− βξwνw (1− β)

)(

1− ξwξw

)

ϑh = 1 + κw

13. The interest rate rule:

Rt = ρRRt−1 + (1− ρR) [φππt + φx (xt − x∗t )] + φdX [(xt − xt−1)− (x∗

t − x∗t−1)] + ηmp,t

14. GDP:

xt = yt −rkk

yut

15. The resour e onstraint:

c

yct +

i

yit +

rkk

yut =

1

gyt −

1

ggt

16. The de�nition of gw,t:

gw,t = wt −(

νLt + bt − λt

)

Stru tural Sho ks In terms of log deviations from steady state values, there are 7 stru tural

sho ks, ontaining (i) the te hnology sho k zt, (ii) the monetary sho k ηmp,t, (iii) the government

spending sho k gt, (iv) the preferen e sho k bt, (v) the investment sho k µt, (vi) the pri e mark-up

sho k λp,t, and (vii) the wage mark-up sho k: λw,t.

Wedge Pro esses In terms of log deviations, there are 6 wedges in our model, ontaining (i)

the labor demand wedge τl,t, (ii) the apital demand wedge τk,t, (iii) the onsumption wedge τc,t,

(iv) the bond market wedge: τr,t, (v) the labor market wedge τh,t, and (vi) the intermediate good

wedge τy,t.

33

Tables

Table 1. The DGP Parameter Values and the Prior Setting of Simulation Exer ises

Parameter Des ription DGP Prior

Type Mean Std.

β Subje tive dis ount fa tor 0.99 - - -

γ Risk aversion oe� ient 1.50 - - -

ϕ Inverse Fris h elasti ity 1.00 - - -

ξ Calvo parameter 0.67 B 0.70 0.05

γπ Interest rate poli y rule: in�ation 1.80 G 2.00 0.05

γy Interest rate poli y rule: output 0.50 G 0.40 0.05

ρr Persisten e oef. of interest rate 0.70 B 0.75 0.10

ρz Persisten e of te hnology sho ks 0.70 B 0.75 0.10

ρν Persisten e of monetary sho ks 0.70 B 0.75 0.10

ρg Persisten e of spending sho ks 0.70 B 0.75 0.10

ρχ Persisten e of labor supply sho ks 0.70 B 0.75 0.10

σz Std. of innovation of te hnology sho ks 0.10 I 0.30 2.00

σν Std. of innovation of monetary sho ks 0.10 I 0.30 2.00

σg Std. of innovation of spending sho ks 0.10 I 0.30 2.00

σχ Std. of innovation of labor supply sho ks 0.10 I 0.30 2.00

ρl Persisten e of labor wedge (τl,t) - B 0.50 0.20

ρc Persisten e of onsumption wedge (τc,t) - B 0.50 0.20

ρb Persisten e of bond market wedge (τb,t) - B 0.50 0.20

σl Std. of innovation of labor wedge - I 0.10 2.00

σc Std of innovation of onsumption wedge - I 0.10 2.00

σb Std of innovation of bond market wedge - I 0.10 2.00

Notes to the Table. The table des ribes the parameter values used in the simulation of the model

( olumn labeled "DGP") as well as the distribution ("Type"), the mean ("Mean") and the standard

deviation ("Std.") of the prior used in the estimation ( olumns labeled "Prior"). The distribution

an be either Gaussian ("G"), Beta ("B"), Gamma ("G") or Inverse Gamma ("I").

34

Table 2: Fore ast Error Varian e De ompositions

Panel (a): Model M0

Variable z ν g χ τl τc τb

Hours 9.15 57.78 10.06 21.41 0.44 0.89 0.26

Output 9.64 57.47 10.01 21.29 0.44 0.89 0.26

In�ation Rate 3.56 87.81 0.10 8.17 0.29 0.00 0.07

Interest Rate 21.21 27.99 1.20 47.53 1.16 0.21 0.70

Panel (b): Model Ml

Variable z ν g χ τl τc τb

Hours 9.40 60.67 8.93 − 19.73 1.13 0.15

Output 9.51 60.59 8.92 − 19.70 1.13 0.15

In�ation Rate 3.39 88.29 0.06 − 8.23 0.01 0.03

Interest Rate 21.25 30.04 1.51 − 46.62 0.27 0.31

Panel ( ): Model Mm

Variable z ν g χ τl τc τb

Hours 11.96 14.87 8.55 22.18 0.50 0.95 40.99

Output 9.66 15.26 8.77 22.76 0.51 0.97 42.06

In�ation Rate 6.54 22.84 0.19 15.10 0.62 0.02 54.69

Interest Rate 2.94 3.81 0.06 6.97 0.14 0.02 86.06

Panel (d): Model Mg

Variable z ν g χ τl τc τb

Hours 9.46 59.89 − 22.49 0.51 6.59 1.05

Output 9.04 60.17 − 22.60 0.51 6.63 1.06

In�ation Rate 3.37 87.25 − 8.66 0.33 0.04 0.36

Interest Rate 18.64 27.56 − 47.04 1.23 1.19 4.33

Notes to the table. The table reports the median of the FEVD (in per entage) for the models we

estimated. The fore ast horizon is 12 periods.

Table 3. Marginal Likelihood Values of Models

M0 Ml Mm Mg

All wedges -1412.3 -1415.4 -1497.1 -1412.2

Remove τl -1413.0 -1429.1 -1497.6 -1412.6

Remove τc -1411.8 -1414.4 -1496.4 -1413.3

Remove τb -1411.7 -1414.6 -1506.7 -1412.6

Notes to the table. The table shows values of the log marginal likelihood, al ulated via the modi�ed

harmoni mean. Lower values of the likelihood denote models that are the most at odds with the

data. Thus, the wedges whose removal are asso iated with the lowest likelihood are the wedges that

are deemed the most ne essary to explain the data.

35

Table 4. Prior and Posterior Distributions of the Ben hmark Model

Parameter Des ription Prior Posterior

Type Mean Std Median Std 5% 95%

α Capital share Normal 0.30 0.05 0.17 0.01 0.16 0.18

ιp Pri e indexation Beta 0.50 0.15 0.19 0.08 0.09 0.31

ιw Wage indexation Beta 0.50 0.15 0.10 0.03 0.04 0.15

100γ SS Te hnology growth rate Normal 0.50 0.03 0.48 0.03 0.44 0.53

~ Consumption habit Beta 0.50 0.10 0.83 0.05 0.78 0.89

λp SS pri e mark-up Normal 0.15 0.05 0.23 0.04 0.17 0.29

λw SS wage mark-up Normal 0.15 0.05 0.14 0.05 0.06 0.22

100 log(Lss) 100 times log hours Normal 0.00 0.50 0.12 0.51 -0.76 0.99

100(π − 1) SS in�ation rate Normal 0.50 0.10 0.63 0.12 0.48 0.79

100(β−1 − 1) Dis ount fa tor Gamma 0.25 0.10 0.14 0.05 0.07 0.21

ν Inverse Fris h elasti ity Gamma 2.00 0.75 3.84 0.80 2.58 5.03

ξp Calvo pri es Beta 0.66 0.10 0.85 0.02 0.81 0.88

ξw Calvo wages Beta 0.66 0.10 0.72 0.05 0.63 0.81

χ Capital utilization osts Gamma 5.00 1.00 5.33 0.99 3.80 7.11

S′Investment adjustment osts Gamma 4.00 1.00 2.23 0.50 1.37 3.12

φπ Taylor rule: in�ation Normal 1.70 0.30 2.03 0.17 1.69 2.39

φx Taylor rule: output Normal 0.13 0.05 0.06 0.02 0.03 0.09

φdX Taylor rule: output growth Normal 0.13 0.05 0.24 0.02 0.20 0.28

ρR Interest rate smoothing Beta 0.60 0.20 0.85 0.02 0.81 0.88

Stru tural Sho ks

ρmp Monetary poli y Beta 0.40 0.20 0.11 0.06 0.02 0.21

ρz Te hnology growth Beta 0.60 0.20 0.24 0.06 0.13 0.35

ρg Government spending Beta 0.60 0.20 1.00 0.00 1.00 1.00

ρµ Investment Beta 0.60 0.20 0.60 0.05 0.51 0.73

ρp Pri e mark-up Beta 0.60 0.20 0.94 0.02 0.90 0.97

ρw Wage mark-up Beta 0.60 0.20 0.99 0.01 0.98 1.00

ρb Intertemporal preferen e Beta 0.60 0.20 0.27 0.13 0.11 0.45

θp Pri e mark-up MA Beta 0.50 0.20 0.47 0.10 0.30 0.69

θw Wage mark-up MA Beta 0.50 0.20 0.90 0.03 0.85 0.94

100σmp Monetary poli y InvGamma 0.10 1.00 0.23 0.01 0.21 0.25

100σz Te hnology growth InvGamma 0.50 1.00 0.87 0.05 0.80 0.95

100σg Government spending InvGamma 0.50 1.00 0.35 0.02 0.32 0.37

100σµ Investment InvGamma 0.50 1.00 5.24 0.89 3.47 7.24

100σp Pri e mark-up InvGamma 0.10 1.00 0.06 0.02 0.03 0.08

100σw Wage mark-up InvGamma 0.10 1.00 0.09 0.02 0.05 0.12

100σb Intertemporal preferen e InvGamma 0.10 1.00 0.10 0.02 0.06 0.12

Wedge Pro esses

ρl Homogenous labor Beta 0.50 0.20 0.52 0.28 0.20 0.84

ρk Capital Beta 0.50 0.20 0.53 0.28 0.21 0.85

ρc Consumption Beta 0.50 0.20 0.52 0.28 0.17 0.82

ρr Bond Beta 0.50 0.20 0.93 0.28 0.90 0.96

ρq Intermediate good Beta 0.50 0.20 0.39 0.30 0.10 0.71

ρh Heterogenous labor Beta 0.50 0.20 0.74 0.28 0.51 0.94

100σl Homogenous labor InvGamma 0.10 1.00 0.06 0.02 0.03 0.11

100σk Capital InvGamma 0.10 1.00 0.05 0.02 0.03 0.09

100σc Consumption InvGamma 0.10 1.00 0.07 0.02 0.03 0.15

100σr Bond market InvGamma 0.10 1.00 0.17 0.02 0.10 0.24

100σq Intermediate good InvGamma 0.10 1.00 0.11 0.03 0.07 0.14

100σh Heterogenous labor InvGamma 0.10 1.00 0.35 0.02 0.28 0.41

36

Table 5. Prior and Posterior Distributions: The Ben hmark Model with No Wedges

Prior Posterior

Parameter Des ription Type Mean Std Median Std 5 P t 95 P t

α Capital share Normal 0.30 0.05 0.17 0.01 0.16 0.18

ιp Pri e indexation Beta 0.50 0.15 0.22 0.08 0.10 0.34

ιw Wage indexation Beta 0.50 0.15 0.11 0.03 0.06 0.17

100γ SS Te hnology growth rate Normal 0.50 0.03 0.48 0.03 0.44 0.52

~ Consumption habit Beta 0.50 0.10 0.81 0.05 0.74 0.88

λp SS pri e mark-up Normal 0.15 0.05 0.23 0.04 0.17 0.29

λw SS wage mark-up Normal 0.15 0.05 0.15 0.05 0.07 0.22

100 log(Lss) 100 times log hours Normal 0.00 0.50 0.19 0.51 -0.61 0.97

100(π − 1) SS in�ation rate Normal 0.50 0.10 0.64 0.12 0.50 0.79

100(β−1 − 1) Dis ount fa tor Gamma 0.25 0.10 0.14 0.05 0.06 0.21

ν Inverse Fris h elasti ity Gamma 2.00 0.75 3.93 0.80 2.68 5.24

ξp Calvo pri es Beta 0.66 0.10 0.83 0.02 0.80 0.87

ξw Calvo wages Beta 0.66 0.10 0.68 0.05 0.58 0.76

χ Capital utilization osts Gamma 5.00 1.00 5.32 0.99 3.78 7.02

S′Investment adjustment osts Gamma 4.00 1.00 2.72 0.50 1.91 3.60

φπ Taylor rule: in�ation Normal 1.70 0.30 2.08 0.17 1.80 2.39

φx Taylor rule: output Normal 0.13 0.05 0.07 0.02 0.04 0.10

φdX Taylor rule: output growth Normal 0.13 0.05 0.24 0.02 0.20 0.28

ρR Interest rate smoothing Beta 0.60 0.20 0.82 0.02 0.77 0.85

Stru tural Sho k

ρmp Monetary poli y Beta 0.40 0.20 0.16 0.06 0.06 0.27

ρz Te hnology growth Beta 0.60 0.20 0.25 0.06 0.15 0.34

ρg Government spending Beta 0.60 0.20 1.00 0.00 1.00 1.00

ρµ Investment Beta 0.60 0.20 0.71 0.05 0.64 0.79

ρp Pri e mark-up Beta 0.60 0.20 0.94 0.02 0.89 0.98

ρw Wage mark-up Beta 0.60 0.20 0.98 0.01 0.96 1.00

ρb Intertemporal preferen e Beta 0.60 0.20 0.58 0.12 0.43 0.74

θp Pri e mark-up MA Beta 0.50 0.20 0.72 0.08 0.57 0.83

θw Wage mark-up MA Beta 0.50 0.20 0.92 0.03 0.88 0.96

100σmp Monetary poli y InvGamma 0.10 1.00 0.22 0.01 0.20 0.25

100σz Te hnology growth InvGamma 0.50 1.00 0.89 0.05 0.80 0.97

100σg Government spending InvGamma 0.50 1.00 0.35 0.02 0.32 0.38

100σµ Investment InvGamma 0.50 1.00 5.84 0.89 4.24 7.34

100σp Pri e mark-up InvGamma 0.10 1.00 0.14 0.01 0.12 0.16

100σw Wage mark-up InvGamma 0.10 1.00 0.21 0.02 0.18 0.24

100σb Intertemporal preferen e InvGamma 0.10 1.00 0.05 0.01 0.03 0.07

37

Table 6. Wedges and Sho ks: Summary Table

Wedge Des ription Innovation Referen e Equation

Panel A. Wedges

τl Homogeneous labor market wedge εl Eq. (10)

τk Capital market wedge εk Eq. (9)

τc Consumption wedge εc Eq. (15)

τr Bond market wedge εr Eq. (12)

τq Intermediate good demand wedge εq Eq. (5)

τh Household labor wedge εh Eq. (19)

Panel B. Sho ks

ηmp Monetary poli y sho k εmp Eq. (21)

z Te hnology sho k εz Eq. (8)

g Government spending sho k εg Eq. (20)

µ Investment sho k εµ Eq. (14)

λp Pri e mark-up sho k εp Eq. (4)

λw Wage mark-up sho k εw Eq. (18)

b Intertemporal preferen e sho k εb Eq. (16)

Notes to the table. The table summarizes the sho ks and the wedges in the model onsidered in this

paper.

38

Table 7. The Fore ast Error Varian e De omposition: Ben hmark Model (All Wedges in luded)

Sho ks Wedges

Series εmp εz,t εg,t εµ,t εp,t εw,t εb,t εl,t εk,t εc,t εr,t εq,t εh,t

Fore ast Horizon: H = 1

Output growth 5.79 12.41 10.04 51.79 0.83 0.06 13.71 0.00 0.00 0.00 3.32 0.03 0.03

Consumption growth 1.84 15.00 1.46 0.05 0.09 1.94 77.88 0.00 0.00 0.00 1.20 0.01 0.02

Investment growth 4.93 4.59 0.02 85.30 0.92 0.24 0.38 0.00 0.00 0.00 2.67 0.03 0.04

Hours 5.29 20.32 9.23 47.06 0.73 0.06 12.23 0.00 0.00 0.00 3.01 0.03 0.04

Wage growth 0.27 8.46 0.00 0.48 11.36 11.70 0.10 0.01 0.00 0.00 0.22 4.43 61.78

In�ation rates 2.83 13.53 0.10 1.19 36.27 15.11 0.42 0.06 0.00 0.00 4.78 19.60 1.15

Interest rates 53.29 6.49 0.62 12.39 1.05 0.96 13.65 0.03 0.00 0.00 6.22 1.75 0.36

Fore ast Horizon: H = 4

Output growth 6.16 22.05 7.75 43.64 2.05 0.80 10.78 0.00 0.00 0.00 3.81 0.03 0.06

Consumption growth 2.10 26.60 2.57 0.34 0.26 4.48 60.75 0.00 0.00 0.00 1.47 0.01 0.02

Investment growth 5.56 8.72 0.02 78.04 2.31 0.27 0.41 0.00 0.00 0.00 3.20 0.03 0.06

Hours 10.90 7.11 4.44 54.39 3.62 1.16 7.77 0.00 0.00 0.00 7.09 0.01 0.13

Wage growth 0.32 26.54 0.00 0.77 13.90 7.38 0.07 0.01 0.00 0.00 0.26 3.72 45.40

In�ation rates 4.83 14.36 0.13 1.48 26.74 26.26 0.58 0.03 0.00 0.00 8.68 9.05 0.77

Interest rates 21.75 9.69 0.70 27.93 1.50 3.83 8.75 0.02 0.00 0.00 18.92 0.48 0.34

Fore ast Horizon: H = 20

Output growth 6.10 22.57 6.90 42.63 2.47 2.65 10.02 0.00 0.00 0.00 3.63 0.03 0.08

Consumption growth 1.82 28.59 2.66 1.71 0.38 6.21 54.87 0.00 0.00 0.00 1.30 0.01 0.02

Investment growth 5.58 8.44 0.03 77.05 2.81 0.97 0.42 0.00 0.00 0.00 3.09 0.03 0.07

Hours 8.82 7.20 2.38 23.18 9.87 32.63 2.70 0.00 0.00 0.00 7.47 0.01 0.11

Wage growth 0.32 30.70 0.00 0.88 14.18 6.80 0.07 0.01 0.00 0.00 0.27 3.36 41.78

In�ation rates 5.29 9.52 0.13 1.02 16.63 41.26 0.54 0.02 0.00 0.00 12.85 5.25 0.49

Interest rates 10.07 5.70 0.46 19.56 1.15 10.53 4.54 0.01 0.00 0.00 40.14 0.22 0.17

Notes to the table. The table reports the median of the FEVD (in per entage) for the model with

wedges. The FEVD is al ulated over the MCMC draws we do not dis ard (therefore, the sum of

the per entages in a given row does not ne essarily equal 100).

39

Table 8. The Fore ast Error Varian e De omposition:

The Ben hmark without Wedges

Series εmp εz,t εg,t εµ,t εp,t εw,t εb,t

Fore ast Horizon: H = 1

Output growth 3.01 9.69 8.41 68.75 0.96 0.50 7.85

Consumption growth 1.81 17.20 2.29 1.75 0.23 2.79 72.90

Investment growth 1.60 2.44 0.01 94.23 0.72 0.01 0.70

Hours 2.89 12.86 8.14 66.08 0.90 0.52 7.57

Wage growth 0.30 10.82 0.00 2.25 21.72 63.68 0.33

In�ation rates 1.92 12.00 0.11 7.34 54.08 21.44 1.15

Interest rates 46.28 6.04 0.67 24.84 3.51 1.22 14.77

Fore ast Horizon: H = 4

Output growth 3.00 16.24 6.45 63.18 1.66 1.90 6.31

Consumption growth 1.72 27.03 3.60 1.74 0.37 5.70 58.16

Investment growth 1.68 4.32 0.01 91.16 1.29 0.24 0.84

Hours 4.32 3.44 3.05 76.16 2.64 2.90 6.24

Wage growth 0.34 34.50 0.00 3.82 19.09 40.63 0.25

In�ation rates 3.56 12.96 0.15 11.96 29.93 35.86 1.90

Interest rates 15.49 6.89 0.53 55.86 2.38 3.60 12.59

Fore ast Horizon: H = 20

Output growth 3.04 16.19 5.80 62.66 1.78 2.97 6.52

Consumption growth 1.44 25.19 3.33 7.66 0.33 6.17 53.53

Investment growth 1.69 4.04 0.01 90.89 1.39 0.59 0.86

Hours 3.41 4.09 1.98 48.17 5.01 32.53 2.94

Wage growth 0.33 37.58 0.01 4.30 18.44 37.64 0.31

In�ation rates 3.95 9.64 0.14 10.71 20.65 49.70 1.91

Interest rates 9.03 5.01 0.41 62.14 1.64 9.91 9.18

Notes to the table. The table reports the median of the FEVD (in per entage) for the model without

wedges. The FEVD is al ulated over the MCMC draws we do not dis ard (therefore, the sum of

the per entages in a given row does not ne essarily equal 100).

40

Table 9. Log Marginal Likelihood Values: Variants of the Ben hmark Model

Models Log Marginal Likelihood Ranking

All wedges -1167.62 4

Remove all wedges -1166.99 3

Remove τl wedge -1169.13 7

Remove τk wedge -1166.19 1

Remove τc wedge -1170.86 8

Remove τr wedge -1166.66 2

Remove τq wedge -1168.87 6

Remove τh wedge -1168.25 5

Remove {τr, τq, τh} wedges -1171.01 9

Notes to the table. The table shows values of the log marginal likelihood, al ulated via the modi�ed

harmoni mean. Lower values of the likelihood denote models that are the most at odds with the

data. Thus, the wedges whose removal are asso iated with the lowest likelihood are the wedges that

are deemed the most ne essary to explain the data.

41

Table 10. Log Marginal Likelihood Values: Model with a

Simple AR Pro ess for Wage Mark-up Sho k

Models Log Marginal Likelihood Ranking

All wedges -1173.93 6

Remove all wedges -1192.49 7

Remove τl wedge -1170.18 1

Remove τk wedge -1171.52 3

Remove τc wedge -1170.19 2

Remove τr wedge -1172.53 4

Remove τq wedge -1173.09 5

Remove τh wedge -1207.99 8

Notes to the table. The table shows values of the log marginal likelihood, al ulated via the modi�ed

harmoni mean. Lower values of the likelihood denote models that are the most at odds with the

data. Thus, the wedges whose removal are asso iated with the lowest likelihood are the wedges that

are deemed the most ne essary to explain the data.

42

Figure 1. Wedges Over Time (Smoothed)

1960 1970 1980 1990 2000−4

−2

0

2

4x 10

−3 τ l,t

1960 1970 1980 1990 2000−4

−2

0

2

4x 10

−4 τk,t

1960 1970 1980 1990 2000−3

−2

−1

0

1

2

3x 10

−3 τc ,t

1960 1970 1980 1990 2000−1.5

−1

−0.5

0

0.5

1τr,t

1960 1970 1980 1990 2000−0.2

−0.1

0

0.1

0.2

0.3τq ,t

1960 1970 1980 1990 2000−1

−0.5

0

0.5

1

1.5τh,t

Note to the �gure. The �gure plots the time series of the estimated wedges over time. Shaded areas

indi ate U.S. re essions.

Figure 2. Wedge Innovations Over Time (Smoothed)

1960 1970 1980 1990 2000−4

−2

0

2

4

6x 10

−3 εl , t

1960 1970 1980 1990 2000−4

−2

0

2

4x 10

−4 εk , t

1960 1970 1980 1990 2000−2

−1

0

1

2x 10

−3 εc , t

1960 1970 1980 1990 2000−0.4

−0.2

0

0.2

0.4εr, t

1960 1970 1980 1990 2000−0.2

−0.1

0

0.1

0.2

0.3εq , t

1960 1970 1980 1990 2000−1

−0.5

0

0.5

1

1.5εh,t

Note to the �gure. The �gure plots the time series of the estimated innovations to the wedges over

time. Shaded areas indi ate U.S. re essions.

43