Discussion Paper No.308
Zero Lower Bound and Parameter Biasin an Estimated DSGE Model

Yasuo Hirose
Atsushi Inoue
September, 2013

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY

27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANZero Lower Bound and Parameter Biasin an Estimated DSGE Model∗
Yasuo Hirose†

Atsushi Inoue‡

September 2013

Abstract
This paper examines how and to what extent parameter estimates can be biased in a
dynamic stochastic general equilibrium (DSGE) model that omits the zero lower boundconstraint on the nominal interest rate. Our experiments show that most of the parameter estimates in a standard sticky-price DSGE model are not biased although some biases
are detected in the estimates of the monetary policy parameters and the steady-state realinterest rate. Nevertheless, in our baseline experiment, these biases are so small that theestimated impulse response functions are quite similar to the true impulse response functions. However, as the probability of hitting the zero lower bound increases, the biases
in the parameter estimates become larger and can therefore lead to substantial diﬀerences
between the estimated and true impulse responses.

Keywords: Zero lower bound, DSGE model, Parameter bias, Bayesian estimation
JEL Classification: C32, E30, E52∗

We would like to thank Xiaoshan Chen, Günter Coenen, Andrew Foerster, Nikolay Iskrev, Fabio Milani,

Tatevik Sekhposyan, and participants at the International Conference on Computing in Economics and Financeand the North American Summer Meeting of the Econometric Society for insightful comments and discussions.
†‡

Faculty of Economics, Keio University. E-mail: yhirose@econ.keio.ac.jp
Department of Economics, Southern Methodist University and Graduate School of Economics, Tohoku

University. E-mail: ainoue@mail.smu.edu

11

Introduction

Dynamic stochastic general equilibrium (DSGE) models have become a prominent tool forpolicy analysis. In particular, following the development of Bayesian estimation and evaluation techniques, estimated DSGE models have been extensively used by a range of policy
institutions, including central banks. At the same time, the zero lower bound constraint on
the nominal interest rates has been a primary concern for policymakers. Much work has been
devoted to understand how the economy works and how policy should be conducted in thepresence of this constraint from a theoretical perspective.1 However, empirical studies that
estimate DSGE models including the interest-rate lower bound are still scarce because of computational diﬃculties in the treatment of nonlinearity arising from the bound,2 and hence mostpractitioners continue to estimate linearized DSGE models without explicitly considering thelower bound.
This paper examines how and to what extent the estimates of structural parameters canbe biased in an estimated DSGE model where the existence of the interest-rate lower bound isomitted in the estimation process. Suppose that there is a zero lower bound constraint in theeconomy and that an econometrician ﬁts a model without taking into account this constraint.
We would then expect the parameter estimates in the model to be biased. If signiﬁcant biases
were detected, it would cast some doubt on the common practice in which the zero lower boundconstraint is not explicitly taken into account and would motivate researchers toward the useof an estimation procedure that deals with the constraint. Conversely, if the biases involved are
negligible, it would at least assure practitioners that their common practice leads to reliable
estimates in its own way.
More speciﬁcally, we construct artiﬁcial time series simulated from a standard sticky-priceDSGE model that incorporates an occasionally binding constraint on the nominal interest rate.
1See, for instance, Eggertsson and Woodford (2003), Jung, Teranishi, and Watanabe (2005), Adam andBilli (2006, 2007), Christiano, Eichenbaum, and Rebelo (2011), Aruoba and Schorfheide (2013), Bodenstein,
Guerrieri, and Gust (2013), Braun, Körber, and Waki (2012), Erceg and Lindé (2012), Fernández-Villaverde,
Gordon, Guerrón-Quintana, and Rubio-Ramı́rez (2012), and Gavin, Keen, Richter, and Throckmorton (2013).
2

A remarkable exception is Gust, López-Salido, and Smith (2012) who estimate a nonlinear DSGE model inwhich the interest-rate lower bound is occasionally binding. Based on the estimated model, they quantify theeﬀect of the zero lower bound constraint on the recent economic slump in the US.

2

The parameters calibrated in this data generating process (DGP) are regarded as true values.
The solution algorithm follows from Erceg and Lindé (2012) and Bodenstein, Guerrieri, andGust (2013).3 In their algorithm, if the model-implied nominal interest rate falls below zero, asequence of contractionary monetary policy shocks is added in both the current and anticipatedperiods so that the contemporaneous and expected interest rates at the lower bound are zero.
Then, using the simulated data, the model is estimated without including the interest-rate
lower bound. In the estimation, we employ Bayesian methods, which are now extensively used
to estimate DSGE models. We set the prior means to the true parameter values and assess
the parameter biases from neglecting the lower bound by comparing the posterior means andBayesian credible intervals with the true values.
One could argue that the DGP in our analysis should be replaced with a fully nonlinear model as in Braun, Körber, and Waki (2012), Fernández-Villaverde, Gordon, GuerrónQuintana, and Rubio-Ramı́rez (2012) or Gust, López-Salido, and Smith (2012). However, if
the DGP is characterized by a fully nonlinear model, we cannot identify the source of parameter
bias: whether the exclusion of the interest-rate lower bound or the linearization of the model.
In this regard, the solution method employed in this paper is compatible with the standard
solution algorithms for linear rational expectations models, such as Blanchard and Kahn (1980)
and Sims (2002). Therefore, our analysis points to parameter bias only resulting from omitting
the zero lower bound constraint.
Our main results are summarized as follows. In the baseline economy, where the true value
for the steady-state real interest rate is calibrated such that the probability of hitting the zero
lower bound is about ﬁve percent, we ﬁnd that most of the parameter estimates are not biased,
although some biases are detected in the estimates of the monetary policy parameters and thesteady-state real interest rate. We demonstrate that these biases in parameter estimates do
not amount to substantial biases in the estimates of the impulse response functions.4 However,if the true parameter values in the DGP are altered such that the probability of binding at
the constraint increases, we show that the biases in the parameter estimates become larger
3

These studies apply the method in Laséen and Svensson (2011), who propose a convenient algorithm toconstruct policy projections conditional on alternative anticipated policy rate paths in linearized DSGE models.
4

Our robustness analysis shows that these results hold even if we use a larger-scale DSGE model à la Smets

and Wouters (2003, 2007).

3and that the estimated impulse response functions can then substantially diﬀer from the true
functions. This ﬁnding suggests that researchers in a very low interest rate environment shouldmake use of an estimation procedure that explicitly takes into account the zero lower bound.
The remainder of the paper proceeds as follows. Section 2 describes the stylized DSGE
model used for our analysis and the solution algorithm that incorporates the occasionallybinding constraint on the nominal interest rate into the linear rational expectations system.
Section 3 explains the experimental design used for the analysis and presents our results.
Section 4 conducts a robustness analysis with a larger-scale DSGE model. Section 5 is theconclusion.

2

The Model and Solution Method

This section describes the DGP used in our analysis. The DGP consists of a small-scale DSGE
model with sticky prices and a monetary policy rule. In order to incorporate the zero lower
bound constraint on the nominal interest rate, the model is solved using the method employedin Erceg and Lindé (2012) and Bodenstein, Guerrieri, and Gust (2013).

2.1

The Model

In the model economy, there are households, perfectly competitive ﬁnal-good ﬁrms, monopolistically competitive intermediate-good ﬁrms that face price stickiness, and a monetary authority.
For empirical validity, the model features habit persistence in consumption preferences, priceindexation to recent past and steady-state inﬂation, and monetary policy smoothing. A similar
model is used in Dennis (2004, 2009) and Milani and Treadwell (2012), among others.
The log-linearized equilibrium conditions are summarized as follows:5γ
1−γ
1Et Ỹt+1 +Ỹt−1 −
(r̃n − Et π̃t+1 ) + ztd ,1+γ1+γ
σ (1 + γ) t
β
ι
π̃t =
Et π̃t+1 +π̃t−1
1 + βι
1 + βι
[()](1 − ξ) (1 − ξβ)
σ
σγ
+
χ+
Ỹt −
Ỹt−1 + ztp ,ξ (1 + βι)
1−γ
1−γ
(
)
n
r̃tn = max[ϕr r̃t−1+ (1 − ϕr ) ϕπ π̃t + ϕy Ỹt + ztr , − (r̄ + π̄)]
Ỹt =

5

See the Appendix for the full description of the model.

4

(1)

(2)
(3)

Eq. (1) is the spending Euler equation, where Ỹt , r̃tn and π̃t are output, the nominal interest
rate, and inﬂation in terms of the percentage deviations from their steady-state values, ztd isa shock to households’ preferences, interpreted as a demand shock, γ ∈ [0, 1] is the degree of
habit persistence, and σ > 0 is the inverse of the intertemporal elasticity of substitution.
Eq. (2) is the New Keynesian Phillips curve, where β ∈ (0, 1) is the subjective discountfactor determined by the steady-state relationship β = π/r, where π and r are the steady-stategross inﬂation and real interest rate respectively, ι ∈ [0, 1] is the weight of price indexation
to recent past inﬂation πt−1 relative to steady-state inﬂation π, ξ ∈ (0, 1) is the so called
Calvo parameter, which measures the degree of price stickiness, and χ > 0 is the inverse of theelasticity of labor supply. ztp is a cost-push shock.
Eq. (3) is a Taylor (1993) type monetary policy rule with the zero lower bound on thenominal interest rate, where ϕr ∈ [0, 1) is the degree of interest rate smoothing, ϕπ > 1 andϕy > 0 represent the degrees of interest rate policy responses to inﬂation and output gap
respectively, ztr is a monetary policy shock, and π̄ = 100(π − 1) and r̄ = 100(r − 1) are thesteady-state values for the real interest rate and inﬂation, respectively. As r̃tn is expressed as
the deviation from the steady-state nominal interest rate r̄ + π̄, the max function constrains
the level of the nominal interest rate to be greater than or equal to zero.
Each shock ztx , x ∈ {d, p, r} is governed by a stationary ﬁrst-order autoregressive process:x
ztx = ρx zt−1
+ εxt ,

(4)

where ρx ∈ [0, 1) is an autoregressive coeﬃcient and εxt is a normally distributed innovation
with mean zero and standard deviation σxn .
In the DGP, the percentage deviation of output from the steady state 100 log(Yt /Y ), theinﬂation rate 100 log πt , and the nominal interest rate 100 log rtn are assumed to be observable.
Then, these observables are related to the model-implied variables by the following observation
equations:









100 log(Yt /Y )100 log πt



 
 
=
 

100 log rtn

5


0




Ỹt

 

 

+
π̄   π̃t  .
 r̄ + π̄
r̃t

(5)

2.2

Parameter Setting

The model is parameterized according to standard choices in the literature in order to makeour test economy as representative as possible. Table 1 summarizes our baseline parametersetting.
The inverse of intertemporal elasticity of substitution σ is set to be unity, so that thehouseholds’ preferences are characterized by the log-utility function. We assign the inverse of
the elasticity of labor supply χ = 1. The habit persistence and price indexation parameters
are both set at 0.5. As is often the case in the New Keynesian literature, we set the Calvo
parameter ξ = 0.75, which implies that the average duration of prices is four quarters. The
monetary policy parameters (ϕπ = 1.5, ϕy = 0.125) that represent the degrees of interest rate
responses to inﬂation and output gap follow from the coeﬃcients in the original Taylor (1993)
rule, adapted to a quarterly frequency. The policy-smoothing parameter ϕr is set to be 0.5. The
steady-state inﬂation rate π̄ is set at 0.5, implying that the central bank’s target inﬂation rateis two percent annually. The steady-state value of the real interest rate (r̄ = 0.5) is almost thesame as the average of the ex-post real interest rate calculated from the three-month Treasury
bill rate and changes in the GDP implicit deﬂator in the post-1980s US sample.
For each shock, moderate persistency is assumed: ρd = 0.5, ρp = 0.5, ρr = 0.5. The
standard deviation of each shock (σd = 0.25, σp = 0.25, σr = 0.25) is calibrated in line withthe prior mean often used in the literature that estimate similar DSGE models using Bayesian
methods.

2.3

Solution Method

The solution algorithm follows from Erceg and Lindé (2012) and Bodenstein, Guerrieri, andGust (2012). They apply the method in Laséen and Svensson (2011), who propose a convenientalgorithm to construct policy projections conditional on alternative anticipated policy ratepaths in linearized DSGE models. This solution method is compatible with the standard
solution algorithms for linear rational expectations models, such as Blanchard and Kahn (1980)
and Sims (2002), and hence yields a straightforward interpretation about the implications of
the zero bound constraint.

6

To be speciﬁc, Eq. (3) that includes the max operator is replaced with:()
nnr̃t = ϕr r̃t−1 + (1 − ϕr ) ϕπ π̃t + ϕy Ỹt + ztr + mt ,where mt is the contractionary monetary policy shock that enforces the zero lower bound()
n +(1 − ϕ ) ϕ π̃ + ϕ Ỹ +z r
constraint. If the unconstrained nominal interest rate r̄+π̄+ϕr r̃t−1
r
π t
y t
t
(in terms of percentage level) falls below zero, a positive mt is endogenously determined so that()
n + (1 − ϕ ) ϕ π̃ + ϕ Ỹrr̄ + π̄ + ϕr r̃t−1
rπ t
y t + zt + mt = 0. We call this mt a binding shock.
mt should be treated not only as an unanticipated shock but also as an anticipated shock
given agents facing large negative shocks in the economy expect that the nominal interest rateshould be at the zero lower bound for some periods in the future. To take such expectation
channel into account, mt is extended as follows:
mt = m1t−1 + νt0
m1t = m2t−1 + νt1
m2t = m3t−1 + νt2
..
.
K−1
mK−1
= mK
t−1 + νttKmKt = νt ,where K is the maximum number of future periods in which the unconstrained monetary policy
rule implies the negative nominal interest rate. Based on this speciﬁcation of the bindingnshock, each νtk , k = 0, 1, . . . , K has an eﬀect on Et r̃t+ksince Et mt+k = νtk . Therefore, ifnnr̄ + π̄ + Et r̃t+k< 0 without the binding shocks, r̄ + π̄ + Et r̃t+k= 0 can be enforced by adjustingνtk . As these binding shocks are chosen depending on the state of the economy, the expectednduration of zero lower bound periods is endogenously determined. In practice, νtk aﬀects Et r̃t+k′

for k ′ ̸= k because of the dynamic structure of the model, but we can exactly ﬁnd a set of νtk
for k = 0, 1, . . . , K that ensures the zero lower bound since there are as many binding shocks
as there are periods for the zero nominal interest rate.

3

Econometric Experiments

In this section, we conduct econometric experiments to examine how the parameter estimates
can be biased if the existence of the interest-rate lower bound is excluded in the estimation7

process. Moreover, we investigate the eﬀects of these parameter biases on the model properties
by comparing the estimated impulse responses with those based on the true parameter values.

3.1

DesignOur experiments proceed as follows. First, we generate an artiﬁcial time series of output(deviation from the steady state), inﬂation, and the nominal interest rate from the DGP as
described in the preceding section. Thus, the simulated series can be regarded as those thatreﬂect the existence of the zero lower bound constraint. The simulated sample size is 200
observations, which corresponds to quarterly observations over a period of 50 years. This
sample size is chosen because it is comparable to that with which many researchers estimate
DSGE models in practice. According to a simulated sample using the baseline parametersetting presented in Table 1, our model economy is at the zero lower bound for 5.5 percent
of quarters (11 periods of 200). This probability of hitting the interest-rate lower bound isalmost identical to that simulated in Fernández-Villaverde, Gordon, Guerrón-Quintana, andRubio-Ramı́rez (2012).
Next, using the simulated data, we estimate the DSGE model that consists of Eq. (1)–(4)together with the observation equations (5) without imposing the zero lower bound on thenominal interest rate. In the estimation, we employ Bayesian methods. The prior distributionsof the model parameters are given in Table 2. As each prior mean is set to the correspondingtrue parameter value used in generating the data, we can evaluate the parameter bias frommissing the lower bound constraint by examining how the resulting posterior mean and Bayesiancredible interval diﬀer from the true value. To obtain the posterior distributions, we generate
500,000 draws using the random-walk Metropolis–Hastings algorithm and discard the ﬁrst halfof these draws.6

3.2

Results for Baseline Experiment

The second to fourth columns in Table 3 compare the true parameter values with the posterior means and the 90 percent credible intervals for the estimated parameters in the baseline
6

The scale factor for the jumping distribution in the Metropolis–Hastings algorithm is adjusted so that theacceptance rate of candidate draws is approximately 25 percent. We use the Brooks and Gelman (1998) measure
to conﬁrm the convergence of the posterior distributions.

8

experiment. We can see that most of the estimated parameters are not biased as the posteriormean estimates are very close to the true parameter values and the 90 percent credible intervals
contain the true values. This may be good news for many researchers in that the structuralparameters relating to preferences and nominal rigidities, the so-called deep parameters, are
not much aﬀected by ignoring the interest-rate lower bound. However, we do detect some biases
in the estimates of parameters that characterize the monetary policy rule, ϕπ and ϕr , and thesteady-state real interest rate r̄ in the sense that the credible intervals for these parameters do
not include the true values.
An intuitive explanation for these biased estimates is the following. In the DGP, themonetary policy reaction function has a kink where the interest-rate lower bound becomesbinding, that is, the reaction function has positive slopes with respect to inﬂation, output, andthe lagged nominal interest rate if the unconstrained nominal interest rate is positive, but theslopes become ﬂat if it is negative. However, when such a kink is omitted in the estimation
process, as is the case in our experiment, the estimated slopes are approximated to lie between
the positive and ﬂat slopes in the DGP, and thus the parameters in the monetary policy rulecan be underestimated. In our experiment, the biases emerge as a change in the estimate of
ϕπ and ϕr , although the estimate of ϕy could also be potentially aﬀected. The reason for the
upward bias in the estimate of r̄ is also straightforward. The presence of the interest-rate lower
bound forces the nominal interest rate to be equal or greater than zero. However, if the model
estimation fails to consider the zero lower bound, the nominal interest rate can be negative,
and the mean of the model-implied nominal interest rate declines. Then, the estimate of thesteady-state nominal interest rate must rise to adjust the diﬀerence between the mean of themodel-implied nominal interest rate and that of the corresponding series simulated from the
DGP with the lower bound, given the model is log-linearized around the steady state. In thisexperiment, such an adjustment has mostly emerged as a change in the estimate of r̄,7 rather
than π̄. These causes of the parameter biases could have some inﬂuence on the estimates ofthe other structural parameters that characterize the Euler equation and the Phillips curve as
we apply the system-based estimation approach. However, our experiment has revealed thatsuch inﬂuence is quite limited.
7

A change in r̄ aﬀects the subjective discount factor β through the steady-state relationship β = π/r and thedeﬁnition r = r̄/100 + 1, but the resulting change in β is quite marginal in its magnitude.

9Our baseline experiment is based on the posterior estimates for a sample of 200 observations.
We cannot deny the possibility that the parameter biases described above may arise becauseof the small sample, and that sampling variability could lead to diﬀerent biases, regardless ofthe exclusion of the interest-rate lower bound. To mitigate the eﬀect of small sample bias,
we conduct an experiment using a large sample of 5,000 observations generated from the same
DGP. The estimation results are presented in the last two columns in Table 3. As in the baseline
experiment, signiﬁcant biases are found in the estimates of the monetary policy parameters, ϕπ
and ϕr , and the steady-state real interest rate r̄, and the credible intervals for these parameters
do not contain the true values. The other structural parameters are not much biased in thatthe posterior mean estimates are quite similar to the true parameter values.8 Therefore, thebaseline results are not altered in this experiment.
An important issue is whether the parameter biases detected in the present experimentcan lead to a sizeable diﬀerence in the implications of the model. To investigate this issue, we
compare the impulse response functions estimated without considering the interest-rate lower
bound with those computed using the true parameter values. Figure 1 depicts the impulse
responses of output, inﬂation, and the nominal interest rate to one standard deviation shocks
in demand, cost-push, and monetary policy. The responses are expressed in terms of thepercentage deviation from the steady state in order to focus on the changes in the transmissionof shocks.9 In each panel, the solid thick line represents the true response, and the solid thinline and dashed lines are respectively the posterior mean and 90 percent credible interval forthe estimated response. Throughout the ﬁgure, although small diﬀerences are found between
the mean estimates and the true response, all the credible intervals include the true responses.
Therefore, we can conclude that the parameter biases in the baseline experiment do not amountto substantial biases in the impulse response functions.
8

The credible intervals for some of the parameters do not include their true values, but the mean estimatesthemselves are not much diﬀerent from the true values. Thus, we do not assess that these parameter estimates
are signiﬁcantly biased in this experiment.
9

The estimated responses of the nominal interest rates in level terms are obviously diﬀerent from the true

responses as the estimate of the steady-state real interest rate is biased.

10

3.3

Results for Alternative Experiments

According to the simulated time series in our baseline experiment, the nominal interest rateis bounded at zero for 5.5 percent of quarters. As the probability of hitting the zero lower
bound increases in the DGP, it could be possible that parameter biases from excluding thebound in the estimation become large. To analyze such a possibility, we change the parametersassigned to the DGP so that the model economy is more frequently constrained by the interestrate lower bound. We then conduct the same estimation exercises as the baseline experimentusing the data simulated by the DGP with the alternative parameter settings. Speciﬁcally,
we consider two cases. One is where the true steady-state real interest rate falls by 0.25 (onepercent annually) so that r̄ = 0.25. The other is where the standard deviation of the demand
shock increases to twice its size so that σd = 0.5. The latter case increases the number of zerointerest rate periods because large negative shocks are more likely to depress the economy andto lower the nominal interest rate. In both cases, the probability of hitting the lower bounddoubles, that is, they are now 11 percent of the simulated sample.
The middle two columns in Table 4 present the posterior means and the 90 percent credibleintervals for the parameters in the experiment where the true parameter value for the realinterest rate decreases to r̄ = 0.25. As is the case with the baseline experiment, the estimatesof the monetary policy parameter ϕπ and the steady-state real interest rate r̄ exhibit substantial
biases. In contrast, the posterior mean estimate of the policy smoothing parameter ϕr is almost
the same as the true value. Instead, signiﬁcant biases are found in the autoregressive coeﬃcientρr and the standard deviation σr of the monetary policy shock. However, the other estimates
are not very biased in the sense that the posterior credible intervals include their correspondingtrue values. Figure 2 graphs the estimated and true impulse responses in order to examine theeﬀects of these parameter biases on the model properties. Compared with the responses in thebaseline experiment, the diﬀerences between the mean estimates and the true responses becomelarge in most of the panels. In particular, the credible intervals for the output responses to the
cost-push and monetary policy shocks and the inﬂation response to the monetary policy shock
do not contain the true responses in the ﬁrst ﬁve periods.
The last two columns in Table 4 provide the posterior means and 90 percent credibleintervals for the parameters in the experiment where the true standard deviation of the demand
shock increases to σd = 0.5. The estimates of the monetary policy parameters, ϕπ and ϕr , and11

the steady-state real interest rate r̄ exhibit larger biases than those in the baseline experiment.
In addition, signiﬁcant biases are found in ϕy , ρr and σr , which are all related to monetary
policy, and the credible intervals for these parameters do not include the true values. As in
the previous experiment, the estimates of the other parameter estimates are not very biased.
According to the impulse responses presented in Figure 3, the true responses are outside thecredible intervals in the ﬁrst few periods regarding the response of the nominal interest rate to
the demand shock and the responses of output and inﬂation to the cost-push shock.
On the basis of the two experiments above, the parameters estimated without considering
the zero lower bound constraint may not be suﬃciently accurate to replicate the true economic
dynamics.

4

Robustness Analysis

While we have employed a small-scale DSGE model in our analysis, an increased number ofpolicy institutions are actively developing and estimating larger-scale DSGE models à la Smets
and Wouters (2003, 2007). Thus, as a robustness analysis, we conduct a similar exercise as in
the preceding section using a version of the Smets and Wouters (2007) model.

4.1

Smets–Wouters Type Model

The model employed in this section is a slightly simpliﬁed version of the Smets and Wouters(2007) model. This version diﬀers from their original model in the following three aspects. First,
for the purpose of comparison with our baseline experiment, the monetary policy rule is replacedwith the one speciﬁed in our baseline model whereas Smets and Wouters employ a generalized
Taylor rule in which the policy rate is adjusted in response to the level and changes in thetheoretical output gap (i.e., the gap between real output and output that would be obtainedin the absence of nominal rigidities) in addition to inﬂation. Second, our structural shocks
are all governed by stationary ﬁrst-order autoregressive processes and their disturbances aremutually uncorrelated with each other. In the Smets–Wouters model, the exogenous spending
disturbance is aﬀected by a contemporaneous innovation to total factor productivity (TFP)
and the price and wage markup disturbances follow ARMA(1,1) processes. Third, as in thebaseline experiment, the binding shocks mt are incorporated into the monetary policy rule in12

order to take account of the zero lower bound constraint on the nominal interest rate in the
DGP.
The model consists of the following log-linearized equations.10 In what follows, all thevariables are expressed in terms of their percentage deviation from the steady-state balanced
growth path.
)λ/γ1(σc − 1)wh l/c (˜c̃t−1 +
Et c̃t+1 +lt − Et ˜lt+1
1 + λ/γ
1 + λ/γσc (1 + λ/γ)()
1 − λ/γr̃t − Et π̃t+1 + εbt ,−σc (1 + λ/γ)
βγ 1−σc
1
1
ı̃
+
E
ı̃
+
q̃t + εit ,
ı̃t =
t−1
t
t+1
1−σ
1−σ
2
1 + βγ c
1 + βγ c
γ φ(1 + βγ 1−σc )
(
)
1−δ
rk
k
q̃t = kEt q̃t+1 + k
Et r̃t+1− r̃t − Et π̃t+1 + εbt ,r +1−δ
r +1−δ
[]ỹt = ϕp αk̃ s + (1 − α)˜lt + εa ,
c̃t =

t

t

k̃ts = z̃t + k̃t−1 ,

(6)
(7)
(8)
(9)
(10)

1−ψ k
r̃ ,
ψ t
(
)
]
1−δ
1−δ [
k̃t =
k̃t−1 + 1 −ı̃t + γ 2 φ(1 + βγ 1−σc )εit ,
γ
γ
z̃t =

ỹt = cy c̃t + iy ı̃t + rk ky z̃t + εgt ,()µ̃pt = α k̃ts − ˜lt − w̃t ,

(11)
(12)
(13)
(14)

ιp
βγ 1−σcπ̃
+
Et π̃t+1
t−1
1 + βγ 1−σc ιp
1 + βγ 1−σc ιp(1 − ξp )(1 − βγ 1−σc ξp )−µ̃p + εpt
ξp (1 + βγ 1−σc ιp )[(ϕp − 1)εp + 1] t()
r̃tk = − k̃ts − ˜lt + w̃t ,
[
]}
{
1
λ
w
˜
µ̃t = w̃t − σl lt +c̃t − c̃t−1 ,1 − λ/γ
γ
1−σ
c
1
βγ
1 + βγ 1−σc ιw
w̃t =
w̃
+
(E
w̃
+
E
π̃
)
−
π̃t
t−1
t
t+1
t
t+1
1 + βγ 1−σc
1 + βγ 1−σc
1 + βγ 1−σcιw
(1 − ξw )(1 − βγ 1−σc ξw )+π̃−µ̃w + εw
t−1
t ,
1−σ
1 + βγ cξw (1 + βγ 1−σc )[(ϕw − 1)εw + 1] tπ̃t =r̃t = rr r̃t−1 + (1 − rr ) (rπ π̃t + ry ỹt ) + εrt + mt

(15)
(16)
(17)

(18)
(19)

Eq. (6) is the consumption Euler equation, where c̃t denotes consumption, ˜lt is labor input,
10

See the appendix to Smets and Wouters (2007) for a full description of the model.

13r̃t is the short-term nominal interest rate, π̃t is the inﬂation rate, εbt captures the risk premium
shock in the return on assets held by households, λ is the degree of external habit persistence
in consumption preferences, γ is the steady-state growth rate, σc is the degree of relative risk
aversion, and wh l/c is the steady-state value of labor relative to consumption. Eq. (7) is theinvestment adjustment equation, where ı̃t represents investment, q̃t denotes the real value ofthe existing capital stock, εit represents the shock to investment eﬃciency, β is the subjective
discount factor, and φ is the steady-state elasticity of investment adjustment costs. Eq. (8) is
the no-arbitrage condition for the value of capital, where r̃tk is the real rental rate of capital,
δ is the depreciation rate of capital, and rk is the steady-state real rental rate of capital.
Eq. (9) is the Cobb–Douglas production function with ﬁxed costs where ỹt denotes output,k̃ts is eﬀective capital services, εat represents the TFP shock, ϕp is one plus the share of ﬁxed
costs in output, α is the share of capital in production. Eq. (10) gives the eﬀective capital
services used in production, where z̃t and k̃t−1 denote the capital utilization rate and capitalinstalled in the previous period. Eq. (11) is the condition for the capital utilization rate,
where ψ is determined by a function of the steady-state elasticity of the rate adjustment costs.
Eq. (12) is the capital accumulation equation. Eq. (13) is the aggregate resource constraint,
where εgt represents the exogenous spending shock, cy , iy , ky are the steady-state output ratiosof consumption, investment, and capital. Eq. (14) is the equation for the price markup µ̃pt ,where w̃t is the real wage. Eq. (15) is the New Keynesian Phillips curve, where εpt represents
the price markup shock, ξp and ιp are the degrees of price stickiness and price indexation topast inﬂation, (ϕp − 1) is the steady-state goods market markup, and εp is the curvature of
the Kimball (1995) goods market aggregator. Eq. (16) is the condition for capital and labor
inputs in production. Eq. (17) is the equation for the wage markup µ̃w
t , where σl is the inverse
elasticity of labor supply. Eq. (18) is the wage equation, where εwt represents the wage markup
shock, ξw and ιw are the degrees of wage stickiness and wage indexation to past inﬂation,(ϕw − 1) is the steady-state labor market markup, and εw is the curvature of the Kimball(1995) labor market aggregator. Eq. (19) is the monetary policy rule, where rr is the degreeof interest rate smoothing, rπ and ry represent the degrees of interest rate policy responses toinﬂation and output gap, respectively, εrt is the monetary policy shock, and mt is the binding
shock.
There are seven structural shocks in the model. Each of them εxt , x ∈ {a, b, g, i, r, p, w}

14

follows
εxt = ρx εxt−1 + ηtx ,ηtx ∼ i.i.d. N (0, σx2 )where ρx is an autoregressive coeﬃcient and ηtx is a normally distributed innovation with mean
zero and standard deviation σx .
The observation equations are100∆ log Yt




γ̄



ỹt − ỹt−1 + εat



 

 

 

 

 100∆ log Ct   γ̄   c̃t − c̃t−1 + εat 
 

 

 

 

 100∆ log It   γ̄   ı̃t − ı̃t−1 + εat 
 

 


 
 

 100∆ log Wt  =  γ̄  +  w̃t − w̃t−1 + εat  ,

 
 

 

 

˜lt
 100 log lt   ¯l  


 
 

 

 

 100∆ log Pt   π̄  

π̃
t
 

 

100 log rtr̄r̃t
where γ̄ = 100(γ−1), ¯l is the steady-state hours worked, π̄ = 100(π−1), and r̄ = 100(β −1 γ σc π−
1).

4.2

Parameter Setting and Prior Distribution

The true parameter values that characterize the DGP basically follow from the prior means
used in Smets and Wouters (2007), but some are changed so that the resulting probability ofhitting the interest-rate lower bound is comparable (12 periods of 200) to that in our baseline
experiment. To be speciﬁc, the steady-state inﬂation rate π̄ is changed from 0.625 to 0.5, andthe standard deviations of the structural shocks σx , x ∈ {a, b, g, i, r, p, w} are all changed from
0.1 to 0.3.
Then, the model without the zero lower bound constraint is estimated using the artiﬁcial
time series generated from the DGP presented above. The second to fourth columns in Table
5 summarize the prior distributions of parameters used for the estimation in the present experiment. As in the preceding experiments, each prior mean is set to the corresponding true
parameter value. For the standard deviations of the prior distributions, we use the same values
as set in Smets and Wouters (2007).

15

4.3

Results for the Robustness Experiment

The last two columns in Table 5 show the posterior means and 90 percent credible intervals forthe estimated parameters in the Smets–Wouters type model. As in the baseline experiment,
most of the posterior mean estimates are almost the same as the true parameter values. However, substantial biases are detected in the estimates of the elasticity of investment adjustment
costs φ and the monetary policy response to inﬂation rπ , that is, the 90 percent credible intervals for these two estimates do not contain their corresponding true values. The intuition
about the bias in φ is as follows. Without considering the interest-rate lower bound in theestimation, the model-implied interest rate series is biased. As the transmission eﬀect of thechange in the interest rate on investment is quite strong, the parameter φ that characterizes
the relationship between investment and the interest rate has to be biased in order to ﬁt to theinvestment series generated from the DGP. Regarding the biases in rπ , the same explanation
as provided in the previous section applies.
Figure 4 compares the estimated and true impulse responses of output, inﬂation, and thenominal interest rate to one standard deviation shocks regarding the TFP, the risk premium,external demand, investment, monetary policy, the price markup, and the wage markup. As
with the preceding experiments, in each panel, the solid thick line represents the true response,
the solid thin and dashed lines are respectively the posterior mean and 90 percent credibleinterval for the estimated response, and the responses are shown in terms of the percentage
deviation from the steady state. A remarkable ﬁnding here is that as shown in almost all of thepanels, the true responses lie in the estimated credible intervals, although some of the mean
estimates diﬀer from the true responses. Therefore, we reach the same conclusion as in thebaseline experiment that the parameter biases from ignoring the zero lower bound constraint
only have quite marginal eﬀects on the properties of the model.

5Conclusion

This paper has investigated the parameter biases in an estimated DSGE model that excludes theexistence of the zero lower bound constraint on the nominal interest rate. To this end, we haveconducted econometric experiments using a standard sticky-price DSGE model. According
to the results in our baseline experiment, some biases are detected in the estimates of the16monetary policy parameters and the steady-state real interest rate, but the estimates of theother structural parameters are not biased. However, we have demonstrated that the parameter
biases become large with the increased probability of hitting the zero lower bound. The lastﬁnding cautions researchers against the common practice of estimating linearized DSGE models
without considering the lower bound.
However, as Braun, Körber, and Waki (2012) and Gust, López-Salido, and Smith (2012)suggest, the solution algorithm in which nonlinearity is considered only in monetary policy rules
but the remaining equilibrium conditions are linearized may lead to an inaccurate assessment of
the lower bound constraint. In this regard, it is an important research agenda to examine how
parameter estimates in linearized DSGE models can be aﬀected by ignoring the true economic
structures characterized by fully nonlinear models.

17

Appendix
This appendix presents the full description of the model. In the model economy, there are acontinuum of households, a representative ﬁnal-good ﬁrm, a continuum of intermediate-good
ﬁrms, and a monetary authority.
Each household h ∈ [0, 1] consumes ﬁnal goods Ch,t , supplies labor lh,t , and purchases
one-period riskless bonds Bh,t so as to maximize the utility function
[
]
1+χ
∞
1−σ
∑
l
(C
−
γC
)
t−1
h,t
h,t
βt
E0
−
1−σ
1+χ
t=0

subject to the budget constraintn
Pt Ch,t + Bh,t = Pt Wt lh,t + rt−1
Bh,t−1 + Th,t ,where Et is the expectation operator conditional on information available in period t, β ∈(0, 1) is the subjective discount factor, σ > 0 is the inverse of the intertemporal elasticity of
substitution, γ ∈ [0, 1] is the degree of external habit persistence in consumption preferences,
χ > 0 is the inverse of the elasticity of labor supply, Pt is the price of ﬁnal goods, Wt is thereal wage, rtn is the gross nominal interest rate, and Th,t is the sum of a lump-sum public
transfer and proﬁts received from ﬁrms. The ﬁrst-order conditions for optimal decisions on
consumption, labor supply, and bond-holding are identical among households and therefore
becomeΛt = (Ct − γCt−1 )−σ ,
Wt =

ltχ
Λt

,

1 = Et β

(A.1)
(A.2)

Λt+1 rtn,
Λt πt+1(A.3)

where Λt is the marginal utility of consumption and πt = Pt /Pt−1 denotes gross inﬂation.
The representative ﬁnal-good ﬁrm produces output Yt under perfect competition by choos∫1
ing a combination of intermediate inputs {Yf,t } so as to maximize proﬁt Pt Yt − 0 Pf,t Yf,t df(∫)1+λp
1 1/(1+λp )subject to a CES production technology Yt = 0 Yf,tdf
, where Pf,t is the price ofintermediate good f and λp ≥ 0 denotes the intermediate-good price markup. The ﬁrst-ordercondition for proﬁt maximization yields the ﬁnal-good ﬁrm’s demand for intermediate goodf , Yf,t = Yt (Pf,t /Pt )−(1+λ )/λ , while perfect competition in the ﬁnal-good market leads to(∫)−λp
1 −1/λp
Pt = 0 Pf,t
df
.
p

p

18

Each intermediate-good ﬁrm f produces one kind of diﬀerentiated goods Yf,t under monopolistic competition by choosing a cost-minimizing labor input lt given the real wage Wt subject
to the production function
Yf,t = At lf,t ,where At represents the exogenous technology level. The ﬁrst-order condition for cost minimization shows that real marginal cost is identical among intermediate-good ﬁrms and is given
by
mct =

Wt
.
At

(A.4)

In the face of the ﬁnal-good ﬁrm’s demand and marginal cost, the intermediate-good ﬁrms
set prices of their products on a staggered basis as in Calvo (1983). In each period, a fraction
1 − ξ ∈ (0, 1) of intermediate-good ﬁrms re-optimize their prices while the remaining fraction
ξ indexes prices to a weighted average of past inﬂation πt−1 and steady-state inﬂation π. The
ﬁrms that re-optimize their prices in the current period then maximize expected proﬁt
]
[
j
∞
jΛ
∑
∏
(
)
P
β
t+j
f,t
ι
Et
ξj
πt+k−1π 1−ι − mct+j Yf,t+j
Λt
Pt+j
j=0

k=1

subject to the ﬁnal-good ﬁrm’s demand[
Yf,t+j = Yt+jj)
Pf,t ∏ ( ι
πt+k−1 π 1−ι
Pt+jp]− 1+λλp,k=1where ι ∈ (0, 1) denotes the weight of price indexation to past inﬂation relative to steady-state
inﬂation. The ﬁrst-order condition for the reoptimized price Pto is given byp []− 1+λ


λp
j (
)
o


∏
ι


Pt
πt+k−1πj Λt+j




(βξ)
Y
∞
t+j


∑
Λt
Pt
π
πt+k
k=1
Et
[
]  = 0.
j (

)
o
∏


ι
j=0 
Pt
π
πt+k−1− (1 + λp ) mct+j
 × P

π
π
t

t+k

k=1

Moreover, the ﬁnal-good’s price Pt =(∫1(
1 = (1 − ξ)

(A.5)

−1/λp

0 Pf,t

Pto
Pt

)−λp

df

)− 1p

[(

λ

+ξ

can be rewritten as] 1
πt−1 )ι π − λp
.
π
πt

(A.6)

The ﬁnal-good market clearing condition is
Yt = Ct ,
19

(A.7)

while the labor market clearing condition leads to
Yt dt
=
At
where dt =∫ 1
lf,t df = lt ,

(A.8)

0

∫1

−(1+λp )/λp
df represents price dispersion across the intermediate-good
0 (Pf,t /Pt )ﬁrms. Note that this dispersion is of second-order under the staggered price setting and thatits steady-state value is unity.
A monetary policy rule is speciﬁed as
[
(
)
]
πt
Yt
n
n
n
r
log rt = max ϕr log rt−1 + (1 − ϕr ) log r + ϕπ log+ zt , 0 .
+ ϕy log
π
Y

(A.9)

In the absence of the zero lower bound on the nominal interest rate, the monetary authority
adjusts the interest rate following a Taylor (1993) type monetary policy rule where ϕr ∈ [0, 1)
is the degree of interest rate smoothing, rn is the steady-state gross nominal interest rate,and ϕπ , ϕy ≥ 0 are the degrees of interest rate policy responses to inﬂation and output. ztr is amonetary policy shock, which captures unsystematic components of monetary policy.
The equilibrium conditions are (A.1)–(A.9). Log-linearizing the equilibrium conditions
around the steady state and rearranging the resulting equations yields Eq. (1)–(3). The demand
shock ztd and cost shock ztp have been introduced in a reduced form manner, and interpreted
as a shock to consumption preferences and time-varying price markup of intermediate-goods
respectively.

20

References[1] Adam, Klaus, and Roberto M. Billi, 2006. “Optimal Monetary Policy under Commitment
with a Zero Bound on Nominal Interest Rates.” Journal of Money, Credit and Banking,38(7), 1877–1905.
[2] Adam, Klaus, and Roberto M. Billi, 2007. “Discretionary monetary policy and the zero
lower bound on nominal interest rates.” Journal of Monetary Economics, 54(3), 728–752.
[3] Aruoba, S. Borağan, and Frank Schorfheide, 2013. “Macroeconomic Dynamics Near the
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[6] Braun, R. Anton, Lena M. Körber, and Yuichiro Waki, 2012. “Some Unpleasant Properties
of Log-Linearized Solutions When the Nominal Rate is Zero.” Working Paper 2012-05a,
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[7] Brooks, Stephen P., and Andrew Gelman, 1998. “General Methods for Monitoring Convergence of Iterative Simulations.” Journal of Computational and Graphical Statistics, 7(4),434–455.
[8] Calvo, Guillermo A., 1983. “Staggered Prices in a Utility-Maximizing Framework.” Journalof Monetary Economics, 12(3), 383–398.
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[10] Dennis, Richard, 2004. “Specifying and estimating New Keynesian Models with Instrument
Rules and Optimal Monetary Policies.” Working Paper 2004-17, Federal Reserve Bank of
San Francisco.
21[11] Dennis, Richard, 2009. “Consumption Habits in a New Keynesian Business Cycle Model.”Journal of Money, Credit and Banking, 41(5), 1015–1030.
[12] Eggertsson, Gauti B., and Michael Woodford, 2003. “The Zero Bound on Interest Rates
and Optimal Monetary Policy.” Brookings Papers on Economic Activity 34(1), 139–235.
[13] Erceg, Christopher J., and Jesper Lindé, 2012. “Is There a Fiscal Free Lunch in a Liquidity
Trap.” Journal of European Economic Association, forthcoming.
[14] Fernández-Villaverde, Jesús, Grey Gordon, Pablo A. Guerrón-Quintana, and Juan RubioRamı́rez, 2012. “Nonlinear Adventures at the Zero Lower Bound.” NBER Working Paper
18058, National Bureau of Economic Research.
[15] Gavin, William T., Benjamin D. Keen, Alexander Richter, and Nathaniel Throckmorton,
2013. “Global Dynamics at the Zero Lower Bound.” Working Papers 2013-007, Federal
Reserve Bank of St. Louis.
[16] Gust, Christopher, David Lopez-Salido, and Matthew E. Smith, 2012. “The Empirical
Implications of the Interest-Rate Lower Bound.” Finance and Economics Discussion Series
2012-83, Board of Governors of the Federal Reserve System.
[17] Jung, Taehun, Yuki Teranishi, and Tsutomu Watanabe, 2005. “Optimal Monetary Policy
at the Zero-Interest-Rate Bound.” Journal of Money Credit and Banking, 37(5) 813–835.
[18] Kimball, Miles S., 1995. “The Quantitative Analytics of the Basic Neomonetarist Model.”Journal of Money Credit and Banking, 27(4) 1241–1277.
[19] Laséen, Stefan, and Lars E.O. Svensson, 2011. “Anticipated Alternative policy Rate Paths
in Policy Simulations.” International Journal of Central Banking, 7(3), 1–35.
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22

[23] Smets, Frank, and Rafael Wouters, 2003. “An Estimated Dynamic Stochastic General
Equilibrium Model of the Euro Area.” Journal of the European Economic Association,
1(5), 1123–1175.
[24] Smets, Frank, and Rafael Wouters, 2007. “Shocks and Frictions in US Business Cycles: A
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23

Table 1: Parameter setting
Parameter
σ
χ
γ
ι
ξ
ϕπ
ϕy
ϕr
π̄
r̄
ρd
ρp
ρr
σd
σp
σr

Economic interpretation

Assigned value

Inverse of intertemporal elasticity of substitution
Inverse of elasticity of labor supply
Habit persistence
Price indexation
Price stickiness
Policy response to inﬂation
Policy response to output
Interest rate smoothing
Steady-state inﬂation rate
Steady-state real interest rate
Persistence of demand shock
Persistence of cost shock
Persistence of policy shockStandard deviation of demand shockStandard deviation of cost shockStandard deviation of policy shock

1.000
1.000
0.500
0.500
0.750
1.500
0.125
0.500
0.500
0.500
0.500
0.500
0.500
0.250
0.250
0.250

24

Table 2: Prior distributions
Parameter
σ
χ
γ
ξ
ι
ϕπ
ϕy
ϕr
π̄
r̄
ρd
ρp
ρr
σd
σp
σr

Distribution

MeanStandard deviation

Gamma
Gamma
Beta
Beta
Beta
Gamma
Gamma
Beta
Gamma
Gamma
Beta
Beta
Beta
Inverse Gamma
Inverse Gamma
Inverse Gamma

1.000
1.000
0.500
0.750
0.500
1.500
0.125
0.500
0.500
0.500
0.500
0.500
0.500
0.250
0.250
0.250

0.150
0.150
0.150
0.150
0.150
0.150
0.150
0.150
0.150
0.150
0.150
0.150
0.150
2.000
2.000
2.000

25

Table 3: Posterior distributions of parameters in baseline experimentParameter
σ
χ
γ
ξ
ι
ϕπ
ϕy
ϕr
π̄
r̄
ρd
ρp
ρr
σd
σp
σr

True value
1.000
1.000
0.500
0.750
0.500
1.500
0.125
0.500
0.500
0.500
0.500
0.500
0.500
0.250
0.250
0.250

Baseline
Mean 90% interval

Large sample
Mean 90% interval

1.054
1.002
0.439
0.757
0.495
1.293
0.156
0.404
0.474
0.594
0.470
0.498
0.518
0.265
0.240
0.281

1.041
1.005
0.467
0.785
0.479
1.216
0.125
0.455
0.470
0.539
0.516
0.466
0.498
0.237
0.228
0.253

[0.829, 1.278][0.752, 1.240][0.303, 0.582][0.686, 0.826][0.288, 0.693][1.107, 1.472][0.000, 0.273][0.340, 0.466][0.344, 0.595][0.514, 0.673][0.322, 0.623][0.346, 0.647][0.412, 0.623][0.217, 0.310][0.187, 0.295][0.240, 0.319][0.950, 1.132][0.761, 1.250][0.434, 0.501][0.765, 0.806][0.413, 0.541][1.158, 1.272][0.094, 0.156][0.440, 0.469][0.442, 0.498][0.522, 0.556][0.482, 0.550][0.415, 0.515][0.474, 0.522][0.227, 0.247][0.216, 0.240][0.245, 0.261]

Notes: In the baseline, the model is estimated using a sample of 200 observations. In the large sample case, thesame model is estimated using a sample of 5,000 observations.

26

Table 4: Posterior distributions of parameters in alternative experiments

Parameter
σ
χ
γ
ξ
ι
ϕπ
ϕy
ϕr
π̄
r̄
r̄
ρd
ρp
ρr
σd
σd
σp
σr

True value
1.000
1.000
0.500
0.750
0.500
1.500
0.125
0.500
0.500
0.250
0.500
0.500
0.500
0.500
0.500
0.250
0.250
0.250

Case of low r̄
Mean 90% interval

Case of large σdMean 90% interval

1.086
0.997
0.421
0.762
0.432
1.279
0.410
0.505
0.413
0.343
0.527
0.486
0.642
0.266
0.253
0.426

0.953
0.994
0.432
0.764
0.391
1.197
0.055
0.329
0.471
0.722
0.516
0.615
0.642
0.438
0.203
0.302

[0.855, 1.326][0.761, 1.248][0.290, 0.546][0.701, 0.818][0.269, 0.596][1.053, 1.492][0.000, 0.823][0.409, 0.598][0.266, 0.560][0.231, 0.461][0.415, 0.643][0.349, 0.622][0.583, 0.702][0.220, 0.310][0.194, 0.315][0.337, 0.515][0.731, 1.175][0.746, 1.232][0.308, 0.558][0.723, 0.807][0.237, 0.541][1.035, 1.346][0.023, 0.077][0.264, 0.391][0.307, 0.628][0.617, 0.831][0.403, 0.634][0.498, 0.733][0.557, 0.727][0.355, 0.516][0.162, 0.243][0.264, 0.338]

Notes: In the case of low r̄, the true values are set at r̄ = 0.25 and σd = 0.25. In the case of large σd , the true
values are set at r̄ = 0.5 and σd = 0.5.

27

Table 5: Prior and posterior distributions of parameters in the Smets–Wouters type modelParameter

Prior distribution
Distribution Mean S.D.

Posterior distribution
Mean 90% interval

φ
σc
λ
ξw
σl
ξp
ιw
ιp
ψ
ϕp
rπ
ρ
ry
π̄
100(β −1 − 1)
¯l
γ̄
α
ρa
ρb
ρg
ρi
ρr
ρp
ρw
σa
σb
σg
σi
σr
σp
σw

NormalNormalBeta
Beta
Normal
Beta
Beta
Beta
Beta
NormalNormalBeta
NormalGamma
Gamma
NormalNormalNormal
Beta
Beta
Beta
Beta
Beta
Beta
Beta
Inv. Gamma
Inv. Gamma
Inv. Gamma
Inv. Gamma
Inv. Gamma
Inv. Gamma
Inv. Gamma

4.852
1.468
0.728
0.503
1.502
0.500
0.420
0.459
0.482
1.231
1.208
0.721
0.093
0.500
0.191
-0.210
0.399
0.287
0.557
0.499
0.481
0.489
0.482
0.520
0.445
0.314
0.318
0.303
0.278
0.290
0.312
0.308

4.000
1.500
0.700
0.500
2.000
0.500
0.500
0.500
0.500
1.250
1.500
0.750
0.125
0.500
0.250
0.000
0.400
0.300
0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.300
0.300
0.300
0.300
0.300
0.300
0.300

28

1.500
0.375
0.100
0.100
0.750
0.100
0.150
0.150
0.150
0.125
0.250
0.100
0.050
0.100
0.100
2.000
0.100
0.050
0.200
0.200
0.200
0.200
0.200
0.200
0.200
2.000
2.000
2.000
2.000
2.000
2.000
2.000

[4.071, 5.641][1.304, 1.622][0.685, 0.769][0.442, 0.559][0.741, 2.279][0.449, 0.553][0.284, 0.557][0.343, 0.574][0.424, 0.540][1.182, 1.279][1.015, 1.384][0.674, 0.767][0.033, 0.152][0.363, 0.626][0.087, 0.291][-0.673, 0.241][0.397, 0.402][0.259, 0.315][0.468, 0.649][0.416, 0.590][0.374, 0.583][0.384, 0.590][0.379, 0.583][0.406, 0.642][0.339, 0.544][0.286, 0.343][0.266, 0.370][0.276, 0.328][0.237, 0.320][0.262, 0.317][0.266, 0.356][0.263, 0.352]

Figure 1: Impulse responses in baseline experiment(1) Demand shock
OutputInflation

Nominal interest rate

0.6

0.3

0.3

0.4

0.2

0.2

0.2

0.1

0.1

0

0

0

-0.2
0

5

10

15

-0.1
0

5

10

15

-0.1
0

5

10

15

(2) Cost-push shock
OutputInflation

Nominal interest rate

0.2

0.6

0.6

0

0.4

0.4

-0.2

0.2

0.2

-0.4

0

0

-0.6
0

5

10

15

-0.2
0

5

10

15

-0.2
0

5

10

15

(3) Monetary policy shock
OutputInflation

Nominal interest rate

0.4

0.1

0.3

0.2

0

0.2

0

-0.1

0.1

-0.2

-0.2

0

-0.4
0

5

10

15

-0.3
0

5

10

15

-0.1
0

5

10

15

Notes: The ﬁgure shows the impulse responses of output, inﬂation, and the nominal interest rate (in terms of
the percentage deviation from the steady state) to one standard deviation demand, cost-push, and monetary
policy shocks. The solid thick lines represent the true responses, and the solid thin lines and dashed lines arerespectively the posterior means and 90 percent credible intervals for the estimated responses.

29

Figure 2: Impulse responses in alternative case I (r̄ = 0.25)(1) Demand shock
OutputInflation

Nominal interest rate

0.6

0.3

0.3

0.4

0.2

0.2

0.2

0.1

0.1

0

0

0

-0.2
0

5

10

15

-0.1
0

5

10

15

-0.1
0

5

10

15

(2) Cost-push shock
OutputInflation

Nominal interest rate

0.2

0.6

0.6

0

0.4

0.4

-0.2

0.2

0.2

-0.4

0

0

-0.6
0

5

10

15

-0.2
0

5

10

15

-0.2
0

5

10

15

(3) Monetary policy shock
OutputInflation

0.5

Nominal interest rate

0

0.3

-0.2

0.2

-0.4

0.1

-0.6

0

0

-0.5

-1
0

5

10

15

-0.8
0

5

10

15

-0.1
0

5

10

15

Notes: The ﬁgure shows the impulse responses of output, inﬂation, and the nominal interest rate (in terms of
the percentage deviation from the steady state) to one standard deviation demand, cost-push, and monetary
policy shocks. The solid thick lines represent the true responses, and the solid thin lines and dashed lines arerespectively the posterior means and 90 percent credible intervals for the estimated responses.

30

Figure 3: Impulse responses in alternative case II (σd = 0.5)
(1) Demand shock
OutputInflation

Nominal interest rate

1.5

0.6

0.6

1

0.4

0.4

0.5

0.2

0.2

0

0

0

-0.5
0

5

10

15

-0.2
0

5

10

15

-0.2
0

5

10

15

(2) Cost-push shock
OutputInflation

Nominal interest rate

0.2

0.6

0.6

0

0.4

0.4

-0.2

0.2

0.2

-0.4

0

0

-0.6
0

5

10

15

-0.2
0

5

10

15

-0.2
0

5

10

15

(3) Monetary policy shock
OutputInflation

Nominal interest rate

0.2

0

0.3

0

-0.1

0.2

-0.2

-0.2

0.1

-0.4

-0.3

0

-0.6
0

5

10

15

-0.4
0

5

10

15

-0.1
0

5

10

15

Notes: The ﬁgure shows the impulse responses of output, inﬂation, and the nominal interest rate (in terms of
the percentage deviation from the steady state) to one standard deviation demand, cost-push, and monetary
policy shocks. The solid thick lines represent the true responses, and the solid thin lines and dashed lines arerespectively the posterior means and 90 percent credible intervals for the estimated responses.

31

Figure 4: Impulse responses in robustness analysis(1) TFP shock
OutputInflation

Nominal interest rate

0.2

0.05

0.1

0.1

0

0.05

0

-0.05

0

-0.1

-0.1

-0.05

-0.2
0

5

10

15

-0.15
0

5

10

15

-0.1
0

5

10

15

(2) Risk premium shock
OutputInflation

Nominal interest rate

1.5

0.6

0.6

1

0.4

0.4

0.5

0.2

0.2

0

0

0

-0.5
0

5

10

15

-0.2
0

5

10

15

-0.2
0

5

10

15

(3) External demand shock
OutputInflation

0.6

Nominal interest rate

0.04

0.04

0.02

0.02

0

0

0.4
0.2
0
-0.2
0

5

10

15

-0.02
0

5

10

15

-0.02
0

5

10

15

(4) Investment shock
OutputInflation

Nominal interest rate

0.4

0.15

0.15

0.3

0.1

0.1

0.2

0.05

0.05

0.1

0

0

0
0

5

10

15

-0.05
0

5

10

32

15

-0.05
0

5

10

15

(5) Monetary policy shock
OutputInflation

0.5

Nominal interest rate

0.4

0.3

0.2

0.2

0

0.1

-0.2

0

0

-0.5

-1
0

5

10

15

-0.4
0

5

10

15

-0.1
0

5

10

15

(6) Price markup shock
OutputInflation

Nominal interest rate

0.2

0.6

0.3

0

0.4

0.2

-0.2

0.2

0.1

-0.4

0

0

-0.6
0

5

10

15

-0.2
0

5

10

15

-0.1
0

5

10

15

(7) Wage markup shock
OutputInflation

Nominal interest rate

0.4

0.3

0.15

0.2

0.2

0.1

0

0.1

0.05

-0.2

0

0

-0.4
0

5

10

15

-0.1
0

5

10

15

-0.05
0

5

10

15

Notes: The ﬁgure shows the impulse responses of output, inﬂation, and the nominal interest rate (in terms of
the percentage deviation from the steady state) to one standard deviation shocks in the TFP, the risk
premium, external demand, investment, monetary policy, the price markup, and the wage markup. The solid
thick lines represent the true responses, and the solid thin lines and dashed lines are respectively the posteriormeans and 90 percent credible intervals for the estimated responses.

33