The Natural Rate of Interestin a Nonlinear DSGE Model∗
Yasuo Hirose†

Takeki Sunakawa‡

September 2021
Abstract
This paper investigates how and to what extent nonlinearities, including the zero lower
bound on the nominal interest rate, affect the estimate of the U.S. natural rate of interestin a dynamic stochastic general equilibrium model. The estimated natural rate in a nonlinear
model is substantially different from that in its linear counterpart after the global financial crisis
because of the zero lower bound. Other nonlinearities such as price and wage dispersion, fromwhich a linear model abstracts, play a negligible role in identifying the natural rate.
JEL Classification: C32, E31, E43, E52
Keywords: Natural rate of interest, Nonlinearity, Zero lower bound, Particle filter, Inversion
filter∗

The authors would like to thank Kosuke Aoki, Borağan Aruoba (the editor), Guido Ascari, Florin Bilbiie, Mark

Bognanni, Hess Chung, Todd Clark, Damjan Pfajfar, Matthieu Darracq Pariès, Martin Ellison, Jesús FernándezVillaverde, Andrea Ferrero, Cristina Fuentes-Albero, Ippei Fujiwara, Chris Gust, Michel Julliard, Timothy Kam,
Munechika Katayama, Ed Knotek, Takushi Kurozumi, David Lopez-Salido, Taisuke Nakata, Ed Nelson, Patrick
Pintus, Sebastian Schmidt, Mototsugu Shintani, Georg Strasser, Hitoshi Tsujiyama, Takayuki Tsuruga, Kozo Ueda,
Yuichiro Waki, Maik Wolters, Francesco Zanetti, two anonymous referees, and seminar and conference participantsat Australian National University, European Central Bank, Federal Reserve Bank of Cleveland, Federal Reserve
Board, Goethe University Frankfurt, University of Mannheim, Sophia University, University of Oxford, University ofTokyo, CIGS Year-End Macroeconomics Conference, DSGE Conference, CEF, International Workshop on Monetary
Policy When Heterogeneity Matters, Second Workshop for Heterogeneous Macro Models, and Summer Workshop
on Economic Theory for their insightful comments and discussions. Hirose is supported by grants-in-aid from JSPS
Scientific Research C (19K01560) and Tokyo Center for Economic Research. Sunakawa is supported by grants-in-aid
from JSPS Scientific Research for Young Scientists (18K12743), Zengin Foundation for Studies on Economics andFinance, ISM Cooperative Research Program (2021-ISMCRP-0003), and Fondation France-Japon de l’EHESS.
†

Faculty of Economics, Keio University. E-mail: yhirose@econ.keio.ac.jp‡

Graduate School of Economics, Hitotsubashi University. E-mail: takeki.sunakawa@gmail.com

1

1

Introduction

The natural rate of interest—the equilibrium real interest rate that yields price stability (Wicksell,1898)—has been a key concept for monetary policy analysis. In particular, a modern New Keynesianframework relates the concept of the natural rate to intertemporally optimizing agents and makes
it relevant for social welfare (Woodford, 2003; Galı́, 2008). The level of the natural interest rate inthis framework is a useful indicator for policymakers because it is a benchmark as to whether policy
is too tight or too loose from a welfare perspective.1 However, the natural rate is unobservableand must be estimated. Whereas the literature has developed various empirical methods to infer
the natural rate, an increasing number of researchers have estimated the natural rate measures
based on New Keynesian dynamic stochastic general equilibrium (DSGE) models.2 Examples for
the U.S. economy include Andrés, López-Salido, and Nelson (2009), Barsky, Justiniano, and Melosi(2014), Cúrdia (2015), Cúrdia, Ferrero, Ng, and Tambalotti (2015), Del Negro, Giannone, Giannoni,
and Tambalotti (2017), Edge, Kiley, and Laforte (2008), Justiniano and Primiceri (2010), and Neissand Nelson (2003).
This paper estimates the natural rate of interest in the U.S. economy using a nonlinear New
Keynesian DSGE model with a zero lower bound (ZLB) constraint on the nominal interest rateand examines how and to what extent nonlinearities affect the estimates of the natural rate andits driving forces. Whereas the previous studies estimate the DSGE-based natural rate only in a
linear setting that abstracts from the ZLB, this paper is one of the first to estimate the naturalrate in a fully nonlinear and stochastic setting that incorporates the ZLB.3
1

Closing the gap between the actual real interest rate and the natural rate is not necessarily optimal in theeconomy where “divine coincidence” (Blanchard and Galı́, 2007) does not hold. However, Barsky, Justiniano, andMelosi (2014) demonstrate that, even in such a circumstance, a central bank would be able to stabilize both inflationand the welfare-relevant output gap to a considerable degree by tracking the natural rate using an estimated New
Keynesian model.
2

Another stream of the literature estimates the long-run natural interest rate based on semi-structural or reduced-

form models. See, for instance, Holston, Laubach, and Williams (2017), Johannsen and Mertens (2021), Kiley (2015),
Laubach and Williams (2003, 2016), Lubik and Matthes (2015), Pescatori and Turunen (2016), and Williams (2015).
3

A contemporaneous paper by Iiboshi, Shintani, and Ueda (2020), which evolved independently from our work,

estimates a nonlinear small-scale New Keynesian model for Japan and extracts the sequence of the natural rate.

2

Our analysis is motivated by the following two strands of literature. First, Fernández-Villaverde
and Rubio-Ramı́rez (2005) and Fernández-Villaverde, Rubio-Ramı́rez, and Santos (2006) demonstrate that the level of likelihood and parameter estimates based on a linearized model can besignificantly different from those based on the original nonlinear model. The same may be true forthe estimation of unobservable state variables, including the natural rate. If substantial differencesin the estimates of the natural rate arise between linear and nonlinear models, it will cast doubt
on the common practice in which the natural rate is estimated based on a linear model. Overor under-estimation of the natural rate would be misleading in evaluating the stance of monetary
policy.
Second, the recent experience of the global financial crisis and the extremely low interest rate
period that followed has led researchers to conduct empirical analyses based on nonlinear DSGE
models in order to take the ZLB into consideration. For instance, Gust, Herbst, López-Salido, andSmith (2017) incorporate the ZLB into a medium-scale DSGE model similar to those developed by
Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2007), and estimate the modelin a nonlinear setting using U.S. macroeconomic time series. Plante, Richter, and Throckmorton(2018) and Richter and Throckmorton (2016a, 2016b) estimate a nonlinear version of a prototypical
New Keynesian model with the ZLB for the U.S. economy, and Iiboshi, Shintani, and Ueda (2020)
estimate a similar model for the Japanese economy. Aruoba, Cuba-Borda, and Schorfheide (2018)consider Markov switching between the targeted-inflation and deflation regimes in a New Keynesianframework with the ZLB and estimate the probabilities of the U.S. and Japan having been in either
the targeted-inflation or deflation regime using a nonlinear filtering technique. The present paper
contributes to this strand of the literature by focusing on the estimation of the natural rate.
In estimating the natural rate of interest, we follow a two-step approach. First, to parameterize
the model, we estimate a piecewise linear version of the model, in which the ZLB constraint is
imposed but all the equilibrium conditions are linearized, using the OccBin toolbox developed by
Guerrieri and Iacoviello (2015) and the inversion filter following Guerrieri and Iacoviello (2017).
Regarding this estimation strategy, Atkinson, Richter, and Throckmorton (2020) demonstrate thatpiecewise linear and fully-nonlinear approaches give rise to similar parameter estimates. Thus, thepiecewise linear approach enables us not only to avoid a computational burden that would increase3

exponentially in the estimation of a fully nonlinear model, but also to obtain reliable estimates ofparameters.
Next, given the estimated parameters, we solve the model in a fully nonlinear and stochastic
setting with the ZLB and apply a nonlinear filter to extract the sequence of the natural interest
rate. The literature (e.g., Boneva, Braun, and Waki, 2016; Fernández-Villaverde, Gordon, GuerrónQuintana, and Rubio-Ramı́rez, 2015; Gavin, Keen, Richter, and Throckmorton, 2015; Gust, Herbst,
López-Salido, and Smith, 2017; Nakata, 2016, 2017; Ngo, 2014; and Richter and Throckmorton,
2016a) has emphasized the importance of considering nonlinearities and related uncertainty effects
in assessing the quantitative implications of New Keynesian models that include the ZLB. The
natural rate estimated in the present paper takes account of these important features. Moreover,
our analysis is based on an empirically richer DSGE model than the prototypical New Keynesianmodel. The model features habit persistence in consumption preferences, price and wage stickiness,backward-looking components in price and wage settings, and monetary policy smoothing.
The main results are summarized as follows. Comparing the estimated natural interest rate
based on the nonlinear model with that based on the linear counterpart, we find that the former ishigher than the latter by up to 1.7 percent after the global financial crisis, when the nominal interest
rate was constrained by the ZLB. This difference is non-negligible; if we relied on the linear model,
we would overestimate the tightness of monetary policy due to the ZLB to a substantial degree.
The difference is ascribed to a contractionary effect arising from the ZLB, which is considered onlyin the nonlinear model. Although such a contractionary effect lowers expected output and inflation,
actual output and inflation are pegged to the corresponding observables in the filtering process.
Then, shocks to aggregate demand must be identified upward in order to satisfy the household’sintertemporal Euler equation that equates the marginal utility of consumption today and the expected discounted one in the future. As a consequence, the estimated natural rate increases in the
nonlinear setting. Although price and wage dispersion potentially affect the identification of shocks
and the estimate of the natural rate, their effects turn out to be negligible.
These findings allure researchers to use a piecewise linear model because such a model is easier
to solve than a fully nonlinear model. In this regard, we demonstrate that the piecewise linear
model can well replicate the natural interest rate based on the nonlinear model in the aftermath of4the global financial crisis, although it can slightly underestimate the natural rate due to ignoring
uncertainty at the ZLB.
The remainder of the paper proceeds as follows. Section 2 describes the model used in our
analysis and a strategy for estimating the natural interest rate. Section 3 presents our empiricalresults. Section 4 is the conclusion.

2

Model and Estimation Strategy

This section begins by describing the model used in our analysis. In the model economy, there arehouseholds, perfectly competitive final-good firms, monopolistically competitive intermediate-goodfirms, and a central bank that faces the ZLB constraint on the nominal interest rate. To ensure abetter fit to the macroeconomic time series, the model features habit persistence in consumption
preferences, price and wage stickiness, backward-looking components in price and wage settings,
and monetary policy smoothing.4 In the model, the natural rate of interest is defined as the realinterest rate that would prevail if prices and wages were fully flexible without any markup shocks.5
To obtain the estimates of the natural interest rate, we implement a two-step approach. First,
we estimate a piecewise linear version of the model, in which the ZLB constraint is imposed but
all the equilibrium conditions are linearized using the OccBin toolbox and a inversion filter. Next,
given the estimated parameters, we solve the model in a fully nonlinear and stochastic setting with
the ZLB using a projection method and apply a particle filter to extract the sequence of the naturalrate.
4Regarding modeling choice, we consider nominal wage rigidities as in Erceg, Henderson, and Levin (2000) as

well as price rigidities to incorporate additional nonlinearities through a wage dispersion term and its variability. A
medium-scale model with capital accumulation could take account of more nonlinearities, but we leave an analysis
using such a larger-scale model for future work.
5

This definition follows from Barsky, Justiniano, and Melosi (2014) and is the most commonly used in the literaturethat estimates the natural rate based on DSGE models. Cúrdia, Ferrero, Ng, and Tambalotti (2015) estimate theefficient interest rate, which is defined as the real interest rate under flexible prices and perfect competition with zero
markups.

5

2.1

The model

2.1.1

Households

Each household h ∈ [0, 1] consumes final goods Ch,t , supplies labor lh,t =R1

0 lf,h,t df to intermediate-

good firms f ∈ [0, 1], and purchases one-period riskless bonds Bh,t so as to maximize the following
utility function
E0

∞
X

β

t

t=0

t
Y

!−1 "
dk

log (Ch,t − γCt−1 ) −

k=1

1+η
lh,t

1+η

#
,

subject to the budget constraintnn
Pt Ch,t + Bh,t = Wh,tlh,t + Rt−1
Bh,t−1 + Th,t ,where β ∈ (0, 1) is the subjective discount factor, γ ∈ [0, 1] is the degree of external habit persistence
in consumption preferences (Ct−1 is the aggregate consumption in period t − 1), η ≥ 0 is the inverse
n is the nominal wage for householdof the labor supply elasticity, Pt is the price of final goods, Wh,th, Rtn is the gross nominal interest rate, and Th,t is the sum of a lump-sum public transfer andprofits received from firms. Following Christiano, Eichenbaum, and Rebelo (2011), a shock to thediscount factor dt affects the weight of the utility in period t + 1 relative to the one in period t. In
the present model, this shock is broadly interpreted as a shock to aggregate demand. The log ofthe discount factor shock follows an AR(1) process
log dt = ρd log dt−1 + εd,t ,(1)

where ρd ∈ [0, 1) is an autoregressive coefficient and εd,t is a normally distributed innovation
with mean zero and standard deviation σd . The first-order conditions for optimal decisions on
consumption and bond-holding are identical among households, and therefore become
1,
Ct − γCt−1
βΛt+1
Λt = Rt Et
,
dt
Πt+1
Λt =

where Λt is the marginal utility of consumption, and Πt = Pt /Pt−1 denotes gross inflation.

6

(2)
(3)

2.1.2

Wage setting

A labor packer collects differentiated labor {lf,h,t } from each household h and resells a labor packiθw /(θw −1)hR
1 (θw −1)/θw
dh
to intermediate-good firms
age augmented by a CES aggregator lf,t = 0 lf,h,t
indexed by f , where θw > 1 represents the elasticity of substitution among labor varieties. Givenn , cost minimization yields a set of labor demand schedules
the nominal wage for each household Wh,tR
1/(1−θw )

−θw
1
n =
n 1−θw dh
n /W n.
landtheaggregate
wage
index
W
W
lf,h,t = Wh,t
f,t
t
t
h,t
0

Given the demand for labor by the labor packers, labor unions representing each household hset nominal wages on a staggered basis, as in Erceg, Henderson, and Levin (2000). In each period,a fraction 1 − ξw ∈ (0, 1) of labor unions reoptimizes their nominal wages, whereas the remaining
fraction ξw indexes nominal wages to the economy’s trend growth γa and a weighted average ofpast inflation Πt−1 and steady-state inflation Π̄. The labor unions that reoptimize their nominal
wages in the current period then maximize expected utility as follows

Et

∞
X

j j
ξw
β

j=0

j
Y

dk

!−1 " jjn Y
γa Wh,t

k=1

Pt+j

w
Πιt+k−1Π̄1−ιwΛh,t+j lh,t+j −

k=1

1+η
lh,t+j1+η

#,subject to the labor demandlf,h,t+j =where lh,t =R1

0 lf,h,t df

" j
j
n Y
γa Wh,t
n
Wt+j

#−θw
w
Π̄
Πιt+k−1
1−ιw

lf,t+j ,k=1

is the amount of labor supplied by each household h, and ιw ∈ [0, 1) is

the weight of wage indexation to past inflation relative to steady-state inflation. The first-ordercondition for the reoptimized wage Wtn,o is given by−1 Q
−(1+η)θw
P∞ j j Qj
γaj Wtn
j
1+η
ιw
1−ι
w
 n,o 1+ηθw
Et j=0 ξw β
ld,t+jn
k=1 dkk=1 Πt+k−1 Π̄
Wt+j
Wt
θw

,
=
−1 Qnjn 1−θwWtn
θw − 1 P∞ j j Qj
Wt+j
γ
W
j
ιw
a
t
1−ιw
Et j=0 ξw β
Λt+j Pt+j ld,t+j
k=1 dkk=1 Πt+k−1 Π̄
Wn
t+j

(4)
R1
where ld,t = 0 lf,t df is the total labor demand.
R1/(1−θw )1n 1−θw dhW
can be written ash,t0

The aggregate nominal wage index Wtn =h1−θw i 1−θ1 w
nw
Wtn = (1 − ξw ) (Wtn,o )1−θw + ξw Πιt−1
Π̄1−ιw γa Wt−1
.

7

(5)

2.1.3Firms

The representative final-good firm produces output Yt under perfect competition by choosing a
R1
combination of intermediate inputs {Yf,t } so as to maximize profit Pt Yt − 0 Pf,t Yf,t df , subject to aiθp /(θp −1)hR
1 (θ −1)/θp, where Pf,t is the price of intermediatedf
CES production technology Yt = 0 Yf,tp
good f and θp > 1 denotes the elasticity of substitution among the variety of intermediate goods.
The first-order condition for profit maximization yields the final-good firm’s demand for each in1/(1−θp )
R
1 1−θ
.
termediate good Yf,t = (Pf,t /Pt )−θp Yt and the aggregate price index Pt = 0 Pf,t p df
Each intermediate-good firm f produces a differentiated good Yf,t under monopolistic competition by choosing a labor input lf,t , given the real wage Wt = Wtn /Pt , and subject to the production
functionYf,t = At lf,t ,where At represents total factor productivity. The log of the productivity level follows a nonstationary stochastic processlog At = log γa + log At−1 + at ,(6)

where log γa represents the steady-state growth rate of productivity and at is a shock to the productivity growth. The productivity shock follows an AR(1) process
at = ρa at−1 + εa,t ,(7)where ρa ∈ [0, 1) is an autoregressive coefficient and εa,t is a normally distributed innovation withmean zero and standard deviation σa . Assuming the existence of a shock to real marginal cost zt ,which is interpreted as an inefficient cost-push shock, the first-order condition for cost minimization
is given by6
M Ct =

Wt
zt .
At

(8)

The log of the cost-push shock follows an AR(1) process
log zt = ρz log zt−1 + εz,t ,6(9)

The first-order condition also indicates that the real marginal cost M Ct is identical across the intermediate-goodfirms.

8where ρz ∈ [0, 1) is an autoregressive coefficient and εz,t is a normally distributed innovation withmean zero and standard deviation σz .
In the face of the final-good firm’s demand and marginal cost, the intermediate-good firms set
the prices of their products on a staggered basis, as in Calvo (1983). In each period, a fraction
1 − ξp ∈ (0, 1) of intermediate-good firms reoptimizes their prices, whereas the remaining fraction
ξp indexes prices to a weighted average of past inflation Πt−1 and steady-state inflation Π̄. The
firms that reoptimize their prices in the current period then maximize expected profit as follows
#
!−1
"
j
j
∞
X
Y

Λt+j Pf,t Y ιp
1−ιp
j j
dk
Et
ξp β
Πt+k−1 Π̄− M Ct+j Yf,t+j ,Λt Pt+j
j=0

k=1

k=1

subject to the final-good firm’s demandj"
Pf,t Y ιp
Yf,t+j =Πt+k−1 Π̄1−ιp
Pt+j

#−θp
Yt+j ,

k=1

where ιp ∈ [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
−1
P∞ j j Qj
Qj h Πt+k−1 ιp Π̄ i−θpΛt+j
M Ct+j Yt+j
Et j=0 ξp β
k=1 dk
k=1
Λt
Πt+k
Π̄
Pto
θ

 .
=
−1
Q
h

i1−θpP∞ j j Qj
Pt
θ−1
Λt+j
Πt+k−1 ιp Π̄j
Et j=0 ξp β
Yt+j
k=1 dk
k=1
Λt
Πt+k
Π̄
The final-good’s price Pt =R

1/(1−θp )1 1−θpdf
P
can be written as
0 f,t

1
h
1−θp i 1−θιpp.
Pt = (1 − ξp ) (Pto )1−θp + ξp Πt−1
Π̄1−ιp Pt−1

2.1.4

(10)

(11)

Market clearing conditions

The final-good market clearing condition is
Yt = Ct ,(12)

whereas the labor market clearing condition leads tolt =where lt =∆p,t ∆w,t Yt
,
At

R1R1
0

(13)

0 lf,h,t df dh is the aggregate labor input, ∆p,t =across the intermediate-good firms, and ∆w,t =

R1
0

9

n /W nWh,tt

R10 (Pf,t /Pt )
−θw

−θp

df is price dispersion

dh is wage dispersion across thelabor unions. Equation (13) can be rewritten in terms of ld,t =R1

0 lf,t df as∆p,t Yt.
At

ld,t =(14)

In the present model, the price and wage dispersion evolve according to θp Pto −θpΠtΠt−1 −ιp θp∆p,t = (1 − ξp )+ ξp∆p,t−1 ,
Pt
Π̄
Π̄
θw 


 n,o −θw
Πt−1 −ιw θw
Πt W tWt∆w,t−1 .
∆w,t = (1 − ξw )
+
ξ
w
Wtn
Π̄γa Wt−1
Π̄


2.1.5

(15)
(16)

Flexible wage and price equilibrium

Natural output Yt∗ and the natural rate of interest Rt∗ are defined as the levels that would prevail
if both wages and prices were perfectly flexible with no cost-push shocks. Such a flexible wage andn = W n, P
price equilibrium is obtained with ξw = ξp = 0, Wh,tf,t = Pt , and zt = 1 for all h, f , andtt in the model above and is characterized by the following equations:∗Yt∗ − γYt−1





Yt∗
At

η
= µAt ,

(17)


∗ −1
Yt∗ − γYt−1
Et ∗,β
Yt+1 − γYt∗

(18)

dt
Rt∗ =
θ −1

where µ = θwθw−1 pθp is the product of price and wage markups. Thus, the law of motion for natural
output Yt∗ is determined by (17), given the sequence of total factor productivity At . The naturalrate of interest Rt∗ is determined by (18), given the sequences of natural output Yt∗ and the discountfactor shock dt .
2.1.6

Central bank

A monetary policy rule is specified as
btn , 1],
Rtn = max[Rwhere"bn = (Rbn )φr R̄Π̄
R
t
t−1



Πt
Π̄

φπ 

Yt
At

 φy Yt
γa Yt−1(19)

φ∆y #1−φrexp(εr,t ).

(20)

bn denotes the notional nominal interest rate that the central bank would set according to a Taylor
Rt(1993) type monetary policy rule in the absence of the ZLB constraint, where R̄ is the steady-state10

gross real interest rate, φr ∈ [0, 1) is the policy-smoothing parameter, and φπ ≥ 0, φy ≥ 0, andφ∆y ≥ 0 are the degrees of the interest rate policy response to inflation, detrended output, andoutput growth. εr,t is a monetary policy shock, which is normally distributed with mean zero andstandard deviation σr . The max function in (19) constrains the nominal interest rate to be greater
btn > 1, the ZLB constraint is not imposed, i.e., Rtn = R
btn . If R
btn ≤ 1, thethan or equal to zero. If R
ZLB is binding, i.e., Rtn = 1.
2.1.7

Equilibrium

An equilibrium is given by the sequences {Yt , Ct , Λt , Wt , Wtn , Wtn,o , lt , ld,t , M Ct , Πt , Pt , Pto , ∆p,t , ∆w,t ,bn , dt , At , at , zt }∞ satisfying the equilibrium conditions (1)–(20) and two definitional
Yt∗ , Rt∗ , Rtn , Rt
t=0equations, Wt = Wtn /Pt and Πt = Pt /Pt−1 .
Because total factor productivity At is nonstationary, as specified by (6), we rewrite the equilibrium conditions in terms of stationary variables detrended by At , as follows: yt = Yt /At , ct = Ct /At ,λt = Λt At , wt = Wt /At , wtn = Wtn /At , wtn,o = Wtn,o /At , mct = M Ct /At , and yt∗ = Yt∗ /At , so that
we can derive a nonstochastic steady state for the detrended variables.

2.2

Estimation of parameters

To parameterize the model, we estimate a piecewise linear version of the model, in which the ZLB
constraint is imposed but all the equilibrium conditions are linearized. More specifically, such apiecewise linear model is solved using the OccBin toolbox developed by Guerrieri and Iacoviello(2015), and the likelihood function is evaluated with the inversion filter following Guerrieri and
Iacoviello (2017). While it is computationally very intensive to estimate a fully nonlinear modelusing a projection method and a particle filter, the piecewise linear approach considerably reduces
the computational burden. Regarding estimation accuracy, Atkinson, Richter, and Throckmorton (2020) demonstrate that the piecewise linear and fully-nonlinear approaches result in similarparameter estimates.
The model is fitted to four U.S. quarterly time series: the per capita real GDP growth rate(100∆ log GDPt ), the inflation rate of the GDP implicit price deflator (100∆ log P GDPt ), the11federal funds rate (F Ft ), and the log of hours worked (100 log Ht ).7 Following Wolters (2018), thedata on hours worked is adjusted for low-frequency movements due to sectoral and demographic
changes so that the data is consistent with the model. The sample period is from 1987:III to
2019:IV. The beginning of the sample period is set at the time when Alan Greenspan became the
Chairman of the Fed, because thereafter, the style of the Fed’s policy conduct seems consistentand stable. The end of the sample is determined to exclude the COVID-19 pandemic period. The
linearized equilibrium conditions and observation equations are presented in Appendix A.
The parameters are estimated using Bayesian methods. The prior distributions of parameters
are presented in the second to fourth columns of Table 1. For most of the parameters, each priormean is set at the corresponding prior mean used in the literature including Smets and Wouters(2007). The prior mean of the policy-smoothing parameter φr is set at 0.5, which is lower thanthat in Smets and Wouters (2007) because a higher value of the estimated φr would lead to anonconvergence problem in the next step for solving our nonlinear model.8 As for the steady-statevalues of output growth, inflation, and real interest rates and hours worked (ā, π̄, r̄, ¯l), the priors are
centered at the sample mean. The prior mean of the AR(1) coefficient for the discount factor shockρd is 0.75, whereas those for the productivity and cost-push shocks (ρa , ρz ) is 0.5. For the standard
deviations of the shocks (100σd , 100σa , 100σz , 100σr ), we assign inverse-gamma distributions witha mean of 0.5 and a standard deviation of 2.0.
Following Guerrieri and Iacoviello (2017), 50, 000 posterior draws are generated using theRandom-Walk Metropolis-Hastings algorithm, and the first 10, 000 draws are discarded. The posterior mean and 90 percent credible interval for each parameter are reported in the middle columnsof Table 1, labeled “Piecewise linear.” For the sake of comparison, we estimate a linear versionof the model that omits the ZLB constraint, and the results are shown in the last two columnsof Table 1. The parameter estimates are very similar to each other, in contrast to the finding ofHirose and Inoue (2016), who conduct a Monte Carlo analysis and demonstrate that omitting the
ZLB in estimation causes biased estimates of parameters.
7

The series of hours worked is demeaned.

8

For the same reason, relatively tight priors are used for the parameters that determine the persistency of endoge-

nous variables in the model.

12

Table 1: Prior and posterior distributions of parameters
Prior

Posterior
Piecewise linear

Linear (No ZLB)Parameter

Distribution

Mean

S.D.

Mean

90% interval

Mean

90% interval

γ

Beta

0.500

0.050

0.664

[0.609, 0.716]

0.654[0.597, 0.708]

η

Gamma

2.000

0.250

1.763

[1.372, 2.212]

1.824[1.438, 2.249]

ξw

Beta

0.500

0.050

0.681

[0.561, 0.756]

0.687[0.598, 0.752]

ιw

Beta

0.500

0.050

0.488

[0.407, 0.568]

0.492[0.409, 0.576]

ξp

Beta

0.500

0.050

0.868

[0.817, 0.913]

0.841[0.792, 0.887]

ιp

Beta

0.500

0.050

0.451

[0.360, 0.545]

0.490[0.404, 0.579]

φπ

Gamma

1.500

0.250

1.575

[1.353, 1.825]

1.608[1.361, 1.876]

φy

Gamma

0.500

0.100

0.168

[0.143, 0.194]

0.141[0.120, 0.164]

φ∆y

Gamma

0.125

0.050

0.238

[0.134, 0.350]

0.265[0.155, 0.381]

φr

Beta

0.500

0.050

0.750

[0.706, 0.789]

0.803[0.770, 0.836]ā

Normal

0.356

0.100

0.348

[0.250, 0.452]

0.361[0.267, 0.453]π̄Normal

0.529

0.100

0.576

[0.489, 0.670]

0.536[0.443, 0.631]

r̄

Gamma

0.276

0.100

0.215

[0.146, 0.293]

0.240[0.162, 0.322]¯lNormal

0.000

0.100

-0.058[-0.146, 0.037]

0.016[-0.018, 0.058]

ρd

Beta

0.750

0.050

0.831

[0.794, 0.862]

0.851[0.819, 0.881]

ρa

Beta

0.500

0.050

0.418

[0.352, 0.487]

0.412[0.345, 0.477]

ρz

Beta

0.500

0.050

0.573

[0.481, 0.663]

0.610[0.525, 0.697]

100σdInv. Gamma

0.500

2.000

0.404

[0.311, 0.523]

0.359[0.274, 0.466]

100σaInv. Gamma

0.500

2.000

0.558

[0.502, 0.618]

0.559[0.503, 0.622]

100σz

Inv. Gamma

0.500

2.000

7.612

[3.613, 15.18]

5.256[2.870, 9.135]

100σrInv. Gamma

0.500

2.000

0.119

[0.104, 0.136]

0.107[0.096, 0.121]

Note: Each posterior mean and 90% credible interval are calculated from 50, 000 draws (the first 10, 000 draws are
discarded) generated using the Metropolis-Hastings algorithm.

13

In the subsequent analysis, the parameters are fixed at the posterior mean estimates of thepiecewise linear model.

2.3

Nonlinear solution and filtering

Given the posterior mean estimates for the piecewise linear model obtained above, the model issolved in a fully nonlinear and stochastic setting with the ZLB constraint using a projection method.
The model has seven endogenous state variables (output yt−1 , inflation Πt−1 , the real wage wt−1 , thebn , price dispersion ∆p,t−1 , wage dispersion ∆w,t−1 , and natural
notional nominal interest rate R
t−1∗ ) and four exogenous shocks (the discount factor shock d , the productivity shock a ,output yt−1tt

the cost-push shock zt , and the monetary policy shock εr,t ). The policy functions satisfying thedetrended equilibrium conditions can be written as
St = g(St−1 , τt ),bn , ∆p,t−1 , ∆w,t−1 , y ∗ ]0 and τt = [dt , at , zt , εr,t ]0 .
where St−1 = [yt−1 , Πt−1 , wt−1 , R
t−1
t−1
To compute the policy functions, we employ a projection method with an index function approach as in Aruoba, Cuba-Borda, and Schorfheide (2018), Gust, Herbst, López-Salido, and Smith(2017) and Nakata (2017). For interpolating the policy functions within a time iteration algorithm,
we adapt a standard linear interpolation to reduce approximation errors that tend to be large particularly when the ZLB binds due to large negative shocks to the economy.9 The details of thesolution method are described in Appendix B.
According to an artificial sample of 40,000 periods simulated from the nonlinear solution of themodel, the economy is at the ZLB for 3.8 percent of quarters, and the average duration of ZLB
spells is 3.4 quarters. These statistics are in line with the simulation results in the previous studies
that employ nonlinear New Keynesian models. Fernández-Villaverde, Gordon, Guerrón-Quintana,and Rubio-Ramı́rez (2015) simulate a small-scale model calibrated for the U.S. economy and showthat the economy spends 5.5 percent of quarters at the ZLB and that the average duration at the
ZLB is 2.1 quarters. Gust, Herbst, López-Salido, and Smith (2017) estimate a medium-scale modelin a nonlinear setting using U.S. data from 1983:I to 2014:I, and the simulation of their estimated9

Because of the high dimensionality of state variables, we use a supercomputer system to utilize parallel computing.

14

model demonstrates that the economy is at the ZLB for about 4 percent of quarters on averageand that the average duration of the ZLB spells is just over 3.5 quarters.
We apply a particle filter as in Fernández-Villaverde and Rubio-Ramı́rez (2007) to extract thesequence of the state variables and then compute the estimates of the natural interest rate.10 The
data used for filtering is the same as those used for the parameter estimation in Section 2.2. To
facilitate the use of the particle filter, measurement errors are added in the observation equations.
The size of the measurement errors of output growth, inflation, the nominal interest rate, and hours
worked are respectively set to be one percent of the standard deviations of the data so that we canreduce the effect of measurement errors on the filtered estimates of the natural interest rate and its
related variables. We use 100,000 particles and confirmed that any further increase in the numberof particles delivered almost the same results as those presented below.

3

Results

This section presents the estimate of the natural interest rate based on the nonlinear model andcompares it with that based on its linear counterpart. To understand the source of the difference
between the two estimates, we investigate how the natural rate of interest is identified in each case.
Moreover, we consider the piecewise linear model, which is used for estimating model parameters,
to examine whether it can be a useful substitute for estimating the natural rate accurately.

3.1

Estimated natural rate of interest

In Figure 1, the thick solid line shows the filtered mean estimate of the natural rate of interest on
an annualized basis, based on the nonlinear model. The gray shaded area is the 90 percent interval
obtained from the distributions of particles in each period. The estimated natural rate measure
peaked more than 10 percent at the end of 1980s and the beginning of 2000, then fell to about −10
percent in the aftermath of the global financial crisis, and thereafter increased to slightly positive10

For a textbook treatment of a particle filter, see Fernández-Villaverde, Rubio-Ramı́rez, and Schorfheide (2016)

or Herbst and Schorfheide (2015).

15

Figure 1: Natural rate of interest

15
LinearNonlinear

10

5

0

-5

-10

-15
1990

1995

2000

2005

2010

2015

Notes: The figure shows the filtered mean estimate of the natural interest rate, in annualized percentage terms, basedon the nonlinear model (thick solid line) with its 90 percent interval (shaded area) and that based on the linear model(thin solid line). The yellow area indicates the period when the federal funds rate was constrained by the ZLB.

values toward the end of the sample period.11 The overall cyclical movements and variability of thenatural rate are similar to those estimated by Barsky, Justiniano, and Melosi (2014), who employ
a medium-scale New Keynesian DSGE model with capital accumulation.
In the present framework, the natural interest rate is driven by structural shocks and hence
exhibits short-run fluctuations. To make it comparable to the estimates of the long-run natural rate
11

In an earlier version of this paper (Hirose and Sunakawa, 2017), the estimates of the natural rate were too volatile,

ranging from −15 to 15 percent. This issue was resolved by employing the hours data constructed by Wolters (2018).
As mentioned in Section 2.2, the data is adjusted for sectoral and demographic changes, and hence it is less volatile
than the original data.

16

Figure 2: Long-run natural rate of interest

4
Laubach-Williams
10-year

3

2

1

0

-1

-2
1990

1995

2000

2005

2010

2015

Notes: The figure shows the filtered mean estimate of the 10-year natural interest rate, in annualized percentage
terms, based on the nonlinear model (solid line) with its 90 percent interval (shaded area) and the filtered estimateof the natural interest rate based on Laubach-Williams (2003) model (dashed line). The yellow area indicates theperiod when the federal funds rate was constrained by the ZLB.

based on a semi-structural or reduced-form model as in Laubach and Williams (2003), we calculate
the 10-year natural rate Rt∗10Y according to the expectation hypothesis of the term structure ofinterest rates
39

log Rt∗10Y =1 X∗
Et log Rt+i
.
40
i=0

Figure 2 compares the filtered mean estimate of the 10-year natural rate based on the nonlinear
model (solid line) with the filtered estimate of the natural rate based on Laubach-Williams (2003)
model (dashed line, henceforth, the LW estimate).12 Two notable differences emerge. First, cyclical12

The latter estimate is available at the website of the Federal Reserve Bank of New York.

17

patterns are somewhat different from each other. Second, our estimate of the 10-year natural rate
is lower than the LW estimate throughout the sample period, whereas these two estimates are close
to each other in the last few years. The second difference arises because of the difference in theestimation sample. The sample for the LW estimate begins in 1961:I, and hence the steady-state
real interest rate in their model is estimated to be higher than that in our model.
The primary objective of this paper is to examine how and to what extent nonlinearities,
including the ZLB, affect the estimates of the natural interest rate. To this end, we estimate thenatural rate using a linear counterpart of the model, as in the previous studies, and compare it withthe one obtained above. The thin solid line in Figure 1 shows the filtered estimate of the naturalinterest rate based on the linearized version of the model with the same parameters and data set as
used in the nonlinear case.13 The figure indicates that the natural rate based on the linear model
is mostly very similar to that based on the nonlinear model, except for the period after 2009, when
the actual nominal interest rate was constrained by the ZLB. The difference amounts to 1.7 percent
during 2009 and remains substantial thereafter.
This difference matters when we evaluate the Fed’s monetary policy stance after the global
financial crisis. In the present framework, the natural interest rate is a benchmark for whether
monetary policy is too tight or too loose from a welfare perspective. Such a policy stance ismeasured by the interest rate gap—the difference between the actual (ex-post) real interest rate
and the natural rate. A larger value of the interest rate gap indicates a tighter policy stance. As
shown in Figure 3, both measures of the interest rate gap were around zero from 2002 to 2008,
but thereafter jumped to about 10 percent. More precisely, while the gap in the nonlinear modelincreased to 9.5 percent in 2010:I, its linear counterpart increased to 11.1 percent. These jumps
are due to the existence of the ZLB; that is, the Fed was not able to lower the policy rate any
more in response to the severe economic downturn, and hence the policy stance was evaluated to13

In the linear setting, the model is solved by the standard linear solution method and the filtered estimates ofmodel variables are computed using the Kalman filter with zero measurement errors. As a preliminary analysis,
we applied a particle filter to the linear model and obtained a very similar estimate of the natural rate. Thus, we
confirmed that the difference in the filtering methods with and without measurement errors is not the major sourceof the difference in the estimated natural rates between the linear and nonlinear models.

18

Figure 3: Interest rate gap

15
LinearNonlinear

10

5

0

-5

-10
1990

1995

2000

2005

2010

2015

Notes: The figure shows the real interest rate gap, in annualized percentage terms, based on the nonlinear model(thick solid line) with its 90 percent interval (shaded area) and that based on the linear model (thin solid line). The
yellow area indicates the period when the federal funds rate was constrained by the ZLB.

be too tight. In this context, if we relied on the linear model, the estimated natural rate andthe resulting interest rate gap would overestimate the tightness of the policy stance by nearly 17
percent (; (11.1 − 9.5)/9.5).

3.2

What accounts for the difference?

To understand what causes the difference between the two estimates of the natural interest rate, we
consider how the natural rate is identified in each case. As addressed in Section 2.1, equation (17), ∗∗i.e., Yt∗ − γYt−1(Yt /At )η = µAt , determines natural output Yt∗ , given the sequence of total factor
productivity At (or, equivalently, the productivity shock at ). The natural rate Rt∗ can be traced

19



∗ )/(Y ∗ − γY ∗ ) −1 , given the sequences ofout from equation (18), i.e., Rt∗ = dt /β Et (Yt∗ − γYt−1tt+1natural output Yt∗ and the discount factor shock dt . Thus, the natural rate is pinned down byidentifying the two shocks, at and dt .
In the linear model, the productivity shock at is explicitly identified by the data on output andhours worked because detrending and log-linearizing the labor market clearing condition (13) yields
ỹt = ˜lt and because the associated observation equations are 100∆ log GDPt = ā+ ỹt − ỹt−1 +at and
100 log Ht = ¯l + ˜lt , where ā and ¯l are the steady-state growth rate and hours worked, respectively,
and the variables with ˜ represent percentage deviations from their steady-state values. In the
nonlinear model, however, equation (13) contains the price and wage dispersion, ∆p,t and ∆w,t ,and can be written as yt = lt /(∆p,t ∆w,t ) in detrended terms. These dispersion terms fluctuate
so that ∆p,t ≥ 1 and ∆w,t ≥ 1, as the price and wage deviate from the steady state. Thus, yt
becomes lower than in the linear case where the dispersion terms are suppressed. Consequently,
to satisfy the observation equation for output growth, at is identified to be larger in the nonlinear
case. From equations (17) and (18), higher productivity raises natural output and results in thehigher estimate of the natural rate.
Identification of the discount factor shock dt is more complicated and influenced by the whole
structure of the model. However, taking account of the finding that the two estimates of thenatural interest rate differ from each other during the period when the nominal interest rate wasbounded at zero, the existence of the ZLB, from which the linear model abstracts, possibly affects
the identification of dt in the nonlinear model. The literature has established that the ZLB has
a contractionary effect on the economy not only when the nominal interest is already binding atzero, but also when uncertainty exists about whether the ZLB will bind in the future.14 Althoughsuch a contractionary effect lowers expected output and inflation, the filtering procedure pegs
contemporaneous output and inflation to the corresponding observables, which are the same in thelinear and nonlinear cases. Then, in the nonlinear case, the discount factor shock dt must increase
to satisfy the consumption Euler equation that equalize the marginal utility of consumption todaywith the expected discounted one in the future. As a result, the estimated natural rate becomes
14

Hills, Nakata, and Schmidt (2016) quantify such an uncertainty effect on inflation in the face of the interest rate

lower bound.

20

Figure 4: Estimated shocks

Discount factor shock4

LinearNonlinear

2
0
-2
-4

1990

1995

2000

2005

2010

2015

2010

2015

Productivity shock

2
1
0
-1
-2

1990

1995

2000

2005

Notes: The figure shows the filtered mean estimates of the discount factor shocks dt and the productivity shocks at ,in percentage terms, based on the nonlinear model (thick solid lines) with their 90 percent intervals (shaded areas)
and those based on the linear model (thin solid lines). The yellow areas indicate the period when the federal funds
rate was constrained by the ZLB.

higher.
To quantify the differences in the sequences of identified shocks, Figure 4 shows the filteredmean estimates of the discount factor shocks dt and the productivity shocks at , in percentage
terms, based on the nonlinear model (thick solid lines) and its linear counterpart (thin solid lines).
The gray shaded areas are the 90 percent intervals for the filtered estimates in the nonlinear model.
The sequence of dt identified in the nonlinear model is remarkably different from that in the linear
model after the global financial crisis. In particular, the difference is pronounced in 2009–2010,
bn sharply fell below zero, as shown in Figure 5. Thus,
when the notional nominal interest rate Rt21

Figure 5: Notional rate

Notional rate
6

4

2

0

-2

-4

-6
2008

2010

2012

2014

2016

2018

Notes: The figure shows the filtered mean estimate of the notional nominal interest rate and its 90 percent interval(shaded area), in annualized percentage terms. The yellow area indicates the period when the federal funds rate wasconstrained by the ZLB.

whether the ZLB constraint is imposed or not substantially affects the identification of dt .
On the other hand, the movements of the productivity shocks at are very similar between thetwo estimates although temporary deviations are found occasionally. The finding of the small
difference in at implies that the price and wage dispersion terms, ∆p,t and ∆w,t , play a minor rolein the nonlinear model. Indeed, as shown in Figure 6, the estimates of ∆p,t and ∆w,t based on
the nonlinear model fluctuate little, i.e., 0.14 percent at most, even though they exhibit cyclicalmovements over the sample period.

22

Figure 6: Price and wage dispersion

Price dispersion
0.15

0.1

0.05

0

1990

1995

2000

2005

2010

2015

2010

2015

Wage dispersion
0.1
0.08
0.06
0.04
0.02
0

1990

1995

2000

2005

Notes: The figure shows the filtered mean estimates of the price and wage dispersion and their 90 percent intervals(shaded areas) in terms of percentage deviation from the steady state. The yellow areas indicate the period when
the federal funds rate was constrained by the ZLB.

3.3

The natural rate of interest based on the piecewise linear model

The analysis thus far suggests that the existence of the ZLB constraint plays a crucial role inidentifying the natural interest rate in a nonlinear setting, but that the role of price and wage
dispersion is negligible. These findings tempt researchers to exploit a piecewise linear model, in
which the ZLB constraint is imposed but all the equilibrium conditions are linearized, for estimating
the natural rate measures, because such a model is easier to solve than a fully nonlinear model.
While the literature argues that the solution for this sort of piecewise linear models can give rise to23

Figure 7: Natural rate of interest: Comparison with piecewise linear model

15
LinearNonlinear
Piecewise linear

10

5

0

-5

-10

-15
1990

1995

2000

2005

2010

2015

Notes: The figure shows the filtered mean estimate of the natural interest rate, in annualized percentage terms, basedon the nonlinear model (thick solid line) with its 90 percent interval (shaded area) and those based on the linear
model (thin solid line) and the piecewise linear model (dashed line). The yellow area indicates the period when thefederal funds rate was constrained by the ZLB.

an inaccurate assessment of the ZLB,15 Atkinson, Richter, and Throckmorton (2020) demonstratethat the piecewise linear and fully-nonlinear approaches lead to similar results with regard toparameter estimates. In what follows, we examine whether the piecewise linear model can be auseful substitute for estimating the natural rate.
As in the estimation of parameters described in Section 2.2, the piecewise linear version of15

See Boneva, Braun, and Waki (2016), Fernández-Villaverde, Gordon, Guerrón-Quintana, and Rubio-Ramı́rez(2015), Gavin, Keen, Richter, and Throckmorton (2015), Gust, López-Salido, and Smith (2017), Nakata (2016,2017), Ngo (2014), and Richter and Throckmorton (2016a).

24

the model is solved using the OccBin toolbox, and the filtered estimates are computed using theinversion filter.16 Figure 7 depicts the filtered estimate of the natural interest rate in the piecewise
linear setting (dashed line) along with the estimates in the fully nonlinear (thick solid line) andlinear (thin solid line) settings. The estimate based on the piecewise linear model coincide withthat based on the linear model before the global financial crisis, but in the aftermath, it is very
close to that based on the fully nonlinear model. Thus, the higher estimate of the natural rate in
the nonlinear model from 2009 to 2011 is well replicated by the piecewise linear model.
However, the natural rate in the piecewise linear model moves closer to that in the linear model
after 2012, when the notional rate shown in Figure 5 approached toward zero. This is because
the OccBin solution, by its construction, does not take into account uncertainty about the future.
An uncertainty effect enhances the contractionary effect of the ZLB because agents expect thatmonetary policy will be unable to accommodate negative shocks to the economy. As addressed in
the previous subsection, the contractionary effect of the ZLB leads to the higher estimate of thenatural rate in the nonlinear model. This effect is reduced in the absence of uncertainty, and hence
the piecewise linear model slightly underestimates the natural rate.17
These mechanisms are confirmed by the upper panel in Figure 8. The estimated series of the
discount factor shock dt in the piecewise linear model (dashed line) is substantially larger than thatin the linear model (thin solid line) for a few years after the global financial crisis, whereas it issmaller to a lesser extent than that in the nonlinear model (thick solid line) until the end of thesample period.
The lower panel in Figure 8 compares the estimates of the productivity shock at . The piecewise
linear model abstracts from price and wage dispersion as does the linear model, and hence theestimated productivity shocks in the two models exactly coincide with each other.
In summary, a piecewise linear model incorporating the ZLB can be a possible substitute for a
fully nonlinear model in estimating the natural rate measures, although it can slightly underestimate
the natural rate due to ignoring uncertainty at the ZLB.
16

To run the inversion filter, the measurement errors are set to zero as in the Kalman filter for the linear model.

17

If the probability of binding at the ZLB were higher, the uncertainty effect would be larger, which would resultin a larger difference in the natural rate between the nonlinear and piecewise linear models.

25

Figure 8: Estimated shocks: Comparison with piecewise linear model

Discount factor shock4

LinearNonlinear
Piecewise linear

2
0
-2
-4

1990

1995

2000

2005

2010

2015

2010

2015

Productivity shock

2
1
0
-1
-2

1990

1995

2000

2005

Notes: The figure shows the filtered mean estimates of the discount factor shocks dt and the productivity shocks at ,in percentage terms, based on the nonlinear model (thick solid lines) with their 90 percent intervals (shaded areas)
and those based on the linear model (thin solid lines) and the piecewise linear model (dashed line). The yellow areas
indicate the period when the federal funds rate was constrained by the ZLB.

4

Concluding Remarks

This paper has estimated the natural rate of interest in a nonlinear New Keynesian model using
U.S. macroeconomic data and compared it with the one estimated with the model’s linear counterpart. We have found that the natural rate based on the nonlinear model is substantially higher
than that based on the linear model during the period when the nominal interest rate was boundedat zero. This difference is explained by a contractionary effect of the ZLB, which is omitted in thelinear model. Although the existence of the price and wage dispersion terms potentially affects the26estimate of the natural rate in the nonlinear setting, their effects are negligible.
Whereas the present paper employs an empirically richer DSGE model than the prototypical
New Keynesian model, existing studies, including Barsky, Justiniano, and Melosi (2014), Cúrdia,
Ferrero, Ng, and Tambalotti (2015), Edge, Kiley, and Laforte (2008), Del Negro, Giannone, Giannoni, and Tambalotti (2017), and Justiniano and Primiceri (2010) estimate the natural interest
rate using medium-scale DSGE models with capital accumulation in a linear setting. Our analysis
could be extended to exploit such a medium-scale model so that the estimated natural rate would
be comparable to the rates obtained in these studies. We conjecture that our results regarding
the higher estimate of the natural rate in a nonlinear setting would still hold, even if we extended
our model to a larger scale, because the main mechanism through which the ZLB can affect theidentification of the natural rate remains unchanged.

27

Appendix
A

Linearized Equilibrium Conditions and Observation Equations

Log-linearizing the detrended equilibrium conditions around the nonstochastic steady state, andrearranging the resulting equations, yields
ỹt =

γa
γ
γa − γ n(Et ỹt+1 + Et at+1 ) +(ỹt−1 − at ) −(R̃ − Et Π̃t+1 − d˜t ),γa + γγa + γγa + γ tw̃t = w̃t−1 − Π̃t + ιw Π̃t−1 − at + β(Et w̃t+1 − w̃t + Et Π̃t+1 − ιw Π̃t + Et at+1 )(1 − ξw ) (1 − ξw β) ˜1+η lt +(γa ỹt − γ ỹt−1 + γat ) − w
et ,ξw (1 + ηθw )γa + γỹt = ˜lt ,ιp(1 − ξp ) (1 − ξp β)β
Et Π̃t+1 +Π̃t−1 +(w̃t + z̃t ),1 + βιp
1 + βιpξp (1 + βιp )γỹt∗ =(ỹ ∗ − at ),γa (1 + η) − γη t−1hinR̃tn = φr R̃t−1+ (1 − φr ) φπ Π̃t + φy ỹt + φ∆y (ỹt − ỹt−1 + at ) + εr,t ,Π̃t =d˜t = ρd d˜t−1 + εd,t ,at = ρa at−1 + εa,t ,z̃t = ρz z̃t−1 + εz,t ,where the variables with ˜ represent percentage deviations from their steady-state values.
The observation equations are

100∆ log GDPt





ā





 

 
 
 

 100∆ log P GDPt   π̄  
 

 
+
=

 
 

F Ft

  r̄ + π̄  
 
 

¯l
100 log Htỹt − ỹt−1 + at
Π̃t
R̃tn
˜lt






,




where ā = 100 log γa , π̄ = 100 log Π̄, r̄ = 100 log R̄(= 100 log(γa /β)), and ¯l are, respectively, thesteady-state growth rate, the inflation rate, the real interest rate, and hours worked.

28

B

Nonlinear Solution Method

B.1

Recursive forms of the price and wage setting equations

After detrending, the equilibrium conditions (10) and (4) can be written in the following recursive
forms:
Sp,t
Pto
=
Pt
Fp,t
  −ιp #θpΠΠt
yt+1 λt+1
t+1
Sp,t = θp wt zt + ξp βd−1
Sp,t+1 ,
t Et
yt λt
Π̄
Π̄
"
  −ιp #θp −1ΠΠt
yt+1 λt+1
t+1
Fp,t = (θp − 1) + ξp βd−1
Fp,t+1 ,
t Et
yt λt
Π̄
Π̄
"



Wtn,o
Wtn

1+ηθw
=

Sw,t
Fw,t

  −ιw #(1+η)θw
ld,t+1 λt+1
Π
Πt
w,t+1
η
−1
exp(at+1 )Sw,t+1 ,Sw,t = θw ld,t
λ−1
t + ξw βdt Et
ld,t λtΠ̄Π̄"  −ιw #θw −1
ld,t+1 λt+1
Π
Πt
w,t+1
exp(at+1 )Fw,t = (θw − 1)wt + ξw βd−1
Fw,t+1 ,t Et
ld,t λtΠ̄Π̄"where Πw,t = Πt wt /wt−1 and ld,t = ∆p,t yt .

B.2

Solution algorithm

In what follows, we drop the time subscript and use −1 and 0 for previous- and next-period variables,
respectively. To solve for the policy functions on each grid point of the state space (S−1 , τ ), where
bn , ∆p,−1 , ∆w,−1 , y ∗ ]0 and τ = [d, a, z, εr ]0 , we follow an index-function
S−1 = [y−1 , Π−1 , w−1 , R−1−1
approach as in Aruoba, Cuba-Borda, and Schorfheide (2018), Gust, Herbst, López-Salido, andSmith (2017) and Nakata (2017).18 First, regime-specific expectation functions are defined as
18

See also Hirose and Sunakawa (2019) for details about the solution algorithm with an example of a prototypical

New Keynesian model with the ZLB.

29

follows:λ01
Φ(τ 0 |τ )dτ 0 ,eλ,s (S−1 , τ ) ≡ βd R0 ) Π0
γ
exp(a
0
a
τ


Z " 0  −ιp #θp 0 0 0 
y λ SpΠΠ
esp,s (S−1 , τ ) ≡ θp wz + ξp βd−1
Φ(τ 0 |τ )dτ 0 ,
yλ 
Π̄
Π̄
τ0


Z " 0  −ιp #θp −1 0 0 0
y
λ
F
Π
Π
p
Φ(τ 0 |τ )dτ 0 ,ef p,s (S−1 , τ ) ≡ θp − 1 + ξp βd−1


yλ
0
Π̄
Π̄
τ


 −ιw #(1+η)θw 0 0 0 
Z " 00l
λSΠΠw exp(a )wdesw,s (S−1 , τ ) ≡ θw ldη λ−1 + ξw βd−1
Φ(τ 0 |τ )dτ 0 ,


l
λ
0
Π̄
Π̄
d
τ
"

 −ιw #θw −1 0 0 0 
Z  0
0)
l
λ
F
Π
exp(a
Π
w
w
d
ef w,s (S−1 , τ ) ≡ (θw − 1)w + ξw βd−1
Φ(τ 0 |τ )dτ 0 ,


l
λ
0
Π̄
Π̄
d
τ
−1

n

Z 

where the index s ∈ {NZLB, ZLB} is associated with the interest-rate regime in which the notionalbn implied by its unconstrained policy function g bn
nominal interest rate R(S , τ ) is either
R ,NZLB −1
above or below the lower bound. Then, the expectation functions are constructed as weighted
averages of the regime-specific functions
ex (S−1 , τ ) = ex,NZLB (S−1 , τ )1{Rbn >1} + ex,ZLB (S−1 , τ )1{Rbn ≤1} ,where 1{D} is the indicator function that equals one if the condition D is true and zero otherwise.
We obtain the policy functions by a time iteration method, which takes the following steps.
(0)

1. Make an initial guess for the expectation functions es

=



(0)

(0)

(0)

(0)

(0)

eλ,s , esp,s , ef p,s , esw,s , ef w,sfors ∈ {NZLB, ZLB}.
2. For i = 1, 2, ... (i is an index for the number of iterations), taking as given the expectation(i−1)

functions previously obtained es, solve the relevant equations to obtain the policy functions


(i)
(i)
(i)
(i)
(i)
(i) (i)(i)(i)
gs = gΠ,s , g∆p ,s , gΠw ,s , g∆w ,s , gy,s , gw,s , gld ,s , g bn .
R ,s

(i)

(i)

3. Update the expectation functions es by interpolating the policy functions gs .
(i)

(i−1)

4. Repeat Steps 2-3 until es − esis small enough.
(i−1)

In Step 2, taking as given the values of ex,s (S−1,j , τm ) for x ∈ {λ, sp, f p, sw, f w} at each grid30

point indexed by (j, m) and each regime s ∈ {NZLB, ZLB}, we have1

1−θp θp −1


(i−1)
Πjms  −1
e
(S
,
τ
)
Π−1,j ιp

−1,j m −1  sp,s,= ξp + (1 − ξp ) (i−1)
Π̄Π̄
ef p,s (S−1,j , τm )


−θp

 

(i−1)
esp,s (S−1,j , τm ) Πjms θp Π−1,j −ιp∆p,jms = (1 − ξp ) (i−1)+ ξp∆p,−1,j ,Π̄Π̄
ef p,s (S−1,j , τm ) 1−θw  θw1−1



1+ηθw
(i−1)
Πw,jms exp(a)  −1
−1  esw,s (S−1,j , τm ) = ξw + (1 − ξw ) (i−1)
Π̄
ef w,s (S−1,j , τm )

 −θw
1+ηθw
(i−1)
e
(S
,
τ
)
sw,s
−1,j
m




∆w,jms = (1 − ξw )(i−1)

ef w,s (S−1,j , τm )(i−1)

yjms = eλ,s (S−1,j , τm )−1 ++ ξw

Πw,jms exp(am )
Π̄



Π−1,j
Π̄

θw 

ιw

Π−1,j
Π̄

,

−ιw
∆w,−1,j ,γyj,−1γa exp(am )wjms = w−1 Πw,jms /Πjms ,ld,jms = ∆p,jms yjms ,
"
φπ 

φy #1−φr
y
exp(a
)
Π
jms
m
jms
n
n
φ
r
bjms = (Rb−1,j )RR̄Π̄
exp(εr,m ).
y−1
Π̄


bn
Then, we can evaluate Πjms , ∆p,jms , Πw,jms , ∆w,jms , yjms , wjms , ld,jms , R
jms at each grid point
(i)

bn }
(j, m) and each regime s and the policy functions gx,s (S−1 , τ ; θ) for x = {Π, ∆p , Πw , ∆w , y, w, ld , R
parameterized by a vector of polynomial coefficients θ for computing the values off the grid points.
Note that this procedure does not rely on any numerical optimization routines to solve the nonlinear
equations.
In Step 3, the expectation functions are updated by
)
Z (
(i−1)
0 ; θ)
e
(S,
τ
1
(i)
n
λ
Φ(τ 0 |τ )dτ 0 ,eλ,s (Sj,−1 , τm ) = βd−1m Rjms0(i)
τ 0 γa exp(a ) g (S, τ 0 ; θ) Π!
−ιp #θp
Z " (i)0 ; θ) 
g
(S,
τ
Π
jms
−1
Π
e(i)
sp,s (Sj,−1 , τm ) = θp wjms zm + ξp βdm

0
Π̄
Π̄
τ
)
(i)
(i−1)
(i−1)
gy (S, τ 0 ; θ)eλ (S, τ 0 ; θ)esp (S, τ 0 ; θ)×Φ(τ 0 |τ )dτ 0 ,(i−1)
yjms eλ (Sj,−1 , τm ; θ)

31


!
−ιp #θp −1
Z " (i)0 ; θ) 
g
(S,
τ
Π
jms
(i)
−1
Π
ef p,s (Sj,−1 , τm ) = θp − 1 + ξp βdm

0
Π̄
Π̄
τ

(i)
(i−1)
(i−1)
0
0
0
gy (S, τ ; θ)eλ (S, τ ; θ)ef p (S, τ ; θ)
×Φ(τ 0 |τ )dτ 0 ,(i−1)
yjms eλ (Sj,−1 , τm ; θ)!
−ιp #θp
Z " (i)0 ; θ) exp(a0 ) 
g
(S,
τ
Π
jms
η
Πw
−1
−1
e(i)
sw,s (Sj,−1 , τm ) = θw ld,jms λjms + ξw βdm
Π̄
Π̄
τ0

(i)
(i−1)
(i−1)
gld (S, τ 0 ; θ)eλ (S, τ 0 ; θ)esw (S, τ 0 ; θ)
×Φ(τ 0 |τ )dτ 0 ,(i−1)
ld,jms eλ (Sj,−1 , τm ; θ)

!
 #θp
Z " (i)gΠw (S, τ 0 ; θ) exp(a0 )Πjms −ιp(i)−1
ef w,s (Sj,−1 , τm ) = (θw − 1)wjms + ξw βdm
Π̄
Π̄
τ0

(i)
(i−1)
(i−1)
gl (S, τ 0 ; θ)eλ (S, τ 0 ; θ)ef w (S, τ 0 ; θ)
× dΦ(τ 0 |τ )dτ 0 ,(i−1)
ld,jms eλ (Sj,−1 , τm ; θ)(i)n0where the values (Πjms , Πw
jms , yjms , wjms , Rjms ) and the policy functions gx (S, τ ; θ) evaluatedat the next period’s state (S, τ 0 ) are obtained in the previous step. Note that we interpolate(i)bn } off the grid points (or equivalently e(i−1)
gx (S, τ 0 ; θ) for x ∈ {Π, Πw , y, ld , R(S, τ 0 ; θ) for x ∈x{λ, sp, f p, sw, f w}) by piecewise linear interpolation. Numerical integrals are computed with regard
to τ 0 .
For interpolating the policy functions at each regime, we set the number of grids as 5 foreach endogenous state variable, 15 for the discount factor shock dt , and 3 for each of the othershocks. Thus we have 56 × 15 × 33 = 6, 328, 125 grid points in total.19 The integrals over τ 0 are
approximated by the Gauss–Hermite quadrature formula with three nodes for each shock. To deal
with such a large number of grid points and quadrature nodes, we use a supercomputer system torun programming codes with parallel computing.

19

We can drop the natural output from the endogenous state variables in the time iteration as it is relevant only

for the natural rate.

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