The Natural Rate of Interest in a
Non-linear DSGE Model∗
Yasuo Hirosea and Takeki Sunakawabab

Keio University
Hitotsubashi University

This paper investigates how and to what extent nonlinearities, including the zero lower bound on the nominalinterest rate, aﬀect the estimate of the U.S. natural rate ofinterest in a dynamic stochastic general equilibrium model.
The estimated natural rate in a non-linear model is substantially diﬀerent from that in its linear counterpart after the
global ﬁnancial crisis because of the zero lower bound. Other
non-linearities such as price and wage dispersion, from whicha linear model abstracts, play a negligible role in identifying
the natural rate.
JEL Codes: C32, E31, E43, E52.

∗
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 participants at 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 of Tokyo, 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 and Finance, ISM Cooperative Research Program (2021-ISMCRP-0003), andFondation France-Japon de l’EHESS. Author e-mails: yhirose@econ.keio.ac.jp
and takeki.sunakawa@gmail.com.

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Introduction

The natural rate of interest—the equilibrium real interest rate thatyields price stability (Wicksell 1898)—has been a key concept formonetary 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 (Galı́2008; Woodford 2003). The level of the natural interest rate in thisframework is a useful indicator for policymakers because it is abenchmark as to whether policy is too tight or too loose from awelfare perspective.1 However, the natural rate is unobservable andmust 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 et al. (2015), Del Negro et al. (2017), Edge, Kiley,and Laforte (2008), Justiniano and Primiceri (2010), and Neiss andNelson (2003).
This paper estimates the natural rate of interest in the U.S. economy using a non-linear New Keynesian DSGE model with a zero
lower bound (ZLB) constraint on the nominal interest rate andexamines how and to what extent non-linearities aﬀect the estimates of the natural rate and its 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

1

Closing the gap between the actual real interest rate and the natural rate
is not necessarily optimal in the economy where “divine coincidence” (Blanchard and Galı́ 2007) does not hold. However, Barsky, Justiniano, and Melosi(2014) demonstrate that, even in such a circumstance, a central bank would be
able to stabilize both inﬂation and the welfare-relevant output gap to a considerable degree by tracking the natural rate using an estimated New Keynesianmodel.
2Another 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 andTurunen (2016), and Williams (2015).

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estimate the natural rate in a fully non-linear and stochastic settingthat incorporates the ZLB.3
Our analysis is motivated by the following two strands of literature. First, Fernández-Villaverde and Rubio-Ramı́rez (2005) andFernández-Villaverde, Rubio-Ramı́rez, and Santos (2006) demonstrate that the level of likelihood and parameter estimates basedon a linearized model can be signiﬁcantly diﬀerent from those basedon the original non-linear model. The same may be true for the estimation of unobservable state variables, including the natural rate.
If substantial diﬀerences in the estimates of the natural rate arise
between linear and non-linear models, it will cast doubt on the common practice in which the natural rate is estimated based on a linear model. Over- or 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 theextremely low interest rate period that followed has led researchers
to conduct empirical analyses based on non-linear DSGE models inorder to take the ZLB into consideration. For instance, Gust et al.
(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 model in a nonlinear setting using U.S. macroeconomic time series. Plante, Richter,
and Throckmorton (2018) and Richter and Throckmorton (2016a,
2016b) estimate a non-linear version of a prototypical New Keynesian model with the ZLB for the U.S. economy, and Iiboshi, Shintani,and Ueda (2022) 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 Keynesian framework 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 non-linear ﬁltering technique. The present
paper contributes to this strand of the literature by focusing on theestimation 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
3
A contemporaneous paper by Iiboshi, Shintani, and Ueda (2022), which
evolved independently from our work, estimates a non-linear small-scale New
Keynesian model for Japan and extracts the sequence of the natural rate.

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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 thisestimation strategy, Atkinson, Richter, and Throckmorton (2020)
demonstrate that piecewise linear and fully non-linear approaches
give rise to similar parameter estimates. Thus, the piecewise linear
approach enables us not only to avoid a computational burden thatwould increase exponentially in the estimation of a fully non-linear
model, but also to obtain reliable estimates of parameters.
Next, given the estimated parameters, we solve the model ina fully non-linear and stochastic setting with the ZLB and apply
a non-linear ﬁlter to extract the sequence of the natural interest
rate. The literature (e.g., Boneva, Braun, and Waki 2016; FernándezVillaverde et al. 2015; Gavin et al. 2015; Gust et al. 2017; Nakata
2016, 2017; Ngo 2014; and Richter and Throckmorton 2016a) has
emphasized the importance of considering non-linearities and related
uncertainty eﬀects in assessing the quantitative implications of New
Keynesian models that include the ZLB. The natural rate estimatedin the present paper takes these important features into account.
Moreover, our analysis is based on an empirically richer DSGE model
than the prototypical New Keynesian model. The model features
habit persistence in consumption preferences, price and wage stickiness, backward-looking components in price and wage settings, andmonetary policy smoothing.
The main results are summarized as follows. Comparing the estimated natural interest rate based on the non-linear model with thatbased on the linear counterpart, we ﬁnd that the former is higher
than the latter by up to 1.7 percent after the global ﬁnancial crisis,
when the nominal interest rate was constrained by the ZLB. Thisdiﬀerence is non-negligible; if we relied on the linear model, we would
over-estimate the tightness of monetary policy due to the ZLB toa substantial degree. The diﬀerence is ascribed to a contractionaryeﬀect arising from the ZLB, which is considered only in the nonlinear model. Although such a contractionary eﬀect lowers expected
output and inﬂation, actual output and inﬂation are pegged to thecorresponding observables in the ﬁltering process. Then, shocks toaggregate demand must be identiﬁed upward in order to satisfy thehousehold’s intertemporal Euler equation that equates the marginalVol. 19 No. 1

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utility of consumption today and the expected discounted one in the
future. As a consequence, the estimated natural rate increases in the
non-linear setting. Although price and wage dispersion potentiallyaﬀect the identiﬁcation of shocks and the estimate of the naturalrate, 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 non-linear model.
In this regard, we demonstrate that the piecewise linear model canwell replicate the natural interest rate based on the non-linear modelin the aftermath of the global ﬁnancial crisis, although it can slightly
under-estimate 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 are households, perfectly competitive ﬁnal-good ﬁrms, monopolistically competitive intermediategood ﬁrms, and a central bank that faces the ZLB constraint on
the nominal interest rate. To ensure a better 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
4Regarding modeling choice, we consider nominal wage rigidities as in Erceg,
Henderson, and Levin (2000) as well as price rigidities to incorporate additional
non-linearities through a wage dispersion term and its variability. A medium-scalemodel with capital accumulation could take more non-linearities into account, 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 themost commonly used in the literature that estimates the natural rate based on
DSGE models. Cúrdia et al. (2015) estimate the eﬃcient interest rate, which is

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To obtain the estimates of the natural interest rate, we implement
a two-step approach. First, we estimate a piecewise linear versionof the model, in which the ZLB constraint is imposed but all theequilibrium conditions are linearized using the OccBin toolbox anda inversion ﬁlter. Next, given the estimated parameters, we solve
the model in a fully non-linear and stochastic setting with the ZLB
using a projection method and apply a particle ﬁlter to extract thesequence of the natural rate.

2.1

The Model
2.1.1

Households

Each household h ∈ [0, 1] consumes ﬁnal goods Ch,t , supplies labor1lh,t = 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

∞

t=0

β

t

 t


−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] isthe degree of external habit persistence in consumption preferences(Ct−1 is the aggregate consumption in period t − 1), η ≥ 0 is theinverse of the labor supply elasticity, Pt is the price of ﬁnal goods,nWh,tis the nominal wage for household h, Rtn is the gross nominal interest rate, and Th,t is the sum of a lump-sum public transfer
and proﬁts received from ﬁrms. Following Christiano, Eichenbaum,and Rebelo (2011), a shock to the discount factor dt aﬀects theweight 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 todeﬁned as the real interest rate under ﬂexible prices and perfect competition withzero markups.

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aggregate demand. The log of the 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
become1,
Ct − γCt−1
Λt+1
β,Λt = Rt Et
dt
Πt+1
Λt =

(2)
(3)

where Λt is the marginal utility of consumption, and Πt = Pt /Pt−1
denotes gross inﬂation.
2.1.2

Wage Setting

A labor packer collects diﬀerentiated labor {lf,h,t } from each household h and resells a labor package augmented by a constant elasticity
θw /(θw −1) 1 (θw −1)/θw
of substitution (CES) aggregator lf,t = 0 lf,h,t
dh
to intermediate-good ﬁrms indexed by f , where θw > 1 represents
the elasticity of substitution among labor varieties. Given the nomn
inal wage for each household Wh,t, cost minimization yields a set of−θwnlabor demand schedules lf,h,t = Wh,t
/Wtn
lf,t and the aggre1/(1−θ)
w
1n 1−θwgate wage index Wtn = 0 Wh,tdh.
Given the demand for labor by the labor packers, labor unions
representing each household h set nominal wages on a staggeredbasis, as in Erceg, Henderson, and Levin (2000). In each period, afraction 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 of past
inﬂation Πt−1 and steady-state inﬂation Π̄. The labor unions that

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reoptimize their nominal wages in the current period then maximize
expected utility as follows:
 j
−1
∞


j j
ξw
β
dk
Et
j=0

k=1


1+η
j
n 
l

γaj Wh,t
h,t+j
w
,
×
Πιt+k−1Π̄1−ιw Λh,t+j lh,t+j −
Pt+j
1+η


k=1

subject to the labor demand
−θw

j
n 

γaj Wh,tιw
lf,h,t+j =Πt+k−1 Π̄1−ιw
lf,t+j ,
n
Wt+j
k=1

1

where lh,t = 0 lf,h,t df is the amount of labor supplied by eachhousehold h, and ιw ∈ [0, 1) is the weight of wage indexation to past
inﬂation relative to steady-state inﬂation. The ﬁrst-order condition
for the reoptimized wage Wtn,o is given by


Wtn,o
Wtn

1+ηθw

θw
θw − 1−1 
jn −(1+η)θw

j
1+η
ιw
j j j1−ιw γa Wt
ld,t+jEt ∞j=0 ξw β
k=1 dkk=1 Πt+k−1 Π̄
Wn

×
Et

=

∞

j
j=0 ξw βjj

k=1 dk−1 

t+jjn
1−ιw γa Wt
n
Wt+j

j
ιw
k=1 Πt+k−1 Π̄

,
1−θw
n
Wt+j
Λt+j Pt+j
ld,t+j(4)

where ld,t =1

lf,t df is the total labor demand. The aggregate nom1/(1−θw ) 1 n 1−θwdh
can be written as
inal wage index Wtn = 0 Wh,t0nw
Π̄1−ιw γa Wt−1
Wtn = (1 − ξw ) (Wtn,o )1−θw + ξw Πιt−1

1−θw 1−θ1 w

.

(5)
2.1.3

Firms

The representative ﬁnal-good ﬁrm produces output Yt under perfect competition by choosing a combination of intermediate inputs

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The Natural Rate of Interest309

1{Yf,t } so as to maximize proﬁt Pt Yt − 0 Pf,t Yf,t df , subject to a CES
θp /(θp −1)
 1 (θ −1)/θp
df, where Pf,tproduction technology Yt = 0 Yf,tp
is the price of intermediate good f and θp > 1 denotes the elasticity
of substitution among the variety of intermediate goods. The ﬁrstorder condition for proﬁt maximization yields the ﬁnal-good ﬁrm’s−θdemand for each intermediate good Yf,t = (Pf,t /Pt ) p Yt and the
1/(1−θ
) 1 1−θpaggregate 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 non-stationary stochastic process,log 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 with mean zero and standard deviation σa .
Assuming the existence of a shock to real marginal cost zt , which isinterpreted 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-good ﬁrms.

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where ρz ∈ [0, 1) is an autoregressive coeﬃcient and εz,t is a normally
distributed innovation with mean 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 ona 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:Et∞ξpj β j

 j


j=0

−1
dk

k=1

Λt+j
Λt


j

Pf,t  ιp1−ιpΠt+k−1 Π̄− M Ct+j Yf,t+j ,
×
Pt+j


k=1

subject to the ﬁnal-good ﬁrm’s demand
Pf,t  ιp
Yf,t+j =Πt+k−1 Π̄1−ιp
Pt+j
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
Pto
θ
=
Pt
θ−1


−1


−θp

j
Λt+j
Πt+k−1 ιp
j j jΠ̄ξβdM Ct+j Yt+j
Et ∞
k
p
j=0
k=1
k=1
Λt
Πt+k
Π̄

.
×


−1


1−θp

Λt+j
Πt+k−1 ιpjj j j
Π̄
Et ∞
ξ
β
d
Y
t+j
p
k
j=0
k=1
k=1
Λt
Πt+k
Π̄
(10)

The ﬁnal-good’s price Pt =1−θp

Pt = (1 − ξp ) (Pto )

1

1−θ

p
P
df
0 f,t

ι

1/(1−θp )p+ ξp Πt−1Π̄1−ιp Pt−1

can be written as11−θp 1−θ
p

.

(11)

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311

Market Clearing Conditions

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

whereas the labor market clearing condition leads tolt =Δp,t Δw,t Yt
,
At

(13)

11
where lt = 0 0 lf,h,t df dh is the aggregate labor input, Δp,t =1−θ(Pf,t /Pt ) p df is price dispersion across the intermediate-good
0
−θw
1
n
/Wtn
dh is wage dispersion acrossﬁrms, and Δw,t = 0 Wh,t
the labor unions. Equation (13) can be rewritten in terms of ld,t =1
l df as
0 f,t
ld,t =

Δp,t Yt.
At(14)

In the present model, the price and wage dispersion evolve according to−ιp θp
−θp θp Πt
Πt−1
Pto
Δp,t = (1 − ξp )+ ξpΔp,t−1 , (15)PtΠ̄Π̄ n,o −θw
Wt
Δw,t = (1 − ξw )Wtn
−ιw θwθw 
Πt W tΠt−1
Δw,t−1 .
(16)+ ξw
Π̄γa Wt−1
Π̄


2.1.5

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 and pricenequilibrium is obtained with ξw = ξp = 0, Wh,t
= Wtn , Pf,t = Pt ,

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and zt = 1 for all h, f , and t in the model above and is characterized
by the following equations:∗Yt∗ − γYt−1

dt
Rt∗ =
β







Yt∗
At

η
= μAt ,∗Y ∗ − γYt−1
Et t∗
Yt+1 − γYt∗

(17)

−1
,

(18)

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 natural rate of interest Rt∗ is determined by (18), given the sequencesof natural output Yt∗ and the discount factor shock dt .
2.1.6

Central Bank

A monetary policy rule is speciﬁed as
n , 1],
Rtn = max[R
t

(19)

where

n )φr R̄Π̄
tn = (R
R
t−1



Πt
Π̄

φπ 

Yt
At

φy 

Yt
γa Yt−1

φΔy 1−φrexp(εr,t ).
(20)

tn denotes the notional nominal interest rate that the central bankRwould set according to a Taylor (1993) type monetary policy rule in
the absence of the ZLB constraint, where R̄ is the steady-state grossreal interest rate, φr ∈ [0, 1) is the policy-smoothing parameter, andφπ ≥ 0, φy ≥ 0, and φΔy ≥ 0 are the degrees of the interest rate
policy response to inﬂation, detrended output, and output growth,
respectively. εr,t is a monetary policy shock, which is normally distributed with mean zero and standard deviation σr . The max function in (19) constrains the nominal interest rate to be greater thann > 1, the ZLB constraint is not imposed, i.e.,or equal to zero. If Rtn . If Rn ≤ 1, the ZLB is binding, i.e., Rn = 1.
Rtn = Rttt

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2.1.7

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313

Equilibrium

An equilibrium is given by the sequences {Yt , Ct , Λt , Wt , Wtn , Wtn,o, lt ,n , dt , At , at , zt }∞ satld,t , M Ct , Πt , Pt , Pto , Δp,t , Δw,t , Yt∗ , Rt∗ , Rtn , Rt
t=0
isfying the equilibrium conditions (1)–(20) and two deﬁnitional equations, Wt = Wtn /Pt and Πt = Pt /Pt−1 .
Because total factor productivity At is non-stationary, 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 non-stochastic
steady state for the detrended variables.

2.2

Estimation of Parameters

To parameterize the model, we estimate a piecewise linear versionof the model, in which the ZLB constraint is imposed but all theequilibrium conditions are linearized. More speciﬁcally, such a piecewise linear model is solved using the OccBin toolbox developed
by Guerrieri and Iacoviello (2015), and the likelihood function isevaluated with the inversion ﬁlter following Guerrieri and Iacoviello(2017). While it is computationally very intensive to estimate a fully
non-linear model using 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 andfully non-linear approaches result in similar parameter estimates.
The model is ﬁtted to four U.S. quarterly time series: theper capita real GDP (gross domestic product) growth rate
(100Δ log GDPt ), the inﬂation rate of the GDP implicit price deﬂator (100Δ log P GDPt ), the federal funds rate (F Ft ), and the logof hours worked (100 log Ht ).7 Following Wolters (2018), the dataon hours worked are adjusted for low-frequency movements due tosectoral and demographic changes so that the data are consistent
with the model. The sample period is from 1987:Q3 to 2019:Q4.
The beginning of the sample period is set at the time when Alan
Greenspan became the Chairman of the Board of Governors of the7

The series of hours worked is demeaned.

314International Journal of Central Banking

March 2023

Federal Reserve System, because thereafter, the style of the Fed’s
policy conduct seems consistent and 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 priordistributions of parameters are presented in the second to fourthcolumns of Table 1. For most of the parameters, each prior mean isset 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 than that in Smets andWouters (2007) because a higher value of the estimated φr would
lead to a non-convergence problem in the next step for solving our
non-linear model.8 As for the steady-state values of output growth,
inﬂation, and real interest rates and hours worked (ā, π̄, r̄, ¯l), thepriors are centered at the sample mean. The prior mean of theAR(1) coeﬃcient for the discount factor shock ρd is 0.75, whereas
those for the productivity and cost-push shocks (ρa , ρz ) are 0.5. For
the standard deviations of the shocks (100σd , 100σa , 100σz , 100σr ),we assign inverse-gamma distributions with a mean of 0.5 and astandard deviation of 2.0.
Following Guerrieri and Iacoviello (2017), 50,000 posterior draws
are generated using the random-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 columns of Table 1, labeled “Piecewise Linear.” For thesake of comparison, we estimate a linear version of the model thatomits the ZLB constraint, and the results are shown in the last twocolumns of Table 1. The parameter estimates are very similar toeach other, in contrast to the ﬁnding of Hirose and Inoue (2016),
who conduct a Monte Carlo analysis and demonstrate that omitting
the ZLB in estimation causes biased estimates of parameters.
In the subsequent analysis, the parameters are ﬁxed at the posterior mean estimates of the piecewise linear model.

8

For the same reason, relatively tight priors are used for the parameters that
determine the persistency of endogenous variables in the model.

0.500
2.000
0.500
0.500
0.500
0.500
1.500
0.500
0.125
0.500
0.356
0.529
0.276
0.000
0.750
0.500
0.500
0.500
0.500
0.500
0.500

Beta
Gamma
Beta
Beta
Beta
Beta
Gamma
Gamma
Gamma
Beta
NormalNormalGamma
Normal
Beta
Beta
Beta
Inv. Gamma
Inv. Gamma
Inv. Gamma
Inv. Gamma

γ
η
ξw
ιw
ξp
ιp
φπ
φy
φΔy
φr
ā
π̄
r̄
¯l
ρd
ρa
ρz
100σd
100σa
100σz
100σr

0.050
0.250
0.050
0.050
0.050
0.050
0.250
0.100
0.050
0.050
0.100
0.100
0.100
0.100
0.050
0.050
0.050
2.000
2.000
2.000
2.000

S.D.
0.664
1.763
0.681
0.488
0.868
0.451
1.575
0.168
0.238
0.750
0.348
0.576
0.215
–0.058
0.831
0.418
0.573
0.404
0.558
7.612
0.119

Mean
[0.609, 0.716][1.372, 2.212][0.561, 0.756][0.407, 0.568][0.817, 0.913][0.360, 0.545][1.353, 1.825][0.143, 0.194][0.134, 0.350][0.706, 0.789][0.250, 0.452][0.489, 0.670][0.146, 0.293][–0.146, 0.037][0.794, 0.862][0.352, 0.487][0.481, 0.663][0.311, 0.523][0.502, 0.618][3.613, 15.18][0.104, 0.136]

90% Interval

Piecewise Linear

0.654
1.824
0.687
0.492
0.841
0.490
1.608
0.141
0.265
0.803
0.361
0.536
0.240
0.016
0.851
0.412
0.610
0.359
0.559
5.256
0.107

[0.597, 0.708][1.438, 2.249][0.598, 0.752][0.409, 0.576][0.792, 0.887][0.404, 0.579][1.361, 1.876][0.120, 0.164][0.155, 0.381][0.770, 0.836][0.267, 0.453][0.443, 0.631][0.162, 0.322][–0.018, 0.058][0.819, 0.881][0.345, 0.477][0.525, 0.697][0.274, 0.466][0.503, 0.622][2.870, 9.135][0.096, 0.121]

90% Interval

Linear (No ZLB)
Mean

Posterior

The Natural Rate of Interest

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

Mean

Distribution

Parameter

Prior

Table 1. Prior and Posterior Distributions of Parameters
Vol. 19 No. 1
315

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2.3

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Non-linear Solution and Filtering

Given the posterior mean estimates for the piecewise linear modelobtained above, the model is solved in a fully non-linear 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 , the notional nominal interest raten , price dispersion Δp,t−1 , wage dispersion Δw,t−1 , and naturalR
t−1∗
output yt−1
) and four exogenous shocks (the discount factor shock
dt , the productivity shock at , the cost-push shock zt , and the monetary policy shock εr,t ). The policy functions satisfying the detrended
equilibrium conditions can be written as
St = g(St−1 , τt ),n , Δp,t−1 , Δw,t−1 , y ∗ ] and τt =where St−1 = [yt−1 , Πt−1 , wt−1 , R
t−1
t−1

[dt , at , zt , εr,t ] .
To compute the policy functions, we employ a projection methodwith an index function approach as in Aruoba, Cuba-Borda, andSchorfheide (2018), Gust et al. (2017), and Nakata (2017). For interpolating the policy functions within a time iteration algorithm, we
adapt a standard linear interpolation to reduce approximation errorsthat tend to be large particularly when the ZLB binds due to large
negative shocks to the economy.9 The details of the solution method
are described in Appendix B.
According to an artiﬁcial sample of 40,000 periods simulated
from the non-linear solution of the model, 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 resultsin the previous studies that employ non-linear New Keynesian models. Fernández-Villaverde et al. (2015) simulate a small-scale model
calibrated for the U.S. economy and show that the economy spends
5.5 percent of quarters at the ZLB and that the average duration
at the ZLB is 2.1 quarters. Gust et al. (2017) estimate a mediumscale model in a non-linear setting using U.S. data from 1983:Q1 to
2014:Q1, and the simulation of their estimated model demonstrates
9

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

Vol. 19 No. 1

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that the economy is at the ZLB for about 4 percent of quarters on
average and that the average duration of the ZLB spells is just over
3.5 quarters.
We apply a particle ﬁlter as in Fernández-Villaverde and RubioRamı́rez (2007) to extract the sequence of the state variables andthen compute the estimates of the natural interest rate.10 The data
used for ﬁltering are 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 1
percent of the standard deviations of the data so that we can reduce
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 number of particles
delivered almost the same results as those presented below.
3.

Results

This section presents the estimate of the natural interest rate basedon the non-linear model and compares 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 rateaccurately.

3.1

Estimated Natural Rate of Interest

In Figure 1, the thick solid line shows the ﬁltered mean estimateof the natural rate of interest on an annualized basis, based on the
non-linear 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 endof 1980s and the beginning of 2000, then fell to about −10 percent in
10

For a textbook treatment of a particle ﬁlter, see Fernández-Villaverde, RubioRamı́rez, and Schorfheide (2016) or Herbst and Schorfheide (2015).

318International Journal of Central Banking

March 2023

Figure 1. Natural Rate of Interest

Note: The ﬁgure shows the ﬁltered mean estimate of the natural interest rate,
in annualized percentage terms, based on the non-linear 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. (For ﬁgures in color, see the online version of thepaper at http://www.ijcb.org.)

the aftermath of the global ﬁnancial crisis, and thereafter increased
to slightly positive values toward the end of the sample period.11
The overall cyclical movements and variability of the natural rate
are similar to those estimated by Barsky, Justiniano, and Melosi(2014), who employ a medium-scale New Keynesian DSGE modelwith capital accumulation.
In the present framework, the natural interest rate is driven by
structural shocks and hence exhibits short-run ﬂuctuations. To makeit comparable to the estimates of the long-run natural rate based ona semi-structural or reduced-form model as in Laubach and Williams
11
In an earlier version of this paper (Hirose and Sunakawa 2017), the estimatesof 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 are adjusted for sectoral and demographic
changes, and hence they are less volatile than the original data.

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319

Figure 2. Long-Run Natural Rate of Interest

Note: The ﬁgure shows the ﬁltered mean estimate of the 10-year natural interest
rate, in annualized percentage terms, based on the non-linear model (thick solidline) with its 90 percent interval (shaded area) and the ﬁltered estimate of thenatural interest rate based on the Laubach-Williams (2003) model (dashed line).
The yellow area indicates the period when the federal funds rate was constrainedby the ZLB.

(2003), we calculate the 10-year natural rate Rt∗10Y according to the
expectation hypothesis of the term structure of interest rates,
1 ∗=
Et log Rt+i.
40 i=039

log Rt∗10Y

Figure 2 compares the ﬁltered mean estimate of the 10-year natural rate based on the non-linear model (solid line) with the ﬁlteredestimate of the natural rate based on the Laubach-Williams (2003)
model (dashed line, henceforth, the LW estimate).12 Two notablediﬀerences emerge. First, cyclical patterns are somewhat diﬀerent
from each other. Second, our estimate of the 10-year natural rate is
12

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

320

International Journal of Central Banking

March 2023

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 the estimation
sample. The sample for the LW estimate begins in 1961:Q1, andhence the steady-state real interest rate in their model is estimatedto be higher than that in our model.
The primary objective of this paper is to examine how and towhat extent non-linearities, including the ZLB, aﬀect the estimatesof the natural interest rate. To this end, we estimate the natural rateusing a linear counterpart of the model, as in the previous studies,and compare it with the one obtained above. The thin solid line in
Figure 1 shows the ﬁltered estimate of the natural interest rate basedon the linearized version of the model with the same parameters anddata set as used in the non-linear case.13 The ﬁgure indicates thatthe natural rate based on the linear model is mostly very similar
to that based on the non-linear 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 is measured 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 toabout 10 percent. More precisely, while the gap in the non-linear
model increased to 9.5 percent in 2010:Q1, 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
13
In the linear setting, the model is solved by the standard linear solution
method and the ﬁltered estimates of model variables are computed using the
Kalman ﬁlter with zero measurement errors. As a preliminary analysis, we applieda particle ﬁlter to the linear model and obtained a very similar estimate of thenatural rate. Thus, we conﬁrmed that the diﬀerence in the ﬁltering methods withand without measurement errors is not the major source of the diﬀerence in theestimated natural rates between the linear and non-linear models.

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Figure 3. Interest Rate Gap

Note: The ﬁgure shows the real interest rate gap, in annualized percentage
terms, based on the non-linear 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.

more in response to the severe economic downturn, and hence thepolicy stance was evaluated to be too tight. In this context, if we
relied on the linear model, the estimated natural rate and the resulting interest rate gap would over-estimate 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 estimatesof 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, theproductivity shock at ). The natural rate Rt∗ can be traced out from

−1
∗
∗
Equation (18), i.e., Rt∗ = dt /β Et (Yt∗ − γYt−1)/(Yt+1− γYt∗ ) ,given the sequences of natural output Yt∗ and the discount factor

322International Journal of Central Banking

March 2023

shock dt . Thus, the natural rate is pinned down by identifying thetwo shocks, at and dt .
In the linear model, the productivity shock at is explicitly identiﬁed by the data on output and hours worked because detrending and log-linearizing the labor market clearing condition (13)yields ỹt = ˜lt and because the associated observation equations are
100Δ log GDPt = ā + ỹt − ỹt−1 + at and 100 log Ht = ¯l + ˜lt , whereā and ¯l are the steady-state growth rate and hours worked, respectively, and the variables with ˜ represent percentage deviations fromtheir steady-state values. In the non-linear model, however, Equation (13) contains the price and wage dispersion, Δp,t and Δw,t , andcan be written as yt = lt /(Δp,t Δw,t ) in detrended terms. These dispersion terms ﬂuctuate so that Δp,t ≥ 1 and Δw,t ≥ 1, as the priceand wage deviate from the steady state. Thus, yt becomes lower thanin 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 non-linear case. From Equations (17)and (18), higher productivity raises natural output and results in
the higher estimate of the natural rate.
Identiﬁcation of the discount factor shock dt is more complicatedand inﬂuenced by the whole structure of the model. However, taking into account the ﬁnding that the two estimates of the naturalinterest rate diﬀer from each other during the period when the nominal interest rate was bounded at zero, the existence of the ZLB,
from which the linear model abstracts, possibly aﬀects the identiﬁcation of dt in the non-linear model. The literature has establishedthat the ZLB has a contractionary eﬀect on the economy not only
when the nominal interest is already binding at zero, but also when
uncertainty exists about whether the ZLB will bind in the future.14
Although such a contractionary eﬀect lowers expected output andinﬂation, the ﬁltering procedure pegs contemporaneous output andinﬂation to the corresponding observables, which are the same inthe linear and non-linear cases. Then, in the non-linear case, thediscount factor shock dt must increase to satisfy the consumption
Euler equation that equalize the marginal utility of consumption

14

Hills, Nakata, and Schmidt (2016) quantify such an uncertainty eﬀect oninﬂation in the face of the interest rate lower bound.

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Figure 4. Estimated Shocks

Note: The ﬁgure shows the ﬁltered mean estimates of the discount factor shocks
dt and the productivity shocks at , in percentage terms, based on the non-linear
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.

today with the expected discounted one in the future. As a result,the estimated natural rate becomes higher.
To quantify the diﬀerences in the sequences of identiﬁed shocks,
Figure 4 shows the ﬁltered mean estimates of the discount factor
shocks dt and the productivity shocks at , in percentage terms, basedon the non-linear 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 non-linear model. The sequence of dt
identiﬁed in the non-linear model is remarkably diﬀerent from thatin the linear model after the global ﬁnancial crisis. In particular,
the diﬀerence is pronounced in 2009–10, when the notional nominal
n sharply fell below zero, as shown in Figure 5. Thus,
interest rate Rtwhether the ZLB constraint is imposed or not substantially aﬀects
the identiﬁcation of dt .

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Figure 5. Notional Rate

Note: 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 was constrainedby the ZLB.

On the other hand, the movements of the productivity shocks
at are very similar between the two estimates, although temporary
deviations are found occasionally. The ﬁnding of the small diﬀerence
in at implies that the price and wage dispersion terms, Δp,t andΔw,t , play a minor role in the non-linear model. Indeed, as shownin Figure 6, the estimates of Δp,t and Δw,t based on the non-linear
model ﬂuctuate little, i.e., 0.14 percent at most, even though they
exhibit cyclical movements over the sample period.

3.3

The Natural Rate of Interest Basedon the Piecewise Linear Model

The analysis thus far suggests that the existence of the ZLB constraint plays a crucial role in identifying the natural interest rate
in a non-linear setting, but that the role of price and wage dispersion is negligible. These ﬁndings tempt researchers to exploit apiecewise linear model, in which the ZLB constraint is imposed but

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325

Figure 6. Price and Wage Dispersion

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

all the equilibrium conditions are linearized, for estimating the natural rate measures, because such a model is easier to solve than afully non-linear model. While the literature argues that the solution
for this sort of piecewise linear models can give rise to an inaccurate assessment of the ZLB,15 Atkinson, Richter, and Throckmorton(2020) demonstrate that the piecewise linear and fully non-linear
approaches lead to similar results with regard to parameter estimates. In what follows, we examine whether the piecewise linear
model can be a useful substitute for estimating the natural rate.
As in the estimation of parameters described in Section 2.2, thepiecewise linear version of the model is solved using the OccBin
toolbox, and the ﬁltered estimates are computed using the inversion
15
See Boneva, Braun, and Waki (2016), Fernández-Villaverde et al. (2015),
Gavin et al. (2015), Gust et al. (2017), Nakata (2016, 2017), Ngo (2014), andRichter and Throckmorton (2016a).

326International Journal of Central Banking

March 2023

Figure 7. Natural Rate of Interest: Comparison
with Piecewise Linear Model

Note: The ﬁgure shows the ﬁltered mean estimate of the natural interest rate,
in annualized percentage terms, based on the non-linear 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 the federal funds rate was constrained by the ZLB.

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 non-linear (thick solid line) and linear (thin solidline) settings. The estimate based on the piecewise linear model coincides with that based on the linear model before the global ﬁnancial
crisis, but in the aftermath, it is very close to that based on the
fully non-linear model. Thus, the higher estimate of the natural ratein the non-linear model from 2009 to 2011 is well replicated by thepiecewise 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 accountuncertainty about the future. An uncertainty eﬀect enhances the16

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

Vol. 19 No. 1

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327

Figure 8. Estimated Shocks: Comparison
with Piecewise Linear Model

Note: The ﬁgure shows the ﬁltered mean estimates of the discount factor shocks
dt and the productivity shocks at , in percentage terms, based on the non-linear
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 (dashedlines). The yellow areas indicate the period when the federal funds rate wasconstrained by the ZLB.

contractionary eﬀect of the ZLB because agents expect that monetary policy will be unable to accommodate negative shocks tothe economy. As addressed in the previous subsection, the contractionary eﬀect of the ZLB leads to the higher estimate of thenatural rate in the non-linear model. This eﬀect is reduced in theabsence of uncertainty, and hence the piecewise linear model slightly
under-estimates 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
17
If the probability of binding at the ZLB were higher, the uncertainty eﬀectwould be larger, which would result in a larger diﬀerence in the natural rate
between the non-linear and piecewise linear models.

328International Journal of Central Banking

March 2023

linear model (dashed line) is substantially larger than that in thelinear model (thin solid line) for a few years after the global ﬁnancial crisis, whereas it is smaller to a lesser extent than that in the
non-linear model (thick solid line) until the end of the sample period.
The lower panel in Figure 8 compares the estimates of the productivity shock at . The piecewise linear model abstracts from priceand wage dispersion, as does the linear model, and hence the estimated productivity shocks in the two models exactly coincide witheach other.
In summary, a piecewise linear model incorporating the ZLB canbe a possible substitute for a fully non-linear model in estimating
the natural rate measures, although it can slightly under-estimate
the natural rate due to ignoring uncertainty at the ZLB.
4.

Concluding Remarks

This paper has estimated the natural rate of interest in a non-linear
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 non-linear model issubstantially higher than that based on the linear model during theperiod when the nominal interest rate was bounded at zero. Thisdiﬀerence is explained by a contractionary eﬀect of the ZLB, whichis omitted in the linear model. Although the existence of the priceand wage dispersion terms potentially aﬀects the estimate of thenatural rate in the non-linear 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 et al.
(2015), Del Negro et al. (2017), Edge, Kiley, and Laforte (2008),
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 sucha medium-scale model so that the estimated natural rate would becomparable to the rates obtained in these studies. We conjecturethat our results regarding the higher estimate of the natural rate ina non-linear setting would still hold, even if we extended our model toa larger scale, because the main mechanism through which the ZLB
can aﬀect the identiﬁcation of the natural rate remains unchanged.

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Appendix A. Linearized Equilibrium Conditions andObservation Equations
Log-linearizing the detrended equilibrium conditions around thenon-stochastic steady state, and rearranging the resulting equations,yieldsỹt =γaγ(Et ỹt+1 + Et at+1 ) +(ỹt−1 − at )γa + γγa + γγa − γ n(R̃ − Et Π̃t+1 − d˜t ),−γa + 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) ˜1η lt +(γa ỹt − γ ỹt−1 + γat ) − w+
t ,ξw (1 + ηθw )γa + γỹt = ˜lt ,βιp(1 − ξp ) (1 − ξp β)
Et Π̃t+1 +Π̃t−1 +(w̃t + z̃t ),1 + βιp
1 + βιpξp (1 + βιp )γỹt∗ =(ỹ ∗ − at ),γa (1 + η) − γη t−1

Π̃t =

n
R̃tn = φr R̃t−1+ (1 − φr ) φπ Π̃t + φy ỹt + φΔy (ỹt − ỹt−1 + at ) + εr,t ,d˜t = ρd d˜t−1 + εd,t ,at = ρa at−1 + εa,t ,z̃t = ρz z̃t−1 + εz,t ,where the variables with a tilde (∼) represent percentage deviations
from their steady-state values.
The observation equations are
⎤ ⎡
⎤ ⎡
⎤
ỹt − ỹt−1 + atā
100Δ log GDPt
⎢ 100Δ log P GDPt ⎥ ⎢ π̄ ⎥ ⎢
⎥
Π̃t
⎥+⎢
⎢
⎥=⎢
⎥,
n
⎣
⎦ ⎣ r̄ + π̄ ⎦ ⎣
⎦
F Ft
R̃t
¯
˜
100 log Ht
l
lt
⎡

330

International Journal of Central Banking

March 2023

where ā = 100 log γa , π̄ = 100 log Π̄, r̄ = 100 log R̄(= 100 log(γa /β)),and ¯l are, respectively, the steady-state growth rate, the inﬂation
rate, the real interest rate, and hours worked.
Appendix B. Non-linear 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:
Pto
Sp,t
=
Pt
Fp,t
Sp,t = θp wt zt + ξp βd−1
t Et
×



yt+1 λt+1
Sp,t+1 ,yt λt

Fp,t = (θp − 1) + ξp βd−1
t Et
×


Wtn,o
Wtn

1+ηθw
=

Πt+1
Π̄





Πt+1
Π̄

Πt
Π̄



−ιp θpΠtΠ̄

−ιp θp −1

yt+1 λt+1
Fp,t+1 ,
yt λt

Sw,t
Fw,t

η
−1
Sw,t = θw ld,t
λ−1
t + ξw βdt Et−ιw (1+η)θwΠw,t+1exp(at+1 )Π̄

ld,t+1 λt+1
Sw,t+1 ,ld,t λtΠw,t+1−1
Fw,t = (θw − 1)wt + ξw βdt Et
exp(at+1 )Π̄ −ιw θw −1
ld,t+1 λt+1
Πt
Fw,t+1 ,×
ld,t λt
Π̄
×

Πt
Π̄

where Πw,t = Πt wt /wt−1 and ld,t = Δp,t yt .

Vol. 19 No. 1

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331

B.2. Solution Algorithm
In what follows, we drop the time subscript and use −1 and  forprevious- and next-period variables, respectively. To solve for thepolicy functions on each grid point of the state space (S−1 , τ ), wheren , Δp,−1 , Δw,−1 , y ∗ ] and τ = [d, a, z, εr ] ,S−1 = [y−1 , Π−1 , w−1 , R−1−1
we follow an index-function approach as in Aruoba, Cuba-Borda,
and Schorfheide (2018), Gust et al. (2017), and Nakata (2017).18
First, regime-speciﬁc expectation functions are deﬁned as follows:

 
λ
1
−1 n
eλ,s (S−1 , τ ) ≡ βd R
Φ(τ  |τ )dτ  , ) Π
γ
exp(a

a
τ
esp,s (S−1 , τ )⎫⎧ ⎨   −ιp θp    ⎬y λ SpΠΠΦ(τ  |τ )dτ  ,≡ θp wz + ξp βd−1
yλ ⎭
Π̄
Π̄
τ⎩

ef p,s (S−1 , τ )⎫⎧ ⎨   −ιp θp −1    ⎬y λ FpΠΠΦ(τ  |τ )dτ  ,≡ θp − 1 + ξp βd−1
⎭
⎩
yλ
Π̄
Π̄

τ

esw,s (S−1 , τ ) ≡ θw ldη λ−1⎫⎧
 −ιw(1+η)θw    ⎬ ⎨ 

Π
Π
exp(a
)
l
λ
S
w
w
d
Φ(τ  |τ )dτ  ,+ ξw βd−1
⎭
⎩
l
λ
Π̄
Π̄

d
τ
ef w,s (S−1 , τ ) ≡ (θw − 1)w⎫⎧
 −ιw θw −1    ⎬ ⎨ 

Π
Π
exp(a
)
l
λ
F
w
w
d
Φ(τ  |τ )dτ  ,+ ξw βd−1
⎭
⎩
l
λ
Π̄
Π̄

d
τ
where the index s ∈ {NZLB, ZLB} is associated with the interest
n implied
rate regime in which the notional nominal interest rate R
by its unconstrained policy function gRn ,NZLB (S−1 , τ ) is either above18

See also Hirose and Sunakawa (2019) for details about the solution algorithmwith an example of a prototypical New Keynesian model with the ZLB.

332

International Journal of Central Banking

March 2023

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{Rn >1} + ex,ZLB (S−1 , τ )1{Rn ≤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, whichtakes the following steps.
(0)(i) 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 for s ∈ {NZLB, ZLB}.

=(ii) For i = 1, 2, ... (i is an index for the number of iterations),
taking as given the expectation functions previously obtained(i−1)
es, solve the relevant equations to obtain the policy func(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
tions gs = gΠ,s , gΔp ,s , gΠw ,s , gΔw ,s , gy,s , gw,s , gld ,s , gRn ,s .
(i)(iii) Update the expectation functions es by interpolating the(i)policy functions gs .
&
&
& (i)
(i−1) &
(iv) Repeat Steps (ii)–(iii) until &es − es
& is small enough.
(i−1)

In Step (ii), taking as given the values of ex,s (S−1,j , τm ) forx ∈ {λ, sp, f p, sw, f w} at each grid point indexed by (j, m) andeach regime s ∈ {NZLB, ZLB}, we have⎛Πjms ⎝ −1= ξp + (1 − ξp−1 )Π̄Δp,jms = (1 − ξp )(i−1)

esp,s (S−1,j , τm )
(i−1)

⎞

1−θp θp −1 ι(i−1)
Π−1,j pesp,s (S−1,j , τm )
⎠
,
(i−1)
Π̄
ef p,s (S−1,j , τm )−θpef p,s (S−1,j , τm )
θ 
−ιp

Πjms p Π−1,j+ ξp
Δp,−1,j ,Π̄Π̄

1Vol. 19 No. 1

The Natural Rate of Interest

Πw,jms exp(a)
Π̄
⎛



−1
−1
+ (1 − ξw
)
= ⎝ξw



(i−1)
esw,s (S−1,j , τm )(i−1)
ef w,s (S−1,j , τm )(i−1)

esw,s (S−1,j , τm )Δw,jms = (1 − ξw )333

11−θw ⎞ θw −1
 1+ηθ
ιw
Π−1,j w⎠,
Π̄−θw
 1+ηθ

w(i−1)

ef w,s (S−1,j , τm )
θ 
−ιw

Πw,jms exp(am ) w Π−1,j+ ξw
Δw,−1,j ,Π̄Π̄
γyj,−1(i−1)
yjms = eλ,s (S−1,j , τm )−1 +γa exp(am )wjms = w−1 Πw,jms /Πjms ,ld,jms = Δp,jms yjms ,
n
R
jms



n )φr R̄Π̄
= (R
−1,j



Πjms
Π̄

Then, we can evaluate
nld,jms , R
jms

φπ 

yjms exp(am )y−1

φy 1−φrexp(εr,m ).

Πjms , Δp,jms , Πw,jms , Δw,jms , yjms , wjms ,at each grid point (j, m) and each regime s and the

(i)
n }
policy functions gx,s (S−1 , τ ; θ) for x = {Π, Δp , Πw , Δw , 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 notrely on any numerical optimization routines to solve the non-linear
equations.
In Step (iii), the expectation functions are updated by(i)eλ,s (Sj,−1 , τm )n
= βd−1m Rjms

 +
τ

,
(i−1)
(S, τ  ; θ)eλ1Φ(τ  |τ )dτ  ,γa exp(a ) g (i) (S, τ  ; θ)Π334International Journal of Central Banking

March 2023

e(i)sp,s (Sj,−1 , τm ) = θp wjms zm⎧−ιp θp ⎨ (i)gΠ (S, τ  ; θ)Πjms
−1
+ ξp βdm
Π̄
Π̄
τ⎩
,
(i)
(i−1)
(i−1)
gy (S, τ  ; θ)eλ(S, τ  ; θ)esp (S, τ  ; θ)Φ(τ  |τ )dτ  ,×(i−1)
yjms eλ(Sj,−1 , τm ; θ)(i)ef p,s (Sj,−1 , τm ) = θp − 1⎧−ιp θp −1 ⎨ (i)

g
(S,
τ
;
θ)
Π
jms
Π
+ ξp βd−1
m
⎩
Π̄
Π̄

τ
(i)

×

,
(S, τ  ; θ)Φ(τ  |τ )dτ  ,(i−1)
yjms eλ(Sj,−1 , τm ; θ)gy (S, τ  ; θ)eλ(i−1)

(S, τ  ; θ)ef p

(i−1)

η
−1
e(i)
sw,s (Sj,−1 , τm ) = θw ld,jms λjms⎧−ιp θp ⎨ (i)


g
(S,
τ
;
θ)
exp(a
)
Π
jms
Πw
+ ξw βd−1
m
Π̄
Π̄
τ⎩
(i)

×

,
(S, τ  ; θ)Φ(τ  |τ )dτ  ,(i−1)
ld,jms eλ(Sj,−1 , τm ; θ)gld (S, τ  ; θ)eλ(i−1)

(S, τ  ; θ)esw

(i−1)

(i)

ef w,s (Sj,−1 , τm ) = (θw − 1)wjms⎧
−ιp θp ⎨ (i)


g
;
θ)
exp(a
)
(S,
τ
Π
jms
Π
w
+ ξw βd−1
m
Π̄
Π̄
τ⎩
(i)

×

,
(i−1)
(S, τ  ; θ)ef w (S, τ  ; θ)Φ(τ  |τ )dτ  ,(i−1)
ld,jms eλ(Sj,−1 , τm ; θ)gld (S, τ  ; θ)eλ

(i−1)

n
where the values (Πjms , Πw
jms , yjms , wjms , Rjms ) and the policy

functions gx (S, τ  ; θ) evaluated at the next period’s state (S, τ  ) are(i)
obtained in the previous step. Note that we interpolate gx (S, τ  ; θ)n } oﬀ the grid points (or equivalently
for x ∈ {Π, Πw , y, ld , R(i)Vol. 19 No. 1

The Natural Rate of Interest

335

ex(S, τ  ; θ) for x ∈ {λ, sp, f p, sw, f w}) by piecewise linear interpolation. Numerical integrals are computed with regard to τ  .
For interpolating the policy functions at each regime, we set the
number of grids as 5 for each endogenous state variable, 15 for the
discount factor shock dt , and 3 for each of the other shocks. Thus wehave 56 × 15 × 33 = 6, 328, 125 grid points in total.19 The integrals
over τ  are approximated by the Gauss–Hermite quadrature formulawith three nodes for each shock. To deal with such a large number ofgrid points and quadrature nodes, we use a supercomputer system
to run programming codes with parallel computing.
(i−1)

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