Trend Inflation in the Japanese pre-2000s:
A Markov-Switching DSGE Estimation
Ryo Kato
Junior Maih
Shin-Ichi Nishiyama
September 2022
Discussion Paper No. 2212

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANTrend In‡ation in the Japanese pre-2000s:
A Markov-Switching DSGE Estimation
Ryo Katoy Junior Maihz Shin-Ichi Nishiyamax
September 12, 2022

Abstract
In Japan, the in‡ation rate declined to near-zero, whereas the monetary policy faceda Zero Lower Bound (ZLB) in the 1990s. We examine whether trend in‡ation has fallen
to near-zero prior to the ZLB. For this purpose, we estimate Japanese pre-2000 trendin‡ation developing a Markov-Switching New Keynesian Dynamic Stochastic General
Equilibrium (DSGE) model in which non-zero trend in‡ation is explicitly incorporated
as a Markov chain state. Our estimation results indicate that the trend in‡ation
remained broadly stable at 2.0–4.0 percent from the 1960s to the late 1970s when it
fell somewhat. Up until 1997 when the ZLB was hit, the trend in‡ation hovered well
above zero, mostly at near 1.0 percent.
.

JEL Classi…cation: E31, E52, C54
Keywords: Trend In‡ation, Markov-Switching Dynamic Stochastic General Equilibrium Model, De‡ation, Monetary Policy, Lost Decade of Japan.
Kato gratefully acknowledges the …nancial support from JSPS Grants-in-Aid for Scienti…c Research(19K23219). Nishiyama gratefully acknowledges the …nancial support from JSPS Grants-in-Aid for Scienti…cResearch (18K01687). All remaining errors are our own.
yFaculty of Economics, Asia Universityz
Norges Bankx
Graduate School of Economics, Kobe University: nishiyama@econ.kobe-u.ac.jp1

1

Introduction

An underlying motivation for our work is to investigate the lost decade of Japan –a protracted
stagnation of the economy since the 1990s. A seminal work in the literature is Hayashi and
Prescott (2002), who argued that real and supply-side factors of the economy, speci…callya slowdown in technological progress and reduction in labor hours, can broadly account for
low GDP growth rates during the periods.1 In contrast, Nishizaki, Sekine and Ueno (2014)
identify multiple factors, including declines in in‡ation expectations. Debate remains ongoingover the major driving factors that gave rise to Japan’s slow growth since the 1990s.
Looking at the nominal aspects of the economy, de‡ation in Japan began in the late
1990s. Figure 1 presents monthly year-on-year changes in the consumer price index (CPI).
CPI in‡ation was 4.3 percent on average before 1960, 9.1 percent in the 1970s and fell tonear one percent remaining low for 1980–2005. In the 2000s, CPI in‡ation hovered aroundzero. In the meantime, nominal interest rates declined, hitting the zero lower bound (ZLB)
in the late 1990s.
We examine whether trend in‡ation has fallen to near-zero prior to ZLB. Although thedata suggest possible declines in trend in‡ation pre-dating the 2000s, timing and rate of suchdeclines remain issues for exploration. To this end, we focus on pre-2000s macroeconomic
time series data, including high in‡ation eras since the 1960s, particularly the 1970s, during
which CPI in‡ation reached 25 percent in the wake of global oil crises.
Speci…cally, we estimate a small-scale New Keynesian model using pre-2000s’Japanesedata. However, with higher than 5 percent average in‡ation, linearizing the model around thezero-in‡ation steady-state could elicit signi…cant estimation bias, as argued by Cogley and
Sbordone (2008) and Ascari and Sbordone (2014, hereafter denoted as AS 2014). Because
our study covers transition periods from high- to low-trend in‡ation eras, we estimate a
Generalized New Keynesian (GNK) model that explicitly allows non-zero trend in‡ation asa Markov-switching unobserved state.
Our main results indicate that trend in‡ation remained broadly stable between 2.0–4.0
percent from 1960 to the late 1970s. From the late 1970s onward, trend in‡ation is likely tohave declined to somewhere around one percent. There is, however, little evidence that trend
1

See also Kaihatsu and Kurozumi (2014) for similar arguments based on an estimated structural macroeconomic model with …nancial market frictions. Kato and Nishiyama (2005) suggest the importance ofmonetary policy in the 1990s.

2in‡ation fell near-zero percent, or even to the negative, prior to 1996, when the ZLB was hit
in Japan. In addition, our results detect regime switches regarding monetary policy stanceand supply-shock volatility. The estimated regime-switching monetary policy rule suggeststhat monetary policy stance has become “hawkish” in the sense that the short-term policy
rate responded more to in‡ation since the 1980s than in the earlier periods. The results
clearly show that the supply-shock volatility surged at the time of two global oil crises in1973 and 1979 while remaining low and stable for the entire non-crisis periods.
Figure 1: Consumer Price Index in Japan
Our work is related to the three strands of literature. First, because we estimate a(G)NK dynamic stochastic general equilibrium (DSGE) model using data in which trendin‡ation is likely to be signi…cantly above zero, we apply a non-zero trend in‡ation model
proposed by AS (2014) rather than the standard NK models.2 Second, regarding the estimation procedure, we apply a Markov-Switching Rational Expectations framework which has
been developed in the literature of Bayesian inference for structural macroeconomic models.
Speci…cally, we follow the framework introduced by Maih (2014) and later demonstrated by
Bjønland, Larsen, and Maih (2018, hereafter denoted as BLM 2018). Third, many studies
estimate NK-DSGE models using Japanese data, including Sugo and Ueda (2008), Aruoba,
Cuba-Borda, and Schorfheide (2018), Abe, Fueki and Kaihatsu (2019), Hirose (2020), andIiboshi, Shintani, and Ueda (2020), among others. Most of the earlier studies rely on nonregime-switching DSGE models.3 The most closely related studies are Aruoba, Cuba-Borda
and Schorfheide (2018) and Abe, Fueki and Kaihatsu (2019), both of which allow their models to regime-switch, but they do not consider the non-zero trend in‡ation discussed by AS(2014) and this paper.
The remainder of this paper is organized into four sections. Section 2 presents the GNK
model applied to this study. Section 3 establishes the estimation procedure used. Section 4
details the results of the analyses and Section 5 concludes.
2
See Woodford (1999) and Clarida, Galí, and Gertler (1999), among many others, for the standard NK
models3
See also, Ichiue, Kurozumi and Sunakawa (2013), Kaihatsu and Kurozumi (2014) and Fueki et al. (2016).
All these DSGE estimations use the data starting later than 1980, thus exlcuding the high in‡ation periods.

3

2

A New Keynesian model with non-zero trend in‡ation

As shown in Figure 1, given the fact that average in‡ation was quite high – far above twopercent – before the 1970s, we consider a small-scale GNK-DSGE model developed by AS(2014), which explicitly incorporates non-zero trend in‡ation

t . Formally, trend in‡ation isde…ned in terms of the in…nite horizon forecast,t =lim Et t+j :j!1

In the context of our GNK-DSGE model, trend in‡ation is the steady state in‡ation under
the rational expectations equilibrium.
Our model is a log-linearized version of the full model developed by AS (2014). We …rst
articulate the supply-side of the economy by presenting a generalized Phillips curve. The
four equations of the generalized Phillips curve are derived from the log-linearized optimal
conditions of the Calvo-price setting …rms and the standard resource constraint. The model’s
set-up is presented in the Appendix.

2.1

Aggregate supply side: GNK Phillips curve

A standard small-scale NK model consists of three equations, that is, the NK Phillips curve,
the consumption-output Euler equation, and a monetary policy rule. By contrast, in ourmodel, the generalized Phillips curve is expressed by four separate equations. Let Pt be theprice index of …nal goods and we de…nesteady statet = Pt =Pt 1 andt as gross in‡ation rate and thet , i.e, trend in‡ation, respectively. In contrast to early studies, we allow

vary over time, such that

t =

t to

(St ) ; in which St will be de…ned later combined with otherunobserved states. Further, yt denotes output which is equal to consumption in this model,

and ztAS indicates the aggregate supply shock. In the remainder of this paper, the variables
with tilde denote log-deviations from steady-state values of the variable, i.e., x~t = ln(xt =x).

4

The four equations of the GNK Phillips curves are as follows:~t =1a ( t)a ( t)~ t = f1
+
~

t

= f1
+

s~t =

(~t

~ ) + ~t 1;tb ( t )gb ( t)(1)

s~t + (1 + ) y~t

Et ~ t+1 + "Et ~ t+1

ztAS
(2)

"~ t ;a ( t )g (1

) y~t
n
~
1) Et ~ t+1 + (1a ( t ) Et t+1 + ("~t 1 +b ( t) s" t a ( t)~t
1a ( t)"b ( t) ~t 1;o") ~ t ;(3)(4)

where ~ t and ~ t represent auxiliary variables inherited from the …rst-order condition of the
Calvo-price setting …rms. In (4), s~t denotes the degree of the real distortion stemming fromprice dispersion. AS (2014) demonstrated that (i) this additional state variable s~t shows
up only in the “generalized” NK Phillips curve and (ii) plays an important role in creatingricher dynamics if the trend in‡ation is away from zero.
Several remarks on notations are in order:and;indicate the discount factor, thedegree of risk aversion and inverse labor supply elasticity. " > 1 denotes the elasticity
of substitution among intermediate goods.

2 [0; 1) represents the degree of in‡ationindexation by non-optimizing price-setting …rms. In the model,b ( t)"(1t)

wherein(" 1)(1ta ( t))

andis a Calvo-parameter representing the fraction of …rms thatcannot change prices in the period. We note that the model is linear in terms of variables
whereas the parameters are nonlinear functions in trend in‡ation. It can be con…rmed thatsetting t = = 1 with = = 0 and = 1 in (1)–(4) eliminates ~ , ~ and s~t , resultingt

in a standard NKPC, such that ~ t = Et ~ t+1 +

2.2

1

(1

)(1

) y~t

t

ztAS .

Aggregate demand side and structural shocks

The model closes with the otherwise standard consumption-output Euler equation and amonetary policy rule, such thaty~t = Et y~t+1
~{t =

MP

1

~{t 1 + 1(~{t

Et ~ t+1 ) + ztIS ;MP

5( y y~t +MP;;t ~ t ) + et

(5)
(6)

where ~{t denotes nominal interest rate and,y , andMP

2 [0; 1) represent monetary

policy response parameters. Following many early works, we allow
that

;t =

2.3.

to regime-switch such

StM P : The Markov-switching state StM P is de…ned with others in subsection

The model contains three structural shocks; that is, the aggregate supply shock ztAS ,PP
the IS shock ztIS , and the monetary policy shock eM. While monetary policy shock eMistt

distributed as N (0;2
M P ); the other two follow stationary …rst-order autoregressive processes

such as:
ztx =
where

x

x x
zt 1 + ext ;

ext

N (0;

2
x );

2 [0; 1) and x 2 fAS; ISg: The aggregate supply (AS) shock contains shocksa¤ecting the supply-side of the economy, such as technology shocks and various markup
shocks, including oil price shocks. IS shock could be a composite of preference shock, …scal
policy shock, and shocks to the natural rate of interest.

2.3

Markov-switching states

In the main model, we stipulate the following four Markov chains,St

2 fHigh; Lowg ;(7)

StM P 2 fHawkish; Dovishg ;

(8)

Stdy

2 fHigh growth; Low growthg ;(9)

StAS 2 fHigh volatility; Low volatilityg :(10)First, we allow trend in‡ationt to change according to a Markov chain that moves among thetwo regimes fHigh; Lowg in the main model. Second, we allow two monetary policy regimes

given by (8). We de…ne a “hawkish”regime as the periods during which the Bank of Japan
responds more sensitively to in‡ation. Speci…cally,in (6) follows the monetary policy

chain StM P . Third, we also allow the steady-state real per-capita GDP growth denoted bydyt to follow the macroeconomic growth chain Stdy given by (9). We explicitly consider Stdy

because our data include the 1950s–60s, the era known as the post-war Japanese “economic
miracle.”4 However, we note that Stdy does not a¤ect any parameters in the model presented
4

See Patrick and Rosovsky (1976) for discussions on the Japan’s rapid growth in the 1960s.

6

in this section, but dyt , which appears only in the observation equation to be presented later.
We also allow the volatility of the supply shock
chain StAS : In our small-scale NK model,

2AS

to change according to a Markov

2
AS includes shocks arising from global oil markets

and we note that BLM (2018) emphasized the importance of considering the role of oil pricevolatility in accounting for business cycles. In the same spirit, we allow

2
AS =

2
AS
AS (St ) to

Markov-switch because (i) our sample periods include the two global oil crisis experiencesin the 1970s and, (ii) Japan’s economy was heavily dependent on imported crude oil during
this period.

3

Estimation procedure

3.1

A Markov switching rational expectations framework

Our model can be cast in a general Markov-Switching DSGE (MS DSGE) framework expressed as,
Et
where dSt is an ndX

pSt ;St+1 dSt [xt+1 (St+1 ) ; xt (St ) ; xt 1 ; et ] = 0;(11)

1 vector of functions with their arguments xt+1 (St+1 ) ; xt (St ) ; xt 1 ; andet . Note that in our log-linearized model setting, function dSt is cast as a set of linear
functions. St = 1; 2; :::h; is the regime at time t, xt is an nx
variables. et is an n"

1 vector of all the endogenous

1 vector of Gaussian shocks with et

N (0; In" ): pSt ;St+1 is thetransition probability for moving from regime St in period t to St+1 = 1; 2; :::h in the nextPperiod such that hSt+1 =1 pSt ;St+1 = 1: In our main model, we have h = 16. This number
follows from the model, wherein we speci…ed two trend in‡ation states (high and low), twomonetary policy states (hawkish and dovish), two macroeconomic growth states (high andlow), and two supply-shock volatility states (high and low), yielding a total of 2 2 2 2 = 16
possible regimes.
In general, no analytical solution to (11) exists, even though dSt is linear. In this paper, weapply Maih’s (2014) perturbation technique for solving MS DSGE models. For the stability
condition of our MS DSGE models, we rely on the concept of mean-square stability (MSS)
following Farmer, Waggoner, and Zha (2011) among others.5
5

In implementing the Maih’s (2014) perturbation method with the stability condition to solve (11), we
use the RISE toolbox for Matlab, which is available at https://github.com/jmaih/RISE_toolbox/.

7

3.2

Data and Bayesian inference

Our dataset is quarterly frequency, consisting of real per-capita GDP (yt ), CPI, excluding
imputed rent (Pt ), and the o¢ cial discount rate (logRt ). The real GDP and CPI, excluding
imputed rent, are taken from 68SNA and from CPI (2015 Base Year index). The o¢ cial
discount rate is taken from the Bank of Japan’s Time Series Data Search site. The per-capita
real GDP is computed as the real GDP divided by total labor force. Log di¤erences of thereal per-capita GDP and CPI are used for the estimation. All data, including the o¢ cial
discount rate, are shown in annualized rates.
Our estimation framework includes the following observation equations, summarized as,
23

2

400 log yt
6
7 6
6
7 6
6 400 log Pt 7 = 64
5 4
100 log Rt
where rt = 400 (1=

1)+

1

3

dyt
t
t + rtdyt and

2

7 6
7 6
7+6
5 4

4 y~t

y~t 1 + ztAS
4~ t
4~{t

3

2

7 6
7 6
7+6
5 4

y
t

3

7
7
;
t 7
5
0

(12)

obs
obs
t is a measurement error distributed as N (0; V ar( t ));where obs 2 fy; g: As noted in the previous section, we assume that the steady-state real
per-capita GDP growth rate dyt = dy (Stdy ) and trend in‡ationt=(St ) change ac-cording to di¤erent Markov chains given by (9) and (8). For notational convenience, we
rede…ne net annual trend in‡ation rate as
porthigh = 100

f (St = High)t

1g and= 100
low = 100

( t

1). Accordingly, we will re-f (St = Low)

1g in line withdata counterpart 400 log Pt in (12). In the same spirit, we note 4~{t = 100 log(Rt =Rt ) whichconnects the nominal interest rates in the model and in the data.6
The sample period of our dataset is from 1958Q2 to 1997Q1. Because our interest lies inthe transition of the Japanese economy from high- to low-trend in‡ation eras, including the1970s when the in‡ation was historically high, assessing this period is indispensable. The
end of the sample period is chosen for the following three reasons. First, our key question iswhether Japan’s trend in‡ation declined before the ZLB. Nominal short-term interest rates,
including the o¢ cial discount rate and the Call rate, were cut to 0.5 percent in 1996, whichwere then considered the e¤ective lower bound (ELB). Moreover, including data beyond 1997
can seriously distort the estimation results due to the nonlinearity arising from ZLB or ELB.7
6

See A.4 in the appendix for greater details.
A number of early studies, such as Aruoba, Cuba-Borda, and Schorfheide (2018), Inoue and Okimoto(2008), and Hirose (2020) argue that the ZLB/ELB gave rise to a structural break in Japan in the late 1990s.
78The second reason arises from the hike of the consumption tax rate from three percent to …ve
percent in April 1997. Re‡ecting this exogenous shock, CPI in‡ation reveals a blip in 1997Q2
in Figure 1, providing a ground for excluding the data after 1997Q2. Third, because we are
estimating a regime-switching model, ensuring consistent and continuous time series data is
critically important. One advantage of our dataset is that all variables remain continuous on
the same o¢ cial basis, and neither arti…cial discontinuation nor connection of di¤erent timeseries is found, such as revision of the base-year, sampling method, and data de…nitions. In
particular, longitudinal time series of real GDP covering the 1950s to the 2010s do not exist
due to base year changes and other statistical revisions.
Before discussing the priors of the parameters to be estimated, we calibrate two parameters for avoiding identi…cation issues. We calibrated the discount factor= 0:999 based on

the sample medians of ex-post real interest rate and output growth rate. We set the inverse
labor supply elasticity= 2 following Hirose (2020). Table 1 summarizes the prior distri-butions of the parameters. Most of the priors for the structural parameters ("; ; ; ) aretaken from Hirose (2020). The main parameters of our interest are trend in‡ation,
low of which priors are set to 3.0 and 1.0 percent, respectively.

to gamma distribution forhigh andWe note that, in contrast

high ; normal distribution is assumed for

low not to over-restrict

the domain of lower bound of trend in‡ation. Assuming normal distribution forlow ‡exibly

allows that negative trend in‡ation, if it is the case, could be estimated.
Regarding the priors for the Taylor rule parameters, we set
1:5 andy = 0:5 and(Hawkish) =(Dovish) = 1:0: Finally, considering the sample sub-period averages, we setdy (High growth) = 8 and dy (Low growth) = 2 percents.
We compute the posteriors of the parameters combining the likelihood function of themodel with the priors noted above. Exploiting the linear-Gaussian nature of the model, thelikelihood function is evaluated based on Kalman …lter. In the procedure, 10,000 draws fromthe posterior distribution are generated by the Metropolis-Hastings algorithm. Convergenceof posterior distributions has been checked based on trace plots and Gelman et al. (2004)
statistics.
Table 1: Prior Distributions of Parameters

94

Results

We report parameter estimates, state transition probabilities and point estimate of trendin‡ation based on the main model. In Section 4.4, we discuss additional results obtained
from two alternative speci…cations in comparison with the main case.

4.1

Parameter estimates

Table 2 summarizes the parameter estimation results. The main interests of this study are
the two values of trend in‡ation rates in each state. The posterior modes of high and low
trend in‡ation states are 2.4, and 0.9 percent, respectively. Standard errors of each estimate
are small enough, suggesting that each of the two states is well identi…ed. Other important
regime-switching parameters include the Taylor coe¢ cients on in‡ation rate
potential GDP growth rate dy (Stdy ) and the supply-shock volatility(StM P ); theAS
2
AS (St ). When mon-etary policy stance is hawkish, the Taylor coe¢ cient is estimated at 1.8, whereas it is at 0.5
under the dovish state. The higher potential GDP growth rate is 7.7 percent compared with
3.0 percent in the lower state. The supply-shock volatility in terms of standard deviationAS
AS (St ) in the high state is 1.0 while that in the low state is 0.1, each of which is translated

into annual 4.0 and 0.4 percent, respectively.
The remainders are non-switching parameters and their standard errors are also small.
Although the outright comparison is not appropriate because of di¤ering sample periods,
most of the posterior modes and means of the parameters are broadly similar to thosein early studies using Japanese data. Relatively, the Calvo parameter
indexationand the in‡ation

are estimated at lower values compared with early studies. The lower

impliesmore ‡exible price adjustments. This is consistent with the higher average and volatility ofactual in‡ation in our pre-2000 sample periods as shown in Figure 1. While the in‡ationindexation has been a controversial “structural”parameter in the literature, AS (2014) claimthat applying the GNK model can eliminate estimation bias and yield lower estimates for.
Table 2: Posterior Estimation Results

10

4.2

Smoothed state probabilities

Another important output of our estimation is the smoothed state probabilities. The upper
panel in Figure 2 presents the probability for being in the high trend in‡ation state. Because
there are only two states for each Markov chain, the ‡ipside is the probability of being low.
The shaded areas in the …gure indicate recession periods identi…ed by the Economic andSocial Research Institute (ESRI) of the Japanese government. From the early 1960s to 1977,
the high trend in‡ation state was likely to be dominant. In the late 1970s, the probabilityof the high trend in‡ation state declines and remains at zero up to 1996, except for a shortbout in the late phase of the “bubble”period of 1989–1991.
The second panel of Figure 2 presents the probability for being in the hawkish monetary
policy state. The …gure indicates that monetary policy was dovish in the early 1960s and formost of the 1970s. From 1980 up until 1996, monetary policy stance remained hawkish. The
high probability for being in the hawkish state since the 1980s is in line with the …ndings in theliterature of Great Moderation in the U.S. context.8 The third panel shows the probability
for being in the high supply shock volatility state. In the …gure, two spikes reaching 100
percent clearly identify the well known …rst and second global oil crisis episodes: the 1973–74
OPEC embargo and the 1978 Iranian revolution. This result suggests that the two recessions
in the Japanese 1970s were precipitated by the elevated global oil price volatility.
Figure 2: Smoothed Transition Probabilities

4.3

Point estimate of the trend in‡ation

By combining the posterior modes of (St ) and the smoothed probabilities for each state,
the point estimate for trend in‡ation can be calculated over time. Figure 3 shows the point
estimate of the trend in‡ation rate. The …gure suggests that the Japanese trend in‡ation was
stable, between 2.0–2.5 percent from 1958 to 1977. Even in the midst of the …rst oil crisis
in 1973, trend in‡ation was notably stable at 2.4 percent while actual in‡ation reached 25
percent. Except for the previously noted occasion of the bubble period, the trend in‡ation islower, at around one percent, in the 1980s and in the 1990s, and little evidence is found thatit fell anywhere close to zero percent. Detecting that trend in‡ation declined to negative
territory prior to the ZLB period is even harder.
8

See Clarida, Galí, and Gertler (2000), among others.

11

Figure 3: Trend In‡ation

4.4

Alternative speci…cations

We consider two alternative speci…cations for checking the robustness of the main estimation
results. Speci…cally, we report the cases of (i) constant supply-shock volatility and (ii) threestates in trend in‡ation.
4.4.1

Constant supply-shock volatility

Because StAS is the only Markov-chain which a¤ects the second moment of exogenous shocks,

here we examine whether the estimation results noticeably change if the aggregate supply
shock is assumed to be homoskedastic. Table 2 presents the posterior estimates of the modelin the case that2
AS is non-switching. In this case, the estimatedhigh andlow are 2.3 and

1.4, respectively, in comparison to 2.4 and 0.9 in the main case. In the case of non-switching2
AS ,low is estimated even higher than one percent whilehigh remains almost the sameacross the two speci…cations. Figure 4 compares the point estimate of the trend in‡ation.
The dotted line shows the trend in‡ation in the case that2
AS is non-switching.

The trendin‡ation moderately increases at the times of the …rst and second oil crises. This result is
intuitive: Because2
AS is assumed to be constant, the times of oil crises are identi…ed as theperiods when the levels of trend in‡ation are elevated, instead of its volatility. Our overall
assessment is that allowing

2
AS to follow a Markov chain gives a better …t of the model tothe data.

4.4.2

Allowing three states in trend in‡ation

A substantial extension is to allow St to follow a Markov chain with three states,
St 2 fHigh, Medium, Lowg :The outright motivation of this extension is to detect whether a more in‡ationary state
would arise in or before the 1970s and thus, whether such an additional state may a¤ect thelevels of other states in St . As indicated by Figure 3, the deviation of actual in‡ation fromthe estimated trend in‡ation is the largest in pre-1980 and the direction of the deviation12

is upward. Re‡ecting this upward deviation, we set the priors
f5:0; 2:5; 1:0g so that, if any,highhigh ;

medium ;low=would be additionally identi…ed in the high in‡ation

periods. The posteriors of the three-statemodel are reported in the right column in Table

2. The posterior modes of the trend in‡ation states arehigh ;

medium ;low= f3:9; 2:8; 0:8g

and each of the three levels is reasonably identi…ed. We emphasize that other parameters
are estimated at close values of those in the main case. In Figure 4, the thin solid line shows
the point estimate of the three-statemodel. Three observations can be noted; (i) trendin‡ation dipped in the mid-1970s and fell sharply in the following years, the same timing
as that in the main case. (ii) The level of trend in‡ation before the sharp decline in thelate 1970s is higher at 3–4 percent than under the main case. Finally, (iii) after the sharp
decline, the trend in‡ation remains at its lowest slightly less than one percent and stable up
to 1997.
Figure 4: Trend In‡ation under Alternative Speci…cations

5

Concluding Remarks

Our estimation results broadly indicate that, in Japan, the trend in‡ation was unlikely todecline to a near- or even below-zero level prior to 1996 when the ZLB was hit. The fact
is that, Japan’s de‡ation began around the year 2000 and continued for protracted periods.
One implication is that, even if trend in‡ation is well above zero— somewhere around one totwo percent— it is possible for a central bank to quickly get caught by the ZLB; hence, it canlose some, if not all, control over in‡ation. This implication needs to be assessed with a few
caveats. One is that the estimation is not based on the real-time data but on the historical
data. As of 1997, which is the end of our sample period, policymakers were observing thereal-time data which were later revised. Estimation using real-time data may elicit morenuanced implications. Another limitation of our work is that our estimation is based on asmall-scale model. Obviously, medium-scale models including more variables, particularly…nancial sector and asset market variables, would provide richer information regarding trendin‡ation as well as for possible state transitions. These are remaining issues to be exploredin future studies.

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[6] Clarida, Richard, Jordi Galí, and Mark Gertler, 2000, “Monetary Policy Rules andMacroeconomic Stability: Evidence and Some Theory,” Quarterly Journal of Economics, 115(1), pages 147–180.
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Journal of Central Banking, 12(1), pages 1–32.
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14[11] Maih, Junior, 2014, “E¢ cient Perturbation Methods for Solving Regime-Switching
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15

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A

Set-up of the model with non-zero trend in‡ation

Our model is a version of the GNK DSGE model developed by Ascari and Sbordone (2014).
This appendix provides the basic setup of our model.

A.1

Households

A representative household exhibits a utility function which is separable in consumption Ct
and labor supply Nt :U (Ct ; Nt ) =

Ct1
1

v

Nt1+
;
1+

where v is a constant parameter. With the period-by-period budget constraint given by,
Pt Ct + Rt 1 Bt = Wt Nt + Dt + Bt 1 ;(13)

where Bt ; Rt ; Wt and Dt denote one-period bond holdings and its (gross) interest rate,
nominal wage and distributed dividend, respectively, her utility maximization yields the…rst-order conditions as follows,
Et Rt

Pt
Pt+1

Ct+1
Ct

= 1;

vNt Ct

=

Wt
:
Pt

(14)
(15)

The consumption-output Euler equation corresponds to (14), whereas (15) eliminates real
wages in the …rms’…rst order conditions.

16

A.2

Production

An intermediate goods producer i, has a linear production function in which labor is the
only input:
Yi;t = At Ni;t ;where At denotes productivity that follows a stationary stochastic process. Then, the aggregate labor demand is
Nt =

Z 1

Ni;t di =

0

Z 1
|

"

Pi;t
Pt
{z

0

st

Yt
st Yt
=
;
di
At
At
}

(16)

where st denotes the price dispersion arising from the Calvo pricing.
In the economy, the …nal good producer aggregates intermediate goods Yi;t according to,

Yt =

Z 1

" 1
"

"
" 1

Yi;t di;0where " represents the elasticity of substitution among intermediate goods.

A.3

Firms’pricing

In each period, a fraction 1of …rms re-optimize their prices denoted as Pi;t . The rest of

the …rms index their prices to the previous period’s in‡ation rate such that Pi;t =
where

t

= Pt =Pt 1 andrepresents the degree of indexation. The pro…t maximization

problem for the …rms is given by,
max : Et
Pi;tt Pi;t 1

1
X

Dt;t+j j

Pi;t

t 1;t+j 1

Pt+j

j=0

Yi;t

Wt+j Yi;t+j
Pt+j At+j

subject to the demand constraint,

Yi;t+j =Pi;t"
t 1;t+j 1

Pt+j

17

Yt+j ;;

where Dt;t+j is a stochastic discount factor andindicates cumulative in‡ation fromt;t+jperiod t to t + j such that,t;t+j =

Pt+1 Pt+2
Pt Pt+1

Pt+j;
Pt+j 1for t = 1:
Let pi;t = Pi;t =Pt and wt = Wt =Pt . Then, the …rst-order condition for the …rms’ pricesetting can be written as
pi;t =

"

t

"

1

1
j Yt+j

wt

;

t

where
Et

t

1
X

(

)

j=0

Et

t

1
X

"
t 1;t+j 1

At

;

t+j
1 "

(

t 1;t+j 1

1)j Yt+j

;

t+j

j=0

which result in (1), (2), and (3) combined with (15) and st de…ned in (16).

A.4

Monetary policy

Recall that the (gross) o¢ cial discount rate denoted by Rt in (12) is the data counterpart ofthe one-period risk-free interest rate in (13). Let 4~{t = 100 log(Rt =Rt ). Then, in the model,monetary policy follows a standard Taylor rule given by
Rt
=
R

Rt 1
R

MP

t

;t

t

which corresponds to (6).

18

Yt
Yt

y

1

MP

P
exp(eM);t

Table 1: Calibration and Prior Distributions of Parameters
Parameter
ε
θ
σ
αy
ρIS
ρAS
ρM P
ρ
σIS
σM P

Definition
Elasticity of substitution among intermed. goods
Calvo-Yun parameter
Inverse intertemporal elasticity of substitution
MP reaction cofficient of output
Persistence of IS shock
Persistence of AS shock
Interest rate smoothing
Inflation indexation
S.D. of IS shock (Q%)
S.D. of MP shock (Q%)

∗
πhigh

Trend inflation when Stπ = High

∗
πlow
M
P
απ (St= Hawkish)
απ (StM P = Dovish)dy ∗ (Stdy∗ = High growth)dy ∗ (Stdy∗ = Low growth)σAS (Stoil = High volatility)
σAS (Stoil = Low volatility)

Trend inflation when Stπ = Low
Taylor coefficient when StM P = Hawkish
Taylor coefficient when StM P = Dovish
Potential GDP growth when Stdy∗ = High growth
Potential GDP growth when Stdy∗ = Low growth
S.D. of tech. shock when Stoil = High volatility
S.D. of tech. shock when Stoil = Low volatility

Dist.
G
B
G
G
B
B
B
B
IG
IG

Mean
8.00
0.66
3.00
0.50
0.80
0.80
0.80
0.50
0.125
0.125

S.D.
1.50
0.10
1.00
0.30
0.10
0.10
0.10
0.20
0.125
0.125

G
N
G
G
G
G
IG
IG

3.00
1.00
1.50
1.00
8.00
2.00
1.00
0.25

1.00
1.00
0.50
0.30
2.00
1.00
1.00
0.25

0.05
0.05
0.20
0.20
0.20
0.20
0.025
0.20

0.025
0.025
0.10
0.10
0.10
0.10
0.05
0.10

p_{High,Low}p_{Low,High}p_{Dovish,Hawkish}p_{Hawkish,Dovish}
p_{High growth,Low growth}

Transition prob. from High π̄ to Low π̄
Transition prob. from Low π̄ to High π̄Transition prob. from Dovish to Hawkish
Transition prob. from Hawkish to Dovish
Transition prob. from High growth to Low growthp_{Low growth,High growth}p_{Low vol.,High vol.}p_{High vol.,Low vol.}Transition prob. from Low growth to High growth
Transition prob. from Low vol. to High vol.
Transition prob. from High vol. to Low vol.

B
B
B
B
B
B
B
B

stderr_η ystderr_η π

S.D. of measurement error for GDP growth
S.D. of measuremnt error for inlation

IG
IG

0.50
0.10

0.50
0.10

β
χ

Quarterly discount factor
Inverse labor supply elasticity

Calib.
Calib.

0.999
2.00

–
–

NOTE: MP stands for monetary policy, S.D. stands for standard deviation, and Calib. stands for calibrated parameter. B,
G, IG, and N stand for Beta, Gamma, inverse Gamma and normal distribution, respectively. Unit of trend inflation andpotential GDP growth rate is annual percent.

Table 2: Posterior Estimation Results
Main Model
Mode Mean
S.D.
7.409 7.512 0.063
0.437 0.529 0.017
4.073 4.390 0.027
0.529 0.457 0.006
0.915 0.941 0.006
0.902 0.887 0.007
0.881 0.864 0.010
0.205 0.203 0.020
0.021 0.019 0.003–––
0.091 0.083 0.007

Constant σAS Model
Mode Mean
S.D.
7.962 7.640 0.023
0.789 0.827 0.014
5.583 5.284 0.023
0.570 0.531 0.009
0.843 0.849 0.013
0.944 0.942 0.009
0.860 0.846 0.005
0.022 0.020 0.007
0.051 0.050 0.009
0.143 0.165 0.021
0.044 0.040 0.006

3-state π ∗ Model
Mode Mean
S.D.
7.376 7.281 0.077
0.475 0.455 0.009
4.948 4.575 0.076
0.487 0.454 0.024
0.888 0.918 0.008
0.891 0.888 0.013
0.880 0.897 0.004
0.247 0.209 0.030
0.026 0.030 0.005–––
0.090 0.105 0.006

∗
πhigh
∗
πmedium∗πlow
MP
απ (St= Hawkish)
απ (StM P = Dovish)dy ∗ (Stdy∗ = High growth)dy ∗ (Stdy∗ = Low growth)σAS (Stoil = High volatility)
σAS (Stoil = Low volatility)

2.402
–
0.942
1.802
0.504
7.720
3.009
1.008
0.102

2.333
–
1.041
1.902
0.288
7.200
2.621
1.333
0.116

0.042
–
0.024
0.012
0.027
0.045
0.060
0.014
0.039

2.343
–
1.350
1.804
0.351
7.921
2.440
–
–

2.228
–
1.184
1.778
0.285
8.109
2.390
–
–

0.012
–
0.016
0.032
0.020
0.037
0.020
–
–

3.949
2.796
0.820
1.760
0.394
7.816
2.840
1.327
0.178

3.559
2.265
0.957
1.997
0.504
7.779
2.894
1.361
0.137

0.012
0.024
0.018
0.034
0.010
0.028
0.019
0.018
0.008

p_{High,Low}
p_{High,Medium}p_{Medium,High}p_{Medium,Low}p_{Low,Medium}p_{Low vol.,High vol.}p_{High vol.,Low vol.}

0.028
–
–
–
–
0.026
0.047
0.054
0.030
0.020
0.013
0.196

0.049
–
–
–
–
0.024
0.049
0.033
0.053
0.041
0.000
0.160

0.016
–
–
–
–
0.004
0.016
0.009
0.013
0.012
0.002
0.015

0.027
–
–
–
–
0.027
0.078
0.040
0.032
0.021
–
–

0.031
–
–
–
–
0.013
0.091
0.039
0.078
0.020
–
–

0.007
–
–
–
–
0.007
0.014
0.007
0.029
0.006
–
–

0
0.034
0.036
0.021
0.018
0
0.042
0.038
0.036
0.024
0.012
0.188

0
0.021
0.053
0.032
0.014
0
0.031
0.046
0.037
0.039
0.000
0.195

–
0.007
0.012
0.013
0.005
–
0.008
0.011
0.010
0.018
0.002
0.012

stderr_η ystderr_η π

4.110
0.050

4.121
1.338

0.044
0.021

4.056
2.526

4.183
2.703

0.125
0.063

3.999
0.434

4.063
0.420

0.034
0.035

Parameter
ε
θ
σ
αy
ρIS
ρAS
ρM P
ρ
σIS
σAS
σM P

p_{Low,High}p_{Dovish,Hawkish}p_{Hawkish,Dovish}
p_{High growth,Low growth}p_{Low growth,High growth}

NOTE: S.D. stands for standard deviation. Transition probability from High π̄ to Low π̄ and vice-versa are set to be zeroin 3-state π ∗ model.

Yoy changes, percent
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
‐2
‐4

1955
1956
1958
1960
1962
1964
1966
1967
1969
1971
1973
1975
1977
1978
1980
1982
1984
1986
1988
1989
1991
1993
1995
1997
1999
2000
2002
2004

CPI excluding imputed rent, yoy
1950‐69 avg=4.30
1970s avg=9.10
80‐05 avg=1.25

Note: Monthly year‐over‐year changes in Consumer Price Index, excluding imputed
rent.

Figure 1

100%

Recession

Prob. for "High Trend Inflation state"

50%

0%
1958

100%

1961

1964

1967

1970

1973

1976

1979

1982

1985

1988

1991

1994

1970

1973

1976

1979

1982

1985

1988

1991

1994

1973

1976

1979

1982

1985

1988

1991

1994

1973

1976

1979

1982

1985

1988

1991

1994

Prob. for "Hawkish state"

50%

0%
1958

100%

1961

1964

1967

Prob. for "High Volatility state"

50%

0%
1958

100%

1961

1964

1967

1970

Prob. for "High Growth state"

50%

0%
1958

1961

1964

1967

1970

Notes: The figures graph the mode estimates. The shaded areas correspond to the dated ESRI
recessions.

Figure 2

Quarterly change, annualized percent
10

Exp. Pibar (benchmark)

9

Actual CPI inflation

8
7
6
5
4
3
2
1
0
‐1

Note: The blue bold line is the estimated average trend inflation rate. The thin
red line is the actual CPI inflation. Both are seasonally adjusted quarterly
changes of price levels expressed as anualized percent rates.

Figure 3

1996

1994

1992

1990

1988

1986

1984

1982

1980

1978

1976

1974

1972

1970

1968

1966

1964

1962

1960

1958

‐2

Annualized percent
6
Exp. Pibar (benchmark)Exp. Pibar (3‐state)

5Exp. Pibar (const. volatility)
4

3

2

1

1996

1994

1992

1990

1988

1986

1984

1982

1980

1978

1976

1974

1972

1970

1968

1966

1964

1962

1960

1958

0

Note: The blue bold line is the point estimate of trend inflation rate under benchmark model. Thin
red line and dotted gray line are those under the 3‐state model and the constant volatility model,
respectively.

Figure 4