Discussion Paper Series No.235
The Estimation of the New Keynesian Phillips Curve inJapan and Its Implication for the Inflation Response to a
Monetary Policy ShockSHIBAMOTO Masahiko

January

Kobe University

2009

The Discussion Papers are a series of research papers in their draft
form, circulated to encourage discussion and comment. Citation anduse of such a paper should take account of its provisional character.
In some cases, a written consent of the author may be required.

The Estimation of the New Keynesian Phillips Curve inJapan and Its Implication for the Inflation Response to a
Monetary Policy Shock∗Masahiko Shibamoto†Research Institute for Economics and Business Administration, Kobe University
January 19, 2009

Abstract
The New Keynesian Phillips Curve (NKPC) is a key building block in many modern
macroeconomic models. This study assesses the empirical ﬁt of the NKPC in Japanby estimating a variety of its speciﬁcations. Some empirical results suggest that introducing nominal interest rates into the pure forward-looking NKPC, which implies theexistence of a cost channel for monetary policy, helps improve their ability to explain
Japanese inﬂation dynamics. In addition to the existence of the cost channel for monetary policy, these results show that the use of labor share (real unit labor costs) is
an important factor in estimating the NKPC. As an implication of these ﬁndings, thisstudy proposes that, in the context of the New Keynesian economics, both the existence of the cost channel for monetary policy and the sluggish adjustment of real unit
labor costs can account for the fact that there is a long time lag between a monetary
policy shock and its impact on inﬂation.
JEL Classiﬁcation: C3; E3; E5
Keywords: New Keynesian Phillips Curve; Cost Channel of Monetary Policy; Inﬂation
Responses to a Monetary Policy Shock∗I would like to express my sincere gratitude to Ryuzo Miyao for his extensive help and encouragement inwriting this paper. I am grateful to Hitoshi Inoue, Daisuke Ida, Yoichi Matsubayashi, Kiyotaka Nakashima,
Takayuki Tsuruga, Yoshiro Tsutsui and seminar participants at Kobe University and Osaka University for
their helpful comments and suggestions. I also acknowledge ﬁnancial support from a Grant-in-Aid fromthe Japanese Ministry of Education.
†
Correspondence: Research Institute for Economics and Business Administration, Kobe University,
2–1, Rokkodai, Nada, Kobe, 657–8501, Japan. E–mail: shibamoto@rieb.kobe-u.ac.jp.

11Introduction

This study seeks to understand the dynamics of aggregate price inﬂation and the eﬀects
monetary policy has on it. It explores suitable structural inﬂation dynamics in the sensethat the empirical speciﬁcation provides a better ﬁt, and then draws out some implications
for the time lag in the eﬀect of monetary policy on inﬂation. Speciﬁcally, it highlights
the importance of the role of monetary policy eﬀects on the supply-side as well as thedemand-side.
It is well known that monetary policy aﬀects the demand side of the economy byinﬂuencing the investment and savings decisions of households and ﬁrms through changing
interest rates. On the other hand, some economists, such as Barth and Ramey (2001), haveproposed that there may be important supply-side eﬀects of monetary policy augmenting
the conventional demand-side channel. To the extent that ﬁrms must pay the factors of
production before they receive revenues from selling their products, they rely on borrowing
from ﬁnancial intermediaries. Higher interest rates translate into higher costs of working
capital and then induce a rise in inﬂation. Such an eﬀect is called the cost channel ofmonetary policy.
Recently, many macroeconomists have used a “New Keynesian” (NK) model, which
can be derived from the optimizing decisions of economic agents and nominal rigidities, asa useful framework for the analysis of monetary transmission and monetary policy issues.1
The prominent feature of the NK model is that macroeconomic dynamics are criticallydependent on private agents’ forward-looking expectations about the future state of theeconomy. And the presence of nominal rigidities yields a correlation between inﬂation
dynamics and the real economy and the real impact of monetary policy.
Inﬂation dynamics are one of the important components of the monetary transmission
mechanism. According to Woodford (2003), the New Keynesian Phillips Curve (NKPC)
is used as a model of inﬂation dynamics in the context of the NK model. The NKPC
relates current inﬂation to the present-value of current and future real marginal costs.
The NKPC has emerged as the dominant theoretical attempt to explain the interaction
between inﬂation and real aggregates, and gives a description of the supply-side of the1

See Woodford (2003) for a standard derivation of the NK model.

2economy.
Within the NKPC, measures of real marginal costs are a crucial determinant of theevolution of the inﬂation rate. In estimating NKPC, many empirical studies, e.g., Gali
and Gertler (1999), Sbordone (2002), Rudd and Whelan (2005, 2006, 2007), and Kurmann(2005), use real unit labor costs or detrended output as a proxy of real marginal costs.2
However, in the presence of the cost channel for monetary policy, we can consider nominalinterest rates as one of the key driving variables for inﬂation, in addition to real unit
labor costs or detrended output. Ravenna and Walsh (2006) and Chowdhury et al. (2006)
introduce the cost channel of monetary transmission into an otherwise standard NK model.
As shown by them, when ﬁrms’ costs of working capital increase with interest rates, we
can augment an NKPC by short-run nominal interest rates. And Tillmann (2008) showsthat the cost channel adds signiﬁcantly to the explanation of inﬂation dynamics in the
NKPC for the U.S., U.K., and the Euro area. In particular, in Japan’s ﬁnancial system,
where banks play a highly important role in ﬁnancial intermediations, monetary policy
through this channel may have signiﬁcant eﬀects on inﬂation.
This study examines the empirical performance of the NKPC in explaining Japanese
inﬂation dynamics by estimating a variety of its speciﬁcations. To our knowledge, there
are the following two empirical studies that estimate the NKPC in Japan. First, Muto
(2009) estimates the NKPC in Japan by focusing on the measurement of real marginal
costs. To obtain a better proxy for real marginal costs, he corrects the labor share ofnational income (real unit labor costs) by taking into account two kinds of labor market
frictions: labor adjustment costs and real wage rigidity. He ﬁnds that if a corrected
measure of real marginal costs is used, the NKPC provides a good explanation of Japan’s
inﬂation dynamics, and that the inclusion of a lagged inﬂation term into the NKPC does
not improve its ﬁt. Although we also seek to obtain a better proxy for real marginal costs,
we focus on the role of nominal interest rates for real marginal costs. Second, Chowdhury
et al. (2006) estimate the NKPC including short-run nominal interest rates and lagged
inﬂation, and report that there is less evidence for the existence of an eﬀective cost channel
2
Although the NKPC may hold a central place in the recent theoretical literature on monetary economics, the empirical ﬁt of the NKPC is not always found to be successful. For example, Rudd andWhelan (2005, 2006, 2007) report that neither real unit labor costs or detrended output provide a good ﬁt
for the NKPC. Furthermore, by estimating the so-called “hybrid” class of the NKPC which allows partial
dependence of inﬂation on its own lags, they ﬁnd that the ﬁt of the model is due only to the inclusion oflags of inﬂation.

3

in Japan. Although they employ the generalized method of moments (GMM) regression
approach, the resulting GMM estimators often suﬀer from an econometric problem called“weak identiﬁcation” as Ma (2002) and Mavroeidis (2005) point out.3 In order to avoid
this problem, we estimate the NKPC by using an alternative approach based on the present
value model, which is employed by Rudd and Whelan (2005, 2006, 2007), Muto (2009),
Kurmann (2005), and Tillmann (2008).
Some empirical results in this study suggest that introducing nominal interest rates
into the pure forward-looking NKPC model helps improve their ability to explain Japanese
inﬂation dynamics within the NK economy. In addition to the existence of the cost channel
for monetary policy, these results show that the use of labor share is an important factor
in estimating the NKPC. As an implication of these ﬁndings, this study points out thatboth the existence of the cost channel and the sluggish adjustment of real unit labor costs
account for the fact that there is a long time lag involved in the eﬀect of monetary policy
on inﬂation.
The remainder of the paper is structured as follows. Section 2 summarizes the transmission eﬀects of monetary policy, focusing especially on the time lags with which theshock is transmitted. Section 3 describes the empirical model used to explore the ﬁt of
the NKPC, and reports some empirical results. Section 4 provides the implications of
the empirical results obtained in Section 3 for the time lag of monetary policy eﬀects on
inﬂation. Some concluding remarks are oﬀered in the ﬁnal section.

2

Eﬀects of Monetary Policy Shocks

In order to understand monetary policy eﬀects on aggregate price inﬂation, it is important
to distinguish between labor share (real unit labor cost) and detrended output (outputgap), which is considered to be more closely associated with inﬂation. This section summarizes the transmission eﬀects of monetary policy by using a vector autoregressive (VAR)
approach, which is a standard framework used to study the eﬀects of monetary policy
shocks on macroeconomic variables. The VAR is useful in the sense that we can analyze
the dynamic eﬀects of economic variables without requiring a particular theoretical model.
3

See following subsection 3.5 for a further explanation of the weak identiﬁcation problem.

4

This section provides our speciﬁcation of the VAR model. We then identify a monetary
policy shock and report estimates of how major macroeconomic variables, including labor
share and detrended output, respond to a monetary policy shock, focusing especially on
the time lags with which the shock is transmitted.

2.1

Monetary VAR Model and Identiﬁcation of Monetary Policy Shocks

Consider a six-variable VAR model consisting of a general price inﬂation rate (πt ), sensitive
price inﬂation rate (πtf ), detrended output (yt ), labor share of national income (st ), short

term interest rate (rt ) and money growth (Δmt ).4 Let Xt = [πt , , πtf , yt , st , rt , Δmt ]denote the vector of variables included in the analysis. Assume that the dynamics of Xt
are given as follows:
Xt = a0 + A1 Xt−1 + · · · + Ap Xt−p + t ,(1)

where E(t ) = 0, E(t τ ) = Ω for t = τ and 0 elsewhere. The lag order p is empirically determined by some appropriate statistical information criteria. Following Christiano et al.
(1999), (1) is a reduced form VAR model, which may contain a block recursive restrictionthat the monetary policy variables be ordered ﬁfth in the VAR on the assumption that thenonmonetary variables do not respond to the monetary policy shocks contemporaneously
but the monetary variables do. A Choleski decomposition of the reduced form covariance matrix Ω can be used to orthogonalize the reduced form innovations and to identify
the structural model. The innovations in the equations of rt are treated as monetary
policy shocks. We estimate the parameters a0 , Aj (j = 1, · · · , p) and Ω using ordinary
least squares (OLS). Using these estimates, we compute the impulse responses of Xt to amonetary policy shock.
As for the data on the general price inﬂation rate (πt ), we use two kinds of variables:
the quarterly change of the log GDP deﬂator, and the consumer price index (CPI) in
percentage points. The sensitive price inﬂation πtf is measured as the quarterly change
4
In many monetary VAR studies, it is now standard practice to specify the VAR model that does not
generate the “price puzzle”, the frequent ﬁnding in VAR literature that a contractionary monetary policy
shock is followed by a permanent increase in the price level. Sims (1992) explains that the “price puzzle”
can be interpreted as the result of imperfectly controlling for any information that the central bank may
have about future inﬂation. In actuality, because speciﬁcation with the sensitive price variable often solves
the “price puzzle”, estimation with such a variable generalizes.

5

of the log domestic Wholesale Price Index (WPI) in percentage points.5 The detrended
output measure is the percentage deviation of the log of real GDP from an HP-ﬁlter trend.
The labor share of income st (in percentage point deviation) is st = 100 log(W L/(P Y ))t ,where W L is the System of National Accounts’ (SNA’s) “compensation of employees” andP Y is nominal GDP−“household’s operating surplus”.6 The short-term interest rate rtis the three-month Gensaki rate (average). The money growth is the quarterly change ofthe log M2 + CD in percentage points.
In the estimation, we use quarterly data from 1975:Q1 to 1999:Q1.7 The lag length isset to 5 in the estimation of the reduced-form model (1).8

2.2

Estimating Eﬀects of Monetary Policy Shocks

Figures 1 and 2 show the estimated impulse responses of all variables in Xt to a 1%
monetary policy shock in the cases of GDP deﬂator inﬂation and CPI inﬂation measures,
respectively. The solid line indicates the estimated response. The upper and lower dashedlines plotted in each graph represent 90% bootstrap conﬁdence intervals, based on 10000
bootstrap samples.
Overall, estimated impulse responses look reasonable in both ﬁgures. The results
suggest that after a contractionary monetary policy shock,1. Detrended output immediately falls, peaking after a half year and returning to preshock levels after one year;
5

Hosono et al. (2001) examine four possible candidates for the sensitive price variable when estimating
structural VAR models: the Nikkei commodity price index, the import price index, total and domestic
WPI. The results suggest that, among the four possible candidates, domestic WPI shows the most favorable
results.
6
We cannot use the deﬁnition, which is “compensation of employees” divided by nominal GDP, because
the deﬁnition of “compensation of employees” does not include compensation of employees in self-employedﬁrms. Following Muto (2009), we employ the deﬁnition as above.
7
We restrict the sample period to before the introduction of the period of zero interest rate policy.
Ueda (2002) thinks of the zero interest rate policy as the combination, the zero rate and the commitment
to maintain it until a certain set of conditions are met. Following Ueda (2002), we assume that the Bankof Japan implemented the zero interest rate policy in the period from April 1999, and adopt 1999:Q1 as
the ending point.
8
We perform a modiﬁed likelihood ratio (LR) test proposed by Sims (1980) to check whether taking
5 lags is suﬃcient. Here, the null of 5 lags is tested against the alternative of 6, 7, or 8 lags. The LR
statistics indicate that the null is not rejected by conventional signiﬁcance levels. In addition, we performthe multivariate Lagrange Multiplier (LM) test statistics for residual serial correlation for up to the 1st and4th orders. See Johansen (1995), p. 22, for the formula of the LM statistics. The LM statistics for eachorder indicate that the null hypothesis of no serial correlation is not rejected by asymptotically signiﬁcantlevels.

6

2. Labor share (real unit labor costs) slowly falls, peaking after one year;
3. Interest rate falls for about 2 years.
4. General price inﬂation falls at ﬁve quarters, but the decrease is small; on the other
hand, sensitive price inﬂation falls slowly, peaking after one year, and the decreaseis larger than that of general price inﬂation;
5. Money growth falls at ﬁve quarters.
In Figures 1 and 2, some important features of the eﬀects of a monetary policy shock
are worth emphasizing here. First, we can conﬁrm the stylized fact that there is a longtime lag between a monetary policy shock and its impact on price inﬂation. There is aconsiderable delay in the eﬀect of monetary policy on price inﬂation, unlike detrended
output.
Second, the detrended output and labor share responses are noticeably diﬀerent inresponse to a monetary policy shock. We ﬁnd that the responses for labor share follow
those for price inﬂation very closely.9 As for the demand side eﬀect of monetary policy, an
increase in the nominal interest rate reduces the demand on consumption and investment
because of the inﬂuence of the real interest rate, and then leads to a negative detrended
output. Under monopolistic competition, the reduction of current demand then decreases
real unit labor costs through a reduction in labor demand. However, as the model withwage rigidity implies, deﬂation pressures can impede the eﬀective clearing of labor markets.
With regard to wage rigidity, a gap between detrended output and labor share can arise.
This gap may lead on the demand side eﬀects of monetary policy to a diﬀerence in theinﬂuence between real unit labor costs and detrended output.

3

Japanese Inﬂation Dynamics

This section examines the empirical performance of the NKPC in explaining Japanese
inﬂation dynamics by estimating a variety of its speciﬁcations. In particular, it provides
an empirical model to investigate whether the presence of the cost channel for the monetary
9

Shibamoto (2007) reports that with regard to regular workers, employment responses to a monetary
policy shock in Japan follow those for the price indices very closely, using a factor-augmented VAR approach.

7

policy, in which monetary policy has an eﬀect on the supply-side of the economy, and thepresence of lagged inﬂation aﬀects structural inﬂation dynamics. Then, it reports some
empirical results.

3.1

Interest-Rate Augmented New Keynesian Phillips Curve

Following Woodford (2003) among others, the marginal-costs-based NKPC is described asfollows:
 t + error,π̂t = βEt π̂t+1 + κrmc

(2)

with a subjective discount factor, β. π̂t is a measure of the deviation of the inﬂation ratefrom the steady-state value, rmc
 t denotes a measure of the deviation of real marginal costs
from the steady-state value, and Et is the expectations operator. The composite parameterκ is a function of the structural model parameters. Under rational expectations, equation(2) can be re-written asπ̂t = κ∞β j Et rmc
 t+j + error,

(3)

j=0

Equation (3) states that the inﬂation rate at time t is a fraction of the present-value ofthe expected path of future real marginal costs.10
We now consider the speciﬁcation of the NKPC to allow an impact of the nominalinterest rate on the supply-side of the economy by raising ﬁrms’ marginal costs and, thus,
on the inﬂation rate.11 Following Chowdhury et al. (2006) and Tillmann (2008), the proxy
for the measure of the deviation of real marginal costs from the steady-state value, rmc
 t,
is
rmc
 t = ŝt + ψR r̂t ,(4)where ŝt and r̂t are the deviations of the labor share and the gross nominal interest rate
10

As Roberts (1995) has shown, the rational expectations sticky-price models of Taylor, Rotemberg,
Calvo, and others imply an NKPC in which current inﬂation incorporates a forward-looking component.
11
Christiano and Eichenbaum (1992), Christiano et al. (2005) and Ravenna and Walsh (2006) assume
that factors of production have to be paid before the proceeds from the sale of output are received, tointroduce the costs of working capital into a general equilibrium model.

8

from their respective steady-state values. The parameter ψR measures the strength of thecost channel. This means that its strength may depend on the share of bank-dependentﬁrms and the degree of the pass-through from money market and capital market rates tocorporate lending rates. If ψR = 0, the cost channel is absent.
Considering the cost channel by combining equations (2) and (4) thus leads to thefollowing present-value relation for the inﬂation rateπ̂t = βEt π̂t+1 + κ(ŝt + ψR r̂t ) + error,(5)

and equivalently,

πt = μ0 + μ1∞


β j Et st+j + μ2

j=0

∞


β j Et rt+j + error,

(6)

j=0

where μ0 is constant, which includes the constant steady-state values for inﬂation, labor
share and nominal interest rate, μ1 = κ and μ2 = κψR . Many previous studies, e.g., Gali
and Gertler (1999), Sbordone (2002), Rudd and Whelan (2005, 2006, 2007), and Kurmann(2005), study only the present-value of the labor share as a driving variable of inﬂation,i.e., they assume μ2 = 0. The empirical analysis in this section examines whether theinﬂation rate is driven by the present-value of expected future interest rates in addition
to the labor share.
The NKPC as derived above has frequently been criticized for a lack of inﬂation inertia.12 We now include πt−1 to see whether there is an extra source in the inﬂation rate notpresent in the standard formulation of the NKPC as above.13 If there is some direct eﬀect
12

The notion of inertia refers to the slow adaptation of a variable to unexpected changes in economic
conditions. By this deﬁnition, a variable is inertial if and only if its past levels, or past expectations about
its current level, have a direct inﬂuence on its current level. The notions of stickiness and inertia are usedas synonyms. In contrast to inertia, the notion of persistence refers to the slow transition of a variable toits steady state after the initial impact of the unexpected shock. Note that a variable can theoretically bepersistent even if it is not sticky.
13
Inﬂation inertia models assume some kind of backward looking behavior or bounded rationality. Christiano et al. (2005) model backward looking behavior on the assumption that all ﬁrms apply indexation
to lagged inﬂation in periods when they cannot readjust their prices. Gali and Gertler (1999) specify
backward looking behavior on the assumption that a fraction of ﬁrms are backward looking, and hence
follow a rule of thumb in readjusting their prices. The two assumptions are equivalent in that they yield
the same aggregate relationship, linking current inﬂation rate to lagged inﬂation, expected future inﬂationand some measure of real economic activity. On the other hand, as an alternative to bounded rationality,
Mankiw and Reis (2002) model inﬂation inertia based on sticky information. The assumptions of somekind of backward looking behavior and bounded rationality yield similar implications for inﬂation andoutput dynamics.

9

of lagged inﬂation besides the one that occurs because lagged inﬂation helps to forecast
the labor share and nominal interest rates, ignoring inﬂation inertia could lead to biased
estimates that attribute too much weight to contemporaneous labor share and interest
rates.
Following Christiano et al. (2005) and Woodford (2003), the NKPC in the presence of
price indexation becomes t + error.
π̂t − ρπ̂t−1 = βEt (π̂t+1 − ρπ̂t ) + κrmc

(7)

Solving (7) forward yields
π̂t − ρπ̂t−1 = κ∞

β j rmc
 t+j + error.

(8)

j=0

Using (4), (8) can be re-written as follows:

πt = μ0 + μ1∞


β j Et st+j + μ2

j=0

∞


β j Et rt+j + ρπt−1 + error.

(9)

j=0

Note that (9) nests the purely forward-looking model when ρ is set to zero. The empiricalanalysis in this section also examines whether the inﬂation rate is driven by the lagged
inﬂation.

3.2

Forecasts from VAR projections

We estimate the Japanese NKPC, which is derived as (9). Following Rudd and Whelan(2005, 2006, 2007), Kurmann (2005), Muto (2009), and Tillmann (2008), we employ theestimation used originally by Campbell and Shiller (1987). In this approach, we ﬁrst
forecast the future path of labor share and interest rate based on a VAR, then use it toestimate (9).14


Letting Zt = [1, Xt , Xt−1 , · · · , Xt−p+1 ] , (1) can be represented as a VAR(1) process:Zt = BZt−1 + Ht ,

(10)

14
Campbell and Shiller (1987) propose this framework to assess the ﬁt of forward-looking present-value
models.

10where
⎡
1
⎢ 1×1
⎢
⎢
⎢ a0
⎢ 6×1
⎢
⎢
=⎢ 0
B
⎢ 6×1
(6p+1)×(6p+1)
⎢
⎢ ..
⎢ .
⎢
⎣
0

0



6×1

0

6×1

A1
6×6

· · · Ap−1

I

···

..
.

..

0

···

6×6

6×1

···



6×6

6×6

0

6×6

..
.

.

I

6×6

0



⎤

⎥
⎥
⎡
⎤

⎥
Ap ⎥
0
⎢ 6×1
⎥
6×6 ⎥
⎥
⎢
⎥
⎥
⎢
⎥
I
and
H
=
⎥
⎢
⎥.
0
6×6
⎥
⎢
⎥
(6p+1)×6
6×6 ⎥
⎣
⎦
.. ⎥
0
. ⎥
6(p−1)×6
⎥
⎦
0

6×1

6×6

Hence, Zt is an approximation to the agents’ information set, which can be described byconstant, current and past realizations of price inﬂation; WPI inﬂation; detrended output;
labor share; interest rate, and money growth. Forecasts of the endogenous variables are
given by the multi-period forecasting formula

Et [Zt+j ] = B j Zt .

(11)

Using the matrix version of the summation formula for geometric series, the vector of the
discounted future paths of the variables can be calculated as∞

β j Et Zt+j = (I − βB)−1 Zt ,

(12)

j=0




where I is the 6 × 6 identity matrix. Let es and eR be selection vectors that single out thepresent-value of future labor shares and interest rates, respectively. We map these forecasts
into the present-value representation of the NKPC model (9) to obtain an expression forthe model-consistent inﬂation rate πtm as follows,

πtm = μ0 + μ1 es (I − βB)−1 Zt + μ2 eR (I − βB)−1 Zt + ρπt−1 .

(13)

To calculate this series of fundamental inﬂation from the VAR coeﬃcients and the data,
we need to ﬁx the discount factor β. In accordance with the vast literature, we choose
to calibrate β = 0.99. However, all results are robust to other choices of the discountfactor as long as the discount factor is close to but below unity. Using the present-value of
future labor shares and interest rates based on the VAR, we simply estimate (9) by OLS
11and then calculate the adjusted coeﬃcient of determination (adj-R2 ) to assess the ﬁt of
the NKPC by comparing actual inﬂation πt with the series of fundamental inﬂation πtm
resulting from this estimation.15
Note that using VAR projections disguises the degree of estimation uncertainty, as
Kurmann (2005) points out. The VAR companion matrix B that is needed to derive
fundamental inﬂation according to (13) contains only point estimates. Following Tillmann(2008) to assess the accuracy of the model’s ﬁt, we employ a bootstrap approach that infers
the distribution of our measures of ﬁt by estimating the model with artiﬁcial data. We
obtain conﬁdence intervals by drawing from the residuals of the estimated model andgenerating new observations for the data vector using the estimated companion matrix.
The VAR model is estimated again and a new coeﬃcient matrix B is computed. From this,we compute the series of expected future labor shares and interest rates and regress actualinﬂation on the present-value of these two forcing variables to infer the slope coeﬃcients.
Finally, adj-R2 is computed. Repeating this procedure 10000 times provides us with anempirical distribution for adj-R2 from which an interval that includes 90% of the estimates
can be calculated.

3.3

Estimation of the Discounted Present Value Model

We estimate the NKPC under four diﬀerent scenarios. In the ﬁrst scenario, the costchannel is absent and inﬂation is driven solely by the discounted present value of futurelabor share. This scenario corresponds to a model with μ2 = 0 and ρ = 0. In the second
scenario, as well as in Rudd and Whelan (2005, 2006, 2007) and Kurmann (2005), the costchannel is absent, but lagged inﬂation is included. This scenario corresponds to a modelwith μ2 = 0. In the third scenario, the cost channel is present and inﬂation is driven by
the discounted present value of future labor share and future interest rates, but lagged
inﬂation is not included. This scenario corresponds to a model with ρ = 0. In the fourth
scenario, the cost channel is present, and lagged inﬂation is included.
Figures 3 and 4 compare the series of fundamental inﬂation with the actually observed
inﬂation rates under four scenarios, in cases with GDP deﬂator inﬂation and CPI inﬂation
15

To examine the possibility that Zt correlates with an error term, we have also estimated μ0 , μ1 , μ2 , ρby the instrumental variables method. Using Zt−1 as instruments, we ﬁnd that the result is almost thesame as with OLS.

12

measures, respectively. The solid lines in this ﬁgure are fundamental inﬂation series πtmof the discounted present value generated using an econometric forecasting model. The
dashed lines are actual Japanese inﬂation πt . For convenience, we include shaded regions,which begin at an Economic and Social Research Institute (ESRI) business cycle peak,and end at a trough.
(a) in Figures 3 and 4 reveals a bad performance under the ﬁrst scenario. (b) showsthat, although the ﬁt is better than the one under the ﬁrst scenario, fundamental inﬂation
πtm is quite volatile.16 On the other hand, (c) and (d) show that fundamental inﬂation
πtm tracks the behavior of actual inﬂation πt quite well, and satisfactorily explain thehistorical pattern of inﬂation in the sense of the popular empirical evidence that inﬂation
is procyclical. In particular, we ﬁnd that in the recession phase (1975Q1, 1977Q1–1977Q4,
1980Q1–1983Q1, 1985Q2–1986Q4, 1991Q1–1993Q4, 1997Q2–1999Q1), their fundamental
inﬂation tends to fall.
Below we can interpret the diﬀerences among four scenarios. Table 1 presents baseline
estimation results under four scenarios. In the ﬁrst scenario, we ﬁnd that the present
value of future real unit labor costs turns out to be negatively correlated with inﬂation.
The table shows that its present value of labor share generated from a VAR that ﬁts thedata well has an adj-R2 of only 1.5 and 4.2% in the cases with GDP deﬂator inﬂation and
CPI inﬂation measures, respectively. In the second scenario, we also ﬁnd that the present
value of future real unit labor costs turns out to be negatively correlated with inﬂation.
Although the adj-R2 from the second scenario is relatively larger than that from the ﬁrst,
the ﬁt of this model might be due only to the inclusion of a lagged value of πt . Therefore,
in the NKPC model without the cost channel, not only does inﬂation not appear to equal
the present value of future real unit labor costs, it does not even have the correctly signed
correlation. This implies that the use of labor share alone as a proxy of real marginal costs
is not a suitable driving variable for inﬂation.
On the other hand, in the third scenario, we ﬁnd that the present value of both future
real unit labor costs and interest rates turns out to be positively correlated with inﬂation.17
16

As we show below, an adj-R2 is not as high as (c) and (d).
The parameters μ1 and μ2 are quite small. However, this does not always imply that the fraction ofﬁrms that re-optimize prices within the quarter is small. For example, the strength of strategic complementarity depends on the value of the parameter κ, coeﬃcient of the real marginal costs. See Woodford(2003), for details. In addition, Altig et al. (2005) present models in which capital is ﬁrm-speciﬁc and1713In addition, the adj-R2 is 45.4 and 73.7% in the cases with GDP deﬂator inﬂation and CPI
inﬂation measures, respectively. Furthermore, the presence of the cost channel means asigniﬁcant improvement in the model’s empirical performance, because the 90% conﬁdence
intervals do not include the value of the adj-R2 for the case without the cost channel, i.e.,
0.015 ∈
/ [0.158, 0.579] and 0.042 ∈/ [0.543, 0.809]. These results mean that both labor
share and interest rate as a proxy for real marginal costs are suitable driving variables
for inﬂation. Moreover, in the fourth scenario, we ﬁnd that the estimated ρ coeﬃcient isstatistically insigniﬁcant, which implies that lagged inﬂation has little inﬂuence on current
inﬂation. In addition, the adj-R2 from the fourth scenario is almost the same as that in
the third scenario. These results mean that the ﬁt of NKPC with the cost channel canhardly be attributed to the contribution made by the lagged value of πt .18
It should not be surprising that, in the case with the restriction μ2 = 0, the present
value of future real unit labor costs is negatively correlated with inﬂation. Here we provide
the possible explanation that underspeciﬁcation, without considering the cost channel formonetary policy, leads to a severe problem for the estimation of the coeﬃcient of thepresent value of future real unit labor costs μ1 . The estimator μ∗1 without considering thecost channel is as follows;μ∗1 = μ1 +∂(∂(where r̄ and s̄ denote the average of∞
j=0 Et rt+j − r̄)μ2 ,∞j=0 Et st+j − s̄)∞
j=0 Et rt+jand∞
j=0 Et st+j , respectively.

(14)

(14)

shows that the estimator μ∗1 is produced by the eﬀects of future real unit labor costs on
inﬂation mixing with those of future interest rates through the cost channel. Except in
the cases that (∂(∞j=0 Et rt+j − r̄))/(∂(∞
j=0 Et st+j − s̄)) is equal to zero and/or μ2 is

equal to zero, the estimator μ∗1 is biased and inconsistent.
Figure 5 displays a scatter plot of the present value of future labor share and futureinterest rate estimated by using the auxiliary VAR model (10). As is obvious in Figure 5,(∂(∞
j=0 Et rt+j − r̄))/(∂(∞
j=0 Et st+j − s̄)) is strongly negative. Thus, if μ2 is not equal topredetermined within a period, and show that the parameter κ diminishes despite the microeconomic
empirical evidence that ﬁrms change prices frequently.
18
Muto (2009) ﬁnds that in considering the labor adjustment costs and real wage rigidity as labor market
frictions, the inclusion of a lagged inﬂation term into the NKPC does not improve the ﬁt of the NKPC atall. He points out that the backward-looking component is no longer required to explain Japan’s inﬂation
dynamics if a corrected measure of real marginal costs is used.

14

zero, then the estimator μ∗1 is biased and inconsistent. In addition, if μ2 is positive, thenthe estimator μ∗1 is possibly negative. This implies that the estimation without considering
the cost channel spuriously causes the present value of future real unit labor costs to be
negatively correlated with inﬂation, because of confusion with the eﬀects of future interest
rates on inﬂation.
Next, we use the detrended output, which some empirical researchers (for example,
Rudd and Whelan (2005, 2006, 2007)) often employ as a proxy for real marginal cost in
estimating NKPC, in place of the real unit labor costs. This is because of the assumptionthat real unit labor costs are proportional to the gap between actual and potential output(equilibrium output in the case of fully ﬂexible prices), that is detrended output. With
this assumption and (9), the detrended output-based NKPC with cost channel becomes

πt = λ0 + λ1

∞

j=0

β j Et yt+j + λ2∞


β j Et rt+j + ρ1 πt−1 + error.

(15)

j=0

Table 2 presents estimation results under four scenarios by using the detrended output
in place of the labor share.19 When we use the detrended output instead of the labor share,
the estimated coeﬃcient on the detrended output is negative, even allowing for the costchannel.20 This suggests that the link between real unit labor costs and detrended output
is the problem and not the link between marginal costs and inﬂation.21 To go from realunit labor costs to a detrended output measure, real wages are replaced by the marginal
rate of substitution between leisure and consumption. As Woodford (2003) among others
show, this procedure assumes that nominal wages are perfectly ﬂexible, so that the real
wages can adjust to maintain workers on their labor supply curve. If nominal wages
are sticky, a gap can open between the real wage and the marginal rate of substitution
between leisure and consumption. In fact, as shown in the previous section, the time lags,
with which the monetary policy shock are transmitted, vary somewhat between detrended
output and labor share. Our results imply a problem in the replacing of labor share with
19

Even in using the present values of future detrended output and interest rate generated from a VAR
without the labor share, almost the same results as the following empirical ones are obtained.
20
It is well known that the present value of future detrended output is negatively correlated with inﬂation,in spite of the fact that detrended output is procyclical as well as inﬂation is procyclical. For empiricalevidence in the U.S., see e.g., Rudd and Whelan (2005, 2006, 2007).
21
In contrast to this interpretation, Gali and Gertler (1999) argue that the problem is that “potential
output” is hard to measure, thus, empirical proxies for the detrended actual output may be poor proxies
for the real output gap.

15

detrended output when the NKPC is estimated, in the sense that these two variables
respond to diﬀerent inﬂuences over time.22
In summarizing some empirical results in this subsection, we suggest that both thenominal interest rate and the labor share be used as a proxy for real marginal costs.

3.4

Comparison with the GMM Estimation

As an alternative method for estimating the NKPC, some researchers employ the GMM
regression approach, which is often proposed in the literature for estimating the forwardlooking rational expectation model. Chowdhury et al. (2006) estimate the hybrid NKPC,
including the short-run nominal interest rate in G7 countries, by using the GMM estimation approach. They report that there is less evidence for the existence of an eﬀective costchannel in Japan, relatively in Canada, France, Italy, the U.K., and the U.S. This subsection reports the GMM estimation results similar to those obtained by following Chowdhury
et al. (2006), and discusses the diﬀerence between these results and the results obtained
before the previous subsection.
Consider the cost channel by combining equations (7) and (4), thus leads to the equation for the inﬂation rate

π̂t =

β
κ
ρ
Et π̂t+1 +(ŝt + ψR r̂t ) +π̂t−1 + error
1 + βρ
1 + βρ
1 + βρ

(16)

From (16), we obtain

πt = ζ0 + ζ1 Et πt+1 + ζ2 st + ζ3 rt + ζ4 πt−1 + error,(17)

where ζ0 is a constant that includes the constant steady-state values for inﬂation, the labor
β
κψR
ρ
κ
, ζ2 = 1+βρ, ζ2 = 1+βρ
and ζ4 = 1+βρ
. (17) is
share and nominal interest rate, ζ1 = 1+βρ

a linear GMM regression, with moment conditions of the following form:
Et [Qt (πt − ζ0 − ζ1 πt+1 − ζ2 st − ζ3 rt − ζ4 πt−1 )] = 0,

22

(18)

Blanchard and Gali (2007) show that, under the assumption of the real wage rigidities, the NKPC
becomes the relation between inﬂation, expected inﬂation and both the level and change of the detrended
output.

16

where Qt is a q × 1 vector of instruments. According to moment conditions (18) withand without the restriction ρ = 0 (i.e. ζ4 = 0), GMM estimations are carried out for
the quarterly time-series data. The sample period spans the time interval 1975–1999 asabove. Following Chowdhury et al. (2006), the vector of instruments Qt includes fourlags of general price inﬂation, sensitive price inﬂation, labor share, and short-run interest
rate. To account for possible correlation in the moment conditions and to control for
autocorrelation and heteroskedasticity in unknown form in the weighting matrix, we allow
for the Newey and West (1987) correction up to order 4.
Table 3 reports the estimates of the interest rate augmented NKPC as speciﬁed in (17)with and without the restriction ζ4 = 0 in the cases with two inﬂation measures using
the GDP-deﬂator and the CPI. In all cases, as reported in Chowdhury et al. (2006), theestimated coeﬃcients of the labor share and the nominal interest rate are not statisticallysigniﬁcant. This is inconsistent with the empirical results obtained before the previous
subsection. Furthermore, as is obvious from the results shown in Table 3, the J-statistic
fails to reject the null that the moment conditions (18) are asymptotically satisﬁed at theestimated values ζ̂j for j = 0, · · · , 4. Conventionally, empirical researchers have consideredthat this speciﬁcation works well in the sense that the regression model (17) is correctly
speciﬁed and the instruments Qt are asymptotically uncorrelated with the error term.
However, some applications of GMM have what is known as “weak instruments” or“weak identiﬁcation”; that is, instruments that are only weakly correlated with the included endogenous variables. Ma (2002) and Mavroeidis (2005) point out that we suﬀer
from the weak identiﬁcation problem of the GMM in the context of NKPC. If instruments
are weak, then the sampling distributions of GMM are in general nonnormal, and standard
GMM point estimates, hypothesis tests, and conﬁdence intervals are unreliable. Therefore,
empirical researchers need to check whether or not the instruments are weak. In practice,there are methods more robust to weak instruments than conventional GMM. One central
ﬁnding of Stock and Wright (2000) is that the usual limiting distribution theory does not
apply, and this motivates the use of alternative inference methods. Furthermore, Kleibergen (2005) proposed test statistics (K statistics) that do not depend only on whether or
not the parameter is weakly identiﬁed but also on the number of moment conditions. If
the instruments are weak, the values of the K statistics evaluated at certain parameter
17

values, other than the true values, are very close to the minimized value, and therefore
the conﬁdence sets contain very large sets of parameters. A detailed test method basedon Kleibergen (2005) is presented in Appendix.
Figures 6 and 7 show 90% conﬁdence sets for ζ1 , ζ2 , and ζ3 based on the K statistics
concentrated with respect to ζ0 , ζ4 (shaded) in the GDP deﬂator inﬂation measure withand without the restriction ρ = 0 (i.e. ζ4 = 0), respectively. Figures 8 and 9 show
90% conﬁdence sets for ζ1 , ζ2 , and ζ3 based on the K statistics concentrated with respect
to ζ0 , ζ4 (shaded) in CPI inﬂation measure with and without the restriction ρ = 0 (i.e.
ζ4 = 0), respectively.
In Figures 6, 7, 8, and 9, we ﬁnd that the 90% conﬁdence sets for ζ1 , ζ2 , and ζ3 based on
the K statistics contain a much larger set of parameters than the conventional asymptotic
conﬁdence sets. This implies that it is possible that the instruments are weak. When the
instruments are weak, conventional asymptotic inferences fail even if the sample size islarge. This casts doubt on the results using GMM estimation.

4

Inﬂation Responses to a Monetary Policy Shock

From our empirical results in the previous section, we ﬁnd that (i) the existence of thecost channel for monetary policy and (ii) the use of labor share are key points to estimate
the NKPC. So, can the pure forward-looking NK model with the cost channel explain
the observed inﬂation responses to the monetary policy shock, which are hump-shaped
as shown in section 2?23 This section provides one plausible answer to this question byconsidering some characteristics that inﬂuence the demand-side and supply-side eﬀects ofmonetary policy on real marginal costs over time. We set up four simple endogenous variable dynamic systems to quantitatively investigate the role of cost channels for monetary
policy and the sluggish adjustment of real unit labor costs.
We use the simple NK dynamic general equilibrium model to calibrate the eﬀects of a23

Without the cost channel of the monetary policy, the pure forward-looking NK model always generates
an immediate decline of inﬂation after a contractionary monetary policy shock. This is because current
inﬂation depends on the current and future real unit labor costs, which always decline after a monetary
policy contraction. Then, inﬂation declines sharply back to its original level. Thus, as shown from thefollowing results in Figure 11, inﬂation responses to a monetary policy shock are not hump-shaped. Note
that, in the hybrid NKPC case, the inﬂation responses are hump-shaped, because inﬂation has an inertia
eﬀect. See Woodford (2003) for details.

18

tightening of monetary policy. In the model, the NKPC (16) as introduced in the previous
Section corresponds to the supply side. From (16) with ρ = 0, the aggregate supply side
is characterized by a pure forward-looking NKPC:π̂t = βEt π̂t+1 + κŝt + κψR r̂t .

(19)

We set β = 0.99. And, from the empirical results in section 3, we set κ = 0.01 and ψR = 1
as the baseline. In the case without the cost channel, we set ψR = 0.
The aggregate demand representation is derived from the Euler condition for the representative agent’s decision problem. Following Woodford (2003), we assume that theaggregate demand side is characterized by the following forward-looking IS (NKIS) curve:
yt = Et (yt+1 ) + σr (r̂t − Et π̂t+1 );(20)

and the empirical equation is expressed in the following form:
yt = σ0 + Et (yt+1 ) + σr (rt − Et πt+1 ) + error.

(21)

Many empirical studies add some forms of habit formation to the model in order to matchthe hump-shaped output responses that are apparent in the data. See Woodford (2003)
for details. Therefore, we replace Et (yt+1 ) with a term such as (1 − σy )Et (yt+1 ) + σy yt−1 .
Although empirical estimates of σy with quarterly data are hard to ﬁnd in the literature,
we set σy = 0.3. Under this parameter value, we can replicate the observed hump-shaped
output responses to a monetary policy shock as shown in the following empirical results.
For the NKIS function, the coeﬃcient of the real interest rate is expected to be negative.
The ﬁrst line in Table 4 shows that the estimate of σr in (21) obtained by using GMM isσ̂r = −0.175, which means that the detrended output has the correctly-signed correlation
with the real interest rate.24 Thus, in (20), we set σr = −0.175.
We can show the simple linear relationship between the real unit labor costs and thedetrended output in the standard model with no labor market rigidities.25 However, we
24

The set of instruments used include a constant and lags 1–5 of yt , st , πt , Δmt and rt .
Strictly speaking, in the presence of habit formation, the real unit labor costs relate expected detrended
output at current, future and past periods (see Woodford, 2003).
25

19

want to consider the case in which the real unit labor costs respond sluggishly to labormarket conditions as a result of some unmodeled imperfection or friction in labor markets.
To this end, we assume the partial adjustment model as follows:ŝt = g1 ŝt−1 + (1 − g1 )g2 yt ,(22)

and the empirical equation is expressed in the following form:
st = g0 + g1 st−1 + (1 − g1 )g2 yt + error,(23)

where g1 can be interpreted as an index of the persistent eﬀects of the real unit laborcosts. We view (22) as an admittedly ad hoc but simple way of modeling the sluggish
adjustment of real unit labor costs without taking a stand on what the right model is.
Assume that we cannot obtain accurate information about the current values of the output
gap yt . Therefore, we estimate (22) by using the GMM.26
The second line in Table 4 shows the GMM estimates of the partial adjustment equationof real unit labor costs as speciﬁed in (23). The estimates of the coeﬃcients for the lagged
real unit labor costs and the detrended output are ĝ1 = 0.761 and (1 − ĝ1 )ĝ2 = 0.124,
respectively. Thus, as in the baseline case, we set g1 = 0.761 and g2 = 0.124/(1 − 0.761) =0.519. If there is no persistency of real unit labor costs, we set g1 = 0.
We close the model with a simple monetary policy rule, in our case, the Taylor rule
(Taylor, 1993), an interest rate rule that reacts to deviations of inﬂation and output:r̂t = θr̂t−1 + (1 − θ)(ηy yt + ηπ π̂t ) + vt

(24)

where vt is a monetary policy shock. In (24), we set θ = 0.8, ηy = 0.5 and ηπ = 1.5.
It is instructive to compare the dynamics of the NK structural model (19), (20), (22),and (24) with given parameter values in the baseline to those of a VAR model. Figure 10
shows the responses of these two systems to a monetary policy shock. The VAR impulse
responses from Figure 1 are shown as solid lines and their 90% conﬁdence intervals asdashed lines. The impulse responses of the structural system are shown as solid lines with
26

The set of instruments used include a constant and lags 1–5 of yt , st , πt , Δmt , and rt .

20

an (◦). Although the responses of output are much larger in the model than appears tobe the case in practice, the empirical structural model and VAR impulse responses appear
to be quite similar overall. Thus, the empirical structural model appears to be very usefulas a benchmark, at least, to investigate the roles of cost channel for monetary policy andthe sluggish adjustment of real unit labor costs quantitatively as we will next discuss.
Now we examine the inﬂation responses to a monetary policy shock focusing on theroles of the presence of the cost channel and the sluggish adjustment of real unit labor costs.
Figure 11 shows the inﬂation responses to a monetary policy shock for some alternativevalues of the parameters, g1 , ψR . The solid lines with (◦) represent the simulated impulse
responses in the baseline case (g1 = 0.76, ψR = 1). The dashed lines with () represent
the simulated impulse responses with the restriction, ψR = 0. The solid lines with ()represent the simulated impulse responses with the restriction, g1 = 0. The dashed lines
with (∗) represent the simulated impulse responses with the restrictions, g1 = 0 andψR = 0.
We see from Figure 11 that the model fails to replicate the observed hump-shaped
responses for inﬂation when g1 = 0 and/or ψR = 0. The maximum or minimum responseof inﬂation occurs in the period of the monetary shock. This suggests that, within thepure forward-looking NKPC framework, the existence of the cost channel alone or thesluggish adjustment of real unit labor costs alone cannot generate the observed humpshaped behavior of inﬂation.
In addition, we see that the structural model can replicate the observed hump-shaped
responses for inﬂation only when g1 = 0 and ψR = 0. Here, we consider the roles of theexistence of the cost channel and the sluggish adjustment of real unit labor costs within
the pure forward-looking NKPC for the inﬂation responses to a monetary policy shock.
First, by considering the cost channel for monetary policy, monetary policy has significant eﬀects on the supply-side as well as the demand-side. As the demand-side eﬀect
of monetary policy, an increase in the monetary policy target rate reduces the demandon output, because of the inﬂuence on the real interest rate through the investment andsavings decisions of households and ﬁrms. Under monopolistic competition, the reduction of the current demand then decreases real unit labor costs through a labor demand
reduction. This leads to the exertion of downward pressure on inﬂation. On the other21

hand, with a cost channel of monetary policy, an increase in the monetary policy target
rate will have a direct eﬀect on all ﬁrms that rely on bank borrowing or loans of any kindlinked to short-term money market interest rates. A rise in interest rates, which increases
borrowing costs, puts upward pressure on inﬂation. Thus, changes in the interest rate
have an opposite inﬂuence on inﬂation, at least in the short-run. Second, by considering
the sluggish adjustment of the real unit labor costs, a monetary policy contraction has
persistent eﬀects on inﬂation due to the persistent reduction of the real unit labor costs.
By taking into account the monetary policy eﬀects on inﬂation as above, we can provide
a plausible explanation for the hump-shaped responses of inﬂation to a monetary policy
shock. An increase in the monetary policy target rate exerts downward pressure on current
inﬂation because of the reduction of current and future aggregate demand through theinvestment and saving decisions of households and ﬁrms. On the other hand, an increase in
the interest rate directly causes current and future borrowing costs to increase through thecost channel of the monetary policy. This eﬀect acts as a cost-push shock but endogenously,
and then removes the downward pressure on the current inﬂation on the demand-side.27
Over time, the upward pressure on inﬂation through the cost channel of the monetary
policy gradually disappears as the interest rate moves back to its baseline level. Although
detrended output also moves back to its baseline level, real unit labor costs persistently
decrease, and inﬂation falls because of the reduction of the current and future real unit
labor costs. This implies that the demand-side eﬀects eventually dominate the supply-side
eﬀects.
In this way, by considering the existence of the cost channel and the sluggish adjustment of real unit labor costs within the pure forward-looking NKPC, we can explain theempirical fact that there is a long time lag between a monetary policy shock and its impact
on inﬂation.

27

It is well known that a contractionary monetary policy shock identiﬁed by the VAR model is followed bya small and temporary rise in the price level. See Christiano et al. (1999) and Walsh (2003) for details. We
see from Figure 11 that our structural model even replicates inﬂation increases for 2 quarters. This implies
that the supply-side eﬀects initially dominate the demand-side eﬀects after a monetary policy tightening.
Rabanal (2007) provides some calibration results in the NK model with nominal and real rigidities following
Christiano et al. (2005), and points out that high real-wage stickiness and variable capital utilization rates(which lead to the sluggish adjustment of the production costs) and the presence of the cost channel areneeded to generate a positive response of inﬂation after a monetary policy contraction.

22

5Concluding Remarks

This study examines the empirical performance of the NKPC in explaining Japanese inﬂation dynamics by estimating a variety of its speciﬁcations. It then provides the implicationsof the empirical results for the time lags of monetary policy eﬀects on inﬂation. The main
message of this study is that both (i) the cost channel of the monetary policy and (ii)the sluggish adjustment of real unit labor costs play a critical role in understanding the
dynamics of inﬂation and the monetary transmission on it.
Some empirical results in this paper suggest that introducing nominal interest rates into
the pure forward-looking NKPC model, which implies the existence of the cost channel formonetary policy, helps improve their ability to explain Japanese inﬂation dynamics within
the NK economy. In addition to the existence of the cost channel for monetary policy,
they show that the use of labor share is an important factor in estimating the NKPC. As
an implication of these ﬁndings, we point out that both the existence of the cost channel
for monetary policy and the sluggish adjustment of real unit labor costs account for theempirical fact that there is a long time lag between a monetary policy shock and its impact
on inﬂation
Some avenues related to this study’s empirical ﬁndings are worth pursuing. Although
this study regards the cost channel of the monetary policy as a direct eﬀect of interest
rates on ﬁrms’ marginal costs and thus on inﬂation, we might be able to consider the
cost-side eﬀects of monetary policy through some alternative channels.
As Barth and Ramey (2001) point out, the cost channel may also run through thebroad credit channel due to the existence of ﬁnancial frictions as proposed by the ﬁnancialaccelerator and bank lending channel literature. Given that ﬁnancial intermediaries play
an important role in monetary policy transmission, when the ﬁrms are forced to turnto external funds, an increase in interest rates may aﬀect their marginal ﬁnancial costs
and the short-run productive capacity of the economy. Furthermore, the interest rateeﬀect may reﬂect other factors leading to cost pressure and, thus, to higher inﬂation.28 A
28Tsuruga (2007) proposes an NK general equilibrium model with a dynamic externality into the production function of ﬁrms, and shows that this model explains the hump-shaped response of inﬂation to amonetary policy shock. In his model, in addition to the demand-side eﬀects on inﬂation, contractionary
monetary policy shocks act as cost push shocks through a dynamic externality. Firms decrease their production and lose knowledge in response to an expansionary monetary policy shock. The decreased stockof knowledge causes productivity to decrease and, thus, increases marginal costs. In addition, Aoki (2001)23

detailed discussion about these issues is beyond the scope of this study, but is worthy of
future research.

Appendix: Inference in the presence of weak identiﬁcation
Letting the 2 × 1 identiﬁed parameters vector ψ = (ζ0 , ζ4 ) and the 3 × 1 possibly weakly

identiﬁed parameters vector φ = (ζ1 , ζ2 , ζ3 ) , we rewrite (17) as follows:

πt = ψ Υ1t + φ Υ2t + error,


(25)



where Υ1t = (1, πt−1 ) and Υ2t = (Et πt+1 , st , rt ) . In such circumstances, valid conﬁdence
sets can be based on the minimand of the restricted GMM estimation in which the minimization is performed over the identiﬁed parameter, ψ, conditional on a value for the
weakly identiﬁed parameter, φ.
Let ψ̂T (φ̃) denote the GMM estimator of ψ conditional on φ = φ̃; that is:ψ̂T (φ̃) = arg min ST (ψ, φ̃; ψ̄T (φ̃))|φ=φ̃ ,
ψ∈Ψ

(26)



where ST (ψ, φ̃; ψ̄T (φ̃)) = T gT (ψ, φ̃) ΣT (ψ̄T (φ̃))gT (ψ, φ̃), ψ̄T (φ̃) is a preliminary estimatorof ψ, gT (ψ, φ̃) = T −1T
t=1 gt (ψ, φ̃), gt (ψ, φ̃) = Qt (πt − ψ Υ1t − φ̃ Υ2t ), and ΣT (ψ̄T (φ̃))

is a positive deﬁnite weighting matrix that is computed using ψ̄T (φ̃). Stock and Wright(2000) show that:
ST (φ̃, ψ̂T (φ̃)) → χ2q−2 ,d(27)

where χ2q−2 denotes a chi-square distribution with q − 2 degrees of freedom. Therefore,
they propose the following asymptotically valid 100(1 − a)% conﬁdence sets for φ̃:{φ̃ : ST (φ̃, ψ̂T (φ̃)) < cq−2 (a)},(28)

and Carvalho (2006) model heterogeneity in the frequency of price changes and show that, in the presenceof strategic complementarities in price setting, sectors with lower frequencies of price adjustment have adisproportionate eﬀect on the aggregate price level. In their model, contractionary monetary policy shocks
act as cost-push shocks through a relative price change.

24where cq−2 (a) is the 100(1 − a) percentile of the χ2q−2 distribution. Stock and Wright(2000) refers to such sets as concentrated S-sets.
Although the S-sets are robust to weak identiﬁcation, even if the number of the parameters is ﬁxed, the power of the statistics ST (φ̃, ψ̂T (φ̃)) is reduced as the number ofmoment conditions q increases, so that the statistics ST (φ̃, ψ̂T (φ̃)) have a χ2q−2 limiting
distribution, and the critical value cq−2 (a) then becomes large. Concerning this problem
with regard to the low power of the test statistic, Kleibergen (2005) proposed the K statistic, which does not depend only on whether or not the parameter is weakly identiﬁed but
also on the number of the moment conditions. Under the null hypothesis H0 : φ = φ̃, Kstatistic KT (φ̃, ψ̂T (φ̃)) is as follows:
1
4T



 

∂S((φ , ψ ) )∂(φ̃ , ψ̂T (φ̃) )

−1
D̂T ((φ̃ , ψ̂T (φ̃) ) ) ΣT ((φ̃ , ψ̂T (φ̃) ) )−1 D̂T ((φ̃ , ψ̂T (φ̃) ) )


  





 

 





 

 







 

∂S((φ , ψ ) )∂(φ̃ , ψ̂T (φ̃) )

,

(29)

 

where D̂T ((φ̃ , ψ̂T (φ̃) ) ) = (D̂0,T ((φ̃ , ψ̂T (φ̃) ) ) · · · D̂4,T ((φ̃ , ψ̂T (φ̃) ) )) and1
D̂j,T ((φ̃ , ψ̂T (φ̃) ) ) =
T


T

 

t=1

−

T


1
T t=1



 



 

∂gt ((φ̃ , ψ̂T (φ̃) ) )∂ζj∂gt ((φ̃ , ψ̂T (φ̃) ) )
∂ζj



 



 



 

gt ((φ̃ , ψ̂T (φ̃) ) ) ΣT ((φ̃ , ψ̂T (φ̃) ) )−1 gT ((φ̃ , ψ̂T (φ̃) ) )for j = 0, · · · , 4. Kleibergen (2005) propose the following asymptotically valid 100(1− a)%conﬁdence sets for φ̃.
{φ̃ : KT (φ̃, ψ̂T (φ̃)) < c3 (a)}(30)

where c3 (a) is the 100(1− a) percentile of the χ23 distribution. If the instruments are weak,the values of the K statistics KT (φ̃, ψ̂T (φ̃)) evaluated at certain parameter values φ̃, otherthan the true values, are very close to the minimized value, and so the conﬁdence setsdeﬁned by (30) contain very large sets of parameters.

25

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29

Table 1: Estimates of NKPC Based on the Labor Share
πt

Model
No Interest RateGDP
DeﬂatorInﬂation
Rate

Hybrid:No Interest Rate
IncludingInterest Rate
Hybrid: IncludingInterest Rate
No Interest Rate

CPI
Inﬂation
Rate

Hybrid:No Interest Rate
IncludingInterest Rate
Hybrid: IncludingInterest Rate

μ1
-0.002(0.001)-0.001(0.001)0.007(0.001)0.007(0.002)-0.003(0.001)-0.001(0.001)0.014(0.001)
0.013(0.002)

μ2

ρ

0.426
(0.083)0.009(0.001)
0.009(0.001)

0.013(0.096)0.682(0.062)0.014(0.001)
0.013(0.002)0.075(0.094)adj-R20.015[-0.007, 0.524]
0.222[0.051, 0.538]
0.454[0.158, 0.579]
0.448[0.135, 0.581]
0.042[-0.006, 0.676]
0.579[0.377, 0.740]
0.737[0.543, 0.809]
0.736[0.550, 0.817]

Notes: Figures in parentheses are standard errors. Figures in brackets are 90% conﬁdence
intervals from bootstrapping.

30

Table 2: Estimates of NKPC Based on the Detrended Output
πt

Model
No Interest RateGDP
DeﬂatorInﬂation
Rate

Hybrid:No Interest Rate
IncludingInterest Rate
Hybrid: IncludingInterest Rate
No Interest Rate

CPI
Inﬂation
Rate

Hybrid:No Interest Rate
IncludingInterest Rate
Hybrid: IncludingInterest Rate

λ1
-0.007(0.012)0.000(0.011)-0.032(0.010)-0.027(0.011)-0.003(0.001)
0.003(0.011)-0.040(0.011)-0.012(0.011)

λ2

ρ1

0.438
(0.083)0.006(0.001)0.006(0.001)

0.101(0.097)0.701(0.064)0.006(0.001)
0.003(0.001)0.493(0.082)

adj-R2-0.007[-0.010, 0.410]
0.214[0.040, 0.469]
0.389[0.113, 0.578]
0.389[0.130, 0.584]
0.031[-0.009, 0.547]
0.573[0.377, 0.709]
0.483[0.275, 0.801]
0.624[0.451, 0.802]

Notes: Figures in parentheses are standard errors. Figures in brackets are 90% conﬁdence
intervals from bootstraping.

31

Table 3: GMM Estimates of NKPC Based on the Labor Shareπt
GDP
DeﬂatorInﬂation
Rate
CPI
Inﬂation
Rate

Model
IncludingInterest Rate
Hybrid: IncludingInterest Rate
IncludingInterest Rate
Hybrid: IncludingInterest Rate

ζ1
0.804
(0.158)0.974(0.229)0.943(0.089)0.815(0.117)

ζ2
0.026(0.025)0.021(0.028)0.015(0.021)-0.003(0.020)ζ3
0.032(0.033)0.019(0.040)0.019(0.018)0.015(0.020)ζ4-0.113(0.090)0.226(0.100)J
7.396[0.881]6.677[0.878]
10.576[0.646]
11.500[0.487]

Notes: Figures in parentheses are Newey and West (1987) heteroskedasticity and autocorrelation consistent standard errors corrected with four lags. J denotes the test statistic of
the model’s overidentifying restrictions, with p-value in brackets. The set of instruments
used included a constant and lags 1–4 of πt , πtf , st and rt .

32

Table 4: Estimates of NKIS and Partial Adjustment Equation of Real Unit Labor Costs
Model
(21)
Model
(23)

σr
−0.175(0.012)g1
0.761(0.012)

g1 (1 − g2 )0.124
(0.039)J
15.953[0.890]J
13.698[0.935]

Notes: Figures in parentheses are Newey and West (1987) heteroskedasticity and autocorrelation consistent standard errors corrected with four lags. J denotes the test statistic of
the model’s overidentifying restrictions, with p-value in brackets. The set of instruments
used included a constant and lags 1–5 of yt , st , πt , Δmt , and rt .

33

Response of GDP Deflator Inflation

Response of WPI Inflation
0.5

0.2

0

0
-0.5

-0.2
-0.4

0

4

8

12

-1

16

0

Response of Output Gap

4

8

12

16

12

16

12

16

Response of Labor Share
1

0.2
0

0

-0.2
-0.4
-0.6

0

4

8

12

-1

16

0

Response of Interest Rate

48

Response of Money Growth

1

0.2

0

-0.2

0
-0.4
-1

0

4

8

12

-0.6

16

0

4

8

Figure 1: Impulse Response to a Monetary Policy Shock (GDP deﬂator). Note: The solidlines are the estimated impulse responses. The dashed lines are 90% conﬁdence intervals
about the impulse responses based on bootstrapping.

34

Response of CPI Inflation

Response of WPI Inflation
0.5

0.2

0

0
-0.5

-0.2
-0.4

0

4

8

12

-1

16

0

Response of Output Gap

4

8

12

16

12

16

Response of Labor Share

0.5

1

0
0
-0.5
-1

0

4

8

12

-1

16

0

Response of Interest Rate

48

Response of Money Growth

1.5

0.5

1
0.5

0

0
-0.5

0

4

8

12

-0.5

16

0

4

8

12

16

20

Figure 2: Impulse Response to a Monetary Policy Shock (CPI). Note: The solid lines are
the estimated impulse responses. The dashed lines are 90% conﬁdence intervals about theimpulse responses based on bootstrapping.

35

(a) No Interest Rate(b) Hybrid: No Interest Rate

3

3
Recession
Phase
Fit of NKPC
GDP deflator

2.5
2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1
1975

1980

1985

1990

-1
1975

1995

(c) Including Interest Rate

1980

1985

1990

1995

(d) Hybrid: Including Interest Rate
3

3
Recession
Phase
Fit of NKPC
GDP deflator

2.5

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

1980

1985

1990

Recession
Phase
Fit of NKPC
GDP deflator

2.5

2

-1
1975

Recession
Phase
Fit of NKPC
GDP deflator

2.5

-1
1975

1995

1980

1985

1990

1995

Figure 3: Japanese fundamental (solid line) and actual (dotted line) inﬂation (GDP deﬂator)

36

(a) No Interest Rate(b) Hybrid: No Interest Rate
3

3
Recession
Phase
Fit of NKPC
CPI

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1
1975

1980

1985

1990

-1
1975

1995

(c) Including Interest Rate

1980

1985

1990

1995

(d) Hybrid: Including Interest Rate

3

3
Recession
Phase
Fit of NKPC
CPI

2.5

2

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

1980

1985

1990

Recession
Phase
Fit of NKPC
CPI

2.5

2

-1
1975

Recession
Phase
Fit of NKPC
CPI

2.5

-1
1975

1995

1980

1985

1990

1995

Figure 4: Japanese fundamental (solid line) and actual (dotted line) inﬂation (CPI)37

GDP deﬂator inﬂation
㪈㪌㪇

㪈㪇㪇

㪌㪇

㪇
㪄㪈㪌㪇

㪄㪈㪇㪇

㪄㪌㪇

㪇

㪌㪇

㪈㪇㪇

㪈㪇㪇

㪈㪌㪇

㪄㪌㪇

㪄㪈㪇㪇

㪄㪈㪌㪇

CPI inﬂation

㪈㪌㪇

㪈㪇㪇

㪌㪇

㪇
㪄㪈㪇㪇

㪄㪌㪇

㪇

㪌㪇

㪄㪌㪇

㪄㪈㪇㪇

㪄㪈㪌㪇

㪄㪉㪇㪇

Figure 5: Scatter Plot: Present Value of Future Labor Share (deviation from the average, horizontal axis) and Future Interest Rate (deviation from the average, vertical axis)
estimated by using the auxiliary VAR model (10)

38

39

Figure 6: 90% Conﬁdence Sets for ζ1 , ζ2 and ζ3 based on the K statistics, sliced in ζ1 dimension: NKPC Including Interest Rate (GDP deﬂator
Inﬂation)

40

Figure 7: 90% Conﬁdence Sets for ζ1 , ζ2 and ζ3 based on the K statistics, sliced in ζ1 dimension (shaded): Hybrid NKPC Including Interest
Rate (GDP deﬂator Inﬂation)

41

Figure 8: 90% Conﬁdence Sets for ζ1 , ζ2 and ζ3 based on the K statistics, sliced in ζ1 dimension: NKPC Including Interest Rate (CPI Inﬂation)

42

Figure 9: 90% Conﬁdence Sets for ζ1 , ζ2 and ζ3 based on the K statistics, sliced in ζ1 dimension: Hybrid NKPC Including Interest Rate (CPIInﬂation)

Response of Inflation

Response of Output Gap
0.5

0.2
0.1

0

0
−0.1

−0.5

−0.2
5
10
15
Quarters After ShockResponse of RULC

51015Quarters After ShockResponse of Interest Rate

0.5

1

0

0.5
0

−0.5

−0.5
−1

5
10
15
Quarters After Shock5
10
15
Quarters After Shock

Figure 10: Impulse Responses for VAR and Structural Model of the inﬂation, detrended
output, real unit labor costs and interest rates. Note: The solid lines with (◦) represent
the simulated impulse responses in the baseline. The solid lines and dashed lines are from
Figure 1.

43Response of Inflation
0.01

0.005

0

−0.005g =0.76, ψ =1

−0.01

1

R

g1=0.76, ψR=0
g1=0, ψR=1

−0.015g1=0, ψR=0
−0.02

2

4

6
8
10
Quarters After Shock

12

14

16

Figure 11: Comparion of the Simulated Inﬂation Responses to a Monetary Policy Shock inthe estimated NK Model: Note. The solid lines with (◦) represent the simulated impulse
responses in the baseline. The dashed lines with () represent the simulated impulse
responses with the restriction, ψR = 0. The solid lines with () represent the simulated
impulse responses with the restriction, g1 = 0. The dashed lines with (∗) represent thesimulated impulse responses with the restrictions, g1 = 0 and ψR = 0.

44