Discussion Paper Series No.174
Estimating a Hybrid New Keynesian Phillips Curve for Japan
Kazuhiko Nakahira
Associate Professor, Department of Business Administration and Information
Tokyo University of Science, Suwa

December 2011

THE INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Estimating a Hybrid New Keynesian Phillips Curve for Japan

Kazuhiko Nakahira
Associate Professor, Department of Business Administration and Information
Tokyo University of Science, Suwa

Abstract
This paper evaluates the applicability of the hybrid variant of the New Keynesian Phillips curve(NKPC) for Japan since 1980s through GMM estimation. The result of the empirical study leads us toconclude that the pure forward-looking hypothesis is rejected. With regard to the estimated structuralparameters, the coefficient on the degree of backwardness in price setting is larger for the period after
the collapse of the bubble economy than before. The coefficient on the degree of price rigidity is also
larger after the bubble crash leading to the longer average duration of price adjustment. The
estimated values of the discount factor are very close to the theoretically postulated level.

Keywords : New Keynesian Phillips curve; Inflation; GMM
JEL classification : E31, E52

11. Introduction
To study the evolution of aggregate price is one of the prominent topics in empiricalmacroeconomics, and a clear understanding of the inflationary process is necessary in effectiveplanning of a monetary policy for price stabilization. The so-called New Keynesian Phillips curve(NKPC), which is well established by microeconomic foundations with the New Keynesian DSGE
(Dynamic Stochastic General Equilibrium) framework, is recently the most canonical apparatus toinvestigate this kind of issue. In other words, the shift in recent emphasis from the traditional Phillips
curve to the New Keynesian Phillips curve is due to the inability of the former to grasp thedevelopments of today’s inflationary processes in several countries. Actually, it is often reported thatsome countries with high levels of economic activities are accompanied by relatively low levels of
inflation that cannot be explained by the traditional framework.
The New Keynesian Phillips curve describes the link between inflation and economic activities
based on the firms’ price-setting behaviours, marginal costs, and various economic activities.
Concretely, it incorporates two significant factors: (1) The forward-looking character of inflation which
depends on the firm’s price-setting manner with their expectations of demands and costs in the future,(2) The linkages between inflation, real economic activity, and marginal cost.
Concomitantly, literature on the New Keynesian Phillips curve continues to increase. For instance,
Woodfood (1996), Goodfriend and King (1997), and Clarida, Galí, and Gertler (1999) study theaggregate inflation through short-run nominal rigidity. Coenen, Levin, and Chilistoffel (2007)
investigate the interaction between real and nominal rigidities. Galí and Gertler (1999) and Galí,
Gertler, and López-Salido (2005) describe the importance of the lagged inflation term in their model
considering the gradual response of inflation to the monetary policy shocks. Sbordone (2002), Galí andGertler (1999), and Galí, Gertler, and López-Salido (2001) insist that real marginal cost is the key
element to analyze inflation dynamics in the U. S. and the Euro area by their NKPC. From the aspectof model’s specification, Zhang and Clovis (2010) conclude that further lags of inflation are necessary
in the hybrid-type NKPC to rule out serial correlation. Christiano, Eichenbaum, and Evans (2005)
incorporate the dynamic inflation indexation as a modification to the Calvo-type sticky and staggered
price setting, while Yun (1996) applies the indexation to the inflation in steady-state. Furthermore,
Smets and Wouters (2003) and Giannoni and Woodford (2005) utilize the partial dynamic inflationindexation. On the other hand, some of the recent studies deal with the flattening of the NKPC. For
instance, Kuester, Müller, and Stölting (2009) insist that the NKPC looks flatter than it actually is byconsidering the estimated pass-through of marginal costs.
This paper proceeds to investigates whether the hybrid version of NKPC, which incorporates bothbackward- and forward-looking elements, provides a good description of the evolution of inflation in
Japan since 1980s. Empirical study is conducted by GMM (Generalized Method of Moments)
estimation for the periods before and after the collapse of Japan’s bubble economy happened in theearly 1990s.

2

The reminder of this paper is organized as follows. Section 2 explains the basic formulation of the
New Keynesian Phillips curve. Section 3 examines the result of GMM estimations, and Section 4
presents the concluding remarks.

2. The Structure of New Keynesian Phillips Curve
2.1 The Basic Formulation of New Keynesian Phillips Curve
We are able to derive NKPC by proceeding as follows.1 The business sector is assumed to be acontinuum of monopolistic competitor indexed bywith a nominal price, and produces a differentiated good

. Firm faces an isoelastic demand curve given by

production function for firm
, where. The

is given by a special type of Cobb-Douglas technology:

is a technological factor,

is the fixed firm specific capital stock, andis the employment.
Households are assumed to be paid the nominal wage
cost of production. The Dixit-Stiglitz-type aggregate price
,

(1)

,

(2)

, and each firm faces the same nominaland output

are represented by:where is the constant price elasticity of demand. In this model, because investment and foreign
trade are abstracted, output

equals consumption

Without any price frictions, firms would set price level.
which maximizes real profit at any

given time. The optimization framework gives the markup equation:, whererepresents the fixed markup and mc is the log nominal marginal cost. In this framework,firms set nominal prices in the Calvo (1983)-type staggered fashion facing constraints on thefrequency of price adjustment. With this specification, the probability that a firm resets the price inany period t is

, denoting as a measure of the degree of price rigidity. Since this probability is

time-independent, the mean lag (or duration) of price adjustment becomes. Therefore, a measureof producers reset their prices, while a fraction remains unchanged. By applying the propertyof law of large numbers and log linearization of the price index around the steady state of zeroinflation, we have the following expression for the evolution of log pricethe log of lagged price level.
1

and the log of newly optimized priceas a convex combination of:(3)

See Goodfriend and King (1997), Galí, Gertler, and López-Salido (2001), or Scheufele (2010) for an explicit

derivation.

3

All firms that reset price in period t choose the same value ofsince there are no firm-specific statevariables. In addition, with the given technology, factor prices, and the constraint on price adjustment,, a firm which resets its price in period t tries to maximize the expected

and the reset probability

discounted profits. Considering these elements, the Calvo-type optimized reset price can be described
as2
,

(4)

where is a subjective discount factor andmeans the logarithm of nominal marginal cost attime t+k of a firm which last change its price at time t. This specification implies that firms which
reset prices in period t will take into consideration the each expected future stream of nominal
marginal cost expressed in percent deviation from the steady state value with the chance that newly
reset price might be subject to the adjustment constraints in the future. Thus, prices are expected toremain unchanged for an extended period, and firms place more weight on expected marginal costs
when they set current prices as increases.
The next problem is to find a plausible expression of marginal cost in equation (4) as an observablemeasure. If we assume a simple Cobb-Douglas production function, we have,

(5)

where Yt is production, At refers to technology, Kt denotes capital, and Nt is labor. A Cost
minimization with this technology implies that the real marginal cost equals the real wage divided by
the marginal product of labor. Therefore, the real MC at time t+k for a firm which optimally sets priceat time t is given by:
(6)
where

represents output,

indicates employment, and is the curvature of the production

function for a firm which has set its price in period t at the optimal value. From the aspect of thefact that the real MC of individual firm is unobservable, it is helpful to define the average marginal
cost depending only on aggregates:
,
where

(7)

is the labor share (or real unit labor costs).3 Letting lower case letters describe

percentage deviations from each steady-state value, it becomes.

(8)

Making the assumption of Cobb-Douglas technology with isoelastic demand curve following Woodford(1996), Galí, Gertler, and López-Salido (2001), and Sbordone (2002), we have the log-linear connection
betweenand:2

The fixed markup ( ) is disappeared because all variables are expressed in deviation from steady state.

3

Equation (7) is derived as

.

4

,
where

and

(9)

are the deviation in logarithm of

andfrom their steady-state values.4 Combination of equations (3), (4), and (9) gives the basic formulation of (marginal-costbased) New Keynesian Phillips curve (NKPC):5
,

(10)

where
.

(11)

The slope coefficient is decreasing in (the frequency of price adjustment). Thus, a smaller fractionof firms resetting their prices implies inflation will less sensitive to the evolutions of marginal cost.
Since it is also decreasing in (the elasticity of substitution between factor inputs or the curvature of
the production function) and (the elasticity of demand), the larger and lead the more sensitive
marginal cost to the deviation of price from the average level.
2.2 The Hybrid Model of New Keynesian Phillips Curve
The basic New Keynesian Phillips curve expressed in equation (10) postulates relatively low
persistence of inflation. It is, however, not always consistent with actually observed inflation dynamics
or not data coherent due to price rigidities. An alternative formulation of the NKPC considering thisfactor proposed by Galí and Gertler (1999) and Galí, Gertler, and López-Salido (2001) incorporates the
backward-looking component or lagged dependence of inflation, as well as the forward-looking
element.6 The derivation of this “hybrid model” starts with the modification of the Calvo-type contract
by introducing two kinds of firms. A subsample of firms
behavior, while the remaining fraction

has forward-looking price-setting

set their prices with a backward-looking rule of thumb.

Therefore, the aggregate price level is given by the equation:,
where

represents the index of prices at time t such that
,

where

(12)
(13)

is the price for backward-looking rule of thumb andis the price for forward-looking firms

which behave just as basic Calvo-type sectors. Thus, the behavior of forward-looking firms can be
described as
4

In the case of linear technology or constant returns to labor (), all firms are confronted with the same

marginal cost.
5

Real marginal cost can be expressed as a related variable of the output gap. Following this condition, the output-

gap-based New Keynesian Phillips curve can be derived. For the concrete discussions, see Walsh (2010), Galí(2008), and Woodford (2003).
6

This kind of specification is regarded as a “hybrid-type” NKPC in the sense that it incorporates both forward-

and backward-looking components.

5

.

(14)

Galí and Gertler (1999) assume that backward-looking firms follow a rule of thumb behavior based onrecent aggregate pricing. In this sense,.

can be expressed as:(15)

Since forward-looking firms set prices as the markups over their marginal costs and fix prices
probably more than one period, their decisions over prices are based on expected future streams of
marginal costs. On the other hand, backward-looking firms fix prices by referring to the equilibrium
levels in the previous period.
Totally, combination of equations (10) through (15) derives the reduced-form specification of the(marginal-cost-based) hybrid NKPC:
,

(16)

where
,
,

(17)
(18)

,

(19)
.

(20)

This hybrid specification can be regarded as a special case of the basic formulation of NKPC describedby equations (10) and (11) with no backward-looking element ().

3. Empirical Results
The section 3 is for our fundamental challenge with the estimations of the parameters
incorporated in the hybrid NKPC discussed in the previous section by utilizing the Japanesequarterly data spanning the period 1980:1 to 2010:4. Our data set is constructed by the following
variables.7
Yr: real GDP (quarterly, chain-linked estimates, first preliminary estimates, reference year: 2000,
seasonally adjusted, billion yen)
Df: GDP deflator (quarterly, first preliminary estimates, reference year: 2000, seasonallyadjusted)
Cp: consumer price index (monthly, excluding fresh food, whole Japan, total, reference year: 2005)

7 The data on “GDP deflator” and “compensation of employees” are obtained from the Economic and Social

Research Institute, Cabinet Office’s website (in English) (http://www.esri.cao.go.jp/en/sna/). The “employed person”
and “employee” are retrieved from the “Portal Site” of Official Statistics of Japan administered by the Ministry ofInternal Affairs and Communications, Statistics Bureau, Director-General for Policy Planning (Statistical
Standards)&

Statistical

ResearchandTraining

stat.go.jp/SG1/estat/eStatTopPortalE.do).

6

Institute(inEnglish)(http://www.e-Cn: compensation of employees (quarterly, chain-linked estimates, first preliminary estimates,
reference year: 2000, seasonally adjusted, billion yen)
Ee: employee (monthly, whole Japan, total, seasonally adjusted)
Ep: employed person (monthly, whole Japan, total, seasonally adjusted)
Wp: nominal wage per capita (= Cn / Ee)
Lp : labor productivity (= Yr / Ep)
Uc: unit labor cost (= Wp / Lp)
Ls: labor share (or real unit labor costs) (= Uc / (Df /100))
Lc: trend component of Ls obtained by the Hodrick-Prescott filter8 setting the penalty parameter= 1600
Lg: proxy variable for) = log(Ls) － log(Lc)

The monthly data on “consumer price index”9, “employee”, and “employed person” were converted intoquarterly series by taking three-months averages. As to the inflation rate constituted by the consumer
price index, four-quarter moving averageis adopted. One problem in our estimation is how we have the proxy for themarginal cost in equation (16) (and (8)). We utilize “Lg” indicated above as ourwith considerationof the characteristic of Japanese data regarding the trend component estimated by the HodrickPrescott filter could be the proxy for the steady-state value. Another problem that we confront is thecorrelation due to the causal relationship between the variables. The expected inflation
obliged to be replaced by actually observedisunder the assumption of rational expectation since it

is unobservable. Thus, we set( : expectational error). However, thistreatment may cause the correlation between the error term and the explanatory variable. To copewith this problem, GMM (Generalized Method of Moments) is applied to our estimation.
The reduced-form coefficient expressed in equation (17) is a function of, but wecannot estimate all these structural parameters because of the identification restriction. One plausible
strategy to deal with this problem is as follows. Let us define

as a function ofand . Next, suppose the special case of constant , in other words, the case of constant returns toscale or constant marginal costs across firms.10 If we take advantage of this assumption following Galí,
Gertler, and López-Salido (2001) and Maturu, Kisinguh, and Maana (2007), we can regard
Plugging

as 1.

into equation (17), we have.

(21)

8 See Hodrick and Prescott (1997) for a concrete discussion.
9

Seasonally non-adjusted series of consumer price index was converted into a seasonally adjusted series by

Eviews (Ver. 6.1) applying X-12-ARIMA. The spec file for X-12-ARIMA was adjusted as close as possible to those
applied to the indices of industrial production by the Ministry of Economy, Trade and Industry. See theinterpretive article at (http://www.meti.go.jp/english/statistics/tyo/syoudou/pdf/h2snotee.pdf).
10

In this case, capital is assumed to be mobile freely across firms.

7

With this specification, we are able to estimate the parameters , , and . The corresponding
orthogonality condition for our estimation is constructed as:
,
where

(22)

denotes the vector of instrumental variables.

Instrumental variables dated t–1 and earlier are set to construct

by the following two reasons: (і)

The public may not utilize all the current information when they form their expectations, (іі) Certain
level of measurement errors of

may exist, but the errors may not be correlated with lagged

instruments (as the past information). In addition, small number of lags of the proxy forthan theones of inflation rate is chosen to minimize the potential estimation bias following Galí, Gertler, andLópez-Salido (2001).11 Specifically,

includes five lags of inflation rate and four lags of the proxy for

marginal cost (Lg).
The estimation periods are divided into two categories – the periods before and after the collapse ofJapan’s bubble economy in the early 1990s. It is not easy to definitely define the end of bubble
economy. However, the peak of the 11th business cycle determined by the Working Group of Indexesof Business Conditions at the Economic and Social Research Institute, Cabinet Office (Government ofJapan) is February 1991. Taking into account this definition, the first quarter of 1991 is regarded as
the end of the bubble economy and the second quarter of 1991 is set as the start date of the period“after the bubble” in this study for the sake of convenience.
Our model of the hybrid NKPC allows us to estimate the reduced-form and the structuralparameters. In particular, the estimation of the latter enables us to examine the impacts of structuralparameters on inflation dynamics. The estimation results are summarized in Table 1 and Table 2.
The null hypotheses of over-identification for all GMM estimations cannot be rejected by the Hansen’s
tests. (See each test statistic in notes under each table.)
Table 1 displays the result of estimation for the period before the collapse of the Bubble Economy.
Concerning the reduced-form parameters, the estimate for the reduced-form coefficient
inflation is significant. Also,

on futureon lagged inflation is significant. The fact that the estimated value ofthe latter coefficient is larger than that of the former implies the backward-looking behaviour is
predominant over inflationary process in the period we concern. This finding seems to reject the pureforward-looking specification. The coefficienton the marginal cost (or the slope of NKPC) issignificant, and this means the marginal cost is a kind of leading indicator of inflation.
The structural parameter , the degree of backwardness in price setting, is significant. This result
is consistent with the significance ofon backward-looking component in reduced-form estimation.

With respect to , which is for the measure of the price stickiness (or for the fraction of firms thatkeeps price constant), is also significantly estimated. The average duration of a price remaining fixed(in quarters) corresponding to the estimate of is 3.724520. It means that the high frequency in priceadjustment is not always observed so far as this estimation period is concerned. On the other hand,

11 Galí, Gertler, and López-Salido (2001) insist that this bias is generated in small number of samples when there

are too many overidentifying restrictions.

8

Table 1: GMM Estimation of Inflation before the Collapse of the Bubble Economy
reduced-form parameter
coefficient
standard error
p-value
0.433501
0.013659
0.00000.566344
0.013630
0.00000.024628
0.007877
0.0036
structural parametervariablecoefficient
standard error
p-value
0.559501
0.043282
0.00000.731509
0.026377
0.00000.999379
7.84E-05
0.0000duration
3.724520
Notes (reduced-form parameter): J-statistic = 3.734937, p-value = 0.712495, Included
observations = 37 (after adjustments). Convergence achieved after 7 weight matrices, 8 totalcoefficient iterations. S.E. refers to Newey-West heteroscedasticity and autocorrelation consistent
standard error (Kernel: Bartlett, Bandwidth: Fixed (3)).
variable

Notes (structural parameter): J-statistic = 3.732653, p-value = 0.712802, Included observations =37 (after adjustments). Convergence achieved after 12 weight matrices, 13 total coefficient
iterations. S.E. refers to Newey-West heteroscedasticity and autocorrelation consistent standard
error (Kernel: Bartlett, Bandwidth: Fixed (3)).

Table 2: GMM Estimation of Inflation after the Collapse of the Bubble Economy
reduced-form parameter
coefficient
standard error
p-value
0.475463
0.006563
0.00000.524530
0.006562
0.00000.012156
0.008122
0.1386
structural parametervariablecoefficient
standard error
p-value
0.690140
0.052879
0.00000.761407
0.051795
0.00000.999956
5.37E-05
0.0000duration
4.191238
Notes (reduced-form parameter): J-statistic = 7.830030, p-value = 0.250822, Included
observations = 78 (after adjustments). Convergence achieved after 14 weight matrices, 15 totalcoefficient iterations. S.E. refers to Newey-West heteroscedasticity and autocorrelation consistent
standard error (Kernel: Bartlett, Bandwidth: Fixed (3)).
variable

Notes (structural parameter): J-statistic = 7.829751, p-value = 0.250843, Included observations =78 (after adjustments). Convergence achieved after 15 weight matrices, 16 total coefficient
iterations. S.E. refers to Newey-West heteroscedasticity and autocorrelation consistent standard
error (Kernel: Bartlett, Bandwidth: Fixed (3)).

the estimated value of the discount factor is 0.999379, and it is very close to the theoretically
postulated level.12
Table 2 indicates the estimated parameters with respect to the period after the collapse of bubble
economy. As to the reduced-form parametersand, the estimated value of the latter are larger

than that of the former. This implies the backward-looking component is relatively dominant than the12

For instance, Christiano, Eichenbaum, and Evans (2005) regard this as = 1.03－0.25. This can be interpreted as β

= 1.03－0.25 ≈ 0.99. Erceg, Henderson, and Levin (2000), Giannoni and Woodford (2003), Steinsson (2003), Walsh(2003), and Christiano, Eichenbaum, and Evans (2005) also assume = 0.99.

9forward-looking one in shaping inflation dynamics. In this sense, the pure forward-looking
specification is rejected again. The coefficient estimated on is not significant. This implies that the
impact of marginal cost on inflationary process in Japan after the bubble economy is very low or
negligible.
The significantly estimated coefficient of the backward-looking price setting measured by the size ofis 0.690140, which corresponds to the estimates of reduced-form parameters

over 0.5. What ismore, the estimated is little bit larger than the one for the period before the bubble crash leading to
the comparatively longer average duration of price adjustment around 4.2 quarters. This value
suggests that price remain unchanged for roughly 13 months. Lastly, the estimate of , the discountfactor, is 0.999956, and this is again close to the usually postulated level.
Comparing the results of estimations for the periods before and after the collapse of the bubble
economy, we know that the reduced-form coefficient(on lagged inflation) as well as(on future

inflation) are significantly estimated in both cases, and this finding seems to deny the pure forwardlooking hypothesis in the recent two decades. Moreover, the influence of marginal cost on inflation
dynamics after the bubble economy is weaker than that before the collapse of the bubble. From
another aspect, the slope of the NKPC might become flatter after the collapse of the bubble economy.
With respect to the structural parameters,(the degree of backwardness in price setting) is larger inthe latter period, and the (the measure of the price stickiness) is also larger in the period after the
bubble leading to the longer average duration of price adjustment. The estimated values of discountfactor are around 0.99 in both periods.

4. Concluding Remarks
This paper evaluates the applicability of the hybrid variant of the New Keynesian Phillips curve(NKPC) for Japan since 1980s through GMM estimation. By the comparison between the results of
estimations for the periods before and after the collapse of bubble economy, it is apparent that the
reduced-form coefficient on lagged inflation as well as the one on future inflation is significantly
estimated in both periods, and this finding virtually rejects the pure forward-looking hypothesis in the
recent two decades. In addition, the impact of marginal cost on inflation dynamics for the former
period is significant, while the one for the latter period is negligible. From another aspect, the slope ofthe hybrid NKPC might become flatter after the collapse of the bubble. With regard to the structuralparameters, the estimated coefficient on the degree of backwardness in price setting is larger after the
bubble crush and the one on the price stickiness is also larger in the latter period leading to the longer
average duration of price adjustment. The estimated values of discount factor are around 0.99 inboth periods.
Taken as a whole, it can be concluded that the estimated hybrid New Keynesian Phillips curve
provides a good description of the Japanese inflation dynamics since the 1980s. The rejection of thepure forward-looking specification gives us the policy implication that the determination process ofmonetary policy should contain not only the forward-looking view but also the backward-looking

10perspective. Moreover, the estimated larger price stickiness and longer average duration of priceadjustment for the period after the collapse of the bubble economy might be the reflection of recent
recession. In this sense, the more active monetary and fiscal policies would be required to escape fromthe prolonged stagnant economy.

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