Volume 31, Issue 2
Testing the "Catching up with the Joneses" Model with Consumption
Externality in Japan
Akihiko Noda
Faculty of Economics, Toyo University

Abstract
The purpose of this paper is to test Abel's (1990, 1999) "Catching up with the Joneses" model with a consumption
externality using Japanese financial data. It is found that the model is rejected in Japan when it is estimated using
generalized empirical likelihood (GEL) estimators.

I would like to thank Mikio Ito, Colin McKenzie, Taisuke Otsu, Makoto Yano and an anonymous referee for their helpful comments andsuggestions.
Citation: Akihiko Noda, (2011) ''Testing the "Catching up with the Joneses" Model with Consumption Externality in Japan'', Economics
Bulletin, Vol. 31 no.2 pp. 1648-1658.
Submitted: May 10 2011. Published: June 06, 2011.

Economics Bulletin, 2011, Vol. 31 no.2 pp. 1648-1658

1

Introduction

Mehra and Prescott (1985) point to a puzzle, namely, the inability of standard intertemporal economic models such as the consumption-based capital asset piricing model(CCAPM) to rationalize the statistics that have characterized U.S. ﬁnancial markets overthe past century. They show that the models fail to explain the diﬀerence between the average returns of risky and safe assets in U.S. ﬁnancial markets. This puzzle, called the equity
premium puzzle, comes from an equation concerning the intertemporal rational behaviorof participants in ﬁnancial markets. We can easily verify the puzzle using several statistics
calculated from ﬁnancial data and estimates of the discount factor and the degree of risk
aversion. Inspired by the equity premium puzzle, Weil (1989) points to another puzzle,
namely the risk-free rate puzzle. In turn, economists confront the inability of the models
to explain the average return of the safe asset. The puzzles are still puzzles for U.S. andother industrialized countries, including Japan (see Kocherlakota (1996) and Mehra and
Prescott (2003) for details).
In order to resolve the discrepancy between the CCAPM’s predictions and empirical
data, a number of economist have modiﬁed CCAPM theoretically by introducing additional settings (see also Noda (2011) for details). One of these modiﬁcations is a habit
formation approach. We can classify research on the habit formation approach into twogroups depending on the eﬀect of an investor’s own decisions on the future levels of habit.
Constantinides (1990) and Sundaresan (1989) propose an internal habit model in which
the habit level depends on the investor’s own past consumption, and the investor takes this
into account while making decisions about his/her current consumption choice. Abel (1990,
1999), Gali (1994) and Campbell and Cochrane (1999) propose an external habit modelin which the habit level depends on past aggregate consumption and is not determined byany one investor’s decision. Campbell (2003) insists on the importance of habit formation
to resolve the puzzles. For instance, Campbell and Cochrane (1999) report that using thehabit formation approach, they are able to partially resolve the puzzles in the U.S. stock
market.
There are, however, only a few studies about whether habit formation resolves the
puzzles in Japan. In particular, there is little consensus regarding whether Abel’s (1990,
1999) or Gali’s (1994) model, which consider a consumption externality in the standard
CCAPM, performs well empirically. Ikeda and Tsutsui (1996) estimate the parametersof Gali’s (1994) model using consumption data by income quantile, and conclude that a
consumption externality is corroborated in high income quantiles. Bakshi and Naka (1997)
and Baba (2000) evaluate the validity of the Abel’s (1990, 1999) model, and report that themodel is rejected when tested using the volatility bounds test of Hansen and Jagannathan(1991) and the speciﬁcation error test of Hansen and Jagannathan (1997). However, Hansen
and Jagannathan’s (1991, 1997) tests have problems in small samples because those tests
are based on generalized method of moments (GMM) estimates (see also Cochrane andHansen (1992), Burnside (1994), Balduzzi and Kallal (1997) and Ahn and Gadarowski(2004)).
While we pay attention to empirical methods, few economists have been concerned
about using alternative methods to estimate the parameters of the underlying asset pricing models. Since Hansen and Singleton (1982), GMM has been used to estimate theparameters of CCAPMs. When we apply GMM to estimate the parameters of CCAPMs,
two important drawbacks have been reported: (i) the problem of weak identiﬁcation and(ii) the problem of many moment conditions. We can understand both problems in the1649Economics Bulletin, 2011, Vol. 31 no.2 pp. 1648-1658

context of the small sample properties of the GMM estimator; the GMM estimator has a
non-negligible small sample bias when we fail to choose appropriate instruments, and useof too many moment conditions allow us to fail to extract information from the available
data (see also Stock et al. (2002) for details).
In an eﬀort to improve on the poor performance of GMM in small samples, a numberof alternative estimators have been suggested. The class of generalized empirical likelihood(GEL) estimators is attracting the attention of many econometricians because of their
better small sample performance compared to the GMM estimator. The class of GEL
estimators have the same asymptotic distributions as GMM, but Newey and Smith (2004)
demonstrate the theoretical advantage of GEL estimators by comparing the higher order
asymptotic biases of the GEL and GMM estimators. Some other economists have also
reported the advantage of GEL-based estimators. For instance, when Ito and Noda (2010)
employ the standard CCAPM and use Japanese data, they report that the GEL estimates
are incomparably better than the Two-step GMM (2S-GMM) estimator in terms of thehigher order mean squared error of Donald and Newey (2001). Following them, instead of
the 2S-GMM estimator, this paper employs the GEL estimator to estimate the parametersof the Abel’s (1990, 1999) model. The main results of this paper are: (i) Abel’s (1990,
1999) model is rejected when estimated using Japanese ﬁnancial data, (ii) the returns in
the Japanese stock market can be explained by the standard CCAPM with power utility,and (iii) the GEL estimates are stable regardless of the model speciﬁcation, but the GMM
estimates are not.
The rest of the paper is organized as follows. Section 2 presents a review of Abel’s(1990, 1999) “Catching up with the Joneses” model with a consumption externality and
the GEL estimator. Section 3 provides details of the data used, while section 4 presents
the empirical results. Section 5 contains some brief concluding remarks.

2

Model and Empirical Method

In this section, Abel’s (1990, 1999) “Catching up with the Joneses” model with aconsumption externality is presented together with details of the empirical methods used
to estimate the parameters of the model.

2.1“Catching up with the Joneses” Models

Abel (1990) incorporates a consumption externality into the standard CCAPM with power
utility. Following Abel (1990, 1999), we assume that a representative investor at time tchooses his/her life-time consumption and holdings of several assets in order to maximize
his/her expected life time utility subject to the budget constraint. The maximization
problem is given by∞∑
Max Etβ j u(ct+j , vt+j ), 0 < β < 1,(1)
j=0

s.t. ct +N∑
i=1

pi,t Ai,t =N∑[pi,t + di,t ]Ai,t−1 + Yt , i = 1, 2, · · · , N,

(2)

i=1

where the subscript t indicates time, ct is investor’s own real consumption, vt is the benchmark level of consumption, pi,t is the price of the ith asset, di,t is the dividend on the ith
asset, Ai,t is the amount of the holdings of the ith asset, Yt is real labor income, N is

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the number of assets, β is the subjective time discount factor, and Et [·] is the expectation
operator conditional on the information available at time t.
Let us deﬁne the period utility function u(·) with vt as(ct /vt )1−γ
, γ > 0,
1−γ
[
]
1−D κ
, κ ≥ 0, D ≥ 0,
vt := cDt−1 Ct−1u(ct , vt ) :=

(3)
(4)

where γ is the degree of relative risk aversion, κ is the degree of time non-separability andCt−1 is the per capita aggregate consumption level at time t−1. Here if we specify; (i) D = 0
and κ ≥ 0, then the benchmark level of consumption, vt , depends only on the lagged levelof aggregate consumption per capita; and (ii) D = 1 and κ ≥ 0, then the benchmark levelof consumption, vt , depends only on the investor’s own past consumption. These respective
speciﬁcations are, (i) the relative consumption model; and (ii) the internal habit model ofConstantinides (1990) and Sundaresan (1989). The benchmark level of consumption canbe speciﬁed to generate an internal or an external habit. Since there is a representative
investor, in equilibrium; aggregate consumption equals the investor’s own consumption(ct−1 = Ct−1 ), that is,κ
vt = Ct−1.
(5)This is internal habit formation which is similar to Bakshi and Naka (1997) and Baba
(2000). Therefore, we assume that the habit is completely endogenous to the investor.
Solving the above utility maximization problem, we can derive the following Euler
equations:
[ (
]
)−γ (
)κ(γ−1)
Ct+1
Ct
Et β
(1 + ri,t+1 ) − 1 = 0, i = 1, 2, · · · , N,
(6)
Ct
Ct−1
where ri,t+1 is the real return of the ith asset at time t + 1, which is deﬁned as
ri,t+1 =pi,t+1 + di,t+1− 1, i = 1, 2, · · · , N.
pi,t

(7)

When κ = 0, the Euler equation (6) reduces to the case of standard CCAPM with power
utility
[ (
]
)−γ
Ct+1
Et β
(1 + ri,t+1 ) − 1 = 0, i = 1, 2, · · · , N.
(8)Ct
At this stage, we do not assume any data generating process for Ct and ri,t . Later inthis section, we will show how to estimate the parameters in this Euler equation using theconditional expectation operator and how to conduct statistical inference using estimatesof these parameters.

2.2

Moment Restriction ModelI present a framework, called the moment restriction model, that allows us to cope generallywith statistical models where the distribution of the data is not speciﬁed. Many elaborate
estimators, such as Newey and Smith’s (2004) GEL estimator can be discussed in termsof the moment restriction model. In particular, we transform equation (6) or (8) into one
without a conditional expectation operator in order to estimate the unknown parameters.

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Deﬁne an N × 1 error vector ut+1 (θABEL ) and ut+1 (θCCAP M ) as
[ (
]
)−γ (
)κ(γ−1)
Ct+1
Ct
ut+1 (θABEL ) = β(1 + rt+1 ) − 1,
Ct
Ct−1
[ (
]
)−γ
Ct+1
ut+1 (θCCAP M ) = β(1 + rt+1 ) − 1,
Ct

or

(9)

(10)

where rt+1 = (r1,t+1 , r2,t+1 , · · · , rN,t+1 )0 , 1 = (1, 1, · · · , 1)0 , θABEL = (β, γ, κ)0 and θCCAP M =(β, γ)0 . Let zt be an K vector of instruments known at time t, and deﬁne an N K × 1 vectorgt (θ) as
gt (θ) = ut+1 (θ) ⊗ zt .
(11)
Then the Euler equation implies
E[gt (θ)] = 0,(12)where E[·] is the unconditional expectation operator. We call this equation a moment
restriction model.
Generally, let yt , (t = 1, 2, · · · , T ), denote observations on a ﬁnite dimensional process,which are usually assumed to be stationary and strongly mixing (see Smith (2004)). The
right-hand side of equation (11), gt (θ), is called a moment indicator, which is a function
with respect to the parameters concerned, but also depends on the data, yt , and potentiallyon the instruments, zt . For economy of notation, we omit the dependence of g on yt andzt .
In the next subsection, I present the GEL estimator used to estimate the parameters
in moment restriction models.

2.3

GEL

It is widely known that the GMM estimator of Hansen (1982) has poor small sample
properties. Many econometricians have tried to improve on the GMM estimator’s small
sample properties and have suggested several alternative estimators. These include theempirical likelihood (EL) estimator of Owen (1988), the continuous updating estimator(CUE) of Hansen et al. (1996), and the exponential tilting (ET) estimator of Kitamura
and Stutzer (1997).
As shown by Newey and Smith (2004), all these estimators belong to the class of GEL
estimators. Following Newey and Smith (2004), this subsection provides a brief review ofthe estimators of a moment restriction model. The model presented in subsection 2.2 is
one with m moment restrictions. Before reviewing the estimators of the parameters in sucha model, let us deﬁne some notation. xt , (t = 1, 2, · · · , T ) denotes i.i.d. observations of
the data, and p (equal to 3 or 2 in our model) denotes the number of parameters to be
estimated. We sometimes write g(x, θ) as an N K vector of functions of the data and theparameters. We assume that m ≥ p and that the model has a true parameter θ0 satisfying
the following condition:
E[g(x, θ0 )] = 0,
where E is the expectation taken with respect to the distribution of the xt ’s. In order toexplain the GEL estimator, let gt (θ) = g(xt , θ), and ρ(v) be a concave function on a real

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open interval V containing zero. The GEL estimator is deﬁned asθ̂GEL = arg minθ∈Θ supλ∈Λ̂t (θ)

T∑ρ(λ0 gt (θ)),

(13)

t=1

where Θ denotes the parameter space, and Λ̂t (θ) = {λ : λ0 gt (θ) ∈ V}. Alternative estimators to the GMM estimator can be obtained by specifying ρ(v). As Kitamura and Stutzer(1997) show, in the case of the EL estimator, ρ(v) = ln(1 − v), and in the case the ET
estimator, ρ(v) = −ev . Furthermore, when ρ(v) = −(1 + v)2 /2, the GEL estimator isequivalent to the CUE estimator,
θ̂CU E = arg minθ∈Θ ĝ(θ)0 Ω̂(θ)− ĝ(θ),(14)∑
where Ω̂(θ) = T −1 Tt=1 gt (θ)gt (θ)0 and A− denotes any generalized inverse of the matrix
A. (See Theorem 2.1 in Newey and Smith (2004)).
For convenience, we impose a normalization on ρ(v) as follows. Let ρj (v) = ∂ j ρ(v)/∂v jand ρj = ρj (0) for each j. We assume that ρ1 = ρ2 = −1. Associated with each GEL
estimator there are implied probabilities for the observed data. Since these probabilities
are used in our empirical analysis, we brieﬂy review them. Consider ρ(v), an associated
GEL estimator θ̂, and ĝt = gt (θ̂). The implied probabilities are given byρ1 (λ̂0 ĝt )π̂t = ∑T,
0 ĝ )
ρ
(
λ̂
1
s
s=1

t = 1, 2, · · · , T,

(15)∑
where λ̂ = arg maxλ Tt=1 ρ(λ0 ĝt )/n. For any function f (x, θ) and GEL estimator of θ, an∑eﬃcient estimator of E[f (x, θ0 )], Tt=1 π̂t f (xt , θ̂), can be derived, as shown in Brown and
Newey (1998).
Similar to the J-statistic in GMM estimation, we can use a J-statistic to test the overidentifying restrictions when a GEL estimator is employed. The J-statistic for using the
GEL estimator is computed using the kernel-smoothed moment indicator (see Section 4
in Smith (2004) for details). Under the null hypothesis that equation (12) is true, thetest statistic is asymptotically distributed as χ2m−p , where p is the number of parameters
estimated.

3

Data

In this paper, quarterly data from 1980Q3 to 2009Q4 are used in estimating Abel’s
model. The returns on short-term instruments are employed as the return on the riskfree asset and these are obtained from Nikko Financial Intelligence.1 The Fama-French’s
market portfolio returns are treated as the returns on the risky asset and these are obtained
from Nikkei Portfolio Master.2 Per capita consumption is computed as “Nondurable goods
plus service consumption (benchmark year 2000)” divided by the estimates of the total
population reported in the Annual Report on National Accounts in Japan. The per capita
consumption data are seasonally adjusted using the X-12 ARIMA procedure.
1

The returns on shor-term instruments contains interest-bearing instruments with maturities of threemonths or less (call, bill, gensaki, CD, CP and government short-term securities (excluding securites hel
by Bank of Japan and the Government)).
2
Fama-French’s market factors in Japan are calculated by following Kubota and Takehara (2007).

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To deﬂate all series, the “Nondurable plus service consumption” deﬂator published in
the Annual Report on National Accounts is used.3 Lagged values of the real return on
the risk-free asset and the real return on the market portfolio, and the current and lagged
values of the real consumption growth rate are used as instruments. For the GEL estimator,
all variables that appear in the moment conditions should be stationary. The ADF testof Dickey and Fuller (1981) is used to check whether the variables satisfy the stationarity
condition. Table I provides some descriptive statistics and the results of the ADF tests.
For all the variables, the ADF test rejects the null hypothesis that the variable contains aunit root at conventional signiﬁcance levels.

Table I: Descriptive Statistics and Unit Root Tests

CGt
CGt−1
rtf
rtm

Mean
1.00381.00360.00490.0120SD
0.00900.00910.00630.1045Min
0.97700.9770-0.0143-0.3335Max
1.0312
1.03120.02070.2331

ADF-10.9889-10.7147-6.1741-7.5044

Lag
0
0
0
0

N
118

“CGt ” denotes the gross real per capita consumption growth rate, “CGt−1 ” denotes the ﬁrst lagged
gross real per capita consumption growth rate, “rtf ” denotes the real return on risk-free asset, “rtm ”
denotes the real return on the market portfolio, “SD” denotes the standard deviation, “ADF”
denotes the Augmented Dickey-Fuller (ADF) test statistics, “Lag” denotes the lag order selectedby an information criterion of Akaike (1973), and “N ” denotes the number of observations. In
computing the ADF test, a model with a time trend and a constant is assumed. The critical value
at the 1% signiﬁcance level for the ADF test is “-3.99”. The null hypothesis that each variable hasa unit root is clearly rejected at the 1% signiﬁcance level.

4

Empirical Results

Table II presents the empirical results of estimating Abel’s (1990, 1999) model using the
2S-GMM and the GEL estimators (CUE, EL and ET). In GEL estimation, the truncatedkernel proposed by Kitamura and Stutzer (1997) is used to smooth the moment function(that is, equation (13) in this case) because Anatolyev (2005) demonstrates that, in thepresence of correlation in the moment function, the smoothed GEL estimator of Kitamura
and Stutzer (1997) is eﬃcient.4 In addition, the heteroskedasticity and autocorrelationconsistent (HAC) covariance matrix of Andrews (1991) is employed to reduce estimationbiases in the 2S-GMM and the GEL estimators.

3

The “Nondurable plus service consumption” deﬂator is a weighted deﬂator computed using deﬂatorsof “Nondurable goods” and “Services” that are also published in the Annual Report on National Accounts.
4
I employ the smoothed GEL estimator, but the optimal kernel weights do not exceed one. Thissuggests that the kernel smoothing has no eﬀect.

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Table II: Empirical Results

β̂
γ̂
κ̂
pJ
DF

2S-GMM
0.9669[0.0028]-4.7167[0.4956]
0.5152[0.1056]
0.5110
7

ABEL
CUE
EL
0.9991
0.9989[0.0021] [0.0021]
0.79060.8143[0.3918] [0.3925]
1.44481.6098[3.6000] [4.4160]
0.28940.282110
10ET
0.9990[0.0021]0.8077[0.3923]
1.5823[4.2060]
0.285910

2S-GMM
0.9972[0.0007]0.5600[0.1883]
0.9179
6

CCAPM
CUE
EL
0.99850.9987[0.0008] [0.0009]
0.85020.8969[0.2306] [0.2403]
0.62270.61848
8

ET
0.9981[0.0008]0.8026[0.2204]
0.61428“β̂” denotes the estimate of the subjective discount rate, “γ̂” denotes the estimate of degree of therelative risk aversion, “κ̂” denotes the estimate of the degree of time non-separability, “pJ ” denotes
the p-value for Hansen’s J test and “DF ” denotes the degrees of freedom for this J test. Andrews(1991) adjusted standard errors for each of the parameter are reported in brackets. R version 2.13.0
was used to compute the estimates. The starting values of the parameters are set equal to β = 1,
γ = 1 and κ = 1 (β = 1 and γ = 1).

The GEL (CUE, EL and ET) estimates of β and γ are statistically signiﬁcant at conventional levels. The GEL estimates of β are about 0.99; and the GEL estimates of γ areabout 0.8. The estimates of κ are not statistically diﬀerent from zero. The p-values for
Hansen’s J test are large enough that we cannot reject the null hypothesis that the moment
conditions hold. I have conﬁrmed that the GEL estimation results are robust to changes of
the initial starting values. When I employ the 2S-GMM estimator, all parameter estimates
are statistically signiﬁcant at conventional levels, but the estimate of γ in Abel’s model isabout −4.7 which is not economically realistic and violates the models assumptions. In
contrast to the GEL, I have conﬁrmed that the 2S-GMM estimation results are not robust
to changes of the initial starting values.
Table II also presents the empirical results of estimating the standard CCAPM model.
The GEL estimates of β and γ are statistically signiﬁcant at conventional levels. The
estimates of β are about 0.99; and the estimates of γ range from 0.8 to 0.9. The p-valuesfor Hansen’s J test are large enough that we cannot reject the null hypothesis that the
moment conditions hold. I have conﬁrmed that the GEL estimation results are robust tochanges of the initial starting values. When I employ the 2S-GMM estimator, the estimatesof β and γ are also statistically signiﬁcant at conventional levels. The estimates of β isabout 0.99; and the estimate of γ is about 0.6 which is higher than the estimates from Abel’s
model. The p-values for Hansen’s J test are large enough that we cannot reject the null
hypothesis that the moment conditions hold. In contrast to the GEL, I also have conﬁrmedthat the 2S-GMM estimation results are unstable to changes of the initial starting values.
These results suggest that: returns in the Japanese stock market can be explained by thestandard CCAPM with power utility, and the GEL estimates are robust regradless of themodel speciﬁcation, but the 2S-GMM estimates are not.
I ﬁnd that the GEL estimates are similar regardless of the model speciﬁcation, but 2SGMM estimates are quite diﬀerent. As a result, we conclude that the 2S-GMM estimates
are unreliable in small sample cases. I also ﬁnd that Abel’s (1990, 1999) habit model isrejected for Japan, and that returns in the Japanese stock market can be explained by thestandard CCAPM with power utility.

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5

Concluding Remarks

The purpose of this paper is to investigate whether Abel’s (1990, 1999) “Catching up
with the Joneses” model with consumption externality performs well in Japan. In orderto estimate the parameters of the relevant Euler equation, the GEL estimators which
improve on the poor performance of Hansen’s (1982) 2S-GMM estimator small samples are
employed.
As a result, it is found that: (i) Abel’s (1990, 1999) “Catching up with the Joneses”
model with consumption externality does not perform well empirically in Japan, (ii) returns
in the Japanese stock market can be well explained using the standard CCAPM with power
utility, and (iii) the GEL estimates are robust regradless of the model speciﬁcation, but
the 2S-GMM estimates are not.

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