Volume 30, Issue 1
Measuring the Intertemporal Elasticity of Substitution for Consumption: Some
Evidence from Japan
Akihiko Noda
Graduate School of Business and Commerce, Keio
University

Shunsuke Sugiyama
Graduate School of Economics, Keio University

Abstract
The purpose of this paper is to present improved estimates of the intertemporal elasticity of substitution (IES) forJapan assuming a constant relative risk aversion (CRRA) utility function. The estimates of the IES we obtain range
from 0.2 to 0.5 when we use quarterly consumption data and the Continuous Updating Estimator (CUE). We find that
the IES is weakly identified when we employ the two-step GMM estimator, while the CUE can identify the IES.
Moreover, we also find that using consumption data of different frequencies leads to quite different estimates of the
IES.

We thank Colin McKenzie, Masao Ogaki, Taisuke Otsu, Makoto Yano and an anonymous referee for their helpful comments. We also thankMikio Ito, Shin-ichi Nishiyama and seminar participants at Keio University and Meiji University for helpful discussions. All data and programs
used for this paper are available from Noda on request. Noda also acknowledges that this research was supported by Keio University/Kyoto
University Joint Global COE Program Raising Market Quality-Integrated Design of "Market Infrastructure".
Citation: Akihiko Noda and Shunsuke Sugiyama, (2010) ''Measuring the Intertemporal Elasticity of Substitution for Consumption: Some
Evidence from Japan'', Economics Bulletin, Vol. 30 no.1 pp. 524-533.
Submitted: Sep 02 2009. Published: February 12, 2010.

1

Introduction

In general, it is expected that a household will change its intertemporal consumption
allocation in response to changes in real assets returns. This relationship can be explainedby the intertemporal elasticity of substitution (IES) or relative risk aversion (RRA). Many
empirical studies still have an interest in the appropriate value of IES (see, for example,
Oshio and Kobayashi (2009)). Therefore, measuring a consumer’s IES (RRA) is one of
the most important economic problems economists face. Estimates of the IES (RRA) areoften based on the constant relative risk aversion (CRRA) utility function. Although there
is a consensus that the estimates of the IES (RRA) based on the CRRA utility function
cannot adequately explain movements in U.S., the same statement cannot be made forJapanese data. For example, Hamori (1992, 1996) and Baba (2000) use Japanese monthlyor quarterly consumption data, and conclude that estimates of the IES (RRA) using theGeneralized Method of Moments (GMM) estimator perform well empirically, while Nakano
and Saito (1998) show that using Japanese semiannual consumption data the estimate of
the IES (RRA) is not significantly different from zero.
In the context of this previous empirical work on estimating the IES (RRA), the main
purpose of this paper is to resolve those two opposite stances in Japan. In particular, there
are two issues we consider in obtaining estimates of the IES (RRA) based on the CRRA
utility function. First, we investigate whether or not the frequency of consumption data
used affects estimates of the IES (RRA). This is because there is a possibility that consumption data of a different frequency may produce different empirical results (for example,
Nakano and Saito (1998) and Baba (2000)). Thus, we use not only quarterly consumption
data, but also semiannual consumption data when we estimate the IES (RRA).
Secondly, we take into account the fact that the IES (RRA) is often estimated using a
two-step GMM estimator. However, it is widely known that the two-step GMM estimator
have weak instrument problems and poor small sample property problems. As for those
problems of the two-step GMM estimator, Yogo (2008) uses the Continuous Updating
Estimator (CUE) proposed by Hansen, Heaton, and Yaron (1996) because Newey andSmith (2004) show that this estimator performs better in finite samples than the two-step
GMM estimator. Thus, we employ not only the two-step GMM estimator, but also the
CUE.
The estimates of the IES we obtain range from 0.2 to 0.5 when we use quarterly consumption data and employ the CUE. We find that the IES is weakly identified when we
employ the two-step GMM estimator, while the CUE can identify the IES. The main reason for this differing result may be that the CUE utilize more information of the dataset
than the two-step GMM estimator. Therefore, these results suggest that the estimates of
the IES in Japanese earlier studies may be not efficient and we can improve on the estimates of the IES by using the CUE. Moreover, we obtain the same results as Nakano andSaito (1998) that the estimate of IES is not significantly different from zero regardless ofthe empirical method when we use semiannual consumption data. These results suggestthat consumption data of different frequency leads to quite different estimates of the IES.
However, we conclude that one plausible reason for these different results is that the larger
sample size available with quarterly data improves the efficiency of estimator.
This paper is organized as follows. In section 2, we describe our model and empiricalmethod. In section 3, we explain the data that our study uses. In section 4, we present
our empirical results. In the final section, we conclude with some remarks.

12Model and Empirical Method

In this section, we explain the model and empirical method used in this paper. The
representative consumer at time 0 is assumed to choose his/her life-time consumption andasset holdings to maximize his/her expected utility subject to the budget constraint. The
consumer’s optimization problem is summarized as follows:"∞#
X Ct1−γ − 1
Max E0βt, 0 < β < 1, 0 < γ,
(1)
1
−
γ
t=0
s.t. Ct + pt At = [pt + dt ]At−1 + Yt ,(2)

where Ct is real per capita consumption at time t, pt is the price of the asset at time t, dt
is the dividend of the asset at time t, At is the amount of the per capita asset held at timet, Yt is real per capita labor income at time t, β is the subjective time discount factor, γ isthe relative risk aversion (RRA), and Et [·] is the expectation operator conditional on theinformation available at time t. In equation (1), we assume that the utility function is of
the CRRA class. In CRRA utility functions, the IES is the reciprocal of the coefficient of
RRA.
By solving the above utility maximization problem, we can derive the following Euler
equation:
Ct+1 −γEt [β() (1 + rt+1 ) − 1] = 0,
(3)
Ct
where rt+1 is the real return of the asset at time t + 1, which is defined as
rt+1 =

pt+1 + dt+1
− 1.
pt

(4)

In order to estimate the parameters in the Euler equation, we employ the GMM estimator
proposed by Hansen (1982). Define the error term ut+1 (θ) as
ut+1 (θ) = β(Ct+1 −γ) (1 + rt+1 ) − 1,Ctwhere θ = (β, γ)0 . Let zt be an R × 1 vector of instrumental variables known at time t,
and define an R × 1 vector gt (θ) as
gt (θ) = ut+1 (θ)zt ,(5)

E[gt (θ)] = 0,

(6)

Then, the Euler equation implies
where E[·] is the unconditional expectations operator. Moreover, if we define gT (θ) asTgT (θ) :=1X
gt (θ),
T t=1(7)

where T is the sample size, then the GMM estimator of θ, θ̂GM M , minimizes the quadratic
form:
(8)
θ̂GM M = arg min gT (θ)0WT gT (θ),θ

2where WT is an R × R weighting matrix, which is assumed to be positive definite for any
finite T .
We can obtain the most efficient GMM estimator by choosing the weighting matrix
WT = S−1 , where S−1 is the inverse of the asymptotic covariance matrix of T 1/2 gT (θ).
However, because we cannot observe the true value of θ, we cannot know S−1 either.
Therefore, we need to adopt a two-step GMM. In order to estimate S, we use the estimator
proposed by Newey and West (1987). Following Newey and West (1994), for the lag
selection criteria of the Newey-West estimator, we use int(4(T /100)2/9 ), where int(·) is theinteger part of the argument.
Furthermore, in order to analyze the goodness-of-fit of the model, we adopt Hansen’s(1982) J test of overidentifying restrictions. Under the null hypothesis that equation (6) istrue, T times the minimized value of equation (8) is asymptotically distributed as χ2R−K ,where K is the number of parameters (that is, two in our case).
However, it is widely known that the two-step GMM estimator has poor small sample
properties. For example, Newey and Smith (2004) show that the CUE performs betterin finite samples than the standard two-step GMM estimator. Thus, we also employ the
CUE in addition to the two-step GMM estimator when we estimate the IES. The CUE
minimizes
θ̂CU E = arg min gT (θ)0[S(θ)]−1 gT (θ),
(9)
θ

−1

where [S(θ)] = WT . For the CUE, the weighting matrix is continuously updated as θchanges in the minimization process. The CUE θ̂CU E has the same asymptotic distribution
as the two-step GMM estimator.

3

Data

We construct three datasets labeled “Dataset 1” through “Dataset 3” in this paperthat differ in their choices of consumption. In all datasets long-term government bondis treated as the asset in equation (2), and its total asset return (LGB) is obtained from
Ibbotson Associates. To compute the inflation rate, we use the total consumption deflator
published in the Annual Report on National Accounts. “Dataset 1”, quarterly data is used,
and for per capita consumption, we use “Total consumption (Benchmark year is 2000)”
divided by population which is reported in the Annual Report on National Accounts inJapan. This per capita consumption data is seasonally adjusted using the X-12 ARIMA
procedure. As instrumental variables, we use the lagged values of the real return on theasset, the consumption growth rate, and the growth rate of the deflator. The real asset
return and real consumption series are both computed using the total consumption deflator.
The sample period is from 1980Q2 to 2008Q4.
“Dataset 2” differs from “Dataset 1” only in the frequency of the data used. “Dataset
2” uses semiannual data which is also used in Nakano and Saito (1998) when they estimate
the IES. “Dataset 3” differs from “Datasets 1” only in the measure of consumption used.
“Dataset 3” uses quaterly data, but uses “Nondurable goods plus services” as consumptiondata.
For both the two-step GMM and CUE, all variables that appear in the moment conditions should be stationary. To check whether the variables satisfy stationarity, we use
the Augmented Dickey and Fuller (1981) (ADF) test. Table 1 provides some descriptive
statistics and the results of the ADF tests. For all the variables, the ADF test rejects the3null hypothesis that the variable contains a unit root at conventional significance levels.
Although not reported, the Phillips and Peron (1988) test leads to similar results.
(Table 1 around here)

4

Empirical Results

Table 2 presents the empirical results using “Dataset 1” through “Dataset 3”. Table
2 shows that the CUE estimates of β and γ are statistically significant at conventional
levels only when we use “Dataset 1” or “Dataset 3”. The estimates of β range from 0.9895
to 0.9990, which are economically realistic; and the estimates of γ range from 1.9082 to
5.9981, implying estimates of the IES in the range 0.1667 to 0.5240. The p values for the
J test are large enough that we cannot reject the null that the moment conditions hold.
We have confirmed that the CUE estimation results are robust to changes of the initialstarting values. When we employ the two-step GMM, the estimates of β range from 0.9870
to 0.9923, which are economically realistic. The estimates of γ are not signficantly different
from zero regardless of the type of the dataset. In constrast to the CUE, we have confirmedthat the two-step GMM estimation results are not robust to changes of these initial values.
Then, we must explain the reason why the two-step GMM estimation does not work.
(Table 2 around here)
We can consider one plausible reason is that γ is weakly identified. To see this point,
Figures 1 and 2 provide three dimensional plots of the two-step GMM and CUE objectivefunctions (see, Hall (2005), for a similar example).
(Figures 1 and 2 around here)
Figure 1 suggests that the two-step GMM objective function be flat in the direction ofγ. On the other hand, in Figure 2 we can not ascertain whether the CUE is flat in the
direction of γ. Then, in Figures 3 and 4, two dimensional plots of the two-step GMM and
CUE objective functions as β and γ are varied to investigate whether these parameters are
weakly identified. We find that β is clearly identified regardless of the empirical method.
Menanwhile, we also find that the two-step GMM objective function is flat in the directionof γ, but the CUE objective function is not.1 This figure supports our view that γ is weakly
identified when we employ the two-step GMM estimator, while the CUE can identify γ.
These results appear to be similar to Stock and Wright (2000). They show that if evenone parameter is weakly identified, all other parameters do not satisfy the consistency andthe asymptotic normality on two-step GMM estimation. Moreover, the main reason forthese differing results may be that the CUE utilize more information of the dataset than
the two-step GMM estimator.2 Therefore, this suggests that the estimates of the IES inearlier studies be not correct and we can improve the estimates of the IES using the CUE.
(Figures 3 and 4 around here)1

We also attempt to test whether this result is changed when we adapt the different instrument variables
such as “per capita real investment growth rate”. The two-step GMM objective function is still found to
be flat in the direction of γ.
2
Newey and Smith (2004) show that the CUE is one special case of the Generalized Empirical Likelihood(GEL) estimator proposed by them.

4

Table 2 also shows that the estimates of IES (RRA) are not significantly different fromzero regardless of empirical method when we use “Dataset 2”. These results are the sameas Nakano and Saito (1998). This suggests that consumption data of different frequency
leads to different estimates of the IES. One plausible reason for these differing results isthat the use of quarterly data increases the sample size leading to more efficient estimates.

5

Concluding Remarks

This paper has estimated the IES for Japan. We have specified the long-term government bond as the asset that the representative consumer invests in, and estimated theEuler equation using the two-step GMM estimator and CUE. The estimates of the IES
we obtain range from 0.2 to 0.5 when we use quarterly consumption data and the CUE.
We find that the IES is weakly identified when we employ the two-step GMM estimator,
while the CUE can identify the IES. Moreover, we also find that using consumption dataof different frequencies leads to quite different estimates of the IES.

5

References
Baba, N. (2000): “Exploring the Role of Money in Asset Pricing in Japan: Monetary Considerations and Stochastic Discount Factors,” Monetary and Economic Studies, 18(2),159–198.
Dickey, D., and W. Fuller (1981): “Likelihood Ratio Statistics for Autoregressive
Time Series with a Unit Root,” Econometrica, 49(4), 1057–1072.
Hall, A. (2005): Generalized Method of Moments. Oxford University Press, USA.
Hamori, S. (1992): “Test of C-CAPM for Japan: 1980-1988,” Economics Letters, 38(1),67–72.
(1996): “Consumption Growth and the Intertemporal Elasticity of Substitution:
Some Evidence from Income Quintile Groups in Japan,” Applied Economics Letters,
3(8), 529–532.
Hansen, L. (1982): “Large Sample Properties of Generalized Method of Moments Estimators,” Econometrica, 50(4), 1029–1054.
Hansen, L., J. Heaton, and A. Yaron (1996): “Finite-Sample Properties of Some
Alternative GMM Estimators,” Journal of Business and Economic Statistics, 14(3), 262–
280.
Nakano, K., and M. Saito (1998): “Asset Pricing in Japan,” Journal of the Japaneseand International Economies, 12(2), 151–166.
Newey, W., and R. Smith (2004): “Higher Order Properties of GMM and Generalized
Empirical Likelihood Estimators,” Econometrica, 72(1), 219–255.
Newey, W., and K. West (1987): “A Simple, Positive Semi-Definite, Heteroskedasticity
and Autocorrelation Consistent Covariance Matrix,” Econometrica, 55(3), 703–708.
(1994): “Automatic Lag Selection in Covariance Matrix Estimation,” Review of
Economic Studies, 61(4), 631–653.
Oshio, T., and M. Kobayashi (2009): “Income Inequality, Area-level Poverty, Perceived
Aversion to Inequality, and Self-rated Health in Japan,” Social Science & Medicine,69(3), 317–326.
Phillips, P., and P. Peron (1988): “Testing for a Unit Root in Time Series Regression,”
Biometrika, 75(2), 335–346.
Stock, J., and J. Wright (2000): “GMM with Weak Identification,” Econometrica,
68(5), 1055–1096.
Yogo, M. (2008): “Asset Prices under Habit Formation and Reference-Dependent Preferences,” Journal of Business and Economic Statistics, 26(2), 131–143.

6

Table 1: Descriptive Statistics and Unit Root Tests
D/S

Variable Mean
SD
Min
Max
ADF
CV
CGt
1.0033 0.0107 0.9569 1.0321 -10.8501
1
rt
0.0153 0.0367 -0.1072 0.1157 -9.7831 -3.9900
πt1.0018 0.0062 0.9916 1.0301 -7.1070
CGt
1.0069 0.0142 0.9668 1.0392 -5.9480
2rt
0.0129 0.0378 -0.1059 0.0768 -5.0237 -3.1500
πt1.0038 0.0100 0.9878 1.0444 -3.2865
CGt
1.0027 0.0093 0.9759 1.0293 -11.5397
3
rt
0.0142 0.0366 -0.1094 0.1132 -9.7101 -3.9900
πt1.0029 0.0058 0.9907 1.0333 -7.4078“D/S” denotes the dataset, “CGt ” denotes the gross real per capita consumption growth, “rt ” denotes
the real return on long-term government bonds, “πt ” denotes the inflation rate, “SD” denotes thestandard deviation, and “ADF” denotes the Augmented Dicky-Fuller (ADF) test statistic. In computing the ADF test, we assume a model with a time trend and a constant. “CV” for “Dataset 1” and“Dataset 3” denote the critical values at the 1% significance level for the ADF test for each sample
size. “CV” for “Dataset 2” also denotes the critical values at the 10% significance level for the ADF
test for sample size. The null hypothesis that each variable has a unit root is rejected at the 1% or
10% significance level, respectively.

Table 2: Empirical Results of two-step GMM and CUE
E/M

D/S

Lag
β̂
SE(β̂)γ̂SE(γ̂) p-value DFN
1
0.9870 0.0043 0.8046 0.9886 0.1293
2
114
1
2
0.9923 0.0037 2.4930 0.7290 0.3856
5113
3
0.9908 0.0034 2.0854 0.6475 0.5440
8112
10.9837 0.0063 -0.5468 0.7467 0.8471
2
56
GMM
2
2
0.9834 0.0051 -0.7591 0.6910 0.9917
555
3
0.9846 0.0032 -0.3710 0.3132 0.4542
8541
0.9908 0.0050 1.8440 1.4922 0.2809
2
114
3
2
0.9920 0.0034 2.1630 1.0020 0.1380
5113
30.9917 0.0033 2.1049 0.8279 0.3543
8112
1
0.9895 0.0043 1.9082 0.9284 0.1158
2
114
1
2
0.9928 0.0039 2.8379 0.8056 0.3941
5113
3
0.9931 0.0041 3.0717 0.8407 0.5457
8112
1
0.9837 0.0063 -0.5585 0.7457 0.8476
2
56
CUE
2
2
0.9837 0.0052 -0.7218 0.6934 0.9895
555
3
0.9887 0.0041 0.3822 0.3812 0.5203
854
1
0.9967 0.0049 4.5588 1.7642 0.1450
2
114
3
2
0.9990 0.0051 5.8717 2.0190 0.2765
5113
3
0.9990 0.0051 5.9981 1.8190 0.5417
8112“E/M” denotes the empirical method, “D/S” denotes the dataset used, “Lag” denotes the number oflags of the instruments used, “β̂” denotes the estimate of the subjective discount rate, “γ̂” denotes
the estimate of the relative risk aversion (RRA), “SE(·)” denotes the Newey-West adjusted standard
error of “β̂” or “γ̂”, respectively, “p-value” denotes the p-value for Hansen’s (1982) J test statistics,“DF ” denotes the degrees of freedom for the J test, and “N ” denotes the sample size. To computethe two-step GMM and CUE estimates, R version 2.10.1 was used. In our study, we assume thatutility function is of the CRRA class. Therefore, the intertemporal elasticity of substitution (IES) is
the reciprocal of the coefficient of relative risk aversion (RRA). The starting values of the parameters
set equal to β = 1, γ = 1.

7

GMM Objective

0.10

0.05

2.0
1.5

0.0

0.5
1.0
gamm
a

0.5

1.5
2.0

1.0
ta
be

0.0

Figure 1: GMM Objective Function for Dataset1 with Lag=1

bjective
CUE O

0.20
0.15
0.10
0.0
2.0

0.5
1.5
1.0
ta
be

a
m

m

ga

1.0
1.5

0.5
2.0 0.0

Figure 2: CUE Objective Function for Dataset1 with Lag=1

8

0.20
0.15
0.10
0.00

0.05

Objective Function

GMM Objective
CUE Objective

0.0

0.5

1.0

1.5

2.0

beta

0.10

GMM Objective
CUE Objective

0.00

0.05

Objective Function

0.15

0.20

Figure 3: Objective Function for Dataset1 with Lag=1 (γ is fixed)

0

1

2

3

gamma

Figure 4: Objective Function for Dataset1 with Lag=1 (β is fixed)

9