Discussion Paper No. 275
The Cross-Euler Equation Approach in Estimating
the Elasticity of Intertemporal Substitution for Foodand Non-Food Consumption in Japan
Shin-Ichi Nishiyama
Masao Ogaki
September 2011

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY
KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANThe Cross-Euler Equation Approach in Estimating the Elasticityof Intertemporal Substitution for Food and Non-Food
Consumption in Japan 1
This Version: September, 2011
First Version: June, 2005
Shin-Ichi Nishiyama

Graduate School of Economics andManagement, Tohoku University

Masao Ogaki

Department of Economics,
Keio University

Abstract
We use the standard two-good version of the life cycle/permanent income model inanalyzing the intratemporal and intertemporal aspect of food and non-food
expenditure in Japan. The empirical dilemma in identifying and estimating theparameters governing the intertemporal elasticity of substitution (IES) is
addressed.

In overcoming this empirical dilemma we employ the Cross-Euler

equation approach proposed by Nishiyama (2005).

The IES parameters are

estimated by exploiting the cointegration restriction implied by the Cross-Euler
equation and also from the standard Euler equation using GMM.

Further, bycomparing the IES estimates from the Cross-Euler equation to those from thestandard Euler equation, we formally test the hypothesis whether food andnon-food expenditure in Japan is affected by some factors that cause
misspecification in the standard Euler equation approach, such as liquidity
constraints or habit formation.
Key words: Cointegration, Elasticity of Intertemporal Substitution, Euler Equation
JEL Classification: C22, E21
We would like to thank the comments and suggestions made by Ippei Fujiwara,
Yuuichi Fukuta, Koichi Futagami, Chiaki Hara, Takuji Kawamoto, Kazuo Mino,
Toshikatsu Okubo, seminar participants at Osaka University, session participants of
the JEA Fall Meeting 2004 at Okayama University, and the staff members at theInstitute for Monetary and Economic Studies. We would like to especially thank CY
Choi for graciously sharing his GAUSS code. Needless to say, remaining errors are
our own. The first version of the paper was written while the first author was at theInstitute for Monetary and Economic Studies, Bank of Japan. The views expressed inthis paper are those of the authors and do not necessarily reflect those of the Bank ofJapan or the Institute for Monetary and Economic Studies.

11. Introduction
Previous empirical studies often estimated the intertemporal elasticity of
substitution (IES) of Japanese consumer in the context of single goods and foundthe IES to be considerably high, 2 implying that the consumers are willing tosubstitute current consumption for future consumption in response to small change
in real interest rate.

If these high estimates of IES are indeed true, then weshould be observing a significant decrease in the consumption growth rate during
an expansionary monetary policy and significant increase during a contractionary
monetary policy in Japan. However, in reality, we do not observe this kind of
phenomenon in Japan and, therefore, the counter-intuitively high estimates of IES
from the previous literature remains to be a puzzle.

In this paper, we estimate

and test intertemporal and intratemporal implications of a two-good model withfood and non-food goods with a representative consumer, using Japanese data.
The preference specification of the two-good model follows Atekeson andOgaki (1996). In this model, the intertemporal elasticity of substitution (IES) for
total consumption expenditure increases as the level of wealth increases.

Theintuition behind this is that the consumer is less willing to substitute the foodconsumption than the non-food consumption over time.
good is smaller than that for the non-food good.

Thus, the IES for the food

The IES for the total consumption

expenditure is a weighted average of the IES for the food and the IES for the
non-food good, with the weights being the budget shares.

Because the budget

share for the food is larger when the consumer is poor, the IES is small when the
consumer is poor. The IES increases as the wealth increases in this model.

Also,

the model has various implications on how macroeconomic variables behave as theJapanese economy grows out of the destruction during the World War II.

For

example, the model typically predicts that the saving rate in Japan was very low
immediately after the World War II, and then started to increase as the economy
grew (see, e.g., Ogaki, Ostry, and Reinhart (1996) for a description of the savingrate behavior of the model with wealth-increasing IES).

Thus it is important toinvestigate the extent to which the model is consistent with the Japanese data.
As for the methodological strategy of this paper, we apply Nishiyama’s(2005) Cross-Euler Equation approach to test intertemporal and intratemporal
For instance, Hamori (1996) reports the IES of Japanese consumer to be well above
10.

2

2

implications of the model.
two-good models.

This is a new approach to test Euler equation in

In many applications of two-good models, researchers have

faced a methodological problem in estimating intratemporal first order condition
and Euler equations. The Cross-Euler equation approach 3 can be a solution tothis methodological problem.
The methodological problem in a two-good model with time separable
preferences arises because the intratemporal first order condition holds without
any forecasting error in the model.

In order to use the first order condition toestimate and test the model, it is necessary to add measurement errors or
preference shocks to the model.

However, adding measurement errors or

preference shocks causes problems for the Euler equation approach based on the
Generalized Method of Moments (GMM) as pointed out by Garber and King (1983).
Because of the nonlinearity of Euler equations, measurement errors or preference
shocks make GMM estimators inconsistent.

Nishiyama (2005) solved thisproblem by focusing on first order conditions involving a good at time t and theother good at time t+1. He calls such a first order condition a cross-Euler equation.
A cross-Euler equation involves a forecasting error unlike the intratemporal firstorder condition.

Hence statistical methods can be applied to the cross-Euler

equation without adding measurement errors nor preference shocks to the model.
The first step in his approach is to derive a long-run restriction from a cross-Euler
equation.

This long-run restriction implies a relationship between variables

called cointegration. The cointegration relationship allows one to use a regression
to estimate preference parameters.
Regarding the cointegration regression and the test of coingetration,
Nishiyama (2005) used Park’s (1992) Canonical Cointegration Regression (CCR)
estimator, and Park’s (1990) tests for the null hypothesis of cointegration in the
CCR framework.

In this paper, we use Stock and Watson’s (1993) Dynamic

Ordinary Least Squares (DOLS) estimator and Choi, Hu, and Ogaki’s (2005)
Hausman-type test for the null hypothesis of cointegration.

If the parameterized

endogeneity correction used in the DOLS is a good approximation, these methods
Further, Nishiyama (2005) showed that the cointegration relationship implied by
the cross-Euler equation is robust to many factors such as liquidity constraints andtime non-separability of preferences. These factors are often pointed out aspossible causes of empirical rejections of standard Euler equations. Thus, bycomparing estimates from cross-Euler equations and those from standard Euler
equations, it is possible to formally test the hypothesis that liquidity constraints or
non-separability of preferences casue the misspecification of the standard Euler33have better small sample properties than the nonparametric CCR methods. On theother hand, CCR is expected to perform better when the prameterized endogeneity
correction is misspecified. Because of a relatively small sample we used in thispaper, we prefer to use the DOLS estimator. In the DOLS framework, only the
Hausman-type cointegration test and Shin’s (1994) test are available at this point(it should be noted that the popular augmented Dickey-Fuller test should beapplied to the static OLS residual rather than the DOLS residual.)

The

Hausman-type test has better small sample properties than Shin’s test as shown in
Choi, et. Al.(2005).
The rest of the paper is organized as follows.

In Section 2, we describe thetwo-goods version of life cycle/permanent income model (LCPIM) for food andnon-food consumption.

In Section 3, we explain the Cross-Euler equation

approach and derive the cointegrating restriction among the forcing variables.

In

Section 4, we estimate the IES parameters exploiting the cointegration restriction
implied by the Cross-Euler eqation.

We also estimate the IES parameters fromthe standard Euler equations using GMM.

In Section 5, we formally compare the

IES estimates from the Cross-Euler to standard Euler equation.

Section 6

summarizes this paper with a tentative conclusion.

2. Modeling Food and Non-Food Consumption Behavior
2.1. Setup of the Model
This

paper

adoptsthestandard

two-goods

version

of

the

life

cycle/permanent income model following Atkeson and Ogaki (1996).

A

representative agent is assumed to maximize his expected lifetime utility under his
lifetime budget constraint. The dynamic optimization problem is formulated asfollows,∞
max Et  ∑ β iU ( Ft , NFt )  i =0

s.t. Wt +1 =(1 + rt ) (Wt − Pt F Ft − Pt NF NFt ) + Yt +1(1)

where Ft stands for food expenditure at period t, NFt for non-durable non-food
expenditure, Wt for assets held by the agent, Yt for the stochastic labor income of
the agent, rt for the real interest rate from period t to t+1, Pt F for the price ofequations.
4

food, and Pt NF for the price of non-food, respectively.

β stands for the agent's

discount rate.
We assume that period-by-period utility is time separable for this agent
and also assume additive-separability between durable goods and non-durable
goods.

As for the period-by-period utility function, we employ a standard addi-log

function following Houthakker (1960):=
U ( Ft , NF
t)

Ft1−α
NFt1−γ
.
+K
1−α
1− γ

(2)

Under this specification, the FOCs will then be as follows:

Pt F
1 Ft −α=Pt NF K NFt −γ(3)  F  −αPF 
1Et  β  t +1  (1 + rt ) t F  =Pt +1   Ft (4)  NF  −α
P NF 
1
Et  β  t +1  (1 + rt ) t NF  =Pt +1   NFt (5)It should be noted that under this addi-log specification, 1/ α and 1/ γ can beinterpreted as the intertemporal elasticity of substitution (IES) of food andnon-food expenditure, respectively.
Some remarks should follow for these FOCs. As was pointed out by Ogaki
and Park (1997), the specification of the intratemporal relationship eq. ( 3 ) turns
out to be robust to several kinds of nuisance factors, such as liquidity constraint
and/or habit formation in the utility function. However, the specification of Euler
equations is very sensitive to the presence of liquidity constraint or habit formation.
In other words, specification of the intratemporal relationship is robust, but the
specification of Euler equations is not. Conversely, if by any method we can findevidence that the Euler equation is correctly specified, this will be strong evidence
against the presence of liquidity constraint or habit formation.
If the model is correct, then the intratemporal optimality condition ( 3 )and Euler equations ( 4 ) and ( 5 ) will be correctly specified. Therefore, parameter
estimates α and γ from eq. ( 3 ) and Euler equations ( 4 ) and ( 5 ) should bereasonably close. If the statistical test concludes that parameter estimates are
significantly different from each other, then, by contrapositive logic, we canconclude that some of the assumptions we had made (i.e. addi-log type utility
function, additive separability of durable and non-durable goods, non-existence of5liquidity constraints, non-existence of habit formation, etc.) are implausible.
Conversely, if a statistical test does not reject the null hypothesis that parameter
estimates are equal, it will support or, at least, leave some possibility open for thejoint assumption of addi-log utility specification without the presence of habit
formation or liquidity constraints.
2.2. Complications in Estimating IES parameters
The empirical task is to first obtain the IES parameter estimates from theintratemporal relationship and from Euler equations. However, as Nishiyama(2005) pointed out, when the utility function is of the addi-log type, there is anempirical complication in estimating parameters.
A complication arises from the deterministic relationship of theintratemporal FOC ( 3 ).

A natural way to estimate the IES parameters from theintratemporal relationship is to log-linearize eq. ( 3 ) as follows

ln

Pt F
+ const. + α ln Ft − γ ln NFt =
0
Pt NF

(6)

Provided that (log) relative price, (log) food expenditure, and (log) non-food
expenditure all follow the difference stationary process, one may be tempted to
exploit the cointegration restriction by adding the ad-hoc stationary error term on
RHS of eq. ( 6 ).

But then, in order to maintain coherence of the error structure

within the model, the ad-hoc error term should also be incorporated in the Euler
equations ( 4 ) and ( 5 ).

Indeed, Nishiyama (2005) shows that, by introducing

new kind of error term to the model, the conditional moment conditions implied by
the Euler equations to be non-standard – i.e.,  F  −αPF Et  β  t +1  (1 + rt ) t F  ≠ 1 andPt +1   Ft −α  NF 
P NF 
Et  β  t +1  (1 + rt ) t NF  ≠ 1 .
Pt +1   NFt As such, in the presence of ad-hoc error term, eq. ( 4 ) and ( 5 ) are no longer
correctly specified and, therefore, the GMM estimation based on those misspecified
conditional moment conditions yields inconsistent estimates of the IES parameters.
This is the point where one experiences an empirical difficulty inestimating the IES parameters.

In order to overcome this difficulty, Nishiyama(2005) proposed the Cross-Euler equation approach in estimating the IES

6parameters. By exploiting the cointegrating restriction implied by the cross-Euler
equation, it becomes possible to estimate the IES parameters from both standard
Euler and cross-Euler equation without altering the error structure of the model.

3. The Cross-Euler Equation Approach
3.1. Economic Interpretation of the Cross-Euler Equation
In this section, we derive the Cross-Euler equations based on atime-separable utility function. Since the time-horizon of a representative agent's
optimization problem is infinite, we can reformulate the problem using Bellman
equation as follows,=
V (Wt ) max {U ( Ft , NFt ) + β EtV (Wt +1 )}Ft , NFt

s.t. Wt +1 =(1 + rt ) (Wt − Pt F Ft − Pt NF NFt ) + Yt +1.

First, the FOCs with respect to Ft and NFt will be

respectively.

U F ( Ft , NFt )= β (1 + rt ) EtV '(Wt +1 ) and
Pt F

(7)

U M ( Ft , NFt )= β (1 + rt ) EtV '(Wt +1 ) ,
Pt NF

(8)

Next, invoking the envelope theorem on the Bellman equation, we

obtain the following relationship between the current and future shadow price ofwealth;V '(W=β (1 + rt ) EtV '(Wt +1 )t)

(9)

Substituting eq. ( 9 ) for eq. ( 7 ) and eq.( 8 ), the FOCs for Ft and NFt can berearranged as

U F ( Ft , NFt )= V '(Wt ) and
Pt F( 10 )U NF ( Ft , NFt )= V '(Wt ) .
Pt NF( 11 )

Conventionally, the standard Euler equation can be derived by updatingeq. ( 10 ) (or eq. ( 11 )) and substituting back to eq. ( 7 ) (or eq. ( 8 )) which was thecase in the previous section. Instead, we derive the Cross-Euler equation by
updating eq. ( 11 ) and substituting it for eq. ( 7 ). After some manipulation, theCross-Euler equation can be shown to be7 U ( F , NFt +1 )PF Et  β NF t +1
(1 + rt ) tNF  =1.
Pt +1 
 U F ( Ft , NFt +1 )( 12 )

By the same token, another type of Cross-Euler equation can be derived by
updating eq. ( 10 ) and substituting it for eq. ( 8 )as follows, U F ( Ft +1 , NFt +1 )Pt NF Et  β(1 + rt ) F  =
1.
U
(
F
,
NF
)
P
+
1
NF
t
t
t
 ))))( ))( ΘΩ( 13 )

Now, let us attempt to make an economic sense of the Cross-Euler
equation taking the case of eq. ( 13 ). First, notice that the term Θ in eq. ( 13 )stands for the cross-intertemporal marginal rate of substitution (CIMRS) 4 between
goods NFt and Ft +1 . In other words, the term Θ represents the agent's perceived
trade-off between current non-food goods and future food goods. Second, let us turnto the term Ω in eq. ( 13 ).
obtaining Ft +1 in terms of NFt .

The term Ω stands for the opportunity cost of
The logic is as follows.

By selling one unit of

NFt at period t, the agent can obtain Pt NF amount of numeraire goods -- i.e., Wt
in this context.

By saving all of these numeraire goods at period t, the agent canobtain (1 + rt ) Pt NF of numeraire goods at period t+1.

By using all of these tobuy Ft +1 , the agent can buy (1 + rt ) Pt NF / Pt +F1 units of Ft +1 .
cost of Ft +1 in terms of NFt is (1 + rt ) Pt NF / Pt +F1 .

Thus, the opportunity

Finally, if the agent is optimallytrading NFt to Ft +1 , then the agent is equalizing the opportunity cost to CIMRS
between Ft +1 and NFt , yielding the above Cross-Euler equation ( 16 ). This is the
economic intuition behind the Cross-Euler equation.
The concept of cross-intertemporal marginal rate of substitution (CIMRS) is a key
ingredient of the cross-Euler equation and is defined as follows. For more elaboration
regarding the concept of CIMRS, see Nishiyama (2005).
Definition: Let V ( x11 , ... , x1K , ... , xT1 , ... , xTK ) be a utility function defined upon K

4goods with T periods and let x be a KT × 1 vector such that x= ( x11 , ... , x1K , ... , xT1 , ... , xTK ) .
Then we call the following expression,∂V ( x) / ∂xti+1,−∂V ( x) / ∂xtj
as the cross-intertemporal marginal rate of substitution (CIMRS) between goods xt +1i

and xtj , where i ≠ j and t = 1, …, T-1.
8

In order to understand the structure of Cross-Euler equation further, it isuseful to decompose the terms Θ and Ω .

Decomposing the Cross-Euler

equation ( 16 ), we obtain the following relationship



NF
NF 
 U
P U Ft +1 Pt +1
NFt +1=
Et  β(1 + rt ) t NF
1.
( 14 )
Pt +1 U NFt +1 Pt +F1  U NFt

)
)
(

))
(

Ω2 
Ω1
Θ2
 Θ1

It is possible to decompose CIMRS into IMRS component and MRS
component.

Exploiting this property, the term Θ can be decomposed to IMRS

portion (denoted Θ1 in the above equation) and MRS portion (denoted Θ 2 in theabove equation).

Turning to the term Ω , which stands for the opportunity cost of

Ft +1 in terms of NFt , it is also possible to decompose it into two parts; theopportunity cost of NFt +1 in terms of NFt (denoted Ω1 in eq. ( 14 )) and theopportunity cost of Ft +1 in terms of NFt +1 (denoted Ω 2 in eq. ( 14 )). Here,notice that IMRS Θ1 and opportunity cost Ω1 constitutes a standard Euler
equation in the conventional context. Further, notice that MRS Θ 2 andopportunity cost Ω 2 constitutes an intratemporal FOC at period t+1. Thus, in thissense, the Cross-Euler can be interpreted as the composite optimal condition thatembeds both intertemporal and intratemporal optimality conditions into one
equation.
3.2. Cointegration Relationship Implied by the Cross-Euler Equation
Next, under certain assumptions, we show that the Cross-Euler equation
implies the cointegrating restriction among the forcing variables.

Further, weshow that the Cross-Euler equation approach can overcome the empirical dilemma.
In other words, the Cross-Euler equation approach allows us to estimate andcompare the IES parameters to those from the standard Euler equations without
altering the error structure of the model.
In what follows, we assume the following stochastic processes for eachvariable.
Assumption 1:

Log food and non-food goods expenditure follow the I(1) process.

Assumption 2:

Log price index for food and non-food goods follow the I(1) process

and they are not cointegrated.
Assumption 3: Real interest rate follows the I(0) process.
We parametrize the Cross-Euler equation ( 12 ) using Houthakker's (1960)9addi-log utility function;


NF −γ
PF 
1.
Et  β K −tα+1 (1 + rt ) tNF  =FtPt +1 ( 15 )In a similar fashion, we parametrize the another version of the Cross-Euler
equation ( 13 ) as follows, β Ft +−1α
Pt NF 
+
(1
)
1.
Et 
r
=
t
−γ
Pt +F1  K NFt( 16 )

The forecast error from the Cross-Euler equation ( 15 ) can be defined asfollows,=
et +1 β K=γ
NFt +-Pt F
NFt +1γ
Pt F 
1
(1
)
(1
)
+
+
β
r
E
K
r

t
t 
t
aa
Ft -Pt +NF1
Ft
Pt +NF1 β (1 + r )K -a t F
F / Pt)t )(
non-stochastic at period tγ
 NFt +- NFt +1γ  
1
E
 NF
t 
NF  
Pt +1
 Pt +1  ))))))(stochastic at period t

It is useful to notice where the stochasticity of et +1 is arising. Rearranging thedefinition of the forecast error as in the second line of the above equation, we seethat et +1 is composed of non-stochastic portion and stochastic portion as of period t.
As can be seen, the stochasticity of et +1 arises from the discrepancy between therealized marginal utility of NFt +1 and expected marginal utility of NFt +1 . This
discrepancy is discounted to the present value and denominated by the marginalutility of Ft , which is non-stochastic as of period t.

In this sense, the forecast error

in the context of the Cross-Euler equation measures the magnitude of expectation
error in future marginal utility from goods j in terms of current marginal utility
from goods i.
From the definition of the forecast error, the Cross-Euler equation ( 15 )can be rewritten as

βK

NFt +−1γ
Pt F
(1
+
r
)
=
1 + et +1 .
t
Ft −α
Pt +NF1( 17 )Taking logarithm on both sides of eq. ( 17 ) will yieldPF 
const. + ln (1 + rt ) tNF  − γ ln NFt +1 + α ln Ft =ln(1 + et +1 ) .
Pt +1 

Under the assumption that growth rate of food and non-food expenditure, the realinterest rate, and the growth rate of the price level of both food and non-food arestationary, Nishiyama (2005) showed that ln(1 + et +1 ) to be stationary.
Exploiting the I(0) process of ln(1 + et +1 ) , we obtain the following
cointegrating restriction among the forcing variables,
10
Pt F 
ln (1 + rt ) NF  + α ln Ft − γ ln NFt +1 ~ I (0)Pt +1 ( 18 )

By the same token, the Cross-Euler equation ( 16 ) implies the following
cointegrating restriction,

Pt NF ln (1 + rt ) F  − α ln Ft +1 + γ ln NFt ~ I (0)Pt +1 ( 19 )

Thus, we have derived the legitimate cointegrating restriction among theforcing variables based on the forecast error alone.

Further, since we have notintroduced any ad-hoc error structure to the model, such as an optimization error,
measurement error, preference shock, the specification of the standard Euler
equation ( 4 ) and ( 5 ) remain intact.

In the sense that we now can compare the

IES estimates from Cross-Euler equation to those from the standard Euler
equations without altering the error structure of the model, our proposed approachsuccessfully overcomes the empirical dilemma.
In addition, the log-linearized Cross-Euler equations can be shown to berobust against several nuisance factors (see Appendix). In particular, even in theexistence of liquidity constraints or a certain type of habit formation, cointegration
relationships ( 18 ) and ( 19 ) yield super-consistent estimates for α and γ , while
Euler equations ( 4 ) and ( 5 ) are not guaranteed to yield consistent estimates. On
the other hand, in the absence of liquidity constraints or habit formation, bothlog-linearized

Cross-Euler

equationsandstandard

Euler

equationsyield

super-consistent and consistent estimates of α and γ , respectively. This latter
proposition, which basically states that the estimates of IES parameters fromcointegration analysis and GMM to be close in the absence of liquidity constraints
or habit formation, is particularly important since we can formally test thishypothesis using statistical methods such as Cooley and Ogaki's (1996) LR typetest.

4. Estimating IES Parameters from the Cross-Euler Equations
4.1 Data Description
The data used in this paper is based on the Family Income andExpenditure Survey (FIES) from 1982 to 2004.

In order to preserve enoughnumber of observations, we focused on the expenditure behavior of both
metropolitan and rural households excluding single-person households and11agricultural households. 5

To smooth out the haphazard monthly movements, theexpenditure data is transformed into quarterly data by summing monthly
observations and the price data is transformed into to quarterly data by taking
quarterly averages.

The seasonal factor in both expenditure and price data havebeen removed using X-12 ARIMA method. 6 As a result of these adjustments, wehave total of 90 observations (1982Q1 to 2004Q2) in estimating the IES
parameters.
Although it is possible to define food expenditure in various ways, weconstruct food expenditure data by excluding ‘alcoholic beverages’ from ‘all food’ in
FIES expenditure category.

As Unayama (2003) reports, alcoholic beverages inJapan are well characterized as luxury goods rather than necessity goods.

Whileother food related goods are characterized as necessity goods, the characteristic of
alcohol beverages stands in sharp contrast to other food related goods. In order toretain the homogeneity among the food related goods so as to avoid the goods
aggregation bias, we have deliberately excluded alcoholic beverages from the foodcategory.
In constructing the non-food expenditure data, we have combined
expenditure categories ‘apparel’ and ‘other goods’ in FIES.

Again, Unayama(2003) found some empirical evidence that suggest ‘apparel’ and ‘other goods’ to beluxury goods.

Based on this finding, the goods aggregation bias from combing‘apparel’ and ‘other goods’ does not seem to be too problematic.
The price indexes for food and non-food have been constructed by taking
the weighted average within the same category.

The weight for each item

corresponds to real expenditure share of each item within the category.

Thenthose price indexes are used to deflate the nominal expenditure of food andnon-food in order to convert them into real expenditure.

Finally, the real

expenditure for food and non-food are further adjusted to per-capita base.

The

Figure 1 and Figure 2 show the time-series plots for each variable.
4.2. Preliminary Analysis: Testing for Unit Root
As for a preliminary analysis before the cointegrating regression andFamily Expenditure and Income Survey started to include the single-personhouseholds and agricultural households in the survey starting from January 2000.
6 The consumption tax, which was initially set at 3%, was introduced in April 1989 andsubsequently raised to 5% in April 1997. The effects of consumption tax on the pricesof goods have been removed by including the level shifts in X-12 ARIMA method.
5

12

testing, we test the null of unit root non-stationarity against the alternative of(trend) stationarity for the variables included in the cointegrating regression. In
particular, we test the unit root non-stationarity of the following four variables:
log food expenditure ( ln Ft ), log non-food expenditure ( ln NFt ), opportunity cost of
current

food

consumption

againstfuturenon-food

consumption( ln(1 + rt ) * Pt F / Pt +NF1 ), and opportunity cost of current non-food consumption against
future food consumption ( ln(1 + rt ) * Pt NF / Pt +F1 ).

For the sake of visualization, thetime series plot of the log food and non-food expenditure are provided in Figure 1,
while the time series plot of the opportunity costs are provided in Figure 3.

In

Figure 3, the legend `Opp1` stands for the opportunity cost of current foodconsumption against future non-food consumption and `Opp2` stands for theopportunity cost of current non-food consumption against future food consumption.
The test results are reported in Table 0. For the tests setting the null as
unit root non-stationarity, we have conducted Said and Dickey’s (1984) augmented
Dickey-Fuller (ADF) test and Phillips and Perron’s (1988) PP test.

The test

results for ADF test and PP test are reported on the left hand side of Table 0. As
can be seen from the table, the tests were not able to reject the null of unit root
non-stationarity at the 5% significance level for all variables relevant tocointegrating regression.

As for confirmatory analysis, we have tested the null of(trend) stationarity using KPSS method proposed by Kwiatkowski et al. (1992).
The test results are reported on the right hand side of Table 0. The test rejects
the stationarity of log food expenditure and `Opp1` -- conforming to the test resultsfrom ADF and PP tests --, but was not able to reject the stationarity of the log
non-food expenditure and `Opp2`.

We have also tested the null of trend

stationarity using KPSS test. Here, the test rejects the trend stationarity for 3
out of 4 variables.

Although, the hypothesis regarding the unit root

non-stationarity regarding 4 variables are somewhat mixed, considering that ADF
and PP tests were not able to reject the unit root hypothesis for all the cases andalso based on the several rejections of (trend) stationarity from KPSS test, fromthis point forward, we regard four variables considered in the cointegration
regression to be I(1) variables.
4.3. Dynamic Regressions and the Hausman-type Cointegration Test

13

In this subsection, we use Stock and Watson’s (1993) DOLS estimator inestimating the IES parameters from the log-linearized Cross-Euler equations 7 .
We test the null hypothesis of cointegration.with Choi, Hu, and Ogaki’s (2005)
Hasusman-type Cointegration test in the DOLS framework.
The DOLS estimator is well known, but the Hausman-type cointegration
test is not.

Therefore, we mainly explain the idea of the Hausman-type

cointegration test in this subsection. Consider the following dynamic regression
where y t is an I(1) variable, xt is a 2-dimensional vector of I(1) variables, and k

stands for the order of leads and lags.
2

k

i =1

j =1

y t = β ' xt + ∑ ∑ (γ i , j ∆xi ,t − j ) + et

( 20 )The leads and lags of the first difference of the regressors are added in thisregression in order to correct for the endogeneity problem.

Following Stock andWatson (1993), we assume that the endogeneity correction of adding leads and lags
perfectly eliminates the endogeneity problem in that et is strictly exogenous withrespect to the regressors in ( 20 ) in this paper 8.
Now, if the error term et is I(0), then this is a dynamic cointegrating
regression.

As shown by Stock and Watson (1993), the OLS estimator for thisregression is super-consistent and asymptotically efficient under their regularity
conditions.
On the other hand, if the error term et is I(1), then regression ( 20 ) is aspurious regression and, the Dynamic OLS is inconsistent for the coefficient β asshown in Choi et al. (2005).

The Hausman-type cointegration test utilized theseproperties to discriminate between the situation in which et is I(0) and that inwhich et is I(1).

In order to understand the idea of this test, it is useful to startwith an analysis of the DOLS estimator by the Gauss-Markov Theorem, using
Ogaki and Choi’s (2001) framework.

For this purpose, we consider a special casein which et is serially uncorrelated.

In this case, Ogaki and Choi’s conditional

probability version of the Gauss-Markov Theorem applies 9, and the OLS applied to
Nishiyama (2005) adopted Park’s (1992) Canonical Cointegration Regression (CCR)
in estimating the IES parameters. If the parametric form of the endogenerity
correction is a good approximation, then the Dynamic OLS Regression is more efficientthan the CCR. .
8 To ease our exposition, we assumed that the endogeneity correction is complete. We
can relax this assumption as shown by Saikkonen (1991) under some regularity
conditions.
9 We implicitly assume that the error has finite second moments and that the design
matrix is of full column rank given the realization of the regressors.
7

14

( 20 ) is the Best Linear Unbiased Estimator (BLUE) given the realization of theregressors.
Now consider the case in which et is a random walk.

Then all the

assumptions of the condional probability version of the Gauss-Markov Theorem
hold except for the spherical variance assumption.

In this case, the OLS applied

to (3.1) is unbiased (since we are assuming strict exogeneity), but is not efficient.
In this case, we can apply the Generalized Least Squares (GLS) to ( 20 ) to obtain
the BLUE. Applying GLS to (3.1) basically means that we apply OLS after taking
the first difference of ( 20 );
2

k

i =1

j =1

∆y t = β ' ∆xt + ∑ ∑ (γ i , j ∆2 xi ,t − j ) + ∆et( 21 )

Choi et al. (2005) call this estimator the GLS corrected estimator.
In more general cases in which the error is serially uncorrelated, Choi etal. (2005) note that asymptotic theory shows that (a) the DOLS estimator is
asymptically efficient if

et is I(0), (b) the GLS corrected estimator is consistent,
but is not as efficient as the DOLS estimator if et is I(0), (c) the DOLS estimatoris inconsistent if et is I(1), and (d) the GLS corrected estimator is consistent if
et is I(1).
These observations naturally lead to the idea of testing for cointegrationby comparing the DOLS estimates and the GLS corrected estimates for β. Let theHasuman-type cointegration be defined by

(

)

(

)

H T = T βˆ dgls − βˆ dols 'Vˆβ −1 βˆ dgls − βˆ dols ,where T stands for the sample size, βdols

stands for DOLS estimator in level

regression, βdgls stands for GLS corrected estimator in differenced regression, andVβ stands for a consistent estimator for the asymptotic variance of()

T βˆ dgls − β .

Under the null hypothesis that error term is I(0), both estimators βdols and βdgls are
consistent and, therefore, they should be ‘close’ to each other. The test statistic, HT,
has an asymptotic chi-square distribution with 2 degrees of freedom. On the other
hand, under the alternative hypothesis that the error term is I(1), the level
regression will be spurious and, therefore, only the differenced regression will beconsistent.

Therefore, the estimates from these two estimators will be very

different with a large probability. The test statistic, HT, diverge in this case.
4.4. Estimation Results

15

Having

explainedthedynamic

regression

andHausman-type

Cointegration test, we are now in the position to estimate the IES parameters fromthe log-linearized Cross-Euler equation.

The estimation results of the IES

parameters are reported in Table 1.
First, we have estimated the IES parameters α and γ from eq. ( 18 ) using
the Dynamic OLS.

In order to check for the robustness of the estimate, we report

the estimation results for several lag specifications.

As can be seen from theleft-side panel of the Table 1, level regression estimates of α and γ havetheoretically ‘correct’ signs.

The estimates for α ranges from 1.281 to 1.424, while

γ ranges from 0.382 to 0.435. We have also estimated the IES parameters α and γ
from the differenced regression.

In a sharp contrast, the differenced regression

estimates of α and γ turned out to be conspicuously smaller than those from thelevel regression.

The estimates for α ranged from 0.148 to 0.515, while theestimates for γ ranged from 0.013 to 0.343. Although the difference between theestimates from level regression and differenced regression are evident, we formallyconduct a statistical test using Hausman-type Cointegration test. As can be seenfrom the test results reported in Table 1, for all the lag specifications, theHausman-type test rejects the null hypothesis of cointegration.

Taking this test

result for a face value, this implies that the log-linearized Cross-Euler eq. ( 18 ) is aspurious regression and, therefore, the estimates of α and γ from the level
regression are likely to be inconsistent.

As such, for the purpose of recovering thestructural parameters from eq. ( 18 ), it seems to be reasonable to rely on theestimates from the differenced regression.
Second, by the same token, we have estimated the IES parameters α and γ
from eq. ( 19 ) and the estimation results are reported on the right-side panel ofTable 1. As can be seen from the table, both level and differenced regression yieldtheoretically ‘correct’ signs for α and γ. Turning to estimation results from thelevel regression, the estimates for α ranges from 0.167 to 0.519, while the estimatesfor γ ranges from 0.163 to 0.298. Next, turning to the estimation results from thedifferenced regression, the estimates for α ranges from 0.215 to 0.446, while theestimates for γ ranges from 0.1 to 0.337. Giving a cursory look at the estimation
results, there seems to be no discernible difference between the estimates from thelevel regression and differenced regression.

Again, to formally back thisobservation, we conduct the Hausman-type Cointegration test.

Conforming to ourguess, the Hausman-type test did not reject the null hypothesis of cointegration for

16

all lag specifications, supporting the cointegration relationship of eq. ( 19 ).
Provided this test result, for the sake of recovering the structural parameters fromeq. ( 19 ), it seems to be reasonable to rely on the level regression, which is known
to be more efficient than the differenced regression.

Thus, thanks to theHausman-type Cointegration test, we now have some guidance in which type ofregression to rely on. In other words, for the Cross-Euler eq. ( 18 ), it seems to bereasonable to rely on the differenced regression, while for the Cross-Euler eq. ( 19 ),it seems to be better to rely on the level regression.
Now, the big puzzle remains.

If indeed the permanent income /life-cycle

model described in Section 2 is correct, we should be expecting both log-linearized
Cross-Euler equation to be cointegrated.

Likewise, if some assumptions

pertaining to the cointegration restriction are violated, then it is natural for us to
expect both log-linearized Cross-Euler equation to be spurious. However, the test
results from the Hausman-type Cointegration test were perplexing. That is, forthe log-linearized Cross-Euler eq. ( 18 ), the test rejected the null of cointegration,
implying the regression to be spurious, while for the log-linearized Cross-Euler eq.
( 19 ), the test did not reject the null of cointegration.

How should we interpreting

this contradicting results?
It is indeed difficult to find a clear-cut answer to the above question.
Although this deep puzzle remains, however, based on the observations that 1)
differenced regression estimates for α and γ from both Cross-Euler equation are
relatively similar, and 2) that the level regression from the Cross-Euler eq. ( 19 )yields estimates that are close to the differenced regression estimates 10, in thispaper, we simply regard the log-linearized Cross-Euler ( 18 ) to be spurious anddisregard their estimates based on the level regression.

5. Estimating IES parameters from the Euler Equations
In this section, we conduct Hansen's (1982) GMM on eq. ( 4 ) and ( 5 ).
Before reporting the GMM estimation results, we discuss the choice of
instrumental variables (IV).
5.1. Choice of Instruments and Lag Order

This “closeness” is statistically confirmed by the Hausman-type Cointegration test
which is reported on the right-panel of Table 1.
1017

As was pointed out by Hall (1993) and Ogaki (1993), it is well known thatthe estimate of GMM is very sensitive to the choice of instrumental variables.

To

test for the robustness of the estimates vis-a-vis the choice of instruments, weestimated the parameters under several types of instruments with varying timelags.

The family of instrumental variables was chosen following the convention inapplied GMM literature.

The following table summarizes the choice of

instrumental variables.
IV Type
IV0
IV1
IV2
IV3
IV4
IV5
IV6

Euler Equation ( 4 )Const., F-lag, PF-lag
Const., F-lag, Int-lag
Const., F-lag, NF-lag
Const., PF-lag, PNF-lag
Const., F-lag, PF-lag, Int-lag
Const., F-lag, NF-lag, PF-lag, PNF-lag
Const.,
F-lag,
NF-lag,PF-lag,
PNF-lag, Int-lag

Euler Equation ( 5 )Const., NF-lag, PNF-lag
Const., NF-lag, Int-lag
Const., F-lag, NF-lag
Const., PNF-lag, PF-lag
Const., NF-lag, PNF-lag, Int-lag
Const., F-lag, NF-lag, PF-lag, PNF-lag
Const., F-lag, NF-lag, PF-lag, PNF-lag,Int-lag

Note: Following the convention in applied GMM literature, the instrumental variables
have been constructed by lagging the forcing variables.

Namely, constant (const.),lagged food consumption growth rate (F-lag), lagged non-food consumption growth rate
(NF-lag), lagged price change in food (PF-lag) and non-food (PNF-lag), and lagged realinterest rate (Int-lag).

Another issue in conducting GMM estimation is to choose the lag order ofthe error term when estimating the variance-covariance matrix of GMM
disturbance terms.

According to the rational expectation hypothesis, the forecast

error will be serially uncorrelated.

Since our model is based on the representative

agent with rational expectation, economic theory suggests a lag order of zero.
Nevertheless, taking into account the time aggregation problem which was pointed
out by Grossman et al. (1987) and Heaton (1995) among others, we choose a lag
order of one in estimating the variance-covariance matrix of GMM disturbanceterms following Hansen and Heaton (1996).

Also, to be consistent with time

aggregation issues, we have lagged instrumental variables for at least two periods
when conducting GMM estimations.
5.2. Estimation Results
GMM

estimation was

conducted

18

using a family of

conventionalinstruments. GMM estimation results of food Euler eq. ( 4 ) are summarized inTable 2.

Similarly, GMM estimation results of non-food Euler eq. ( 5 ) aresummarized in Table 3.

Hansen’s J-statistics for each GMM estimation are also

reported.
Let us first turn to Table 2.

As can be seen from the table, the estimatesof α are close to zero for the most cases with occasional negative estimates.

Also,

for the estimates which were relatively apart from zero, there had been a tendency
for those estimates to be accompanied by relatively high standard errors.
Literally interpreting this estimation result, this implies that the representative
agent is nearly risk-neutral with regard to the food consumption.

Also, as for theintertemporal substitution of the food consumption, low estimates of α implies
high IES.

Even worth, the negative estimates of α implies the representative

agent to increase current consumption by intertemporally substituting future
consumption in response to a rise in real interest rate.

In other words, thenegative estimates of IES implies that the income effect from the rise of realinterest rate dominate the substitution effect, which is quite unlikely to happen inpractice. Finally, as for the specification check of the Euler equation, we now turnto Hansen’s J-statistics.

The test rejected the specification of the Euler equationfor 12 out of 21 cases. Considering this frequent rejections by the Hansen’s J-test,
this can be considered as empirical evidence against the specification of the Euler
equation for the food consumption.
Next, let us turn to Table 3.
some negative estimates.

The estimates of γ are close to zero with

Interpreting this result, this implies that the

representative agent to be risk-neutral with regard to the non-food expenditure.
Also, as for the intertemporal substitution of the non-food consumption, lowestimates of γ implies high IES.

Again, literally interpreting, high IES for

non-food expenditure means that the representative agent is willing to substitutecurrent non-food consumption for future non-food consumption in response to aminiscule change in the real interest rate.

Turning to J-test results, we found 14

rejections out of 21 cases which can be considered as evidence against the
specification of the non-food consumption.
From our conventional wisdom, risk-neutral preference or extremely highvalue of IES (or even negative IES) is counter-intuitive and therefore it is hard toaccept this GMM estimation result for face value.

In addition, taking into accountthe strong evidence against the specification of the Euler equations for both foodand non-food consumption from Hansen J-test, there seems to be little ground to19

believe that the GMM estimates for α and γ are consistent.

However, at the sametime, we should be aware of the possibility of size distortion since we have onlyused 90 sample periods in estimating the IES parameters.

Due to the small

sample size, it may well be the case that the J-test over-rejected the specification of
the Euler equation.

In order to verify the specification of the Euler equation

further, we use likelihood ratio type test in the next section.
5.3.1. Further Specification Test
In the previous subsection, we have solely relied on Hansen J-test as forthe specification check of the Euler equations. In this subsection, for the sake ofadditional specification check, we conduct the likelihood ratio type test proposed by
Cooley and Ogaki (1996).

The idea of the likelihood ratio type test is as follows.

If the model is correct under the assumption that there is no liquidity
constraint or habit formation, log-linearized Cross-Euler equations will be correctly
specified with cointegrating restriction.

At the same time, standard Euler

equations will also be correctly specified. Consequently, under the null hypothesisthat the model is correctly specified, parameter estimates of α and γ fromcointegration regression and GMM estimation should be statistically close.

Under

the test, the null hypothesis will be

H 0 :α Euler = α Cross − Euler αnd γ GMM = γ Cross − Euler .
The rejection of the null implies that there exists at least one assumption that isviolated.

Unfortunately, the rejection of the null does not provide us muchinformation about which assumption has been violated. On the other hand, thenon-rejection of the null supports or, at least, leaves some possibility open for theplausibility of the joint hypothesis such that 1) the representative agent's utility
function is of addi-log type, 2) the agent does not face liquidity constraints, and 3)
the agent does not form habit.
5.3.2. Test Results
Here, we report the results of Cooley and Ogaki's (1996) LR-type test.

In

conducting the test, the same instruments from GMM estimation were used forboth restricted and unrestricted GMM.
hypothesis.

We basically tested two types of null

The first null hypothesis is αEuler=αCross-Euler and results are reported in20

Table 4. The second null hypothesis is γEuler=γCross-Euler and results are reported inTable 5.
First, let us see the results under the null of αEuler=αCross-Euler.

Turning toTable 4, which reports the LR-type test results based on the DOLS estimate 11 fromeq. ( 19 ), we found 9 rejections out of 21 cases. Considering this frequent rejectionof the null hypothesis, the LR-type test does not seem to be supporting thespecification of the Euler equation for food consumption.

This test result is

consistent with the test result from Hansen J-test.
Next, we turn to results under the null of γEuler=γCross-Euler.

Examining

Table 5, which reports the test results based on the DOLS estimate, 12again, fromeq. ( 19 ), we found 8 rejections out of 21 cases.

Again, conforming to the result

from Hansen J-test, the LR-type test frequently rejected the null hypothesis ofγEuler=γCross-Euler, giving little support for the specification of the Euler equation for
non-food consumption.
Thus, even for the alternative specification test utilizing LR-type test, wefound an evidence against the specification for both food and non-food Euler
equations.

Based on the test results from Hansen J-test and LR-type test, there

seems to be no strong ground in believing that the Euler equations are correctly
specified.

Since the Euler equations are unlikely to be correctly specified,consequently, the IES estimates from GMM estimation are unlikely to beconsistent as well.

6. Concluding Remarks
In this paper, we adopted the Cross-Euler equation approach in estimating
the IES following Nishiyama (2005).

In particular, based on the two-goods (foodand non-food consumption) version of the LCPIM, we exploited the cointegrating
restriction implied by the Cross-Euler equation in estimating the IES of the
The estimate for αCross-Euler is based on the DOLS estimation of the log-linearized
Cross-Euler eq. ( 19 ) with the lag order of four. The reason why we have relied solely
on the Cross-Euler eq. ( 19 ) is because the regression of the log-linearized Cross-Euler
eq. ( 18 ) is deemed spurious. Regarding the lag order of the DOLS estimation, we
have noticed an influence possibly from the endogeneity problem. As can be seen fromthe Table 1 right-hand side panel, the DOLS estimate for α increases as the lag orderincreases from zero to four. Since the endogeneity bias can be mitigated by including
higher order of lags, we have decided to pick lag order of four in this particularexercise.
12 For the same reason, the estimate for γCross-Euler is based on the DOLS estimation of
the log-linearized Cross-Euler eq. ( 19 ) with the lag order of four.
11

21

representative consumer.
We used Stock and Watson’s (1993) dynamic OLS in estimating thecointegrating regression.

From the level regression estimates, the IES for foodconsumption was around 0.76 (i.e., estimates of α was around 1.3) from one
Cross-Euler specification and was around 3.33 (i.e., estimates of α was around 0.3)
from another Cross-Euler specification.

As for the IES for non-food consumption,

the estimates were around 2.5 for one Cross-Euler specification and were around 5
for another specification.

Thus, we encountered significantly different sets of IES

estimates depending upon the specification of the Cross-Euler equations.

In orderto discern which set of estimates are more reliable, we conducted Hausman-type
Cointegration test proposed by Choi et al. (2005).

According to the test results, itturned out that one of the Cross-Euler equation specifications to be spurious andthe other to be cointegrated.

Simply disregarding the estimation results from a

spurious regression, the sensible estimates of IES seems to be around 3.33 for foodconsumption and around 5 for non-food consumption.

Although these IES

estimates are not as large as the previous estimates of the IES, however, they still
remain to be oddly high.
For the sake of comparison, we also estimated the IES for food andnon-food consumption based on the standard Euler equation using GMM
estimation method, which has been a popular method in the preceding studies.
Conforming to the preceding studies, the IES estimates for both food and non-foodconsumption turned out to be conspicuously high (i.e., CRRA coefficients to be close
to zero).

Further, Hansen J-test and Cooley and Ogaki’s LR-type test frequently

rejected the specification of the Euler equations.
Taking into account the possible miss-specification of the standard Euler
equation based on the test results from J-test and LR-type test, it is likely that
GMM estimation for IES parameters to be inconsistent.

Given this test result,instead of relying on those GMM estimation results, one may be naturally tempted
to count on the IES estimation results from the cointegration regression.
Considering the robustness of the log-linearized Cross-Euler equation against theexistence of liquidity constraint or habit formation, there seems to be a good reason
to count on the IES estimates from the cointegration regression. However, at thesame time, one should bear in mind the fact that one of the specification of the
Cross-Euler equation turned out to be spurious in this paper.

If indeed theassumptions pertaining to LCPIM model adopted in this paper were all plausible,
then we should be expecting both specifications of the Cross-Euler equation to be22

cointegrated.

The mixed evidence from Hausman-type Cointegration test is

perplexing and may be implying that some of the assumptions we made in thispaper to be violated. For this reason, the IES estimates from the cointegration
regression reported in this paper should be interpreted with caution.

23

Appendix: Robustness of the Cointegration Relationship impliedby the Cross-Euler Equation
Unlike the standard Euler equation which specifies the intertemporal
optimality condition in terms of the growth rate of the forcing variables, thelog-linearized

Cross-Euler

condition in terms of level.

equation

specifies

theintertemporal

optimality

By the virtue of long-run equilibrium relationship interms of level, the cointegration regression on ( 18 ) and ( 19 ) can be shown to berobust against certain types of nuisant factors -- i.e., the cointegration regression
yields a super-consistent estimates for the IES parameters.

In this appendix,

based on Nishiyama (2005), we show the robustness of cointegration relationshipagainst two types of nuisant factors discussed in the consumption literature:liquidity constraints and habit formation.
A.1.

Robustness against Liquidity Constraints
Among the various types of liquidity constraints, we adopt the simplest

form -- i.e., a borrowing constraint that requires the net worth of the agent to bealways non-negative.

Specifically, following Zeldes (1989), we impose theconstraint such that Wt − Pt F Ft − Pt NF NFt ≥ 0 to the agent's dynamic optimization
problem. Then, the Bellman equation can be reformulated as follows{}=
V (Wt ) max U ( Ft , NFt ) + mt (Wt − Pt F Ft − Pt NF NFt ) + β EtV (Wt +1 )(1 + rt ) (Wt − Pt F Ft − Pt NF NFt ) + Yt +1
s.t. Wt +1 =(A. 1)

where Lagrangian multiplier µt takes a positive value when the liquidity
constraint is binding and becomes zero when the constraint is not binding.
As a preliminary step in showing the robustness of a cointegration
relationship, we first show that the Lagrangian multiplier is stationary.

The

Euler equations for food and non-food goods can be shown to be

U F ( Ft , NFt )U ( F , NF )=µt + β (1 + rt ) Et F t +1F t +1 and
F
Pt
Pt +1
U NF ( Ft , NFt )U ( F 1 , NFt +1 ),=µt + β (1 + rt ) Et NF t +NF
NF
Pt
Pt +1
respectively.

Now, following the treatment by Zeldes (1989), we normalize theLagrangian multiplier into two types:
24

µtF ≡µt
β (1 + rt ) Et U F ( Ft +1 , NFt +1 ) / Pt +F1 µtNF ≡andµt

β (1 + rt ) Et U NF ( Ft +1 , NFt +1 ) / Pt +NF1 .

As can be seen, µtF stands for the Lagrange multiplier denominated by thepresent-valued marginal utility derived from Ft +1 , and µtNF stands for anothernormalization in terms of NFt +1 .

Here, it should be noted that µtF and µtNF are

both non-negative and inside the information set available at period t, since thedenominators used for the normalization are positive and non-stochastic as ofperiod t.
Using the new normalization of the Lagrangian multiplier and withaddi-log utility function, the Euler equation for food and non-food goods can beconveniently rearranged asF 
β (1 + rt )  t +1  Ft 

−α NF 
β (1 + rt )  t +1 
 NFt Pt F
(1 + µtF ) =1 + ε tF+1 and
F
Pt +1
−γ

Pt NF
(1 + µtNF ) =1 + ε tNF
+1 .
NF
Pt +1

(A. 2)(A. 3)

where ε tF+1 and ε tNF+1 represent the forecast errors arising from the discrepancy
between the expected and realized marginal utility. By assuming that the forecast
errors are stationary over time, the Lagrangian multipliers µtF and µtNF can beshown to be stationary as well.
Given the stationarity of the Lagrangian multipliers, we next demonstrate
the robustness of the cointegration relationship. The specification of the
Cross-Euler equation in the presence of liquidity constraint can be shown to be

U F ( Ft +1 , NFt +1 )U NF ( Ft +1 , NFt +1 )
and
=
µ
+
β
(1
+
r
)
E
t
t
t
Pt F
Pt +NF1U NF ( Ft +1 , NFt +1 )U F ( Ft +1 , NFt +1 )
.
=
µ
+
β
(1
+
r
)
E
t
t
t
Pt NF
Pt +F1(A. 4)(A. 5)For convenience, let us take the case of eq. (A. 4) to show the robustness of thecointegration relationship against the liquidity constraint. Dividing both side of25eq. (A. 4) by β (1 + rt ) Et U NF ( Ft +1 , NFt +1 ) / Pt +NF
1  and rearranging the equation yields
the following relationship

β (1 + rt )Pt F U NF ( Ft +1 , NFt +1 )(1 + µtNF ) =1 + e t +1
Pt +NF1 U F ( Ft , NFt )where µtNF stands for the new normalization of the Lagrangian multiplier defined
above, and e t +1 stands for the forecast error. Taking the logarithm on both sides
and by the assumption of addi-log utility function, it follows thatPt F
ln(1 + rt ) NF − γ ln NFt +1 + α ln Ft = ln(1 + e t +1 ) − ln(1 + µtNF )Pt +1(A. 6)

Now, since the forecast error follows the I(0) stochastic process and the normalized
Lagrangian multiplier follows the I(0) process as well, the RHS of eq. (A. 6) will be
I(0).

As a consequence of this stationarity restriction, the LHS of eq. (A. 6) will

also be I(0), which is exactly the cointegration relationship shown in eq.( 18 ).
Thus, under the addi-log utility function, the cointegration relationship ( 18 ) turns
out to be robust against liquidity constraints and, therefore, allows us to estimate
the IES parameters super-consistently even in the presence of liquidity constraints.
A similar argument holds for the cointegration relationship ( 19 ).
The intuition behind the robustness of the cointegration relationship is asfollows.

The Lagrangian multiplier term, which is an unobservable variable,

enters the regression as an error term in addition to the forecast error. However,since the Lagrangian multiplier term is stationary over time, its presence does notaffect the cointegration relationship among I(1) variables -- i.e., ln NFt +1 , ln Ft , andthe opportunity cost ln(1 + rt ) Pt F / Pt +NF
1 .

Putting it another way, the perturbation

resulting from the Lagrangian multiplier term is of I(0) stochastic order which
distorts the growth rate of optimal consumption (which is the case for the standard
Euler equation), but remains relatively innocuous vis-a-vis the level of the optimal
consumption (which is the case for the log-linearized Cross-Euler equation).
Of course, in the small sample, there is an endogeneity problem in
estimating the cointegrating relationship, since the Lagrangian multiplier µtNF
and future consumption

NFt +1are obviously correlated.

However, thisendogeneity problem could be handled by an estimation method such as Phillips

26and Hansen's (1991) FM-OLS or Park's (1992) CCR.

Further, since the estimators

in cointegrating regression will be super-consistent, i.e. OP (T −1 ) consistent, theendogeneity problem will not matter asymptotically.

Thus, despite theendogeneity problem stemming from the presence of liquidity constraints, we canstill obtain consistent estimates for α and γ from the cointegration relationship( 18 ) and ( 19 ).

A.2.

Robustness against Habit Formation
Next, we show the robustness of the cointegration relationship ( 18 ) and( 19 ) in the presence of habit formation in the agent's preference.

Among thevarious types of habit formation, we follow the specification used by Amano andWirjanto (1996).

Assuming that ln Ft , ln NFt , and ln( Pt NF / Pt F ) follows the I(1)process and that the agent has habit forming preference as follows∞
Et  ∑ U ( Ft *+i , NFt *+i )  i =0
where U ( Ft *+i , NFt *+i ) takes the addi-log type utility function and habit formation
variable defined as Ft * ≡

∞

∞

j =0

j =0

∑ δ j Ft − j and NFt* ≡ ∑ δ j NFt − j , Amano and Wirjanto(1996) shows the stochastic relationship among the intratemporal relative price,current food and non-food goods as follows,

1 Pt NF Ft −α~ I (0)K Pt F NFt − γ
Taking the logarithm on both sides of the above stochastic relationship, theintratemporal relative price, food and non-food goods reveal the following
cointegration relationship,

const. + (ln Pt NF − ln Pt F ) − α ln Ft + γ ln NFt ~ I (0)Now, by adding γ ln NFt +1 and ln Pt +NF
on both sides and subtracting γ ln NFt ,1

ln Pt NF , and ln(1 + rt ) from both sides, we obtain the following relationship27

const. − ln(1 + rt ) − (ln Pt F − ln Pt +NF1 ) − α ln Ft + γ ln NFt +1~ γ (ln NFt +1 − ln NFt ) + (ln Pt +NF1 − ln Pt NF ) − ln(1 + rt ) + I (0)By assumption, the growth rate of NFt , Pt NF and the real interest rate is
stationary and, therefore, the RHS of the above relationship is I(0).

By thestationarity restriction on the RHS, consequently, the LHS will follow the I(0)process which is exactly the cointegration relationship we derived in eq. ( 18 ).
Thus, cointegration relationship ( 18 ) turns out to be robust even in the presence ofhabit formation in the agent's preference.
cointegration relationship ( 19 ).

28

A similar argument holds forReferences
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Elasticities of Substitution: Evidence from Panel and Aggregate Data,” Journal of

Monetary Economics 38, 507-534.
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Regressions and a Hausman-type Cointegration Test," Rochester Center for Economic
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Cooley, Thomas F., and Masao Ogaki (1996), “A Time Series Analysis ofReal Wages, Consumption, and Asset Returns: A Cointegration-Euler equation
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Hall, AR. (1993), “Some Aspects of Generalized Method of Moments
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Heaton, John C. (1995), “An Empirical Investigation of Asset Pricing with
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20, 841-872.

29

Ogaki, Masao (1993), “Generalized Method of Moments: Econometric
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Ogaki, Masao, Jonathan D. Ostyr, Carmen M. Reinhart (1996), “Saving
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Staff Papers 43, 38-71.
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30

Figu re 1 : Log Food an d Non - Food Expe n ditu re

11.45
11.4
11.35
11.3
11.25
11.2
11.15
11.1
11.05
11
10.95

Sep-04

Jun-03Mar-02

Dec-00

Sep-99Jun-98Mar-97

Dec-95

Sep-94Jun-93Mar-92

Dec-90

Sep-89Jun-88Mar-87

Dec-85

Sep-84Jun-83Mar-82log Foodlog Non-foodFigu re 2 : Re al in te re st rate , Food an d Non - Food Pric e In de x
1.08
1.06
1.04
1.02
1
0.98
0.96
Real interest rate
Price Index Food
Price Index Non-food

0.94
0.92

31

Mar-04Mar-02Mar-00Mar-98Mar-96

Mar-94Mar-92Mar-90Mar-88Mar-86

Mar-84Mar-82

0.9

32

0.1

March-02
March-03
March-04

March-99
March-00
March-01

March-96
March-97
March-98

March-93
March-94
March-95

March-90
March-91
March-92

March-87
March-88
March-89

March-84
March-85
March-86

March-82
March-83

Figure 3: Opportunity Costs

0.12

Opp1
Opp2

0.08

0.06

0.04

0.02

0

-0.02

-0.04

-0.06

Table 0
Null of Unit RootADF test

Null of Stationary

PP testKPSS test

cst.

cst.&trend

cst.

cst.&trend

cst.

cst.&trend

ln Ft-2.257-2.310-2.257-2.0320.702*

0.242**

ln NFt-1.383-0.783-1.201-0.8720.315

0.301**


Pt F 
ln  (1 + rt ) NF 
Pt +1 

-1.164-2.819-1.149-3.0381.086**0.077

Pt NF ln  (1 + rt ) F 
Pt +1 


-2.703-2.682-2.612-2.5890.160

0.167*Note:
Lag order used for ADF test was chosen based on Schwartz Information Criteria.
* denotes the rejection of null hypothesis at 5% level.
** denotes the rejection of null hypothesis at 1% level.

Table 1
Cross-Euler Equation Ver.1
DOLS (level)
α
γ

Lag 0
Lag1
Lag2
Lag3
Lag4

GLS-corrected (difference)
α
γ

1.314

0.382

0.333

0.013

(0.184)(0.03)(0.005)(0.001)

1.381

0.414

0.313

0.248

(0.252)(0.038)(0.042)(0.006)

1.424

0.435

0.515

0.343

(0.349)(0.049)(0.072)(0.016)

1.384

0.427

0.343

0.275

(0.483)(0.062)(0.075)(0.011)

1.281

0.413

0.148

0.168

(0.556)(0.060)(0.135)(0.018)Note:
Number in brackets represents the estimated standard error.
* denotes the rejection of null hypothesis at 5% level.
** denotes the rejection of null hypothesis at 1% level.

Cross-Euler Equation Ver.2
Hausman-Test

189.82**26.54**
11.37**
13.44**
9.46**

DOLS (level)αγ

GLS-corrected (difference)
α
γ

0.167

0.163

0.329

0.1

(0.068)(0.011)(0.008)(0.003)

0.23

0.191

0.297

0.288

(0.069)(0.010)(0.019)(0.009)

0.351

0.239

0.386

0.289

(0.064)(0.009)(0.054)(0.021)

0.48

0.289

0.215

0.29

(0.083)(0.011)(0.101)(0.027)

0.519

0.298

0.446

0.337

(0.128)(0.015)(0.112)(0.025)

Hausman-Test

3.13
0.22
0.02
0.69
0.04

Table 2
GMM Result:
Food Euler eq.(2.4)

IV Type

IV0

Lag
(-2)
(-3)
(-4)
(-2)

IV1

(-3)
(-4)
(-2)

IV2

(-3)
(-4)
(-2)

IV3

(-3)
(-4)
(-2)

IV4

(-3)
(-4)
(-2)

IV5

(-3)
(-4)
(-2)

IV6

(-3)
(-4)

  F  −α

Pt F
t +1



Et β
(1 + rt +1 ) F − 1 = 0
Pt +1   Ft 
β
0.978

α
-0.538(0.003)(0.175)[0.220]

0.979

0.665

2.262

(0.003)(0.210)[0.132]

J statistics
1.501

0.979

1.847

0.831

(0.003)(0.543)[0.361]

0.979-0.04420.219**(0.003)(0.191)[0.000]

0.981

0.681

12.763**(0.003)(0.226)[0.000]

0.979

2.342

5.319*(0.003)(0.493)[0.021]

0.978-0.3100.246(0.003)(0.288)[0.619]

0.979

0.536

1.068

(0.003)(0.257)[0.301]

0.978

0.084

2.702

(0.003)(0.189)[0.100]0.978-0.9725.638*(0.004)(0.742)[0.017]0.970-2.7694.131*(0.008)(1.114)[0.042]

0.972

3.857

0.680

(0.007)(1.078)[0.409]0.979-0.14120.809**(0.003)(0.146)[0.000]

0.981

0.667

12.711**(0.003)(0.210)[0.001]

0.979

2.304

5.603

(0.003)(0.489)[0.060]0.980-0.3756.170(0.003)(0.272)[0.103]

0.982

0.413

11.986**(0.003)(0.222)[0.007]

0.978

0.124

11.899**(0.003)(0.172)[0.007]

0.981

0.314

18.068**(0.003)(0.188)[0.001]

0.983

0.473

15.820**(0.003)(0.218)[0.003]

0.982

0.127

18.447**(0.002)(0.168)[0.001]

Note:
Number in brackets represents the estimated standard error.
Number in square brackets represents the p-value.
* denotes the rejection of null hypothesis at 5% level.
** denotes the rejection of null hypothesis at 1% level.

D.F.
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
3
3
3
4
4
4

Table 3
GMM Result:Non-food Euler eq. (2.5)

IV Type

IV0

Lag
(-2)
(-3)
(-4)
(-2)

IV1

(-3)
(-4)
(-2)

IV2

(-3)
(-4)
(-2)

IV3

(-3)
(-4)
(-2)

IV4

(-3)
(-4)
(-2)

IV5

(-3)
(-4)
(-2)

IV6

(-3)
(-4)

  NF −γ
PNF 
Et  β  t +1  (1 + rt +1) t NF − 1 = 0
Pt +1
  NFt 

β
0.952

γ
2.994

(0.010)(0.718)[0.034]

0.981

0.433

4.523*(0.002)(0.208)[0.033]0.979-0.171

6.588*(0.002)(0.199)[0.010]

0.973

1.069

6.128*(0.003)(0.225)[0.013]

0.978

0.762

5.994*(0.002)(0.148)[0.014]

J statistics
4.482*

0.979

0.413

23.447**(0.002)(0.144)[0.000]0.971-1.9180.247(0.011)(1.284)[0.619]

0.979

0.386

0.261

(0.002)(0.273)[0.609]0.979-0.5620.255(0.003)(0.383)[0.613]

0.972

1.533

0.000

(0.003)(0.363)[0.981]

0.982

0.132

6.664**(0.002)(0.435)[0.009]

0.977

0.362

8.092**(0.002)(0.080)[0.004]

0.972

1.075

9.225**(0.003)(0.220)[0.009]

0.979

0.696

7.679*(0.002)(0.140)[0.021]

0.979

0.399

23.835**(0.002)(0.143)[0.000]

0.972

1.331

2.532

(0.003)(0.310)[0.469]

0.982

0.371

8.363*(0.002)(0.227)[0.039]

0.979

0.345

11.295*(0.002)(0.062)[0.010]

0.974

1.039

4.000

(0.003)(0.200)[0.405]

0.980

0.619

11.777*(0.002)(0.155)[0.019]

0.980

0.343

16.715(0.002)(0.062)[0.149]Note:
Number in brackets represents the estimated standard error.
Number in square brackets represents the p-value.
* denotes the rejection of null hypothesis at 5% level.
** denotes the rejection of null hypothesis at 1% level.

D.F.
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
3
3
3
4
4
4

Table 4

Table 5LR-type Test Result:LR-type Test Result:

H0 : αEuler = αCross− Euler
IV Type
IV0

IV1

IV2

IV3

IV4

IV5

IV6

Lag
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)

Restricted
15.03413.24417.38015.03413.24417.380
5.118
0.549
5.618
7.843
9.090
5.950
15.04413.65418.437
12.2099.244
12.36515.66713.88318.456

Unrestricted
1.501
2.262
0.831
20.21912.763
5.319
0.246
1.068
2.702
5.638
4.131
0.680
20.80912.711
5.603
6.170
11.986
11.89918.06815.82018.447Note:* denotes the rejection of null hypothesis at 5% level.
** denotes the rejection of null hypothesis at 1% level.

H0 : γ Euler = γ Cross− Euler
QLR
13.533**10.982**
16.549**-5.1850.481
12.061**4.872*-0.5192.916
2.205
4.959*
5.27*-5.7650.943
12.834**6.039*-2.7420.466-2.401-1.9370.009

IV Type
IV0

IV1

IV2

IV3

IV4

IV5

IV6

Lag
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)
(-2)
(-3)
(-4)

Restricted
2.513
4.056
5.734
15.44615.08715.489
1.172
0.278
3.773
14.076
6.115
7.278
16.81715.23916.39815.6497.548
10.95318.32116.45417.585

Unrestricted
4.482
4.523
6.588
6.128
5.994
23.447
0.247
0.261
0.255
0.000
6.664
8.092
9.225
7.679
23.8352.532
8.363
11.2954.000
11.77716.715Note:* denotes the rejection of null hypothesis at 5% level.
** denotes the rejection of null hypothesis at 1% level.

QLR-1.969-0.467-0.8549.318**
9.093**-7.958
0.925
0.017
3.518
14.076**-0.549-0.8147.592**
7.56**-7.437
13.117**-0.815-0.34214.321**
4.677*
0.870