Discussion Paper No. 273
The Cross-Euler Equation Approach to testing for
the Liquidity Constraint: Evidence from Macro andMicro Data
Shin-Ichi Nishiyama

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 to testing for the Liquidity
Constraint: Evidence from Macro and Micro Data1
Shin-Ichi Nishiyama2
Graduate School of Economics and Management,
Tohoku University
This Version: September 2011
First Version: September 2002

1

My deepest gratitude goes to Masao Ogaki for his advice and encouragement in pursuing thisproject. I would also like to thank Steve Cecchetti, C.Y. Choi, Paul Evans, Ryo Kato, Nelson Mark,
Kiyoshi Matsubara, Hajime Miyazaki, Donggyu Sul, Kiyoshi Taniguchi, and Takayuki Tsuruga for
their valuable comments. All remaining errors are my own.
227-1 Kawauchi, Aoba-ku, Sendai-shi, Miyagi-ken, 980-8576, JAPAN, Tel.: +81-22-795-3471,
Email: nishiyama@econ.tohoku.ac.jp

Abstract
In this paper, we propose a new empirical method in testing for the existence of liquidity
constraint utilizing the concept of Cross-Euler equation. The Cross-Euler equation represents
the optimal consumption pattern of a good in the current period to another good at afuture period. It can be interpreted as the composite optimal condition that embeds bothintertemporal and intratemporal optimal consumption relationships into one equation. The
Cross-Euler equation has an advantage over the standard Euler equation, in the sense thatthe cointegrating relationship is maintained even when the liquidity constraint is present in
the agent’s decision problem. Thus, by comparing the preference parameter estimates fromthe Cross-Euler equation to those from the standard Euler equation, it is possible to detect
the existence of a liquidity constraint. We adopt standard two goods version of Life-Cycle
model to study the consumption behavior of necessity goods and luxury goods. First, basedon the aggregate data, the test rejects the null of no liquidity constraints for necessity goods,
while accpeting the null for luxury goods. Since, by construction, large share of necessity
goods are consumed by poor households, it is possible to interpret the results as evidence
that poorer households are likely to be liquidity constrainted. Next, we construct synthetic
panel data from the Consumer Expenditure Survey where households have been classi…ed tocohorts by their ages and educational attainments. Again, taking the Cross-Euler equation
approach in testing for the liquidity constraints, the test rejected the null of no liquidity
constraint for low-education cohorts, while accepting the null for high-education cohorts.
Taking an education level as a proxy for permanent income, the test results were consistent
with the view that poorer agents are more likely to be liquidity constrained.

1

Introduction

Whether there are liquidity constraints in the economy or not is an important question. It
is important because …scal and monetary policy implications from Life-Cycle model will beconsiderably altered in the presence of liquidity constraints. As pointed out by Altonji andSiow (1987) and Shea (1995), consumption growth may respond asymmetrically to change intransitory income when liquidity constraints are binding. This, in turn, implies the asymmetric …scal policy e¤ect. Also, when liquidity constraints are binding, change in real interest
rate will not cause any intertemporal substitution in consumption that expansionary monetary policy may not have any e¤ect in inducing consumers to spend more. Ironically, sinceconsumers’ net wealth are often seriously damaged during the economic downturn - making liquidity constraints even more compelling -, monetary policy may be ine¤ective during
the recession when the policy stimulus is desperately needed. In the presence of liquidity
constraints, monetary policy will face a serious problem during an economic downturn. In
this connection, the asymmetric monetary policy e¤ect reported by Choi (1999) and Weise(2000) may well be stemming from the presence of liquidity constraints in the economy.
Thus, whether there exist liquidity constraints or not is a serious question for policy makers.
Re‡ecting the importance of liquidity constraints, not surprisingly, considerable amountof research have been devoted in testing for the liquidity constraints. Based on aggregate
data, Flavin (1981) and Campbell and Mankiw (1989,1999), among others, have conducted an
excess sensitivity test and found that consumption growth rate to be signi…cantly correlatedwith lagged or predicted income growth, which can be interpreted as evidence of liquidity
constraint. Turning to the excess sensitivity test based on panel data, Hall and Mishkin(1982), Shapiro (1983) and Hayashi (1985) all found some evidence that lagged income changeor real disposable income change to be signi…cantly correlated with consumption growth.
Mariger (1987), Altonji and Siow (1987) and Zeldes (1989) speci…cally take into account for
the Kuhn-Tucker condition emerging from the liquidity constraint. Mariger (1987), basedon cross-sectional surveys, reports that 19.4% of the households are liquidity constrained.
Zeldes (1989), based on Panel Study of Income Dynamics (PSID), have found that lesswealthy households to be more vulnerable to liquidity constraints. Runkle (1991), basedon PSID, tests the over-identifying restrictions implied by Euler equation and reports little
evidence of liquidity constraints. Attanasio and Webber (1993) and Meghir and Webber(1996) studied the validity of Life-Cycle model using the Consumer Expenditure Survey(CES) and report some evidence that younger households are liquidity constrained. DeJuan
and Seater (1999), also based on CES, exploits asymmetric response of consumption growth
to income change and reports little evidence of liquidity constraints. Recently, Gross andSouleles (2001), using unique panel data on credit card accounts, estimates the marginal
propensity to consume out of liquidity and found their estimates to be higher for people
whose credit limit is low, which they interpret as evidence of liquidity constraints.
In this paper, we propose a new empirical method in testing for the existence of liquidity
constraint utilizing the concept of Cross-Euler equation. The Cross-Euler equation repre-1sents the optimal consumption pattern of a good in the current period to another good at afuture period. It can be interpreted as the composite optimal condition that embeds both intertemporal and intratemporal optimal consumption relationships into one equation. Underaddi-log type period-by-period utility function, we show that the Cross-Euler equation has anadvantage over the standard Euler equation, in the sense that the cointegrating relationshipis maintained even when the liquidity constraint is present in the agent’s decision problem.
Thus, by comparing the preference parameter estimates from the Cross-Euler equation to
those from the standard Euler equation, it is possible to detect the existence of a liquidity
constraint.
As long as the Life-cycle model is constructed in the context of multiple goods1 , theCross-Euler equation approach is potentially feasible. In this paper, in order to explore thepotential of the Cross-Euler equation approach in testing the liquidity constraints we adopt
two goods version of Life-cycle model. Speci…cally, we study the consumption behaviors ofnecessity goods and luxury goods of the agents. Choice of necessity and luxury goods are,to some extent, arbitrary. However, this classi…cation has its own motivation, especially inthe context of aggregate data. In aggregate data, by construction, relatively larger share of
luxury goods are consumed by ”rich” agents, while relatively larger share of necessity goods
are consumed by ”poor”agents in the economy. Thus, by studying the behavior of standard
Euler equations for both goods and also studying the behavior of the Cross-Euler equation
linking both goods, there is a good possibility that we can infer which type of agents are
more vulnerable to liquidity constraints even from the aggregate data. Naturally, since thepoorer agents tend to be more vulnerable to the liquidity constraint, we expect that the Euler
equation for the necessity goods to be misspeci…ed, but the Euler equation for the luxury
goods to be speci…ed. Empirical evidence in this paper, though yielding mixed results,
seemed to have supported this prediction.
However, the empirical results based on the aggregate time-series data implicitly assumes
the existence of the representative agent. The existence of the representative agent requiresthat all the agents in the economy to share the identical and homothetic preferences, which
is obviously a too strong assumption (See Kirman (1992) and Stoker (1993)). In an attempt
to overcome this aggregation problem and to legitimately test for the liquidity constraints,
we also conduct cohort analysis proposed by Deaton (1985) as a complement to the empirical
study from aggregate data. Under the cohort analysis, the households are aggregated into
cohorts who share the similar taste and, therefore, it has an advantage over the representative model in the sense that it is relatively free from the aggregation problem. Basedon the Consumer Expenditure Survey from 1984 to 1998, we classify the households into
cohorts by their age and educational attainment following Attanasio and Browning (1997).
Classi…cation by the age are necessary because the households are expected to have hetero1 One good version of Life-cycle model implicitly assumes that Hicks’Composite Condition to hold for all

consumption goods (See, for instance, Chapter 2 of Deaton (1996) for details). This assumption is obviously
too strong and is unlikely to be met in reality. In this sense, multiple good version of Life-cycle model has
natural advantage over one good version.

2

geneous consumption patterns over the life-cycle. Classi…cation by education attainments
was adopted as a mean to classify the households by their life-time income (i.e. permanent
income). Then we estimate the preference parameters for each education cohorts based on
their Cross-Euler and standard Euler equations. As a mean to formally compare the parameters estimates from both Euler equations, we conduct Cooley and Ogaki’s LR type test
(1996). Empirical evidence based on cohort analysis suggested that low-education cohort tobe liquidity constrained, but not for the high-education cohort. Taking the education levelas a proxy for lifetime income, the test results seems to support the view that poorer agentsare more likely to be liquidity constrained.
Another novelty of this paper is that we have adopted cointegration analysis in the settingof pseudo panel data. In part due to its relative youth, cohort technique developed by
Deaton (1985) have traditionally found an application in I(0) data setting, but not in I(1)data setting. However, as long as number of observations within the cohort is su¢ ciently
large, there is no problem in appealing to the asymptotic theory already established in thepanel cointegration literatures. By exploiting the panel cointegration relationship in pseudo
panel data, researchers can enjoy the reward of super-consistency (Engel and Granger (1987),
Phillips and Moon (2000)) in estimating the preference parameters. More pragmatically,
considering that time dimension of Consumer Expenditure Survey is quite limited, the notionof super-consistency seems to be extremely valuable. In this paper, in particular, we will
estimate the preference parameters from (pseudo) panel cointegrating relationship using Markand Sul’s (2001) Panel Dynamic OLS estimation method.
The rest of paper is organized as follows. Section 2 describes the standard two goods
version of the life-cycle model to study the consumption behavior of necessity goods andluxury goods. Section 3 describes the Cross-Euler equation approach in testing the liquidity
constraints. In Section 4, we apply the Cross-Euler equation approach in testing the liquidity
constraints using the aggregate data. In Section 5, we construct the cohort data set from
Consumer Expenditure Survey and apply the Cross-Euler equation approach in testing forthe liquidity constraints. Section 6 provides the concluding remark.

2

Model Description

This paper adopts the standard two-goods version of Life Cycle/ Permanent Income Model(LCPIM) as in Ogaki (1992). Representative agent is assumed to maximize his expected
lifetime utility under his lifetime budget constraint. Stating mathematically,
max E01
Xt

U (Nt ; Lt )

(1)

t=0

s:t: At = (1 + rt )At 1 + Yt

PtN NtPtL Ltfor 8t0(2)where Nt stands for necessity goods at period t, Lt stands for luxury goods, At stands forthe asset holding of the agent, Yt stands for the labor income of the agent, rt stands for the3real interest rate from period t 1 to t, PtN stands for the price of a necessity good, and PtL
stands for the price of an luxury goods. Finally, we parameterize agent’s subjective discountrate as constant .
We have assumed that period-by-period utility is time separable for this agent and have
implicitly assumed the additive separability between durable goods and non-durable goods.
Solving above optimization problem yields the following …rst order conditions (FOC).

E0
E0

PtN
UNt
=
for 8t 0
ULt
PtL
UNt+1
PtN
(1 + rt+1 ) N
1 = 0 for 8t
UNt
Pt+1
ULt+1
PL
(1 + rt+1 ) Lt
ULt
Pt+1

1 =0

for 8t

(3)
0

(4)

0

(5)

Eq. (3) represents the contemporaneous FOC for this representative agent. These FOC’s
follow if the agent is maximizing his utility given the contemporaneous price ratio of necessity
and luxury goods. In other words, representative agent will equalize his contemporaneous
marginal rate of substitution (MRS) to current price ratio of two goods. Eq. (4) represents
the intertemporal FOC, the Euler equation, of necessity goods. This FOC will follow, if theagent is maximizing his expected utility over time given the discounted expected price ratioof necessity goods at period t to period t + 1. The Euler equation for luxury goods (eq. (5))
holds by the parallel logic.
Next, in order to make the model econometrically estimable, we are going to parametrize
the utility function. We specify the utility function as a standard addi-log function following
Houthakker(1960).
1
1
(Lt )
(Nt )
+K
(6)
U (Nt ; Lt ) =1
1
This addi-log speci…cation was used in Ogaki (1992). Houthakker’s addi-log speci…cation
reveals the non-homothetic preference of the agent in general, but contains the homothetic
preference as a special case when = . This non-homothetic preference is crucial in ourmodel since we try to capture intertemporal aspects of necessity goods (which by de…nition
requires the income elasticity to be smaller than 1) and luxury goods (which requires theincome elasticity to be greater than 1). Homothetic preference, such as CES utility function,
reveals the unit income elasticity that it is not an appropriate utility speci…cation to adopt
when modeling the behavior of necessity and luxury goods consumption at the same time.
It should be noted that under this addi-log speci…cation, 1= and 1= can be interpreted as
the intertemporal elasticity of substitution (IES) of Nt and Lt respectively2 .
Under this speci…cation, FOC will then be as follows.
1 (Nt )PtN=
L
K (Lt )
Pt

for 8t0

2 This will not be the case if a utility function is time non-separable (e.g.

(7)

allowing habit formation) or
goods non-separable (e.g. CES type function). This was pointed out by Constantinides (1991). The generalformula for deriving IES under a time non-separable utility function was shown by McLaughlin (1995).

4

E0

E0

"

"

Nt+1
Nt

PtN
(1 + rt+1 ) N
Pt+1

Lt+1
Lt

PL
(1 + rt+1 ) Lt
Pt+1#

1 = 0 for 8t

0

(8)

#

1 = 0 for 8t

0(9)Given these speci…cations, we are now ready to actually estimate and test the implication of
the model.
Some remarks should follow for these FOC’s. As was pointed out by Amano and Wirjanto(1996) and Ogaki and Park (1998), the speci…cation of the intratemporal relationship eq.(7)turns out to be robust to several kinds of nuisant conditions, such as liquidity constraint andhabit formation in utility function. However, the speci…cation of Euler equations are very
sensitive to the presence of liquidity constraint or habit formation. In other words, speci…cation of the intratemporal relationship is robust, but the speci…cation of Euler equations arenot. Conversely, if for any method we can …nd the evidence that Euler equation is speci…ed,that will be a strong evidence against the presence of liquidity constraint or habit formation.
This speci…cation issue of Euler equations will be the central focus of the rest of our paper.

3

The Cross-Euler Equation Approach

If our model is correct, then the intratemporal relationship eq.(7) and Euler equations eq.(8)and eq.(9) will be speci…ed. Therefore, the parameter estimates and from eq.(7) andEuler equation (8) and (9) should be reasonably close. Thus, the main goal of this paper is
to test whether these estimates from di¤erent equations are statistically close enough or not.
If the statistical test concludes that parameter estimates are signi…cantly di¤erent from eachother, then, by the contrapositive logic, we should conclude that some of the assumptions
we had made (i.e., addi-log type utility function, additive separability of durable and nondurable goods, non-existence of liquidity constraint, non-existence of habit formation etc.)
are illegitimate. Unfortunately, statistical test will not be informative as to say exactly whichassumption is wrong. Conversely, if the statistical test does not reject the null hypothesisthat parameter estimates are equal, it will support or, at least, leaves some possibility open forthe joint assumption of addi-log utility speci…cation without the presence of habit formation
or liquidity constraint.
Thus, the empirical task is to …rst obtain the parameter estimates from contemporaneous
relationship (i.e. eq.(7)) and from Euler equations (i.e. eq.(8) and eq.(9)). Predecessors
in this line of research have used cointegration analysis and/or GMM in estimating theparameters. However, when the utility function is a standard one, there will be an empiricalcomplication in estimating parameters. The following section will address this complicationand propose a remedy for this problem.

5

3.1

An Empirical Dilemma

Let us turn back to the three FOC’s (i.e. eq. (7), (8), and (9)) implied by the model. Since
the conditional moment condition is established for eq. (8) and (9), there is no problem inapplying GMM on these equations. If indeed these Euler equations are well speci…ed, thenthe GMM will yield OP (T 1=2 ) consistent estimate of and . However, unfortunately, thecomplication will arise from contemporaneous relationship eq.(7).
A natural way to estimate the parameters from intratemporal relationship is to loglinearize and rearrange the eq. (7) as follow.
ln Lt + const:

1

ln

PtN
PtL

a

ln Nt = 0

If the forcing variables ln Lt , ln Nt , and ln(PtN =PtL ) follows the I(1) process, one is tempted
to introduce some I(0) disturbance terms on RHS of the above equation in order to conduct
the cointegration analysis.
ln Lt + const:

1

ln

PtN
PtL

a

ln Nt = "t where "tI(0) and E("t ) = 0(10)This error term can be an optimization error, measurement error, preference shock3 ,etc. However, in so doing, one should also include this newly introduced error term into
the existing Euler equations (8) and (9). In other words, in addition to the forecast error
embedded in Euler equations, one is now introducing another kind of error term which is
intrinsically di¤erent type of an error. It can be shown that when a new error term isintroduced to Euler equations, no longer eq.(8) and eq.(9) are speci…ed (See Appendix 1).
This is the point where one experience the dilemma. The objective is to estimate the parameters from both intratemporal relationship and Euler equations. If one introduces somearbitrary error term to the contemporaneous relationship in order to conduct the cointegration analysis, this newly introduced error term will a¤ect the speci…cation of Euler equation.
Conducting GMM on standard Euler equations (8) and (9) will no longer yield a consistentestimates forand . On the other hand, if one does not introduce any error term tocontemporaneous relationship, Euler equation (8) and (9) will remain to be speci…ed and
GMM on these equations will yield consistent estimates assuming that the model is correct.
But then, since the error term is not present for the contemporaneous relationship, one faces
a illegitimacy in conducting the cointegration analysis.

3.2

The Cross-Euler Equation ApproachThis section propose a remedy to the above empirical dilemma. The idea is to …rst de…nethe concept called cross intertemporal marginal rate of substitution (CIMRS) and then to
3 Clarida (1994) and Amano and Wirjanto (1996) adopts preference shock in their model, making cointegration analysis possible under mild conditions. However, it should be noted that if one adopts preference
shock in their model, then as a trade-o¤, GMM estimation of Euler equation will not be implementable unlessthey have a data available for preference shock.

6derive the corresponding …rst order condition which we will call the Cross-Euler equation.
In this subsection, we will show how the concept of Cross-Euler equation can be a remedy
for the above empirical dilemma.
Step 1: De…ning CIMRS and deriving the Cross-Euler equations
1
K
De…nition 1 (CIMRS) Let V (x11 ; :::; xK
1 ; :::; xT ; :::; xT ) be a utility function de…ned upon
K goods with T periods. Then we call the following expression as@V ( )=@xit+1@V ( )=@xjt
the cross intertemporal marginal rate of substitution (CIMRS) between goods xit+1 and xjt
where i 6= j and t = 1; :::; T 1.
The concept of CIMRS is just a simple extension of IMRS. It can be easily conceptualized
as the IMRS de…ned upon di¤erent goods instead of same goods.4 From the concept of
CIMRS and from our model described, we can derive the “alternative”FOC. For convenience
we will call the following FOC as the Cross-Euler equation.
"#
PtN(Lt+1 )
E0 K
1 =0
(11)
a (1 + rt+1 ) L
Pt+1
(Nt )
We can see the intuition of the above equation by thinking of the situation where the agent istrading Nt to Lt+1 . Now the marginal rate of substitution between Lt+1 and Nt (or CIMRS inaour terminology) is de…ned as
ULt ;t+1 =UNt ;t and takes the form of
K (Lt+1 ) = (Nt )under addi-log utility function. Next, let us consider the opportunity cost of obtaining Lt+1
in terms of Nt . By selling one unit of Nt at period t, agent can obtain PtN of numeraire goods.
By saving all of these numeraire goods at period t, agent can obtain (1+rt+1 ) PtN of numeraireL
goods at period t + 1. By using all of these to buy Lt+1 , agent can buy (1 + rt+1 ) PtN =Pt+1
L
units of Lt+1 . Thus the opportunity cost of Lt+1 in terms of Nt is (1 + rt+1 ) PtN =Pt+1. If
the agent is optimally trading Nt to Lt+1 , then the agent is equalizing the opportunity cost
to CIMRS between Lt+1 and Nt , yielding the above Cross-Euler equation5 .
By similar fashion, we can derive the another version of Cross-Euler equation as follow.
"#a(Nt+1 )
PtL
1 =0
(12)
E0
(1 + rt+1 ) N
K (Lt )Pt+1
In deriving the above Cross-Euler equation, we have equalized the CIMRS between Nt+1aand Lt (i.e. ( =K)((Nt+1 ) = (Lt ) )) to the opportunity cost of Nt+1 in terms of Lt (i.e.
N).
(1 + rt+1 )PtL =Pt+1
Step 2: Cointegration relationship implied by Cross-Euler equationj
4 Another way of saying is that IMRS of goods i is a special case of CIMRS between xi
t+1 and xt wherei = j.
5 For a formal derivation, see Appendix 2.

7

Returning to the Cross-Euler eq. (11), it follows thatK

(Lt+1 )(Nt )a (1 + rt+1 )PtN= 1 + et+1
L
Pt+1

(13)

where we de…ned et+1 as
et+1

PtN
K
(1
+
r
)
t+1
a
L
Pt+1
(Nt )
(Lt+1 )

E0

"

PtN
K
(1
+
r
)
t+1
a
L
Pt+1
(Nt )
(Lt+1 )#Taking the log6 on both side of eq.(13) will yield
const: + ln (1 + rt+1 )

PtN
L
Pt+1

ln Lt+1 +ln Nt = ln(1 + et+1 )Assuming that the growth rate of both domestic and imported non-durable goods consumption (i.e. Nt+1 =Nt and Lt+1 =Lt ), real interest rate (i.e. rt ) and the growth rate of the priceN
L
level of both domestic and imported non-durable goods (i.e. Pt+1
=PtN and Pt+1
=PtL ) are
stationary7 , it can be shown that ln(1 + et+1 ) will also be stationary (See Appendix 3).
Exploiting the I(0) process of ln(1 + et+1 ), we can obtain the following cointegrating
relationship.
PN1ln (1 + rt+1 ) Lt
ln Nt I(0)(14)
ln Lt+1 + const:
Pt+1
By similar fashion, we can derive the following cointegration relationship from eq.(12).
ln Lt + const:

1

ln

N
Pt+1
1
1 + rt+1 PtL

ln Nt+1

I(0)

(15)

Given these cointegrating relationships of log-linearized Cross-Euler equations, combinedwith GMM-estimable standard Euler equations (8) and (9), we now have a …rm ground incomparing the estimates of and . To summarize, under the weaker assumption whichallows for the existence of liquidity constraint and/or certain type of habit formation, loglinearized Cross-Euler equation (14) and (15) will yield a super-consistent estimates forand , while Euler equation (8) and (9) are not guaranteed to yield consistent estimates.
Under the stronger assumption which does not allow for the presence of liquidity constraint
or habit formation, both log-linearized Cross-Euler equations and standard Euler equations
will yield super-consistent and consistent estimates ofand , respectively. This latter
proposition, which basically states that the estimates of IES parameters from cointegration
6 As was pointed out by Carrol (1997), ideally speaking, it is preferable to estimate the parameters without
the log-linearization. However, since the possibly I(1) processes are present inside the conditional moment
equations (11) and (12), again, it is likely that the fundamental assumption of GMM estimation method to
be violated. This forced us to log-linearize the Cross-Euler equations.
7 Empirical evidences seems to support this assumption. See, for instance, Clarida (1994), Amano et al(1996) and de la Croix et al (1998).

8

analysis and GMM to be close under the stronger assumption, is particularly important for
us since we can formally test this proposition using statistical method such as Cooley andOgaki’s (1996) LR type test. The following table summarizes the main idea of this section.

Log-linearized Cross-Euler Equation

Standard Euler Equation(M etho d: Cointegration)(M etho d: GM M )Under weaker assum ption

Sup er-consistent estim ates forandInconsistent

Under stronger assum ption

Sup er-consistent estim ates forandConsistent estim ates forandSome remarks should follow for cointegration results. Since we are constructing I(0) error
terms by leading the variables ln Lt ; ln Nt ; ln P L ; ln P N , obviously there will be an endogeneity
problem when estimating these cointegrating relationship. However, this problem could be
handled by the estimation method such as Phillips 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), the endogeneity problem will not matter asymptotically. Thus,
despite this endogeneity problem, we still can get consistent estimates forand from
eq.(14) or eq.(15).

4Empirical Evidence from Macro Data

4.1

Data Description

The data we use in this paper are quarterly and seasonally adjusted U.S. non-durable goods
consumption data covering the period from 1959 Q1 to 200 Q2 (166 observations). Under
the classi…cation of National Income and Product Accounts (NIPA), we use real per adultpersonal consumption expenditure (PCE) for food and tobacco8 and real per adult PCE
for non-durable goods excluding food and tobacco in estimating and testing of the model’s
implications. PCE for food and tobacco is set to be a proxy for necessity goods and PCEof non-durables excluding food and tobacco is set to be a proxy for luxury goods. Table 1
summarizes how we categorized the components of non-durable goods into necessity goods
and luxury goods in this paper.
For a price measure of each goods (i.e. P N and P L ), we adopted chain-type price index(base year 1996) reported in NIPA Table 7.5. “Chain-Type Price Indexes for PCE by Typeof Product”9 . In constructing real per adult consumption for food and tobacco (i.e. Nt ), wede‡ated PCE for food and tobacco by the chain-type price index for PCE food and tobaccoand further divided by U.S. population above 20 years old. It should be noted that by this
8 Following Ogaki (1992), we have excluded alcohol beverages from food consumption expenditure.
9 Chain-type price index of PCE for food and tobacco are published separately by BEA.

In constructing
the composite price index for food and tobacco, we simply computed the weighed average of two price index,
where weight taken according to the food nominal expenditure and tobacco nominal expenditure. Details of
this data manipulation can be found in Data Appendix of this paper.

9

Table 1: Categorizing Non-Durable Goods
Non DurableGoods
Necessity Goods (Nt )Luxury Goods (Lt )Food ex. alcoholic beverages

Clothing and shoes

TobaccoAlcoholic beverages

News and magazine
Entertainment
Other goods
Note: To b e accurate energy goods stands for “Gasoline, fuel oil and other energy goo ds.”.

De…nition of the typ eof consum ption goods in this table follows NIPA’s Table 2.6. “PCE by Typ e of Product”.

manipulation, consumption of food and tobacco is now captured as the real quantity index
rather than real expenditure. This manipulation is crucial because what we try to capturein the model is quantity of consumed goods rather than expenditure. The real per adultconsumption for non-durable goods excluding food and tobacco (i.e. Lt ) was constructedin a similar fashion. Finally, real interest rate (i.e. r) was constructed based on quarterly
average of 3-month U.S. Treasury Bill Rate subtracting in‡ation rate, where the in‡ation
rate was calculated from quarterly average of Consumer Price Index10 .
As for the preliminary step for the cointegration analysis, we tested the null of di¤erence
stationarity against the null of (trend) stationarity for the variables included in the cointegrating regressions. To be speci…c, we tested the di¤erence stationarity of following four
variables: log of necessity goods (i.e. ln Nt ), log of luxury goods (i.e. ln Lt ), log opportunityL
cost of current necessity goods in terms of future luxury goods (i.e. ln (1 + rt+1 )PtN =Pt+1)
and log opportunity cost of future necessity goods in terms of current luxury goods (i.e.
Nln (1 + rt+1 ) 1 Pt+1
=PtL ). The results of the unit root tests are reported on Table 2.
We used Said and Dickey’s (1984) Augmented Dickey-Fuller (ADF) test, Phillips andPerron’s (1988) PP test and Park and Choi’s (1988) J test in testing for the null of di¤erence
stationarity. As can be seen from Table 2, the tests do not reject the null of di¤erence
stationarity against the null of stationarity at 10% signi…cance level for all variables. Further,
the tests do not reject the null of di¤erence stationarity against the null of trend stationarity
at 10% signi…cance level for both log necessity and luxury goods. Thus, log of necessity
and luxury goods may well be thought of as stochastic processes containing unit root withpossible drift. However, for the log opportunity costs, the test results were rather mixed. It
is not clear whether the log opportunity cost follows a di¤erence stationary process or trend
stationary process from this result. Keeping in mind the possibility of trend stationarity
in log opportunity cost, we now proceed to the cointegration analysis of the log-linearizedcross-Euler equations..
1 0 Thus, real interest rate used in this paper is actually an ex-post real interest rate.

10

Table 2: Unit Root Test
Variable

ADF test
cst.

ln Nt
ln Lt
L
ln (1 + rt+1 )PtN =Pt+1
N
ln (1 + rt+1 ) 1 Pt+1
=PtL

PP testJ-testcst. & trd.

cst.

cst. & trd.

J(0,3)

J(1,5)-1.616-1.822-1.5250.378-1.3060.765-1.707

1.240

1.804

-0.63767.771

3.063

0.197

-3.0470.084-5.284**25.8650.670-0.722-2.999-0.661-3.0768.864

0.406*

Note: Lag order used for ADF test and PP test was four. The 10% critical values of ADF test and PP test with aconstant is -2.576 and with constant and trend is -3.143. The 5% critical values are -2.879 and -3.438, resp ectively.
The critical values are due to M acKinnon (1991).

For J(0,3) test and J(1,5) test, 10% critical values are 0.577

and 0.452 and 5% critical values are 0.338 and 0.295, resp ectively..
(1988).

The critical values are due to Park and Choi

It should b e noted under the J-test, the null of di¤erence stationarity is rejected when the statistics are

sm aller than the critical value.

* denotes the rejection of null hyp othesis at 10% level.

** denotes the rejectionof null hyp othesis at 5% level.

4.2

Estimation

4.2.1

Canonical Cointegration Regression

In this section we will explain the step in applying Park’s (1992) Canonical Cointegration
Regression (CCR) on log-linearized Cross-Euler eq. (14) and (15). The model implies thefollowing cointegrating restriction for each equation:
L
Cross-Euler eq. (11): ln Ct+11

ln (1 + rt+1 )

PtN
L
Pt+1

ln CtN

I(0)

(16)

N
Pt+1
1
N
ln Ct+1I(0)(17)
1 + rt+1 PtL
Following the result of unit root pretesting in the previous subsection, log necessity and luxury
goods will be assumed to be a di¤erence stationary process. However, for the log opportunity
costs, the pretest results gave a mixed signal of di¤erence stationary process and trend
stationary process. Therefore, two cases must be considered in conducting the cointegration
analysis: the case when deterministic trend is absent in the cointegrating system (i.e. log
opportunity cost follows di¤erence stationary process) and the case when deterministic trend
is present in the cointegrating system (i.e. log opportunity cost follows the trend stationary
process).

Cross-Euler eq. (12): ln CtL

1

ln

Case 2 (Deterministic trend is not present inside the cointegrating system)
Let yt be a scalar of di¤ erence stationary process and let xt be the k 1 vector of di¤ erence
stationary process whose components are not stochastically cointegrated. If yt and xt satis…es
the deterministic cointegration restriction, then the cointegrated system can be expressed as
yt =
where

c+

0
x xt + "tc is a scalar and "t is a stationary process with mean zero.

11

(18)

In our model, yt can be thought of as the log luxury goods and xt can be thought ofvector containing log opportunity cost and log necessity goods. Under the case that deterministic trend is absent in the log opportunity cost, the model implies the deterministic
cointegration among the variables ln C L ; ln (1 + r) P N =P L and ln C N with cointegrating vector (1; 1= ; = )0 . This sets the ground for applying the CCR in the above regression form.
Some remarks are in order regarding to the CCR estimator. As is the case for any
cointegrating regression, CCR will yield a super-consistent estimate of the parameters. In
addition, by the non-parametric correction for the long-run variance of ( xt ; "t )0 , CCR is
known to be asymptotically e¢ cient and does not require the strictly exogeneity assumption in ( xt ; "t )0 . This latter property is crucial for our purpose since the regressor xt isconstructed by the leads lags of ln P N ; ln P L ; and ln C N and "t also consists of leads andlags of similar variables. For instance, applying OLS estimator to above regression form
following Engle and Granger’s (1987) method, which assumes the strict exogeneity, will yield
asymptotically biased, though consistent estimates of 1= and = (see Phillips and Durlauf(1986) and Stock (1987)). Thus, applying the CCR in the above regression form will yielda super-consistent estimate of the intertemporal substitution parameters.
By applying Park’s (1990) G(p,q) test on the residuals, we can obtain the Park’s H(p,q)
statistics. Under the null of cointegration, Park showed that H(p,q) statistics is asymptotically 2 distributed with q p degrees of freedom. Since we are interested in both deterministic and stochastic cointegration relationship, we conducted H(0,q) and H(1,q) tests inthis paper. The results of Park’s CCR estimates11 are reported in Table 4. for cointegrating
equation 14 and in Table 3 for cointegrating equation 15. In order to check the deterministic
cointegration relationship, we have applied H(0, q) test for both equation. Also, to checkfor the stochastic cointegration relationship, H(1,q) test was also conducted.
Let us …rst turn to the estimation result of equation 14. Parameter estimate for
was 0.9924 and was 0.7663. Thus, IES for necessity goods (i.e. 1= ) was 1.0076 andIES for luxury goods (i.e. 1= ) was 1.3049. The test generally rejected the implicationof the deterministic cointegration relationship, but was not able to reject the stochastic
cointegration relationship.
Next, turning to the estimation result for equation 15, estimate for
was 0.9492 andfor was 0.7504. Therefore the implied IES for necessity goods was 1.0535 and for luxury
goods was 1.3324. We found that estimates of to be reasonably close between eq. 14 and15. For the deterministic cointegration relationship, H(0,1) and H(0,3) test rejected the null
hypothesis. For the stochastic cointegration relationship, only H(1,3) test rejected the null
hypothesis of stochastic cointegration.
1 1 In estimating the long-run covariance matrix of error term, we used Andrews and Monahan’s (1992) VAR

prewhitened HAC estimator. The choice of kernel was QS kernel as suggested by Andrews (1991). Following
the Monte Carlo study of Han (1996), third stage CCR estimates are reported and H(p,q) statistics are basedon fourth stage CCR.

12

Table 3: CCR ResultsL
ln Lt+1 = const: + 1 ln (1 + rt+1 )PtN =Pt+1+

Estimates
const.
1==-2.3451.304
1.295(1.220) (0.042) (0.147)
Test Statistics
H(0,1) H(0,2) H(0,3)
H(1,2)
5.479* 6.073*
7.355
0.593
[0.019] [0.047] [0.061][0.441]ln Nt + I(0)Implied Estimates
0.992

0.766

H(1,3)
1.875[0.391]

H(1,4)
2.168[0.538]

Note: Numb ers in parenthesis stand for the estim ated standard error.

Numb ers in square brackets stand for p-value. * denotes the rejection of null of cointegration at 5% level. ** denotes the rejection of null of cointegration
at 1% level.

Table 4: CCR Resultsln Lt = const: +1Nln (1 + rt+1 ) 1 Pt+1
=PtL +

Estimates
const.
1==-2.096
1.332
1.264(1.031) (0.036) (0.124)
Test Statistics
H(0,1) H(0,2) H(0,3)
H(1,2)
2.780
2.966
9.395*
0.185[0.095] [0.226] [0.024][0.666]ln Nt+1 + I(0)Implied Estimates
0.949

0.750

H(1,3)
6.614[0.036]

H(1,4)
7.470[0.058]

Note: Numb ers in parenthesis stand for the estim ated standard error.

Numb ers in square brackets stand for p-value. * denotes the rejection of null of cointegration at 5% level. ** denotes the rejection of null of cointegration
at 1% level.

4.2.2

GMM

In this section, we will conduct Hansen’s (1982) GMM on eq. 8 and 9. Parameters andwill be estimated under single equation and system equation context. We will also discuss
the choice of instrumental variables (IV) in this paper. Hansen’s J test will also be reported.
As it was pointed out by Hall (1993) and Ogaki (1993), it is well known that the estimateof GMM is very sensitive to the choice of instrumental variables. To test for the robustness ofthe estimates against the choice of instruments, we estimated the parameters under several
types of instruments with varying time lags. First family of the instrumental variables
was chosen following the convention in applied GMM literature. As can be seen from thefollowing table, six types of instrument sets were chosen.

13IV Type
IV1
IV2
IV3
IV4
IV5
IV6

Euler Equation 8

Euler Equation 9

CNconst., Ct+1
N
t
N
Pt+1
const., P N
t

L
Ct+1
CtL
PL
const., Pt+1
Lt

const.,

const., rt+1

const., rt+1

CN
const., Ct+1
N , rt+1
t
N
Ct+1
PN
const., C N , Pt+1
N
t
t
N
N
Ct+1
Pt+1
const., C N , P N , rt+1

L
Ct+1
, rt+1
CtL
L
Ct+1
PL
const., C L , Pt+1
L
t
t
L
L
Ct+1
Pt+1
const., C L , P L , rt+1tt

const.,tt

The next issue in conducting GMM estimation is to choose the lag order of the error termwhen estimating the variance-covariance matrix of GMM disturbance terms. According
to the rational expectation hypothesis, it is known that the forecast error will be serially
uncorrelated. Since our model is based on the representative agent with rational expectation,
the economic theory suggests the lag order of zero. Nevertheless, taking into account for thetime aggregation problem which was pointed out by Heaton (1995), we choose the lag orderof one in estimating the variance-covariance matrix of GMM disturbance terms12 . Also, tobe consistent with the time aggregation issues, we have lagged the instrumental variables fortwo periods when conducting GMM estimations.
4.2.3

Result

GMM estimation was conducted using family of conventional instruments. The GMM estimation results for Euler equation 8 is summarized under Table 5. Similarly, the GMM
estimation result for Euler equation 9 is summarized under Table 6. Hansen’s J-statistics
for each regression are also reported.
Let us …rst interpret the estimation result of Euler equation for necessity goods consumption. We …rst observe the large variance in the estimates of . The estimates for ranges
from -11.917 to 15.136. This wide dispersion can also be con…rmed from the estimatedstandard error for the estimator ^ . We can think of two possibilities that have contributed
to these odd estimation results. First possibility is the weak instruments problem, i.e. if
the instruments and the forcing variables in the regression are weakly correlated, the variance of the estimator will be large. It might be the case that in our GMM estimation, theconventional instruments were weakly correlated to the forcing variables.
Second possibility comes in when Euler equation is misspeci…ed. The easiest way tocheck for the misspeci…cation is to look at Hansen’s J statistics. However, to our surprise,
Hansen’s J test does not reject the null hypothesis that Euler equation 8 is speci…ed for allcases. Does this mean that Euler equation (8) is correctly speci…ed? Statistically speaking,we cannot deny this possibility. But then the odd estimates ofin Table 5. does notconform with the result of Hansen’s J test. Or it might be the case that the low power of
1 2 Since the lag order was explicity chosen, we will use HAC estimator with truncated kernel when estimating

the variance-covariance matrix of the GMM disturbance terms.

14

Table 5: GMM Results for Necessity GoodsEt
IV Type
IV0
IV1
IV2
IV3
IV4
IV5
IV6

0.991
(0.007)0.991(0.009)0.971(0.035)0.991(0.009)0.984(0.005)0.988(0.008)0.988(0.007)

Nt+1
Nt

PN

(1 + rt+1 ) P Nt

1 =0

t+1

9.745
(16.023)
15.136(7.346)-11.917(22.504)
15.485(7.348)-0.791(3.957)12.740(5.934)11.779(4.606)

J-statistics--

D.F.
Just Identi…ed--

Just Identi…ed--

Just Identi…ed

0.066[0.797]1.911[0.588]1.570[0.456]2.031[0.730]1
124Note: All instrum ents are lagged for two p erio ds. Numb ers in parenthesis represent the estim ated standard errors.
Numb ers in brackets represent the p-values.

* denotes the rejection of null at 5% level.

** denotes the rejectionof null at 1% level.

Hansen’s J test resulted in the under-rejection of the null. As such, we propose to use thelikelihood ratio type test proposed by Cooley and Ogaki (1996), which will be the topic of
the next section.
To the sharp contrast to the estimation result of Euler equation for necessity goods, theestimation results of Euler equation for luxury goods have an intuitive result. As can beseen from Table 6, we can observe the relative tightness in the estimates of . The estimatesof range from 0.368 to 3.778, with an exception of 16.780 under IV1. This observationis consistent with the conspicuously small estimated standard error of ^ compared to thatof ^ . However, turning to Hansen’s J test, the test rejected the speci…cation of the Euler
equation 9 for 3 out of 4 cases, which is counter-intuitive given the stable estimates of ^ . Or
it may well be the case that the rejection came from the size distortion of Hansen’s J test.
As such, we will rely on Cooley and Ogaki’s LR type test in testing the speci…cation of theEuler equation 9.

4.3

Test of Liquidity Constraint

In this section, we will discuss why Cooley and Ogaki’s (1996) test best suits for our purpose
and also report the result of the test. Before we discuss Cooley and Ogaki’s LR type test, it

15

Table 6: GMM Results for Luxury GoodsEt
IV Type
IV0
IV1
IV2
IV3
IV4
IV5
IV6

0.985
(0.007)1.062(0.065)1.002(0.009)1.003(0.006)0.991(0.006)0.991(0.006)0.991(0.006)

Lt+1
Lt

PL

1 =0

(1 + rt+1 ) P Lt

t+1

0.368
(1.114)16.780(15.111)
3.778(1.864)2.800(1.037)1.355(0.960)1.003(0.969)0.913(0.966)

J-statistics--

D.F.
Just Identi…ed--

Just Identi…ed--

Just Identi…ed

13.357[0.000]3.088[0.588]17.616[0.000]18.430[0.001]1
124

Note: All instrum ents are lagged for two p erio ds. Numb ers in parenthesis represent the estim ated standard errors.
Numb ers in brackets represent the p-values.

* denotes the rejection of null at 5% level.

** denotes the rejectionof null at 1% level.

may be useful to review the standard LR type test in the GMM literatures. For simplicity,
we impose some linear restriction on the GMM estimator. In the most general linear form,
the null hypothesis can be expressed as follow.
Ho : R^GM M = q
where q is q 1 vector of constant and R is some q k matrix. Then the LR type statistics,
denoted as QLR, is de…ned as follow and can be shown that it will be asymptotically 2
distributed with q degrees of freedom.
QLR = T Jrestricted

T Junrestricted

d

!

2

(q)

where T stands for the number of observations and J stands for the minimized objectivefunction under GMM. Now, it should be noted that under the standard LR type test, q was
simply a vector of constants.
The punch line of Cooley and Ogaki’s LR type test is that they replaced q with theestimator of cointegrating vector q^coint . By exploiting the super-consistency of q
^coint , theyshow that QLR will again be asymptotically 2 distributed with q degrees of freedom13 .
1 3 If, instead, the estimator q^ were only consistent (i.e.

16

O(T

1=2 ) consistent), then one have to calculate

Restating mathematically,dHo : R^GM M = q^coint and QLR !

2

(q).

Since our model involves the cointegration analysis and GMM in estimating the parametersand , Cooley and Ogaki’s LR type test seems to be the best candidate for our speci…cation
test.
We basically tested two types of null hypothesis. First null hypothesis is H01 : ^ GM M =^ coint and results are reported under Table 7. Second null hypothesis is H02 : ^ GM M = ^ cointand results are reported in Table 8. Note again, if indeed eq.7, eq. 8 and eq. 9 are all well
speci…ed, then the test is likely to accept all of the above null hypotheses. We will interpretthe results under three di¤erent nulls one by one.
Table 7: LR-type Test Results: Necessity Goods
H0 : ^ CCR = ^ GM M
IV Type
QLR statistics
IV0
1.660
IV1
15.396**IV2
4.214*
IV3
16.101**
IV4
2.267
IV5
14.590**IV6
14.502**

P-value[0.197][0.000][0.040][0.000][0.132][0.000][0.000]Note: * denotes the rejection of null at 5% level. ** denotes the rejection of null at 1% level.

First let us turn to the results under the null of H01 : ^ GM M = ^ coint . As can be seenfrom Table 7, QLR statistics exceeds the critical value for most of the cases, which implies
the rejection of the null hypothesis. Indeed, the test rejects 5 out of 7 cases. This evidence
suggests that eq. 7 and/or eq. 8 are misspeci…ed.
Next we will turn to the results under the null of H02 : GM M = coint . To the sharp
contrast to the former test, as one can see from Table 8, the test is not able to reject thenull hypothesis, except for the case when we use the instrumental variable set, IV1. Except
for this case, the QLR statistics are below the critical values. According to this result, thetest seems to support the hypothesis that both eq. 7 and eq. 9 are speci…ed.
the covariance of ^GM M and q^ in order to conduct the statistical inference. For details, see Ogaki (1993).

17

Table 8: LR-type Test Results: Luxury Goods

IV Type
IV0
IV1
IV2
IV3
IV4
IV5
IV6

H0 : ^ CCR = ^ GM M
QLR statistics
0.130
15.302**
2.666
2.460
0.193
0.239
0.592

P-value[0.717][0.000][0.102][0.116][0.660][0.624][0.441]

Note: * denotes the rejection of null at 5% level. ** denotes the rejection of null at 1% level.

5

Empirical Evidence from Consumer Expenditure Survey

5.1

Motivation for Cohort Analysis

In the previous section, we have assumed an existence of representative agent and usedaggregate data in estimating the preference parameters. However, when estimating thepreference parameters consistently from the aggregate data, extremely stringent requirements
have to be met. For instance, in order for the representative agent to exist, all the agentsin the economy need to share the identical preference and also their utility function haveto be homothetic (See Kirman (1992) and Stoker (1993) for more details). In reality, it isvery unlikely that these conditions are met. Turning to micro data, several rich data set,such as Panel Study for Income Dynamics (PSID), Consumer Expenditure Survey (CES) andFamily Expenditure Survey (FES), are available. However, these micro data set, despite its
richness, contain some serious defects when testing for life-cycle model. For instance, PSID
only tracks the consumption expenditure of individuals, which is not too useful especiallywhen we are interested in the consumption pattern of non-durable goods and services as awhole. On the other hand, CES and FES, albeit its richness in the categories of consumption
goods, are cross-sectional survey data that it is does not allow researchers to track same
individuals over time. In reconciling this dilemma, Deaton (1985) proposed a method intracking ’cohorts’from time series of cross-sectional survey. Cohort is a subset of households
sharing the identical qualities that are unchanged over time. These qualities, for instance,
can be year of birth, race, gender and so on. By classifying the cross-sectional observations
into these cohorts and tracking them over time, Deaton (1985) showed that it is possible
to construct synthetic panel (or pseudo panel) data that allows researchers to estimate the18cohort-speci…c parameters consistently14 . Moreover, since the households are aggregatedinto cohorts who share the similar taste, it is relatively free from the aggregation problem
compared to representative agent model. Inspired by the virtues, not surprisingly, the cohort
technique has been widely used in the empirical life-cycle literatures. The predecessor in theline of empirical cohort analysis have constructed the cohorts in several ways. Browning,
Deaton, and Irish (1985), Mo¢ t (1993), Attanasio and Webber (1993, 1995), have classi…ed
the households according to their ages. Attanasio et al. (1999) have classi…ed the households
according to their ages and education attainments.
In this paper, we adopt this cohort technique in estimating the preference parameters andtesting for the liquidity constraints. In constructing the cohort data, careful attention must
be exerted. As pointed out by Verbeek (1996), legitimate cohorts are constructed only iftaste and demographic observations of households in each cohort are drawn from the identicalprobability distribution. Thus, the observational di¤erence among each observation within
the same cohort must be idiosyncratic and should be ”averaged out” after taking average.
To comply strictly with this requirement, households should better be classi…ed into cohorts
distinguished by the …nest information available from the data, so that non-idiosyncratice¤ects are isolated. However, due to the limitation of sample observations from the CES approximately 5,000 household observations each quarter -, there exist a trade-o¤ between
the construction of …ner cohorts and sample observations per cohort. Some judgment must
be exerted to strike the ”optimal” balance between two requirements
In constructing the cohorts, we basically follow Attanasio et al. (1999) - that is to classify
the households by ages and educational attainments. Speci…cally, I classi…ed agents into 10
cohorts by their age and education level. This classi…cation can be justi…ed as follows:
1) Age classi…cation is necessary because the agents reveal the heterogeneous consumption
patterns (i.e. life-cycle in consumption) depending on their age. 2) Classi…cation by educationlevel is adopted as a mean to classify the agents by their life-time income (i.e. permanent
income). By the construction of LCPIM, an agent will face a liquidity constraint when his
net wealth decreases to some certain level. Naturally, an agent with low permanent income(i.e. poor agent) is more likely to face this bound over his course of life than an agent withhigh permanent income (i.e. rich agent). Thus, in order to be perfectly consistent withthe theory implication, it is ideal to classify agents by their permanent income. However,since the permanent income of each agent is not directly observable from the data, I havecompromised to proxy the permanent income by the education level. The cohort data foreach type of agent have been constructed from the series of Consumer Expenditure Survey
from 1984 to 1998.
Finally, whenever making a statistical inference in this section, we assume that number ofcross-sectional cohorts is …xed in pseudo panel (i.e. H is …xed), assume that household observations within the cohort is large (i.e. Nc ! 1) and assume that time series observations arelarge (i.e. T ! 1). Under these assumption, consistency of the pooled estimates in pseudo
1 4 For the survey regarding the cohort technique, see Verbeek (1996). For recent development, see Verbeek
and Nijman (1993), Mo¢ t (1993), and Collado (1997).

19

panel are known to yield consistent estimates (See Verbeek (1996) for more discussion).

5.2

Data Description

5.2.1

Construction of Cohorts

As a …rst step, the households are classi…ed into 5 categories by their ages at 1984. Ageintervals for each categories are 25-30, 31-36, 37-42, 43-48, and 49-54, respectively. Here,
we are taking intervals of 6 years for each categories, which is slightly di¤erent from theconvention previously used by Browning et al. (1985) and others. This practice has been
adopted in consideration to the limitation on sample observations, but we believe that theheterogeneity biases relative to conventional intervals are minor.15 . As a second and …nal
step, we further classify the households into two categories according to their educational
attainments. Households who dropped out from a high school or whose highest educational attainment is high school has been classi…ed to the category called ”High School.” A
Household who at least holds a college degree or higher has been classi…ed to the category
called ”College.” It should be noted that we have intentionally omitted households who have
dropped out from a college. This practice has been adopted in order to preserve the sharp
contrast between the ”High School” and ”College” categories. Thus, as a consequence of
this two-step classi…cation, we have constructed 10 cohorts. The following tables summarize
the average observations per quarter for each cohort.
Table 9: Average Cohort Size per Quarter: 1984Q1 to 1999Q1

HS1
322.43High School*
HS2
HS3
HS4
273.21
255.77234.61HS5
231.34Col1
242.02

College**Col3
196.43

Col5
95.82

Note:

Col2
253.41

Col4
120.41*) Highest educational attainm ent of a household head is less than or equivalent to high scho ol diplom a.

W ithin this educational cohort, households were further classi…ed by their ages. HS1 was b orn b etween ’54 to ’59,
HS2 b etween ’48 to ’53, HS3 b etween ’42 to ’47, HS4 b etween ’36 to ’41 and HS5 b etween ’30 to ’35.
educational attainm ent of a household head is m ore than or equivalent to college degree.

**) Highest

Col1 was b orn b etween’54 to ’59, Col2 b etween ’48 to ’53, Col3 b etween ’42 to ’47, Col4 b etween ’36-to ’41 and Col5 b etween ’30 to ’35.

The top panel of the Table 9 reports the average cohort size for High School cohorts.
HS1 stands for the High School cohort who was 25-30 years old, HS2 stands for the cohorts
who was 31-36 years old, HS3 for the cohorts who was 37-42 years old, HS4 for the cohorts
who was 43-48 years old, and HS5 for the cohorts who was 49-54 years old at 1984. The
1 5 We have also tried 5 years intervals.

However, the average observations for some cohorts, especiallyfor older cohorts, were smaller than 100. In order to ensure that each cohort size is over or around 100
observations, we decided to classify the households by 6 years of interval.

20bottom panel of Table 9 reports the average cohort size for College cohorts. Col1 stands
for the College cohorts who was 25-30 years old, Col2 for the cohorts who was 31-36 years
old, and so on. As can be seen, the cohort size for most of the cohorts were more than orapproximately 200, except for Col4 and Col5. This apparently small cohort size for Col4
and Col5 is probably due to the low college enrollment for older generations16 .
For each cohort, information regarding consumption and demographics have been extracted and were used in constructing the average cohort data17 . Following Section 3,
luxury goods were composed of alcohol beverages, apparel, transportation excluding vehicle,
entertainment, reading and other goods. Necessity goods were composed of food and tobacco. In calculating the real expenditure, we have used the common price index used in
Section 3. In other words, all cohorts are assumed to face the common opportunity cost
intratemporally and intertemporally18 . In addition to expenditure information, we have also
calculated an average number of adults and children per cohort. The descriptive statistics
regarding the consumption growth is reported in Table 10.
5.2.2

Age Pro…le and Preliminary Testing

In order to visualize the consumption pattern of the cohorts, so called ”Age Pro…le”regarding
luxury goods and necessity goods expenditure have been laid out in Figure 1.
Age pro…le plots the life-cycle consumption pattern of same education of cohorts withdi¤erent age groups on the same diagram. The left-top panel plots the log real luxury goods
expenditure and the righ-top panel plots the log real necessity goods expenditure of highschool cohorts. As can be seen from these …gures, consumption pattern for both goods
reveals a hump over the life-cycle. For the luxury goods, consumption expenditure peaks
out in the mid 40’s and reveals a sharp decline towards the end of the life-cycle. In a similar
fashion, necessity goods expenditure peaks out during late 30’s to early 40’s, although themagnitude of a hump seems to be milder than that of luxury goods. The left-bottom panel
shows the age pro…le of the log real luxury goods expenditure and the right-bottom panel
shows the age pro…le of the log real necessity goods expenditure of college cohorts. Again,
hump shape in the consumption pattern can be clearly observed. For the college cohorts,
luxury goods and necessity goods expenditure seem to peak out in the late 40’s. Interestingly,
1 6 Col4 is the genration who was born between 1936-1941 and Col5 is the generation who was born between
1930-1935. Considering the e¤ect of Great Depression and WWII, probably only the wealthy families wereable to send their children to colleges. Small cohort size for Col4 and Col5 will apparently introduce alarge standard error when constructing a cohort’s average data. As explained in Deaton (1985) and Verbeek(1996), by rendering number of cross-sectional cohorts to in…nity (i.e. H ! 1) with …xed cohort size (i.e. Nc
is …xed), it is possible to obtain the consistent estimator in the presence of standard error in average cohort
data. But then, due to the limitations in total household observations, one face a same dillemma in claiming
a large sample in terms of H. Thus, obviously there is a trade-o¤ in choosing H ! 1 or Nc ! 1. In thispaper, we assume N c ! 1.
1 7 All the average cohort data have been seasoanlly adjusted using seasonal dummies.
1 8 Though this assumption o¤ers us a simplicity in calculating the real expenditure for each cohort, thisassumption may be too strong. Constructing Stone Price Index for each cohort is often favored solution.
See Attanasio and Webber (1995) for instance.

21

Table 10: Descriptive Statistics of Real Expenditure Growth: 1984Q1 to 1999Q1

Mean
Std. Dev.
Mean
Std. Dev.

Mean
Std. Dev.
Mean
Std. Dev.

High School
HS1
HS2
HS3
HS4
HS5
Luxury Goods Expenditure Growth (% per Quarter)
0.367
0.106
-0.289-0.739-0.755
6.298
6.391
6.103
7.393
6.826
Necessity Goods Expenditure Growth (% per Quarter)
0.532-0.004-0.311-0.370-0.397
4.030
3.565
4.373
4.703
3.728
College
Col1
Col2
Col3
Col4
Col5
Luxury Goods Expenditure Growth (% per Quarter)
0.519
0.119
-0.232-0.553-0.802
5.898
5.609
7.178
8.819
11.53
Necessity Goods Expenditure Growth (% per Quarter)
0.925
0.461
-0.155-0.434-0.569
4.655
5.007
6.152
6.366
7.678

Note: Real exp enditures are seasonally adjusted using seasonal dum m ies. Exp enditure growth rates not adjusted
for adult equivalent scale.

necessity goods expenditure reveals a sharp increase in the beginning of the life-cycle andstays persistent toward the end of life-cycle, while increase and decrease in luxury goods
seems to be fairly symmetric.
The main characteristic to be noted here is the clear violation of consumption smoothing,which is one of the main implications of life-cycle model. Is this hump shape in the consumption pattern solely induced by liquidity constraint? Not likely. Thus, before we canlegitimately test for the existence of liquidity constraints, the …rst task is to account for thisobvious hump shape over the life-cycle. As pointed out by Attanasio and Webber (1995 and1997) and Attanasio et al. (1999), one of the main cause for this hump-shaped consumption pattern is due to demographic factors. As presented in their papers, the number ofadults and children in the household shows similar humped-shape over the life-cycle. Mostly
likely, as they argue, the hump-shape in consumption pattern can be attributed to thesedemographic factors. As for a quick and ad-hoc adjustment, we have adjusted consumption
expenditure by the number of adult equivalent members19 in the household. The age pro…les
1 9 The number of adult equivalent members in the household has been calculated according to the following

formula.
# of adult equivalent = 1 + 0:75

# of adults + 0:4# of childrenChild is de…ned as a household member with age below 18 years old and adult is de…ned as a household
member with age above 18 years old. The forumula is a composition of the scale used by Attanasio (1995)
and Blundell, Browning and Meghir (1994).

22

Figure 1: Age Pro…le of Luxury and Necessity Goods (Demographic Unadjusted)

Source: Consumer Expenditure Survey 1984-1998

of consumption expenditure per adult equivalent have been laid out in Figure 2.
As can be seen from Figure 2, the adjustment by adult equivalent scale have considerably
dissipated a hump shape in the age pro…le. Now, the age pro…les for both education cohorts
reveal a fair magnitude of consumption smoothing over the life-cycle. For the necessity goods,
both high school and college cohorts tend to increase their expenditure over the life-cycle.
Interestingly, for the luxury goods, both education cohorts tend to decrease their spending
toward the end of the life-cycle. Of course, the adjustment of consumption expenditure by
adult equivalent scale is rather ad-hoc. It is not our intention to claim this ad-hoc adjustment
will solve the hump shape puzzle in consumption pattern, but rather wanted to illustratethat demographic adjustment is important whenever scrutinizing the consumption behavior,especially on the cohort level. In the later section, we will adopt a more sophisticated23Figure 2: Age Pro…le of Luxury and Necessity Goods (Demographic Adjusted)

Source: Consumer Expenditure Survey 1984-1998

method to account for hump shape in consumption by allowing demographic taste shift in
the household’s utility function. The bottom line is that it is only possible to test for the
existence of liquidity constraint after taking into account for the demographic factors.
Finally, as for a preliminary step for cointegration analysis in the following section, we
have conducted a unit root test for consumption expenditure of each cohort. Table 11
reports the results from ADF test20 . For the case of HS3, HS5 and Col5, the test results
have indicated that log luxury goods expenditures are trend stationary. For the case of HS2,
the test gave a mixed results rejecting the unit root without trend, while not rejecting theunit root with trend. For the log necessity goods expenditure, the null of unit root were
2 0 We have also conducted PP test and J(p,q) test, but the results were similar and will not be reported
here.

24

Table 11: Unit Root Test of Luxury and Necessity Goods Expenditure

ln Lt
ln Nt
AdulttChildtln Lt
ln Nt
AdulttChildtcst.
cst. & trd.
cst.
cst. & trd.
cst.
cst. & trd.
cst.
cst. & trd.

cst.
cst. & trd.
cst.
cst. & trd.
cst.
cst. & trd.
cst.
cst. & trd.

HS1-2.690-3.184-2.215-0.978-7.151**-7.086**-6.232**-7.184**High School
HS2-3.084*-2.983-1.793-0.951-6.378**-6.319**-7.306**-7.527**HS3-1.421-3.561*-0.861-2.698-6.974**-7.519**-7.471**-7.424**HS4
0.006-2.717-0.455-2.097-7.167**-7.185**-5.851**-6.245**HS5-1.551-3.613*-0.696-2.346-5.657**-5.655**-6.750**-7.056**Col1-2.074-2.347-1.881-1.337-8.450**-8.375**-5.540**-5.971**

CollegeCol2-2.399-1.913-2.092-1.597-5.779**-5.726**-4.570**-5.063**Col3-1.894-2.236-1.580-1.871-6.460**-6.688**-6.867**-6.824**Col4-0.574-2.3690.012-1.273-6.138**-6.093**-6.640**-6.747**Col5-1.150-4.528**-0.639-2.824-6.701**-6.649**-6.676**-6.694**

Note: Results are based on ADF test. Lag order was …xed at 2 for all of the cases. * denotes rejection of the nullof unit root non-stationarity at 5% level. ** denotes the rejection of the null at 1% level.

not rejected for any cases. Overall, the null of unit root non-stationarity for consumption
expenditures were accepted, setting the ground for the cointegration analysis in the next
subsection. In addition to consumption expenditures, we have also tested the stationarityof change in demographic factors (i.e. number of adults and children in household) over thequarter. As we will see later, the stationarity in change of demographic factors are necessary
when conducting Panel cointegration analysis and Panel GMM estimation. As can be clearlyseen from Table 11, the null of unit root non-stationarity has been rejected for any cases,
supporting the stationarity of change in demographic factors. These preliminary results set
the ground for estimation and testing in the following section.

5.3

Estimation and Test

5.3.1

Parametrization and Cross-Euler, Euler speci…cation

In dealing with the cohort data, we slightly modify the shape of utility function to accountfor the socioeconomic factors - number of adults in the household and number of children in25

the household21 . Speci…cally, each cohort’s period-by-period utility is speci…ed as follows:"#hh(Nth )1(Lht )10 hhh
U (Lt ; Nt ) = exp( h Zt )+K
for h 2 fHS1; :::; HS5; Col1; :::; Col5ghh11
where h is a vector of coe¢ cients and Zth represents the socioeconomic factors. To put it
another way, we have modi…ed the Houthakker’s addi-log utility function allowing for “taste
shifter.” When the liquidity constraints not binding, the Euler equation for luxury and necessity goods are:
2

Lht+1
Lht

2

h
Nt+1
Nth

Et 4 hEt 4 hhexp( 0hhexp( 0h

3

PL
h
Zt+1
)(1 + rt+1 ) Lt 5
Pt+1
3

PtN 5
h
Zt+1
)(1 + rt+1 ) N
Pt+1and Cross-Euler equations can be derived as:"#h(Lht+1 )
PtN
0
h
Et K
exp( h Zt+1 )(1 + rt+1 ) L
Pt+1(Nth ) h
"
#
h
h
(Nt+1
)
PtL
0
h
exp( h Zt+1 )(1 + rt+1 ) N
Et
K (Lht ) h
Pt+1

=

1

(19)

=

1

(20)

=

1

(21)

=

1:

(22)

Following the same procedures as in Section 3, by log-linearizing the Cross-Euler equations,
we obtain the following cointegrating restrictions.
ln Khln Lht+1 +hln Nth +0hZt+1 + ln (1 + rt+1 )

PtN
L
Pt+1

I(0)

(23)

ln

h

h
ln Nt+1+h0h

Zt+1 + ln (1 + rt+1 )

PtL
N
Pt+1

I(0):

(24)

K

ln Lht +But since constant terms and Zt+1 are deemed stationary based on the unit root test, theabove cointegration restrictions can be further transformed.
ln (1 + rt+1 )
ln

PtN
L
Pt+1

N
Pt+1
1
1 + rt+1 PtLhln Lht+1 +hln Lht +hhln Nth

I(0)

(25a)

h
ln Nt+1

I(0):

(25b)

In principle, the estimation of the preference parameters (i.e. h ; h and 0h ) can bedone allowing for the heterogeneity among all cohorts. In that case, the estimation can be
2 1 As Attanasio and Webber (????) claims, assuming that marginal utility of consumption is not indepen-

dent from leisure, it is appropriate to account for husband and spouse job status in the socioeconomics factor.
However, since we have assumed that leisure is separable from consumption, we have not included them.

26

implemented by single-by-single equation manner or by SUR method. However, in thispaper, we are going to restrict the preference parameters to be the same among each education cohort. In other words, we restrict the preference parameters of High School cohort
as ( HS ; HS ; HS )0 = ( HS1 ; HS1 ; HS1 )0 = ::: = ( HS5 ; HS5 ; HS5 )0 and College cohort
as ( Col ; Col ; Col )0 = ( Col1 ; Col1 ; Col1 )0 = ::: = ( Col5 ; Col5 ; Col5 )0 . This restriction
can be motivated at least from two reasons. First, considering that time series observation
per cohort is mere 61 observations, estimating preference parameters in a single equation
context or even in a SUR context will be vulnerable to small sample distortion. By restricting the preference parameters to be equal among the education cohort, it will enable
us to estimate the parameters in the (pseudo) panel setting o¤ering a better small sample
property. Second, among the same education cohort, the major portion of di¤erence in the
consumption pattern is probably generated by the life-cycle motives (i.e. age pro…le of thenumber of adults and children in the household). It is hard to believe that the di¤erence in
consumption pattern over time is coming from the shift in the taste parameters. Indeed, as
we have already seen in Figure 2, the consumption pro…le adjusted by adult equivalent haverevealed a considerable smoothness among the same education cohort, supporting the viewthat fundamental preference parameters are stable over the life-cycle.
5.3.2

Panel Dynamic OLS

As a preliminary step for the cointegration analysis, we test for the cointegration restriction
implied in eq.(25a). We conducted Park’s H(p,q) test for each cohort22 and the results arepresented in Table 12.
As can be seen from Table 12, the null of deterministic cointegration was not rejected
for most of the test, except for older college cohorts. It is likely that the rejections are dueto the large sampling errors under Col4 and Col5 whose average cohort size were mere 120
and 98, respectively. For the rest of cohorts, where average cohort size are over 200, H(p,q)
tests were not able rejected the null of deterministic cointegration. Based on this result,
there is good reason to believe that implications of deterministic cointegration by eq. (25a)and (25b)are holding. We proceed to estimate the cointegrating vectors, assuming that thedeterministic cointegration restriction is holding.
In estimating the preference parameters from cointegrating restrictions implied in eq.
(25a) and (25b), we adopt Mark and Sul’s (2001) Panel Dynamic OLS (PDOLS) estimation
method. PDOLS estimation method are in principle equivalent to Panel FM-OLS method,
but it has an advantage of computational simplicity. Obviously, by the virtue of larger degrees
of freedom in estimating the parameters, PDOLS enjoys a better small sample property
compared to CCR estimation in the single equation context.
2 2 Here we have applied H(p,q) test in the single-equation context.

For the sake of power, it is more desirble
to test the cointegrating restriction in the panel context - i.e. panel cointegration test. (Cite literature HERE.)

27

Table 12: Test of Cointegrating Restriction under the Null of Cointegration

H(0,1)
H(0,2)
H(0,3)

HS1
0.527[0.468]1.153[0.572]
4.149[0.245]High School
HS2
HS3
0.021
0.368
[0.884][0.544]0.068
0.376[0.966][0.828]2.186
1.950[0.534][0.582]HS4
0.126[0.722]0.144[0.930]2.244[0.523]HS5
0.366[0.545]0.453[0.797]3.917[0.270]Col44.188*[0.040]4.358[0.113]6.040[0.109]Col5
1.572[0.209]6.040*[0.046]6.270[0.099]College
H(0,1)
H(0,2)
H(0,3)

Col1
1.393[0.237]1.404[0.495]3.282[0.350]Col2
0.155[0.693]0.247[0.883]2.375[0.498]Col3
0.257[0.612]0.378[0.827]0.947[0.814]

Note: Numb ers in the brackets represent p-values. * denotes rejection of the null of cointegration at 5% level.

The regression forms for each education group are speci…ed as follows..
High School Pooled Regression
yth= consth +

HS h
x1;t

HS h
x2;t + I(0) for h = HS1; :::; HS5(26)

College Pooled Regression
0
yth

0

= consth +Col h0x1;t

Col h0x2;t + I(0) for h0 = Col1; :::; Col5(27)

L
where yth = ln (1 + rt+1 )PtN =Pt+1, xh1;t = ln Lht+1 and xh2;t = ln Nth . We pool the timeseries data of each cohort by education category. In order to allow for the cohort-speci…c…xed e¤ect, we allow the constant term to di¤ers among each cohort. As we have seen, in
eq. (23) and (24), the constant term in each cohort’s cointegrating restriction depend upona discount factor. Thus, by …xed e¤ect approach, we are in a sense allowing for young andold cohorts to have di¤erent discount rate depending upon their stage in life-cycle.
Table 13 shows the result of the PDOLS estimation. In this subsection, we will simply
report the results of the estimates and defer interpretation until the end of this section.
Estimated preference parameter on luxury goods were 0.003 for high school cohorts and0.006 for college cohorts. The standard error for the estimator has been estimated based on
Andrew’s (1990) pre-whitening method. As can be inferred from the results, the parameter
estimates on luxury goods are not signi…cantly di¤erent from zero. Turning to the preference
parameter on necessity goods, the estimates under high school cohorts was 0.172 and 0.169
for college cohorts. Both estimates for the necessity goods are signi…cantly di¤erent from28

Table 13: PDOLS Results for Cross-Euler EquationsiPNhln (1 + rt+1 ) P Lt
ln Lht+1 + h ln Nth I(0)ht+1High School
HS

Pooled Estimates
S.E.

Pooled Estimates
S.E.
h
i
PN
ln 1+r1t+1 Pt+1
L
t

0.003
(0.015)
CollegehPooled Estimates
S.E.

0.172
(0.022)

Col

Col

0.006
(0.017)0.169(0.021)ln Lht +hhln Nt+1I(0)High School
HS

Pooled Estimates
S.E.

HS

0.009
(0.014)
CollegeHS

0.074(0.019)

Col

Col

0.013
(0.017)0.073(0.020)Note: Standard error based on Andrew’s (1990) Pre-whitening m etho d..

zero, showing a sharp contrast to the results in luxury goods.
5.3.3

Panel GMM

In this subsection, we conduct Panel GMM in estimating the preference parameters fromstandard Euler equations. For each of the education cohorts, standard Euler equations are
speci…ed for luxury and necessity goods. Preference parameters pertaining to each goods are
estimated in the context of pooled conditional moment restrictions with …xed cross-sectional
observations of 5 cohorts. Following the procedure in PDOLS estimation, we allow for the…xed e¤ect in each cohort’s discount factor. For convenience, conditional moment restrictions
for each education group and goods are presented below.
High-School Conditional Moment Restrictions
Luxury Goods:
3
2
HS
h
L
L
P
t+1
h
Et 4 hexp( 0HS Zt+1)(1 + rt+1 ) Lt
15 = 0 for h = HS1; :::; HS5
Pt+1
Lht

29

Necessity Goods2
h
Nt+1
Et 4 hNthHSexp( 0HS

PtN
h
Zt+1
)(1 + rt+1 ) N
Pt+1

College Conditional Moment Restrictions
Luxury Goods:
2
! Col
0
0
Lht+1
PtL
0
h
h0
exp(
Z
)(1
+
r
)
Et 4
t+1
0
Col
t+1
L
Pt+1
Lht

Necessity Goods:
2
!
h0
0
N
t+1
Et 4 h0NthCol

exp( 0Col

PtN
h
Zt+1
)(1 + rt+1 ) N
Pt+1
0

3

15 = 0 for h = HS1; :::; HS5

3

15 = 0 for h0 = Col1; :::; Col53

15 = 0 for h0 = Col1; :::; Col5

The choice of instrumental variables in Panel GMM were chosen according to the following
principle.
Constant term and lagged real interest rate are common for all goods and cohorts.
For the moment restriction of goods X and cohort H, lagged growth rate of goods X ofcohort H and lagged real price change in goods X are chosen as instrumental variables.
Since all cohort representative agents are forward looking and form the future expectation
by exhausting all the information available at period t, life-cycle model implies that forecast
errors are serially uncorrelated. Thus, when conducting panel GMM estimation, we have set
the lag order of GMM disturbance terms to be zero. Although lagged instrumental variablesof any order are considered to be valid since they are inside the information set at periodt, we have simply adopted the lagged instrumental variables of order one23 . Estimation ofvariance-covariance matrix of GMM disturbance terms was based on Andrews and Monahan’s(1992) HAC estimator with truncated kernel. However, since the lag order of disturbanceterms was set to zero, HAC estimator is equivalent to White’s (1980) HC estimator.
The results of Panel GMM estimation is shown in Table 14. As can be seen fromthe results, preference parameter on luxury goods was not statistically signi…cant for botheducation cohorts. Estimation from high school cohorts even showed a negative estimates.
Turning to preference parameter on necessity goods, the estimates have been statisticallysigni…cant for both education cohorts, though the estimated standard errors tend to belarger than those from PDOLS estimates. Probably this is partially due to the di¤erencein the rate of convergence. Estimates of coe¢ cient on socioeconomic factors turned out
to be not robust in general. None of the coe¢ cients were statistically signi…cant and were
2 3 This treatment is partially due to a consideration of weak correlation between the forcing terms andinstruments. Our preliminary inspection (which is not reported here) revealed that correlation between
forcing terms and instruments to be weaker, as lag order of instruments increased.

30

Table 14: Panel GMM Results: Euler Equations for Luxury and Necessity Goods
High School
Luxury Goods
Estimates
S.E.
J-statistics
P-value

Necessity Goods

HS

HS
1

HS
2

HS

HS
1

HS
2

-0.0154(0.016)-0.0136(0.022)13.500[0.333]-0.0021(0.005)0.2251(0.108)0.0002(0.0004)
9.627[0.648]0.0002(0.0003)

College
Luxury Goods
Estimates
S.E.
J-statistics
P-value

Necessity Goods

Col

Col
1

Col
2

Col

Col
1

Col
2

0.0047(0.008)0.0259(0.010)20.729[0.054]-0.0086(0.008)0.3094(0.110)-0.0009(0.001)6.474[0.890]0.0002(0.001)

Note: Degrees of Freedom for J-statistics was 12 for all cases.
of adults and1 stands for the co e¢ cient on change in numb ers

2 stands for the coe¢ cient on change in numb er of children.

* denotes rejection of the null

hyp othesis as 5% level.

in general very close to zero. Finally, Hansen’s J-statistics were not able to reject any of
the over-identifying restriction of the standard Euler equations, though the call was pretty
close for college cohort’s luxury goods. Often, researchers have claimed non-existence ofliquidity constraint solely based upon the non-rejection of the over-identifying restrictionsof the standard Euler equations. There at least two arguments against this interpretation.
First of all, it should be borne in mind that the null hypothesis of Hansen’s J-test is thatconditional moment restrictions are valid, nothing less, nothing more. In other words, thenull of the test is joint in nature. Thus, although it is appropriate to claim the overall validityof standard Euler equation based on the test, it is not logical to claim the non-existence ofliquidity constraint directly from there. Second, in the …nite sample, there is always an issueof size distortion. It may well be the case that J-test was simply lacking the power that thenull was not rejected. Apprehensive of these issues, we will not hasten to give any inference
regarding the existence of liquidity constraint based on the result of J-test, but rather we
propose to compare the parameter estimates from Cross-Euler and standard Euler equationin making an inference regarding the existence of liquidity constraint.
5.3.4

Testing Liquidity Constraint

In this subsection, we report the results from Cooley and Ogaki’s LR type test. Once again,
the motivation of this test is to check whether the parameter estimates from cointegration
restriction implied by log-linearized cross-Euler equation is close enough to the estimates31

from standard Euler equations. Under the null that two parameter estimates are equal,
the LR type test statistics (denoted QLR in this paper) is asymptotically 2 distributed.
Following the same procedure as in Section 4, we used same instruments were used for both
restricted and unrestricted GMM. For the restricted GMM, the preference parameters wererestricted based on the parameter estimates from PDOLS. Two null hypothesis were tested
for each education cohorts - i.e. H01 : ^ P DOLS = ^ GM M and H02 : ^ P DOLS = ^ GM M . The
test results are reported in Table 15.
Table 15: LR-type Test Results: Luxury and Necessity Goods

QLR statistics
P-value

QLR statistics
P-valueHigh School
Luxury GoodsHS
H0 : ^ HS
P DOLS = ^ GM M
3.683
[0.054]
College
Luxury GoodsCol
H0 : ^ Col
P DOLS = ^ GM M
1.066
[0.301]

Necessity GoodsHS
H0 : ^ HS
P DOLS = ^ GM M
3.846*[0.048]
Necessity GoodsCol
H0 : ^ Col
P DOLS = ^ GM M
3.371
[0.066]

Note: * denotes rejection of the null hyp othesis at 5% level.

First, let us turn to the results under the null of H01 : ^ P DOLS = ^ GM M . The LR type
test was not able to reject the null hypothesis for both education cohorts, though the call
was close for high school cohorts. QLR was 3.683 for high school cohorts and was 1.066 forcollege cohorts. Next, turning test the test results under the null of H02 : ^ P DOLS = ^ GM M ,the LR type test rejected the null hypothesis for high school cohorts at 5% level, but was notable to reject it for college cohorts. QLR was 3.846 for high school cohorts and was 3.371
for college cohorts.

5.4

Interpretation

Based on the above estimation and test results, some interpretations are in order. First ofall, we compare the preference parameter estimates from aggregate data and cohort data.
While the estimates from aggregate data were, for the majority of the cases, over or close toone, to the sharp contrast, the estimates from cohort data were very close to zero. Since theparametrization was almost identical - except for the treatment in socioeconomic factor - thediscrepancy can be mostly attributed to the aggregation problem, though some portion of thediscrepancy is, for sure, stemming from the di¤erence in data collection methodology. Thesekind of discrepancy in the estimates from aggregate data and cohort data have been widely
observed by the predecessors (See Attanasio and Webber (1993, 1997)). The estimation
results of this paper can be considered as, yet, another case that exempli…es the presence of32

aggregation problem.
Second, we observed that preference parameter estimates for both luxury and necessity
goods to be extremely close to zero. Literally interpreting, the estimates implies that IES ofnecessity goods to be around 5 or so, and IES of luxury goods to be around 200! Comparing
with the past literatures, the IES estimates of this paper are way higher than those reported
in Hall (1988), Campbell and Mankiw (1989), Cooley and Ogaki (1996) and Attanasio andWebber (1997). Considering the past evidence, the IES estimates of this paper are hard
to believe. One reason that may have contributed to this unaccountable estimates can be
attributed to the restrictive assumption of Houthakker’s addi-log utility function. Addi-log
utility function assumes additive separability among goods, rendering marginal utility of one
good to be independent from another. It o¤ers a virtue of parsimony in the number ofparameters when the assumption is holding, but at the same, its restrictive parametrization
can mar the estimation when the assumption is not holding24 . Thus, in the context of thispaper, if luxury and necessity goods are non-separable with each other or if they are notseparable from durable goods and/or leisure, IES is no longer a simple inverse of or , butwill be a function of luxury and necessity goods. If this is the case, parameter estimates in
Section 5 are merely parameters that a¤ect the intertemporal substitution and should not
be directly linked to the interpretation of IES for each goods.
Apart from the abnormally low estimates of the preference parameters, we made one
important observation regarding the existence of liquidity constraints. As have argued
in Section 3, assuming that aggregation problem is treated appropriately and each agent’s
preference is intertemporally additive-separable, the parameter estimates from Cross-Euler
equation and standard Euler equation are expected to yield a same estimates. However, as
we have seen in the LR type test results in Table 15, the null hypothesis H02 : ^ P DOLS =^ GM M has been rejected for high school cohorts. Also, we have seen that the call for thenull hypothesis H01 : ^ P DOLS = ^ GM M was pretty close for high school cohorts. Under
the assumptions we have made earlier, the main culprit of the rejection is the presence ofliquidity constraints. Taking education level as a proxy of life-time income, the results canbe interpreted as an evidence that ”poor”agents are more likely to be liquidity constrained,
which conforms with the …nding from the aggregate data in Section 4. The test results arealso consistent with the past …ndings by Hall and Mishkin (1982), Zeldes (1989), Meghir andWebber (1996) who report some evidence of liquidity constraint for ”poor”agents. However,
one should not hasten to conclude that ”poor”agents are liquidity constrained, while ”rich”
agents are not from the test results in Section 5. Turning to the test results for college cohorts(which can be thought of as a proxy of high life-time income agents), the call for the null
hypothesis H02 : ^ P DOLS = ^ GM M was very close - p-value of 0.066 - while the null hypothesisof H01 : ^ P DOLS = ^ GM M was accepted, signaling the mixed evidence whether the ”rich”
agent are liquidity constrained or not. This observation is does not exactly conform with
2 4 The robust remedy for this defects is to rely on more ‡exible functional form such as translog utility

function adopted by Meghir and Webber (1996).
future research.

This remedy seems promising and will be kept for the

33

the evidence reported, for instance, in Meghir and Webber (1996) that liquidity constraints
are not binding for the ”rich” agents, which is the standard view in the literature.
One factor, which should be borne in mind whenever interpreting the test result of thispaper, is the power of the test based on Cross-Euler equation approach. The approach takenin this paper is structural and therefore inherently parametric. Moreover, considering thatutility function has been tightly parametrized by the addi-log function, some sort of misspeci…cation in utility function will probably contribute to the higher value of QLR statistics.
Thus, unless the utility function is correctly speci…ed, the LR type test will probably be more
powerful than one desire to be. The general observation of high value of QLR statistics bothfor high school and college cohorts, may well be stemming from the misspeci…cation of theutility function. That said, still the relative di¤erence in QLR statistics between high school
cohorts and college cohorts cannot be satisfactorily explained by the misspeci…cation of theutility function. Under auxiliary assumptions that aggregation problem is treated appropriately and the agent’s preference is time-separable, as far as we are concerned, it seemreasonable to call for the liquidity constraint factor to account for this relative di¤erence in
QLR statistics between high school and college cohorts. To summarize, we interpret theresult of the LR type test in this section as an evidence, albeit weak, that agents with low
life-time income is liquidity constrained.

6

Conclusion

In this paper, we adopted standard two goods version of the life-cycle model to study the
consumption behavior of necessity goods and luxury goods under addi-log utility function
which allows for the non-homothetic preference. We proposed a new empirical method in
testing for the existence of liquidity constraint utilizing the concept of Cross-Euler equation. The Cross-Euler equation represents the optimal consumption pattern of a good in thecurrent period to another good at a future period. It can be interpreted as the composite
optimal condition that embeds both intertemporal and intratemporal optimal consumption
relationships into one equation. The Cross-Euler equation has an advantage over the standard Euler equation, in the sense that the cointegrating relationship is maintained even when
the liquidity constraint is present in the agent’s decision problem. Thus, by comparing thepreference parameter estimates from the Cross-Euler equation to those from the standard
Euler equation, it is possible to detect the existence of a liquidity constraint.
In testing for the existence of liquidity constraints, we applied the Cross-Euler equation
approach to both aggregate data and synthetic panel data constructed from Consumer Expenditure Survey. For the aggregate data, by construction, signi…cant portion of the luxury
goods expenditure comes from the richer agents in the economy, while signi…cant portionof the necessity goods expenditure comes from the poorer agents. Since the poorer agentstend to be more vulnerable to the liquidity constraint, we expect that the Euler equation
for the necessity goods to be misspeci…ed, but the Euler equation for the luxury goods to be
speci…ed. Indeed, the empirical results presented in Section 4 of this paper supported this34

view. We conducted LR type test on the null hypothesis that the IES parameter estimatesfrom the Cross-Euler equations and the standard Euler equations are equal. We rejected
null hypothesis for the necessity goods frequently, while that of luxury goods was not. Thisempirical results implies that the Euler equation for necessity goods is misspeci…ed, but keeps
the possibility open for the Euler equation of the luxury goods to be speci…ed - empiricalevidence that the liquidity constraint is a serious factor in rendering the Euler equation to be
misspeci…ed, but only for the poor agents. This result can be thought of as empirical evidence
from the aggregated time series data that supports the existence of liquidity constraint in
the U.S. economy.
However, the empirical results in Section 4 were based on the aggregate time-series data,
implicitly assuming the existence of the representative consumer. The existence of therepresentative agent requires that all the agents in the economy to share the identical andhomothetic preferences, which is obviously a too strong assumption. In order to control
for this aggregation problem and to legitimately test for the liquidity constraints in theeconomy, we conducted cohort analysis in Section 5. Based on Consumer Expenditure
Survey from 1984 to 1998, the households have been classi…ed into 10 cohorts by their age andeducational attainment. Age classi…cation was necessary because the households revealed theheterogeneous consumption patterns (i.e. life-cycle in consumption) depending on their age.
Classi…cation by education attainments was adopted as a mean to classify the householdsby their life-time income (i.e. permanent income). In order to compare the result with theempirical evidence from aggregate data, we aggregated the consumption goods into necessity
and luxury goods. Then we estimated the preference parameters for each education cohorts
exploiting the cointegrating restriction implied by the Cross-Euler equations using (pseudo)
Panel Dynamic OLS estimation method. Further, preference parameters has been estimatedfrom standard Euler equations using (pseudo) Panel GMM.
Two major features emerged from this cohort analysis. First, the preference parameter
estimates from the Consumer Expenditure Survey were signi…cantly lower than those basedon aggregate time-series data. Di¤erence in estimates can be interpreted, in some extent,
as the severity of aggregation problem in the aggregate data sets. Second, the LR type test
rejected the null hypothesis for high school cohorts for some cases, while the null hypothesis
for college cohorts were not. Taking the educational attainment as proxy for permanent
income, the test results were consistent with the view that poorer agents are more likely to beliquidity constrained. Thus, despite the large di¤erence in estimates stemming possibly fromthe aggregation problem, the hypothesis that the poorer agents are more liquidity constrained
was both supported by the evidence from aggregate data and from disaggregated data.
Obviously, the main drawback in this paper was an adoption of tightly parametrizedaddi-log utility function. It is interesting to see whether the empirical results found in thispaper will also be con…rmed under other functional form such as trans-log utility functionin Meghir and Webber (1996) or ‡exible functional form proposed by Attanasio and Webber(1997). The Cross-Euler equation approach to testing for the liquidity constraints undermore ‡exible functional form will be left for future agenda.

35

A

Appendix 1

In this Appendix 1, we will prove the proposition that when some kind of an error is introduced to the intratemporal relationship eq. (?), then at least one of the Euler eq. (?) or (?)
will be misspeci…ed. For notational simplicity, let us de…ne the terms in eq.(?), (?) and (?)
as follows.
a

PtN
PtL

At

1 (Nt )K (Lt )

Bt+1

Nt+1
Nt

(1 + rt+1 )

t+1

Lt+1
Lt

(1 + rt+1 )

PtN
N
Pt+1

PtL
L
Pt+1

Observe that under the optimization behavior of the representative agent,
At

=

1

Bt+1

=

1 + "B
t+1

t+1=

1 + "C
t+1

where
"B
t+1

Nt+1
Nt

PtN
(1 + rt+1 ) N
Pt+1

Et

"t+1

Lt+1
Lt

PL
(1 + rt+1 ) Lt
Pt+1

Et

"

"

#
PtN
(1 + rt+1 ) N
Pt+1
#
PtL
(1 + rt+1 ) L
Pt+1

Nt+1
Nt
Lt+1
Lt

Before we prove the proposition, it is useful to prove the following lemma.
Lemma 1 Let X and Y be any two random variables. Then, in general1 ,XY

6== E (X) E

1
Y

E

E (X)
.
E(Y )

Proof. Let us …rst observe thatE

X
Y

+ Cov X;

1
Y.

We prove the lemma by way of contradiction. Suppose the statement E (X=Y ) = E (X) =E(Y )is true. Then the following relationship must be true.
E (X) E

1
Y

+ Cov X;

1
Y

=

E (X)
E (Y )

Dividing both side by E (X), we get the following expression.
E

1Y

1 Except for the special case when Cov+

Cov (X; 1=Y )
1
=
E (X)
E (Y )X; Y1= E(X)

36

h

1
E(Y )

E

1
Y

i
.

Now, this equation have to be true for any two random variables X and Y . In particular, let
us choose the case when random variables X and Y are independent with each other. Then
Cov (X; 1=Y ) = 0. This implies the following equation.
1
Y

E

=

1
E (Y )

But by Jensen’s Inequality, the above equation cannot be true. A contradiction.

A.1

Case 1: Introducing Additive Error to At

Proposition 3 When additive stationary error term is introduced to At , i.e. At = 1 + etwhere etI(0), at least one moment condition of E (Bt+1 ) = 1 or E( t+1 ) = 1 will beviolated.
Proof. Su¢ ce to show one inequality. Let us …rst note the following algebraic relationship
t+1

At
Bt+1
At+1
(1 + et )(1 + "Bt+1 )(1 + et+1 )==Applying conditional expectation operator Et ( ) on both side,
Et ( t+1 ) = (1 + et )Et

1 + "Bt+1
1 + et+1

By lemma 1, the following inequality holds.
Et ( t+1 ) 6=6=

Et 1 + " B
t+1
Et (1 + et+1 )1 + et
1 + Et (et+1 )(1 + et )Applying unconditional expectation operator E ( ) on both side,
1 + et
1 + Et (et+1 )E(1 + et )1 + E(et )=E [1 + Et (et+1 )]1 + E(et+1 )E ( t+1 ) 6= E
6=Now by the stationarity of et , [1+E(et )]=[1+E(et+1 )] = 1. Therefore, in general, E( t+1 ) 6=
1.

37

A.2

Case 2: Introducing Multiplicative Error2 to At

Proposition 4 When multiplicative error term is introduced to At , i.e. At = et whereet I(0), at least one moment condition of E(Bt+1 ) = 1 or E( t+1 ) = 1 will be violated.
Proof. Su¢ ce to show one inequality. Again, by the algebraic relationship,
t+1

At
Bt+1
At+1
et
(1 + "B
t+1 )
et+1

=
=

Applying conditional expectation operator Et ( ) on both side,
Et ( t+1 )1 + "B
t+1
et+1= et Et
6= et

Et (1 + "B
et
t+1 )
=
Et (et+1 )Et (et+1 )Applying unconditional expectation operator E ( ) on both side,etEt (et+1 )E(et )E(et )==1
E [Et (et+1 )]E(et+1 )E ( t+1 ) 6= E
6=Thus, in general, E( t+1 ) 6= 1.

BAppendix 2

In this Appendix 2, we will show how to derive the Cross-Euler equations formally.
convenience, let us restate the representative agent’s problem.
max Et

1
X

i

For

U (Nt+i ; Lt+i )

i=0

s:t: At+1+i=(1 + rt )At+i + Yt+i

N
Pt+i
Nt+i

L
Pt+i
Lt+i for 8i0

By constructing a lifetime budget constraint from period-by-period budget constraints, we
can reformulate the above optimization problem as follows.
max Et1
X

i

U (Nt+i ; Lt+i )

i=0

s:t: At +

1
X
i=0

0
@

i
Y

1

1
X

0

i
Y

1

1
1
N
L
A Yt+i =@A Pt+i
Nt+i + Pt+i
Lt+i
1
+
r
1
+
r
t+j
t+j
j=0
i=0
j=o

2 The import demand model with preference shocks adopted by Clarida (1994) and Amano et al.

is a special case of this multiplicative error.

38(1996)

Left-hand side of the constraint can be considered as the present value of a life-time wealthof the agent, while right-hand side of the constraint can be considered as the present valueof a life-time consumption. Formulating the Lagrangian,
L = Et1
X

i

U (Nt+i ; Lt+i )i=02+ 4At +

1
X
i=0

0

1

i
Y

1
X

1
A Yt+i
1
+
rt+j
j=0

@

0

1

i
Y

3

@

1
N
L
A Pt+i
Nt+i + Pt+i
Lt+i 5
1
+
r
t+j
j=o

UNt = PtN

(B.1.)

ULt = PtL

(B.2.)

i=0

Then, the FOC’s for Nt , Lt , Nt+1 , and Lt+1 will be as follows.

Et

UNt+1 =

1PN
1 + rt+1 t+1(B.3.)

Et

ULt+1 =

1PL
1 + rt+1 t+1(B.4.)

From eq.(B.1.) and (B.4.), we get,
Et

ULt+1
PN
(1 + rt+1 ) Lt
=1
UNt
Pt+1

which is the Cross-Euler equation (11) in the paper. From eq.(B.2.) and (B.4.), we get,
Et

UNt+1
PL
=1
(1 + rt+1 ) Nt
ULt
Pt+1

which is the Cross-Euler equation (12) in the paper.

C

Appendix 3

In this Appendix 3, we will prove the (strict) stationarity of the forecast error embeddedin the Cross-Euler equation (11). The strict stationarity of the forecast error from the
Cross-Euler equation (12) is similar and therefore will be omitted.
N
Proposition 5 Let Nt+1 =Nt , rt , and Pt+1
=PtN be strictly stationary processes. The forecast
error et is de…ned as
et = t Et 1 ( t )

where
t

K

(Lt )
CtN 1

a (1 + rt )

PtN 1
PtL

Then, ln(1 + et ) is a strictly stationary process with E(et ) = 0 and E(et et j ) = 0 for 8j 6= 0.
39

Proof. First, let us prove the proposition E(et ).
operator E ( ) on both side of et = t Et 1 ( t ),E(et )

Applying unconditional expectation= E( t )E [Et 1 ( t )]= E( t )E( t ) = 0

Thus, E(et ) = 0.
Next, let us prove the prove the proposition E(et et j ) = 0 for 8j 6= 0. Without lossof generality, consider the case where j
1. Since Et 1 (et ) = 0 and et j is inside theinformation set available at period t 1 for any j 1, this will imply Et 1 (et et j ) = 0 for any
j 1. Applying unconditional expectation operator E ( ) on both side of Et 1 (et et j ) = 0,
E [Et 1 (et et j )] = 0) E(et et j ) = 0 for 8j

1

Finally, let us prove the strict stationarity of ln(1 + et ). Since the logarithmic function is acontinuous and monotone function, it su¢ ces to show the strict stationarity of et Recallingthe de…nition of At and Bt+1 from Appendix 1, we can observe the following algebraic
relationship.
Bt
t =
At 1
N
By the strict stationarity assumption of Nt+1 =Nt , (1 + rt ) and Pt+1
=PtN , Bt is strictly
stationary. Also, since At 1 = 1 from Appendix 1, this implies the strict stationarity ofNow, since et = t Et 1 ( t ) and by the strict stationarity of t , et will be strictlyt.
stationary.

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44