Consumption and asset prices

An analysis across income groups

H.J. Smoluk*, Raymond P. Neveu

School of Business, University of Southern Maine, 96 Falmouth Street, PO Box 9300,

Portland, ME 04104-9300, USA

Received 29 May 2000; received in revised form 5 November 2000; accepted 12 December 2001

Abstract

Employing aggregate consumption data to test the consumption-based capital asset pricing model

(CCAPM) is likely to lead to a specification error since a significant portion of consumers live from
paycheck-to-paycheck and, therefore, are constrained in their ability to intertemporally allocate
consumption. Furthermore, these consumers lack the savings needed to directly influence an
equilibrium between consumption expenditures and asset returns. Using consumption expenditures
grouped by consumer income, this paper examines the issue of whether the CCAPM is more consistent
with the consumption of unconstrained (high-income) consumers as compared to constrained (low-
income) consumers. Several traditional methods of analyzing the CCAPM are explored utilizing five
time series of consumption expenditures delineated by consumer income. This approach allows us to
indirectly test whether liquidity constraints affect the CCAPM without imposing additional specification
on the model. Overall, the tests fail to find any discernible patterns across income groups that are
consistent with the idea that liquidity constraints bind lower income consumers.

D 2002 Elsevier

Science Inc. All rights reserved.

1. Introduction

The theoretical models that bridge microeconomics and finance have been the subject of

numerous empirical studies which, for the most part, find an inconsistent relationship

1058-3300/02/$ – see front matter

D 2002 Elsevier Science Inc. All rights reserved.

PII: S 1 0 5 8 - 3 3 0 0 ( 0 2 ) 0 0 0 3 7 - X

* Corresponding author. Tel.: [phone]; fax: [phone].
E-mail address: [e-mail] (H.J. Smoluk).

Review of Financial Economics 11 (2002) 47 – 62

between US consumption data and asset returns.

1

The inconsistency stems from the finding

that equity risk, as measured by the covariance between stock returns and consumption
growth within the context of the consumption-based capital asset pricing model (CCAPM),
is too small to warrant the large average risk premium on stocks. In other words, the
CCAPM shows that aggregate investors, or the representative single agent, must be
extremely averse to consumption risk (corresponding to a large coefficient of relative risk
aversion) to demand such a large equity premium. The disparity between the predictions of
these models and the data are so pronounced that the literature has continued to refer to this
phenomenon as the ‘‘equity premium puzzle,’’ based on the work of Mehra and Prescott
(1985), who first brought the issue to light.

Attempts to rescue the CCAPM or solve the equity premium puzzle are bountiful. They

seek to unravel the assumptions employed by Mehra and Prescott (1985) in hopes of ex-
plaining this puzzle. The assumptions, as well as the literature attacking these assumptions,
are typically categorized into three (not necessarily mutual exclusive) areas: consumer
preferences, the representative agent, and market frictions. In brief, papers focusing on the
consumer preference assumption seek to generalize the very convenient time-separable power
utility function employed by Mehra and Prescott. Papers focusing on the representative agent
assumption examine the implications of using aggregate consumption data to test the
CCAPM. And finally, papers studying market frictions examine transaction costs, the im-
plication of incomplete markets, and economically constrained consumers.

The purpose of this paper is to examine the implications of economically constrained and

unconstrained consumers on the standard CCAPM. It therefore spans the area of surrounding
the representative agent assumption, as well as, market frictions (see Campbell, Lo, &
MacKinlay, 1997, p. 317 for an in-depth review of these issues). More specifically, this
paper seeks to determine whether real asset returns are more consistent with the consumption
patterns of higher income consumers than lower income consumers. The hypothesis is that
the consumption patterns of lower income consumers, who live from paycheck-to-paycheck,
should bear a weaker relation to real asset returns than higher income consumers since they
lack the ability to save and manipulate consumption. This hypothesis suggests that
traditional tests of the CCAPM based on national per capita consumption are misspecified
since a significant portion of consumers either do not save or have negligible savings rates.
Furthermore, even if these consumers have some savings, their awkward economic position
would indicate that they probably do not manipulate their consumption patterns based on
expected real asset returns. An important element of the tests employed in this paper is that
they do not impose any form of liquidity constraint on the CCAPM, thus eliminating the
possibility of additional misspecification.

The paper is organized as follows. Section 2 reviews the CCAPM literature and its

relation to the equity premium puzzle of Mehra and Prescott (1985). Section 3 covers the

1

For theoretical models, see Breeden (1979, 1986), Lucas (1978), and Rubenstein (1976). For the empirical

studies, see Grossman and Shiller (1981), Kocherlakota (1996), and Mehra and Prescott (1985) for an in-depth
literature review. Our discussion follows Campbell et al. (1997).

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

48

tests used to measure the differences in the relationship between consumption growth and
asset returns for low-income and high-income consumers. Specifically, the generalized
method of moments (GMM) is employed to exploit the orthogonality conditions implied by
the model. In Section 4, the data are described and the coefficients of relative risk aversion
across income groups are estimated using methods that rely on rather strong distribution
assumptions for consumption growth and asset returns as well as distribution-free methods
such as the GMM. The overall results indicate that there appears to be no clear pattern
between consumption and asset returns across income quintiles. In other words, the
consumption-based CAPM specification does not unambiguously become stronger as
income rises. Consequently, categorizing consumption expenditures by income quintiles
appears to provide little, if any, information about the equity premium puzzle in that the
relationship between consumption and asset returns shows no discernible systematic
patterns. Section 5 concludes.

2. Background

2.1. Literature review

The lack of consumption data sorted by consumers’ income level makes testing of the

constrained/unconstrained hypothesis difficult, and consequently, there appear to be few
papers that directly address this issue within the context of the CCAPM. For example,
Mankiw and Zeldes (1991) examine this issue indirectly, without consumption sorted by
income. They examine a unique data set that distinguishes between the consumption of
stockholders and nonstockholders.

2

Their objective is to test the hypothesis of whether

stockholders’ consumption is more consistent with asset returns as compared to nonstock-
holders. Their data set, however, has severe limitations in that it only covers food con-
sumption. The use of food consumption has both positive and negative implications for the
CCAPM. On the positive side, food consumption probably comes closest to matching the
assumption of time-separable utility since consumers are assumed not to store goods in-
tertemporally under the standard power utility function. On the negative side, Attanasio and
Weber (1995) indicate that food is considered a necessity and is not likely to change
proportionally to income changes. Furthermore, the use of only food consumption data
requires the assumption that the utility derived from food is separable from the utility derived
from other goods and services.

Mankiw and Zeldes (1991) conclude that the food consumption of stockholders differs

from that of nonstockholders. Food consumption data of stockholders are more volatile and
consistent with the CCAPM. Specifically, they find that food consumption is more highly

2

The data Mankiw and Zeldes employ is from the Panel Study of Income Dynamics (PSID). The survey,

starting in 1984, questions consumers as to the market value of their holdings in publicly traded corporate shares,
mutual funds, and stocks held in IRAs.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

49

correlated with excess returns of the S&P 500 over 3-month T-bills. Their conclusions are
tenuous, at best, however. Mankiw and Zeldes’s work is plagued by severe data limitations
not only because they use just food consumption data, but also because they have only 13
observations in which to draw their conclusions.

In this paper, we attempt to provide more conclusive evidence concerning the relationship

between consumption and investment behavior.

3

Specifically, we examine the relationship

between consumption spending and asset returns across income classes as designed by the
Consumer Expenditures Survey: Quarterly Data from the Interview Survey (CES), published
by the Bureau of Labor Statistics (BLS). The CES provides 51 quarterly observations across
five different income groups.

The BLS publishes household consumption and income data quarterly.

4

Income is divided

into quintiles where the first quintile corresponds to consumers with the lowest 20% of
reported income and the fifth quintile denotes the top 20% of consumers in terms of reported
income. The assumption made in this paper is that there are five independent representative
economic agents in the economy, one corresponding to each income quintile. Each agent
faces the same set of expected asset returns; however, the lowest quintile representative agent
is hypothesized not to invest since he or she lives from paycheck-to-paycheck and hence has
no savings. Representative agent 2, corresponding to the second income quintile, may have
some savings, but is generally limited in his or her ability to consistently invest in assets. The
other representative agents can be categorized accordingly, with the fifth agent consistently
saving income depending on expected asset returns. The preferences of each agent are as-
sumed to be characterized by the power utility function.

An examination of consumption across higher income quintiles should show stronger and

more consistent results with the CCAPM, since higher income consumers are assumed to
have the capacity to alter consumption patterns with the use of financial assets. The re-
presentative agent corresponding to the lowest income quintile, on the other hand, does not
have the luxury of manipulating consumption patterns intertemporally, and therefore, has no
direct influence on asset returns. Since there is a significant proportion of consumers in the
United States living from paycheck-to-paycheck, such a finding may be relevant to the equity
premium puzzle and the general failure of the consumption-based CAPM based on national
per capital consumption.

3

The CCAPM is often tested by employing a Markov chain to estimate, for example, consumption growth

with the assumption that dividends from a major stock index equals consumption (see, e.g., Cecchetti, Lam, &
Mark, 1990, 2000). We have decided not to follow this method for three reasons. First, since we are dealing with
five different consumption streams (by income group) and only one dividend series, from the S&P 500, the
assumption that dividends equals consumption is difficult to overcome. Second, estimating a Markov chain for
five different consumption streams would be difficult under one set of assumptions because parameter estimates
may or may not be significant from one stream to the next. Lastly, estimating a Markov chain for each con-
sumption stream adds another layer of subjectivity to the analysis making comparisons across income groups
much more difficult.

4

The CES also contains information such as the average number of members in the household and the average

respondent’s age.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

50

2.2. Consumption-based capital asset pricing model

A brief review of the consumption-based CAPM and its assumptions will provide the

background necessary to understand the issues involved. The equity premium puzzle was
first brought to light by Mehra and Prescott (1985) and focuses on the tightly para-
meterized intertemporal equilibrium imposed by a standard CCAPM. The standard Euler
equation is

u

0

ðC

t

Þ ¼ bE

t

½ð1 þ R

i;t

þ1

Þu

0

ðC

t

þ1

Þ

ð1Þ

where

b is the rate of time preference. Higher values of b imply that consumers place a higher

value on future consumption, meaning that they would prefer to save today and to consume
tomorrow. The term R

i,t + 1

denotes the expected real return on asset i for the period ending

t + 1. Assuming that the model is tested with some measure of aggregate consumption, u

0

(C

t

)

denotes the marginal utility from aggregate consumption measured at time t. Eq. (1) states
that the representative agent will forego a dollar of consumption today and invest it in asset i
so long as the expected additional discounted marginal utility that it provides is greater than
today’s marginal utility. Eq. (1) is parameterized with a state-independent and time-separable
power utility function

5

u

ðC

t

Þ ¼

C

1

g

t

1

g

ð2Þ

where g > 0 is the constant relative risk aversion coefficient (CRRA). When g = 1,
u(C

t

) = log(C

t

). Marginal power utility in this case is u

0

(C

t

) = C

t

g

so that when Eqs. (1)

and (2) are combined, the first-order conditions are

1

¼ bE

t

ð1 þ R

i;t

þ1

Þ

C

t

þ1

C

t

g

ð3Þ

where the term (C

t + 1

/C

t

) denotes consumption growth. An economically plausible

b

coefficient should be less than 1.0 indicating that consumers discount future consumption.
According to Mehra and Prescott (1985), economically plausible estimates for g are greater
than zero and less than or equal to 10.

6

Coefficients of relative risk aversion that are around

10 or more suggest extremely risk-averse consumers who prefer low consumption growth
(less volatile consumption over time). Coefficients of less than zero contradict Eq. (2) by
indicating nonconvex preferences (increasing marginal utility) and the nonexistence of an
intertemporal equilibrium for asset prices.

Nearly all of the empirical studies that examine Eq. (3) suffer from serious data limitations:

the weight of these studies employ aggregate nondurable goods and/or services consumption

5

State independence implies that the utility an individual receives from consumption is independent of the

state of the world, that is, the utility function is the same regardless of whether times are good or times are bad.
Time separable implies that utility today does not influence utility tomorrow, that is, habits are disallowed.

6

Mehra and Prescott (1985) base their range on estimates derived from microeconomic literature; for a listing

of this literature, see their paper, p. 154.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

51

data from the National Income and Product Accounts (NIPA) published by the Bureau of
Economic Analysis (BEA). By assuming homogeneous preferences and identical investment
portfolios, a representative agent is said to exist and allows Eq. (3) to be tested using per
capita consumption data. However, consumption data in the aggregate are very smooth over
time causing many to wonder whether the aggregation process eliminates the volatility
needed to satisfy the restrictions imposed by the Euler equation (3). This may be one
explanation for the equity premium puzzle.

Eqs. (1 –3) assume that all consumers (the representative agent) invest in financial assets.

However, there is a significant portion of lower income consumers who are constrained in
their ability to invest since they tend to live paycheck-to-paycheck. These consumers have no
direct impact on the return of financial assets, and suggests that a significant proportion of
national per capita consumption may not bear a direct relation to real asset returns. Empirical
tests of Eq. (3) using national per capita consumption, therefore, are likely to fail. In this
context, the failure of the model may help explain the equity premium puzzle in that lower
income groups may be distorting the economic relationship consumption and asset returns.

3. Generalized method of moments

Hansen and Singleton (1982) test Eq. (3) by employing Hansen’s (1982) GMM

Q

ðqÞ ¼ ½gðˆq

T

Þ

0

ˆ

W

1

T

½gðˆq

T

Þ:

ð4Þ

GMM selects the parameter vector

qˆ

T

= (

b,g)

0

so that the sample average of Eq. (3), stacked

by asset and denoted by g(

qˆ

T

), minimizes Eq. (4).

7

W

ˆ

T

1

denotes an N

r

N

r

optimal

weighting matrix so that the GMM estimates are consistent and asymptotically normal. Q(

q)

is

c

2

distributed with (N

r

N

g,

b

) degrees of freedom, where N

r

represents the number of

orthogonal conditions and N

g,

b

the number of parameters estimated in Eq. (4), respectively. If

the number of orthogonal conditions exceeds the number of parameters to be estimated, Q(

q)

represents a test statistic for the overidentifying restrictions imposed on the model. To start
the process of minimizing Q(

q), the initial weighting matrix is the identity matrix. An

iterating process is employed which results in an optimal weighting matrix that also is used
to estimate the standard errors of the GMM parameter estimates. The main advantage to
using the GMM is that it makes no assumptions concerning the distribution of real asset
returns and consumption.

While the parameters in the GMM do not rely to distribution assumptions, Hansen and

Singleton (1983), nevertheless, discuss the empirical difficulties and possible measurement
errors in attempting to specify the exact timing of each variable in Eq. (3). Consequently, they
develop an alternative method to estimating Eq. (3) by assuming that asset returns and con-
sumption data are jointly homoskedastic and lognormal. By transforming Eq. (3) in log form,

7

In other words, GMM is used to estimate a nonlinear system of simultaneous equations, where the number of

stacked equations is equal to the number of different assets employed in Eq. (3).

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

52

they derive the following expression that relates the log real excess return on asset i to the
coefficient of relative risk aversion times the covariance between the log real returns on asset i
and log real consumption

E

t

½r

i;t

þ1

r

f ;t

þ1

þ

s

2
i

2

¼ gs

i;c

ð5Þ

The term

s

i

2

/2 is a Jensen’s inequality adjustment and lower case variables are in log form. All

variables are readily determined by the data, except for g which is solved for algebraically.

A third method of testing the relationship between asset returns and consumption growth is

instrumental variables (IV) regression estimation. Following Hall (1988) and Hansen and
Singleton (1983), Campbell et al. (1997) examine the following IV (two-stage) regression

c

t

þ1

c

t

¼ n

i

þ pr

i;t

þ1

þ x

i;t

þ1

ð6Þ

where

x

i,t + 1

=

p{(E

t

[r

i,t + 1

]

r

i,t + 1

) + g(c

t + 1

/c

t

)

gE

t

[c

t + 1

/c

t

]} and

p denotes the asymptotic

reciprocal of the coefficient of relative risk aversion. The first stage regresses consumption
growth onto a constant, real Treasury bill rates, real S&P 500 total returns, and real
consumption growth. Each dependent variable is lag 1 or lag 1 and 2. The second stage
regresses real Treasury bill rates onto a constant and the fitted consumption growth values
from the first-stage regression. Since

p represents the reciprocal of the coefficient of relative

risk aversion, economically plausible estimates should lie in the range .1 and higher.

4. Data and empirical results

4.1. Data

All data are sampled quarterly from first quarter 1984 to third quarter 1996. The

consumption and income data are from two separate sources, the CES published by the
BLS and NIPA published by the BEA. The CES is produced with the goal of periodically
revising the Consumer Price Index (CPI) and is collected in independent samples that are
representative of the United States population. The survey is composed of two separate parts
that are later integrated: a diary survey and interview survey. The diary captures small and
frequently purchased items, whereas the interview survey captures large and ‘‘regularly
recurring’’ purchased items. With the interview survey, a rotating panel method is employed
where each panel is interviewed every 3 months for five consecutive quarters then dropped so
that about 20% of the consumer units are new each quarter. Approximately 5000 consumer
units are sampled with about 95% of all expenditures covered.

8

The CES reports both total income before incomes taxes and expenditure data by quintiles

based on complete income reporters. Complete income reporters are ranked in ascending

8

Roughly, a consumer unit represents all members of a particular household.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

53

order by total income before taxes reported by the unit consumer.

9

Since the models

employed in this paper depend upon time-separable utility functions, only nondurable goods
and services (NDS) and food are examined, rather than longer lasting durable goods. Blinder
and Deaton (1985) argue that NDS is a better time series than durable goods and the use of
NDS and food also follows a convention generally employed by the literature.

The CES does not report NDS separately, and must be estimated. In keeping with the spirit

of the NIPA, NDS are computed from the CES by subtracting an estimate of durable goods
expenditures from total expenditures. Durable goods are considered to be the CES line items
‘‘House Furnishing and Operations’’ plus a fraction of ‘‘Other Transportation Expenses’’
which includes expenditures on motor vehicles and parts.

10

The derived NDS series includes

food expenditures; however, for comparative purposes with Mankiw and Zeldes (1991), food
expenditures also are examined separately. Food expenditures are a separate line item in both
the CES and the NIPAs.

The CES data has several, possibly severe, limitations. First, the data used in this paper

exclude consumer units that failed to report their income, but nevertheless responded to
expenditure questions. In this paper, it is assumed that the impact from this under-reporting of
income uniformly affects each quintile so that there are no systematic biases from one quintile
to another. Second, the CES only surveys about 7000 consumer units per year which is
relatively small compared to other national economic surveys. Finally, according to the BLS,
the data are preliminary and may be subject to minor revisions.

Reliable consumer expenditure data, sorted by income, are notoriously difficult to obtain

for extended sample periods. The only other conventional data set used in this type of
analysis, besides the CES, is the Panel Study of Income Dynamics (PSID). However, the only
consumer expenditure line item sorted by income is food making it substantially less
appealing than the CES. CES data are used in many papers including Attanasio and Weber
(1995), Caroll (1992), and Lusardi (1992).

The implicit price deflator for NDS from the BEA is used to adjust all nominal variables.

The index is set equal to 100 in the first quarter of 1984. Per capita figures for the CES are
computed by dividing average household NDS consumer expenditures by the number of unit
individuals in the household. The BEA publishes per capita expenditures by dividing
aggregate NDS expenditure by population estimates published by the Bureau of Census.

The real interest rate is computed from the nominal monthly auction-average for the

3-month US and inflation based on the NDS implicit price deflator. The T-bill data are
published by the Board of Governors of the Federal Reserve in Statistical Release H.15. Total
stock returns are based on the S&P 500 index with dividends immediately reinvested.
Following Campbell et al. (1997), the log dividend to price ratio is employed as an instrument

10

Both furniture/household equipment and motor vehicles are considered durable goods by the BEA. The

CES, however, does not separate the portion of line ‘‘other transportation expenses’’ between durables and
nondurables. The fraction of ‘‘other transportation expenses’’ considered durable goods expenditures is derived
from the percentage of durables transportation expenses found in the NIPA.

9

Complete income reporters are respondents who provide at least one of the major sources of income such as

wages and salaries, self-employment income, and Social Security income. See the ‘‘Technical Notes’’ section of
the CES for details on the definition of a consumer unit.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

54

in the IV regressions. This ratio is constructed using S&P 500 dividends divided by the
beginning of the period S&P 500 index.

The timing of the data implicitly assumes that consumers make economic decisions that

coincide with the sampling interval of the data. Consumers make their consumption decisions
at the beginning of the current quarter, based on knowledge of their investment returns for
that quarter (see Campbell et al., 1997, p. 308). The 3-month T-bill returns are from the first
month of the current quarter (as consumers earn this rate during the quarter) and total stock
returns are based on the current quarter ending S&P 500 index (including reinvested
dividends). Consumers are assumed not to intertemporally store purchased items.

4.2. Results

The GMM results based on Eq. (4) are presented in Table 1 for NDS and Table 2 for food

consumption. The instruments used for both tables are lagged and represent information that

Table 1
Generalized method of moments. Nondurable goods and services consumption growth by income. First quarter
1984 to third quarter 1996

Consumer expenditures survey

NIPA

NLAG

CG1

CG2

CG3

CG4

CG5

NCG

2

g

a

.1343

(.1017)

.0930

(.0725)

.0018

(.0280)

.0222

(.0162)

.0143

(.0059)

.6982

(.2957)

b

a

.9833

(.0047)

.9831

(.0032)

.9831

(.0018)

.9839

(.0018)

.98367

(.0017)

.9878

(.0049)

J(12)

5.379

[.9441]

14.119

[.2932]

14.256

[.2846]

11.343

[.4997]

11.548

[.4826]

12.477

[.4081]

4

g

a

.0094

(.0073)

.0038

(.0130)

.0061

(.0085)

.0105

(.0088)

.0041

(.0051)

.1330

(.0706)

b

a

.9777

(.0013)

.9826

(.0015)

.9840

(.0015)

.9823

(.0014)

.9804

(.0013)

.9810

(.0019)

J(24)

27.512

[.2812]

21.437

[.6128]

24.836

[.4147]

20.777

[.6519]

23.614

[.4838]

25.023

[.4045]

6

g

a

.0058

(.0060)

.0862

(.0149)

.0094

(.0068)

.0177

(.0052)

.0034

(.0041)

.4128

(.0935)

b

a

.9746

(.0008)

.9708

(.0011)

.9732

(.0011)

.9831

(.0005)

.9772

(.0007)

.9680

(.0012)

J(36)

34.076

37.461

36.257

29.799

30.980

40.046

[.5604]

[.4020]

[.4566]

[.7573]

[.7061]

[.2953]

g is the coefficient of relative risk aversion where economically plausible estimates are suggested to be between
0 and 10.

b is the rate of time preference. Smaller values of b imply that consumers place smaller weight on future

events. J{2(3NLAG + 1)

2} is Hansen’s test statistic for the number of overidentifying restrictions implied by

the model and is chi-square distributed. The term NLAG represents the number of quarters in which the
instruments are lagged. The instruments are log consumption growth, the log risk free rate of return, the log total
return on the S&P 500, and a constant.

a

Asymptotic standard errors are in parentheses. P values for the

c

2

statistic are in brackets; larger values

suggest stronger evidence against the model.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

55

is easily accessible to consumers. They are real consumption growth, the real risk-free rate
of return, the real total return on the S&P 500, and a constant.

11

The coefficients of

constant relative risk aversion, which should lie within the range of 0 < g

10, are

frequently negative, but statistically insignificant. Furthermore, the model appears to show
larger (less negative) coefficients for wealthier consumers (consumers in Quintiles 4 and 5)
in all three lag structures, but are statistically insignificant in most cases. The rates of time
preference, which should be less than 1.0, are consistent with the model and are statistically
significant. The J-statistics, generally fail to reject the overidentifying restrictions on the
model with no real tendency for the model to perform better for higher income consumers.

12

Table 2
Generalized method of moments. Food consumption growth by income. First quarter 1984 to third quarter 1996

Consumer expenditures survey

NIPA

NLAG

FCG1

FCG2

FCG3

FCG4

FCG5

FNCG

2

g

a

.0160

(.0098)

.0280

(.0303)

.0370

(.0224)

.0469

(.0202)

.0049

(.0072)

.2188

(.3791)

b

a

.9834

(.0019)

.9826

(.0020)

.9827

(.0019)

.9826

(.0020)

.9833

(.0014)

.9822

(.0035)

J(12)

9.501

[.6597]

12.140

[.4345]

10.431

[.5782]

9.473

[.6621]

7.368

[.8324]

13.137

[.3592]

4

g

a

.0038

(.0045)

.0063

(.0112)

.0414

(.0152)

.0239

(.0098)

.0049

(.0072)

.0781

(.0955)

b

a

.9832

(.0011)

.9848

(.0012)

.9856

(.0014)

.9799

(.0012)

.9833

(.0014)

.9796

(.0015)

J(24)

21.257

[.6235]

24.435

[.4370]

21.258

[.6235]

21.210

[.6263]

22.681

[.5387]

20.725

[.6549]

6

g

a

.2534

(.0186)

.0675

(.0150)

.0260

(.0084)

.0333

(.0059)

.0015

(.0037)

.2517

(.0236)

b

a

.9452

(.0028)

.9663

(.0012)

.9793

(.0009)

.9796

(.0009)

.9795

(.0009)

.9821

(.0004)

J(36)

37.252

39.000

28.109

28.478

32.029

33.539

[.4113]

[.3364]

[.8233]

[.8097]

[.6580]

[.5862]

g is the coefficient of relative risk aversion where economically plausible estimates are suggested to be between
0 and 10.

b is the rate of time preference. Smaller values of b imply that consumers place smaller weight on future

events. J{2(3NLAG + 1)

2} is Hansen’s test statistic for the number of overidentifying restrictions implied by

the model and is chi-square distributed. The term NLAG represents the number of quarters in which the ins-
truments are lagged. The instruments are log consumption growth, the log risk free rate of return, the log total
return on the S&P 500, and a constant.

a

Asymptotic standard errors are in parentheses. P values for the

c

2

statistic are in brackets; smaller values

suggest stronger evidence against the model.

11

Lags of 2, 4, and 6 quarters are used as in Hansen and Singleton (1982, 1983). NDS consumption growth is

used in Table 1 and food consumption growth is used in Table 2.

12

Since there are more orthogonal conditions than parameters in this model, the J-statistic tests whether all the

sample moments of g(

qˆ

T

) are statistically equal to zero.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

56

Consequently, the model fits the data reasonably well; however, it suggests that con-
sumers are risk neutral and there is no strong pattern confirming the constrained/uncon-
strained hypothesis.

It is worth mentioning that these results are consistent with other studies. For example, our

economically implausible estimates for the CRRA are consistent with other studies such as
Campbell et al. (1997, p. 312) and Hansen and Singleton (1982, 1983 Errata). In addition, our
plausible and statistically significant time preference estimates are consistent with Hansen
and Singleton.

With the NIPA data in Table 1, the estimated CRRAs are positive and statistically sig-

nificant at the 5% level for lags 2 and 6 and significant at the 10% level for lag 4. The time
preference factors are also in line with expectations. The J-statistics generally fail to reject the
overidentifying restrictions and vary considerably across income levels and lag lengths.

The GMM results for the food consumption data from the CES in Table 2 are mixed

overall. The CRRAs are negative and insignificant at the lag 2 specification. For lags 4 and 6,
the CRRAs are positive and significant in several cases; however, these results tend towards
the lower income quintiles, thereby providing less support for the constrained/unconstrained
consumer hypothesis. The

b coefficients, like the NDS results, all fall within the economically

plausible range of less than 1.0. The J-statistics fail to reject the overidentifying restrictions
on the model and show no real pattern supporting the constrained/unconstrained hypothesis.
For the NIPA data in Table 2, the lags 4 and 6 show positive and statistically significant
CRRA coefficients. The

b coefficient is consistent with the model, while the J-statistics fail to

reject the overidentifying restrictions.

Overall, for both the NDS data and the food data, the GMM estimates provide mixed results

for Eq. (3). The parameter estimates for the CRRA are questionable yet the time preference
parameter appears economically reasonable. The overidentifying restrictions tend to show no
real pattern towards confirming the constrained/unconstrained hypothesis. For food consump-
tion, the results across quintiles are not consistent with the constrained/unconstrained
consumer hypothesis especially at lag 6 — the results contradict the hypothesis by showing
positive statistically significant CRRAs in low-income quintiles. The J-statistics indicate only
a slight pattern toward rejecting the model. The NIPA data are much more consistent with the
model’s predictions on a coefficient by coefficient basis and the overidentifying restrictions are
not rejected.

The GMM estimates presented are very sensitive to the timing assumptions made when

specifying consumer behavior relative to financial asset returns. In an effort to sidestep these
potential measurement errors, Table 3 presents estimates of the implied CRRA based on
Eq. (5) for NDS. The constrained/unconstrained hypothesis suggests that the CRRA should
decrease as income increases. While the results tend to show that the implied CRRAs
corresponding to the top two quintiles are smaller than the bottom two quintiles, the
estimates do not decrease steadily in progressively higher income groups. The most notable
exception to the hypothesis is Quintile 3 which produces a negative implied CRRA. While
the overall results for this test are somewhat consistent with the constrained/unconstrained
hypothesis, the inconsistencies across the income groups weaken any strong conclusions that
can be made.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

57

Table 4 presents the implied CRRA statistics for food consumption. Overall, the CRRAs

for the CES data are somewhat more consistent with the lognormal/power utility model (5)
than NDS data in Table 3. The exception is the fourth income quintile with a CRRA
coefficient equal to

83.6. The results shown in Table 4 make it difficult to discern any

patterns across income quintiles that are consistent with the constrained/unconstrained
hypothesis. The two extreme income quintiles, however, with CRRAs of 7.8 and 7.92, are
within the range (less than 10) that Mehra and Prescott (1985) consider economically
plausible. The results for the NIPA food consumption data show an implausibly high CRRA
of 334.6.

The instrumental variable results are presented in Table 5 for NDS and Table 6 for food

consumption.

13

Overall, the IV estimates reject the CCAPM. The instruments used are the

lagged real 3-month T-bill rate, the lagged real total return on the S&P 500, lagged real
consumption growth, and the lagged log dividend to price ratio.

14

The NDS results in Table 5

for both the CES and NIPA data show that the estimated

p coefficients are generally positive

with the exception of the S&P 500 instrument for lags 1 and 2. Since

p

ˆ is the reciprocal of gˆ

(the CRRA), economically plausible estimates range above .1 since smaller estimates of

p

imply larger gˆ. All of the estimates for

p are statistically insignificant so that a meaningful

interpretation is difficult to develop within the context of the constrained/unconstrained
consumer hypothesis. The R

2

statistic measures the joint explanatory ability of the instruments

about the residuals from the first stage. The R

2

statistics should be small since the instruments

should be uncorrelated with the residuals from the IV regression. The significance levels for

13

These tables are designed to be directly comparable to Table 8.2 (bottom), p. 312 of Campbell et al. (1997).

14

These instruments are frequently employed in the CCAPM literature. The logged dividend-to-price ratio is

found in the literature to forecast excess returns (see Campbell et al., 1997).

Table 3
Summary sample statistics for nondurable goods and services consumption growth by income, stock returns, real
returns, and risk premium. First quarter 1984 to third quarter 1996

Consumer expenditures survey

NIPA

CG1

CG2

CG3

CG4

CG5

NCG

SR

RR

RP

Mean

.0006

.0035

.0085

.0092

.0102

.0162 .1137 .0221 .0916

Standard deviation

.2348

.1881

.1861

.1848

.2167

.0244 .2906 .0169 .2890

Correlation (CG, SR)

.0243

.0138

.0223

.1921

.0499

.0814

Covariance (CG, SR)

.0016

.0007

.0012

.0101

.0031

.0006

Implied CRRA

83.6

191.2

111.5

13.2

43.2

223.0

Consumption growth is the annualized change in log real per capita consumption of nondurable goods and
services. Data from the Consumer Expenditure Survey are split into quintiles by income rank, where CG1 is the
consumption growth associated with the lowest income quintile through CG5, the highest income quintile. SR is
the annualized log of the real total return to the S&P 500. RR is the real risk-free rate of return estimated from the
3-month T-bill. RP is the market risk premium. The implied coefficient of relative risk aversion, CRRA, assumes
power utility with joint conditional lognormality of asset returns and consumption, and is computed as
CRRA

¼ ðRP þ :5s

2
SR

Þ=ðs

CG;SR

Þ. See Eq. (5) where CRRA = g and RP = r

i

r

f

.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

58

the test statistic (T

R

2

) are in brackets. The results generally fail to reject the overidentifying

restrictions in many instances at the 10% significance level, but there is no discernable pattern

Table 5
Instrumental variables estimation. Nondurable goods and services consumption growth by income. First quarter
1984 to third quarter 1996

Consumer expenditures survey

NIPA

Return (lag)

CG1

CG2

CG3

CG4

CG5

NCG

T-bill (1)

p

ˆ

.2347

3.8182

1.4019

2.4676

1.7472

.2300

(S.E.)

(3.2664)

(2.4248)

(2.5603)

(2.5883)

(3.0823)

(.3425)

R

2

.1611

.0918

.3190

.0927

.0072

.0114

df = 4

[.1313]

[.4005]

[.0072]

[.3955]

[.9886]

[.9733]

S&P 500 (1)

p

ˆ

.0906

.4816

.5428

.0831

.0697

.0039

(S.E.)

(.5812)

(.4697)

(.5997)

(.4112)

(.4896)

(.0297)

R

2

.1552

.0437

.1432

.1020

.0128

.0200

df = 4

[.1453]

[.7501]

[.1777]

[.3440]

[.9670]

[.9272]

T-bill (1 and 2)

p

ˆ

1.4420

1.7641

2.7430

4.0814

.2023

.2622

(S.E.)

(3.5378)

(2.7584)

(2.9701)

(3.0270)

(3.4757)

(.3974)

R

2

.5078

.2676

.3410

.3763

.6056

.0607

df = 8

[.0111]

[.2356]

[.1019]

[.0658]

[.0027]

[.9676]

S&P 500 (1 and 2)

p

ˆ

.8963

.1266

.2651

.7341

1.1951

.0176

(S.E.)

(.6342)

(.2955)

(.3373)

(.3703)

(.4262)

(.0254)

R

2

.2105

.2809

.2624

.1613

.0713

.0552

df = 8

[.4134]

[.2043]

[.2490]

[.6145]

[.9473]

[.9760]

pˆ is the reciprocal of the estimated coefficient of relative risk aversion and is from the second-stage regression
with standard errors in parentheses; see Eq. (6). Economically plausible estimates of

pˆ are suggested to range

above .1. R

2

is from the regression of the residuals (from the Stage 1 regression) onto the instruments as in Hansen

(1982). The significance levels for the test statistic (T

R

2

) are in brackets; significant levels less that .10 reject the

overidentifying restrictions implied by the model at the 10% level based on a chi-square distribution.

Table 4
Summary sample statistics for food consumption growth by income, stock returns, real returns, and risk premium.
First quarter 1984 to third quarter 1996

Consumer expenditures survey

NIPA

FCG1

FCG2

FCG3

FCG4

FCG5

FNCG

SR

RR

RP

Mean

.0065

.0027

.0070

.0055

.0034

.0085 .1137

.0221 .0916

Standard deviation

.2549

.1756

.1561

.1614

.2022

.0275 .2906 .0169 .2890

Correlation (FCG,SR)

.2368

.0966

.1426

.0345

.2917

.0556

Covariance (FCG,SR)

.0172

.0048

.0063

.0016

.0169

.0004

Implied CRRA

7.8

27.9

21.2

83.6

7.9

334.6

Consumption growth is the annualized change in log real per capita consumption of food. Data from the Consumer
Expenditure Survey are split into quintiles by income rank, where FCG1 is the food consumption growth associated
with the lowest income quintile. SR is the annualized log change in the real total return to the S&P 500. RR is the
real risk-free rate of return estimated from the 3-month T-bill. RP is the market risk premium. The implied
coefficient of relative risk aversion (CRRA) assumes power utility with joint conditional lognormality of asset
returns and consumption, and is computed as CRRA=(RP + .5

s

2

SR

)/

s

FCG,SR

. See Eq. (5) where CRRA = g and

RP = r

i

r

f

.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

59

among the CES data other than most of the rejections that occur for the T-bill instrument with 1
and 2 lags. For the NDS NIPA data, the results for

p

ˆ indicate that they are all statistically

insignificant; however, the low R

2

s and their corresponding high test statistics indicate that the

overidentifying restrictions are not rejected.

Table 6 presents the results for food consumption. For the CES data,

p

ˆ is frequently

negative and in all cases is statistically insignificant. The overidentifying restrictions are
generally not rejected at the 10 percent level for one lag, but are generally rejected for lags 1
and 2. For the NIPA data, again, the

p

ˆ

are small and positive, but statistically insignificant.

The data fail to reject the overidentifying restrictions of the model.

5. Conclusion

Utilizing US aggregate consumption data poses problems for testing the CCAPM since

lower income (constrained) consumers live paycheck-to-paycheck, and therefore do not
influence asset returns directly through savings. Most CCAPM research, however, neglects to
consider this notion. Consequently, this paper seeks to determine whether the CCAPM is
more consistent with the consumption of higher income consumers than with the consump-

Table 6
Instrumental variables estimation. Food consumption growth by income. First quarter 1984 to third quarter 1996

Consumer expenditures survey

NIPA

Return (lag)

FCG1

FCG2

FCG3

FCG4

FCG5

FNCG

T-bill (1)

p

ˆ

1.8677

1.3089

1.4247

1.0239

.0034

.2259

(S.E.)

(3.5515)

(2.4944)

(2.2080)

(2.2906)

(2.854)

(.3767)

R

2

.0700

.0506

.0389

.0260

.1439

.0071

df = 4

[.5445]

[.6945]

[.7887]

[.8873]

[.1756]

[.9891]

S&P 500 (1)

p

ˆ

.1286

.1047

.2230

.0590

.1790

.0271

(S.E.)

(.3025)

(.2439)

(.3105)

(.3364)

(.3659)

(.0676)

R

2

.0749

.0544

.0387

.0292

.1232

.0092

df = 4

[.5098]

[.6641]

[.7901]

[.8643]

[.2467]

[.9821]

T-bill (1 and 2)

p

ˆ

1.0796

1.4443

.5504

.6230

1.4912

.2070

(S.E.)

(4.1422)

(2.8766)

(2.5306)

(2.6312)

(3.2548)

(.4234)

R

2

.4409

.1933

.4890

.4659

.6323

.0399

df = 8

[.0281]

[.4799]

[.0145]

[.0200]

[.0018]

[.9917]

S&P 500 (1 and 2)

p

ˆ

.0042

.1032

.0974

.1212

.0093

.0237

(S.E.)

(.3050)

(.2281)

(.2934)

(.3015)

(.3180)

(.0584)

R

2

.4425

.1874

.4378

.4271

.6291

.0363

df = 8

[.0275]

[.5037]

[.0293]

[.3389]

[.0019]

[.9940]

pˆ is the asymptotic reciprocal of the estimated coefficient of relative risk aversion and is from the second-stage
regression with standard errors in parentheses; see Eq. (6). Economically plausible estimates of

p

ˆ are suggested

to range above .1. R

2

is from the regression of the residuals (from the Stage 1 regression) onto the instruments

as in Hansen (1982). The significance levels for the test statistic (T

R

2

) are in brackets; significant levels

less that .10 reject the overidentifying restrictions implied by the model at the 10% level based on a chi-
square distribution.

H.J. Smoluk, R.P. Neveu / Review of Financial Economics 11 (2002) 47–62

60

tion of lower income consumers. An important aspect of this paper is that it eliminates the
possibility of additional misspecification error by not modeling a liquidity constraint.

Overall, using consumption expenditure data split into quintiles based on consumer

income, this paper shows that the CCAPM behaves erratically across income groups. Since
we have not employed a specific model for the liquidity constraint, the rejections found in this
paper are a function of the data rather than a misspecification of the constraint itself. The
distribution-free GMM tests performed fails to show any discernible patterns across income
groups for both NDS and food consumption, but in general, the data fail to reject the model.
The implied coefficient of relative risk aversion exercise, which assumes that consumption
and asset returns are jointly homoskedastic and lognormal, shows results that are slightly
more consistent with the constrained/unconstrained consumer hypothesis. Any conclusions
based on these results, however, would be untenable since the results are not consistent across
income groups. Finally, the estimated coefficients under the instrumental variables tests are
generally statistically insignificant so that conclusions about consumption patterns across
income groups are absent. Our constrained/unconstrained consumer analysis, therefore, fails
to shed light on the equity premium puzzle.

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