NONDURABLES AND SERVICES RESPONSIVENESS

IN A NONLINEAR MODEL:

MORE EVIDENCE OF BINDING BORROWING CONSTRAINTS

by

Thomas G. Moehrle,

U.S. Bureau of Labor Statistics

(202) 606 - 6900

For comment and review

April 5, 1995
2

ABSTRACT

Recent empirical analyses using the Euler Equation approach have repeatedly

rejected the stochastic implications of the Rational Expectations-Life Cycle

Hypothesis. Several authors have argued that empirical results using aggregated

data are inconclusive because of aggregation bias. Statistical conclusions from panel

studies using food expenditure proxies are suspect as well. This study reexamines

the stochastic implications of the hypothesis using a broader expenditure definition

constructed from the Consumer Expenditure Survey. The study constructs sample

orthogonality conditions from the underlying nonlinear Euler equations, and

thereby relaxes the linear assumptions applied in previous studies. If the error

structure of the model is conditionally heteroscedastic, it is shown that linearizing

can produce bias and inconsistent estimation. For this study, hypothesis tests and

parameter estimates are attained using the numerical solution of the generalized

method of moments minimization problem. The hypothesis-testing methodology is

flexible in that it tests the "pure" Rational Expectations Hypothesis against a

borrowing constraint hypothesis through a split sample technique. Various test

results from this study consistently show that borrowing constraints do affect

consumption growth of financially strapped households. These conclusions are not

as sharp when using food expenditure data alone, however.
3

I. INTRODUCTION

In recent years economists have tested numerous propositions concerning

intertemporal consumer behavior using cross-sectional and panel data sets. Many of

these tests are based on direct estimation of the first-order conditions, or Euler

equations, of the consumer's optimization problem. One common practice used to

simplify this estimation has been linearization of the Euler equations. This

procedure induces two econometric problems that have yet to be addressed. First,

the linear approximation of the underlying nonlinearity may be so poor as to

substantively distort inferences. The second problem is less obvious, but potentially

more harmful. I show in the paper that any conditional heteroscedasticity of the

error in the original, nonlinear, Euler equation is transformed by the linearization

process into a correlation between the regressors and the error of the resulting

linear equation. This induces bias and inconsistency in the least squares and

related methods that have been used to estimate the linearized models. I address

these two problems by directly estimating the original, nonlinear Euler equation

using the numerical solution of the relevant generalized method of moments (GMM)

minimization problem.

An additional concern addressed in this paper is the choice of the consumption

proxy. Cross-sectional and panel data sets of households provide limited

information on consumption, and thereby force researchers to settle on an

approximation of actual consumption. The Panel Survey of Income Dynamics

(PSID) has been a widely used survey. This annual household survey tracks

families over multiple years, collecting a wealth of information including household

characteristics, income, and food expenditure data. Because the expenditure section
4

of the PSID is limited to food spending estimates, intertemporal analyses that use

these data are forced to assume that the utility function is separable between food

consumption and all other consumption items. If the separability assumption is

valid, the econometric tests of the intertemporal theory can proceed. To this point,

no clear evidence has been given to justify this separability assumption.1 To shed

some light on the validity of food expenditures as a consumption proxy, this study

uses data from the Consumer Expenditure Survey (CES) that allows a more

comprehensive measure of consumption by including expenditures on nondurables

and services. Results using this more comprehensive measurement are compared to

the food expenditure proxy estimates. The comparisons presented here are mixed,

but in general do not support the argument that food is an appropriate consumption

proxy within intertemporal models.

The theory of intertemporal optimization of consumers' lifetime well-being has

been the generally accepted explanation of consumers' intertemporal saving and

consumption decisions. According to the theory, consumers will determine their

current levels of consumption based on expected lifetime wealth, while current

income, apart from the new information it brings, is inconsequential. Alternative

consumption models such as the Keynesian-myopic model, which sets current

consumption equal to a proportion of current income, have been strongly refuted

over the years. Still, intertemporal models have been questioned in light of the

repeated failures of empirical studies to confirm some of the fundamental liquidity

restrictions of the hypothesis. Namely, liquidity variables including past levels of

 1 The most compelling argument is that of the limitations to human caloric intake reflected

by the Engel relationship between food expenditures and income.
5

income have not been found to be orthogonal to the expectations error of the

stochastic Euler equations. Frequently, borrowing constraints are blamed for the

orthogonal test failures, particularly in aggregate data studies.

The studies by Zeldes (1989) and Runkle (1991) are of interest here. The two

studies, using PSID data, have explicitly tested the intertemporal model under the

condition that borrowing constraints may be binding for a subset of the population.

Zeldes uses a linearlized versions of the Euler equations and a two-stage least

squares estimator, and finds evidence of borrowing constraint behavior. In contrast,

Runkle (1991), who also fits PSID data to a linearlized model but uses a GMM

estimator, finds no evidence that a fraction of the population is constrained. He

explains that the failure of aggregate data studies to confirm the stochastic

properties of the intertemporal model is the fault of aggregation bias and not of

borrowing constraints. He further contends that micro data studies that have not

econometrically accounted for the potential measurement error in consumption data

have estimated the wrong covariance matrix, and therefore have made incorrect

statistical inferences.

I reexamine the stochastic implications of a nonlinear Rational ExpectationsLife Cycle Hypothesis (RE-LCH) model while addressing the econometric concern of

serial correlation resulting from measurement error. The econometric tests of the

pure RE-LCH are performed by splitting the sample of families in two groups: an

unconstrained and a borrowing constrained subsample. Families are sub-setted

conditional on their financial health. If the pure RE-LCH is to be accepted,

stochastic implications of the hypothesis should hold for each subsample as well as

the full sample of families. The nonlinear Euler equation results, while using the

comprehensive consumption proxy, concur with Zeldes’ findings that a fraction of
6

sampled consumer units are borrowing constrained, and thus the pure RE-LCH can

not be unconditionally accepted for the population. The parameter estimates from

this study, however, are more sharply estimated then those shown in the Zeldes

study. Additionally, the nonlinear fit allows direct estimation of the subjective rate

of time preference, which, along with the standard orthogonality tests of the

stochastic implications, provide a more convincing argument that binding borrowing

constraints are present among a substantial fraction of American consumer units.

Here, the higher rates of time preference estimated from the financially strapped

units reflect the wedge driven between the intertemporal consumer equilibrium

condition. The model is also estimated using the food expenditure proxy and the

results compared. Evidence is offered to suggest that the food proxy is a relatively

noisy series that prevents sharp parameter estimates and conclusive results from

the orthogonality tests.

The paper is organized as follows. The life-cycle model with constraints are

discussed in section II, econometric issues are addressed in section III, the data are

described in section IV, empirical results are presented in section V, and the paper is

concluded in section VI.

II. THE RATIONAL EXPECTATIONS-LIFE CYCLE MODEL

The "pure" RE-LCH describes the intertemporal consumption decisions of

agents with multiple period planning horizons. It is "pure" in the sense that all the

underlying assumptions of the standard intertemporal model remain valid,2 

including the intrinsic assumption that capital markets which facilitate

 2 King (1983) provides a comprehensive survey of the intertemporal theory.
7

intertemporal trade are perfectly competitive. If it is not valid to make these strong

market assumptions, it is important to examine the stochastic implications of the

hypothesis with the perfectly competitive market assumption relaxed.

By splitting the sampled families into two subsamples conditional on the

likelihood of facing a binding borrowing constraint, a distinction can be made of

whether prior rejections of the pure RE-LCH are caused by the explicit assumption

that claim markets are perfectly competitive, when indeed they may not be, or

whether other model assumptions—underlying or auxiliary—are the cause for

rejection.3 

Life Cycle Model with Complete Capital Markets:

Assuming that the utility function is time separable, the standard

intertemporal optimization problem for each consumer is to maximize:

U E UC it it i

t

it it t

T

, , ,, = (,) ∑ = β η 0 (1)

subject to the sequence of wealth constraints:

A r A YC it k it k it k it k it k , , , ,, ( ) ( ), + +− +− + + =+ + − 1 1 1 (2)

along with the deprivation assumption placed on consumption, and the terminal

condition placed on end-of-life assets:4 

C A k Tt it k iT , , and , for ,..., + ≥ =− 0 0 (3)

 3 For instance, Runkle (1991) contends that the auxiliary assumption that preferences are

homogeneous is invalid, and therefore have resulted in aggregation bias.

4 The deprivation assumption is implicitly applied if it is assumed that the marginal utility

at zero consumption is infinitecMUb0g= ∞h. The constraint on end-of-life assets negates

any intergenerational resource allocation. Since this is a period-to-period analysis, this

need not be a concern here.
8

where

Et = mathematical expectation conditional on Ii,t,

Ii,t = information available to consumer unit i at time t,

βi = ith consumer unit's discount rate, or ( ( )) 1 1+ δ i ,

δi = ith consumer unit's subjective rate of time preference,

T = Expected economic lifetime

U(.) = instantaneous utility function (strictly concave),

Ci,t = ith consumer unit's consumption in period t,

ηi t, = ith consumer unit's specific preference shocks,

Yi,t = ith consumer unit's income in period t,

Ai,t = ith consumer unit's nonhuman wealth in period t, and

ri,t = ith consumer unit's one-period ex post real after-tax rate of return.

It is further assumed that the instantaneous utility function is additively

separable in consumption and leisure. The preference shock vector appearing in the

utility function may contain observed and unobserved factors that shift the

instantaneous utility function. Seasonality or changes in the demographic

characteristics of the consumer units are examples of factors that can shift the

function.

It can be shown that the maximization of (1) subject to (2) gives the following

set of standard stochastic first-order conditions:

MU

MU

r e

i t

i t

i it it

,

,

, , ( ) + + −= +

1

1 β 1 1 (4)

where ei,t+1 represents the expectational or forecast error with the property

Ee I it t ( |) , +1 = 0, and MU is the first derivative of U. The forecast error results from
9

new information received between the periods t and t+1 about labor income, the real

after-tax rate of return, and preference shocks.

Life Cycle Model with Incomplete Capital Markets:

Once the perfect competitive market assumption is relaxed, a distinction must

be made between Keynesian-myopic behavior and life-cycle behavior under

borrowing constraints. Although the two behaviors are theoretically different,

observationally each will predict a strong response in consumption resulting from a

change in current income.5 To empirically distinguish the two behaviors, the

sample is split between likely and unlikely borrowing constrained families, and the

intertemporal model fitted and tested for each subsample. If anything is to be

salvaged from the RE-LCH, it should be shown that an identifiable fraction of the

population unconditionally obeys the dictates of the intertemporal theory. If this

fraction consists entirely of identifiable, nonborrowing constrained consumers, it will

in itself provide strong evidence that prior rejections are the result of these

constraints and not Keynesian-myopic behavior.

The perfect competitive capital market assumption is relaxed by introducing

an exogenous borrowing constraint. Specifically, I assume a quantity constraint in

the form:

Ai,t ≥ Bi,t (5)

where Ai,t represents an individual's liquid asset holdings at time t, and Bi,t is an a

priori floor set on the level of liquid assets necessary before the borrowing constraint

 5 See Zeldes (1989) for discussion about differences between testing hypotheses of the

Keynesian model and the life-cycle models with binding borrowing constraints.
10

binds. It is reasonable to argue that Bi,t should be set to zero to trigger the

constraint. This would seem particularly true if consumers face rising interest

schedules that differentiate borrowing rates from saving rates. Whenever current

income falls short of the planned level of consumption, an optimizing consumer will

prefer disinvesting all liquid assets and consuming the proceeds before contracting

for borrowed funds.6 However, for the split-sample test it is not imperative to

exactly identify when the constraint is triggered. What is necessary is a criterion

that will identify potential borrowing constrained units so as to purge them from the

full sample of families under investigation. Consequently, we can allow Bi,t to take

on positive values, while identifying borrowing constrained units who, at any

particular time in the sampled period, may hold some liquid balances for

transactions.

Including the additional exogenous borrowing constraint into the standard

intertemporal optimization problem of (1), the following set of augmented stochastic

Euler equations can be written:

MU r

MU

e

it i it

i t

it it

, ,

,

, ,

( ) ( ) +

+

+

+ −= ′ 1

1

1

1 1 β λ , (6)

 6 Although the constraint is assumed exogenous in this study, an endogenous constraint

would seem more appropriate. The exogenous constraint implies that consumers have no

power in which to influence lending institutions on how much they can borrow, or at what

rate they are charged. In the long-run, this assumption is difficult to justify when many

loan approvals are contingent on an individual's credit and earnings history. Perhaps the

most appropriate method to enter an endogenous constraint is through a variable rate of

interest that is a function of a consumer's debt to asset ratio and loan default history.

However, such information is not available from the data source I use. The Survey of

Consumer Finance of the Federal Reserve Board is a rich source of data on household debt

and assets that provide such information. Unfortunately, the SCF does not collect

consumption information. In the short-run, however, consumers have little if any affect on

their financial health, and thus the assumption of an exogenous constraint would seem

reasonable.
11

where λi t, represents the lagrangian multiplier associated with the borrowing

constraint. This quantity constraint is similar to those imposed by Zeldes (1989),

Runkle (1991), and Whited (1992). The multiplier has the standard economic

interpretation. If the constraint in period t is binding ( ) λi t, >0 , then its value

represents the increase in lifetime utility given a one dollar decrease in the current

borrowing constraint.

III. ECONOMETRIC ISSUES

Econometric Specification:

To make the problem tractable, it is necessary to impose a preference

specification on equations (4) and (6). It is assumed that preferences can be modeled

by the constant relative risk aversion utility function:

Cit it , , exp( ) 1

1

−

−

α η

α

(7)

where α is the Arrow-Pratt parameter of relative risk aversion assumed equal across

the sample.7 The specification is flexible enough to permit observable and

unobservable household-specific preference shift variables to enter the Euler

equations through the vector ηi t, .8 Allowing the subjective rate of time preference

to vary across households and incorporating the preference specification into (4) and

(6), the standard nonborrowing constrained Euler equations become

 7 The constant relative risk aversion utility function has been widely used in similar

studies including those by Mankiw (1981), Hansen and Singleton (1982), Shapiro (1984),

Zeldes (1989) and Runkle (1991).

8 In this particular specification, I follow the leads of Hayashi (1985c) and Zeldes (1989).
12

C

C

r e

i t

i t

i it it it it

,

,

, ,, , ( ) exp( ) +

−

+ +

F

HG IKJ + − −= 1

1 1 1 1

α

β ηη , (8)

and the borrowing constrained Euler equations become

C

C

r e

i t

i t

i it it it it i t

,

,

, ,, , ( ) exp( )( ),

+

−

+ +

F

HG IKJ + − − −= ′ 1

1 1 1 11

α

β ηη λ (9)

The variables of the preference shift vector ηi t, may include both observable

and unobservable variables that can vary across families and time. The observable

variables include the age of the reference person9 (AGER) and the size of the family

(FS). The unobserved factors include a family component that is fixed across time

(FCi), an aggregate component (ACt) that is constant across families but varies

across time, and a white noise component (ui,t) that is orthogonal to the other

unobserved factors. ui,t is assumed to have the properties:

E u

Eu u i jt s

i t

it js u

,

, , , if ,

otherwise.

=

= ==

=

0

0

2 σ (10)

The representation of ui,t is broad. It will capture a shock to the utility function or

will reflect white-noise measurement error in the consumption data. At this

juncture, the precise cause of this white-noise term—preference shock or

measurement error—is inconsequential when addressing the econometric issues of

the Euler equations. After collecting these terms, the preference shift vector

becomes

 9 The reference person is loosely defined as the head of the household. It is the first person

mentioned when asked to "Start with the name of the person or one of the persons who owns

or rents the home."
13

ηit i t it = a AGER a AGER b FS FC AC u ′

0 1 + + +++ 2 (11)

To account for the aggregate component ACt, quarterly time dummy variables

(QTRt) are included. The most obvious aggregate variation in quarterly data that

these time variables will capture is seasonality.10 Notwithstanding the seasonal

correlation, these time variables capture other macroeconomic changes that are

unique to each quarter—but constant across the sample—that the real after-tax

rate of interest fails to capture. Consequently, the inclusion of these time variables

ensure that the expected value of the Euler equations’ residual vector has mean zero

for the short panel sample. Incorporating this information into (8) and (9), the set of

equations to estimate become

C

C

r

d QRT a AGER b FS u u e

i t

i t

i t

t t it it t it it it

,

,

,

, ,, , ,

( )

exp( )

+

−

+ +

F

HG IKJ +

∑ + + + − −=

1

1 1

1

1

α

β

∆

(12)

C

C

r

d QRT a AGER b FS u u e

i t

i t

i t it

t t it it t it it it

,

,

,

, ,, , ,

( )( )

exp( )

+

−

+ +

F

HG IKJ + +

∑ + + + − −= ′

1

1 1

1 1

1

α

β λ

∆

(13)

where ∆FS represents the change in family size.

By taking first differences, note that the potential family fixed time effect

component FCi cancels out. However, the unobserved white noise difference

( ) , , u u it it +1 − remains within these fitted Euler equations, and if present, produces

serially correlated residuals. In addition to this specification, some authors have

 10 The CES collects expenditure information from households who are asked to report

expenditures for the three-month period preceding the month-of-interview. Since this

three-month “collection period” may not coincide with a calendar quarter, applying

estimated seasonal factors is not straightforward.
14

suggested an additional unobserved term that captures persistent household-specific

effects.11 A persistent household effect would include, for example, differences in

the subjective rate of time preference among sampled households. In this instance,

the unobserved persistent term, denote it as ψ i , enters directly into the Euler

equations. Note that ψ i differs from FCi in that it enters into the change in

preferences and not the parent preference vector. As will be discussed later, no

evidence of persistent household-specific effects appeared in the samples examined,

and thus ψ i is not considered in the final estimation.

Conditional Heteroscedastic Error in the linearlized Model:

It may be fruitful at this point to compare the nonlinear estimation of (2) with

the frequently estimated log linear form. The log linear growth equations are

derived by rearranging (12), taking logs on each side of the equations, and

approximating log (1+x) by a second-order Taylor expansion. This produces the

standard log linear consumption growth equation:

ln ln( ) C

C

a rv t

t

o tt

+

+

F

HG IKJ=+ ++ 1

1

1 1 α (14)

 11 See Runkle for the econometric issues involved when persistent household effects are

present.
15

where

a

ve e

Ee I

E e

e

t t et

t t

e t

0

1 1

2

2

1

1 1

2 1

2 1

2

2

1

1

2

1

2

0

= +

= −−

=

=

++ +

+

+

α

α

β σ

σ

σ

(ln ),

( ),

( |) ,

( ).

Note that the log linear Euler equation's residual vi t, +1 becomes

1 1

2 1

2 1

2

2

α σ 1 e e c t ei t + + − −; h, where σ e i;

2 is the variance of the expectational error indexed

for individual i.

12 Conditional heteroscedasticity will emerge if σ σ ei e j ; ;

2 2 ≠ , when

i j ≠ . Hence, linearizing the model forces the identification assumption that

instrumental variables are uncorrelated with this heteroscedastic variance term

residing within the residual.

How valid is this identification assumption? It is easy to argue that the level

of uncertainty about consumption growth rises as the level of household wealth

declines. For instance, less wealthy families who lack financial security are more

uncertain about their futures than more wealthy families, and consequently we

would expect the error variance to increase with the decline in wealth. Other

economic and demographic variables may also play a part in the heteroscedastic

variance. Families headed by younger, less established persons may indeed be more

uncertain about their future than established families headed by middle aged

persons, and consequently the error variance may decline as households mature.

Certainly income will be correlated with the heteroscedastic variance term.

Families with higher incomes will have more budgetary discretion with respect to

consumption and saving decision which will result in larger error variance for higher

 12 The individual indexing is dropped from the other terms to reduce notational clutter.
16

income families. With these seemingly obvious correlations, it is difficult to justify

that the conditional heteroscedastic variance term of the expectations error is

uncorrelated with time t variables. In the presence of the conditional

heteroscedastic variance, the parameter estimates of the linearlized models will be

both biased and inconsistent. To avoid this pitfall, I abandon linear approximations,

and estimate the set of nonlinear Euler equations using a GMM criterion function.

Econometric Procedure and Tests:

To estimate the nonlinear Euler equations, I use the GMM technique

introduced and described in Hansen (1982) and Hansen and Singleton (1982), where

the optimal weighting matrix of the GMM minimization problem follows Newey and

West (1987). In short, the GMM estimator constructs sample orthogonality

conditions implied by the rational expectations hypothesis: Ee x c hit it , , +1 = 0 where xit

is a vector of instrumental variables. These variables are a subset of the variables

contained within the consumer's information set at period t (It). The GMM

estimates of the parameters -α and β along with parameters of the preference shift

vector are those that numerically minimize the weighted sum of squares:

e x We x i t i t it i t i t ,, ,, + +

′ ch ch 1 1 where Wit is an optimally chosen weighting matrix

computed from the covariance matrix of e x c hit it , , +1 .13 

 13 Gallant (1987) provides a SAS illustration of the procedure.
17

Econometric Tests:

The orthogonality tests performed here are twofold. First, I test the pure RELCH for the full sample as well as for each subsample by fitting the Euler equations

of (12) and (13). Second, a more traditional and direct test for binding borrowing

constraints is conducted by determining the statistical significance of the change in

lagged income in each of the samples. Each test is described below.

Test of the Pure RE-LCH:

To accept the pure RE-LCH, the overidentifying restrictions must hold

unconditionally for any subsample drawn from the population. The fitting of the

Euler equations of (12) to the full sample and each split subsample should produce

similar parameter estimates and the model restrictions placed on the data should be

satisfied. The fitting of the two subsamples, conditional on liquidity holdings,

distinguishes whether a rejection of the hypothesis is the fault of borrowing

constraints or by other model assumptions. If the model assumptions remain intact

along with the auxiliary assumptions for the unconstrained subsample but the

perfectly competitive financial market assumption is invalid for the borrowing

constrained subsample, we would expect economically plausible parameter

estimates and an overall acceptance of the model restrictions for the former, while

rejecting for the latter. In contrast, a rejection of the hypothesis for any subsample

drawn would suggest Keynesian-myopic behavior, and the intertemporal theory of

consumption behavior would be unconditionally refuted. There are two steps in the

hypothesis test. First, the rational-expectations model is fit to data from the

subsample of financially healthy families—the unconstrained group. If the model is

accepted for this subsample, then it is reasonable to apply the same auxiliary model
18

assumptions to any defined sample of the population including the subsample of

borrowing constrained units, and perform the hypothesis tests. A rejection of the

model for the borrowing constrained subsample provides strong evidence that

binding constraints are present and affect intertemporal choice.

Test of the Borrowing Constraint Hypothesis:

If borrowing constraints are binding, the correctly specified Euler equations

include the positive lagrangian multiplier λi t, associated with the constraint as

presented in (13). Although the CES data do not permit explicit modeling of λi t, , the

data do allow an indirect test for the presence of the positive λi t, . If the borrowing

constraint is binding, λi t, is strictly positive and negatively correlated with

household financial variables dated in period t. It follows then that the composite

residual term in ei t,

'

+1 (13) will be correlated with lagged financial variables if the

constraint binds. It is straightforward to use the orthogonality test between the

Euler residuals and the instrumental variables that include household financial, or

liquidity, information. That is, any liquidity variable dated in period t or earlier

that relaxes the consumer unit's borrowing constraint will in turn not be orthogonal

to the multiplier λ residing within the composite error term. Consequently, liquidity

variables that place additional restrictions on the sample data will not be

statistically validated if a borrowing constraint is binding.
19

IV. DATA DESCRIPTION AND SAMPLE SPLITTING CRITRIA

The data are drawn from the quarterly Interview portion of the 1990-91

CES.14 The Interview survey collects data from detailed expenditure questions on

approximately 60 to 70 percent of total household expenditures and the remaining

expenditures are collected using global type questions. Because these data are

collected over a three-month period instead of the 12 month PSID method,

measurement error resulting from recall is comparatively low.15 The Interview

survey also collects household characteristic, income, and asset and liability

information. Unlike the three-month expenditure collection method, however,

income data collection requires recall of the prior 12 month period. These questions

are detailed and exhaustive in that they collect all sources of income, whether they

are received on a regular basis, such as wages and salaries, or an irregular basis,

such as proceeds from sale of assets or lottery winnings.

Most data on asset and liabilities are collected in the last interview. The

majority of these questions target the account balances as of the prior month of the

interview, while some alternative questions collect changes in these account

balances over the prior year. Although the asset and liability questions are detailed,

they are not collected frequently enough over the interview period to provide data

for quality panel data analysis of asset-holding behavior.16 The lack of asset and

 14 For a complete description of the Consumer Expenditure Survey, see “BLS Handbook of

Methods,” Chapter 18.

15 PSID data attained annual estimates by extrapolating from a weekly average food

expenditure estimate. Hence, any recall error is magnified.

16 An additional shortfall is that unrealized capital gains are not collected nor are values of

assets that were acquired before the 12 month period preceding the interview. One
20

liability information tracked over the complete four quarters represents the greatest

shortfall of the CES data for life-cycle analyses. For this study, asset levels are

solely used as a measure to select families for split-sample analysis. There is a

caveat. Because these questions determine asset levels in the last of the four

quarters of interview, using these data will risk violating the orthogonality

conditions implied by the rational expectations hypothesis. For consistent

parameter estimates, I assume that the asset levels of the unconstrained group are

relatively stable over the interviewed periods, and known by the consumer unit for

that period. This assumption allows me to use these data to split the sample.

Sample Selection:

The Interview portion of the CES provides as many as four quarterly

interviews per participating consumer unit. Two waves of families are selected for

this study. The first eligible wave consists of families who participated in their first

quarterly interview in the first calendar quarter of 1990, and completed their fourth

and final quarterly interview in the last quarter of 1990. The second wave

experienced their first quarterly interview in the second calendar quarter of 1990

and their last interview in the first quarter of 1991. Placing these two waves

together cover the consumption periods from last quarter 1989 through last quarter

1990. Families are selected for analysis if they participated in four complete

quarterly interviews during this time frame.

Additional restrictions are also applied. Excluded are families headed by

individuals under the age of 25, as well as units who did not provide complete

 

exception are the values of real-estate properties. Respondents are asked to approximate

that value each property would sell for on “today’s market.”
21

reports of their incomes. The under 25 age group are excluded since these consumer

units are characterized by sporadic reporting of income and expenditures. The

sporadic reporting in part may be explained by parental support for these young

households. The exclusion of incomplete income reporters is self-evident. If the goal

is to test the sensitivity of consumption to changes in income, a full measure of

consumer unit income is necessary.17 

This initial selection provides a sample of 1,518 unique families. To reduce the

effect of expenditure and income reporting error, an additional 226 units who had

consumer unit members self-employed are excluded.18 I conjecture here that the

line between consumption and business expenses becomes more blurred for the selfemployed, and thus reducing the accuracy of expenditure reports. Four additional

units are deleted in the sample because of nonpositive expenditures.19 This left a

full sample size of 1,288 unique families, each providing three quarterly expenditure

growth records.

Sample Splitting Criteria:

In order to perform the split-sample tests, the full sample of 1,288 families

must be subdivided into the unconstrained and the borrowing constrained groups.

 

17 Incomplete reporting of income is determined by criteria set by the Bureau of Labor

Statistics when processing income data. An example of an incomplete income reporter is

one who reports receiving wage or salary income but who does not report the income

amounts. Currently, BLS does not make any income imputations on missing data.

18 A consumer unit is identified as self-employed if any of its members received selfemployed income during the interview period.

19 Because medical expenditures are measured as net out-of-pocket costs, a consumer unit

may have zero, or negative expenditures because medical expense reimbursements exceed

all other expenditures.
22

To split the samples, I use the ratio of liquid assets to normal income. This criterion

is chosen because it closely characterizes borrowing constrained units described

within the life-cycle hypothesis. Liquid assets are defined as the sum of checking

and savings account balances, U.S. government savings bonds, and the estimated

market value of marketable securities.20 I define normal income as that part of

income that is received on a regular basis. This includes wage and salary, interest

and rent, pension, unemployment compensation and other such income sources. If a

consumer unit has liquid assets equal to two months worth of annual income, the

consumer unit is selected for the unconstrained subsample. In the empirical

analysis, I allow minimal asset holdings to ensure that only those consumers who

are selected for the nonborrowing constrained group are indeed unconstrained. This

follows since some consumers, constrained or not, hold some liquid assets to make

efficient day-to-day transactions. By this criterion, there are 430 families in the

unconstrained group and 850 families in the constrained group.21 

Proxy for Consumption:

The preferred consumption argument within the Euler equation would contain

expenditures on nondurables and services, as well as the service flow received from

the stock of durables. Although the CES provides expenditure reports on practically

 20 Marketable securities include stocks, non-government bonds, mutual funds and other

such securities. This liquidity information is collected at the last interview. The

respondents are asked to give accounts and market values as of the last day of the previous

month of interview.

21 This suggest that as much as two-thirds the sample is constrained. This sample

proportion of two-thirds constrained is close to what Zeldes finds in his PSID sample.

Although the sample proportions are only suggestive, they are in large disagreement with

the proportional estimates of 20 percent found by Hall and Mishkin (1982).
23

all consumer items, it does not provide the array of information necessary to

compute service flow values received from the stock of durables. Here it is assumed

that durable consumption is separable, within the utility function specified in (1),

from nondurables and services.22 In this paper, comparisons are made using a

comprehensive measure that includes nondurables and services and one that

captures total food and beverage expenditures. Results from these two different

proxies are used to gauge the effectiveness of the food proxy, that has been used in

many micro studies, to mirror changes in total consumption. A complete description

of the consumption proxy variables and other variables of the estimation are

provided in the data appendix.

Instrumental Variables of the Sample Orthogonality Conditions:

The GMM criterion function of the rational expectations model requires a

vector of instrumental variables. These variables should be carefully selected so as

to capture a subset of the CU’s information set that it uses to predict consumption.

Liquidity variables, such as income, will be among the list of valid instrumental

variables if and only if consumer units are unconstrained from optimally borrowing.

To test liquidity responsiveness, two sets of estimates are performed using two

alternative instrumental variables lists. The first list—call it the standard

instrumental variables list—includes non-liquidity variables. These variables

include the number of weeks worked and hours per week worked for the reference

person and spouse, the age of the reference person and spouse, the number of

persons in the consumer unit, the change in the number of persons in the consumer

 22 This follows the lead of Hall (1978), Mankiw (1981), Flavin (1985) and others.
24

unit, quarterly time dummies, and real after-tax rate of returns lagged one, two and

three periods.23 The employment status data and the family composition data are

lagged one period to avoid violating the implied orthogonality conditions. These

variables are relatively well measured and are similar to the set used in both the

Zeldes and Runkle studies. Note that because the ex post rate of return is not

known when forecasts are made, it may be correlated with the forecast error.

Hence, lagged values of this variable are used as a substitute in the instrumental

variables list. Recall that I am using the ex post real after-tax rate of return, and

thus the real market rate of return must be adjusted by family-varying marginal tax

rates. Because these marginal tax rates are not provided from the data source, they

have been computed from information on income and household composition

characteristics. See the data appendix for a description of this computation.

To test the significance of liquidity variables, I augment this standard

instrumental variables list by the change in lagged income.24 Call this list the

augmented instrumental variable list. Income measured here is from all sources—

normal and transitory.25 For this study, transitory income includes lottery

winnings, income from sale of assets, tax refunds and alike. I call these transitory

because they are irregularly received income components. This comprehensive

measure is used to capture any liquidity effects that may relax a binding borrowing

 23 Because the reference person is not necessarily the head of the household, it is

important to use information about the spouse when using CES data.

24 This follows Flavin's (1985) liquidity constraint test where she used unemployment

within her aggregate data study as well as the orthogonal tests of Hall (1978) and those who

followed.

25 As a BLS employee, I have been provided the non-censored data. Because of

confidentiality restrictions, the public-use micro data are censured, or what BLS calls

topcoped. The topcoding rules apply to some expenditures and all income data.
25

constraint. There is one caveat to the income measurement. Income for a

participating consumer unit is measured twice: once at the first quarterly interview

and again at the fourth quarterly interview. At both of these interviews, the

respondent is asked about all sources of income received for the 12 month period

preceding the interview. Because there is this partial overlap in income and

expenditure collection, the orthogonality condition can not be strictly guaranteed

here. That is, income received over the recent months of interview could contain

new information about lifetime wealth that induces a change in the level of

consumption. However, I regard any new information embedded in income to be

known at the beginning of the collection period. This strong assumption permits the

construction of the sample orthogonality conditions to proceed. Based on the

orthogonal tests from the unconstrained sample, I find no evidence to invalidate this

assumption.

V. EMPIRICAL RESULTS

We know from Section III that the presence of either unobserved preference

shift shocks or measurement error can produce serially correlated errors. Presented

in Table 1 are the autocorrelations of residuals $, ei t computed from the Euler

equations fitted to the full sample. Here the only statistically significant

autocorrelation is the first autocorrelation, which in turn supports the earlier

assumption that the Euler equations' errors follow an MA(1) process.26 

Additionally, this MA(1) process supports the assumption that persistent householdspecific effects are not present within these data. If persistent effects are present

 26 The autocorrelations were computed from the unconstrained subsample, and the

MA(1) process was accepted for this sample as well.
26

but not captured by the model, the serial correlation of the residuals would extend

across several periods, and not completely dampen after just one. With this finding,

the weighting matrix Wit of the GMM criterion function is constructed with an

MA(1) error process.

Tables 2 through 4 present the GMM results from the full sample as well as

the unconstrained and borrowing constrained subsamples. In these tables, the

GMM results are shown for the two consumption proxies considered: nondurables

and services (CNDS), and food and beverages (CFD). For each of these proxies,

GMM estimates are computed from the standard and the augmented (liquidity)

instrumental variables lists. Each column contains the chi-square test statistic, and

beneath these test statistics, the corresponding p-value is shown.27 It is this test

statistic which is used to judge the acceptability of the overidentifying restrictions of

the model.

Model Fit for the Pure RE-LCH:

To test the pure RE-LCH conditional on the liquid asset holdings, equation

(12) is fitted to the unconstrained subsample using CNDS. Although this is a

conditional test of the hypothesis, I use this subsample first to confirm the auxiliary

and underlying assumptions of the rational expectations model. Given this model is

accepted, I can apply the same model to sample data where binding constraints are

suspected. I show the unconstrained GMM results for the CNDS proxy in the first

column of Table 3. The model results from the standard instrumental variables list

are accepted based on both the satisfaction of the overidentifying restrictions and

the plausibility of the parameter estimates. With the chi square of 3.4, the

 

27 The p-value is the computed probability that a Xdf

2 random variable is greater

than the computed test statistic under the hypothesis that the overidentifying

restrictions hold.
27

overidentifying restrictions are accepted at the 1, 5 and 10 percent levels of

significance. Moreover, the preference parameter β is estimated at 0.9807 which

implies a subjective rate of time preference of just under 2 percent. Based on the

average annual return for three-month T-bills in 1990, the real rate of return

equaled just over 2 percent. Confidence is placed in this estimate if it is reasonably

within the neighborhood of the real rate of return. The preference parameter -α is

estimated at -0.1741, which translates to an Arrow-Pratt estimate of 0.17. This

estimate is close to but less than the parameter estimate found by Hansen and

Singleton (1982), and its value implies the standard risk aversion and concavity of

the utility function. I compare this with the results from Hansen and Singleton

since they too fit a similar nonlinear model. The Arrow-Pratt parameter estimates

of this study are lower than the range of estimates shown in the Zeldes and Runkle

studies.28 Both of these preference parameters of this study are statistically

significant.

With an accepted model with which to work, I proceed to test the pure RELCH unconditional on the sample selected. I do this by fitting the full sample of

families to the Euler equations of (12) with CNDS as the proxy. The GMM results

are shown in Table 2. For the standard instrumental variables list, the

overidentifying restrictions are clearly rejected for this sample, although the

preference parameter estimates remain plausible. The subjective rate of time

preference is approximately 2.7 percent which is a slightly higher rate of time

preference than the strictly unconstrained sample. If the model rejection is brought

about by a subsample who are constrained, this higher rate of time preference is

 28 The Arrow-Pratt parameter estimate ranged from -0.7 to 2.7 in the Zeldes study

and 1.4 to 4.4 in the Runkle study.
28

expected.29 The parameter -α is estimated with the appropriate negative sign, but

the degree of risk aversion is slightly less than that found in the unconstrained

sample, 0.1421 compared to 0.1741. Both preference parameter estimates are again

found statistically significant.

By rejecting the pure RE-LCH for the full sample, I focus on the cause of the

rejection by fitting the borrowing constrained sample to the same model. These

results are presented in Table 4 and show that the overidentifying restrictions are

overwhelmingly rejected by the reported chi-square of 75.3. For this fit, the

preference parameter estimates are clearly implausible for the pure RE-LCH. Here,

-α is estimated with the wrong sign which suggests that the utility function is not

concave. However, this parameter estimate is found statistically insignificant.30 As

a comparison, the parameter estimates in both the Zeldes and Runkle studies are

statistically insignificant for this group as well. The estimate for β is statistically

significant, but the subjective rate of time preference is the highest among the three

estimates—just under 4 percent. This rate is substantially above the average

annual market rate of return during this period, but it is not unexpected when

binding constraints are present.31 The higher rate of time preference reflects the

 29 For instance, if consumers are constrained today, then the constraint forces a

wedge between today's and tomorrow's level of marginal utility. Hence, an

optimizing, but constrained, life-cycle saver will discount future consumption more

substantially.

30 For the model fit of the borrowing constrained group (selected by the ratio "total

wealth to income"), Zeldes finds similar results. His Arrow-Pratt estimate is -0.68,

though statistically insignificant.

31 Varying rates of time preference are examined by Lawrance (1991) and Moehrle

(1994). Lawrance finds higher rates of time preference when she classifies a

selected PSID sample by income percentiles. Lower income families have higher

rates of time preference than higher income families. Her results are interesting in

that she finds no evidence of borrowing constraints among the low income even

though they exhibit significantly higher rates of time preference. Moehrle finds

varying rates of time preference across age groups, and this variation in rates is

substantially greater for low liquidity wealth families.
29

wedge forced between the standard intertemporal consumer equilibrium when a

binding borrowing constraint is present. Because this time preference rate is not

available from the linearized models of Zeldes and Runkle, no comparisons can be

made with these earlier panel studies. Nevertheless, the higher rate is consistent

with what would emerge in a borrowing constrained environment.

Results Using the Food Consumption Proxy

The rejection of the model for the full and borrowing constrained sample

strongly supports the alternative hypothesis of borrowing constrained behavior. But

before proceeding, it would be convenient at this point to compare these results

using CNDS with those from CFD. The comparison provides a tool to judge the

reliability of prior micro data studies using food expenditures as the consumption

proxy. The testing procedure follows the same steps as before, and the GMM results

from the standard instrumental variables list are shown in the third columns of the

same tables.

The conclusions run parallel to the CNDS results: the overidentifying

restrictions are accepted for the unconstrained while rejected for the full and

borrowing constrained samples. The preference parameters are not as sharply

estimated, however. Except for the borrowing constrained sample, the preference

parameter estimates are smaller than those from the CNDS fits. The summary

statistics of these data do reveal that the sample variance is much larger for CFD

than CNDS, and this may contribute to the looser fit.32 Table 5 shows the sample

statistics for both these proxies across the three samples examined. Notice that for

 

32 This may also help to explain why there are large differences among the

parameter estimates in both the Zeldes and Runkle studies. See footnote 28 for the

range of estimates presented in each.
30

each sample the standard error is larger for the CFD variable than it is for CNDS.

These larger standard errors are also accompanied by larger mean growth rates,

particularly for the borrowing constrained group. This higher variability in the food

sample data may reflect either the survey-respondent-recall error that is associated

with global questions, or a greater sensitivity of food expenditure growth to current

income changes. The latter suggestion obviously weakens the life-cycle theory.

These issues are addressed later in this section.

Tests for the Presence of Borrowing Constraints:

The previous test results using either nondurables and services or food

consumption data suggest that binding borrowing constraints are present among

some families and therefore influence consumption growth. Thus for constrained

families, the multiplier appearing in (13) is positive and reflects the increase in

lifetime utility if the borrowing constraint is relaxed by a one dollar unit. With the

presence of this positive multiplier within the rational expectations model, a

straightforward empirical test for binding borrowing constraints is enabled. If the

constraint is binding, any time period t financial variable that relaxes the constraint

will in turn be correlated with the multiplier, and correspondingly with the

composite residual in which the multiplier resides. Hence, a more direct test for

binding borrowing constraints is a test of whether the change in last period's income

serves as a valid instrumental variable for the fit of the nonlinear model. If

constraints are binding, changes in lagged income will influence consumption

growth by its direct affect on lagged consumption. For a constrained consumer unit,

any increase in income will be immediately and fully used for consumption.33 If

 33 Any lagged liquidity variable that relaxes the constraint is likely to be correlated

with the lagrangian multiplier λi t, and thereby correlated with the composite

residual. For the liquidity variables available from the CES, changes in lagged

income is the only variable best measured for this test.
31

binding constraints are present, this additional liquidity restriction placed on the

sample data will be outright rejected. The results of this test using the augmented

variable list are shown in the second columns of tables 2 through 4. This additional

model restriction which was "excluded" from the standard instrumental variable list

is shown in the table row titled "Chi Square: Exclusion restriction." For the CNDS

fits, the liquidity restriction is satisfied for the unconstrained sample, while it is

clearly rejected for the full and borrowing constrained samples. These results are

consistent with what are found using the standard instrumental variables list, and

further confirm borrowing constraints as the cause of the model rejections for the

full and borrowing constrained samples.

The liquidity restriction test is also performed on the consumption proxy,

CFD. With the previous results in hand, the CFD results suggest that food

expenditures are not an appropriate proxy for consumption. The exclusion

restrictions are not satisfied for the full or unconstrained sample, but are oddly

satisfied for the borrowing constrained sample. Although this is completely in

opposition of what would be expected with borrowing constrained behavior, it is not

surprising given the consumption proxy used. I offer two explanations for this

contradicting result.

First, food expenditures reveal a much greater variability in the sample data

than do the composite expenditures, nondurables and services. Looking at table 5,

the sample mean growth in food expenditures was just under 16 percent (1.159

growth rate) with a standard error of 2.24. In comparison, the growth for

nondurables and services was much less, at 7 percent (1.072 growth rate), and the

standard error , 0.39, was substantially lower. Certainly the relatively large growth

rate of food spending alone is bothersome, but coupled with the size of its standard

 
32

error casts this as a questionable series within an intertemporal model. It is then

an understatement to say that reported food expenditures are a noisy series, and it

would appear that most of this noise is attributed to measurement error.

The cause of the measurement error is by no means elusive. Recall from the

previous section that the majority of expenditure reports from the CES are collected

from detailed expense questions, while food expenditures are collected from global

questions.34 By the simple nature of global questions, survey respondents will be

less accurate in their responses about actual food spending than they will about

expenditures collected from detailed questions. Global questions ask respondents to

estimate their usual expenses over a specified period of time, while detailed

questions ask respondents to recall specific expense items and the period in which

they were made. Because the latter does not require estimation on the part of the

respondent, the quality of the data from detailed-question responses are much

higher.

This large period-to-period variation in observed food expenditure reports is

not unique to the CES. Runkle noted large variation in PSID food expenditures

which led him to smooth his data by selecting only those families who had estimated

expenditure growth that did not exceed 300 percent or declines of more than 75

percent. Even after this smoothing, however, Runkle estimated that as much as 76

percent of the variation in PSID food spending growth resulted from measurement

error. It also should be noted that with the food expenditure data, any small

reporting error is magnified since these data are collected as weekly estimates in

both the CES and PSID, and then multiplied to either obtain quarterly or annual

 34 The global questions for food expenditures begin by asking respondents " ... what

has been your usual weekly expense at the grocery store or supermarket."

Additional questions are asked to determine the food amounts for the weekly

expense estimates. See the data appendix.
33

estimates. In light of the contradicting results between Zeldes and Runkle and the

results presented here using CFD, the extremely noisy nature of household survey

food expenditures appear to substantially cloud the borrowing constraint hypothesis

tests of intertemporal models.

As a second explanation for the contradicting borrowing constraint results,

Hall (1978) concluded that intertemporal maximizing consumers may not optimally

react to market signals instantaneously, but rather may delay the full adjustment

when receiving the new information about lifetime wealth. This adjustment delay

may come about from difficulties experienced by some consumer units to interpret

the economic news. If we accept this hypothesis, it then follows that the more

uncertain consumers are about market signals, the more likely their observed

behavior will appear myopic. Hence, Hall's hypothesis coupled with the

contradicting results between CNDS and CFD implies that the changes in some

components of total consumption may not be solely determined by new information

received about permanent income but by current income levels. Eating out at

restaurants may be an example of one of these components. Putting aside the high

variability in food expenditures for the moment, changes in expenditures on food

away from home, an easily adjusted and high income elastic expenditure, may more

closely track the changes in current income than permanent income.35 If consumers

become less certain of their future prospects, they may quickly adjust these types of

expenditures to hedge against an unpredicted and large fall in liquidity and

consumption. Hence, there may be a strong precautionary savings motive by leery

consumers. If this scenario is true, then the RE-LCH can be best described as

loosely explaining consumption behavior. Given the high variability of food

 35 Published reports from the Consumer Expenditure Survey of BLS do show that

the variation in food away from home expenditures classified by income level is

much more volatile than food at home expenditures.
34

expenditures, however, the empirical findings here that show food expenditures

excessively sensitive to changes in lagged income is more suggestive than evidence

against the model.

VI. SUMMARY

This paper contributes to the voluminous amount of theoretical and empirical

thought that has gone into the study of intertemporal consumption choice. To say

the least, there has been an evolution of analysis in regard to modeling and testing

the fundamental stochastic assumptions of the theory reaching back to Hall’s (1978)

seminal paper. Results from many studies have been contradictory, however. Some

have found little or no statistical evidence to reject the “pure” hypothesis, while

others have found little support for it. Many who have found evidence to reject the

hypothesis have identified either borrowing constraints or aggregation bias as the

cause. For instance, to avoid aggregation bias and thereby focus on borrowing

constraints, Zeldes (1989) and Runkle (1991)—among others—have used micro data

sets within linear models to test the implications of the hypothesis. This study has

taken this approach, but has directly estimated the original nonlinear Euler

equation using a GMM estimator. The direct nonlinear estimation avoids

estimating from a bias and inconsistent estimator that emerges when the error is

characterized as conditionally heteroscedastic. Examples are given in the paper to

strongly suggest that the variance of a family’s forecast error will be related to

variables frequently used as regressors in linearized models. This conditional

variance is then shown to appear in the linearized model’s residual which will

thereby induce the bias and inconsistency of parameter estimates.
35

Comparative evidence from two alternative consumption proxies is also offered

to show that food expenditures are less than suitable in an intertemporal model.

Standard liquidity restriction tests presented in this paper show that the food

expenditure proxy produces economically inconsistent results. The difficulty with

the food series is not necessarily its inability to mirror total consumption movements

but the relative noisiness of the series. This conclusion seems particularly viable

when compared with results derived from the more comprehensive nondurables and

service consumption proxy. The relative noisiness of food expenditure data appear

to result from the method in which the survey household data are collected. As a

survey remedy for the poor Interview food estimates, the Bureau of Labor Statistics

does conduct a Diary expenditure survey which specifically targets food expenditures

and other small frequently purchased items. However, the sample respondents from

the Diary survey are not tracked over multiple periods, and therefore these data are

not aptly usable in this analysis.
36

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 , "Asymptotic Efficiency in Estimation with Conditional Moment

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 , "Excess Sensitivity of Consumption to Current Income: Liquidity

Constraints or Myopia?" Canadian Journal of Economics, 1985, pp. 117-136.

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Hall, Robert E. and Frederic S, Mishkin, "The Sensitivity of Consumption to

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pp. 1269-86.
38

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 , "The Permanent Income Hypothesis and Consumption Durability:

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39

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40

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41

DATA APPENDIX

Variable Descriptions

Variables Descriptions

CFD Food and beverage consumption including food at home, food away from,

alcoholic and nonalcoholic beverages: These data are collected using

global questions such as "Since the first of the month, three months ago,

what has been your usual weekly expense at the grocery store or

supermarket." Because grocery and supermarket expenses will also

include nonfood items, estimates are given by the respondents for the

nonfood items as well.

CNDS Nondurable and service consumption: Definitions are matched as close to

NIPA as possible. CES nondurable expenditures used are as follows: food

and alcoholic beverages, nondurable apparel expenditures (excluding

jewelry and alike), gasoline and other fuels, tobacco, drug preparations

and sundries, and stationary and writing supplies.

The following lists some exceptions in matching the CES with NIPA.

Toilet articles and preparations are not shown separately in CES, but

some expenditures will appear in personal care products. Cleaning and

polishing are void in the Interview survey. Nondurable toys and sport

equipment are not separated in the CES and therefore not used.

Stationary and writing supplies are not all inclusive in the CES; for
42

instance postage stamps are not collected. Net Foreign remittances are

not applicable. Other expenditures of NIPA are also not applicable.

Service expenditures include housing rental equivalence, housing rental

expenditures, other lodging expenditures, utilities and services, domestic

services and household operations, transportation (including vehicle

finance charges, maintenance, repairs, insurance, rents and leases, and

public transportation), medical care, personal care, personal business,

recreation, education, and contributions.

r The ex post real after-tax rate of return: Because of differences in

marginal tax rates, ri,t varies across time and households. ri,t will vary

across households because of differences in marginal tax rates. To

measure the real after-tax rate of return, I deflate the nominal rate of

return on the riskless asset, the 3-month T-Bill, by the change in the

Consumer Price Index. This real rate of return is then adjusted by

individual marginal tax rates (τ it ) to derive a cross-sectional and time

varying rate of return. τ it is the summation of estimated federal, state

and local marginal tax rates. The estimated marginal federal tax rates

are provided by taxable adjusted gross income (AGI) from the U.S.

Treasury, Internal Revenue Service. These marginal tax rates account for

filing status, number of exemptions, and average allowable deductions.

To carry out the rate computations, AGI's are computed for each

consumer unit. I compute AGI based on household composition, income
43

level, and taxable deductions. The IRS produces tables of average

taxable deductions by income levels, and these data are used to estimate

household tax deductions. The state and local marginal tax rates are also

based on AGI. These rates are gathered from the tables provided in the

U.S. Statistical Abstracts published by the Department of Commerce. The

real rate of return can be stated as:

r

R P

P i t

t it t

t

,

( )( ) = + −

+

1 1

1

τ

where Rt is the risk free nominal rate of interest, Pt is a measure of the

price level at time t, andτ it is the estimated marginal tax rate. The

calendar quarter average of the general CPI-U measures price levels (P)

in each quarter.

QRT Time variables assigned to the quarters to capture macroeconomic shocks,

particularly seasonality present in quarterly expenditure data.

AGER Age of the reference person: Reference person is the first member

mentioned by respondent when asked to “Start with the name of the

person or one of the persons who owns or rents the home.” It is with

respect to this person that the relationship of other consumer unit

members are determined.

∆FS Change in the number of members in the consumer unit measured from

the previous quarter.
44

DY Change in total income (liquidity variable): Total 12 months of income

measured from the prior month of the last interview less total 12 months

of income measured from the prior month of the first interview. Total

income includes income received on a regular and irregular bases.

Regular received income includes wages and salary, pensions and social

security payments, interest, rents, royalties, trusts, and supplemental

insurance income and other pubic assistance. Irregular income includes

tax refunds from federal, state, and local governments, receipt from sale

of personal or real properties, lottery winnings, and lumpsum payments

from alimony, child support, trusts, and royalties. Lagged income values

are adjusted by the general Consume Price Index.
45

Table 1: Autocorrelations of residuals $ , ei t

Variable Estimate

Standard

error t-statistic

Constant 0.000134 0.00769 0.1736

$, ei t−1

-0.350430 0.02764 -12.67732

$, ei t−2

-0.004279 0.027565 -0.155236
46

Table II: GMM estimates for full sample

(3,684 observations on 1,288 unique consumer units)

Fit for Euler equations (4.5) and (4.6), 1990 Data

Consumption definition: nondurables & services food & beverages

standard augmented standard augmented

instrumental instrumental instrumental instrument

 variable list variable list variable list variable list 

-a -0.1421* -0.0883*** -0.0743 -0.0575**

 Starting value = -0.50 (0.0822) (0.0315) (0.0675) (0.0231)

b 0.9734*** 0.9714*** 0.9657*** 0.9667***

 Starting value = 0.97 (0.0039) (0.0019) (0.0041) (0.0020)

Age of reference person (a) -0.0002*** -0.0002*** -0.0002*** -0.0002***

 Starting value = 0.00 (4.3E-5) (2.5E-5) (3.4E-5) (2.6E-5)

Change in family size (b) 0.0031 0.0014 -0.0039 0.0028

 Starting value = 0.00 (0.0035) (0.0018) (0.0049) (0.0021)

Time dummy 1 (d1) -0.0145* -0.0096*** 0.0024 0.0018

 Starting value = 0.00 (0.0077) (0.0030) (0.0035) (0.0018)

Time dummy 2 (d2) -0.0118*** -0.0107*** -0.0035 -0.0048**

 Starting value = 0.00 (0.0022) (0.0011) (0.0058) (0.0023)

Time dummy 3 (d3) 0.0043** 0.0037*** 0.0017 0.0009

 Starting value = 0.00 (0.0021) (0.0013) (0.0033) (0.0016)

Chi Square:

 Overidentifying restrictions 52.9119 133.1580 70.6541 117.1110

Degrees of freedom 7 8 7 8

Probability value 5.4E-9 6.3E-25 1.5E-9 1.3E-21

Liquidity variable: None DY None DY

Chi Square:

 Exclusion restriction - 80.2461 - 46.4569

Degrees of freedom - 1 - 1

Probability value - 1.5E-9 - 1.5E-9

Concave utility function? Yes Yes Yes Yes 

Standard errors are in parentheses under coefficients. * indicates significance at the 10 percent level,

** indicates significance at the 5 percent level, and *** indicates significance at the 1 percent level.

All significance levels are computed for values from zero except b which is computed from 1.
47

Table III: GMM estimates for nonborrowing constrained sample

(1,314 observations on 438 unique consumer units)

Fit for Euler equations (4.5) and (4.6), 1990 data

Consumption definition: nondurables & services food & beverages

standard augmented standard augmented

instrumental instrumental instrumental instrument

 variable list variable list variable list variable list 

-a -0.1741* -0.1407** -0.1220 -0.0742**

 Starting value = -0.50 (0.0967) (0.0619) (0.0938) (0.0339)

b 0.9807*** 0.9791*** 0.9658*** 0.9697***

 Starting value = 0.97 (0.0070) (0.0050) (0.0102) (0.0042)

Age of reference person (a) -0.0003*** -0.0003*** -0.0003*** -0.0003***

 Starting value = 0.00 (0.0001) (7.4E-5) (9.3E-5) (4.4E-5)

Change in family size (b) 0.0094 0.0080 0.0192 0.0114

 Starting value = 0.00 (0.0102) (0.0074) (0.0169) (0.0066)

Time dummy 1 (d1) -0.0201* -0.0161** 0.0058 -0.0024

 Starting value = 0.00 (0.0117) (0.0076) (0.0087) (0.0036)

Time dummy 2 (d2) -0.0035 -0.0042 0.0128 -0.0043

 Starting value = 0.00 (0.0052) (0.0038) (0.0175) (0.0068)

Time dummy 3 (d3) 0.0064 0.0060** 0.0008 0.0042**

 Starting value = 0.00 (0.0051) (0.0038) (0.0083) (0.0034)

Chi Square:

 Overidentifying restrictions 3.3885 7.1896 9.51033 38.7150

Degrees of freedom 7 8 7 8

Probability value 0.8469 0.5163 0.2181 5.5E-6

Liquidity variable: None DY None DY

Chi Square:

 Exclusion restriction - 3.8011 - 29.2047

Degrees of freedom - 1 - 1

Probability value - 0.0514 - 6.5E-8

Concave utility function? Yes Yes Yes Yes 

Standard errors are in parentheses under coefficients. * indicates significance at the 10 percent level,

** indicates significance at the 5 percent level, and *** indicates significance at the 1 percent level.

All significance levels are computed for values from zero except b which is computed from 1.
48

Table IV: GMM estimates for borrowing constrained sample

(2,550 observations on 850 unique consumer units)

Fit for Euler equations (4.5) and (4.6), 1990 data

Consumption definition: nondurables & services food & beverages

standard augmented standard augmented

instrumental instrumental instrumental instrument

 variable list variable list variable list variable list 

-a 0.0840 -0.0280* -0.0760 -0.0722**

 Starting value = -0.50 (0.0798) (0.0168) (0.0657) (0.0368)

b 0.9620*** 0.9702*** 0.9661*** 0.9662***

 Starting value = 0.97 (0.0052) (0.0011) (0.0036) (0.0011)

Age of reference person (a) -0.0002*** -0.0003*** -0.0002*** -0.0002***

 Starting value = 0.00 (3.6E-5) (1.0E-5) (5.0E-5) (5.0E-5)

Change in family size (b) -0.0031 0.0005 0.0020 0.0020

 Starting value = 0.00 (0.0027) (0.0007) (0.0032) (0.0025)

Time dummy 1 (d1) 0.0046 -0.0050*** 0.0015 0.0014

 Starting value = 0.00 (0.0065) (0.0015) (0.0031) (0.0028)

Time dummy 2 (d2) -0.0052 -0.0105*** -0.0078*** -0.0079***

 Starting value = 0.00 (0.0039) (0.0009) (0.0027) (0.0023)

Time dummy 3 (d3) 0.0023 0.0027*** 2.3E-5 -6.6E-5

 Starting value = 0.00 (0.0016) (0.0005) (0.0026) (0.0023)

Chi Square:

 Overidentifying restrictions 75.2655 405.8230 36.7756 36.5673

Degrees of freedom 7 8 7 8

Probability value 1.5E-9 1.1E-82 5.2E-6 1.4E-5

Liquidity variable: None DY None DY

Chi Square:

 Exclusion restriction - 330.5575 - 0.2083

Degrees of freedom - 1 - 1

Probability value - 1.5E-9 - 0.6547

Concave utility function? No Yes Yes Yes 

Standard errors are in parentheses under coefficients. * indicates significance at the 10 percent level,

** indicates significance at the 5 percent level, and *** indicates significance at the 1 percent level.

All significance levels are computed for values from zero except b which is computed from 1.
49

Table 5. Summary Statistics for Consumption Proxy, by Sample

Unconstrained Borrowing

Full Sample Sample Constrained Sample

Variable Mean

Standard

error Mean

Standard

error Mean

Standard

error

CFD 1.1593 2.2446 1.1244 0.6970 1.1772 2.7174

CNDS 1.0722 0.3913 1.0979 0.4378 1.0589 0.3643