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WORKING PAPER
ALFRED P. SLOAN SCHOOL OF MANAGEMENT



THE INTERACTION BETWEEN

TIME-NONSEPARABLE PREFERENCES

AND TIME AGGREGATON

by

John Heaton

Latest Revision: December 1991

Previous Version: 3181-90-EFA

Working Paper No. 3376-92-EFA



MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139



THE INTERACTION BETWEEN

TIME-NONSEPARABLE PREFERENCES

AND TIME AGGREGATON

by

John Heaton

Latest Revision: December 1991

Previous Version: 3181-90-EFA

Working Paper No. 3376-92-EFA



f FEB ? 1 1992 I



The Interaction Between Time-Nonseparable Preferences
AND Time Aggregation



John Heaton
Department of Economics and Sloan School of Management

M. I.T.



June 1990
Revised December 1991



ABSTRACT

In this paper I develop and empirically analyze a continuous-time,
linear-quadratic, representative consumer model in which the consumer has
time-nonseparable preferences of several forms. Within this framework I
show how time aggregation and time nonseparabilities in preferences over
consumption streams can interact. I show that the behavior of both
seasonally adjusted and unadjusted consumption data is consistent with a
model of time-nonseparable preferences in which the consumption goods are
durable and in which individuals develop habit over the flow of services
from the good. The presence of time nonseparabilities in preferences is
important because the data does not support a version of the model that
focuses solely upon time aggregation and ignores time nonseparabilities in
preferences by making preferences time additive.



Many helpful comments and suggestions were made by Edward Allen, Toni Braun,
John Cochrane, George Constantinides, Martin Eichenbaum, Ayman Hindy, Chi-fu
Huang, Narayana Kocherlakota, Robert Lucas, Robin Lumsdaine, Masao Ogaki,
Chris Phelan, Paul Romer, Karl Snow, Grace Tsiang, Kei-Mu Yi and and
participants of workshops at Chicago, Carnegie-Mellon, Duke, Harvard,
M.I.T., Minnesota, Northwestern, Princeton, Queen's, Rochester, Stanford,
Toronto, Western Ontario, Wharton and Yale. I would especially like to
thank three referees for their very detailed and helpful comments and
suggestions and Lars Peter Hansen for his many suggestions and his constant
encouragement. I gratefully acknowledge financial support from the
Greentree Foundation, the National Science Foundation, the Sloan Foundation
and the Social Science and Humanities Research Council of Canada. This
paper is a revised version of my University of Chicago Ph.D. dissertation.
Any errors are of course my own.



1. Introduction

There has been an extensive amount of work examining whether aggregate
consumption expenditures are consistent with the restrictions implied by the
permanent income hypothesis. A strict interpretation of this hypothesis
predicts that aggregate consumption should be a martingale as shown by Hall
(1978). This implication has been questioned by a number of authors who
have found that consumption changes are predictable over time (see, for
example, Flavin (1981), Hayashi (1982) and Hall and Mishkin (1982)).
However recently Christiano, Eichenbaum and Marshall (1991) have argued that
there is little evidence against the martingale hypothesis using aggregate
consumption data, once the fact that the data is time averaged is taken into
account.

If consumers are assumed to make consumption decisions quite
frequently, so that the martingale hypothesis applies to consumption at a
much finer interval than observed data, time averaging of the consumption
data implies that consumption changes will be predictable. Under further
assumptions, time averaging the data implies that the first-order
autocorrelation in consumption changes should be 0.25 and information lagged
two periods should not be useful in predicting consumption changes.
Christiano, Eichenbaum and Marshall (1991) show that these implications are
remarkably consistent with quarterly observations of seasonally-adjusted
consumption so that rejections of the martingale hypothesis could be due to
the fact that the data is time averaged. However, I show that these results
are sensitive to the data used in the analysis. In particular, monthly
observations of seasonally adjusted consumption and quarterly observations
of seasonally unadjusted consumption are at odds with the martingale model,



even with an account for the effects of time averaging of the data.

The martingale hypothesis and its implications for time averaged data
that are exploited by Christiano, Eichenbaum and Marshall (1991), are
derived under the strong assumption that there is a representative consumer
with time-separable preferences over consumption in continuous-time. This
assumption about preferences has also been imposed in most other
investigations of the effects of time averaging of the consumption data .
In a continuous-time environment the assumption that preferences are time
separable is far from appealing since it implies that an individual's
preferences over consumption at one instant are unaffected by consumption
the instant before. In this setting it seems more reasonable to assume that

2

preferences are time-nonseparable .

In this paper, I show that with the introduction of time-nonseparable
preferences, the different dynamics of monthly and quarterly seasonally
adjusted consumption data can be easily explained. This occurs because over
short periods of time time nonseparabilities are very important. However as
the data is averaged over longer periods of time, the model's implications
tend to be consistent with a model of time-separable preferences. I also
show that the same type of model is consistent with seasonally unadjusted
consumption data.

In conducting this study, I develop a continuous-time linear-quadratic
permanent income model in which the representative consumer has
time-nonseparable preferences. Although the implications of the model are
derived under very general forms of time-nonseparable preferences, I show
that without further restrictions upon preferences it is not possible to
identify the preferences of the consumer using discrete-time data. As a
result, I examine several specific forms of time nonseparabilities in



preferences.

The first form of time-nonseparable preferences captures the notion
that consumption is substitutable over time or that the consumption goods
are durable. Discrete-time versions of this type of model have been
considered by Dunn and Singleton (1985), Eichenbaum and Hansen (1990),
Hansen (1987) and Ogaki (1988), for example. The second form of time
nonseparability is a model of habit persistence like those studied by
Constantinides (1990), Detemple and Zapatero (1991), Novales (1990), Ryder
and Heal (1973) and Sundaresan (1989). This preference specification
implies that consumption is complementary over time.

Using seasonally adjusted observations of consumption expenditures on
nondurables and services, I show that there is strong evidence for the model
where consumption is substitutable over time (or that the consumption goods
are durable) and that this model reconciles the conflict between monthly and
quarterly consumption data. Further, I find no evidence for habit
persistence alone, however there is some weak evidence for habit persistence
if habit is assumed to develop over the flow of services created by the
durable nature of the consumption goods. Using seasonally unadjusted data I
show that there is also very strong evidence in favor of a model where
consumption is substitutable over time. Also there is evidence for habit
persistence at seasonally frequencies, but again the durable nature of the
goods must be mode ted.

I also apply the model to durable goods expenditures. Using this data
there is strong evidence for habit persistence that forms over the flow of
services from the durable goods. I examine whether the model can help to
explain some of the durable goods puzzles discussed by Mankiw (1982) and
more recently by Caballero (1990). I show that the model provides only



a partial resolution of these puzzles.

The rest of the paper is organized as follows. In section II I examine
the implications of the martingale hypothesis for time averaged data and I
show that the model is inconsistent with monthly seasonally-adjusted data
and quarterly seasonally-unadjusted data. I also discuss whether these
results could be due solely to measurement error in the consumption data.
In section III I develop a model of time-nonseparable preferences and
capital accumulation and develop its implications for consumption. I show
that it is necessary to focus upon several specific parametric forms of the
preferences. In section IV I discuss the implications of two examples of
time-nonseparable preferences that capture notions of substitution and
complementarity over time. In section V I present the empirical results of
applying these examples to consumption data. Section VI concludes the
paper.

II. Time-Separable Preferences and Time Aggregation

Consider a situation in which there is a representative consumer with
time-separable preferences over consumption facing a constant interest rate
that equals the consumer's pure rate of time preference. In this case the
euler equation for the consumer implies that the marginal utility of
consumption is a martingale (see, for example Hall (1978)). Under the
further restriction that preferences are quadratic, with a constant bliss
point, the model implies that consumption, {c( t ) : t=0, 1 , 2, ...}, is a
martingale.

A typical way to investigate this sharp prediction is to construct
observations of c(t+l) - c(t) using consumption data observed at quarterly



frequencies (for example). A test is then conducted of whether £{c(t+l) -
c(t)l?(t)} = where f(t) denotes the consumer's information set at time t
(see. for example, Hall (1978), Flavin (1981) and Hayashi (1982)). A
potential problem with this approach (as emphasized recently by Christiano,
Eichenbaum and Marshall (1991)) is that the available aggregate consumption
data consists of observations of consumption expenditures over a period. In
other words, at time t observed consumption is c(t)=J" c(T)dT. If
consumption within the observation interval is not viewed by the consumer as
being perfectly substitutable, then the use of this time averaged data could
lead to spurious rejections of the martingale implication for consumption.

The fact that the consumption data is time averaged implies that
{c(t )-c(t-l ) : t=0,l,2, ...} follows a first-order moving average process
with first-order autocorrelation of 0.25. This temporal aggregation
problem could help to explain some of the rejections of the martingale
hypothesis for aggregate consumption. This is exactly what Christiano,
Eichenbaum and Marshall (1991) find using quarterly data.

Il.d Tests of the Model Using Seasonally Adjusted Consumption Data

Following Flavin (1982) and Christiano, Eichenbaum and Marshall (1991),

I assume that the model applies to detrended consumption where /j is the

trend parameter . Table 2.1 gives maximum likelihood estimates of the MA(1)

model: c(t) - c(t-l) = 9 c(t) + 6 c(t-l). where £{c(t)^} = 1 and

1

£{c(t)c(T)} = for T * t. The first-order autocorrelation of c(t) -
c(t-l), R(l), is restricted to 0.25 so that the 9 implies 9 . The
estimates are reported for quarterly per capita seasonally adjusted
consumption expenditures on nondurables and services. Estimates of the
MA(1) model with 9 unrestricted are given in table 2.2. In both tables



estimated parameters are reported for the period 1952,1 to 1986,4 and 1959,1
to 1986,4. The latter subsample was used since this matches the period of
the monthly data. Likelihood ratio tests of the restriction that R(l) =
0.25 yield probability values of 0.73 for the data set from 1952,1 to 1986,4
and 0.78 for data from 1959,1 to 1986,4. This occurs because R{1) is very
close to 0.25 in each case.

The model also implies that information lagged two periods should not
be useful in predicting the consumption change today. In particular, if
z(t-2) is a set of instruments that are members of 3^(t), and allowing
c(t)-c(t-l) to have a constant mean value of m, the model implies that:

(2.1) £{[c(t)-c(t-l)-m]z(t-2)} = .

Letting zU) = [ 1 , c( t-2)-c( t-3) , c( t-3)-c( t-4) ,c( t-4)-c( t-5) , c( t-5)-c( t-6) ] ' ,
a test of the restriction (2.1) can be performed using the GMM criterion
function where the parameter m is estimated using GMM and the moment
condition (2.1) . If (2.1) is true, then the minimized GMM criterion
function is (asymptotically) a chi-squared random variable with 4 degrees of
freedom. The resulting P-value of the implied test of (2.1) is 0.065 for
the data set from 1952,1 to 1986.4 and 0.083 for the data set from 1959,1 to
1986,4. As a result, there is not substantial evidence against the
restriction (2.1). This indicates that the time-separable model is
reasonably consistent with quarterly seasonally adjusted consumption, due to
the fact that the data is time averaged. This success was noted by
Christiano, Eichenbaum and Marshall (1991).

Now consider monthly measures of seasonally adjusted consumption

7

expenditures on nondurables plus services . If the only difficulty with the



time-separable model is that the consumption data is time averaged, the
results should be robust to different intervals of time averaging. Table
2.3 reports estimates of the trend and the moving average parameters of
c(t)-c(t-l) with and without the 0.25 restriction on R(l) using monthly
data. A likelihood ratio test of the restriction yields a probability value
of essentially zero. As a result, the model is inconsistent with the

Q

monthly data . This occurs because the first order autocorrelation of the
first difference of monthly consumption is -0.188 with a standard error of
0.052, which is significantly negative.

II. B. Tests of the Model Using Quarterly Seasonally Unadjusted

Consumption Data

The process of seasonal adjustment used to construct seasonally
adjusted data changes the correlation structure of the observed series in a
fundamental way (for a discussion of this issue see, for example, Miron
(1986) and Person and Harvey (1991)). This is a serious issue since a large
part of the success of the time-separable model with seasonally adjusted
quarterly data is due to the fact that R(l) is estimated to be close to
0.25. It is important to examine whether this result is sensitive to the
process of seasonal adjustment.

As a first pass, consider the autocorrelation structure of first
differences of searsonally unadjusted consumption with trend and seasonal
dummies removed^. Table 2.4 gives estimates of the autocorrelation function
using quarterly expenditures on nondurables and services . There are two
important things to notice. First, unlike the seasonally adjusted quarterly
data, the first-order autocorrelation is not close to 0.25. Hence with a
different manner of accounting for seasonality, the time-separable model is



not consistent with the data, even at quarterly frequencies. Second, note
that the autocorrelation value at the fourth lag is significantly positive
and large. This indicates that the use of seasonal dummies does not remove
all of the seasonality in the data and some addition must be made to the
model to account for this behavior.

1 1 I.e. Measurement Error

Before turning to a model based explanation of these different results,
the problem of measurement error in the consumption series must be
addressed. Consider a world in which the discrete-time version of the
time-separable model is correct at monthly frequencies, but the monthly data
is contaminated with i.i.d. measurement error. In this case, observed
consumption differences at monthly frequencies would be given by:



(2.2) c(t) - c(t-l) = u (t) + u (t) - u (t-1

12 2



where u is the model error and u is the i.i.d. measurement error . Notice

1 2

that in this case consumption differences are negatively correlated over
time. Time averaging from monthly to quarterly frequencies eliminates much
of the effect of measurement error and hence would explain the different
results using seasonally adjusted monthly and quarterly data.

However, an i.i.d. model for measurement error in consumption data is
not very reasonable due to the methods used to construct the data. This
point has been stressed in several recent papers Bell and Wilcox (1991) and
Wilcox (1991). I will provide a brief summary of the conclusions from these
studies. A detailed discussion of these issues can be found in Bell and
Wilcox (1991) and Wilcox (1991).



The measure of aggregate consumption used in this paper is personal
consumption expenditures on nondurables and services taken from the U.S.
National Income and Product Accounts. This measure of consumption is
constructed by the Bureau of Economic Analysis in the Department of
Commerce. An important ingredient in the construction of personal
consumption expenditures is the monthly estimates of retail sales
constructed by the Census Bureau. In estimating retail sales, the Census
Bureau receives reports of retail sales from all large retail establishments
every month and from a sample of small retail establishments. The small
companies report in rotating panels every three months. The sampling error
Induced by this sampling scheme is an important source of measurement error.
This sampling error is correlated over time for two reasons.

First, the small retail establishments reporting each month report both
their current month's sales and their sales for the previous month. The
final estimate of a month's retail sales reported by the Census Bureau are
based on the sales reported by the panel of that month and the panel
reporting the following month. Due to the reporting overlap, there is very
strong autocorrelation in the sampling error from one month to the next.
Given their information on the methods used in sampling the small retail
establishments, the Census Bureau has attempted to measure the
autocorrelation in the sampling error induced by the two month reporting of
each panel. Not surprisingly, the autocorrelation is quite close to one
(see Bell and Wilcox (1991)). A second source of autocorrelation in the
sampling error is the use of a rotating panel. This Induces correlation
between the current month's sampling error and the sampling error 3 months
back.

As discussed by Bell and Wilcox (1991), the strong positive



autocorrelation induced by the sampling practices of the Census Bureau
implies that the first-order autocorrelation in c(t) - c(t-l), due to the
presence measurement error in (2.2), is likely to be very small. As a
result, measurement cannot explain the negative correlation of the
first-differences of monthly consumption. Further measurement error cannot
explain the correlation structure of seasonally unadjusted quarterly
consumption data.

III. A Model with Time-Nonseparable Preferences

A problem with the model of preferences that underlies the MA(1) model
for {c(t) - c(t-l ) : t=0, 1 ,2, ...} is the assumption that the consumer has
time-separable preferences over consumption in continuous time. This
assumption implies that the consumer's preferences over consumption at time
t are unaffected (through preferences) by consumption in the instant before
time t. A reasonable alternative to this model is to allow preferences to
be time-nonseparable.

In this section, I present a model of time-nonseparable preferences and
capital accumulation and derive its implications for observations of
time-averaged consumption. Time nonseparability in preferences is
introduced by specifying a mapping from current and past consumption goods
into a process called services. The representative consumer is assumed to
have time-separable preferences over services. The mapping from consumption

12

into services is a type of Gorman-Lancaster technology in which the
consumption goods are viewed as bundled claims to characteristics that the
consumer cares about

I develop the implications of the model under a very general



10



specification of preferences. I show that without further restriction on
preferences, discrete-time data will not reveal the preference structure.
This implies that preferences must be restricted in some way. In the
development of the model I ignore several technical issues. The appendix
provides a discussion of these issues.

III. A. Preferences Over Services and Consumption

The preferences of the consumer are assumed to be time separable over a
stochastic process, s={s( t ) : Ost = A(t). Also (3.5) and (3.6) imply that


1 3 4

Online LibraryJohn HeatonThe interaction between time-nonseparable preferences and time aggregaton [i.e. aggregation] → online text (page 1 of 4)