HD28
.M414
-57
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
THE INTERACTION BETWEEN
TIME-NONSEPARABLE PREFERENCES
AND TIME AGGREGATON
by
John Heaton
December 1991
Latest Revision:
3181-90-EFA
Previous Version:
3376-92-EFA
Working Paper No.
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
THE INTERACTION BETWEEN
TIME-NONSEPARABLE PREFERENCES
AND TIME AGGREGATON
by
John Heaton
December 1991
Latest Revision:
3181-90-EFA
Previous Version:
3376-92-EFA
Working Paper No.
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
develop and empirically analyze a continuous-time,
representative consumer model in which the consumer has
Within this framework I
time-nonseparable preferences of several forms.
aggregation
and
time
nonseparabilities
in preferences over
show how time
show that the behavior of both
consumption streams can interact.
I
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
The presence of time nonseparabilities in preferences is
from the good.
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.
this paper
linear-quadratic,
In
I
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
Duke,
Harvard,
Carnegie-Mellon,
participants of workshops at Chicago,
M.I.T., Minnesota, Northwestern, Princeton, Queen's, Rochester, Stanford,
would especially like to
I
Toronto, Western Ontario, Wharton and Yale.
thank three referees for their very detailed and helpful comments and
suggestions and Lars Peter Hansen for his many suggestions and his constant
gratefully acknowledge financial support from the
encouragement.
I
Greentree Foundation, the National Science Foundation, the Sloan Foundation
This
and the Social Science and Humanities Research Council of Canada.
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
income
permanent
hypothesis.
A
strict
interpretation of
this
hypothesis
predicts that aggregate consumption should be a martingale as shown by Hall
(1978).
have
found
example,
implication has been questioned by a number of authors who
This
that
Flavin
consumption changes
(1981),
Hayashi
predictable
are
and
(1982)
Hall
over
time
(see,
Mishkin
and
for
(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
frequently,
so
are
that
much finer interval
assumed
make
to
consumption
the martingale hypothesis applies
to
decisions
consumption at a
time averaging of
than observed data,
the
data implies that consumption changes will be predictable.
assumptions,
averaging
time
data
the
implies
that
quite
consumption
Under further
the
first-order
autocorrelation in consumption changes should be 0.25 and information lagged
two
periods
should
not
be
useful
Christiano, Eichenbaum and Marshall
remarkably
consistent
with
(1991)
quarterly
consumption
predicting
in
changes.
show that these implications are
observations
of
seasonally-adjusted
consumption so that rejections of the martingale hypothesis could be due to
the fact that the data is time averaged.
are
sensitive
to
the
data used
in
the
However,
analysis.
I
show that these results
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
that
exploited
are
Christiano,
by
implications for time averaged data
Eichenbaum
and
Marshall
are
(1991),
derived under the strong assumption that there is a representative consumer
with time-separable preferences over consumption in continuous-time.
preferences
about
assumption
investigations of
In
a
has
effects of
the
time
continuous-time environment
separable
is
from
far
the
preferences over consumption at one
preferences are time-nonseparable
this paper,
preferences,
the
the
most
in
consumption data
assumption that preferences are
it
implies
that
to
I
2
.
show that with the introduction of time-nonseparable
different
is averaged over
be
show that
dynamics
of
monthly
quarterly
and
seasonally
This occurs because over
longer periods of time,
the model's
However as
implications
consistent with a model of time-separable preferences.
the same
time
individual's
an
short periods of time time nonseparabilities are very important.
tend
.
instant are unaffected by consumption
adjusted consumption data can be easily explained.
the data
other
In this setting it seems more reasonable to assume that
the instant before.
In
imposed
averaging of
since
appealing
been
also
This
type of
model
is
I
also
consistent with seasonally unadjusted
consumption data.
In conducting
permanent
income
this study,
model
I
in
time-nonseparable preferences.
derived under very general
that
without
identify
result,
the
I
further
which
the
representative
forms of
several
the
consumer
has
Although the implications of the model are
time-nonseparable preferences,
restrictions upon preferences
preferences of
examine
develop a continuous-time linear-quadratic
it
is
not
possible
consumer using discrete-time data.
specific
forms
of
time
show
I
As
nonseparabilities
to
a
in
preferences.
that
form
first
The
consumption is substitutable over time or
Discrete-time
durable.
are
considered
Hansen
by
and
(1987)
Constantinides
Heal
Singleton
Ogaki
(1988),
is
model
a
of
Sundaresan
and
of
persistence
implies that consumption
(1989).
complementary
is
model
of
The
have
Hansen
and
second
like
(1991),
notion
the
consumption goods
the
type
example.
habit
captures
Eichenbaum
(1985),
for
that
this
Detemple and Zapatero
(1990),
(1973)
versions
and
Dunn
nonseparability
and
preferences
time-nonseparable
of
form
those
Novales
This
preference
over
time.
been
(1990),
time
of
studied
(1990),
by
Ryder
specification
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
Further,
data.
I
find
evidence
no
for
habit
persistence alone, however there is some weak evidence for habit persistence
if
habit
is
assumed
to
develop over
the
durable nature of the consumption goods.
show
that
there
is
also
flow of
consumption is substitutable over time.
the
Using seasonally unadjusted data
evidence
very strong
services created by
Also
in
favor
there
is
of
a
model
I
where
evidence for habit
but again the durable nature of the
persistence at seasonally frequencies,
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.
explain some of
more
the
I
examine whether the model can help to
durable goods puzzles discussed by Mankiw
recently by Caballero
(1990).
I
show
that
the
model
(1982)
and
provides only
a
partial resolution of these puzzles.
The rest of the paper is organized as follows.
the
model
the
seasonally-unadjusted
quarterly
section
III
develop
I
I
inconsistent with monthly seasonally-adjusted data
is
data.
I
model
a
of
discuss
also
results could be due solely to measurement error
In
examine
I
implications of the martingale hypothesis for time averaged data and
show that
and
In section II
the
in
whether
these
consumption data.
time-nonseparable
preferences
capital accumulation and develop its implications for consumption.
I
and
show
that it is necessary to focus upon several specific parametric forms of the
preferences.
section IV
In
time-nonseparable
preferences
these
examples
discuss the implications of two examples of
that
to
notions
capture
In section V
complementarity over time.
applying
I
consumption
I
substitution
of
and
present the empirical results of
Section
data.
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.
equation
euler
consumption
is
for
a
the
consumer
martingale
implies
(see,
for
that
example
the
model
implies
that
consumption,
case the
the
marginal
utility
of
Hall
(1978)).
Under
the
with a constant bliss
further restriction that preferences are quadratic,
point,
In this
{c( t
)
:
t=0,
1
,
2,
...},
is
a
martingale.
A
typical
way
to
observations of c(t+l)
investigate
-
c(t)
this
sharp
prediction
is
to
construct
using consumption data observed at quarterly
c(t)l?(t)} =
-
where f(t) denotes the consumer's information set at time
t
Hall
example,
for
(see.
then conducted of whether £{c(t+l)
A test
(for example).
frequencies
is
Flavin
(1978),
and
(1981)
Hayashi
A
(1982)).
potential problem with this approach (as emphasized recently by Christiano,
Eichenbaum and Marshall
(1991))
that the available aggregate consumption
is
data consists of observations of consumption expenditures over a period.
words,
other
time
at
consumption
observed
t
is
In
c(T)dT.
c(t)=J"
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.
that
fact
The
{c(t )-c(t-l
)
t=0,l,2,
:
problem
could
hypothesis
for
...}
help
to
explain
aggregate
data
follows
autocorrelation
first-order
with
consumption
the
first-order
a
of
some
the
consumption.
moving
This
0.25.
of
averaged
time
is
is
average
of
that
process
aggregation
temporal
rejections
This
implies
martingale
the
exactly what
Christiano,
Eichenbaum and Marshall (1991) find using quarterly data.
Tests of the Model Using Seasonally Adjusted Consumption Data
Il.d
Following Flavin (1982) and Christiano, Eichenbaum and Marshall (1991),
I
assume
that
trend parameter
model:
c(t)
applies
model
the
consumption where
detrended
to
the
is
/j
Table 2.1 gives maximum likelihood estimates of the MA(1)
.
-
=
c(t-l)
9 c(t)
+
6 c(t-l).
where
£{c(t)^}
=
1
and
1
£{c(t)c(T)}
c(t-l),
R(l),
estimates
consumption
MA(1)
=
model
are
for
is
T
t.
restricted
reported
expenditures
with 9
*
for
on
The
to
autocorrelation
first-order
0.25
so
that
the
9
quarterly
per
capita
nondurables
and
services.
implies
9
seasonally
unrestricted are given in table 2.2.
Estimates
In
c(t)
of
-
The
.
adjusted
of
the
both tables
estimated parameters are reported for the period 1952,1 to 1986,4 and 1959,1
The latter subsample was used since this matches the period of
1986,4.
to
the
Likelihood ratio
monthly data.
tests of
restriction
the
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
This occurs because R{1)
1986,4.
to
is very
close to 0.25 in each case.
information lagged two periods should not
The model also implies that
be useful
in
predicting
z(t-2)
a
set
is
of
the
consumption change
instruments
c(t)-c(t-l) to have a constant mean value
£{[c(t)-c(t-l)-m]z(t-2)} =
(2.1)
zU)
Letting
a
test
[
where
condition
1
,
c( t-2)-c( t-3)
restriction
the
of
function
=
(2.1)
the
(2.1)
of
if
allowing
the model implies that:
of m,
.
using
estimated
is
then
true,
is
and
3^(t),
can be performed using
m
particular,
In
c( t-3)-c( t-4) ,c( t-4)-c( t-5)
(2.1)
parameter
If
.
,
members
are
that
today.
the
,
c( t-5)-c( t-6)
the
GMM
'
,
GMM criterion
and
minimized
]
the
GMM
moment
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
restriction
a
result,
(2.1).
there
This
is
substantial
not
indicates
that
the
evidence
against
time-separable
model
the
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
seasonally
of
expenditures on nondurables plus services
7
.
If
adjusted
consumption
the only difficulty with the
model
time-separable
that
is
results should be robust
0.25
is
time
averaged,
time averaging.
moving average
the
restriction
on
the
Table
parameters
using
R(l)
of
monthly
A likelihood ratio test of the restriction yields a probability value
data.
of
the
intervals of
and
trend
the
with and without
c(t)-c(t-l)
consumption data
different
to
estimates of
reports
2.3
the
essentially
As
zero.
result,
a
model
the
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.
Tests
II. B.
of
Model
the
Using
Quarterly
Seasonally
Unadjusted
construct
seasonally
Consumption Data
process
The
adjustment
seasonal
of
used
to
adjusted data changes the correlation structure of the observed series in a
way
fundamental
(for
a
discussion of
(1986) and Person and Harvey (1991)).
part
of
the
success of
quarterly data
0.25.
It
is
is
due
the
to
this
issue
fact
for
example,
Miron
This is a serious issue since a large
time-separable model
the
see,
that R(l)
is
with seasonally adjusted
estimated
important to examine whether this result
to
be
close
to
sensitive to the
is
process of seasonal adjustment.
As
a
differences
first
of
pass,
consider
searsonally
dummies removed^.
the
unadjusted
autocorrelation
consumption with
structure
trend
first
seasonal
Table 2.4 gives estimates of the autocorrelation function
using quarterly expenditures on nondurables and services
important things to notice.
data,
and
of
.
There are two
First, unlike the seasonally adjusted quarterly
the first-order autocorrelation is not close
different manner of accounting for seasonality,
to
0.25.
Hence with a
the time-separable model
is
consistent with the data,
not
the fourth
the autocorrelation value at
that
is
note
significantly positive
seasonality in the data and some addition must be made to the
the
of
lag
Second,
This indicates that the use of seasonal dummies does not remove
and large.
all
even at quarterly frequencies.
model to account for this behavior.
11
Measurement Error
I.e.
Before turning to a model based explanation of these different results,
problem
the
measurement
of
Consider
addressed.
a
in
the
consumption
which
the
discrete-time
error
world
in
series
must
version
be
of
the
time-separable model is correct at monthly frequencies, but the monthly data
is
with
contaminated
measurement
i.i.d.
error.
In
this
observed
case,
consumption differences at monthly frequencies would be given by:
12
c(t) - c(t-l) = u (t) + u (t) - u (t-1
(2.2)
is the model error and u
where u
in
measurement error
Notice
.
this
case
consumption differences are negatively correlated
over
Time averaging from monthly to quarterly frequencies eliminates much
time.
of
is the i.i.d.
2
1
that
2
the
effect of measurement
error and hence would explain the different
results using seasonally adjusted monthly and quarterly data.
However,
not
an i.i.d.
very reasonable due
model for measurement error
to
the
methods used
to
in consumption data
construct
the
data.
is
This
point has been stressed in several recent papers Bell and Wilcox (1991) and
Wilcox (1991).
studies.
I
will provide a brief summary of the conclusions from these
A detailed discussion
Wilcox (1991) and Wilcox (1991).
of
these
issues can be
found
in
Bell
and
The
measure of aggregate consumption used
consumption expenditures
National
Product
and
Income
nondurables
on
Accounts.
Bureau
constructed
by
Commerce.
An
consumption
expenditures
the
important
This
Economic
of
ingredient
in
constructed by the Census Bureau.
taken
measure
Analysis
in
is
from
estimates
In estimating
the
U.S.
consumption
is
the
Department
of
of
retail
personal
of
construction
the
monthly
the
is
services
and
paper
this
in
of
personal
retail
sales
the Census
sales,
Bureau receives reports of retail sales from all large retail establishments
every month and from a sample of small
retail
establishments.
companies report in rotating panels every three months.
The
small
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.
the small retail establishments reporting each month report both
First,
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
reported
sales
the
reporting the following month.
strong autocorrelation
Given their
in
the
by
the
panel
Census
the
that
month
Due to the reporting overlap,
sampling error from one month
information on the methods used
establishments,
of
Bureau
has
in
and
panel
the
there is very
the
to
next.
sampling the small
attempted
to
retail
measure
the
autocorrelation in the sampling error induced by the two month reporting of
each panel.
(see
Bell
Not
surprisingly,
and Wilcox
sampling error
is
the
(1991)).
quite close
to
one
second source of autocorrelation
in
the
the
A
autocorrelation
use of a rotating panel.
is
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
implies that
by
induced
sampling
the
practices
first-order autocorrelation in c(t)
the
presence
measurement
result,
measurement
error
in
is
(2.2),
explain
cannot
likely
the
first-differences of monthly consumption.
explain
structure
correlation
the
of
be
to
negative
c(t-l),
very
due
Bureau
small.
correlation
the
to
As
a
the
of
Further measurement error cannot
unadjusted
seasonally
of
-
Census
the
quarterly
consumption data.
A Model with Time-Nonseparable Preferences
III.
A problem with the model of preferences that underlies
for
{c(t)
-
c(t-l
time-separable
)
:
t=0,
1
...}
,2,
preferences
the
is
over
assumption that
consumption
the MA(1) model
the
continuous
in
consumer has
This
time.
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
this model
to
is
to allow preferences
to
be time-nonseparable.
In this section,
capital
accumulation
time-averaged
I
present a model of time-nonseparable preferences and
and
derive
consumption.
its
implications
nonseparability
Time
for
observations
of
preferences
is
in
introduced by specifying a mapping from current and past consumption goods
into a process called services.
The representative consumer
have time-separable preferences over services.
services
into
is
a
type
of
is
assumed to
The mapping from consumption
Gorman-Lancaster
12
technology
in
which
the
consumption goods are viewed as bundled claims to characteristics that the
consumer cares about
I
develop
the
implications
of
10
the
model
under
a
very
general
specification of preferences.
show
discrete-time data will
preferences,
This
I
implies
preferences
that
development of the model
that
reveal
not
must
be
further
restriction on
preference
the
restricted
ignore several
I
without
some
in
technical
structure.
way.
In
the
The appendix
issues.
provides a discussion of these issues.
Preferences Over Services and Consumption
III. A.
The preferences of the consumer are assumed to be time separable over a
stochastic
process,
s={s(
t
)
:
Ost<oo}
,
called
services.
representative
The
consumer evaluates s via the utility function:
U(s) s -(l/2)£{X"exp(-pt)(s(t)-b(t))^dt>, p
(3.1)
where
{b(t) Oit<oo}
:
is
a
deterministic process describing
movement and p is the pure rate of time preference
element
of
the
space
£{/ exp(-pt)s(t) dt}
<
>
feasible
of
00
.
services
Under
the
14
I
the
bliss
assume that each
restricted
is
further
deterministic bliss point satisfies / exp(-pt)b(t) dt
assumption
<
oo,
point
such
that
that:
the
the preferences
of the consumer are well defined.
Time nonseparability in the consumer's preferences over consumption is
introduced by making s(t) a linear function of current and past consumption
given by a convolu4:ion between a nonrandom distribution g and consumption
cs{c(t):Ost<co>^^:
s(t) = g»c(t)
(3.2)
where
*
denotes convolution of two distributions.
11
To insure that s(t)
is a
function of current and past consumption, g is assumed to be one sided with
we ight
on
only
convolution
however
in
the
(3.2)
real
given
the
is
For example,
integration.
by
For
line.
integral:
many
s(t)
=
examples,
the
J g(T)c( t-T)dT,
interpretable using standard notions of
need not be
integral
this
nonnegative
the nonseparability could involve a comparison of
current consumption to past consumption as in a model of habit persistence.
g would include a dirac delta function.
In this case,
III.
Capital Accumulation and Equilibrium
B.
The model is completed by assuming that there is a capital accumulation
technology that allows the transfer of consumption over time at the constant
instantaneous rate
p:
Dk(t) = pk(t) + e(t)
(3.3)
where kit)
is
-
c(t),
k(0) given,
the level of the capital stock at time
condition and e(t)
is
the endowment
t,
k{0)
process of the consumption good.
endowment process is also assumed to satisfy £{J exp(-pt)e(t) dt}
assumption and the restrictions on c discussed in the appendix,
£{J"°°exp(-pt)k(t)^dt}
Hansen
(1990),
(1987),
<
CD.
As in Christiano,
Heaton and Sargent
Hansen,
this assumption is used
initial
is an
<
m.
The
This
implies that
Eichenbaum and Marshall (1991),
(1991),
and Hansen and Sargent
instead of a nonnegativity restriction on
the capital stock which would make the model analytically intractable.
Following Hansen (1987) and Sargent (1987)
of
return on capital
is
equal
to
the pure
I
am assuming that the rate
rate of
time preference.
This
restriction on the real rate of interest was also imposed by Flavin (1981)
and Hayashi
(1982).
Christiano, Eichenbaum and Marshall (1991) discuss this
12
J
assumption in the context of the time separable version of this model.
show
assumption
this
that
is
necessary
for
the
model
consumption and capital in a deterministic world.
directly
to
consumption
when
model,
this
positive
is
restrictions on
g,
as
I
in
applied
deterministic
a
These results carry over
services.
to
positive
imply
to
They
The
world
restriction
imposes
set
a
that
of
will discuss in section IV.
The problem facing the consumer is to maximize the objective function
subject to the mapping
(3.1)
(3.3),
(3.2)
and the capital accumulation technology
The Lagrangian for this problem is given by:
by the choice of c(t).
1 = -£-|j" exp(-pt)](l/2)[g*c(t)
(3.4)
-
bU)f
{
-
A(t)U(t)
-
k(0) - pJ"*'k(T)dT - j''e(T)dT + /c(T)dT I
^
L
°
where X{t) is the value of the Lagrange multiplier at time
The
tiO,
first-order conditions for
the
choice of kit)
^
1
t.
and
c(t)
for
each
are given by:
(3.5)
£i-g^»[g*c(t)
-
b(t)];?(t)l
+ E|j-"exp(-pt)X(t+T)dT;3^(t)|- = 0,
and
(3.6)
where g
^3^^
A(t) - pE\s'^exp{-pT)\{t+T]dx\^U)}- =
0.
is given by:
gf ^
j
exp(pt)g(-t) if
t
i
otherwise
Although it is not possible to completely characterize the solution to this
13
problem for general forms of
a
g,
simple
implication for services can be
derived.
Notice
that
EW'lg'cit)
The
good
consumption
of
gives
(3.8)
time
at
A(t)
is
a
martingale,
that
so
b(t)];?(t)| = A(t)/p.
-
side
left
that
Also (3.5) and (3.6) imply that
f{A(t+T) ;?(t)> = A(t).
(3.8)
implies
(3.6)
As
t.
the
marginal
result,
a
utility of a unit
the
implies
model
of
the
that
the
marginal utility of consumption is a martingale and:
-
g'"*[s(t+T)
(3.9)
where
£{u( t+x) ^(
1
t
)
}
b(t+T)] = g''*[s(t) - bit)] + U(t+T). T>0
where
and
=
I
have
substituted
in
.
the
fact
that
g*c(t)=s(t).
Under
assumptions
reasonable
on
g
one-sided forward looking inverse denoted g
s(t+T)
(3.10)
f
Since
t)(
t+x)
13-
(
t
martingale.
and
I
=
}
I
s(t)
forward
-
I.e.
looking,
b(t)
for
Applying g
.
2
}
=
2
cr
to
g
has
a
(3.9) yields:
+ g'"*u(t+T)
(3.10)
x>0,
appendix),
so
implies
that
s-b
is
denote the time derivative of (s-b)[t] as:
assume that £{D^(t)
11
- b(t)
b(t+T) = s(t)
is
g
)
-
the
(see
that
£{s(t+T)
a
continuous-time
D(s-b)[t] = D^(t)
dt.
Implications for Time- Averaged Consumption
The assumption that g
f
has a one-sided inverse also implies that there
14
exists a one-sided inverse of g, g
,
so that the convolution in (3.2)
can be
inverted to yield a mapping from services to consumption:
g^»s = c.
(3.11)
Recalling
that
D[s-b]{t)
D^(t),
=
(3.10)
and
(3.11)
imply
the
following
representation for the first difference of consumption:
(3.12)
Observed
c(t) - c(t-l) = g^*/^_jD?(T)
consumption,
{c(t):
+
g^*[b(t) - b(t-l)].
t=0,l,2,
consumption expenditures over a unit of time:
Averaging
over
(3.12)
a
unit
of
consists
...},
c(t) = J
time,
we
of
averages
of
c(t).
obtain
the
following
general representation for observed consumption:
(3.13)
c(t) - c(t-l) = g^*w{t)
where wit) = S^
t-l
= S^
J"^
T-l
+
g^«{J^_^[b(T)
an MA(1)
is a
b(T-l)]dT>
DC(r)dT
(t-T)DC(T) + J-''"^T-t + 2)D?(T)
t-2
t-l
Notice that w(t)
-
time-averaged martingale difference so that w(t) has
structure" with first-order autocorrelation
of
difference in consumption is obtained by "filtering" w(t)
through
-
c(t-l)
g^.
0.25.
+
[b(t)
The first
-
b(t-l)]
In the special case where c(t) = sit) and b is a constant,
= w(t)
and
(3.13)
c(t)
reduces to the MA(1) model examined in section
II.
15
.
Identification of g with Sampled Data
III.D.
identification issues
There are several
using
of
the bliss point
b(t)
in
(3.13).
If
Inferences about g
The first involves the presence
and observed consumption data.
(3. 13)
in making
an unobservable stochastic process
then the dynamics of consumption
for the bliss point were added to be model,
could be completely explained by this bliss point movement and the mapping
g^
in
is
to minimize
the role of unobservables and to try to explain consumption
based upon the mapping g
As a result,
.
in a deterministic fashion.
nonseparabilities
in
I
assume that the bliss point moves
One way to interpret the introduction of time
preferences
movements to observable variables.
is possible
The strategy taken here
could be set to the identity mapping.
(3.13)
to give an economic
as
is
attempt
an
link
to
bliss
point
An advantage to this approach is that it
interpretation to the required bliss point
movement
Even with the strong assumption that the bliss point is deterministic,
there is a further identification issue.
then (3.13)
-
let b(t) be constant,
implies a continuous-time moving average representation for c(t)
c(t-l) of the form:
(3.14)
c(t)
-
c(t-l) = g^«DC(t).
where g" = h*g^, h(t) =
otherwise.
If
the functions g
t
for te[0,l],
a continuous
and g
to
identify
the
h(t)
= 2 -
record of c(t)
t
for
t
€[1,2]
and h(t) =
c(t-l) were available,
-
then
could be exactly identified.
With the discrete-time data,
is
To see this,
{c(t )-c(t-l
discrete-time
moving
time-averaged consumption differences:
16
):
t=0,
1
,...}, the best we can do
average
representation
for
.
c(t) - c(t-l) = I " G^(J)c(t-j)
(3.15)
where £{c(t)^> =
to
map
and £{c(t)c(T)> =
1
sequence
the
{G
possible
(3.15)
identify
to
We would like to be able
describes
that
g
Of course without further restriction
time-nonseparabilities.
be
t.
function
the
to
}
for x *
completely
the
function
will not
it
However,
g*^.
the
note
that
is exactly the relationship that arises in a discrete-time version of
the model in which the decision interval of the consumer is set equal to the
interval
of
data.
the
discrete-time
the
In
model,
reflects the discrete- time time-nonseparabilities in
Marcet
(1991)
function of g
the
finite
g (t-2),
g (t-3)
of
the L
2
the
sequence
To construct this mapping,
.
all
set
g (t-1),
c
how
shows
linear
projection operator.
g (t) at many values of x.
Note that a(t)
function
constructed
is
let A be the closure
-
is
of
the
Projig
G(j)
preferences.
{G (j)>
combinations
Let a = g
the
lA)
functions
(in L
of
as
2
)
t€lR
a
of
:
where Proj denotes
in general
a
convolution of
Marcet (1991) shows that G (j) is given by:
jV(t+j)a(t)dt
G (j) =
r a(t)^dt
= -a»g''(j)/J'"a(t)^dt
°
(3.16)
~^
= -a*h»g^(J).
where a( t
)
= a(-t
If g'^(j)
)
were an average of the function g^(t) for values of
t
near
j,
then we would expect the discrete-time interpretation to be reasonably good.
However
(3.16)
implies that
the g''(j)
17
is
an average of
the function g^(t)
for
values
all
about
of
t
structure
the
positively
are
would
interpretation
discrete-time
lead
correlated
be
consider
positive
is
over
the
observed
since
The
time.
preferences
that
inferences
poor
to
example,
For
G (1)
case,
this
In
differences
consumption
general,
in
preferences.
of
model.
time-separable
will,
and G (j)
are
time
nonseparable, which they are not.
will not in general allow us to obtain a correct
Since the function G
economic
forms
g^
of
that
I
intuitive
summarize
continuous-time form of g
In
.
that
simple
in
allow for the estimation of the
I
outline two basic forms
capture notions of
In section V,
complementarity over time.
several
time-nonseparability
of
the next section
time-nonseparable preferences
upon
focus
next
types
These restricted forms of g
preferences.
of
preferences,
interpretation of
I
substitution and
show that these preferences do a
good job in explaining the difficulties with the time additive model that
pointed out in section
IV.
Two Polar Cases
II.
of Time-Nonseparabilities
Two specific examples of time-nonseparabilities are of
first
example
time.
In
captures
the
second
Throughout this section,
IV. A.
In
the
preferences
I
the
notion
consumption
example
I
consumption
that
is
is
interest.
The
substitutable
over
complementary
over
time.
assume that the bliss point is constant.
Exponent ial Depreciat ion
continuous-time
are
time
substitutable from one
model
of
separable
would
instant
the
to
18
assumption
section
III,
the
imply
that
consumption
next,
so
that
the
is
that
not
consumer cares a
great
deal
about
timing
exact
the
consumption.
of
natural
A
way
to
introduce time-nonseparable preferences is to assume that consumption can be
easily
substituted
over
periods
short
time
of
or
preferences
that
continuous in the dimension of the timing of consumption.
restriction
preferences
on
in
suggested by Huang and Kreps
continuous-time
(1987)
models
This intuitive
recently
has
and Hindy and Huang
are
been
Another
(1991).
reason to consider preference structures where consumption can be
readily
substituted over short periods of time is that most consumption goods are
relatively durable.
series
examined
in
In particular,
section
II
the nondurables and services consumption
includes
several
components
that
should be
thought of as durable (for example clothing and shoes).
let g be given by:
To capture these ideas,
fO
if
t
<
g(t) =
(4.1)
,
\
Iexp(5t)
The
fact
easily
that
g(t)
substituted
requirement
of
is
t
positive
over
Huang
where
5
<
0.
iO
time
t^O,
implies
and
the
model
satisfies
Hindy
and
Kreps
and
consumption can be
for
(1987)
that
the
Huang
continuity
The
(1991).
consumption good can also be interpreted as a durable good where
5
governs
the depreciation of the good.
To apply the results of section III,
found.
the
The Laplace transform of g
Laplace
transform of
g.
In
Laplace transform of the operator
the mapping g
is given by g (<)
this
case
(D-5).
g^(<)
c(t) = Ds(t) - 5s(t)
19
(3.13)
= 1/giC,) where
C,
As a result,
section III that s(t) is a martingale implies that:
(4.2)
=
of
-
5
the
,
which
must be
g(.C,)
is
is
the
implication of
= D?(t)
-
5s(t).
Note that the goods process has a component that is the time derivative of a
As
martingale.
a
result,
process
consumption
the
not
is
17
stochastic process but is a generalized stochastic process.
can tolerate very erratic consumption
in
this model,
a
standard
The consumer
because the consumer
"consumes" a weighted average of past consumption that is relatively smooth.
In (4.1),
implies that services are constant over time.
the model
consumption is also constant over time and when
consumption is positive.
the
In a deterministic world,
is restricted to be less than 0.
6
model
<
through
1
of
4
this situation
implies that
(4.2)
0,
the assumption that
Further,
assumptions
satisfies
6
In
5
the
implies
<
appendix
that
that
are
required to apply the results of section III.
Using (3.13) the exponential depreciation model implies that:
c(t) - c(t-l) = (D-5)w(t)
(4.3)
= J-^l-5T)DC(t-T) - X^[l+5(l-T)]DC(t-l-T)
As
in
consumption
time-separable
the
follow
an
MA(1).
first
case,
In
this
autocorrelation value need not be 0.25
even positive.
.
.
in
however,
the
the
time-averaged
first-order
time-separable case) nor
The first-order autocorrelation of the first differences in
R(l) =
:
^'^^
2 +
R(l)
case
in
.18
consumption is
(4.4)
(as
differences
is plotted
"
^
(2/3)5^
in Figure
1.
Note that as 5 goes to
20
-oo,
the value of R(l)
goes
0.25
to
since,
as
5
is
driven
to
the
-oo,
consumption good
instantly perishable and preferences are time separable.
if 5 = -/6
=
(a
becomes
Notice also that
half life of 0.28 periods for the consumption good) then R(l)
and a discrete-time martingale model
would fit
the
consumption data.
This is a further example of the identification problem discussed in section
III.D.
IV. B.
Habit Persistence
Suppose that the consumer cares about
the
level
of
consumption today
relative to an average of past consumption so that the consumer develops an
acceptable level of consumption over time.
has
been
studied,
for
example,
Habit persistence of this form
Constantinides
by
Detemple
(1990),
Zapatero (1991), Novales (1990), Pollack (1970), Ryder and Heal
Sundaresan
Following
(1989).
Constantinides
(1990)
(1973),
model
I
and
and
habit
persistence by assuming that s(t) is of the form:
(4.5)
s(t) = c(t) - a(.-7)S
exp(rT)c(t-T)dT, y<0, 0<a<l
(0,00)
Note
that
the
term
(-r)J" exp(3'T)c(
t-T)dT
is
a
weighted
average
of
past
consumption, and a gives the proportion of this average that is compared to
current consumption to arrive at the level of services today.
model
of
exponential
depreciation,
consumption is complementary over time.
In this example,
(4.6)
g = A -
g is given by:
ttT}
21
habit
persistence
Unlike the
implies
that
.
where A is the dirac delta function and
if
,
(4.7)
t
given by:
is
tj
i
T7(t) =
expiyt)
t
>
The Laplace transform of g in the case is:
that g^iC,)
=
1/giC,)
the operator
(4.8)
=
[D-'jr]/ [D-
D[c(t)
-
lC,-y]/[C,-il-a)y]
(l-a)y]
.
g(C) =
[C,-{l-a.)y]/[C,-y]
which is the Laplace
,
so
transform of
Using (3.13) this implies that:
c(t-l)] = (l-a)y[c(t) - c(t-l)]
-
+ J"^(l-9'T)DC(t-T)
J'^[l+y(l-T)]DC(t-l-T)
.
This continuous-time autoregressive model for c(t) - c(t-l) implies that the
discrete-time observations {c( t )-c( t-1
model
where
Independent
the
of
the
autoregressive
effects of
)
t
:
=0,1,2,
parameter
time-averaging
...} satisfy an ARMA(1,2)
is
given
by
the
data,
habit
induces smoothness into consumption in the sense that
of consumption is positively autocorrelated
time-averaged reinforces this effect.
depreciation,
19
Also,
the habit persistence model
exp[
(
l-a)^]
persistence
the first difference
The fact that consumption is
.
unlike the case of exponential
implies higher order dynamics for
consumption than the time-separable model.
To satisfy the assumptions of the appendix,
the parameters of the habit
persistence model must be restricted such that y<p/Z and y ={\-a)Tr<p/2.
(4.5)
these parameters are further restricted for two reasons.
world of certainty, s is constant (at level
t
is given by:
22
s,
say)
First
In
in a
and consumption at time
c(t) = iexp(y t) -
(4.9)
)[exp{y
(y/^-
-
t)
l]Vs.
•
Notice that if
y<0.
If
=
y
positive
(for
3^
tends to
then c(t)
0,
(by setting a =
large
only if
t)
•
approximately
3'<0.
If
=
expif
20
t
=
consumption
1,
grows
large
For
)
t,
c(t)
is
to
be
this
•
and r >0 implies that y
linearly
geometrically at the rate y
for
[a/(a-l ]s.
)
the appendix is satisfied as long as y <p/2,
3 of
which is again
jt)
_
•
=
positive only if
is
-
s(l
then,
>
r
-
•
expiy tj(l-y/y )s
(y/y )s which
then c(t)
1),
positive, a must be large that one
a
-
»
<
when
and
<
ail.
When
consumption
grows
which allows
a
>
1,
Assumption
0.
This type of explosive consumption path under
.
extreme habit persistence has been emphasized by Becker and Murphy (1988).
the growth
support
To
consumption,
in
consumption since the capital stock at time
k(t) = J'"exp(-pT)c(t+T)dT
(4.10)
When
the
the
endowment
/ exp(-pT)c(t+T)dT
-
e/p,
stock grows along with
is given by:
t
X°°exp(-pT)e( t+T)dT
-
over
constant
is
capital
the
time
level
the
at
stock
that
the
capital
capital
and
consumption
so
.
grows
e,
k(t)
along
=
with
consumption.
restriction
The
deterministic world
that
is
consistent
with a
>
as
1
However in this situation if there is an initial
model,
they will not die out when a
model
the
,
-
problem
empirically.
The
>
1
.
23
long
as
level
of
positive
y
is
in
a
negative.
services in the
and it would be difficult to analyze
assumption that
21
are
a
<
1
in
(4.5)
avoids
this
.
V.
Empirical Results with Time-Nonseparable Preferences
section
this
In
persistence
habit
and
depreciation
results
the
report
I
models
some
must
account
made
be
Eichenbaum and Marshall
Following Christiano,
in
assume
I
,
and
these
consumption
22
(1991)
adjusted
to estimate
In order
trend
the
for
exponential
the
seasonally
to
seasonally unadjusted consumption expenditures.
models
fitting
of
data.
that
the
«
model
applies to detrended services sit)
s (t)
is
the
by
assuming
Given that
trend exp(jit).
is
ji
bliss
the
and
>
/i
This assumption can be
t.
follows
point
the
geometric
mappings from consumption to services are
the
trending consumption c
parameter
that
where
exp(-jit)s (t)
trending level of services at time
directly modeled
linear,
=
inherits
(t)
estimated along with
the
the
rest
trend exp((it).
of
the
trend
The
parameters
each
for
model
Seasonal ly Adjusted Expenditures on Nondurables and Services
V.A.
The exponential depreciation model implies that R(l) is negative if
<
Vs
(see
so
(4.4)),
differences
that the model
monthly
of
becomes more negative,
fact
that
first
the
consumption
R(l)
investigate
autocorrelated.
To
parameterize
moving
the
depreciation
model
'^
(see
consistent with the fact that first
is
negatively
are
autocorrelated.
becomes positive which would
differences
average
(4.3))
quarterly
of
process
implied
-
=
c(t-l)
are
by
the
model,
exponential
the
+
5
positively
depreciation
ec(t)
As
then explain
consumption
exponential
the
as c(t)
|5|
ec(t-l).
This
1
MA(1)
model
can
then be parameterized
down the parameter 6
in
terms of G
and
5,
where 5
through its affect on R(l) given in (4.4).
gives maximum-likelihood estimates of the MA(1)
24
ties
Table 5.1
model using the seasonally
adjusted monthly expenditures on nondurables and services.
value of 5 is -1.36,
The
estimated
implying that the consumption goods have a half-life of
0.51 months.
If a
quarter is used as the basic time interval, the value of
by the monthly estimation is 3*(-1.358) = -4.07.
implied
5
A likelihood ratio test of
of this restriction on the quarterly data versus an unrestricted MA(1) model
gives a P-value of 0.101 for the data from 1952,1 to 1986,4 and a P-value of
0.281
the data from
for
1959,1
to
1986,4.
As a result,
neither data set
rejects this restriction at the 10% significance level and the monthly and
quarterly first-order autocorrelation values can be reconciled with
exponential depreciation model.
simple
a
The exponential depreciation model fits the
first-order negative correlation in monthly consumption differences and time
averaging the data to quarterly frequencies implies a model consistent with
the quarterly data.
Maximum likelihood estimation
23
of the habit-persistence model using the
quarterly consumption series yielded parameter estimates of a that were not
significantly different
differences
in
from
persistence
Habit
zero.
implies
consumption are positively correlated over
aspect of the data is well accounted for by the fact
first
but
time,
this
the consumption
the habit persistence model
In fitting
data is time averaged.
that
that
to
monthly
data the model did very poorly since the monthly data requires a model that
induces negative co.rrelation in the first difference of consumption.
The nonseparability
that people develop a
the fact
that g(t)
instant
to
specify
the
level
is
not
substitutable
persistence
model
25
the
notion
However
implies that consumption from one
>
t
captures
consumption over time.
of acceptable
is negative for
the next
habit
induced by habit persistence
is
24
.
to
A
potentially better way to
first
define
an
intermediate
denoted
service process,
which
s,
captures
the
fact
that
consumption
is
durable or that consumption can be substituted over short periods of time.
The consumer is then assumed to develop habit over the intermediate services
process.
The intermediate service process is generated according to:
sit) = j"exp(5T)c(t-T)dT, 5
(5.1)
Again,
<
.
habit persistence is modeled using (4.5),
except that habit develops
over s instead of consumption directly:
This model
nests
the
exponential
depreciation model
persistence effects to develop much more slowly.
as
habit
persistence
depreciation
(5
with
close
to
autocorrelation of c(t)
depreciation,
-
0<a<l
y<0,
s(t) = s(t) - a(-3-)X°°exp(yT)s (t-T)dT,
(5.2)
exponential
zero)
the
c(t-l)
is
I
and
slow
For
implies
negative.
the
habit
will refer to this model
depreciation.
model
allows
that
Further,
of
first-order
the
for
rates
low
rates
the model satisfies the requirement of Huang and Kreps
of
(1987)
and Hindy and Huang (1991) that consumption be easily substitutable from one
instant to the next.
As 5 is driven to
-oo,
the model approaches the habit
*
persistence model.
Table 5.2 reports results of
the
estimation of
with exponential depreciation model using monthly data
5.2
to
table 5.1
function
in
notice that
adding
the
there
habit
is
some
persistence
depreciation model.
26
the
25
persistence
In comparing table
.
improvement
effects
habit
to
in
the
the
likelihood
exponential
.
formal
A
test
the
of
exponential
depreciation model
habit persistence with exponential depreciation model,
When a =
for
test
the
in
the parameter y is not
likelihood ratio
test
relative
to
the
test of a =
is a
0.
identified and the regularity conditions
break down.
Davies
(1977)
has
suggested a
this situation where the test statistic is found by evaluating the
likelihood
statistic
ratio
at
all
supremum of the resulting values.
possible
values
of
y
taking
and
the
A lower bound on the P-value of this test
can be found by using a chi-square distribution with one degree of freedom
and the likelihood ratio statistic based upon the results of tables 5.1 and
lower
This
5.2.
persistence
to
bound
is
0.067.
exponential
the
As
a
result,
depreciation
the
model
addition
does
not
of
result
habit
in
a
dramatic improvement in the fit of the model
Table
5.3
reports
estimated
half-life
values
for
the
exponential
depreciation and habit-persistence effects using the parameter estimates of
table 5.2.
The half-life of the habit persistence effect is estimated to be
relatively large possibly reflecting longer-run habit formation, which seems
reasonable.
However,
the half-life
is
not precisely estimated.
It
seems
that the degree of habit persistence is difficult to measure with this data
set
In
sum,
there
depreciation model
.
is
strong
evidence
in
favor
of
the
exponential
There is some weak evidence for habit persistence in
conjunction with the exponential depreciation model.
The lack of evidence
for habit persistence alone is due to the fact that the consumption data is
time averaged and this explains the persistence in the first difference of
consumption that the habit persistence model predicts.
27
Seasonally Adjusted Expenditures on Durables
V.B.
examining time-nonseparable preferences
In
Table
goods.
durable
expenditures
on
exponential
depreciation
reports
5.4
consider
to
estimates
of
seasonally-adjusted
using
model
natural
is
it
the
monthly
Notice that the estimated value of 5 is in fact
expenditures on durables.
more negative than the estimate using nondurables and services
(table 5.1)
which implies that the durable goods are less durable than nondurables and
The half-life of the durable good implied by this estimate of 5
services.
months which seems very unreasonable for durable goods.
is 0.42
was first discussed by Mankiw
this puzzle
time
they account
since
averaged
(1982).
raises
which
for
The results of
the
fact
first-order
the
that
the
This puzzle
table 5.4 reinforce
consumption data
autocorrelation
first
the
in
is
difference of consumption.
The
puzzle
series.
presence
since
habit
of
persistence
habit
Table
persistence
5.5
reports
induces
estimates
could
principle
in
smoothness
of
the
into
explain
consumption
the
persistence
habit
this
with
exponential depreciation model using the monthly expenditures on durables.
Notice
first
that
the
log
likelihood
significantly
persistence is added to the exponential depreciation model
5.4
and
5.5)
so
that
there
is
much
more
evidence
when
improves
in
(compare
favor
of
habit
tables
habit
persistence using durable goods. Also notice that the estimate of 5 is much
lower and the impli-ed half-life for the durable good is 1.50 months which is
somewhat more reasonable, but still too small.
The estimates reported in table 5.4 impose a set of restrictions on the
corresponding model for quarterly data.
In particular the implied quarterly
values of 5 and j are three times the values reported in table 5.5 and the
value of a is a third of
the
value reported
28
in
table 5.5.
A
likelihood
ratio test of these restrictions against an unrestricted habit persistence
with exponential depreciation model using quarterly expenditures on durables
results in a P-value of 0.087 for the data from 1952,1
for
data
the
from
1959,1
to
1986,4.
with
As
depreciation for nondurables and services,
1986,4 and 0.358
to
model
the
the model
of
of
exponential
habit
persistence
with exponential depreciation is consistent with time averaging the monthly
data to form the quarterly data.
Cabal lero
in durables
(1990)
has pointed out that there are higher order dynamics
than that captured by the exponential depreciation model.
The
habit persistence with exponential depreciation model captures some of this
evidenced
as
by
the
improvement
likelihood
the
in
function
when
persistence is added to the model, however the model does not do
see
To
job.
this
consider
table
5.6
which
reports
a
habit
complete
likelihood
ratio
statistics and corresponding P-values of tests of the habit persistence with
exponential depreciation model against higher order ARMA models
with
persistence
ARMA(1,2)
model
exponential
for
c(t)
-
depreciation
model
Notice
c(t-l)).
that
implies
the
models provide significant evidence against the model.
be
due
to
lumpy adjustment
and other
costs
of
a
higher
(the
habit
restricted
order
ARMA
This could perhaps
adjustment
as
argued,
for
example, by Caballero (1991).
V.C.
In
Seasonally Unadjusted Expenditures on Nondurables and Services
this
section,
1
examine whether exponential depreciation model
also consistent with the seasonally unadjusted data examined in section
is
II.
Given the weak evidence for habit persistence found with seasonally adjusted
expenditures on nondurables and services,
I
examine whether the seasonally
unadjusted data provides more evidence for habit persistence.
29
As
noted in section
I
the seasonal pattern of consumption requires
II,
the models of section III
that
and
IV be
part of the model for seasonals that
I
modified in some way.
The first
use is formed by assuming that
the
bliss point follows a deterministic seasonal pattern which induces an exact
pattern
seasonal
consumption.
in
Seasonal
bliss
movement
point
used in a similar way by Miron (1986) and Person and Harvey (1991).
continuous-time
problem
there
since
continuous-time
function.
I
that
is
functions
assume
time
continuous
model
that
with
is
an
being
examined
here
dimensional
infinite
class
an
is
of
In the
aliasing
deterministic
match any deterministic discrete-time
that
will
the
bliss
point
follows
corresponding
frequencies
there
been
has
a
seasonal
exactly
to
pattern
the
in
seasonal
frequencies of the observed data.
Since the seasonally unadjusted data is quarterly,
the bliss point
is
assumed to be governed by:
(5.3)
hU)
where
6,6,8
==
Zl
{<t>
and 6
cosi.inj/2)+IB,sin{tnj/2)}
are constants.
This model of b(t) and (3.13)
imply
that:
:5.4)
c(t) - c(t-l) = g^»w(t) + g •B(t:
where
B(t) = Y.I
ia.
[sin(t7rj/2)
+
sin({t-2)nj/Z)]
-P [cosUnj/2 + cos( (t-2)Trj/2)
=
and where a
J
integers,
2a
/tt
J
=
and 8
J
2S /n.
Note
that
]
}
g *B(t),
J
can be represented using seasonal dummies.
30
observed at
the
In order
to capture
the durable nature of the goods and to capture the
first-order autocorrelation noted
negative
intermediate service process s of (5.2).
exponential
depreciation
model
that
table
in
As
in
I
again use
the
the habit persistence with
considered
I
2.4,
above,
consumer
the
is
assumed to compare the current level of s to an average of past intermediate
services.
this case
In
instead of
letting
the habit
stock be a weighted
average of all past consumption, assume that the habit stock is given by the
level
of
the
intermediate service process of exactly one year ago.
In
other words sit) is given by:
s(t) = s(t) - as(t-4)
(5.5)
where
the
transform
basic
of
time
(D-5)/(l-aL*).
]
which
assumed
is
distribution
the
(^-6)/[l-aexp(-4<)
period
for
g
this
Laplace
the
is
to
be
a
case
quarter.
given
is
transfer
The
by:
g^(C)
the
of
Laplace
=
operator
As a result:
c(t) - c(t-l) = J"^(l-5T)D^(t-T) - J'^[l+5(l-T)]DC(t-l-T)
(5.6)
(1
-
aL*)
(D-5)B(t)
+
(1
The
representation given
time-averaged
consumption
have
-
aL*)
in
(5.6)
two
moving-average with a first-order
implies
pieces.
that
The
first
of
(5.6)
is
correct,
then
is
seasonal-autoregressive
second piece is a set of deterministic seasonal dummies.
model
seasonal
adjustment
differences
first
a
of
first-order
structure.
The
Notice that if the
removes
some
of
the
dynamics of consumption that are due to the preferences of the consumer and
31
applying
model
the
to
adjusted
seasonally
data
inferences about the structure of preferences
expenditures
unadjusted
lead
.
29
of
(5.6)
nondurables
on
misleading
to
28
Table 5.7 gives estimates of the parameters
seasonally
will
and
using quarterly
services.
A
likelihood ratio test of a=0 results in a P-value of 0.01% so that there is
strong evidence for the presence of habit persistence
inclusion of seasonal dummies
the
confirms the findings in table 2.4 that
This
this case.
in
Notice that
does not completely remove all seasonal effects.
(5.6)
implies
that first-differences in consumption follow an ARMA(4,1) where all but the
parameters
autoregressive
fourth-order
are
Estimation
zero.
an
of
unrestricted AFIMA(4,1) model for the random piece of the first difference in
consumption yields a
log
likelihood value of
test of the model given in
(5.6)
indicates
that
of
This
0.184.
150.02.
A
likelihood
ratio
against this alternative yields a P-value
model
the
doing
is
a
reasonable
job
in
representing the seasonally unadjusted consumption data.
The estimated value of 5 in table 5.7 is much closer to zero than the
value
unadjusted
data
0.38
is
quarters
Estimation of
adjusted data.
(5=-oo)
seasonally adjusted data.
with
found
results
in
a
a
versus
The
implied
quarters
0.17
half-life
using
using
seasonally
model with just seasonal habit persistence
value
log-likelihood
of
restriction against the model with 5 unrestricted,
Testing
140.61.
this
results in a P-value of
0.0002 for the likelihood ratio test.
The seasonally unadjusted data at quarterly frequencies indicates that
durability
in
goods
the
habit persistence
30
.
is
important
and
there
is
evidence
seasonal
The model with time-separable preferences does a very
poor job in fitting this quarterly data set.
In section V.A,
against
favor
the
for
time-separable
model
and
32
in
of
the
the evidence
exponential
depreciation model came in the form of negative first-order autocorrelation
in monthly data
Unfortunately,
and
the
changing structure of monthly and quarterly data.
Department
the
seasonally-unadjusted
data,
of
so
Commerce
that
does
publish
not
corresponding
the
monthly,
exercise
with
seasonally unadjusted data cannot be performed.
VI.
Concluding Remarks
Time averaging of aggregate consumption data
since
an
is
important problem
destroys the information structure implied by any economic model
it
and can lead to misleading analysis about the underlying economic structure.
Christiano, Eichenbaum and Marshall
taken into account,
showed that once this problem is
(1991)
there is little evidence in quarterly data against
martingale hypothesis for consumption.
However
there
is
great
a
deal
the
of
evidence against the martingale hypothesis using monthly seasonally adjusted
data and quarterly seasonally unadjusted data.
In
this paper
quarterly
I
have shown that the changing structure of monthly and
consumption
expenditures
on
nondurables
and
services
can
be
reconciled by a model in which consumption is substitutable over time, or in
which the consumption goods are durable.
The model of preferences is also
sensible given the continuous-time perspective that
time-averaged
consumption
data.
The
empirical
I
used
to
findings
analyze
support
the
the
continuous-time theoretical developments of Huang and Kreps (1987) and Hindy
and Huang (1991).
In addition to this model of substitution over time,
evidence for habit formation when habit
flow
of
services
created
by
the
is
durable
33
I
found some weak
modeled as developing over the
nature
of
the
goods.
Using
model
durable goods
on
expenditures
seasonally
Quarterly
preferences.
of
evidence
further
found
I
for
this
unadjusted
type
of
consumption
expenditures on nondurables and services provided stronger evidence for a
model of preferences in which consumption is substltutable over time and in
which there is habit formation at seasonal frequencies.
analysis
the
general
In
paper
the
of
implications
has
for
investigations of the temporal aggregation Issue and models of preferences
First in investigations of the effects of time averaging
In other contexts.
of the consumption data it is not appropriate to assume that preferences are
Also the empirical results with seasonally-unadjusted data
time separable.
indicate that In addition to accounting for time averaging of the data,
it
process
of
is
Important
to
whether
examine
results
robust
are
the
to
seasonal adjustment.
Second
introduces
shown
have
I
serious
a
tlme-nonseparable
that
fact
the
that
problem
identification
preferences
to
consumption
data
the
time
is
fitting
in
models
particular,
In
data.
averaged
of
the
spuriously
be
interpreted as being due to habit persistence in a discrete-time model.
I
smoothness
found
in
weak
consumption
evidence
for
due
habit
to
averaging
time
persistence
could
using
seasonally
consumption expenditures on nondurables and services because
the
fact
that
consumption data
the
is
time
averaged.
I
As
adjusted
accounted for
a
result,
in
fitting models of time nonseparable preferences using aggregate consumption
data,
the fact
that
the consumption data
is
time averaged should be
taken
into account.
The analysis
in
this paper has
two major drawbacks:
constant exogenously given interest rate.
that
the
linearity and a
These assumptions were imposed so
implications of the model for time averaged data could be easily
34
developed.
paper
to
In Heaton
(1991).
an endowment
I
'have extended some of
economy where asset
the analysis of this
returns are endogenous and
representative consumer's period utility is not quadratic
(CRRA).
I
the
have
found implications about the structure of preferences similar to the results
of this paper.
However the model in Heaton (1991) assumes that endowments
are exogenously given and makes strong assumptions on other dimensions.
It
are
would be interesting to determine whether the results of this paper
robust
nonlinear
example.
to
the
introduction
time-nonseparabilities
of
have
endogenous
been
production.
proposed
by
Abel
Also
(1990),
more
for
An extension of the analysis of this to that class of preferences
would also be interesting.
Footnotes
for example, Grossman, Melino and Shiller (1985), Hansen
(1991), Litzenberger and Ronn (1986), Naik and Ronn (1987)
(1988).
See,
Singleton
Hall
and
and
2
See Huang and Kreps (1987) and Hindy and Huang (1991) for theoretical
arguments in favor of time-nonseparable preferences in continuous-time
models.
3
This result was originally derived by Working (1960).
See section III
and Christiano, Eichenbaum and Marshall (1991) for a derivation of this
result.
4
In section V,"
I
discuss this assumption further.
The mean m can be introduced by allowing differences in the bliss
allow for this
point of the consumer to be constant over time.
I
possibility so as to match the results reported in Christiano, Eichenbaum
and Marshall (1991).
35
For a detailed discussion
For a discussion of GMM see Hansen (1982).
Eichenbaum
and Marshall (1991).
see
Christiano,
performed
here
test
of
data
to
perform
this
GMM
estimation
consumption
I
used the
the
detrending
In
MA(1)
estimation
reported
unrestricted
in
table
2.2.
from
estimate
trend
the
^Note that the quarterly and monthly data are consistent in the sense
that quarterly averages of the monthly data yield the quarterly data used in
the paper.
p
This fact was also noted by Ermini (1989).
9
In section V, I show that this is consistent with a
model with a deterministic bliss point.
time
separable
The trend value and seasonal dummies were removed using a likelihood
In table
function in which the residuals were assumed to be white noise.
to
1986,4.
2.4, estimates are reported using the sample period 1959,1
The longer data set was
Seasonally unadjusted data is available from 1946.
not used because the simple trend model used in this paper could not account
for what appears to be a dampening of the variance of the seasonal component
This did not occur over the
of the series over the longer time period.
Modeling the change in the seasonal
shorter time period used in this paper.
behavior of the series is beyond the scope of this paper.
This model also occurs in a situation where the bliss point in the
quadratic preferences of the consumer is stochastic and i.i.d. over time
discuss this issue further in Section III.
I
(see Sargent (1987)).
12
Unlike most analyses
nonnegativity constraints.
of
Gorman-Lancaster
technologies,
I
ignore
13
Discrete-time versions of this specification of preferences have been
See, for example, Dunn and Singleton (1986),
considered by many authors.
Eichenbaum and Hansen (1990), Eichenbaum, Hansen and Richard (1987) and
Hansen (1987).
14
The assumption that the preferences of the consumer can be represented
This assumption
with a quadratic utility function is made for convenience.
accumulation
described
below
with
linear
constraint
on
capital
along
the
variables
that
be
laws
motion
for
the
endogenous
can
implies linear
of
of
the
model
for
implications
In
particular,
the
easily analyzed.
The quadratic
time-averaged data can be studied without much difficulty.
assumption for preferences could be viewed as an approximation to a
different utility function.
For a discussion of this approximation issue
see, for example, Christiano (1990).
36
Notice that s(t) is a process that gives the contribution to services
from consumption purchases from time
forward.
To simplify the exposition,
have set initial conditions for services to zero.
I
The issue of initial
conditions is discussed in the appendix.
For a discussion of this result and a derivation of the representation
for wit) see, for example, Grossman, Melino and Shiller (1987).
17
See Gel'fand and Vilenkin (1964) for a discussion of generalized
stochastic processes.
As it stands, this model does not directly fit within
the framework of section III since consumption is not a standard stochastic
However, with a simple modification the analysis of section III
process.
can be applied.
The idea is to think of the consumer as choosing
accumulated consumption at time t, where accumulated consumption is given
by:
c (t) = J c(T)dT.
The mapping from consumption to services can be
easily modified to be a mapping from accumulated consumption to services.
interpret the capital accumulation problem integrate (3.3) over time to
yield a law of motion for another capital
stock where accumulated
consumption is the control.
This remapping of the problem is discussed
further in Heaton (1989) and Hansen, Heaton and Sargent (1991).
To
12
18
The
—
variance
/ (1+5(1-t)) dx
and
of
of
—
c(t)
c(t-l)
-
autocovariance
the
is
is
given
given
by
by
1
-J"
17
S (1-5t) dx
(l-5x)
(
+
l+5( 1-x) )dx
(see Rozanov (1967)).
19
Sundaresan (1989) and Detemple and Zapatero (1991) investigate whether
habit persistence results in smooth consumption in the sense that habit
persistence may lower the volatility of consumption for a given volatility
in the endowment process.
20
Assuming
persistence.
that
a
i
so
that
we
are
considering
a
model
of
habit
21
In the estimation of the habit persistence model reported in section V
found no evidence
searched for values of a in an unrestricted manner.
I
for values of a > 1.
I
22
Actually Christiano, Eichenbaum and Marshall (1991) follow a slightly
The
different strategy for the trend than the one used here section.
does
not
affect
the
results
presented
here.
difference is very minor and
23
and
habit
persistence
with
exponential
persistence
The
habit
depreciation models were fit using the frequency domain approximation to the
likelihood function suggested by Hannan (1970).
24
This implies that the habit persistence model does not satisfy the
continuity restriction of Huang and Kreps(1987) and Hindy and Huang(1991).
37
Maximum likelihood estimates of the habit persistence with exponential
depreciation durability model for the two quarterly data sets yielded very
marginal improvement in the likelihood function over the pure exponential
As a result, I do not report the results here.
depreciation model.
^^Gallant and Tauchen (1990) and Eichenbaum and Hansen (1990) also find
The results
evidence for durability in the nondurables and services data.
results
since
they
account
for
these
the
fact that
reinforce
reported here
averaged.
data
is
time
the consumption
=
be
s(t)
would
to
set
s (t)
general
model
more
A
[a(l-r)/(l—y^ )]s (t-4) so that the habit stock is a weighted average of
A likelihood ratio test of
services one year ago, two years ago and so on.
yields a P-value of 0.26 (with y allowed to be negative in the
y =
When jr is set to zero, we get the model given in
unrestricted estimation).
Since the more complicated model provides very insignificant
(5.5).
improvement in the likelihood function, it is not discussed.
28
Heaton (1989) also examines models in which the durable good has a
This induces peaks in the spectral density of consumption
finite life.
then
If these peaks are close to the seasonal frequencies,
differences.
seasonal adjustment will also remove dynamics that are induced by the nature
of the consumption goods themselves.
29
The estimated
simplicity.
values
of
the
30
an
seasonal
dummies
are
omitted
for
Osborn (1988) and Person and Harvey (1991) also suggest that there is
important role for habit persistence in fitting seasonally unadjusted
data.
38
Appendix
This appendix provides
technical
support
for
section
and gives a
III
more general development of some of the results.
Let
(n.^^.Pr)
be the underlying probability space.
Information in the
economy is represented by a sequence of sub sigma-algebras of ?:
where
t6[0,oo)}
for
3:(t)£3^(s)
t
<
Let
s.
n* =R^
j""
xQ and
be
Fs{^(t):
product
the
+
sigma-algebra given by 9=xS
+
a measure on
where p
the
denote the product measure given by
I
predictable
(1982))
of
denotes the Borel sets of
sigma-algebra
subsets
of
Q
.
Chung
(see
Services
respect to T and square integrable,
at
t,
and
are
to Lebesgue measure,
PrxA,
as Pr*
Williams
required
to
(1983)
be
Let T be
.
Elliot
or
measurable
an element of £ {Q* ,T,Pr*)
i.e.,
the consumer makes choices over services,
implies that,
Let A be
Ir\
+
that has density exp(-pt) with respect
!R
0.
>
where B
.
with
Since
requiring them to be predictable
the consumer can use only information generated by {?
:
s
S < t>.
The consumer evaluates s
€
£ {Q ,T,Pr
)
with the utility function:
Uis) = -(l/2)£{J'"exp(-pt)[s(t)-b(t)]^dt>
(A.l)
Let
g
be
nonrandom distribution on
a
IR
,
then part
of
services are
generated by:
(A. 2)
s^ =
g*c
The distribution g often puts weight on all of R
defined by setting c to be zero for x
conditions
for
the
service
process,
<
let
39
0.
,
in which case,
In order
s (t)
be
a
to
(A. 2)
is
allow for initial
nonrandom
member
of
}
£^iQ* ,T ,Pr*
]
gives the contribution to services at
s (t)
.
The service process at time
initial stock of services.
s(t) = s^(t)
s^it),
+
t
time
from the
t
then given by:
is
t>0.
Since the preferences of the representative consumer are defined over
elements of £ {Q ,T,Pr
of
some restriction needs to be placed upon the space
),
consumption processes and the distribution g such that
given by
results in a
(A. 2)
member
The first restriction that
member of
(Q
i£
version of
,Pr
,9^
denoted
g,
=
such that
(c
{Be
2
-£
2
+(j
€
1
)
'
where
)}
*
"
g' (w)g' (w)
c'
= ch,
[Dc]'
is
and
^
i
is
I
for
iO and
t
= Dch and so on.
is
restricted
the smallest
essentially bounded and where
integer
g'
is
the
g'
implies
processes into £ [Q
(A. 3)
where h(t) = exp(-ct)
Similarly let
£^(Q* ,T ,Pr*
Fourier transform of
Assumption
define a discounted
The space of admissible consumption processes
1:
C
to be:
the mapping yields a
this restriction,
= gh,
g'
as:
c = p/2.
zero otherwise and
Assumption
g'
convolution
).
insures that
impose
I
To describe
).
£ {Q ,'P,Pr
of
the
,'3'
that
,Pr
(A. 2)
),
maps
space
the
of
admissible
consumption
since:
£{j"exp(-pt)[ g*c(t) ]^dt} = £{/"
=
[g' »€( t
£{/"
]
)
^dt
g' (w)g' (w)
c' (cj)
c' (w) d(j>.
-00
where
c' (w)
is
the Fourier
follows from the Parseval
transform of
formula.
c'
Note
and we have:
40
.
that
The second equality in
(e
+
D)[D^c)' it)
=
(A. 3)
{D^*^c)'
.
£{J'"exp(-pt)[ g»c(t)
(A. 4)
]^dt}
/\
(^^+'^^'
EiSll
Assumption
1
^
g' (w)g' ((!))(D^c)' (w)(Dc^)' (o;)*dw.
and the Holder inequality imply that
To insure that the mapping
the economy,
Assumption
plane:
where
>
The Laplace
2:
cr'
>
p/2}.
satisfies the information structure of
(A. 2)
;g(C)i
,
-
transform of
Further,
gC^),
g,
for each real
some
for
!^ (C)'
compact sets Kc[cr',co) where
analytic in the half
is
p/2 and < =
>
cr'
T
polynomial
+
iw
which depends
on
er
o-eK.
Using Theorem 2.5 of Beltrami and Wohlers
2,
is finite.
(A. 5)
the following additional assumption is imposed upon g:
{C:Re(C)
o-
/\
(1966,
p.
51),
imply that g is one-sided, putting weight only on R
subset of) ^^(n*,y,Pr*) into I^ {7i ,9 ,Pr*
Assumptions
1
and
and maps (possibly a
)
The problem facing the consumer is to maximize the objective function
the mapping
(A.l)
subject
(3.3)
by choice of Dc(t).
to
I
necessarily
not
consumption
(A. 2)
Because the control variable of the problem is
but
itself
convenient to rewrite the mapping
g^
and the capital accumulation technology
is
a
1
as:
s
that
is
(A. 2)
can always be found using a result
derivative
I
= g„*D c.
The
solves
representative consumer
the
following
problem:
(MP)
Max
(-l/2)£{J°'expC-pt)[s(t)-b(t)]^dt>
subject to:
I
2
s(t) = gp*D c + s (t) and
41
p.
c,
it
is
The distribution
analogous to
parts (see, for example, Beltrami and Wohlers (1966,
of
integration by
28)).
resource
allocation
Dk(t) = pk(t)
by choice of a D c process
in
C
+
e(t)
-
c(t).
k(0) given
i=nJ,c
Letting y {t)=D
for j=0,2,
.
(.,
the
Lagrangian for this problem is given by:
£ = -£-|j" exp(-pt)j(l/2)
A (t)
[g^*y^(t)
k(t)
-
kiO)
£-1
r
where A
and A
k
,
A (t)
j=0,l,
y (t)
i-1
s^(t) -
b{t)f
pT'-k(T)dT - j''e(T)dT + f^y (T)dT
-
- y
(0)
j
J
...,
+
-
{T)dT
S y
o-'j +
i
11
are Lagrange multipliers.
J
The first-order conditions for the choice of k(t) and y
for each t^O, are then given by:
(A. 5)
(A.
,
j=0,l,
...
i
£|-gj*[g^»yg(t)
(A. 9)
As in section III,
unit
of
is
a
b(t)]:?(t)|
-
=
the left side of (A. 9}
consumption
the
consumption
s^t)
+
good
martingale.
at
time
To
use
looking inverse for g„ is needed.
A^(t)/p^
and
t
marginal utility of a
is the
this
the
result,
marginal
utility
one-sided
a
of
forward
This inverse exists under the followi ng
restriction on g (see Beltrami and Wohlers (1966)).
Assumption
3:
real
p/2 and
a-'
>
polynomial T
l/g(C)
C,
is analytic
=
cr
+
for
iw where
{C,:
>
cr
which depends on compact sets
ReiC,)
<r'
>
:i/g(^):
,
Further,
p/2}.
Kc[o-',oo)
<
where
This assumption also implies that there is distribution g
= A,
where A is the Dirac delta function.
(A. 10)
c(t+l) - c(t) = g^*s[ jD^(t)
-
+
'.T
(<)
;
for each
for
creK.
such that:
As in section III we have:
g^*[b(t+l) - b(t)]
2
s (t)
is
ignored.
This is justified
(asymtotically) under the following assumption:
4;
g^*g
g^»[s^(t+l) - s^(t)]
In the text the initial condition,
Assumption
some
Ther'e exists an
i^
>
s
exp{vt)g *s
such that lim
t-K»
43
2
(t)
= 0.
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of
1
TABLE 2
.
ESTIMATES OF MA(1) MODEL FOR FIRST-DIFFERENCES
OF PER CAPITA, SEASONALLY-ADJUSTED QUARTERLY CONSUMPTION
EXPENDITURES ON NONDURABLES AND SERVICES.
FIRST-ORDER AUTOCORRELATION RESTRICTED TO 0.25
Parameter Estimates
Parameter
52,
1
to 86,4
59,1 to 86,4
^ (xlO^)
0.475
(0.052)
0.485
(0.060)
e
0.224
(0.016)
0.253
(0.019)
o
Log-Likelihood
67.046
92.491
^Standard errors are in parentheses.
TABLE 2.2
ESTIMATES OF MA(1) MODEL FOR FIRST-DIFFERENCES
OF PER CAPITA, SEASONALLY-ADJUSTED QUARTERLY CONSUMPTION
EXPENDITURES ON NONDURABLES AND SERVICES.
UNRESTRICTED ESTIMATION
Parameter Estimates
Parameter
52,1 to 86,4
yL
(xlO^)
e
e
59,1 to 86,4
0.474
(0.053)
0.487
(0.059)
0.224
(0.016)
0.252
(0.019)
0.062
(0.019)
0.060
(0.023)
1
Log-Likelihood
67.092
92.494
Standard errors are in parentheses.
48
TABLE 2.3
ESTIMATES OF MA(1) MODEL FOR FIRST-DIFFERENCES
OF PER CAPITA, SEASONALLY-ADJUSTED MONTHLY CONSUMPTION
EXPENDITURES ON NONDURABLES AND SERVICES, 59,1 TO 86,12
Parameter
Estimation with 0.25 Restriction
Unrestricted^
on R(l)^
M
(xlO^)
e^
-
0.156
(0.027)
0.164
(0.017)
0.248
(0.015)
0.216
(0.010)
-0.047
(0.011)
e^
Log-Likelihood
211.33
254.54
Standard Errors are in parentheses.
TABLE 2.4
AUTOCORRELATION VALUES FOR FIRST-DIFFERENCES
OF PER CAPITA, SEASONALLY-UNADJUSTED QUARTERLY CONSUMPTION
EXPENDITURES ON NONDURABLES AND SERVICES, 59,1 TO 86,4
TREND AND SEASONAL DUMMIES REMOVED
Autocorrelation
Order of
Autocorelation
"
1
-0.059
(0.087)
2
-0.027
(0.083)
3
0.167
(0.107)
4
0.327
(0.080)
5
-0.073
(0.086)
These were calculated using
^Standard errors are in parentheses.
procedure.
(1987)
Newey-West
in
the
lags
one year of
49
1
TABLE
5.
ESTIMATES OF EXPONENTIAL DEPRECIATION MODEL
USING PER CAPITA, SEASONALLY-ADJUSTED MONTHLY
CONSUMPTION EXPENDITURES ON NONDURABLES AND SERVICES,
59, 1 TO 86,12
Estimated Value
Parameter
11
(x
10^)
e
Log-Likelihood
164
(0.017)
0.215
(0.010)
1.358
(0.161)
0.
254.54
Standard errors are in parentheses
50
TABLE 5.2
MAXIMUM LIKELIHOOD ESTIMATES OF PARAMETERS OF HABIT PERSISTENCE
WITH EXPONENTIAL DEPRECIATION MODEL, USING
PER CAPITA, SEASONALLY-ADJUSTED MONTHLY
CONSUMPTION EXPENDITURES ON NONDURABLES AND SERVICES,
59,1 TO 86, 12
Parameter
H
(x
Parameter Estimates^
10^)
0.
156
(0.024)
(T
0.132 (0.010)
5
-1.558 (0.743)
y
-0.211
(0.414)
0.438 (0.308)
a
256.22
Log-Likelihood
Standard errors are in parentheses
TABLE 5.3
ESTIMATES OF HALF-LIVES IN MONTHS
FOR DURABILITY AND HABIT PERSISTENCE EFFECTS WITH MONTHLY DATA
Depreciation Parameter
Estimates
S
0.656
(0.104)
y
3.293
(6.473)
Standard errors in parentheses
51
TABLE 5.4
ESTIMATES OF EXPONENTIAL DEPRECIATION MODEL
USING PER CAPITA, SEASONALLY- AD JUSTED MONTHLY
CONSUMPTION EXPENDITURES ON DURABLES, 59,1 TO 86,12
Parameter
(x
fi
Estimated Value^
10^)
e
Log-Likelihood
0.389
(0.036)
0.137
(0.009)
•1.647
(0.280)
278.66
Standard errors are in parentheses
TABLE 5.5
MAXIMUM LIKELIHOOD ESTIMATES OF PARAMETERS FOR HABIT PERSISTENCE
WITH EXPONENTIAL DEPRECIATION MODEL, USING
PER CAPITA, SEASONALLY-ADJUSTED MONTHLY CONSUMPTION
EXPENDITURES ON DURABLES, 59,1 TO 86,12
Parameter
fi
(x
Parameter Estimates
10^)
0.390 (0.035)
-
(F
0.047 (0.005)
8
-0.461
r
-4.176 (0.059)
a
0.754 (0.069)
Log-Likelihood
281.87
Standard errors are in parentheses
52
(0.137)
TABLE 5.6
LIKELIHOOD RATIO TESTS OF HABIT PERSISTENCE
WITH EXPONENTIAL DEPRECIATION MODEL
AGAINST UNRESTRICTED ARMA MODELS
USING PER CAPITA. SEASON ALLY- AD JUSTED MONTHLY
CONSUMPTION EXPENDITURES ON DURABLES, 59,1 TO 86,12
ARMA Model
Likelihood Ratio Value
P-Value
ARMA(1,2)
3.52
0.061
ARMA(1,3)
8.32
0.016
ARMA(1,4)
12.4
0.006
ARMA (1,5)
14.6
0.006
ARMA(1.6)
16.3
0.006
53
TABLE 5.7
ESTIMATES OF EXPONENTIAL DEPRECIATION MODEL
WITH SEASONAL HABIT FORMATION
USING PER CAPITA. SEASONALLY-UNADJUSTED
QUARTERLY CONSUMPTION EXPENDITURES ON NONDURABLES AND SERVICES,
59,
1
TO 86,4.
ESTIMATES OF SEASONAL DUMMIES OMITTED
Estimated Value
Parameter
fi
Log-Likelihood
0.408
(0.074)
e
0.127
(0.010)
5
-1.802
(0.414;
a
0.349
(0.087:
(x
10^)
147.60
^Standard errors are in parentheses
54
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