O&WEY
HD28
.M414
~7
IV
OCT
T 1990
-*
ALFRED
P.
^s
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
The Interaction between Time-nonseparable
Preferences and Time Aggregation
John Heaton
Dept.
of Economics and Sloan School of Management
Massachusetts Institute of Technology
WP #3181-90-EFA
June 1990
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
The Interaction between Time-nonseparable
Preferences and Time Aggregation
John Heaton
Dept.
of Economics and Sloan School of Management
Massachusetts Institute of Technology
WP #3181-90-EFA
June 1990
V
\ *2
v^
&
,,
The Interaction Between Time-Nonseparable Preferences
and Time Aggregation
John Heaton
Department of Economics and Sloan School of Management
M.I.T.
June 1990
ABSTRACT
develop and empirically analyze a continuous - time
In this paper I
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
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.
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 Lars Peter Hansen
for his
many suggestions and his constant
encouragment.
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.
,
,
I.
Introduction
In
paper
this
linear-quadratic,
develop
I
show how
time
consumption
streams
can
several
of
aggregation and
analyze
consumer model
representative
time-nonseparable preferences
empirically
and
time
in which
show
I
that
durable
and in which individuals
The presence
from the good.
important because
the
of
in which
consumer has
framework
preferences
in
behavior
the
seasonally adjusted and unadjusted consumption data is
model of time-nonseparable preferences
the
Within this
forms.
nonseparabilities
interact.
continuous- time,
a
of
over
both
consistent with
a
consumption goods are
the
develop habit over the flow of services
time nonseparabilities
in preferences
not support a version of the model
data does
I
is
that
focuses solely upon time aggregation and ignores time nonseparabilities
in
preferences by making preferences time additive.
There has been an extensive
consumption
aggregate
by
implied
the
the data
expenditures
permanent
asset-pricing models.
amount of work examining whether or not
consistent
are
income
hypothesis
with
the
and
consumption-based
restrictions
Often these studies find that the theory does not fit
.
Because of these poor empirical results, many other studies relax some
of the auxiliary assumptions made in previous studies.
A closer examination
of the assumption that the decision interval of economic agents matches the
interval
of
data
collection caused
some
authors
to
replace
assumption that the interval between data observations
actual decision interval.
can
often
Christiano,
with
larger
the
than the
The result is a temporal aggregation problem that
allowed
for
in
Eichenbaum
and
Marshall
be
is
it
estimation
(1987)
and
inference.
find
that
For
rejections
example,
of
the
-
permanent income hypothesis could be due to temporal aggregation problems.
has
there
Concurrently,
been
much
work
examining
the
impact
2
of
time-nonseparable preference specifications upon the consumption-based CAPM
and the permanent income hypothesis.
and
Hansen
(1990)
discrete -time
models
and
Ogaki
for
(1988),
for
allow
that
Dunn and Singleton (1985), Eichenbaum
empirically
example,
durability
in
examine
consumption
the
data.
Although allowing for durability seems to help the fit of the model,
the
results do not provide a complete rationalization of consumption and asset
price data.
The purpose of this paper is to show how time-nonseparable preference
to understand much
specifications along with time aggregation can help us
about
the
behavior of aggregate
consumption.
motivation for
One
such
a
study is the fact that all studies, which examine the empirical relevance of
the temporal aggregation problem,
Within a continuous- time or fine discrete-time
time separable preferences.
this
model,
assumption
conduct the analysis within a framework of
far
is
from
appealing
since
implies
it
that
an
individual's preferences over consumption at one instant are unaffected by
consumption the instant before.
Hindy
and
within
a
Huang
(1989)
argue
In this vein,
that
the
Huang and Kreps
assumption
should be
and
time-separability,
These authors also argue
continuous -time model, should be dropped.
that preferences
of
(1987)
specified such that consumption at dates near
time t should be relatively substitutable for consumption at time
t.
The
argument against using a separable preference specification can also be made
on
the
grounds
consumption
that
of
a
for
measurement
most
problem:
National
classifications have a very durable nature.
clothing are defined as being nondurable.
we
Income
have
and
For example,
observations
Produce
on
Account
expenditures on
In conducting this study,
permanent
income
nonseparabilities
technology with
develop a continuous- time linear-quadratic
with
augmented
model
The
preferences.
in
a
I
general
a
form
assumes
model
of
very
a
time
simple
Although this is a
fixed rate of return to investment.
very severe restriction, my goal is to use this model as a laboratory for
aggregation and nonseparabilities
understanding how temporal
and
interact
for determining what type of preference specifications should or should not
be ruled out before moving on to more complicated environments.
I
apply
examples
several
unadjusted nondurable
of
time-additive version of the model
seasonally
to
consumption
services
plus
model
the
is
data.
adjusted
show
I
not consistent with the
However,
adjusted consumption data at monthly frequencies.
and
that
the
seasonally
the
model
is
consistent with the data if the consumption goods are allowed to be durable.
Further
the
I
form
show that there is some weak evidence for habit persistence (of
studied
Ryder
by
Constantinides (1990)
,
and
Heal
Sundaresan
(1973),
and
(1989)
for example) only if the habit is developed over the
flow of services from the durable nature of the goods.
Using quarterly seasonally unadjusted data
present evidence that the
I
durable nature of the goods may be understated in seasonally adjusted data
and
that
indicates
time- separable
that
there
is
perform
very
poorly
significant
role
for
models
a
.
habit
Also
data
this
persistence
in
understanding the seasonal behavior of consumption.
Along with these general empirical points
temporal
aggregation
and
make
relatively consistent with the data?
negative
also address the following
Can the presence of time-nonseparabilities mitigate the effects
questions:
of
I
autocorrelation
in
a
discrete-time,
time-separable
model
Can temporal aggregation overturn the
consumption
that
is
usually
introduced
with
and make
goods,
durable
a
story
durability
consistent with
positive
the
correlation in quarterly consumption?
rest
The
of
the
paper
structured as
is
In
follows.
Section
2,
I
examine a time-additive model that allows me to focus upon the effects of
aggregation.
temporal
I
data
adjusted
seasonally
quarterly
show that although the model
model
the
inconsistent
very
is
consistent with
is
with
monthly seasonally adjusted data and quarterly seasonally unadjusted data.
In
Section
I
3,
develop
a
with
model
nonseparabilities in preferences.
general
very
forms
of
time
show that with proper restrictions on
I
preferences, a law of motion for consumption can be derived that can be used
to estimate
the preference parameters.
also discuss why it is necessary
I
to focus upon specific parametric forms of the preferences.
Section 4 examines the capital stock and consumption dynamics implied
by
model
the
certainty.
I
in
the
discuss
case
habit
of
persistence,
in
a
world
of
perfect
further restrictions that are necessary so that the
model produces reasonable economic results.
In
Section
time-nonseparable
adjusted
5,
I
empirically
structure
preference
consumption
examine
First,
data.
of
I
several
Section
examine
3,
the
examples
using
effect
of
the
seasonally
of
making
preferences time-nonseparable by assuming that consumption goods are durable
with exponential
allow
for
habit
depreciation.
persistence
I
also
effects.
examine
Section
some
specifications
empirically
6
that
examines
a
version of the model that allows for seasonality in consumption and takes
this model to seasonally unadjusted data.
Section
7
concludes the paper.
II.
A Time-Separable Model
In this section,
income model.
I
examine a continuous- time linear-quadratic permanent
The preferences of the representative consumer in the model
time- separable so that
of this section are assumed to be
upon the time aggregation problem.
that
time
I
show,
aggregation
seasonally-adjusted
did Christian,
as
can
explain
Within the context
Eichenbaum and Marshall
much
aggregate
(s.a.)
can first focus
The model is a continuous -time version
of a model studied by Hansen (1987) and Sargent (1987).
of this model,
I
of
behavior
the
consumption.
quarterly
of
However,
(1989),
I
present
evidence using monthly s.a. consumption and quarterly seasonally unadjusted
(s.u.)
that
data
consumption data
time
3
.
that
inconsistent with
is
aggregation does not provide
the
complete
a
model and suggests
rationalization of the
This motivates the need to look at time-nonseparable specifications.
II. A.
A Continuous -Time Permanent Income Model
Assume first that there is a representative consumer with preferences
*
•
over consumption streams, c
(2.2)
U(c*) - -(1/2)
- [c (t):
£(/%xp(Vt)
<
(c*(t)
where b(t) is assumed to grow at the rate
(2.3)
b (t) - exp(/it)b, m >
t
<«
}
,
of the form:
2
-
b*(t)) dt]
/*:
0.
The geometric trend in b (t) will induce growth in the consumption process
:
,
Consider detrended consumption, c - (c(t): 0<t<«)
chosen by the consumer.
which is given by:
c(t) - exp(-/it)c*(t),
(2.4)
Then the preferences
of
the
t>0.
consumption given in
trending
over
consumer
(2.1) can be rewritten as preferences over detrended consumption:
U(c) - -(1/2) £(/%xp(-pt)
(2.5)
where p as
p
In what follows,
+ 2p.
As
consumption.
I
show
will
2
(c(t)
I
b) dt),
-
will refer to detrended consumption
subsequently,
this
remove all of the nonstationarity in consumption.
consumption process,
with
the
model
of
will have a unit root.
c,
Eichenbaum
Christiano,
and
detrending does
In fact,
This
is
to
Marshall
the
be
(1987)
not
detrended
contrasted
in
which
detrended consumption is stationary.
There is a technology for transferring consumption over time that has a
constant rate of return of
(2.6)
Dfc(t)
p
- pk(t) + e(t)
< t <«)
where (e(t):
-
c(t),
t>0
,
k(0) given,
is stochastic process
that describes the behavior of
endowments in the economy and k(t) gives the level of the capital stock in
the
economy
at
time
t.
uncertainty in the economy.
The
endowment
random
Note that
I
provides
the
source
of
have equated the risk free rate of
return to the discount factor after detrending the consumption process. This
corresponds
to
the
assumption
made
by
Flavin
(1981),
for
example.
.
Christiano,
Marshall
and
Eichenbaum
follow
(1987)
different
a
route
and
equate the discount factor to the rate of return, before detrending.
The
consumer maximizes
constraint (2.6)
.
objective
the
In Section 3,
function
consider solutions
I
subject
(2.3),
to
to
the
general model
a
that nests this model, however it is easy to see that the optimal consumption
process will be a martingale just as in Hall's (1978) model
.
because a unit of the consumption good given up at period
yields exp(pr)
units
period t+r
at
,
via
As
(2.6).
a
result,
the
t
This occurs
consumer
equates
the
marginal utility of the consumption good (muc) today to discounted expected
marginal utility of consumption tomorrow, times exp(pr):
(2.7)
muc(t) - exp(pr)E{exp(-pr)muc(t+l)|?(t)), r>0
- £{muc(t+r) |^(t))
where
!F(t)
gives the information set at time
of consumption at time
(2.8)
t
is
-(c(t)
c(t) - E{c(t+r)|3F(t)},
r
-
>
b)
,
(2.7)
Since the marginal utility
t.
implies that:
0.
The time derivative of consumption then satisfies:
(2.9)
Dc(t) - D£(t)
where D£(t)
is
the
(instantaneous)
innovation in the processes
the permanent value of the endowment shock process at time
the (random) measure induced by D£(t) satisfies £{D£(t)
2
t.
I
2
}
- a dt.
describing
assume that
Time Series Implications for Time -Ave raged Consumption
II. B.
The data are observations of time -averaged consumption expenditure over
a unit
differences
discrete
Jf*
t-i
consumption
time -averaged
in
time,
c(r)dr.
(2.10)
Although consumption is a martingale in continuous time,
of time.
when sampled
the
at
will
integers.
not
To
be
see
a
martingale
this,
let
c(t)
in
•
Then consider:
c(t+l)
-
c(t) - J^
- J^
+1
c(r)dr
+1
-C
[c(0
-
-
J^cCOdr
c(r-l)]dr
[j><r-l +r)dr]dr.
By changing the order of integration in the
last term of (2.10),
c(t+l)
-
c(t) can be expressed as:
(2.11)
c(t+l)
-
c(t) - u(t+l),
where (u(t): t-0,1,2,...) has the first-order moving-average representation:
(2.12)
u(t) - J^_ (t-r)Df (r) + £*(r-t+2)D|(r)
i
,
t-0,1,2
The first-order autocorrelation for u is:
(
2.13)
g<»Ct)u(t-l)) R(l) 2
£{u(t)
}
£&-!
(2/3) a
- 0.25
2
This result was originally derived by Working (1960)
7
In
version
of
matched with
the
discrete- time
a
consumption
data
is
this
where
model,
consumption
the
process
in
observed
the
model,
consumption differences are predicted to be uncorrelated over time [see e.g.
Hall
The
(1978)].
fact
first
the
that
consumption is correlated at the first lag,
difference
time-averaged
could help to explain some of
of the discrete-time permanent
the rejections
in
income hypothesis.
This
is
what Christiano, Eichenbaum and Marshall (1987) find when they use quarterly
data.
II. C
Data
To test the implications of the models in this paper,
I
focused solely
Q
upon consumption data
expenditures
.
The consumption measure
on nondurables
services
and
where
expenditures are those measured by the U.S.
seasonally adjusted (s.a.)
was
used.
The
quarter of 1952
from the
first
seasonally
to
the
quarter
used was per capita real
aggregate
the
adjusted
quarterly
1959
The
the
to
consumption
Department of Commerce.
and seasonally unadjusted (s.u.)
end of 1986.
of
I
data
runs
Both
quarterly data
from
the
first
seasonally unadjusted data runs
end of
1986
g
.
Monthly
seasonally
adjusted data from January 1959 to the end of 1986 was also used.
II .D Tests of the Model Using Quarterly and Monthly,
S.A.
Consumption
Data
The model predicts
order
1.
that c(t)
-
c(t-l)
is
Parameterize this moving average as:
a moving average process of
c(t)
(2.14)
8
is
is
e(t) + ^t(t-l)
9
Q
where £(«(t)
R(l)
c(t-l) -
-
2
-
)
and £{e(t)e(r)) -
1
we can estimate
0.25,
implied by
then
for
*
r
Under the assumption that
t.
and the trend parameter ^.
8
autocorrelation restriction.
the
The parameter
Table
2.1
gives
estimates of the parameters of the model with R(l) restricted to 0.25 using
seasonally-adjusted
Estimates
consumption.
of
given with
are
(2.14)
8
2
unrestricted are given in Table 2.2.
reported for the period 1952,1
are
estimated parameters
In both tables
1986,4
to
1959,1
and
1986,4.
to
The
latter subsample was used since this matches the period of the monthly data.
Log- likelihood values
also
are
reported
each
in
log- likelihood for each sample marginally improves
Likelihood
2.2.
tests
ratio
of
that
the
from Table 2.1 to Table
restriction
0.25
the
Notice
table.
for
yield
R(l)
probability values of 0.73 for that data set from 1952,1 to 1986,4 and 0.78
from
data
for
1959,1
This
1986,4.
to
because
occurs
autocorrelation of differences in detrended consumption
0.276
data
(0.067)
11
for
Hence
set.
longer data set and 0.260
the
the
restriction
on
R(l)
first-order
was estimated to be
(0.081)
implied
the
by
the
for
the
model
shorter
is
very
consistent with the data.
The model also implies
be
useful
in predicting
likelihood ratio
against
higher
and
tests
For
moving- average
models
significance level.
consumption change today.
P-values
of
moving- average
order
differences.
the
that information lagged two periods should not
the
of
longer
orders
of
models
data
2,
tests
set,
3
and
time
the
the
tests
of
4
not
do
set and the moving average model of orders 4 and
5
5
2.3
reports
additive
detrended
for
The moving average model of order
10
Table
consumption
model
reject
model
at
against
the
5%
for the longer data
for the shorter data set
against
evidence
provide more
the
However,
model.
could be
this
due
to
spurious correlation induced through seasonal adjustment or to the presence
cases,
tests
the
12
that are not completely removed.
of seasonal patterns
Table
in
that
indicate
2.3
Other than these
time-additive
the
reasonably consistent with the data as noted by Christiano,
model
is
Eichenbaum and
Marshall (1987).
Now
consider
consumption is
Since
measures
monthly
martingale
a
robust
seasonally
of
continuous
in
different
adjusted
the
time,
consumption.
nature
of
the
results
should
However,
the first-order autocorrelation in first-differences of detrended
monthly
consumption
be
significantly negative
to
estimated
is
The
.
intervals
-0.188
be
to
of
time
aggregation.
which
(0.052),
is
trend was estimated using maximum likelihood
just as was done for the quarterly data.
Table 2.4 reports estimates of the
trend and the moving average parameters
for
with and without
restriction using monthly
0.25
seasonally
adjusted
A likelihood ratio test of the restriction yields a probability value
data.
of
the
difference in consumption
the
essentially zero.
As
result,
a
the
model
does
not
seem
to
be
very
consistent with monthly seasonally adjusted data.
Why
is
this
happening?
Why
is
the
frequencies but not at monthly frequencies?
model
successful
at
In the next section
I
quarterly
relax the
time-separability assumption imposed upon preferences by (2.3) and show that
the
negative
Before
moving
autocorrelation
examined.
correlation
first-order
on
to
this
values,
in
monthly
preference -based
another
explanation
data
explanation
for
this
is
not
for
surprising.
the
finding
changing
must
be
Consider a discrete-time version of this time separable model in
which the model is assumed to be correct at monthly frequencies.
In this
the model implies that monthly consumption is a martingale.
Suppose
case,
11
equal
now that observed consumption is
to
consumption plus
model
the
an
Then
i.i.d. measurement error that is also independent of the model shocks.
observed consumption differences are given by:
c(t)
(2.15)
c(t-l) - u (t) + u (t)
-
where
is
u
model
the
u (t-l)
-
z
2
x
and
error
measurement error.
i.i.d.
the
is
u
2
1
Let E[u (t)
2
)
- a
2
and £(u (t)
2
- a
)
2
Note
.
that
that
implies
(2.15)
consumption differences have a first-order moving average representation and
2
that the first-order autocorrelation value is R(l) - -a /(a
let
a
2
-
then
1,
the
autocorrelation
first-order
for
2
+ a
_
2
)
detrended
.
If we
monthly
l
consumption differences gives us an estimate of a
2
of 0.232.
Given the behavior of consumption at monthly frequencies in (2.15),
it
is easy to show that quarterly consumption also follows a first-order moving
average
R
-
(1)
where
process
(19a
q
12 12
2
2
+
estimated value of a
R (1) of 0.162.
)/(4a
6ff
2
autocorrelation value
first-order
the
2
+
6a
2
-
)
(19
+
2
6a )/(4
+
6a
2
from the monthly data,
this
2
).
given by:
is
Using
the
2
implies
an estimate of
Although this is not equal to the value of 0.276 found with
i
quarterly data from 1959 to 1986,
is equal to 0.162,
level.
As
a
aggregation,
likelihood ratio test of whether R (1)
does not reject this restriction at the 10% significance
result,
is
a
this
capable
measurement
of
error
explaining
model,
the
autocorrelation values in quarterly and monthly data.
along with
changing
temporal
first-order
However, is an i.i.d.
measurement error model for consumption levels reasonable?
As Wilcox
(1988)
has argued,
a
component of the measurement error in
consumption is due to the misallocation of retail sales into the different
categories of consumption by the Commerce Department.
12
Since the allocation
.
correlation
in
measurement
the
measurement error
due
is
the
to
error.
any
If
based
upon
periodic
substantial
positive
is
introduces
this
establishments,
retail
of
surveys
classifications
consumption
into
sales
retail
of
other
component
this will also
introduce positive autocorrelation in the measurement error.
autocorrelation
error
measurement error will
reduce
the
role
Any positive
of measurement
explaining the behavior of consumption at monthly and quarterly
in
frequencies
.
in
the
collection practices of the Department of
and if these practices do not change very often,
Commerce,
of
and will
leave
a
large
role
for
the
economic
model
that
I
,14
,
consider
in the next sections
.
II. E
,
Tests of the Model Using Quarterly S.U. Consumption Data
In this
subsection
I
examine the
has for seasonally unadjusted data.
implications
that
simple model
this
To account for seasonality in the data,
the model must be modified in some manner.
One way to induce seasonality in
consumption is to add a seasonal deterministic bliss point process that will
induce
a
deterministic
pattern
seasonal
in
consumption.
Seasonal
point movements have been used in a similar way by Miron (1982).
6
I
show
that
the
time
additive
with
model
a
deterministic
bliss
In Section
bliss
point
process implies that consumption differences follow the process:
(2.16)
where
3(t)
-
again
autocorrelation
c(t-l) - u(t) + B(t)
u(t)
equal
has
an
to
0.25,
representation
MA(1)
and
B(t)
seasonal dummies
13
is
a
sequence
with
of
first-order
deterministic
consider the autocorrelation structure of the first
As a first pass,
consumption with
s.u.
in
differences
trend
and
seasonal
Table 2.5 gives the autocorrelation values for residuals.
dummies
removed.
The trend value
and seasonal dummies were removed using a likelihood function in which the
residuals
were
assumed
white
be
to
misspecified likelihood function,
Although
noise.
this
may
be
a
it will yield consistent estimates of the
autocorrelation values.
First, unlike the seasonally
There are two important things to notice.
adjusted quarterly data,
first-order
the
autocorrelation is not close
Hence with this different manner of accounting for seasonality,
0.25.
with
data
even
the
time -additive
model
frequencies.
Second, note that the autocorrelation value at the fourth lag
is
significantly
is
consistent
to
not
positive
and
large.
the
indicates
This
Quarterly
at
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.
III.
A Model with Time-Nonseparable Preferences
In this section,
dependencies
modify the model of section
I
preferences
in
over
consumption.
2
and introduce temporal
Nonseparabilities
are
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.
into
services
is
type
a
of
The mapping from consumption
Gorman- Lancaster
technology
in
which
the
consumption goods are viewed as bundled claims to characteristics that the
consumer
durable
cares
goods
about
and
.
habit
The
as
form
of
special
14
nonseparabilities
parametric
nests
examples.
models
I
of
provide
.
sufficient conditions on the mapping from consumption to services such that
the
preferences
consumer's
the
and
accumulation problem are
capital
well
defined.
An important result that is exploited in Sections
assumptions,
these
under
marginal
the
univariate
a
utility
4,
process
and
5
martingale.
a
is
that
is
6
representation for observed consumption
Using this
fact,
derived.
also show that to give this model interesting empirical content,
I
a priori restrictions must be
is
placed upon the form of the nonseparabilities
This provides motivation for looking at specific forms of nonseparabilities
in sections 4, 5 and 6.
A Commodity Space for Services
III. A.
preferences
The
processes that
of
consumer
the
call services
I
.
model will
this
in
defined over
be
In the construction of the commodity space
the first piece that is needed is an information structure for
for services,
the economy.
Let (Q,?,Pr) be the underlying probability space.
economy
represented
is
filtration):
F-(?(t):
events at period
is
not
lost
t.
over
I
by
a
sequence
Let
the
?(t)£?(s) for
assume that
time.
sigma-algebras
sub
gives
?(t)
te[0,«)).
of
then
"ft xfl,
Information in the
t
set
of
x:Q
-*R,
where x(t)
is
the value
of
the
(a
distinguishable
process
+
function,
?
< s, so that information
stochastic
a
of
process at time
t.
is
a
Two
spaces of stochastic processes will be important.
Let ?
be the product sigma-algebra given by ?x£
Borel sets of R
l
.
Let
*
A
be a measure on R
l
where B
denotes the
*
that has density exp(-p t) with
*
respect to Lebesgue measure,
where
p
15
>
0.
I
denote
the
product measure
*
•
PrxX
given by
as Pr
,
The first space of processes that is needed is the
.
space of square- integrable processes:
L
2
4"
ar +
*
,y ,Pr ).
(ft
The second space of processes is used to limit the choice space of the
Let P be
representative consumer and is the commodity space for services.
of subsets of
the predictable sigma- algebra
ft
Services are required to
.
be measurable with respect to T and square integrable,
£
(ft
,T,Pr
Since
).
consumer will
the
be
making
requiring them to be predictable implies that,
only information generated by (f
III.B.
s
choices
at t,
an element of
over
services,
the consumer can use
< t).
Preferences over Services
preferences
The
:
i.e.,
time-nonseparable
of
over
representative
the
consumer
However
consumption.
it
is
are
assumed
convenient
to
to
be
first
assume that the consumer's preferences are time-separable over a stochastic
process,
s
to lie within the space £
The service process is assumed
called services.
*{s (t):0<t<»),
,r ,Pr ).
(ft
representative consumer evaluates s
Given a member of this space, s
,
the
via the utility function:
2
(3.1)
t/(s*)
- -(l/2)£{j^exp('-p*t)(s*(t)-b*(t)) dt}
*
where
{b (t):0<t<«)
is
a
deterministic process
describing the bliss point
*
movement.
(3.2)
To model growth,
I
b*(t) - exp(/it)b(t),
assume that b (t) grows at the rate p:
/i
> 0.
*
When b(t)
is a constant,
the geometric trend in b (t)
16
induces growth in the
chosen by
process
service
consumption process.
consumer
the
Consider
now
and,
as
detrended
growth
result,
a
services,
-
s
in
the
(s(t) :0<t<«)
given by:
s(t) - exp(-/it)s*(t),
(3.3)
t>0.
Then the preferences of the consumer over trending services given in (3.1)
can be rewritten as preferences over detrended services:
2
U(s) - -(l/2)E(J^exp(-pt)[s(t)-b(t)] dt)
(3.4)
where
p
— p
services
+
2fi.
Also,
.
I
In what follows,
I
will refer to detrended services as
will refer to detrended consumption as consumption, where
consumption is detrended in the same manner as in (3.3).
on
Let
A
be a measure
that has density exp(-pt) with respect to Lebesgue measure and let Pr
IR
2
+
+
Note that detrended services lie within 1 (0 ,? ,Pr
- Pr x A.
III. C.
).
Time Nonseparabilities
Time nonseparability in the consumer's preferences over consumption is
introduced by making s(t)
a
linear function current
and past consumption.
The dependence of services on current and past consumption will be modeled
by making services at time
on R
,
g,
and consumption,
1
(3.5)
s
t
a convolution between a nonrandom distribution
c:
- g*c
17
The distribution g will
where * denotes convolution of two distributions.
often put weight on all of R
to be zero for
<
r
in which case,
,
defined by setting c
is
(3.5)
One way to think of this convolutions is to let s (t)
0.
be given by:
(3.6)
s\t) - JJg(r)c(t-r)dr.
However,
the
integral
in
will,
(3.6)
be given by
not
in general
the
contribution
standard
notions of integration.
s
is
a
that
process
gives
consumption purchases from time
conditions
+
£ (n ,? ,Pr
for
the
+
)
s (t)
.
service
s(t) -
Although
s
2
let
2
be
s (t)
services
from
allow for initial
nonrandom member
a
of
gives the contribution to services at time t from the
initial stock of services.
(3.7)
In order to
onwards.
process,
to
s\t) +
The service process at time t is then given by:
2
s (t),
t>0.
could be collapsed into the bliss point process,
b,
it will be
useful to carry along a separate process for initial conditions.
For
a
particular
g,
preferences over consumption.
(3.5)
(3.4),
Below
I
and
(3.7)
give
the
consumer's
show how to restrict g and the space
of consumption processes so that the result in (3.7) is in £
(fl
," ,Pr
).
Examples of g
1.
service
Exponential
process
Depreciation
generated
by
a
of
a
Durable
durable
18
Good.
that
Let
undergoes
s(t)
be
the
exponential
.
In this case, g is given by:
depreciation.
1 9
if t <
g(t) -
(3.8)
exp(6t)
that
Suppose
where
,
\
/ c(r)dr
c (t)
< p/2.
5
>
t
is
a
diffusion
Then
process.
the
first
service process is given simply by:
s\t)
(3.9)
-
J [o
exp(fir)DC (t-r)d7(o)
4
where the integral in (3.9) is defined as a stochastic integral in the usual
Suppose
way.
services
further
in the
that
the
amount ^(0)
at
initial
t-0
condition is
given by a stock of
Then the second service process
.
is
given by:
2
(3.10)
s (t)
- exp(St)?(0)
In
this
initial
fact,
condition could be
process by adding to it a mass at
can
given
be
succinctly
as
a
collapsed
into
in the amount ?(0).
solution
to
the
the
consumption
The service process
stochastic
differential
equation:
(3.11)
Ds(t) - 5s(t) + Dc (t), with s(0) - ?(0)
a
2.
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
Persistence of this form has been studied,
19
consumption
for
example,
over
time.
Habit
by Constantinides
Following Constantinides (1990)
and Sundaresan (1989).
and Heal (1973),
model habit persistence by assuming that
l
- c(t)
s (t)
(3.12)
that
Note
Pollack (1970), Ryder
Detemple and Zapatero (1989), Novales (1990),
(1990),
term
the
-
(
a(-7)J
s (t)
is of the form:
exp(70c(t-r)dr, 7<0
exp(7r)c(t-r)dr
-y)j
I
is
a
,
0<o<l
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.
x(t)
(
-7)Jexp(7r)c(t-r)dr
is
The process
referred to as the habit stock.
In this example, g is then given by:
(3.13)
g - A
-
at)
where A is the dirac delta function and
rj
is
given by:
if t <
'
(3.14)
ij(t)
-
exp(yt)
t
>
The initial condition for services are given by:
(3.15)
2
s (t)
- -oexp(7t)x(0)
where x(0) is the initial value of the habit stock.
Many other examples of time-nonseparable preferences can be mapped into
this notation.
20
:
ResCrictions on g and c
III. D.
Since the preferences of the representative consumer are defined over
),
some restriction needs to be placed upon the space
of consumption processes
and the distribution g such that the convolution
elements of £
given by
,"P
(fi
,Pr
results
(3.5)
in
member
a
of
£
+
(71
,? ,?r
)
desirable that the mapping be from all of £ (n*,P,Pr
Often
will
be
into £ (tf ,?,Pr*)
as
.
+
it
2
)
in the time-additive and habit persistence models.
The first restriction that
member of £
version of
(3.16)
(fl
g,
g'
(Dc)'
ch,
c'
define a discounted
:
- gh
g'
for t >0 and zero otherwise and
where h(t) - exp(-et)
let
impose insures that the mapping yields a
To describe this restriction,
,? ,Pr ).
denoted
I
- Dch and so on.
- p/2.
t
Similarly
The first restriction is in the form of
a restriction upon g and the space of admissible consumption processes:
Assumption
1:
The space of admissible consumption processes
is
restricted
to be
l
C
(3.17)
where
t
>
L
2
- [u c € £ (tf ,?,Pr*))
and
I
is
the smallest integer such that
2
(e
+w
-
2
)
L
g' (cj)g' (w)
is
A
essentially bounded and where
To
see
that Assumption
1
g'
is the fourier transform of g'
implies
that
21
(3.5)
maps
the
.
space of admissible
:
consumption processes into £ (Q ,2 ,Pr
]
]
dt)
Q
E{f~
g'
(w)g' (w)* c' (w) c' (w)*du>)
the fourier transform of
is
c' (w)
2
dt) - E[f° [g' *c(t)
Q
-
where
2
g*c(t)
E[f exp(- P t)[
(3.18)
consider:
),
c'
.
The second equality in (3.18)
Note that
follows from the Parseval formula.
.
+ D)(D J c)'(t)
(e
-
(D
J+1
c)'
.
Hence we have
2
E(f°o exp(-pt)[ g*c(t)
(3.19)
]
dt)
/\
Z
- £{/**
(e +»
" —00
Assumption
1
/\
£
(i)(D c)'
g' (w)g'
(w)(DcV
(u)*du.
and the Holder inequality imply that (3.19) is finite.
process
consumption
The
2 )' 1
information process
the
in
assumed
is
economy.
to
does not destroy the
distribution g so that the mapping (3.5)
the
further restrict
will need to
I
with
compatible
be
the
information
structure in the economy:
The Laplace
Assumption 2:
(f:Re(0 > p/2)
plane:
where
a
>
a'
compact sets Kc[a'
,«)
1^ (f)|
where
g,
g(f).
for each real
Further,
.
^
|g(DI
,
transform of
f° r
some
Z
£ (tf,?,Pr
+
19
¥
p.
51),
putting weight only on R
2
)
> p/2 and
which
in
J"
the half
- a + iw
depends
on
<?eK.
imply that g is one-sided,
a subset of)
a'
polynomial
Using Theorem 2.5 of Beltrami and Wohlers (1966,
2,
analytic
is
+
into 2 (0 ,7\Pr
22
+
)
.
1
Assumptions
1
and
and maps (possibly
Capital Accumulation
III. E.
The model is completed by assuming that there is a capital accumulation
that
technology
constant rate
allows
for
of
consumption over
time
at
the
p:
Dk(t) - pk(t) + e(t)
(3.20)
transfer
the
-
c(t),
k(0) given,
where k(t) is the level of the capital stock at period
t,
k(0) is an initial
condition and e(t) is an endowment process of the consumption good.
an element of £ (0 ,T ,Pr
)
and if c satisfies Assumption
1,
If e is
then k(t)
is
an
+
element of f*(Q*,P,Pr
).
The problem facing the consumer is to maximize the objective function
subject
(3.1)
choice
I
Because
ZJc(t).
of
mapping
the
to
(3.5)
the
necessarily consumption itself but
and
control
is a
the
technology
capital
variable
of
derivative of
c,
the
(3.20)
problem
is
by
not
it is convenient to
rewrite the mapping (3.5) as:
a
1
(3.21)
s
- g *Dc,
t
The distribution
g.
integration by parts
The
can always be found using a result that is analogous to
.
representative consumer solves
problem:
23
the
following resource allocation
2
(-l/2)E{rexpC-pt)[s(t)-b(t)] dt)
Max
(MP)
t
2
s(t) - g.*D c + s (t) and
subject to:
Dk(t) - pk(t) + e(t)
by choice of a
D c
process in C
c(t), k(0) given
-
Letting y (t)-D J c for j-0,2,
.
...
I,
the
Lagrangian for this problem is given by:
t - -(1/2) Elf* exp(-pt)\
[g
Q
X (t)[k(t)
k
-
t
*y (t) +
fc(0)
-
-z^v^h 00
where
A
To
and A
j-0,1,
characterize
problem let
g.
,
g«
2
s (t)
b(t)]
-
2
t
"
-
p/Jk(r)dr
7/ 0)
-
J
-
J%(r)dr +
/^(Odr]
> « (odr ]}
J
l-l are lagrange multipliers.
...,
first-order
the
conditions
for
optimization
this
be a distribution constructed in the following manner.
assigns weight gi(t)
be a distribution that,
weight only to the nonpositive real numbers.
to
-t
Let h be
so
that
the
g.
Let
assigns
function given
by:
n -
(3.22)
exp(-pt)
if
t
<
•
otherwise
The n
g^
is
given
by
g^n.
The first-order conditions for the choice of k(t)
for each t>0, are then given by:
24
and y
,
j-0,1,
...
I
(3.23)
EJ-gJ*[ gi
2
*y (t) +
s (t)
E|j^exp(-pt)A
-
(3.24)
A
E|j^exp(-pr)A
j
(3.25)
b(t)]|?(t)j
-
o
ji
ii
(t+r)dr|?(t)| -
(t+r)dr|3f(t)| - 0,
- 1,2,
j
0,
...
M,
+ £|j^exp(-pr)A (t+r)dr|y(t)l -
A
k
q
and
A (t)
(3.26)
p£|j^exp(-pr)A (t+r)dr|?(t)| -
-
k
Although
this
it
0.
k
problem
possible
not
is
general
for
characterize
to
forms
of
a
g,
complete
the
simple
implication
solution
for
to
services
can be derived
Notice that the solution for
£{A (t+r)|£(t)} -
- -A (t)/p J+
(3.27)
for j-0,1,2,
2
consumption good at
of
to Hall's
(3.27)
time
consumption
(1978)
b(t)]|?(t)j -
-
result.
gives
is
a
then imply that
hence
A (t)
yt)//.
marginal utility of a unit of the
the
Hence we have
t.
a martingale,
is
As a result we have:
4-1.
+ s (t)
(t)
utility
...
o
The left side of
(3.26)
Relations (3.24) and (3.25)
A (t).
£J-gJ*[g£*7
to
A (t),
martingale.
the
result
This
that
result
the
is
marginal
analogous
Because the marginal utility of consumption is a
martingale we have:
gj*[s(t+r)
(3.28)
where
I
£{u(t+r)
have
|SF(t))
-
b(t+r)] - g|*[s(t)
substituted
-
0.
To
in
the
allow
-
fact
for
the
25
b(t)] + u(t+r)
that
,
r>0
g f *y (t)+s (t)-s(t)
inversion
of
the
and
where
convolution
in
(3.28),
impose the following condition on g:
I
Assumption 3:
real
l/g(f)
and
> p/2
a'
polynomial P
Assumption
3
J"
is
analytic for
(
f
Re(D >
:
- a + iu where a >
a'
p/2)
.
|l/g(f)|
,
which depends on compact sets Kc[a'
,«>)
Further,
for each
\? (f)|
for some
<
where a€K.
implies that there is a one-sided forward looking distribution,
e
such that
6
i
g^
t
(3.29)
.
t
- A
where A is the dirac delta function.
s(t+r)
(3.30)
Since g
£
is
-
b(t) + g *u(t+r)
|
-
to (3.28) yields:
f
-
forward looking, and £(u(t+r ) ? (t)
E(s(t+r)
(3.31)
b(t+r) - s(t)
Applying g
b(t+r)|?(t)} - s(t)
-
- 0,
)
b(t),
or that s-b is a continuous -time martingale.
(3.30) implies that:
r>0,
This result can be summarized
in the following Lemma:
Lemma:
then
If
the
the mapping g given in (3.5)
process
(s-b)
chosen
by
the
satisfies Assumption
consumer
is
a
1,
2
and
3,
continuous- time
martingale.
I
will denote the time derivative of (s-b)[t]
26
as:
D(s-b)[t] - D£(t).
I
will assume that £{D£(t)
- a dt.
}
Implications for Time -Averaged Consumption
III.F.
Assumption
3
implies that there is a distribution g
such that:
s
g *g - A.
(3.32)
In fact the Laplace transform of the distribution g
will be useful in the following sections.
is
Using (3.32),
This fact
l/g(f)-
the convolution in
can be inverted to yield a mapping from services to consumption:
(3.5)
g'+s
(3.33)
1
-
c.
Differencing (3.33) between
l
g'*[s (t+l)
(3.34)
Since
s
is
-
and t+1 yields:
s\t)] - c(t+l)
martingale
a
t
and
s
-
c(t)
-
s
-
s
,
(3.34)
implies
the
following
representation for the first difference of consumption:
(3.35)
c(t+l)
S
-
c(t) - g*Sl_VZ(r) + g *[b(t+l)
8
-
Although
(3.5)
completely useful
conditions.
2
g *[s (t+l)
characterizes
for
However
empirical
if
the
-
the
27
b(t)]
s*(t)].
dynamics
work because
initial
-
of
of
conditions
consumption,
the
presence
die
quickly
it
of
is
not
initial
enough
then
asymptotically their effect on the consumption law of motion will be small.
»,_
I
2.
will impose the following condition upon the behavior of g *s (t)
Assumption
Assumption 4
such that lim
There exists an v >
4:
implies
that
Note
geometrically.
the
that
conditions
initial
Assumption
imposes
4
:
exp(ut)g *s (t) - 0.
filtered through
conditions
both
g"
die
on
the
2
initial conditions s (t) and the distribution
g.
Given that the initial conditions and g are such that Assumption 4 is
satisfied,
large
for
first-differences in
t,
the
consumption process are
given approximately given by:
(3.36)
The
c(t+l)
error
term
-
g'^^Cr)
c(t) -
in
the
S
+ g *[b(t+l)
approximation due
conditions is of smaller order than
to
the
-
b(t)]
omission
of
the
initial
t.
In the special case of a time-separable model in which c(t) - s(t) and
where b(t) is
a
constant,
(3.36) gives the familiar result that consumption
follows a (continuous- time)
random walk.
observed consumption as being averages
As
of
in Section 2,
I
interpret the
consumption expenditure
over
a
unit of time so that observed consumption is given by:
(3.37)
c(t) -
/^(r),
t-0,1,2,
..
By averaging (3.36) over a unit of time, we have the following theorem:
28
2
If the mapping g given in (3.5) and s
Theorem:
satisfy Assumptions
1,
and the consumer chooses consumption according to problem (MP)
and 4
the observed consumption process (c(t):
8(t)
(3.38)
c(t-l) - g'*v(t) + g
-
,
- 1,2,3,
t
*(/^ i
[b(r)
-
2,
3
then
satisfies:
...)
b(r-l)]dr) + o(t)
J^/^D^Odr
where v(t) -
- /^(t-r)D^(r) + /^<r-t+2)D£(r).
Since
term
the
o(t)
in
throughout
ignored
rest
the
asymptotically
is
(3.38)
of
time-averaged martingale and that the
obtained by "filtering" v(t) + [b(t)
constant,
be
b(t)
then
average representation for c(t)
where g
c
3
the function g
that
(c(t) -c(t-l)
-
(3.38)
implies
a
consumption is
in
g".
:
continuous- time
a
moving
c(t-l) of the form:
,
h(t) -
t
for te[0,l], h(t) -
If a continuous record of c(t)
otherwise.
With
b(t-l)] through
is
c(t-l) - g *D£(t).
-
- h*g
given
difference
w(t)
be
c
c(t)
(3.39)
first
that
will
it
More Restrictions on g are Needed
III.G.
Let
-
Notice
paper.
the
negligible,
(and as a result g
only
we
t-0, 1
,
.
.
.
)
discrete-time
)
t
for
€[1,2] and h(t) -
2
-
-
c(t-l) were available,
t
could be exactly identified.
have
discrete -time
the
the
best
29
we
can
do
is
However,
observations:
what can we learn about the form of g
data,
then
a
to
?
identify
the
representation
discrete-time moving average
time-averaged consumption
for
differences:
c(t)
(3.40)
where £{f(t)
to
-
)
from
map
c(t-l) - X . G°(j)£(t-j)
J
-
and £{e(t)e(r)} -
1
sequence
the
time-nonseparabilities
be
possible
*
We would like to be able
t.
c
function
the
to
)
r
describes
that
g
the
Of course without further restriction it will not
.
identify
to
{G
for
completely
function g
the
c
However,
.
note
that
(3.40) 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
In
data.
the
of
discrete- time
the
model,
reflects the discrete- time time-nonseparabilities
now
we
use
economic
the
discrete-time
interpretation
given
setting
of
representation
function
the
in preferences.
Would
(3.40).
Suppose
from
preferences
the
G(j)
this
the
economic
interpretation be far wrong?
To answer this question, we can use the analysis of Marcet (1986) which
shows
how
construct
finite
sequence
(G (j)}
this
mapping,
let
linear
g (t-3),
...
operator.
values of
c
the
combinations
Let a - g
.
Note
r.
that
constructed as
is
A be
of
the
functions
the
(3.41)
G (j)
a(t)
is
in
general
/Tg (t+j)a(t)dt
^
2
a(t) dt
c
(3.42)
of
L)
tdR
of
2
- -a*g (j)/Pa(t) dt
30
(
j
a
)
the
of
set
C
1
:
?roj'(g |A) where Proj denotes
Marcet (1986) shows that G
r
(in
c
-
C
c
closure
function
a
g (t-1),
the L
.
of
To
all
C
g (t-2),
projection
convolution of g (r)
is given by:
c
g
at
many
m
- -a*h*g (J)-
(3.43)
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
for
implies
(3.43)
of
all values
that the G°(j)
and G°(j)
t
an average of the
is
will,
general,
in
lead
function g"(t)
poor
to
inferences
about the structure of preferences.
example,
For
consider
model
the
consumption good, as given by (3.8).
l-St if
c
(3.44)
g (t) -
St
exponential
of
In this example,
depreciation
of
the function g
the
c
is:
€[0,1]
t
(1+25) if t e (1,2]
-
otherwise
Further
0(1+6 -St) if te(l,2]
c
(3.45)
Proj(g |A)(t) -
otherwise
where
g
.
-
2
-
l)/(l-5+5 /3)
Further in this
case
the
Notice that if 6--VZ then
.
c
function G (j)
is
zero
for
j
not
and a equal
to
Hence consumption differences have the representation:
zero.
(3.46)
The
2
(S /6
c(t)
c
-
c(t-l) - G (0)e(t).
discrete-time
interpretation
of
this
is
that
preferences
are
time- separable which they are not.
Since the function G
will not in general allow us to obtain a correct
31
]
economic interpretation of preferences,
to
is
the plan for the
that summarize
g*
focus upon several simple forms of
for the estimation of the continuous- time form of g
study the dynamics
intuitive types
These restricted forms of
of time-nonseparability in preferences.
I
rest of the paper
g*
allow
In the next sections,
.
introduced by these preferences and examine how they
perform empirically.
IV.
Habit Persistence
An Illustration:
The
about
assumptions
mapping
the
from
consumption
services
to
introduced so far imply that the process [s-b) is a martingale supported by
stock processes
consumption and capital
In this section,
Assumptions
the
1
I
that are
2
+
of L (0 ,? ,Pr
+
)
.
examine the example of habit and show what restrictions
through 4 place upon the habit model.
economic
violates
potentially
solution
members
I
also examine whether
restrictions
(such
as
nonnegativity of the capital stock and consumption) that are not imposed in
the solution method.
The first assumption to be verified is Assumption
Fourier transform of
1.
Notice that the
for the habit persistence model of (3.12)
g'
is
given
by:
iw
a(-7)
(4.1)
g'(w) -
-
7 +
-
e
a(-7)
1
iu
j +
-
iw-7+€
e
and
w
(4.2)
2
+
w
A
g'
a(-7)
-
7
-
/
+ (e
-
7)
2
-
g'(w)g'(«)
In this case
[e
2
\ 2
A
(w)g' (w)
is
uniformly bounded and the space of consumption
32
consists
processes
of
processes
those
such
that
c
e
+
2
t (Q ,P,Pr
+
).
The
Laplace transform of g is given by:
(4.3)
.
g(
T
-
(1
r
-
a) 7
-
7
and the Laplace transform of g
is
given by:
i'(f) - l/i(f) -
(4.4)
r
- (l-a) 7
Notice that g(f) satisfies Assumption
2
if 7 < p/2 and that Assumption
*
3
requires 7 -(l-a)7 < p/2.
These restrictions allow 7 to be larger than
zero so that the habit effects
Also if 7 <
are
0,
a can be larger that
Although
important.
quite
of past consumption could grow over
time.
so that the effects of the habit stock
1
extreme
these
consistent with the model of Section
it
3,
is
parameter
important
to
settings
are
check whether
they lead to a violation of economic restrictions that are not imposed in
the solution.
To examine this issue, suppose that there is perfect certainty
the service process is a constant, s.
s (t)-0,
consumption at time
t
22
so that
Under the simplifying assumption that
is given by:
s
c(t) - g *s.
(4.5)
If Assumption
plane
a'
[l/g(0)]s.
>
0,
3
were strengthened such that l/g(f)
then as
Hence
the
t
approaches
«,
restriction
33
c(t)
that
is
analytic in the half
would approach
c(t)
be
the
nonnegative
constant
is
the
restriction that l/i(0) >
Assumption
However for many examples, a strengthening of
0.
is not necessary.
3
In the habit persistence model under perfect certainty,
at time
(4.6)
is given by:
t
c(t) - jexp( 7 *t)
-
(
7 /7*)[exp( 7 *t)
then c(t)
0,
If 7
positive only if 7<0.
>
then c(t)
,
then,
exp(7*t) (I-7/7 )s - exp(7 t) [a/(a-l)]s
large that one
23
1]U.
tends to (7/7 )s which is positive only if
(by setting a - 1)
If 7* -
-
*
*
Notice that if 7 <
7<0.
consumption
for
large
t,
7O
-
c(t)
which is again
is
approximately
For this to be positive, a must be
.
*
and 7 >0 implies that 7 <
Hence the economic restriction
0.
which Is a strengthening of
that consumption is positive requires that 7 <
Assumption
- s(l
2.
*
Notice however that Assumption
allows a >
1.
When a —
1,
3
is
satisfied as long as y<p/2.
consumption grows
This
linearly and when a >
1,
*
consumption grows geometrically at the rate 7
.
Although the nonnegativity
restriction on consumption is satisfied in this case,
the restriction that
the capital stock remains positive needs to be checked.
The capital stock at time t is given by:
(4.7)
k(t) - J"%xp(-pr)c(t+r)dr
-
J^exp( -pr )e(t+r )dr
.
Suppose that the endowment is constant over time at the level
(4.8)
k(t) - J"%xp(-pr)c(t+r)dr
-
'e/p
34
.
e,
then:
)
Notice that if 7 >0 (with 7<0) then the capital stock also is positive and
*
grows at the rate 7
*
- 0,
If 7
.
then the capital stock grows linearly over
*
If
time.
then
<
7
stock
capital
the
is
constant
over
time.
As
a
result, once the restriction 7<0 is imposed, no further restrictions besides
those in Assumptions
1,
and
2
are necessary to ensure that consumption and
3
the capital stock have reasonable dynamics.
The last assumption of Section
die
conditions
geometrically as
persistence example
2
8
g *s (t)
2
s (t)
is
3
restriction that the initial
required by Assumption
4.
In
habit
the
- -aexp(7t)x(0) and
- -crx(0)(D-7)J%xp(7%)exp(7(t-r))dr
- -ox(0)(D-7){exp( 7 *t)
(4.9)
the
(7
-
exp(- 7 t)}
-7)
*
- ax (
exp (7
)
t
*
Assumption
4
requires
to
7
be
that
less
zero
or
x(0)-0.
This
additional restriction needed for the empirical work of Sections
5
is
an
and
6,
but it is not a restriction that is implied by the economics of the model.
V.
Empirical Results with Seasonally Adjusted Data
In
this
described
in
section,
I
Section
3,
discussed in Section
b(t)
is
a
2.
empirically
using
the
examine
t.
examples
seasonally-adjusted
In this section
constant for all
some
I
of
the
consumption
model
data
will maintain the assumption that
Under this assumption,
I
show that if the
durable nature of consumption goods is modeled then the model performs much
better than the time -additive model of Section
35
2.
I
also show that there is
.
form of habit persistence
dramatic
evidence
(3.12).
The habit persistence model performs somewhat better if the durable
against
the
pure
However,
nature of the goods is modeled first.
as
given
the model provides
in
little
improvement in the fit of the data over the simple exponential depreciation
model
Exponential Depreciation
V. A.
The nondurables plus services consumption series examined in Section
2
contains several components that should be thought of as being durable.
For
clothing and shoes are components of nondurable consumption.
For
example,
this reason,
Section
2
it is of interest to relax the
time-separability assumption of
and model the durable nature of consumption.
One simple model of durables is given by the nonseparabilities induced
by the mapping (3.8) of Section
applied,
A
g'
A
:
- l/[«
g'( w )g'( w )
that
2
(e
2
2
+ (e-5)
2
]
*
+w )g' (w)g' (w)
As a result,
the solution to
essentially
is
boundedness requirement of Assumption
I.
can be
*
(w)g' (w)
#
Notice
3
the assumed regularity conditions on the mapping g must be checked.
Consider first
(5.1)
Before the analysis of Section
3.
the
1
is
bounded.
Hence
the
satisfied for a negative value of
problem will involve a law of motion
for consumption that is not necessarily a well defined stochastic processes.
This can be seen by considering the Laplace transform of the mapping from
services
Laplace
to
consumption which is
transform of the operator
given by
(D-S).
36
l/g(f)
-
f
-
S,
which
the
is
Hence consumption at time
t
is
given by:
c(t) - Ds(t)
(5.2)
- D£(t)
-
5s(t)
fis(t).
-
Note that the goods process has a component that is the time derivative of a
martingale
and,
as
a
result,
consumption
the
process
is
stochastic process but is a generalized stochastic process.
can tolerate very erratic consumption in this model,
not
a
real
The consumer
because the consumer
"consumes" a weighted average of past consumption that is relatively smooth.
As it stands,
this model does not directly fit within the framework of
Section
3.
However, a simple modification allows us to use the analysis of
Section
3.
Suppose instead of choosing consumption at time
chooses accumulated consumption at time
t,
t,
the consumer
where accumulated consumption, c
is given by:
t
(5.3)
c (t)
a
Notice
that
if
- J c(r)dr
Q
c(t)
satisfies
(5.2),
The mapping from consumption to
then c*(t)
lies
services can be modified
from accumulated consumption to services:
s
- g *c (t).
a
by:
(5.4)
g^
within £
- A + q
where q is the distribution given by:
37
a
to
2
(tf ,P ,Pr*)
be
a
Here g
mapping
is
a
.
given
<0
[0 if t
q(t) -
(5.5)
[5exp(5t)t>0
Replacing consumption by accumulated consumption and the mapping g by g
shown that this
can be
(5.6)
issue
In this case,
problem.
Let
remaining
only
The
-
fc(t)
is
how
the
of section 2.
assumptions
interpret
to
capital
the
accumulation
integrate (3.21) over time to yield:
k(0) - pj>(r)dr + J%(r)dr
pk(r)dr
-
k*(t)
satisfies
structure
it
and
-
e*(t)
-
t
J e(r)dr
c (t).
+
/c(0)
,
then
(5.6)
can
b«
rewritten as:
Dk (t) - pk (t) + e (t)
(5.7)
c (t)
-
a
This is the new capital accumulation constraint that corresponds to letting
c (t)
be
With
variable.
control
the
this
modification
25
the
exponential
a
depreciation model
embedded in the model of Section
is
3
resulting in the
law of motion for consumption given in (5.2).
Using the results of Section
c(t)
(5.8)
-
3,
we have:
c(t-l) - (D-S)w(t)
- /Jd-fir)DC(t-r)
As
in
consumption
the
time-separable
follow
a
case,
first-order
-
J"J[l+5(l-r)]D£(t-l-r)
first
differences
moving- average
process.
autocorrelations beyond the first order are zero. However,
38
in
time-averaged
Hence
all
the first-order
autocorrelation value need not be 0.25 nor even positive.
The first-order
autocorrelation first differences of consumption is:
2/6
*
R(l) -
(5.9)
2
'
1
+ (2/3)5
2
R(l)
is plotted in Figure 1.
goes
to
since,
0.25
as
Note that as
is
S
driven to
S
goes to
consumption good becomes
the
-®,
the value of R(l)
-<*>,
instantly perishable and preferences are time separable.
Notice that if
good)
5
-
then R(l)
- -/6 (a half life of 0.28 periods for the consumption
and
consumption data as noted in Section
idenfication problem.
distinguish
were
time- separable
model.
we
If
in Chrisitiano
equal
fit
able
tried
to
would not be
we
-VZ
to
examined
be
model would
here
and
reject
to
estimate
the
a
able
continuous -time
between
interval
the
to
discrete- time
using a time -separable specification,
(1985))
we would spuriously find that the decision interval is one period when,
fact,
the
Of course, this brings up a serious
3.
model
however
would
We
(as
S
continous-time
the
counterpart.
decisions
If
martingale
discrete-time
a
in
the true model is a countinuous-time one.
Notice also that if
\8\
< /6, R(l) is negative.
This could explain the
fact that monthly first-differences of consumption are negatively correlated
even in the presence of time aggregation. Just
moving
the
8
e(t-l).
where
S
(5.10).
average
process
This MA(1)
ties
down
Table
5.1
the
can
given
then be
parameter
gives
in
8
(5.8)
in
as
as
c(t)
reparameterized
through
its
-
in
5
is
-1.36,
-
c(t-l)
terms
of
affect on R(l)
maximum- likelihood estimates
using the seasonally adjusted monthly consumption data.
of
parameterize
(2.14),
of
the
8
e(t)
and
8
given
MA(1)
+
6,
in
model
The estimated value
implying that the consumption goods have a half- life of 0.51
39
If a quarter
months.
the basic
used as
is
time
interval,
the value of
S
implied by the monthly estimation is 3*(-1.358) - -4.07.
now
Suppose
we
that
quarterly data by
setting
likelihood
Note
model.
can
that
at
Hence
restriction.
values
ratio
test
the
the
against
neither
level,
10%
monthly and quarterly
with
reconciled
be
model
this
of
of
this
for
constrained
Also given in Table 5.2 are the results
estimation are given in Table 5.2.
of a
representation
average
results
The
-4.07.
to
S
moving
the
restrict
a
model
data
an
set
unrestricted MA(1)
would
reject
this
first-order autocorrelation
that
adds
simple
a
exponential
depreciation story.
Habit Persistence
5.B.
Pure Habit Persistence
Following Constantinides
,
services at t are given by (3.12).
way of deriving the mapping from services to consumption that
assume
first
that
2
s (t)
-
that
so
0,
-
s(t)
1
s (t).
I
A useful
need,
Taking
is to
the
time
derivative of (3.12) yields:
(5.10)
2
Ds(t) - Dc(t) +
a-y
Dc(t) - D£(t)
a-rc(t)
f°exp(-iT)c(t-T)dT
-
Q7c(t)
or
(5.11)
since Ds(t) - D£(t).
(1990)
.
-
-
a7
2
rexp(70c(t-r)dr,
This is the same expression derived by Constantinides
The fact that the change in consumption is linked to an average of
past consumption induces a type of smoothness in the consumption process
Using
the
results
of
Section
3
40
the
spectral
density
of
observed
consumption differences
can be
Figure
derived.
plots
2
the
first-order
autocorrelation value for the first difference in time -averaged consumption
implied by this spectral density.
R(l)
plotted as a fuction of 7 for
is
values of 7 from -0.1 to -3, where the parameter a was set such that a(-y)
equals 0.5.
that
Note
time-additive one.
0.2,
the model approaches a
This was done so that as 7 goes to -«,
the
autocorrelation value
starts
out below
moves above 0.25 and then approaches 0.25 as the habit persistence is
driven out of the model and consumption approaches
great
Habit persistence
induces
a
process
sense
that
the
in
correlated
26
Figure
3
gives
differences in consumption.
all values
of
7
and can be
smoothness
of
deal
consumption
in
differences
second-order
the
martingale process.
a
the
consumption
are
positively
autocorrelation value
for
The second-order autocorrelation is small for
negative
small values
for
of 7.
As
7
goes
towards -», this value goes to zero since the consumption process approaches
a time-averaged white noise process.
Maximum likelihood estimation
27
of
habit-persistence model using
this
the quarterly consumption series yielded parameter estimates of a that were
not
significantly different
first-order
autocorrelation
time-additive model
(see
from zero.
in
This
quarterly
because
occurs
data
can
be
easily
For the monthly data,
Section 2).
the
fit
positive
by
the
the model did
not fit well relative to the exponential depreciation model, because of the
observed negative first-order autocorrelation.
Habit Persistence with Exponential Depreciation
The nonseparability induced by (3.12)
may
develop
a
level
of
acceptable
captures the notion that people
consumption
41
over
time
and
introduces
complementarity
defined?
observe,
consumption
is
Given the degree of durability in the consumption goods that we
it
is of
interest to first define an intermediate service process,
which will be denoted
from
flowing
how
However,
time.
over
consumption
in
s*
where
,
stock of
the
gives
s (t)
the
consumption goods
at
of durable services
level
time
Habit
t.
is
then
*
developed over the intermediate service process by feeding s
instead of
,
c,
into the model of habit persistence given in (3.12).
The intermediate service process is generated according to:
s*(t) - J^exp(5r)c(t-r)dr,
(5.15)
Again,
habit
persistence
modeled
using
except
(3.12),
that
habit
is
for
the
instead of consumption directly:
developed over s
s\t) - s*(t)
(5.16)
is
<
5
-
a(-7)J^exp( 7 r)s"(t-r)dr, 7 <
< a <
0,
1
The initial conditions are modified in a similar manner.
Figure 4 plots
the
implied first-order autocorrelation value
first-difference in time-averaged consumption as a function of
and
-1,
was
a
-2.
set
were
found
characteristics
as
depreciation
close to zero)
(5
possible and as
value.
to
However,
tends to
5
1.6.
-1,
This
the
in
Figure
1,
display
plots
namely
slow
with
same
the
rates
of
negative autocorrelation in consumption is
,
the autocorrelation asymptotes
to
a positive
adding habit persistence allows the autocorrelation value
be negative or zero
gamma -
-<*>
These
0.5(-7).
to
and for 7 -
6
for more
substantial amounts
autocorrelation value
allows
durability
with
a
42
is
negative
half
life
if
of
of durability.
1
5
|
0.44
is
With
smaller than
periods
for
an
autocorrelation of zero, which
much larger than the value of 0.13 found
is
without the presence of habit persistence.
Maximum
durability
likelihood
for
model
depreciation model.
the
quarterly
two
habit
the
of
data
function
likelihood
the
in
improvement
estimates
As a result,
persistence
local
yielded very marginal
sets
over
with
pure
the
exponential
do not report the results here.
I
Table 5.3 reports results of the estimation of the habit persistence
with exponential depreciation model using monthly data.
reports
the
depreciation
values
model
of
nested
depreciation model.
This
reported in Table 5.1.
parameter
7
is
not
likelihood
a
within
was
This
the
where
situation
is
and
likelihood
statistic
ratio
test
at
supremum of the resulting values.
in Table 5.3
is
a
the
(1977)
statistic
is
possible
However,
also
the
exponential
with
exponential
when a -
conditions
0,
the
for
the
suggested a test in
has
found
values
5.3
log-likelihood values
regularity
the
Davies
all
of
persistence
test of a-0.
a
identified
the
habit
test
calculated using
likelihood ratio test break down.
this
ratio
Table
by
of
7
evaluating
the
and
the
taking
The value of the test statistic reported
lower bound for Davies
's
statistic and has a chi-square
distribution with one degree of freedom (assuming that 7 is known and set to
the estimated value of Table 5.3).
The P-Value using this distribution is
0.067, hence at the 5% level the addition of habit persistence does not help
the fit of the model.
Table 5.4 reports half- life values for the exponential depreciation and
habit-persistence effects.
0.656 months.
The half -life of the durability of the good is
Notice that this half -life value is higher than the value of
0.51 months found with exponential depreciation alone.
The half- life of the
habit stock is estimated to be relatively large,
it
43
but
is
not precisely
.
measured.
It
seems
that
the
degree of habit persistence
difficult
to
aggregation
is
is
measure with this data set.
The
results
accounted for,
Although
first-order
adding
very weak evidence
is
autocorrelation
requires
in
persistence
habit
to
prices
VI.
unimportant
should be
differences
durability
durability
in
model
found
the
does
at
data.
not
this does not imply that habit
understanding
behavior of asset
the
Empirical Results with Seasonally Unadjusted Data
fit
the
2,
the
continuous- time
time -additive model
seasonally unadjusted consumption data.
examine whether the addition of temporal dependencies
the
the
in
28
As we saw in Section
not
in
persistence
Certainly to allow for the
of
the
time
habit
for
modeling
the
once
consumption
significantly improve the fit of the model,
persistence
that
services consumption series.
frequencies
monthly
indicate
section
this
there
nondurables plus
negative
of
fit
of the model.
The model of preferences
In
this
does
section,
I
in preferences helps
implies
that
consumption
differences display seasonality that can be captured by seasonal dummies and
a
seasonal autoregression.
The first piece of the model for seasonals
the bliss point
formed by assuming that
Seasonal bliss
follows a deterministic seasonal pattern.
point movement has been used in
Before adding this
faced.
is
a
similar way by Miron (1986).
feature to the model,
an aliasing problem must be
Suppose that the discrete- time data follows an exact deterministic
seasonal pattern that could be fit with seasonal dummies.
To model this in
continuous time would require a function that also has this exact seasonal
44
.
pattern
observed
the
at
However,
frequencies.
for
continuous -time
this
model there are an infinite number of unobserved frequencies.
The class of
observationally equivalent continuous- time functions have seasonal patterns
with
least
at
determined by
frequencies
the
discrete-time data.
As a result,
I
period
the
of
the
observed
will assume that the bliss point follows
a seasonal pattern in continuous time with frequencies corresponding exactly
to the seasonal frequencies of the observed discrete -time data.
Assume
that
data
the
is
observed at quarterly
intervals.
The bliss
point is then assumed to be governed by:
^{^cos(t rj/2)+^sin(t rj/2)}
b(t) - X
(6.1)
7
J
j
where
<f>
$
,
and
,
the integers,
(b(t):
are constants.
t
- 0,1,2,
...),
Note that the observations of b at
could be exactly fit by four seasonal
dummies
To
how
determine
this
consumption differences,
(3.38)
(6.2)
model
difference
affects
behavior
the
and average
(6.1)
over
of
observed
time,
and use
to yield:
s
c(t-l) - g"*w(t) + g *B(t)
c(t)
-
B(t)
- £
where
(6.3)
*
{a [sin(tjrj/2)
+ sin( (t-2)irj/2)
]
A
-0[cos( twj/2 + cos((t-2)irj/2)].
and where
a
-
2a /n
and
£
-
2B/n.
45
Note
that
B(t)
observed
at
the
integers can be represented using seasonal dummies and that the sum of these
dummies over the period of one year
consumption differences have
Also note
zero.
Relation
characteristics.
same
these
is
that g *B(t)
that
implies
(6.2)
has
observed
component that has an exact seasonal pattern
a
and a random component with properties that are influenced by the form of
the nonseparabilities.
6. A.
In
capture
Seasonal Habit Persistence and Exponential Depreciation
order
the
to
capture
first
the
durable
nature
autocorrelation noted
first-order
the
of
Table
in
goods
2.5,
and
to
consider
an
*
intermediate service process
(6.4)
As
s
just as in Section
s*(t) - f°exp(Sr)c(t-r)dr
,
6
in the habit persistence model,
current level of s
to
5:
< 0.
the consumer
is
assumed to compare
an average of past intermediate services.
the
In this
case instead of letting the habit stock be a weighted average of all past
consumption,
assume
that
the
habit
stock
is
given by
intermediate service process of exactly one year ago.
29
"
the
level
of
the
In other words s(t)
is given by:
(6.5)
s(t) - s*(t)
-
as*(t-4)
where, again, the basic time period is assumed to be a quarter.
transform of the distribution g for this case is given by:
46
The Laplace
)
i(f) - [1
(6.6)
Since
-
transform of
Laplace
the
aexp(-4f)]/(f
5).
-
g*
is
differences
l/g(f),
of
time-averaged
consumption can be represented as:
c(t)
(6.7)
c(t-l) - (V-6)p
-
(D-5)B(t)
D£(r)
+
(1
-
ll")
(1
-
yL")
or
c(t)
(6.8)
c(t-l) - /Vl-5r)D|(t-r)
-
(1
/J [1+5 (1-r
-
)
]D^(t-l-r
ll")
-
(D-5)B(t)
+
(1
The
representation given
-
lL")
in
time-averaged consumption have
implies
(6.8)
two
pieces.
that
The
differences
first
first
a
is
of
random piece
which has a first-order moving- average structure with a seasonal first order
autoregression.
dummies.
second
The
Notice
that
piece
the
if
is
model
set
of
(6.8)
is
a
of
deterministic
seasonal
correct,
seasonal
then
adjustment removes some of the dynamics of consumption that are due to the
preferences
of
adjusted data will
c
preferences
Table
consumer.
the
lead
to
applying
Hence
misleading
inferences
model
about
seasonally
to
the
structure
of
30
gives
6.1
estimates
the
of
parameters
quarterly nondurables plus services data.
exponential
depreciation
of
habit
restriction a -
is
with
no
persistence
seasonal
in
as
31
of
(6.8)
For comparison Table
estimates of the same model with a set to zero.
inclusion
the
This is
effects
(6.5).
a
6.2
U.S.
gives
model of pure
modeled
First
for
note
through
that
the
the
rejected by the data at the 0.1% level (values for the
47
likelihood ratio test are given in Table 6.1).
in
Table
inclusion
the
that
2.5
seasonal
dummies
random piece of consumption differences
the
does
(after
dummies removed) yields a likelihood value of 150.02.
that
indicates
This
0.184.
completely
trend and seasonal
A likelihood ratio
against this alternative yields a P-value
test of the model given in (6.7)
of
not
Estimation of an unrestricted ARMA(4,1) model
remove all seasonal effects.
for
of
This confirms the findings
the
model
given
in
(6.8)
doing
is
a
reasonable job in representing the n.s.a. consumption data.
Notice that the estimated value of
zero
than
the
value for
found with
value
in Table
5
6.2
is
seasonally adjusted data.
using monthly data seasonally adjusted data
S
much closer to
The
(see
estimated
Table 7)
is
-1.358, which converts to 3x(-1.358) - -5.074 on a quarterly basis, whereas
the
value
corresponding
using
unadjusted
data
is
-1.802.
implied
The
half- life using unadjusted data is 0.38 quarters versus 0.17 quarters using
adjusted data.
Estimation of a model with just seasonal habit persistence
results
(5--00)
in
a
log-likelihood
restriction against the model with
0.0002
for
important
likelihood
the
role
for
ratio
durability
S
value
140.61.
of
unrestricted,
test.
Hence
the
unadjusted
in
Testing
this
results in a P-value of
there
data
evidence
is
even
at
of
an
quarterly
frequencies that does not exist in seasonally adjusted data.
In
sum,
the
unadjusted
seasonally
indicates that durability in the goods
for seasonal habit persistence
32
is
data
at
quarterly
important and there is evidence
The model with time-separable preferences
does a very poor job in fitting this Quarterly data set.
evidence
against
the
time-additive
model
first-order autocorrelation in monthly data.
of
Commerce
does
not
publish
frequencies
monthly,
48
came
in
the
In Section 2,
form
Unfortunately,
of
the
negative
the Department
seasonally-unadjusted
data,
which
makes the corresponding exercise difficult to perform here.
Concluding Remarks
VII.
In this paper
quadratic
model
have developed and empirically investigated a linear
I
which
in
dependent preferences.
representative
the
The model demonstrates
in preferences interact with time aggregation
First,
in
the
consumer
has
temporally
that time nonseparabilities
important
several
in
ways.
the interaction between these two effects produces rich dynamics
consumption
presense
The
series.
of
time
nonseparabilities
in
preferences is important because they help to explain some of the observed
behavior of aggregate consumption of nondurables plus services that cannot
be
by
explained
temporal
aggregation
consumption series is consistent with
a
alone.
The
seasonally
adjusted
model in which the consumption good
There is some weak evidence for habit persistence effects, but
is durable.
only if the durable nature of the consumption goods is modeled first.
Second,
the
presence
of
nonseparabilities
and
continuous -time
a
decision framework may make models which ignore the time aggregation problem
perform
reasonably
identification
well
on
making
problem,
dimensions.
some
difficult
it
decision interval to use in modeling.
severe,
those
since
time
implied by a
The
produces
This
to
determine
serious
the
correct
identification problem is very
nonseparabilities may yield implications
time-separable
a
specification.
This
that
occurs
resemble
due
to
the
simultaneous presence of time aggregation and time-nonseparable preferences.
Third,
the
presence of time nonseparabilities
seasonal behavior.
As
a
result,
if
some
account
can produce
is
to
be
endogenous
taken of time
aggregation, it should probably be done with seasonally unadjusted data.
49
In
as
fact,
quite
I
shown,
between
different
particular,
of
have
inferences
about the structure
unadjusted
seasonally
of preferences
adjusted
and
the use of seasonally adjusted data understates
durability
in
consumption
and
goods
hides
the
the
presence
data.
are
In
importance
of
habit
persistence at seasonal frequencies.
The
features:
the
model
I
examined
in
this
paper
has
two
linearity and a constant interest rate.
behavior
of
consumption
from
the
model
could
important
simplifying
This was done so that
be
easily
developed.
Further work also needs to be done in examining nonlinear environments and
asset
pricing
environments
that
account
preferences and the time aggregation problem.
50
for
time
nonseparability
in
.
Footnotes
Deaton (1986),
Flavin (1981),
See for example:
(1987) and Hansen and Singleton (1982,
See
(1990),
also
and
Melino
Litzenberger and Ronn (1986),
examples
for
Grossman,
of
1983).
Shiller
(1985),
Hansen and Singleton
Naik and Ronn (1987)
temporal
taking account of
Campbell and Deaton
and Hall
(1988)
aggregation in asset pricing
contexts
The fact that the continuous- time time-additive model does not fit the
monthly data was also noted by Ermini (1989)
.
Several other constraints need to be imposed upon the problem to make
it
well-posed economically.
First,
the
consumer must
allowed
not be
to
choose the solution c(t) - b which would result in perverse behavior of the
capital stock.
For instance,
in the case of a sufficiently small endowment
process and small initial capital stock, this would require that the capital
stock
go
to
negative
infinity
at
a
geometric
rate.
To
make
the
model
interesting, some other constraint needs to be imposed to rule this out as a
possibility.
Also,
in
consumption
choosing
at
constrained by the information available at time
time
t.
t,
the
Formal
consumer
is
imposition of
these constraints is discussed in the next section.
See
Hansen
(1987)
and
Sargent
martingale implication arises in
a
(1986)
for
an
examples
of
how
the
discrete- time social planning environment
similar to the one here.
51
.
5
We actually observe averages of trending consumption.
results
averages
observe
assuming that we
of
I
will present
detrended consumption.
The
difference can be accounted for in all of the calculations presented in this
However the difference is very small and would needlessly complicate
paper.
the presentation.
Here, and throughout the paper,
g
If
a
law
define all processes before period
(2.10) is well defined even for t-0 and t-1
As a result,
to be zero.
I
of motion
for
endowment process was
the
posited then the
model would potentially imply a restriction across the law of motion for the
A problem with this approach is that the model
endowment and consumption.
is this paper is of a single
aggregate
labor
income)
consumption goods.
consumption good and observations of e (such as
should be thought of as being split between many
Modeling preferences over many consumption goods is well
beyond the scope of this paper.
See Hansen,
Roberds and Sargent (1989) for
further discussion of these issues.
Q
Seasonally unadjusted data
available
is
from 1946.
The
longer data
set was 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 of the series over the longer time period.
Modeling the change in
the seasonal behavior of the series is beyond the scope of this paper.
The
shorter time period was chosen since this problem does not occur over this
period.
The
trend removed
misspecified
likelihood
in
this
function
case was
in
assumed to be uncorrelated over time.
52
estimated using a
which
consumption
(potentially)
differences
were
Standard errors of the estimates of R(l) are given in brackets.
errors
standard
were
R(l)
for
calculated
using
the
Newey-West
The
(1987)
procedure with one year of lags.
In Section 6,
I
further examine seasonally unadjusted data to try to
account for these effects.
I
here
I
will continue to use c to denote observed consumption even though
am not considering the effects of time aggregation over the month.
From here on
Unlike
will ignore the role of measurement error.
I
most
analyses
of
Gorman- Lancaster
technologies,
I
ignore
nonnegativity constraints.
Discrete-time versions of this specification of preferences have been
considered by many authors.
See for example Eichenbaum and Hansen
Eichenbaum, Hansen and Richard (1987) and Hansen (1987),
(1990),
and the references
therein.
f
is
continuous
(1983)
or
generated by
from
the
Elliot
left
(1982)
the
stochastic
and have
for
a
right
complete
processes
limits.
adapted
See
description
?
to
that
are
Chung and Williams
of
predictable
the
sigma-algebra and predictable processes.
The restriction
19
< p/2 will be explained below.
-
Beltrami and Wohlers (1966) show that if g(£) is analytic in the half
plane (£:Re(£) > 0)
the
6
requirement
on
then g is a one-sided distribution.
the
analyticity
of
g(£)
since
Here
I
distributions that induce growth.
20
See,
for example, Beltrami and Wohlers (1966, p. 28).
53
can
I
can weaken
allow
for
.
21
In Section 4
I
examine the behavior of the capital stock for the habit
persistence example.
22
Under perfect certainty, the habit model is
a linear
quadratic version
of the model considered by Ryder and Heal (1973).
23
that
Assuming
a
>
so
we
that
are
considering
a
model
of
habit
persistence
See,
for
Gel'fand and Vilenkin
example,
(1964)
for a discussion of
generalized stochastic processes.
25
Hansen,
Heaton and Sargent (1990) show that the optimization problem
with this modification is well defined.
2fi
Sundaresan (1989) and Detemple and Zapatero (1989) investigate whether
habit persistence results
in
consumption in the sense that habit
"smooth"
persistence may lower the volatility of consumption for a given volatility
in the endowment process.
27
Throughout
this
section,
the
models
were
fit
using
the
frequency
domain approximation to the likelihood function suggested by Hannan (1970).
28
For
empirical
Ferson (1989)
investigations
and Heaton (1990)
for
of
this
example.
type
See
see
Constantinides
also Abel
discussion of a slightly different form of habit persistence
pricing context.
54
(1990)
in an
and
for a
asset
29
general
more
A
[a(l-y)/(l-yL )]s (t-4)
would
model
so
that
the
be
habit
set
to
stock
services one year ago, two years ago and so on.
7-0
yields a test value of 0.26
unrestricted estimation).
Since
(6.5).
more
the
(with 7
is
a
-
s(t)
*
s (t)
weighted average of
A likelihood ratio test of
allowed to be negative in the
When 7 is set to zero, we get the model given in
complicated
model
provides
very
insignificant
improvement in the likelihood function, it is not discussed.
finite
Heaton
(1989)
life.
This
differences.
If
also examines models
in which
peaks
spectral
induces
these
peaks
the
in
close
are
to
the
durable good has a
density
seasonal
of
consumption
frequencies,
then
seasonal adjustment will also remove dynamics that are induced by the nature
of the consumption goods themselves.
The
estimated
values
of
the
seasonal
dummies
are
omitted
for
simplicity.
32
Osborn (1988) also suggests that there is an important role for habit
persistence in fitting seasonally unadjusted data.
55
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of
TABLE 2.1
ESTIMATES OF MA(1) MODEL FOR FIRST-DIFFERENCES
OF PER CAPITA, SEASONALLY- ADJUSTED QUARTERLY CONSUMPTION
OF NONDURABLES PLUS SERVICES.
FIRST-ORDER AUTOCORRELATION RESTRICTED TO 0.25
Parameter Estimates
Parameter
52,1 to 86,4
fi
(xlO
2
)
Log-Likelihood
59,1 to 86,4
0.475
(0.052)
0.485
(0.060)
0.224
(0.016)
0.253
(0.019)
67.046
92.491
'Standard errors are in parentheses.
60
TABLE 2.2
ESTIMATES OF MA(1) MODEL FOR FIRST -DIFFERENCES
OF PER CAPITA, SEASONALLY-ADJUSTED QUARTERLY CONSUMPTION
UNRESTRICTED ESTIMATION
OF NONDURABLES PLUS SERVICES.
Parameter Estimates*
Parameter
52,1 to 86,4
2
59,1 to 86,4
0.474
(0.053)
0.487
(0.059)
d
0.224
(0.016)
0.252
(0.019)
8
0.062
(0.019)
0.060
(0.023)
H (xlO
)
Log-Likelihood
92.494
67.092
Standard errors are in parentheses.
61
TABLE 2.3
LIKELIHOOD RATIO TESTS OF TIME- SEPARABLE MODEL AGAINST HIGHER ORDER
MOVING AVERAGE MODELS.
SEASONALLY-ADJUSTED QUARTERLY CONSUMPTION OF
NONDURABLES PLUS SERVICES
Likelihood Ratio Value (P-Value)
Order of Nesting Moving
Average Model
52,1 to 86,4
59,1 to 86,4
2
0.091
(0.956)
0.293
(0.864)
3
5.579
(0.134)
7.326
(0.062)
4
8.711
(0.069)
11.882
(0.018)
5
13.703
(0.018)
16.928
(0.005)
62
TABLE 2.4
ESTIMATES OF MA(1) MODEL FOR FIRST-DIFFERENCES
OF PER CAPITA, SEASONALLY- ADJUSTED MONTHLY CONSUMPTION
OF NONDURABLES PLUS SERVICES, 59,1 TO 86 12
Parameter
Estimation with 0.25 Restriction
Unrestricted*
on R(l)*
I*
(xlO
)
9
Log- Likelihood
0.156
(0.027)
0.164
(0.017)
0.248
(0.015)
0.216
(0.010)
0.047
(0.011)
211.33
Standard Errors are in parentheses.
63
254.54
TABLE 2.5
AUTOCORRELATION VALUES FOR FIRST -DIFFERENCES
OF PER CAPITA, NOT SEASONALLY -ADJUSTED QUARTERLY CONSUMPTION
OF NONDURABLES PLUS SERVICES, 59,1 TO 86,4
TREND AND SEASONAL DUMMIES REMOVED
Order of
Autocorelation
Autocorrelation*
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
"Standard errors are in parentheses.
one year of lags in the Newey-West (1987) procedure.
64
calculated using
TABLE 5.1
ESTIMATES OF EXPONENTIAL DEPRECIATION MODEL
FOR FIRST -DIFFERENCES OF PER CAPITA, SEASONALLY -ADJUSTED
MONTHLY CONSUMPTION OF NONDURABLES PLUS SERVICES, 59,1 TO 86,12
Parameter
ft
(x 10
Estimated Value*
2
)
S
0.164
(0.017)
0.215
(0.010)
-1.358
(0.161)
Q
S
Log-Likelihood
254.54
Standard errors are in parentheses
65
.
TABLE 5.2
ESTIMATES OF EXPONENTIAL DEPRECIATION MODEL WITH 5 - 3x(-1.36),
QUARTERLY DATA. ALSO LIKELIHOOD RATIO TESTS
AGAINST UNRESTRICTED MA(1)
Parameter Estimates
Parameter
52,1 to 86,4
I*
(x 10')
0.478
(0.048)
0.492
(0.055)
9
0.225
(0.015)
0.253
(0.019)
Log- Likelihood
against
unrestricted MA(1)
L.R.
59,1 to 86,4
91.14
2.69
(0.101)
"Standard errors are in parentheses
Probability values are in parentheses
66
66.48
1.16
(0.281)
TABLE 5.3
MAXIMUM LIKELIHOOD ESTIMATES OF PARAMETERS FOR HABIT PERSISTENCE
WITH LOCAL DURABILITY MODEL, USING
FIRST- DIFFERENCES OF PER CAPITA, SEASONALLY -ADJUSTED
MONTHLY CONSUMPTION OF NONDURABLES PLUS SERVICES
Parameter Estimates
Parameter
li
(x 10
2
0.156 (0.024)
)
a
0.132 (0.010)
S
-1.558 (0.743)
7
-0.211 (0.414)
a
0.438 (0.308)
Log-Likelihood
256.22
Improvement in
log-likelihood
over pure exponential
depreciation model
3.36
(0.067)
(Table 4.1)
b
Standard errors are in parentheses
Probability value using Chi-square distribution with one degree of
freedom in parentheses.
67
TABLE 5.4
ESTIMATES OF HALF-LIVES IN MONTHS
FOR DURABILITY AND HABIT PERSISTENCE EFFECTS WITH MONTHLY DATA
Estimates
Depreciation Parameter
5
0.656
(0.104)
7
3.293
(6.473)
"Standard errors in parentheses
68
TABLE 6.1
ESTIMATES OF EXPONENTIAL DEPRECIATION MODEL
WITH SEASONAL HABIT FORMATION
FOR FIRST-DIFFERENCES OF PER CAPITA, SEASONALLY-UNADJUSTED
QUARTERLY CONSUMPTION OF NONDURABLES PLUS SERVICES, 59,1 TO 86,4.
ESTIMATES OF SEASONAL DUMMIES OMITTED
Estimated Value*
Parameter
H
2
0.408
(0.074)
8
0.127
(0.010)
6
-1.802
(0.414)
0.349
(0.087)
(x 10
)
Log -Likelihood
147.60
Likelihood Ratio test
of pure exponential
depreciation model
b
(see Table 13)
14.57
(0.0001)
"Standard errors are in parentheses
Probability values are in parentheses
69
TABLE 6.2
ESTIMATES OF EXPONENTIAL DEPRECIATION MODEL
FOR FIRST -DIFFERENCES OF PER CAPITA, SEASONALLY-UNADJUSTED
QUARTERLY CONSUMPTION OF NONDURABLES PLUS SERVICES, 59,1 TO 86,4
ESTIMATES OF SEASONAL DUMMIES OMITTED
Estimated Value
Parameter
p.
(x 10
2
)
8
0.371
(0.060)
0.139
(0.011)
-2.449
(0.927)
o
S
Log-Likelihood
140.32
Standard errors are in parentheses
70
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