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M414
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—MAR 11^*
J
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
An Empirical Investigation of Asset Pricing
with Temporally Dependent Preference Specifications
by
John Heaton
WP.
#3245-91-EFA
February 1991
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
An Empirical Investigation of Asset Pricing
with Temporally Dependent Preference Specifications
by
John Heaton
WP.
#3245-91-EFA
February 1991
An Empirical Investigation of Asset Pricing
With Temporally Dependent Preference Specifications
John Heaton
Department of Economics and Sloan School of Management
M.I.T.
February 1991
Comments Invited
First Draft:
December 1989
ABSTRACT
In this paper, I empirically investigate a representative consumer model in
which the consumer is assumed to have temporally dependent preferences of
several forms.
Comparing the model's implications for asset prices to those
of a time-additive model,
I
find substantial evidence for the local
substitutability of consumption with habit formation occurring over longer
periods of time.
Neither local substitution nor habit formation alone is
found to be significant, so that the interaction between the two effects is
quite important.
Although the model provides a better explanation of the
behavior of asset prices, the model is still statistically rejected.
A
potential explanation for this rejection is that with local substitution the
Hansen and Jagannathan (1990, 1991) bounds on the moments of the marginal
rate of substitution, can be fit only with extreme curvature in the period
utility function.
Keywords:
*
Asset Pricing,
Aggregation
Local
Substitution,
Habit
Persistence,
Temporal
i .
This a revised version of a paper by the same title presented at the 1989
North American Winter Meeting of the Econometric Society, at the conference
on
"Empirical
Applications
of
Structural
Models"
sponsored
by
the
Econometric Society and at the 1990 Canadian Macroeconomics Study Group
sponsored by SSHRCC.
have received many helpful comments and suggestions
I
from a large number of people.
In particular, I would like to thank, George
Constantinides, Eric Ghysels, Ayman Hindy, Chi-fu Huang, Ken Judd, Deborah
Lucas, Andrew Lo, Whitney Newey, George Tauchen and especially Andy Abel and
Lars Hansen.
I
would also like to thank Makoto Saito for research
assistance.
I
gratefully acknowledge financial support from the Provost
fund at M.I.T. and the International Financial Services Research Center at
M.I.T.
Of course any errors in this work are my own.
I.
Introduction
paper,
this
In
model
which
in
consumer
the
those of
to
a
assumed
is
representative
a
temporally
have
to
consumer
dependent
Comparing the model's implications for asset
preferences of several forms.
prices
investigate
empirically
I
time-additive model,
find substantial
I
evidence
in
favor of a preference structure under which consumption at nearby dates is
substitutable
Neither
levels develops slowly.
local
alone is found to be significant,
effects
quite
is
explanation
rejected.
A
substitution
potential
that
is
substitution nor habit persistence
that
so
asset
prices,
needed
to
fit
the
this
many
causes difficulties on other dimensions.
of
In
provides
model
model
the
explanation for
interaction between the two
the
Although
important.
observed
of
and where habit over consumption
subst itution)
(called local
rejection
the
statistically
still
is
is
moments
particular,
better
a
that
of
to
the
asset
local
prices
fit estimated
Hansen and Jagannathan (1991) bounds on the moments of the marginal rate of
substitution,
the
local
substitutability of consumption can be maintained
only if extreme curvature in the period utility function is allowed.
The possibility
explain
some
of
the
that
poor
temporal
dependencies
performance
suggested by a number of authors.
of
asset
in
preferences could help
pricing
models
has
been
For example, Constantinides (1990) argued
that by specifying a preference structure where consumption is complementary
the equity premium puzzle of Mehra and Prescott
over time,
solved
.
In
this model,
an empirical
could be
investigation of the Euler equations implied by
Ferson and Constantinides (1990) found evidence in favor of the
model, using quarterly and annual data
In
(1985)
2
.
the model analyzed by Constantinides
(1990),
habit persistence was
used
to
make consumption complementary over
type
of
specification is that
dates local to time
time
of
t.
violates
it
One problem with
time.
the
this
consumption at
notion that
should be relatively substitutable for consumption at
t
Huang and Kreps (1987), and Hindy and Huang (1989) argued in favor
substitutable.
locally
consumption
making
In
continuous-time
a
environment, Hindy and Huang (1989) showed that when consumption is locally
asset prices will be "smooth"
substitutable,
if
there is no new arrival of
This occurs even if the consumption endowment is very erratic.
information.
This seems to correspond quite well with our intuition about the behavior of
In fact,
prices over short periods of time.
using monthly consumption and
asset market data, Dunn and Singleton (1986), Eichenbaum and Hansen (1989),
Gallant and Tauchen (1989) and Ogaki
all found evidence in favor of
(1988),
consumption being locally substitutable.
aggregation
Temporal
studies
data
monthly
using
is
explanation for
potential
one
evidence
find
for
consumption being complementary over time.
consumption
observed
data,
In
is
average
an
study
a
aggregation
whether
consumption,
frequencies,
time.
temporal
of
of
studies
dynamics,
Heaton
whereas
for
use aggregate
that
problem
quarterly
expenditures
that
found evidence
important
monthly,
at
consumption
consumption
(1990)
an
is
be
it
In
substitution
local
Constantinides
using quarterly data Ferson and
fact
the
over
(1990)
or
because
annual
period
a
showed
of
that
temporal aggregation and temporal dependencies in preferences can interact
in an important way.
as
the
length
of
the
For example,
time
monthly to quarterly data),
if consumption
averaging
it
is
increases
possible
preferences are time additive or even that
course,
a
different
explanation
is
for
there
that
is
locally substitutable,
(for
the
is
going
from
suggest
that
formation.
Of
example
data
habit
to
consumption
is
locally
substitutable,
possible
that
but habit develops much more slowly
3
low frequency data,
when we examine
In
.
the
this case,
it
is
habit effects will
dominate.
A goal
of
this paper
is
to
try to sort out
In studying these issues
these different effects.
which consumption is locally substitutable.
the consumption good is durable.
flow of
services
from
I
investigated a model in
I
modeled this by assuming that
Habit was modeled as developing over the
consumption good and,
the
importance of all of
the
consumption itself develops much more slowly.
I
as
a
habit
result,
over
was careful to take account
of the fact that observed consumption is time averaged.
I
found that habit persistence substantially improves the fit of stock
and bond returns only if local substitution is also present.
The estimated
parameters of the model
imply that consumption is relatively substitutable
over a period of about
a
point.
However,
year,
imposing
the
with habit effects dominating beyond that
requirement
that
consumption
locally
be
substitutable implies that a high degree of curvature in the period utility
function is needed to fit Hansen and Jagannathan (1990,
moments of the marginal rate of substitution.
1991) bounds on the
This difficulty may explain
why the model is still statistically rejected by the data.
The rest of the paper is organized as follows.
preference
the
paper.
4,
I
structures
In section 3,
I
that
are
examined
In section 2,
throughout
I
rest
the
present the asset pricing environment.
outline
of
the
In section
use a diagnostic proposed by Hansen and Jagannathan (1990)
to examine
whether the model of section 3 fits some aspects of the asset market when
local substitution in consumption is maintained.
In section 5
I
present the
results of formal estimation of the entire asset pricing model.
Section 6
concludes the paper.
2.
PREFERENCES
In
this section,
throughout
the
rest
outline the class of preferences that will be used
I
the
of
consumer are assumed
The preferences
paper.
display simple forms of
to
of
representative
the
temporal
dependence.
To
model the temporal dependence in preferences, assume that the preferences of
the representative consumer are time additive over a potentially fictitious
Let
called services.
good
{s(t):
t=0,l,2,
...}.
services at
time
be
t
given by sit)
and
s
*
Preferences over s are then assumed to be given by
the utility function:
Uts) =
(2.1)
t
£{^_ o p u(s(t))>.
Assume further that the period utility function is characterized by constant
relative risk aversion, so that U(s) is given by:
»<*, =
U.2)
-
* ct
eJc/
i'7
.
r > o.
'}
The constant relative risk aversion parameterization of u(-)
in
this
investigation
because
it
makes
it
possible
to
convenient
is
use
simple
transformations of the data that induce stationarity.
To
induce
temporal
dependencies
consumption via:
"
s(t) = I
j
the
services are assumed to be
consumption goods,
(2.3)
in
o
a(j)c(t-j)
consumer's
linked
to
preferences
current
over
and past
I
use several different parametric examples of the function <a(j)> in later
sections:
Durable goods subject to exponential depreciation.
1.
Suppose
that
the
In
this
exponentially.
by
consumption
case
let
good
a(j)
=
A
is
durable
and
depreciates
where 0<A<1 and s(t)
given
is
4
:
sit) = £
(2.4)
ro
j
_oA
c(t-j)
Note that this specification makes the consumption good substitutable over
This specification of durability was also used,
time.
and
Singleton
sections
3,
Eichenbaum and Hansen
(1986),
4 and 5
of temporal aggregation.
seems
reasonable
substitutable.
In
and Ogaki
(1988).
In
assume that the time interval between decisions taken
I
by the consumer is very short.
it
(1990)
by Dunn
for example,
This allows me to account for the potential
When the interval between decisions is very short,
that
fact,
consumption
the
at
adjacent
continuous-time
limit
dates
of
will
this
be
very
preference
structure satisfies the continuity restriction of Huang and Kreps (1987) and
Hindy and Huang
(1989).
This
continuity restriction captures
the
notion
that consumption at relatively adjacent dates should be substitutable.
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 consumption over time.
Habit persistence of this form
has been studied,
(1989),
Sundaresan
for
Constantinides
Following Constantinides
s(t)
(1973),
(1990)
I
(1990)
and Detemple and Zapataro
(1990).
(1970),
model habit persistence by assuming that
is given by:
m
sit) = c(t) - a(l-e)£
(2.5)
j
term
the
that
Note
consumption,
the habit
6 J c(t-l-j), 0«x<l, O<0<1.
o
J
(l-8)£ ™ 8 c( t-l-j)
a
is
weighted
average
of
past
and a gives the proportion of this average that is compared to
current consumption.
as
Ryder and Heal
by Pollack
example,
stock.
The process x(t)
The
J
(l-9)£ " 8 c(t-l-j)
=
referred to
is
introduction of habit persistence effects makes
consumption complementary over time.
3.
Habit Persistence with local durability
With short decision intervals,
(2.2)
and
(2.5)
restrictions.
preference
specification given by
violates the notion that consumers can readily substitute
consumption at adjacent dates.
preferences
the
with
persistence
habit
Instead
In fact,
it
seems
Hindy and Huang (1989) showed that
do
not
reasonable
satisfy
that
habit
their
may
continuity
develop
more
slowly and that the consumer is willing to substitute consumption local to
time
t,
for
c(t).
One way to model
this
is
to
assume
that
the good
is
durable and that the consumer develops habit over the flow of services from
the durable.
(2.6)
In other words:
J
sit) = sit) - a(l-8)£ " G s(t-l-j),
where
(2.7)
sit) = jf
A J c(t-j), 0<A<1.
0<a<l,
0<8<1,
As
demonstrate
I
parameterization
persistence.
Sections
in
capture
to
example,
For
if
and
4
<
at
0,
longer
possible
is
it
substitution
local
A
5,
with
lags
for
this
long-run
the
effect
habit
of
past
consumption on preferences due to the durability of the good will die out
and
the
habit
persistence
effects
will
dominate.
In
this
sense,
the
durability would be local.
In
section
the next
I
discuss the asset pricing implications of this
type of model.
3.
ASSET PRICING MODEL
Suppose now that the representative consumer of section 2 is faced with
a
random payoff of y(t+l) units of consumption at time
t.
Then the time
t
price of this payoff is given by:
;3 .i)
P (t) =
Jp^(tn)
mac It)
1
where muc(t)
preference
structure
consumption at time
(3.2)
ffluc(t)
Because
of
=
'
J
marginal
the
is
t
given
the
by
utility of consumption at
(2.2)
and
(2.3),
the
time
For
t.
marginal
utility
the
of
is given by:
"
£JlT
I
i
o
T
p a(T)s(t+x) *
presence
of
the
|
m)|.
conditional
expectation
testing the Euler equation given in (3.1) is somewhat difficult.
in
(3.2),
However if
observed consumption is used to proxy for consumption in the model, and the
good has a finitely lived impact on services,
restrictions
embodied
and
(3.1)
in
Hansen
and
(1989),
Ferson
Examples
directly.
(3.2)
in Dunn and Singleton (1986),
approach can be found, for example,
and
then it is possible to use the
and
Constantinides
(1990)
this
of
Eichenbaum
Gallant
.
and
Tauchen (1989) follow a different approach and use a nonparametric expansion
of the utility function to estimate the degree of dependence in preferences.
However, Heaton (1990) presents evidence that temporal aggregation can
have important effects upon the estimation of the temporal dependencies in
problem
aggregation
temporal
The
preferences.
occurs
this
in
context
because consumers make decisions at intervals much finer than the observed
frequency of the data and the observed consumption data consists of averages
time.
This issue is
temporally
dependent
of consumption over a period of
context
with
models
of
important within the
preferences
time-averaging consumption induces positive autocorrelation
growth
accounting
Without
rates.
temporal
for
in
because
consumption
aggregation,
this
"smoothness" in consumption could spuriously be interpreted as being induced
persistence
by habit
(1987),
Hall
-important
Heaton
(see
and Hansen and Singleton
(1988),
Given this evidence,
very
short:
one
I
Shiller
also show that
it
is
will assume that the interval between decisions
fourth
the
economic environment
(as
in
and
and
.
of
a
month
complications created by this assumption,
Ferson
(1990)
Melino
for temporal aggregation within the context of models
to account
with time additive preferences
is
Grossman,
(1990)).
is
(i.e.
a
a
week).
(1990),
additional assumptions create an advantage.
handle
the
more complete specification of
needed than in Euler equations
Constantinides
To
for
example).
investigations
However,
these
Keeping muc(t) nonnegative
The marginal utility of consumption in models with temporally dependent
preferences involves forward looking terms as in (3.2).
investigations
where
complete
a
mechanism is not necessary,
utility of
consumption
it
behaved.
of
the
data
to verify that
is not possible
well
is
specification
In Euler equation
This
is
a
potential
generation
the marginal
problem
in
models with habit persistence because the marginal utility of consumption at
time
t
involves expectations of future marginal utilities of services with
negative weights.
Without proper restriction on the preference parameters
the estimated utility function could be decreasing
in
consumption,
at
the
observed consumption levels.
For example suppose that services at time
sit) = cit) - ac(t-l), a
(3.3)
>
t
.
Then the marginal utility of consumption at time
nuc(t) = sit)~ 7
(3.4)
-
a^£{s(t+l)"
are given by:
t
is:
^r
:^ (t)>
:
.
Suppose further that the decision interval of the consumer is a quarter so
that we can use observed quarterly expenditures on consumption to evaluate
the
model.
Under
the
assumption
process:
(3.5)
muc'it) = muc(t)/c(t)~ r
that
c(t)/c(t-l)
is
stationary,
the
Notice that muc(t) is nonnegative if and only if muc (t) is
is stationary.
nonnegative since c(t)£0.
a
=
=
0.9
illustrate
the
Setting
0.9.
dramatically
Constantinides
a
admittedly
is
potential
3=0.95
suppose that y=l,
To illustrate the potential problems,
extreme,
as
was
it
chosen
often obtain parameter estimates for a that
(1990)
and
Ferson
However,
problems.
1/4
to
and
are
of
this order of magnitude.
Using these parameter values the process:
7
T
-o0 [s(t+l)/c(t)l
nuc(t) = [s(t)/c(t)]
(3.6)
is
plotted
in
figure
observations of per
nondurables
is
znuc(t)
sit)
from
through
so
that
forward looking term in
the
chosen
utility
taken
1989,
CITIBASE
from
parameter
function
if
negative
the
of
Notice
.
that
muc(t)
due
are
to
the
Because of the negative values of muc(t),
(3.6).
values
values
7
c.i
For the entire sample
highly erratic and is negative quite often.
positive
is
quarterly
of
seasonally-adjusted aggregate expenditures
capita,
1950
consists
data
consumption
The
3.1.
may
not
be
consistent
consumption
observed
with
a
corresponds
to
well
defined
the
optimal
consumption choice made by the representative consumer when consumption is
freely
disposable.
However,
the
object
of
interest
is
muc (t)
=
Eimucit) iy(.t)}.
As a proxy for the conditional expectations needed to evaluate muc (t),
consider:
(3.7)
muc(t) = [sit)/cit)]~ 7 -
a/3
Proj{
10
[s(t + l )/c(t
7
) ]
I
Hit)
}
where Proj is the projection operator.
generated
{R
R
vw
(t-i)}
vw
(t)
is
by
,
<c(t-j )/c(t-l-j>
constant,
a
where R
(t)
Hit) is taken to be the linear space
2
_
f
2
{R (t-j)>
,
and
return on a 3 month treasury bill and
is the real
j=0
the quarterly real return from the CRSP value-weighted portfolio.
Nominal returns were converted to real returns using the quarterly implicit
deflator
price
Proji [s(t+l )/c(t)
]
!
Hit)
parameters
The
nondurables.
for
were estimated using ordinary
}
governing
least
squares
using the entire sample period of the data.
Figure 3.2 plots the resulting realized values of muc(t).
several values of muc(t)
are negative.
linear projection is proxying for
the
To
true
the
extent
that
conditional
Notice that
the
estimated
expectation,
these
chosen parameters cannot be consistent with the data and should not be part
of
the
admissible
parameter
Although
space.
are
there
relatively
few
observations of the marginal utility process that are negative, the negative
observations are potentially important as can be seen in the plot of the
realized values of the marginal rate of substitution as given in figure 3.3.
Notice
that
there
are
outliers
several
that
are
very
negative.
These
results suggest that it may be important to restrict the estimation of these
models
such
consumption.
(as
is
done
that
the
implied
utility
function
is
nondecreasing
in
Only with a complete description of the economic environment
in
this
paper)
is
it
imposition of the restriction muc(t) i
11
possible
to
determine
is important or not.
whether
the
)
Endowments and Asset Payoffs
The Economic Environment:
To
model
environment
economic
the
more
assume
completely,
that
the
consumer is embedded in an endowment economy with some exogenously given law
of
motion
endowments.
for
a
process
r
markov
{X(t)>
sequence of payoffs {d(x)>
=
p(t)
r
(3.8)
suppose
consumption,
that
the habit stock and the asset payoffs at time
durable stock,
of
particular
In
Elt3
V
mucit *\
mac t
time
[denoted
price
t
are elements
t
p(t)]
of
a
is then characterized by:
_
]
The
.
t=o
the
[p(t +
D
+
'J
d(t + l)]
Given a law of motion for {X(t)>,
X(t)\
I
(
the price
the
of
.
asset
time
at
as
t
a
function of X(t) can be found by finding a fixed point to (3.8) as in Lucas
(1978).
Although it would be possible to use this framework to price a large
number of assets,
it
is of
interest to first see if the model is capable of
pricing a few "aggregate" returns.
dividend
stream
discount bond
Q
.
embedded
As a result,
value
will consider pricing the
I
weighted
in
the
CRSP
Assume that
the
law of motion for
the
and
a
real
logarithm of
real
return,
consumption and dividend growth rates is given by a two dimensional Gaussian
process {/"(t)}
t=o
Monthly observations of the logarithm of the real dividend growth rate
are highly seasonal.
To model
component
of
seasonal
autoregressive
dividends
can
be
this seasonal,
modeled
by
polynomial.
12
assumed that
using
In
consumption and dividends evolve according
I
to:
seasonal
particular,
I
the seasonal
dummies
assumed
and
a
that
1
y(t) =
(3.9)
a
r(L) c (t)
(L:
D(t) +
+
/i
where:
iog[c(t)/c(t-l)
y(t) =
iog[d(t)/d(t-l)
T(L)
is
matrix of polynomials
by 2
2
a
in
the
operator,
lag
a (L)
is
a
s
(seasonal) polynomial in the lag operator,
Gaussian
iid
dummies, a
shocks
=
[u
(E(e(t )e(t
u
12
•
•
•
observed data
The
u
48
],
)'
=
)
I),
{e(t)>
D(t)
is a 2 by
_
is
sequence
a
and u is a vector of means
consists
of
averages
of
observation
finest
interval
I
seasonal
used was monthly
aggregate
this
data.
Dividend data is available at very high frequencies.
the
of
.
since
is
sequence of
consumption and dividends
The observation interval that
over different periods
1
for
consumption
However,
I
used
the data at monthly frequencies to avoid having to model finer details like
the number of weeks in a month.
To
estimate
discussed
Although
by
the
(3.9)
Ingram
model
I
and
could
simulated method of moments estimator
used a
Lee
be
(1987)
fit
and
Duffie
and
Singleton
using Maximum Likelihood
as
(1989).
techniques,
the
simulation method is more convenient to use in this context because fitting
implications
the
asset
pricing
The
data
consists
of
of
observations
the
of
model
{c (t),
requires
d (t)}
simulation as well.
for
am assuming a weekly model) where, for example:
(remember
I
o.io:
s V
=
c (t)
;)
a
3
2Wc(t+x)
13
t=0,
4,
8,
...
.
The simulation estimator could be implemented by first simulating a path for
consumption and dividend levels using
the simulated ratios which would be
the moments of
Application
resulted
technique
this
of
instabilities.
example)
(for
along
the
simulated
followed Grossman Melino
and
Shiller
Singleton
levels
(1990)
approximations
example,
"matched"
severe
very
in
the data.
to
numerical
This occurred because a slight mispecif ication of the growth
properties of consumption
consumption
then averaging and computing
(3.9),
to
and
assumed
arithmetic
resulted
path.
To
(1987),
Hall
geometric
that
averages.
In
very
in
avoid
)
>
this
averages
other
are
words
* iog(c (t))
problem,
and
(1988)
a
that 7 ® ilogicit+x) - iog(c(t-4+x)
large values of
I
-
Hansen
I
and
reasonable
assumed,
for
a
iog(c (t-l
) )
The observed data is of the form:
Y*(t) =
(3.11)
a
a
a
a
iog(c (t))-iog(c (t-4))
iog(d (t))-iog(d (t-4))
for
t
=
0,
4,
8
The first
set
seasonal means of log dividend growth,
of
moments
I
fit were
is
non zero.
I
twelve
and the mean of consumption growth.
Because all of the seasonal dummies in the weekly model of
identified,
the
(3.9)
cannot be
assumed that only the dummy for the first week of each month
Under
this assumption,
the
seasonal dummies and the mean
were estimated directly without using simulation.
parameters are given in table 3.1.
the method of Newey and West
(1987),
The estimates of
\i
these
The standard errors were estimated using
with two years of lags,
serial correlation in the error term of the regression.
14
to correct for
I
parameterized f(L) by setting
a U)
%
=
HI)
B{L)
a (I)
2
where B(L) is a 2 by 2 matrix of first order polynomials in the lag operator
and a
and a (L) are polynomials of order
(Z.)
=
setting a (L)
the
simulated
-
1
method
These parameters were fit using
where s was 48.
a L
moments
of
a (L) was parameterized by
2.
described
estimator
above
using
1344
observations in the simulated sample (approximately four times the observed
sample
r(t)
Let
size).
seasonal dummies and means.
denote
Similarly,
less seasonal dummies and means.
the
following
and £{/(
moments:
a
t
)Y (t-12)
values 12 and
}
residual
the
E<Y*(
Y^tt)
less
estimated
let Y(t) denote the residual of 7(t)
In fitting the parameters of y(t),
a
t
of
)y
)
(
t
'
}
,
£{Y
a
a
(
t
)Y (t-4)
/
}
,
I
used
E{Y*U )Y*(t-8)
'
}
along with the covariance of dividend growth with its
13 months
used in the procedure,
1
in
the
past.
In
estimating the weighting matrix
year of lags in a Newey-West (1987) procedure were
used.
A test of the over identifying restrictions yields a test statistic of
1.23 which implies a P-Value of 0.873.
As a result,
reasonable job of fitting the chosen moments.
the model
is doing a
Given this law of motion for
the exogenous variables of the model, asset prices can now be calculated.
Numerical Solution Method
Even if the law of motion for consumption and dividend growth is taken
as known,
it
is not possible to analytically calculate the marginal utility
15
of consumption or asset prices as a function of
the world at
the approach taken in this paper is to calculate
Instead,
any given date.
the state of
the marginal utility of consumption and asset prices numerically.
a
growing
dynamic
examining
literature
economic
models
the
for
(see,
properties
example,
of
the
numerical
group
of
There is
solutions
papers
in
January 1990 issue of the Journal of Business and Economic Statistics)
solution method that
I
chose was the method suggested by Judd
is based upon methods used to solve differential equations.
a brief outline of the method
Consider,
for
example,
I
I
to
the
The
.
which
(1990)
will provide
used.
the
marginal
utility of
consumption when
consumption good is subject to exponential depreciation.
In this case,
the
the
marginal utility of consumption is given by:
muc(t) =
(3.12)
£{^ =o £
T T
A s(t)"*:?(t)>.
Note that (3.12) is a solution to the difference equation:
r
muc(t) = s(t)' + 0AE{jnuc(t+l)!?(t)}.
(3.13)
The
difference
equation
(3.13)
seems
to
be
relatively
simple
to
solve.
However the problem is that observed consumption is growing over time which
implies
that
the
service flow from consumption is growing over
handle this problem, normalize (3.13) by c(t)
(3.14)
muc*(t) = s*(t)" r
where muc
(t )=muc{ t )/c( t
)
+
time.
To
to give:
9
/3A£{muc*(t + l)[c(t + l)/c(t)r ':? (t)}
A unique stationary solution to
16
(3.14)
exists
if
-r£{iog[c(t+l)/c(t)]}
generated
under
model
the
In
by
^A [see Cochrane (1989), for example].
>
finite-dimensional
a
and dividend growth,
vector,
state
lagged values of
and
at
time
t,
X(t),
that
includes
is
!F(t),
the
current and lagged values of consumption
the habit stock,
stock of durables,
information
consideration,
shock process
the
The
(e(t)}.
solution to (3.14) was approximated by a linear combination of finite-order
polynomials
the
in
vector
state
Let
X(t).
T (X(t))
denote
polynomials of elements of X(t) of order less than or equal
denote
a
vector
approximate muc
with
weights
of
same
the
dimension
(t)
by:
muc (t)
<f>
n
»
Of course muc (t)
= <pT (X(t)).
of
Then
T (X(t)).
~
«
n
satisfy (3.14) exactly.
will not
n
Define a residual vector:
;X(t)] * muc*(t) - s*(t)* - A££{muc*(t + 1
C,14>
as
Let
to n.
n
~
*
(3.15)
vector
a
n
)
[c( t + 1 )/c(
t
\~ 7
)
:?(t)
}.
n
Then the parameters
are chosen such that
<p
the polynomials in the state vector.
(3.16)
;X(t)]r (X(t))> =
£{?[</>
method
That is choose
is are orthogonal
to
<p
n
such that:
o.
was
applied
repeatedly
for
temporal dependencies considered in this paper.
calculate asset prices.
I
;X(t)]
n
n
n
This
C,[<f>
n
the
different
examples
of
The method was also used to
In the calculations presented in sections 4 and 5,
used second order polynomials to approximate the various functions
calculate the inner product given in
1000 observations of X(t).
(3.16)
I
.
To
created a times series with
Sample averages were then used to approximate
the expectation given in (3.16).
17
4.
HANSEN AND JAGANNATHAN DIAGNOSTICS
Before turning to the results of formal estimation and testing of the
model of section
model.
first present
will
I
use
this
section,
I
(1990,
1991)
which
In
Jagannathan
2,
simple
some
developed
diagnostic
a
implications of
restrictions
examines
by
moments
on
marginal rate of substitution implied by asset market data.
the
results
of
estimation
diagnostics presented
results because
they aid
model
are
before
section are useful
this
in
complete
the
of
Hansen
are
useful
the
of
presented.
turning
The
to
these
determining the effects on the asset pricing
in
as
and
In Section 5
environment of the different parts of the preference structure.
diagnostics
the
way
a
whether
determining
of
Also the
numerical
the
solution algorithm discussed in Section 3 is performing reasonably well or
not.
In examining whether
the different
forms of temporal dependencies can
fit the bounds on the moments of the marginal rate of substitution implied
by asset market data,
habit persistence that
and 2,
it
locally
is
found by Heaton
onJy
conclusion
is
important to examine whether the fit is due to
local
in nature.
As
I
argued in the sections
1
seems reasonable to restrict the preference specifications so that
consumption
data
is
it
(1990)
imply
that
substitution
if
local
is
reached
obtained by Heaton
substitutable.
in
(1990)
Section
In
habit
fact,
The
5.
will be useful
section.
18
parameter
persistence fits
consumption
in
the
is
the
present.
estimates
consumption
A
similar
preference parameters estimates
in guiding
the
analysis
in
this
Hansen and Jagannathan Bounds
Consider an asset
that
off y(t+4)
pays
units of consumption at
Then the price of the asset in units of consumption at time
t+4.
t,
time
g(t),
satisfies:
4
g(t) = £{p [muc(t+4)/muc(t)]y(t+4):^(t)}
(4.1)
where muc(t) is the marginal utility of consumption at time
Write (4.1)
t.
as:
q(t) = £{mrs(t+4)y(t+4) !£(t)}
(4.2)
where mrs(t+4) =
(3
[muc( t+4)/muc(
Hansen and Jagannathan
t
)
]
.
1991)
(1990,
(hereafter
H-J)
show how to use
observed asset prices and payoffs to estimate regions in which the mean and
standard deviation for the process,
mean for
the marginal
rate of
mrs,
must lie.
substitution,
it
is
They show that given a
possible
to
estimate a
lower bound for the standard deviation of the marginal rate of substitution.
By changing the hypothetical mean for the marginal rate of substitution,
region for the moments of
the
marginal
rate of substitution
is
created
a
12
.
H-J propose to test different models by asking
After creating this region,
whether the moments of the marginal rate of substitution predicted by the
model lie within the region.
This is the diagnostic
In estimating the H-J region,
I
I
use.
used the one month treasury bill return
and the monthly CRSP value weighted return constructed from daily returns
(from
1962,7
to
1989, 12)
13
.
These
two
19
returns were
multiplied
by
their
lagged values to create 6 asset payoffs and prices to use in estimating the
Figure
region.
gives
4.1
a
plot
I
function plotted in figure 4.1.
is
region.
To
test
in a P-Value
This resulted
zero
region
the
that
indicating
resulting H-J
tested whether the minimum value of the
whether the region has content,
essentially zero
the
of
of
figure
has
4.1
of
empirical
content.
I
now present results of calculating estimated moments for mrs for the
models
different
the
using
estimated in section
law
motion
of
for
the
exogenous
variables
3.
Moments of HRS with Time-Additive Preferences
As
consider
first
benchmark,
a
moments of
the
the
mrs
implied by
a
where the preferences of the representative consumer
time-separable model,
are given by:
(4.3)
£{^"p t [c(t)
y -l]/(l-y)}.
1
Further suppose that we ignore the temporal aggregation issues and set c(t)
equal to the observed monthly consumption of nondurables plus services
setting
time
the
preferences).
of
interval
to
month
one
substitution are
left in increments of
the H-J
assuming
time-separable
The estimated first and second moments of the marginal rate
plotted
(the
m
'
1
.
(3
in
s)
starting from y=l at the point E(mrs) =
is
and
(ie.
1
4.2,
for
and std(.mrs) = 0,
was assumed to be
region of figure 4.1.
figure
Note that
1.
values
of
j
to y=12 on the
Also plotted in figure 4.2
the estimated points
for
the
moments of mrs do not come close to being in the region as y is increased.
20
noted
was
This
by
Hansen
Jagannathan
and
(1991).
As
standard deviation of the mrs is increased but Eimrs}
j
is
increases,
the
pushed very much
out of line.
Suppose that we now shift the time interval back to a week,
maintain
the time-additive preference specification and use the estimated weekly law
Plotted in figure
of motion of section 3 to estimate the moments of muc(t).
4.3 is the H-J region along with the estimated moments for the marginal rate
(the O's) and the monthly model
of substitution for the weekly model
's
as
before).
The
plot
std(mrs)=0 and £(mrs)=l,
for
the
weekly model
again starts at
and moves in increments of
Eimrs) = 0.994 and std(mrs)
1
y=l
to y=20 at
Again £ was set to
= 0.063.
1.
(the
with
the point
Note that the
increase as quickly as in the
second moments estimated for the mrs do not
case of using time averaged data, but that Eimrs) performs much better as a
function of
seem
However,
y.
provide
to
a
accounting for temporal aggregation alone does not
improved
greatly
This
fit.
explain
could
why
investigations that account solely for temporal aggregation problems are not
completely
successful
[see,
for
example,
Grossman,
Melino
and
Shiller
(1987)].
Moments of MRS with Pure Habit Persistence
Constantinides (1990) argued that habit persistence can help to explain
the equity premium puzzle of Mehra and Prescott
of
interest
to
(1985).
As a result,
investigate habit persistence in this context.
it
is
Figure 4.4
plots moments for the habit persistence model given by (2.2) and (2.5) with
6 =
0.9
and
a
outlined above.
=
0.5
and
The value
using
of
8
the
weekly consumption
chosen
21
is
consistent
law
with
of
motion as
the
estimated
half-life for habit effects estimated by Heaton (1990).
triangles
figure
in
give
4.4
with
moments
the
was set to
(3
habit
1.
persistence.
The
The
circles are the same as those in figure 4.3 (except that y ranges only from
1
i ranges from
to 10).
1
to 20 with stdimrs)
Notice that for any given level of
the
case
of
habit
no
utility function,
the
y,
std(mrs)
Hence with
effects.
rising as y rises.
is much larger
curvature
less
than in
the
in
period
required standard deviation of the marginal rate of
substitution could be more easily fit.
This finding
similar
is
the
to
results reported by Gallant, Hansen and Tauchen (1989) using a monthly model
and a monthly law of motion for the exogenous variables of the model.
Figure 4.5 gives the weighting function a(j) of (2.3),
parameter
Notice
settings.
that
this
parameter
implied by these
very
setting puts
small
For any value of y the
negative weight on each of the past consumptions.
volatility of the marginal rate of substitution can be increased by raising
the absolute sum of
way
Another
of
This sum is given by the parameter
these weights.
increasing
the
volatility
the
of
marginal
rate
a.
of
substitution is to increase the absolute value of a(j) for small values of
j.
This
can be
done
by decreasing
0.
Figure
4.6
gives
a
plot
of
the
moments of the marginal
rate of substitution similar to figure 4.4 except
that
Notice
is
set
to
0.5.
deviation of the marginal
that
rate of
for a given value
substitution
is
of
f
the
standard
Figure 4.7
increased.
gives a plot of the a(j) function implied by these parameter setting.
A
difficulty
with
the
result
of
figure
4.6
is
that
it
that
the
improvement in the fit of the moments of the marginal rate of substitution
is to a large extent due to local habit persistence.
22
Moments of MRS with Habit Persistence with Local Durability
Consider now the specification of preferences given by (2.2),
in which the good
(2.7)
is assumed
to be durable and
(2.6) and
the consumer develops
Heaton (1990) shows that
habit over the flow of services from this good.
the observed consumption dynamics are consistent with this specification but
and
also
is
(2.7)
The specification given by
persistence.
not with pure habit
consumption is
consistent with the notion that
(2.6)
(2.2),
locally
substitutable.
I
are
Figure
first set £ =
consistent
4.8
plots
1,
with
estimates
of
moments
the
by
found
estimates
parameter
the
The values for A and
6 = 0.9 and a = 0.5.
A = 0.7,
of
substitution under this preference specification.
Heaton
marginal
the
(1990).
rate
of
squares give plots
The
for the case of habit persistence with local durability as y ranges from
to 20.
Also included in figure 4.8 is the similar plot for the case of pure
habit persistence
the
1
Notice that with the local durability,
(the triangles).
estimated moments are very different
By
persistence.
making
the
than
locally
good
in
case
the
substitutable,
of
pure habit
the
standard
deviation of the marginal rate of substitution is much lower.
At
of
issue is whether the result of figures 4.4 and figure 4.6 play off
local
habit.
increased.
To
examine
that
issue
Figure 4.9 gives a plot
ranges form 0.5 to 0.8.
Notice
this
as
a
consider
similar
to
Again for each value of
increases,
the
closer to fitting the H-J region.
model
with
what
happens
as
figure 4.8 except
a,
local
y varies from
durability
1
a
is
that a
to 20.
comes
much
However very dramatic values of y are
still needed to come close to the region.
In figure 4.10,
I
plot the weights on past consumption in the mapping
23
from
consumption
services
to
implied by
there
substitution
example,
is
with
substitution
local
is
removed from
habit persistence with
local
Notice that the weighting function implies
durability model with a = 0.8.
that
the
the
long-run
model,
the
habit.
As
performs
model
local
the
For
better.
figures 4.11 and figure 4.12 given plots similar to figure 4.9 and
4.10 except with A=0.5 6 = 0.51.
As figure
4.12 demonstrates under
this
parameterization the habit persistence effects dominate quite quickly.
In
this case much lower values of y are needed to come close to the H-J region.
In general
it
only when
the
seems that
large
improvements in the fit of the model occur
substitution
local
is
driven out
in
favor
of
local
habit
persistence.
5.
Estimation Results.
In this section
of section
encounter
2.
I
report results of estimating and testing the models
The results of section 4 indicate that the model is likely to
difficulties
because
of
needed in fitting the H-J regions.
some
evidence
that
the
model
the
extreme
parameter
values
that
are
However the results of section 4 provide
solution algorithm
is
performing reasonable
well since the plots of the moments of the mrs are well behaved as parameter
values are changed.
In estimating and
testing the asset pricing models of section
2,
the
estimated law of motion for consumption and dividends given in tables 3.1
and
3.2 was
preferences,
were
taken
as
given.
Given
stock and bond prices,
calculated using
the
numerical
this
as
a
law
of
motion and
function of
method described
the
in
the
form of
state variable,
Section
3.
The
stock price gives the price of the future dividend stream fitted in Section
24
The bond price is the price of a one month real discount bond.
3.
The preference structures were fit using simulated method of moments 15
I
used observed moments of the vector r (t )=[r
the monthly real
the
monthly
vw
(t
where r
r (t)]'
vw
.
(t)
is
return on the CRSP value weighted portfolio and r f (t)
is
T-bill
real
£<[r(t)-r][r(t)-r]'}
rate
and
,
moments
The
.
)
£{ [r(t )-r] [r(t-l )-r]
'
used
where
}
were
=
r
£{r(t)>,
£{r>.
A
simulation that was three times the size of the observed sample of returns
was used.
The
complete
estimated
model
with
first
weighting
a
estimation
first-stage
durability
local
of
error.
matrix
In
this
habit
and
that
did
initial
correct
not
round
persistence
of
was
for
the
estimation
the
parameters y and £ were driven to regions where the equilibrium in the model
To counter this problem,
does not exist.
I
fixed £ at 0.99
.
of £ implies that an equilibrium exists for all values of y>0.
restriction,
the
rest
the
of
parameters were estimated and
the
This value
Under this
resulting
estimates were used to estimate a weighting matrix that corrects for
estimation error
in
the
law
of
motion for
the
consumption and dividends.
second round of parameter estimation was done for each of the models.
A
In
each case the weighting matrix was fixed at its estimated value implied by
the
first-stage
estimates
of
the
complete
model.
This
allows
a
simple
comparison of the minimized criterion function across the different models.
Table
5.1
reports
the
results of estimating
local durability and habit persistence.
is
the
complete model with
Notice that the curvature parameter
estimated to be quite small and that the estimate of a is quite large
implying a large degree of habit persistence.
The estimates
also quite
large.
nature
consumption
of
is
10.7
imply that
months
25
and
The estimates of A and
the
the
are
half-life of the durable
half-life
of
the
habit
8
stock
persistence
Figure
months.
6.6
is
plots
5.1
governs
shape
The
(2.3)].
form
the
substitutable
for
function
the
of
weeks
51
dependencies
temporal
the
of
year
(one
and
persistence effects dominating beyond 51 weeks
substitutability
of
estimates
degree
the
of
consumption
the
of
durability
of
with
is
habit
the
The relatively long period
goods
with
consistent
is
the
in
[see
consumption
that
month)
one
1
preferences
in
indicates
a(j)
a(j),
Remember that the function
implied by the parameter estimates of table 5.1.
a(j)
function
the
components
other
aggregate
of
nondurables [see Hayashi (1985), for example].
The estimated form of the time-nonseparabilities is consistent with the
a
priori
restriction that consumption is locally substitutable.
of this underlying nonseparability also potentially explains
studies using monthly data
The shape
the fact
that
Gallant and Tauchen (1989) and Eichenbaum
(e.g.
and Hansen (1990)) find evidence for local substitution whereas studies that
have used quarterly data find evidence for habit persistence
and Constantinides (1990)).
(e.g.
Ferson
Data averaged over longer periods of time could
be picking up the presence of long-run habit persistence
19
.
Table 5.1 also reports standard errors for the estimated parameters of
the
model.
parameters governing
The
poorly estimated.
The
may be quite
approximation that
parameters
accuracy
of
(e.g
estimated
a^l
)
.
To
parameter
the
the
standard
nonseparabilities are
errors
because
poor
overcome
this
estimates
can
of
the
problem,
be
done
upon
rely
mean-value
restrictions
on
the
assessment
of
the
an
by
a
seemingly
examining
different
hypothesis about the parameters directly.
A Chi-Square
test
of
the
over
P-Value of 0.0009 so that the model
It
is
perhaps
not
surprising
that
identifying restrictions
is
)
yields a
not very consistent with the data.
this
26
(J
simple
asset
pricing
model
is
.
rejected by the data.
is
It
greater
of
interest
examine whether
to
the
addition of these parameters significantly improves the fit of the model.
Table 5.2 gives parameter estimates of the pure habit persistence model
with
parameter A set
the
Under
zero.
to
this
restriction the
criterion
function deteriorates quite dramatically relative to that reported in table
5.1
The change in the criterion function
for the complete model.
(the J
when this restriction is relaxed has a Chi-Square distribution
statistic)
with one degree of freedom
20
The P-Value of the test that A=0 is
.0005.
Hence there is a great deal of evidence against the pure habit persistence
model
5.2
table
relative
to
conducted
by
parameter
estimates
the
table
to
model
taking
of
the
Again
5.3.
habit
a
the
null
of
the
difference
identified and hence
time
the
in
additive
test
Chi-Square distribution with
1
is
very
little
the
the
of
time
the
additive
time
minimized
(a=0)
additive
criterion
functions
this test
parameter
the
model
durability can be
local
A problem with
model
that
is
9
is
not
However a
has a nonstandard distribution.
degree of freedom can be used to calculate a
lower bound on the P-Value of the test.
there
test
persistence with
values reported in tables 5.2 and 5.3.
under
of
Notice that the criterion function deteriorates very little
specification.
from
presents
5.3
Table
evidence
against
This lower bound is 0.169 so that
the
simple
time
additive
model
relative to a model with habit persistence alone.
The change in the criterion function from the time additive case to the
case
habit
of
persistence
with
local
durability
is
quite
indicates that the addition of the two effects is important.
because
8
is
not
identified
under
the
null
that
A=0
probability value for the test cannot be easily derived.
27
and
large
which
Unfortunately
a=0,
an
exact
Estimation
durability
pure
the
of
resulted
model
in
parameter
a
estimate for A of essentially zero so that the presence long-run habit along
substitution seems
local
with
issue,
to
further
To
important.
be
examine
this
consider figure 5.2 which gives a plot of the criterion function as a
function of the parameter a where the criterion function is minimized over
the
parameters
y,
6
and
quite
deteriorates
function
As
A.
a
toward
driven
is
value
fixed
each
For
rapidly.
criterion
the
zero,
a
of
the
difference between the criterion function value plotted in figure 5.2 and
the value with a unrestricted is asymptotically distributed as a Chi-Square
random variable with one degree of freedom.
probability value of the resulting test
Figure 5.3 gives a plot of the
that
a
the value given on
is
the
There is quite dramatic evidence against values of a smaller than
x-axis.
about 0.7.
The
figure
fu iction
5.4.
disappeared.
a(j),
Notice
under
that
As a result,
the
the
restriction that a = 0.7,
long-run
evident
habit
in
is
figure
given
5.1
in
has
the improvement in the fit of the model is due to
the simultaneous presence of habit and local substitution effects.
To help
interpret this result, consider table 5.4 which reports values of the fitted
moments as implied by the parameter estimates of the different models.
reported in table 5.4 are the sample values of the moments.
persistence
model
and
the
complete
model
do
a
better
Both the habit
job
model
does
a
much
better
job
fitting
of
first-order autocovariance of the risk-free rate.
the
than
the
However the
time-additive model in fitting the level of the risk free rate.
complete
Also
variance
and
The large improvement in
the criterion function with the complete model occurs because these moments
are relatively well measured.
The presence of
local
substitution seems to
be needed to fit the observed smoothness in the risk-free rate,
28
as measured
by its variance and first-order autocorrelation.
Although the habit persistence with local durability model does provide
rationalization of some of the moments of stock and bond returns,
a better
the model
not
is
One potential
complete success.
a
explanation for this
failure is provided by the results of Section 4 which indicate that
substitution
(1990,
makes
difficult
very
to
fit
the
Hansen
Jagannathan
and
1991) bounds on the moments of the marginal rate of substitution.
the mean and
fact
it
local
rate of substitution
standard deviation of the marginal
implied by the estimates of
are
table 5.1
In
0.999 and 0.0021
respectively.
This point is well outside of the Hansen and Jagannathan region examined in
Section
7.
4.
Concluding Remarks
In this paper
the
I
modeled consumption as being durable and
I
assumed that
representative consumer develops habit over the flow of services from
Compared with a simple time additive model
the durable consumption good.
found
that
model
the
observed moments
of
improves
substantially
stock
and
bond
returns.
fit
the
of
occurs
This
substitution and long-run habit are present simultaneously.
the
few
first
only
if
I
local
Neither pure
local substitution nor pure habit persistence significantly improves the fit
of
the
model.
A
difficulty
statistically rejected.
the
local
prices
with
the
results
is
that
the
model
is
A potential explanation for this rejection is that
substitution that
is
needed to fit many of the moments of asset
difficulties
on
other
causes
dimensions.
estimated Hansen and Jagannathan (1990,
29
1991)
In
particular,
to
fit
bounds on the moments of the
marginal rate of substitution, the local substitutability of consumption can
be maintained only if extreme curvature
in
the period utility function
is
allowed.
potentially
several
are
There
limitations
important
the
to
The first is the use of a single
investigation carried out in this paper.
expenditures on nondurables and services.
class of consumption goods:
It
would have been preferable to investigate a model with multiple goods and to
examine durable goods along with nondurables and services as was done by
Dunn and Singleton
(1986)
and
Eichenbaum and Hansen
However this would have made it very difficult
work
necessary
for
asset
the
pricing
(1990),
for
example.
undertake the numerical
to
Investigations
calculations.
with
multiple consumption goods have not been entirely successful when no account
is made for the presence of temporal aggregation in the data.
it
ras
important to first examine a model
temporal
aggregation
and
that
capable of dealing with
is
time-nonseparabilities
As a result,
in
a
world
of
a
single
An analysis of the model with multiple goods is left to
consumption good.
future work.
A
second
seasonally
issue
adjusted
in
this
paper
is
the
consumption data.
It
could
be
that
not
addressed
seasonal adjustment hides some of
preferences
"permanent
evidence
[see
for example Heaton
income"
for
the effects of
model
seasonal
Ferson and Harvey
of
habit
(1990)
(1989)].
consumption
persistence
In
temporal
the
also
Heaton
Osborn
find evidence for seasonal
habit
of
using
process
of
dependencies in
linear
(1990)
found
(1988)].
Also
persistence
Euler equation investigations using seasonally unadjusted data.
model
of
an analysis of a
dynamics,
[see
effect
in
Adding a
seasonality to the other effects considered in this paper would
30
also have dramatically complicated the calculations need for the analysis in
this paper.
An investigation of the model with seasonally unadjusted data
is also left to future research.
31
Footnotes
*Abel
(1990)
also
evidence
presents
that
models
which
in
consumption
is
complementary over time can fit the observed equity premium.
2
Heaton (1990) and Ferson and Harvey (1990) find evidence for seasonal habit
formation using seasonally unadjusted consumption data.
3
Ferson
and
Constantinides
(1990)
argued
that
the
use
quarterly
of
and
annual data helps them to capture habit persistence that develops slowly.
4
The
nonseparabilities are written as depending on
notational convenience.
£ 0,
5
infinite past
for
To make this consistent with consumption only for
set consumption prior to time zero,
Dunn and Singleton
the
(1986)
t
to zero.
and Eichenbaum and Hansen
(1990)
examine models
with multiple goods in which some of the goods may have an infinitely lived
impact on services.
However,
the presence of a least one good with a finite
life is necessary.
6
Christiano,
car
help
to
Eichenbaum and Marshall
explain
some
anomalies
(1989)
in
show that
consumption data
They do not apply the model to asset data however.
32
temporal
and
aggregation
income
data.
7
In later sections of the paper
appendix.
used monthly data as described in the data
I
used quarterly data in this section since this is the data set
I
that tends to produce parameter estimates that imply habit persistence
[see
for Ferson and Constantinides (1990)].
Mehra and Prescott
proceeded by assuming that
(1985)
the
return is the return to holding the aggregate endowment.
link between aggregate consumption and the object
also
by
done
Tauchen
Cechetti,
and
(1986)
Lam
Here,
broke the
I
be priced.
to
Mark
and
aggregate stock
This was
(1990a,
b),
for
example.
9
The
assumption
follow
vector
a
that
the
ARMA
logarithms
process
with
consumption
of
seasonal
dummies
example Cechetti,
Lam and Mark
(1990a,
b)
switching
model
could
be
large
state
would
correlation over
time
found
in
seems
minimal
to
a
is
a
growth
simplifying
very
be
the
state
have
been
For
used a Markov switching model to
model annual consumption and dividend growth.
however
dividend
Other models of dividends and consumption have been used.
assumption.
Markov
and
been
A different model
used
needed
to
in
this
capture
such as a
investigation,
the
degree
of
the monthly data.
The ARMA specification
specification
captures
that
a
few
of
the
important moments in consumption and dividend growth.
A complete description of the data used in this paper can be found in the
appendix.
33
11
I
tried
several
simple
more
models
of
consumption
and
dividends
(for
example with a (L) and a (L) constrained to be equal) that were rejected by
the
data
when
compared
to
the
more
general
alternative
reported
in
the
paper.
12
Due to the large dimension of the state variable X(t)
it
was not possible
to experiment with higher order polynomials in the solution algorithm.
13
Hansen and Jagannathan (1991) show how to create bounds for these moments
both with and without the restriction that the marginal rate of substitution
must be positive.
restriction that
In
the
this paper
marginal
I
will consider those bounds without the
rate of
securities considered in this paper,
substitution be positive.
For
the
the added restriction that the marginal
rate of substitution be positive does not seem to be very important.
See the appendix for a more complete description of the asset return data
used in this section.
This test is described in detail in Hansen and Jagannathan (1990).
A correction to the distribution of the test statistics and the parameter
estimates was done to take account of the estimation error present
estimates of the
law of motion for
consumption and dividends.
the
See Newey
(1986) and Hansen and Heaton (1991) for discussions of this procedure.
34
in
17
In
The
solution the price of a real discount bond was calculated.
the model
comparison
the
to
observed
real
T-bill
rate
valid
is
under
the
assumption that inflation risk is negligible.
1
The estimation reported in this section was also carried out
set
of
The
used.
reported
conditions
moment
in
results
using
paper.
the
in
with a larger
which second-order autocovariances were
second
this
However
some
set
were
problems
similar
were
to
the
also
results
encountered
due
to
difficulties in estimating the very large weighting matrix needed for the
estimation.
As
a
result,
estimation
the
results
with
the
larger
set
of
moments are not reported.
19
Although the half-life of the habit stock is smaller than the half-life of
the durable stock,
when
defined
over
the habit
persistence effects are of a long-run nature
consumption
directly.
This
occurs
because
the
of habit is created from the flow of services from the durable stock.
35
stock
2
°I
tried to assess this
implied
by
estimates
point
the
issue directly by simulating the economy that
of
tables
3.1,
3.1
and
time-averaged the resulting data over periods corresponding
quarters and
(1990)
I
to this data.
result,
This seemed to occur because the underlying model
restrictions
the model
is
not
of
the
model
as
indicted by the
test
table
5.1.
reported
in
of
the
As
a
capable of producing simulated data that closely
resembles the observed data.
21
months and
Unfortunately this resulted in very strange parameter
very much at odds with the true data,
overidentifying
to
then
I
applied the methods and models of Ferson and Constantinides
estimates of the models.
is
5.1.
is
See Eichenbaum, Hansen and Singleton (1988), for example.
36
APPENDIX:
DATA
Real and nominal purchases of nondurables plus services from
Consumption:
the
U.S.
National
Income
Monthly from 1959,1 to 1989,12.
Seasonally
Accounts.
adjusted.
Taken from CITIBASE.
Dividends from the value weighted return series of the New York
Dividends:
Stock Exchange as
Converted
Product
and
real
to
Monthly from
constructed by CRSP.
dividends
using
the
1959,1
price
implicit
to
1989,12.
deflator
for
nondurables plus services.
Population:
The consumption data was converted to a per capita basis using
monthly U.S.
population from 1959,1
to
1989,12.
The data
is
published by
the Bureau of the Census and was obtained from CITIBASE.
Treasury Bill Return:
1959,1
to
1989,12.
One month return on one month
From
the
CRSP bond file.
treasury bills from
Converted
to
real
returns
using the implicit price deflator for nondurables plus services.
Value
weighted
Value
return:
weighted
returns
from
constructed from the CRSP daily value weighted return.
the
NYSE
were
The monthly value
weighted return was not used directly since it assumes that all dividends
are paid at
the
end
of
a
month.
Monthly from
1962,7
to
1989,12.
This
series was converted to real returns using the implicit price deflator for
nondurables plus services.
37
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40
TABLE 3.1
ESTIMATES OF MEANS AND SEASONAL DUMMIES FOR WEEKLY LAW OF MOTION OF THE THE
LOGARITHMS OF REAL CONSUMPTION AND REAL DIVIDEND GROWTH
ljog[c(t)/c(t-l)]
x(t) s
,Jog[d(t)/d(t-l)]
-I
-1
o
1
X(t) =
ru)c(t)
a (L)
+
D(t) +
/j
s
'
Parameter
-
i
r
2
48
Estimate
0.00169
l
s
5
s
9
s
13
s
17
s
21
s
25
s
29
s
33
s
37
s
41
s
45
1
/
_
r
1
„i
Standard Error
TABLE 3.2
SIMULATED METHOD OF MOMENTS ESTIMATES OF
a (L)
1
1
a (L)
a (L)
X(t) = BU)e(t)
2
Parameter
Standard Error
Parameter Estimate
a
1.305
0.063
a
0.575
0.027
1.656
0.034
0.922
0.206
0.547
0.087
0.0089
0.0075
-0.0264
0.0083
0.0165
0.0104
-0.0076
0.0068
-0.0128
0.0027
0.0114
0.0081
-0.0213
0.0120
a
l
a
2
2
a.
48
11
B
i
21
B
i
22
B
i
B
11
1
B
12
1
21
B
1
22
B
1
Parameter estimates based upon 1344 simulated observations and the moments
One-year of lags was used
discussed in the text.
procedure to estimate the weighting matrix.
42
in a Newey-West
(1987)
TABLE 5.1
S.M.M. ESTIMATES OF COMPLETE MODEL
Using means, covariance and
1
autocovariance of stock and bond returns.
Standard Errors
Parameter
Parameter Estimates
y
0.356
0.995
a
0.946
1.300
6
0.974
0.382
A
0.984
0.390
20.794
J
T
P-Value:
8.86 (xl0~
4
)
TABLE 5.2
S.M.M.
ESTIMATES OF HABIT PERSISTENCE MODEL
Using means, covariance and 1 autocovariance of stock and bond returns.
Weighting matrix from habit persistence with local durability model.
Parameter Estimates
Parameter
y
0.0123
0.863
a
0.874
4.573
0.970
0.611
32.816
J
Standard Errors
T
P-Value:
1.
14
(xlO
5
)
43
TABLE 5.3
S.M.M. ESTIMATES OF TIME ADDITIVE MODEL
Using means, covariance and 1 autocovariance of stock and bond returns.
Weighting matrix from habit persistence with local durability model.
Parameter Estimates
Parameter
0.709
y
34.703
J
T
-5
P-Value:
3.028 (xlO
)
44
Standard Errors
0.133
TABLE 5.4
SAMPLE AND FITTED MOMENTS
r(t) =
[
r
vw
f
]',
r (t)
(t)
?(t) = r(t)
£{r(t)}
-
12
11
cr
*
}-,
cr
£{r(t)r(t)'} =
22
21
<T
CT
12
11
cr
cr
l
£{r(t)r(t-l)'} =
l
22
21
cr
cr
l
Moment
l
Sample Value
Fitted
Complete Model
11
1.00256
1.00225
1.00256
1.00094
1.00086
1.00083
1.00113
2.019 (xlO~
1.019 (xlO
)
1.730 (xlO
)
o
12
r
o
22
P
11
r
Time Additive
1.00541
2
r
Habit Model
1.242 (xlO
i.492
(xlO
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-6
)
4
1.179 (xlO
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8.955 (xlO"
-2.515 (xlO
6.116 (xlO
-3.177 (xlO
3.977 (xlO
-4
)
-4
)
-6
5.843 (xlO
-3
)
-3.449 (xlO
)
3.669 (xlO
)
-1.404 (xlO
-6
1.481 (xlO"
)
-6
)
6
]
-4,
l
12
P
1.236 (xlO
-5,
1.199 (xlO
-5,
8.829 (xlO"
)
1.475 (xlO
)
3.425 (xlO
)
1
21
P
1.387 (xlO"
5.980 (xlO"
4.192 (xlO"
3.740 (xlO"
)
3.884 (xlO
-6
-6,
)
1
22
45
-9.116 (xlO
-8
.
-4.622 (xlO
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Lib-26-67
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LIBRARIES DUPL
007D1SE4
6