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SAMPLE PROPERTIES OF SOME
ALTERNATIVE GMM ESTIMATORS
FINITE
Lars Peter Hansen, University of Chicago
John Heaton, Massachusetts Institute of Technology
Amir Yaron, Carnegie-Mellon University
WP#3728-94-EFA
September 1994
Finite
Sample Properties of Some
Alternative
GMM Estimators
*
Lars Peter Hansen! John Heaton+and Amir Yaron^
September 1994
Abstract
We
investigate the small sample properties of three alternative
GMM
estimators of asset
The estimators
that we consider include ones in which the weighting matrix
is iterated to convergence and ones in which the weighting matrix is changed with each
choice of the parameters. Particular attention is devoted to assessing the performance of
the asymptotic theory for making inferences based directly on the deterioration of
pricing models.
GMM
criterion functions.
Iv'.ASSACHUSETTS iNSTiTUTi
OF TECHNOLOGY
DEC 01
1994
LIBRARIES
"We thank Renee Adams, Wen Fang
Monge, .Marc Roston and George Tauchen for helpful
.Marc Roston for assisting with the computations. Financial support from the National Science
Foundation (Hansen and Heaton) and the Sloan Foundation (Heaton and Yaron) is gratefully acknowledged.
.\ilditional computer rfsources were provided by the Social Sciences and Public Policy Computing Center.
Part of this research was conducted while Heaton was a National Fellow at the Hoover Institution.
M.niversity of Chicago. NORC and NBER
Sloan School of Management. MIT and NBER
'Graduate School of Industrial .Administration. Carnegie-Mellon University
comments and
Liu. .\le.\ander
Introduction
1
The purpose
of
of this paper
is
to investigate the small
moments GM.\f) estimators applied
(
initial (consistent)
and ones
parameter value. The
in
which the weighting matrix
last of
is
estimated using an
is
changed
for
but
much
has the attraction of being insensitive to
it
conditions are scaled. In addition,
we study the advantages
how
to basing statistical
inferences directly on the criterion function rather than on quadratic approximations to
We
•
it.
address the following issues:
How
does the procedure for constructing the weighting matrix affect the small sample
behavior of the
•
is
every hypo-
these three approaches has not been used very
in the empirical asset pricing literature,
moment
alternative asymptot-
estimator of the parameter vector, ones in which the weighting matrix
iterated to convergence
the
Our
to asset pricing models.
estimators include ones in which the weighting matrix
ically efficient
thetical
sample properties of generalized method
What
GMM
criterion function?
are the small sample properties of confidence regions of parameter estimators
constructed using the
GMM
criterion function,
compared
to confidence regions con-
structed using standard errors?
•
Is
the small sample over-rejection often found in studies of
when using an estimator
•
How
in
which the weighting matrix
is
GMM
estimators reduced
continuously altered?
are the small sample biases of the GA/.V/ estimators effected by the choice of
procedure
for
constructing the weighting matrix?
Since there has been an extensive body of empirical work investigating the consumption-
based intertemporal capital asset pricing model using
such models as a laboratories
Kocherlakota
(
1990b].
all
for
GMM estimation
our Monte Carlo experiments.
of our experiments
1
come from
single
.\s in
methods, we use
Tauchen (1986) and
consumer economies with
power
to annual
time
Some
experimental design.
flexibility in the
to
Within the confines of these economies, there
utility functions.
series
monthly post war data.
In the
one of the parameters of
nonseparabilities in the
or habit persistence.
Other economies are calibrated
experiments calibrated to annual data and several of
moment
the experiments calibrated to monthly data, the
least
considerable
economies are calibrated
of our experimental
data presuming a century of data.
is still
interest.
conditions are nonlinear in at
Moreover, some of the specifications introduce time
consumer preferences which are motivated by either
In the other experiments calibrated to
conditions are, by design, linear in the parameter of interest.
local durability
monthly data, the moment
Some of
these setups are special
cases of the classical simultaneous equations model. For other setups, the observed data
modeled
as being
is
time averaged, introducing a moving-average structure in the disturbance
terms.
The paper
is
organized as follows. Section 2 describes the alternative estimators we study
and the related econometric
use. Section 4 gives
made based
are
results of the
parameter of
in
literature. Section 3 specifies the
an overview of the calculations including a description of how inferences
directly on the shape of the criterion functions. Section
Monte Carlo experiments
2
-5
then presents the
calibrated to monthly data that are linear in the
interest. Section 6 presents the results calibrated to
which the moment conditions are nonlinear
remarks are
Monte Carlo environments we
in the
annual and monthly data
parameters.
Finally, our concluding
in section 7.
Alternative Estimators and Related Literature
One
ot
tive
GMM
the goals of our study
is
to
compare the
finite
sample properties
of three alterna-
moment
conditions in an
estimators, each of which uses a given collection of
asymptotically efficient manner. Write the
moment
conditions
Ey(X,.3)]=0
as:
(2.1)
where
l3 is
the k dimensional parameter vector of interest. In (2.1) the function
We
coordinates.
distributed
assume that {4fT.J=i'T^i-^t, ^)} converges
in
ip
has n
V
(/i).
Let Vjifi) denote (an infeasible) consistent estimator of this covariance matrix.
typically
is
denoted {bj}. .\n
efficient
made
b that
first
two
l3 is
/3,
then constructed by
minimizes:
T
The
This
operational by substituting a consistent estimator for
GMM estimator of the parameter vector
choosing the parameter vector
k
distribution to a normally
random vector with mean zero and covariance matrix
latter estimator
>
T
GMM estimators
that
we consider
differ in
the
way
in
which
this
is
accom-
plished.
Tu-Q-step Estimator
The
first
moment
estimator, called the two-step estimator, uses an identity matrix to weight the
conditions so that bj
is
chosen to minimize:
T
T
(2-3)
[l.Y.^{Xub)\'[\;Y.^{X,M\-
Let
6y-
denote the estimator obtained by minimizing
(2.2).
Iterative Estimator
The second estimator continues from the two-step estimator by reestimating the matrix
\'{3) using V'(6j~')
or until j attains
and constructing a new estimator
some
large value. Let
6'^
frj-.
This
is
repeated until bj converges
denote this estimator.
Continuous-updating Estimator
Instead of taking the weighting matrix as given in each step of the
we
also consider an estimator in
which the covariance matrix
is
GMM estimation.
continuouslv altered as
b is
changed
in the
minimization. Formally
let bj-
be the minimizer
T
of:
T
[\;Y:AX,M'[yT{h)V[\;Y.^{X,M-
Allowing the weighting matrix to vary with
that
b clearly alters
minimized. While the first-order conditions
is
(2-4)
the shape of the criterion function
minimization problem have an
for this
extra term relative to problems with a fixed weighting matrices, this term does not distort the
limiting distribution for the estimator. [See Pakes and Pollard (1989, pages 1044-1046) for
a
more formal
An advantage
moment
A
discussion and provision of sufficient conditions that justify this conclusion.]
conditions are scaled even
multinomial models
is
that
when parameter dependent
simple example of a continuous-updating estimator
for restricted
two
of this estimator relative to the previous
in
which the
is
a
efficient
three
GMM
estimators have antecedents
yt
at
time
is
t
the parameter of interest.
=
Let
^'xt
Zt
disturbance term
is
is
minimum chi-square estimator used
distance matrix
is
parameter dependent.
the classical simultaneous equations
(2.5)
-f ut
One way
U(.
to estimate
3
is
to use two-
conditionally homoskedastic. and serially uncorrected. In this case the
and hence
that the two-stage least squares estimator
(
is
constructed from
is
our two-step estimator under the additional restrictions that the
iterative estimator converges after two-steps
Hillier
the
denote the vector of predetermined variables
which by definition are orthogonal to
stage least squares which
known
how
Consider estimating a single equation, say
literature.
where 3
in
invariant to
scale factors are introduced.
the probabilities implied by the underlying parameters and hence
The
it is
is
is
the two-step estimator.
It is
well
not invariant to normalization. In fact
1990) criticized the two-stage least squares estimator by arguing that the object that
identified
is
the direction
1
—
.i
but not
its
magnitude.
conventional two-stage least squares estimator of direction
is
Hillier
then showed that the
distorted by
its
dependence on
normalization.
As an alternative, Sargan (1958) suggested an instrumental variables type estimator that
uiiuiimzes
^T
-1
.
(2.6)
by choice of
this
is
b.
Under the additional
restrictions
our continuous-updating estimator.
in (2.6)
and minimize, the solution
is
imposed on the disturbance term above,
Notice that
if
we
ignore the denominator term
the denominator term, Sargan showed that for an appropriate choice of
the (limited information)
quasi-maximum
which as an estimator of direction
The estimation environments
described.
is
By
the two-stage least squares estimator.
~(,
including
the solution
is
likelihood estimator (using a Gaussian likelihood),
invariant to normalization.^
that
we study
more complicated than the one
are
Sometimes the moment conditions are not
linear in the parameters,
just
and the
disturbance terms are often conditionally heteroskedastic and/or serially correlated.
.'Xs
a
consequence, the two-step and iterative estimators no longer coincide. However, the estimation
methods remain limited information
in that
the
moment
conditions used are typically
not sufficient to fully characterize the time series evolution of the endogenous variables.
The second
goal of our analysis
is
to
compare the
reliability of confidence regions
com-
puted using quadratic approximations to criterion functions to ones based directly on the
deterioration of the original criterion functions.
The former approach
used in the empirical asset pricing literature partially because
it
is
is
more commonly
easier to implement.
The
latter
approach exploits the chi-square feature of the appropriately scaled criterion functions.
From
the vantage point of hypotheses testing, the plausibility of an observed deterioration
ot
the criterion function caused by imposing parameter restrictions can be assessed by using
the appropriate chi-square distribution. Using this
^S^e Imbens (1992) for an alternative
mator
IS
constructed to coincide with the
GMM
same
estimator that
maximum
is
insight, confidence regions can
be
invariant to normalization. Imbens' esti-
likelihood estimator
when
the data
is
multinomial
computed by using the appropriate chi-square
distribution to prespecify
some increment
the criterion function and inferring the set of parameter values that imply no
increment. Such confidence regions can have unusual shapes and,
connected.
One
of the key questions of this investigation
is
in"
fact,
more than
may
in
that
not even be
whether or not they lead to more
reliable statistical inferences.
One
performance of criterion-function-based inference
of our reasons for studying the
comes from the work
of
Magdalinos (1994). Within the confines of the
classical
simultaneous
equations paradigm, Magdalinos studied the performance of alternative tests of instrument
admissibility.
As a
result of his analysis,
matrix to embody the restrictions as
tion,
is
Magdalinos recommended altering the weighting
done
in
the continuous-updating method. In addi-
he found that test^-atistics are better behaved lusing the limited infopmation maxykiunT
likelihood estimator than the two-stage least squares estimator. Recall that the former esti-
mator coincides with our continuous-updating estimator and the
later to our two-step
iterated estimators in the classical simultaneous equations estimation
ered by Magdalinos.
Another reason
is
and
environment consid-
Nelson and Startz's (1990) criticism of the use of
instrumental variables methods for studying consumption-based asset pricing models. These
when the
authors were concerned about the behavior of instrumental variables estimators
instruments are poorly correlated with the endogenous variables. Their arguments are based
on analogies to results derived formally
for t-statistics
and over-identifying
The question
restrictions tests
the extent
in
the classical simultaneous equations setting.
to
which criterion-function based inference and continuous updating can help overcome the
of interest to us
is
concerns of Nelson and Startz.
The
finite
sample properties
of the two-step
and iterative
pricing setting have been studied previously by
and Person and Foerster (1991).
GMM estimators
in
an asset-
Tauchen (1986). Kocherlakota (1990b).
These investigators did not study the properties of the
continuous-updating estimator, nor did they study the behavior of criterion function-based
confidence regions.
Further Tauchen (1986) and Kocherlakota (1990b) considered only the
case of time separable preferences for the representative consumer.
Monte Carlo Environment
3
We
consider several
estimators described
Monte Carlo environments
in section 2.
to assess the finite sample properties of the
The data generating mechanisms
are constructed to be
consistent with a representative agent consumption-based asset pricing
model {CCAPA'f).
Estimators of the parameters of the representative agent's utility function are considered
along with tests of the overidentifying conditions implied by the model.
as the basis of our experiments because
model. ^ Further, the
«f
CCAPM
GMM
forms the basis
GMM cohducfced by TaiMen (1986)
and
We use
the
CCAPM
has been used extensively in studying this
for
two studies of the
Koc^rMota
finite
(1990b).
^>
sample properties
^
W
'^ms.
'
Preferences and Euler Equations
In the
model the representative consumer
assumed
is
to
have preferences over consump-
tion given by:
Uo
where
C;
is
consumption
preferences.''
If
^
>
0,
= E
at date
h
t.
,7>0
1-7
The parameter
consumption
is
9 captures
(3.1)
some time nonseparability
durable or substitutable over time.
preferences of the consumer are time additive.
If
^
<
0,
consumption
is
If
^
=
0.
in
the
complementary over
time and the preferences of the representative consumer exhibit habit persistence.
We
consider estimators of the parameters
8,
on implications of the Euler equations. Let must
as the indirect marginal utility for
consumption
7 and 9 as well as tests of the model based
=
(C(
+
9ct-\)~''
"services"" as
which can be interpreted
.
measured by
St
—
Ct
-\-
-See. for e.xample. Dunn and Singleton 1986), Eichenbaum and Hansen 1990), Epstein and 7in
Person and Constantinides (1991), and Hansen and Singleton (1982).
•'For models with time nonseparability in preferences see, for example. Abel 1990), Constantinides
(
(
(
9ct-\
.
(
1991),
(
1990),
Detemple and Zapatero (1991). Dunn and Singleton (1986), Eichenbaum and Hansen (1990), Gallant and
Tauchen (1989). Heaton (1993. 1994), Novales (1990), Ryder and Heal (197.3) and Sundaresan (1989).
Similarly, let muct
set at
time
=
(c,
+ 9ct-i)
The Euler equation
t.
^
+
+ 6ct)
96E[{ct+i
^
where Tt gives the information
J-t]
|
for a representative agent's portfolio allocation decision
is
given by:
muct
where
a gross return on an asset from
/?(+i is
growing over time we divided
equation that
= E{Smuct^iRt+\
we
consider
is
(3.2)
<
to
<
+
A«(^.7.<')
that
when
9
[ct
is
unconditional
+8ct-i)~''
not zero,
moment
+ S6{ct+i
(f)t+2
has an
=
we take
to
is
(3.3)
equation error:
=^-«=^fl,.,
must
MA(1)
(3.4)
must
Notice that E((pt+2
.
\
^t)
By choosing
structure.
=
0-
Further notice
instruments,
r(,
in JF(,
conditions are given by:
(i)t+2
can be expressed
=
E[zt<Pt+2{Sn.9)]^0.
in
terms of consumption ratios and returns, which
(3.5)
to be stationary processes.
One unpleasant
feature of these
be satisfied
degenerate fashion.
inuc'
r)
I
(3.3) results in the Euler
+9ct)~'^
E[^t+2(Sn^9)]
Finally, notice that
Since aggregate consumption
1.
then given by:
Removing conditional expectations from
=
(3-2)
by must to induce stationarity. The (normalized) Euler
i^ = £U
(,=±ifl,„
where muc'
Tt)
\
=
in a
moment
conditions
Suppose that
and the moment conditions are
is
-)
that they can always be
=
trivially satisfied.
and S9
for
the two-step and iterative estimators.
8
—1. then clearly
Without imposing additional
constraints on the parameter vectors, this degeneracy in the
problems
=
made
moment
conditions creates
Of course when time-separability
is
imposed
(0
=
0),
these problematic parameter values cannot be reached.
When
9
is
permitted
to be different
from zero, Eichenbaum and Hansen (1990) were led to divide the moment
conditions by
+ ^^
1
degenerate values.
estimator
is its
in
We
order that the two-step and iterative estimators not be driven to the
will
do
insensitivity to
.An attractive property of the continuous-updating
likewise.
parameter dependent scale factors and hence to the moment
transformation used by Eichenbaum and Hansen (1990).
Moreover, the criterion of the
continuous-updating estimator does not necessarily tend to zero
S$
=
Although the estimated mean
—1.
for {(^(+2}
the vicinity 7
in
becomes small
in the
=
and
neighborhood of
these parameter values so does the estimated asymptotic covariance matrix, and the criterion
function for the continuous-updated estimator plays off this tension.
We
buildiseveral
Monte C^lol^nviconments
that are consistent with (3.3).
tee
simulate returns and consumpti#t-graw^h
These are used to assess the
estimators of section 2 based upon the
moment
finite
sample properties
..
of the
conditions (3.5).
Log-Normal Model
3.1
Time Additive Model. So Time Averaging
In our
first
Monte Carlo environment we model consumption growth and
by assuming that they are jointly log-normally distributed
Let Y{t)
—
[log(ct/c(_i
)
log
R^f
log/?;]'
where
the gross return on a stock index at time
We
assume
e, is
is
=
^i
+
(
1983).
t.R^f is
the gross return on a bond at time
B{L)e,
a normally distributed three dimensional
time, has zero
a
aggregate consumption at time
is
and R{
Hansen and Singleton
t
.
that:
Yt
where
t
C(
as in
returns directly
mean and covariance matrix
matrix of polynomials
in
/.
(3.6)
random vector
and where
fi
is
the
that
mean
is
independent over
of Vj. Further
B(L)
is
the lag operator. To use this assumption about the dynamics of
consumption and returns along with the Euler equation
(3.2)
we assume
that the preferences
of the representative agent are time additive. In this case the Euler equation error for each
return can be written
as:
-
where
E{T}t+i
The
!Fi)
\
=
and
«;
7log(ct+i/Q)
is
a constant
+
logi?(+i
[see, for
relation (3.7) implies a set of restrictions
restrictions
we consider
-
K
=
(3.7)
rjt+i
example, Hansen and Singleton (1983)].
on the law of motion
(3.6).
To impose
these
methods
several finite order parameterizations of B(L) and use the
described by Hansen and Sargent (1991). These parameterizations are of the form:
B(L)
^
=
(3.8)
a(L)
where C(L)
polynomial
a 3 by 3 matrix of polynomials in the lag operator
is
the lag operator.
in
We
restrict the
and a(L)
is
a
1
by
1
polynomial a(L) to be second order and
considered several different orders for C(L).
To estimate the constrained law
Aggregate consumption
is
of motion,
equity return
is
risk free return
CITIBASE. These data were converted
total U.S. population for
the value weighted return from
CRSP, and
from CRSP. Each of these return
For simplicity we removed the sample
and
(3.7) did not
be estimated
is
7.
The
to a per capita
each month, obtained from CITIBASE. The
series
the bond return
was converted into a
the implicit price deflator for nondurables and services from
(3.6)
to 1992,12.
seasonally adjusted real aggregate consumption of nondurables
plus services for the U.S. taken from
measure by dividing by
we used monthly data from 1959,2
mean from
have to be estimated. As a
is
the Fama-Bliss
real return using
CITIBASE.
the vector
Vj so
that the constants in
result, the only preference
results of estimating the law of
motion
parameter to
(3.6) using exact
Likelihood for different orders of the polynomial C(L) are given
in table 3.1.
Maximum
The column
labeled "Unrestricted Log-Likelihood" reports the log-likelihood in the case of unrestricted
estimation of the polynomials
in
the lag operator.
10
The columns
labeled
"Xo Time
.\vg."
report the log-likelihood and the estimated value of 7 under the restrictions implied by (3.7).'
Notice that there
is
a second order polynomial for C(L)
a
first
order polynomial
order polynomial
in
is
needed
in
for
first
C(L)
There
.
is little
improvement
going to a third
in
the unrestricted case.
when the model
restrictions are imposed.
results reported
by Hansen and Singleton (1983).
log-likelihood in
moving from a second order
This
Also there
is
great deterioration in
is
consistent with the
is
little
improvement
to a third order C(L). For this reason
^arlo experiments for the
In assessing the finite
loff'Ttorfnal
model
with.-jaao trime
Monte Carlo samples each with
a sample size of 400.
the size of available monthly consumption data.
upon the Euler equation
We
A
this case
(log)
moment
conditions based
bond and the stock returns simultaneously. For
errors for both the
consumption growth
as instruments since the data
is
we constructed
400 approximates
each Euler equation error we used one period lagged (log) bond and
one period lagged
Monte
size of
sample
constructed
the
^«8e #'
averaging.
GMM estimators in
sample properties of
in
we used
the point estimates from the restricted model with a second order C(L) to conduct our
•500
to
the unrestricted case. This indicates that more than
For both the second and third order polynomial cases there
the log-likehhood
moving from a
substantial improvement in the log-iikelihood in
We
as instruments.
(log) stock returns
and
did not include constants
simulated under the assumption that
it
has a zero mean.
Time Additive Model with Time Averaging
As a further data generating mechanism, we
decision interval of the representative agent
is
also consider an
much
example
^In searching for the
*
larger than
We
-50.
The
maximized log-likeHhood,
is
a
which the
smaller than the interval of the data.
Suppose that there are n decision periods within each observation period.
the representative agent's decision interval
in
week and data
is
For example
if
observed monthly, then n
larger values of the log-likelihood were found for values of
results reported in table 3.1 for the constrained
used the local maximizers for our simulations because they result
11
models correspond
in
to local
plausible values for 7.
maxima.
would be approximately
4.
The
representative agent's utility function at time
1-7
(^^^^)
Ut^E
we maintain the assumption
given by:
(3.9)
J't
that consumption and returns are jointly lognormally dis-
tributed, the Euler equation error
decomposed
is
1
1-7
h=0
If
-
t
again given by (3.7). Moreover, this error can be
is
r]t+i
as:
Vt+i
= X!
h=i
(.3.10)
C+iL
"
where
C(+h
In this
=
EiT]t+i
I
-
J^(+i)
E{T]t+i
3.ii;
^^_^.h^)
I
environment, we presume that observed consumption does not correspond to
the actual point-in-time consumption the representative agent, but instead
is
an average of
actual consumption over one unit of time. Specifically, suppose that observed consumption,
c^, is
a geometric average of actual consumption:
l/n
(3.12)
n^f-i+^
yh=i
and similarly
for
the observed return.
7
log«^^/cn +
7?^.
Averaging
=
log /?:+,
^-
over time implies that:
(3.7)
+ -
E= E
r
l
C,-i + .+
-
[3.131
h=l
Notice that in this case the Euler equation error:
^t+\
= 7
im
:1
is
predictable at time
/.
However
h
=
/",_!)
E{r]'^_^_^
|
c,_i+i+i
(3.14)
l
=
so that instruments can be chosen
from
the information set at time
account
MA(1)
for the
We estimated
t
—
An
1.
structure of the
instrumental variables estimator of this model must
moment
condition.
the log-linear law of motion (3.8) for consumption and asset returns'^ under
the restriction implied by (3.13). Notice that this model imposes a weaker set of restrictions
than does the model that takes no account of time averaging.^
third order polynomial for
C(L) and a second order polynomial
We
consider the case of a
for a(L).
The
results of this
estimation are reported in table 3.1 in the columns labeled "Time Averaged." Notice that
the log-likelihood function improves
is
However the model
ignored.
value of 7
We
is
used
is
somewhat compared
still
substantially at odds with the data.
The estimated
^^OtT
"As
no time averaging, we studied estimators based upon the Euler equations
for
fthis
model
to create-^00
Monte Carl© draws each with a sample
both returns. In this case the instruments were
3.2
where time averaging
slightly larger as well.
in the case of
(log)
to the case
consumption growth
all
(log) stock returns, (log)
size
.
bond returns and
lagged 2 periods.
Discrete-State Models
In our second set of
Monte Carlo environments we
(1990b) and consider a
Markov chain model
.Aggregate consumption
is
assumed
for
follow
Tauchen (1986) and Kocherlakota
aggregate consumption and dividend growth.
to represent the
endowment
sumer and dividends represent the cash flow from holding
stock.
of the representative con-
We
form one-period stock
returns and the returns to holding a one-period (real) discount bond. Each of these returns
"The actual consumption data is an arithmetic average of consumption expenditures over a period.
model to the data we are assuming that geometric averages and arithmetic averages are
appro.ximately the same. This assumption was made by Grossman. .VIelino and Shiller (1987), Hall (1988)
and Hansen and Singleton (1993). Notice also that the returns in (3.13) are time averaged. For simplicity,
in estimating the law of motion (3.8) we use the monthly CRSP series directly. Hansen and Singleton (1993)
constructed time averaged returns from daily data in their analysis of this model. For the consideration of
the finite sample properties of
estimators, this is unlikely to be an important issue.
In fitting the
GMM
'^'There
is
stock return.
should be
a additional restriction on the first-order autocorrelation of the Euler equation error for the
In the limit case of
0.2-5 as
continuous decision making, the first-order autocorrelation of the error
discussed by Grossman. Melino and Shiller (1987) and Hall (1988).
restriction in our estimation.
13
We
do not impose
this
can be represented as functions of the state of the Markov chain. Construction of these
turns in the case of time additive utility
is
re-
described by Kocherlakota (1990b) and Tauchen
(1986).
Annual Model
To
calibrate the
first
Markov chain we used the method described by Tauchen and Hussey
(1991) to approximate a first-order
eters of the
VAR are
6
0.004
lnA(
0.021
1^-
+
0.117
0.414
In
0.017
0.161
lnA(_i
6-1
E(et^[)
e,
chain for
(3.15)
et
•
gross growth rate of U.S. per capita real annual consumption, and
Further
+
the gross growth rate of real annual dividends on the S&:P500, where A(
is
^t
consumption and dividend growth. The param-
taken from Kocherlakota (1990b) and are given by:
In
where
VAR for
is
[if(
assumed
A(]' is
In simulating
=
0.01400
0.00177
0.00177
0.00120
i?y
is
'
the
where
[3.16)
to be normally distributed
and uncorrelated over time. The Markov
chosen to have 16 states.
data from
of the representative
this
model we chose several values
consumer. These are presented
of the preference parameters
Preference setting
in table 3.2.
TSl
was used by Tauchen (1986) and preference setting TS2 was used by Kocherlakota (1990b).
.As
shown by Kocherlakota (1990b), these
model
of
endowments, imply
first
and second moments
sample counterparts. The large value of
an equilibrium
AV'e restrict
in
latter parameters, along with the
6 in
TS2
is
Markov chain
for asset returns that
mimic
their
not inconsistent with the existence of
the model because of the large value of 7 [Kocherlakota (1990a)].
our attention to "moderate" values of 6 and 7
nonseparable preferences, and we consider a range of values of
introduces a modest degree of durability by letting 9
14
=
in
our examination of time
9.
Parameter setting TNSl
1/3 and
TNS2
introduces habit
persistence with d
setting 9
in part
=
= —1/3. TNS3
results in a
The asymmetry
—2/3.
(in
more extreme amount
of habit persistence by
magnitude) across the specifications of 9
guided
is
by the a priori notion that there should only be limited amount of durability
the goods classified as "nondurable" in
NIPA and by
in
Person's and Constantinides's (1991)
empirical evidence for a substantial degree of habit persistence.
Since there
moment
sets of
The
used.
great flexibility in the construction of
is
number
conditions that differ in the
alternative sets are listed in table 3.3.
wa^ multiplied by the
listed
moment
The Euler equation
instruments to construct the
moment
We
^
GMM
as an' instrument in
•bmeht
set
M4
because of
its
we chose
several
and instruments that are
of returns
return to holding the stock and Rf to holding the bond.
4tl Pf
conditions,
for
each listed return
conditions,
i?^
denotes the
used the dividend-price ratio,
ability to predict equity
E&t^ns^
estimates and test statistics were computed for 500 replications of a sample of size
100. This
sample
size
corresponds approximately to the length of most annual data
sets.
Monthly Model
We repeated some of the experiments using
data.
As
in
a first-order
the construction of the
VAR
for
a law of motion calibrated to postwar monthly
Markov chain
for the
consumption and dividend growth.
annual model, we started with
The consumption data used
in
estimating the \'AR was aggregate U.S. expenditures on nondurables and services described
in subsection 3.1.
N\ SE
constructed dividends implied by the monthly
portfolio return.
price deflator for
series
We
is
These dividends were converted to
CRSP
value- weighted
real dividends using the implicit
monthly nondurables and services taken from CITIBASE. This dividend
highly seasonal because of the regular dividend payout policies of most companies.
To avoid modeling
this seasonality
we
let
^,
=
\og{dt/dt-i2)/l'2
represents the one-period dividend growth of the model.
to be stationarv.
15
The
and we assumed that
series {\og(dt/dt-i2)}
^t
appears
The parameters
of the
VAR estimated
6
0.0012
InXt
0.0019
In
+
using this data are given by:
-0.1768
0.1941
0.0267
-0.2150
In
6-1
+
lnAt_i
(3.17)
et
where
E(etc[)
As
0.1438
0.0001
0.0001
0.0145
X 10
-3
(3.18)
annual Markov chain model, we approximated the
in the case of the
(3.18) with a 16 state
Monte Carlo data
=
Markov chain using the methods
of
Tauchen and Hussey
consisted of 500 replications of a sample size of 400.
on the more "moderate" ^reSl^enc£ configuration (TSl) adjusting 6
(More
interval.
precisely,
we used the
we generated Monte Carlo data
with
and
(3.17)
The
(1991).
We focused exclusively
for the she
.eT-^Sipiing
twelfth root of .97 in place of .97 for 6.) In addition,
using nonseparable specification
TNSl and TNS2.
again
adjusted appropriately.
S
Overview of Descriptive
4
VAR of
Statistics for
Monte Carlo
Results
In describing the results of the various
Monte Carlo experiments
in sections 5
and
6,
we
focus most of our discussion on the following calculations:
1.
We
found the
distribution of
minimum
TJj
is
value of the criterion function.
chi-square with degrees of freedom equal to the
ment conditions minus the number
of
distribution to test the overidentifying
2.
We
parameters estimated.
moment
We
The
limiting
number
of
mo-
used this limiting
conditions.
evaluated the criterion function at the true parameter vector. Call this Jj. The
limiting distribution of
of
Call this Jj-
moment
conditions.
TJj
is
With
chi-square with degrees of freedom equal to the
this limiting distribution,
16
number
we characterize the family
of
parameter vectors that look plausible from the standpoint of the moment conditions.
It
is
of interest to see
how
well
th(^
limiting distribution performs in
making
this
assessment.
3.
We
found the
true value.
model,
minimum
value of the criterion function
Call this Jj.
in this
Since 7
is
when 7
is
constrained to be
its
the only parameter estimated in the log-normal
case Jj coincides with Jj.
The
limiting distribution of
T[Jj — Jt)
is
chi-
square with one degree of freedom. This limiting distribution allows us to construct
a confidence region for 7 based on the increments of the criterion function from
unconstrained minimum.
value of 7
4.
We
is
in
By
evaluating TiJy
—
Jt)
we determined whether
its
the true
the resulting interval for alternative confidence levels.
constructed the more standard confidence intervals for 7 based on a quadratic
approximation to the criterion function.
Let 77 be an estimator of 7, and a^ be
the estimated asymptotic standard error" of the estimator 77.
Formally,
we study
T{lT — iYI{(^t)' which has an asymptotic chi-square distribution with one degree
freedom. Notice that this object
true value of the parameter
Our Monte Carlo
is
is
just the
Wald
statistic for the
of
hypothesis that the
7.
calculations are greatly simplified by our knowledge of the true param-
eter vector. In empirical work, the corresponding
computations would be more complicated.
For instance, to construct a confidence interval for 7 based on the original criterion function,
a researcher would have to characterize numerically the hypothetical values of this parameter
that are consistent with a prespecified deterioration in the criterion while concentrating out
all
of the other parameters.
ter vector (in
there are very few remaining components in the parame-
our examples zero, one or two), this concentration
'The standard errors
the sample
When
moment
for the
way
to proceed
tractable. However, this
continuous-updating estimator were based only upon
conditions with respect to the parameters.
are analogous to the standard errors typically constructed in
alternative
is
would be
to base the standard errors
respect to the parameters.
17
first
derivatives of
Standard errors constructed
estimation [see Hansen
GMM
in this
way
(198"2)).
Xn
on derivatives of the criterion function with
may become
approach
In reporting our
very difficult
Monte Carlo
when the parameter
results
we use one
vector
is
large.
methods advocated by
of the graphical
Davidson and McKinnon (1994). For each Monte Carlo setup we computed the empirical
and compare them to the corresponding chi-square
distributions of the statistics
tions.
The
results are plotted
on a
value (depicted on the x-axis),
the fraction of the actual
we computed the corresponding chi-square
computed
Thus the 45 degree
y-axis).
set of figures constructed as follows. For
the quality of the limiting distribution.
plots are referred to as "p-value" plots,
since this
bounds the re^&n
...
)
is
each probability
critical value
and
above that value (depicted on the
statistics that are
line (depicted as
distribu-
the appropriate reference for assessing
Following Davidson and
and we present the
of probability values used in
McKinnon
(1994), these
results for the interval
[0 0.5]
most applications. Although
wJsHise
*
probability values as our basis of comparison, confidence intervals at alternative significance
levels
can be assessed by simply subtracting the probability values from one.
The
figures are organized as follows. For each
figure with four graphs titled as follows:
and "Wald" corresponding
To provide a formal
distributions
and
line
is
TJt. TJj, T(Jj — Jj)
statistical
measure
their theoretical counterparts,
plotted using dotted lines.
first
consider a single
"Minimized", "'True", "Constrained
to the statistics
respectively.
Monte Carlo setup we
This band
-
Minimized"
and Tifj —
,
of the distance
7)^/(<Tj)^,
between the empirical
on each figure a band about the 45 degree
is
a 90 percent confidence region based on
the Kolmogorov-Smirnov Test. This states that the probability that the maximal difference
between the empirical distribution and the theoretical one
percent.
Maximal
significance
In
differences within these
bands are not
will lie
within those lines
statistically significant at the
is
90
10%
level.**
each graph, the dashed
line gives the
Monte Carlo
results for the two-step estimator,
the dot-dash line for the iterated estimator and the solid line for the continuous-updating
estimator. For the minimized criterion function results there
is
a necessary ordering between
^The Kolmogorov-Smirnov confidence region is based on calculating the supremum between the empirical
and the 4-5 degree line over the region [0,1].
distribution
18
When
the continuous-updating and iterative estimator.
the iterative estimator converges,
the value of the criterion function ran also be obtained by the continuous-updating estimator.
Since the continuous-updating estimator minimizes
must he smaller than the
its criterion, this
As a
criterion for the iterative estimator.
minimized criterion of the continuous-updating estimator must
result, the plot for the
below the plot
lie
iterative estimator unless the iterative estimator fails to converge.
minimized value
There
is
for the
no natural
or-
dering between the results for the two-step estimator and the continuous-updating estimator
or between the two-step estimator
To complement our
and the
p-value plots,
we
iterative estimator.
also provide
The
formance of the implied parameter estimators.
esstimates are of interest in tifeir
own
and
right
in
3ome
some
finite
results
summarizing the
per-
sample properties of the point
cases provide additional insight^Kit^
the behavior of the p-value plots.
5
Monte Carlo
Results,
For the case of time-averaged data,
tion 3.1.
To account
for this, the
Log-Normal Model
has an
{i^(}
MA(1)
structure as
we
estimator of Vrib) was computed by
discussed in sec-
first
considering an
estimator of the form:
^T(b)
= ^T.rT{x,.b)^Tix,.by + Y,yT(Xt-,,b)^T{Xt.by + ^T{Xt.b)^T{Xt-,.by]\
^
Kt-l
t=2
)
(o.l)
where ^j( A'(.
6)
=
v(
A',, b)
— j YJ=\
r(-^'(- b).
we used an estimator proposed by Durbin
(1988)].^ Durbin's estimator
order autoregression.
'In the
The
is
When
this
estimator was not positive definite,
(1960) [see also Eichenbaum. Hansen and Singleton
obtained by
first
approximating the MA(1) model with a
finite
residuals from this autoregression are used to approximate the
Monte Carlo experiments with tune averaging,
not necessary at any of the converged parameter estimates.
19
the use of this covariance matri.x estimator
was
Then
innovations.
the parameters of the
MA( 1) model
are estimated by running a regression
of the original time series onto a one-period lag of the "approximate" innovations. Finally,
an estimate of Vj(6)
is
formed using the estimated moving- average
coefficients
and sample
covariance matrix for the residuals. This procedure has the advantage that the finite-order
moving average structure
construction.
However
it
of {(^J
is
imposed and the estimator
does rely on the choice of a
the approximation. In implementing the estimator
finite
is
positive semidefinite by
order autoregression to use in
we ran a 12th order autoregression
in
the
initial stage.
Criterion functions
Figures 5.1 and 5.2 reportJJae properties of the criterion functions for the two Mact|; Carlo
experiments. Figure 5.1
case of time averaging.
is
for the case of
The lower
(approximate quadratic)
criteria.
is
linear in the
The
results for the
and Two-Step estimators.
parameters and the weighting matrix
criteria are different
for the
Wald and "Constrained-Minimized"
is
Jt)- For the continuous-updating estimator the results for the
Minimized"
is
right plots in the figures report the results using the VVald
criteria are identical for the Iterative
model
no time averaging of the data and figure 5.2
This occurs because the
fixed in constructing (Jj
Wald and
—
the "Constrained-
due to the dependence of the weighting matrix on the
hypothetical parameter values.
Notice that the small sample distributions of the minimized criterion functions for the
iterative
and two-step estimators are greatly distorted. The small sample
size of tests of the
overidentifying restrictions based upon the minimized criterion values are too large leading
to over rejections of the
model when using these estimators. The minimized
tion for the continuous-updating estimator
distribution
is
is
much
criterion func-
better behaved and the small sample
very close to being \^ for both Monte Carlo experiments. Tests of the overi-
dentifying restrictions of the models using the minimized value of the criterion function of
the continuous-updating estimator have the correct size for the model without time averag-
20
ing and similarly for the
Even
0.1.
for the
The
for
-y
model with time-averaging
for probability values greater
continuous-updating estimator
finite
than
is
0.1,
for probability values less
the distribution of the minimized criterion
not greatly distorted.
sample coverage probabilities of the two ways of constructing confidence regions
are depicted under the headings "Wald" and "Constrained- Minimized".
these two
methods coincide when they are based on the two-step and
but
when the continuous-updating estimator
differ
than about
is
used.
Recall that
iterative estimators,
The small sample coverage
probabilities are greatly distorted for the intervals constructed with the iterative
and two-
step estimators. In particular, they do not contain the true parameter value as often as
is
to
be expected from the limiting distribution. In the case of the continuous-updating estimator,
coverage rates again are too smali' for confidence intervals built from the Wald cri^fria,*but
the distortion
is
substantially smaller than with the other
two estimators.
Finally, the
coverage rates of the confidence regions implied by the "True-Minimized" criteria for the
continuous-updating estimator accord well with the asymptotic distribution and are clearly
better than the coverage rates for the other three criteria.
These Monte Carlo
results for the log-normal
model support the following remedy
the concerns raised by Nelson and Startz (1990).
From
the standpoint of hypothesis test-
ing and confidence interval construction, use of the continuous-updating criterion
more
reliable than the other
methods we study. The
for
is
much
tests of the overidentifying restrictions
based on the continuous-updating estimator do not reject too often and
in fact are quite well
approximated by the limiting distribution. While confidence intervals based on the Wald
criteria
can be badly distorted, particularly
for the
two-step and iterative estimators, confi-
dence regions constructed from the continuous-updating criteria have coverage probabilities
that are close to the ones implied by the asymptotic theory.
Parameter Estimates
Table
5.1 reports
summaries
of
measures of central tendency
•21
for the three
estimators of
90%
7 along with 10% and
quantiles.
The medians
are considerably lower than the true value of 7 while the
updated estimator
estimator
90%
is
also
quantiles.
timator
is
is
much
more
two-step and iterative estimators
for the
median
smaller. However, the distribution for the continuous-updating
disperse as evidenced by the larger increment between the
Moreover, the Monte Carlo sample means
much more
bias for the continuously-
10% and
continuous-updating
for the
The
severely distorted than they are for the other two estimators.
enormous sample means
for the
continuous-updating estimator occur because
in
es-
the case
no time averaging and of time averaging there were 23 and 31 samples, respectively,
of
which the estimates
the
are, in absolute value, larger
Monte Carlo samples, the sample means
to the true values
than
1^
the
means
than 100.
When
in
these are removed from
of the continuous-updating estimator are closer
for the other
&'
two estimators.
^
Recall that the analog to the two-step and iterative estimator in the classical simultaneous
equations model
estimator
is
is
two-stage least squares and that the analog to the continuous-updating
limited information (quasi)
maximum
likeHhood.
known from
It is
that there are settings in which two-stage least squares has finite
information
(1972)].
maximum
likelihood does not
[e.^.,
In light of these theoretical results
updating estimator
is
Sawa
moments but
and our Monte Carlo
On
moment
limited
(1969) and Mariano and
Sawa
findings, the continuous-
not an attractive alternative to the other estimators
bases of comparison are the (untruncated)
as in Zellner (1978)].
see
the literature
we
consider
if
our
properties [or even relative squared errors
the other hand, Anderson,
Kunitomo and Sawa
(1982) advocated
use the limited information estimator over the two-stage least squares estimator because,
among
other things, the median bias of the former estimator
is
smaller.
We
also find less
distortion in the medians for the continuous-updating estimator in our experiments.^"
.As
its
we noted
previously, one attractive attribute of the continuous-updating estimator
invariance to ad hoc (parameter dependent) transformations of the
'°Even though our model
is
linear in variables
not coincide with limited information
ma.ximum
moment
is
conditions.
and parameters, the continuous-updating estimator does
likelihood in our setting.
heteroskedasticity consistent estimator of Vj{h).
99
.A.mong other things, we use a
For instance, when the object of interest
the continuous-updating estimator
information
is
maximum
is
a "direction"
is
in
the sense of Hillier (1990),
invariant to normalization. Hillier's defense of limited
likelihood over conventional two-stage least squares
a better estimator of direction.''
To
is
that the former
see whether such a conclusion might well extends to
comparisons between the continuously-updating estimator and the other two estimators we
consider,
We
5.4.
axis
we report smoothed
distributions of the estimated "direction" in figures 5.3 and
measure direction by the angle
and the point
identification,
we
we used Gaussian
(1,7). Since there
is still
in radians)
kernel with a bandwidth of
circle.
between the horizontal
a sign normalization that must be imposed for
restrict attention to the interval [— 7r/2, 7r/2]. In
each of the sample point#using a
a^.
measured
(as
0.
1
.
The
smoothing the histogram,
value of the density estimate
The ^ape
of the
is
plotted
smoothed distribution
afengS*
with the mass of the plotted circles provides evidence about the small sample distribution
of the
parameter estimators.
Notice that the primary modes of the continuous-updating
angle estimator are very close to the true parameter values, while the
two angle estimators are distorted. Moreover, the density estimates
larger for the continuous-updating
for the
magnitude estimates
of the other
modal angle are
method. However, the Monte Carlo distributions
continuous-updating angle estimator also have secondary modes near
to large in
modes
of 7 with the
wrong
— 7r/2,
for
the
corresponding
sign.
Continuous-Updating Criterion Function
The
criterion function for the continuous-updating estimator can
treme outliers
for
some
for the
minimizing value of
of the trials as
that there
is
7.
This occurs
we discussed above. To
see
why
in
sometimes lead
to ex-
the two .Monte Carlo experiments
this
can occur suppose
for simplicity
a single return under consideration, no time averaging and several instruments.
GMM
estimator of direction in which the absolute value of
normalized to one instead of one of the elements of that vector. He found that
this alternative estimator \\ss finite sample properties that compare more favorably to those of the limited
"Hillier (1990) considered an alternative
the parameter vector
is
information ma.ximum likelihood estimator.
23
The moment
conditions are constructed using:
<l>{Xt,g)
where
Zt
is
=
[\ogRt+,-g\og{ct+,/ct)]zt
a vector of instruments and E[(i){Xt,f)]
=
'
(5.2)
Since the
[see (3.7)].
moment
conditions are linear in g, the criteria for the iterated and two step estimators are quadratic in
g.
In contrast, the criterion for the continuous-updating estimator converges as g gets large.
To
see this, observe that for a large value of g the
g times the sample
mean
of log(Q^.i/c()^(
and the sample covariance
times the sample covariance of \og(ct+i/ct)zt
is
approximately a quadratic form that
is
sample average of 4>{Xt,g)
Therefore for large
.
g,
is
approximately
approximately g^
is
the criterion function
(4) times the chi-square test statistic for the null
hypothesis that
£[log(Q+i/Q)r,]
As a
result
it
is
=
0.
(5.3)
possible for the minimized criterion for the continuous-updating estimator
to occur for a very large value of g.
To
further illustrate this potential problem, the upper plot in figure 5.5
function for the continuous-updating estimator for a
of g that minimizes the criterion function
is
Monte Carlo draw
673750.4.
the average value of the criterion function over the
The lower
is
in
of the criterion
which the value
plot in figure 5.5 gives
Monte Carlo experiments. These
results
are for the case of no time averaging. Notice that in the upper plot, the criterion function
approaches
its
lowest value as g becomes large in absolute value.
the criterion function asymptotes to a local
minimum
for large
Even
in
the lower plot
negative values of
numerical search used to implement the estimator could be complicated by the
of the criterion function
When
the parameter vector
is
The
sections
and the search routine could end up spuriously searching
direction of very large values of g.
easily
flat
g.
in the
of low dimension, this can
be assessed by gridding the parameter vector and evaluating the criterion function
24
at
the grid points. This
is
what we did
to
make
sure that large estimates of g were not due to
numerical problems. However when the parameter vector
the continuous-updating estimator
Monte Carlo
6
Results,
Time Separable
6.1
First
we consider the
may sometimes
Markov Chain Models
The
For these runs Vr(6)
{9
=
is
computed
0) using the
Markov
as a simple
resulting p- value plots are depicted in figures 6.1 through
6.6 for different combinations of the preference settings
3.3.
implementing
difficult.
time separable preferences where
Chain model calibrated to annual data.
and
of large dimension,
Preferences, Annual Data
results for
covariance matrix estimator.
be
is
and moment
sets given in tables 3.2
Featlires of the Montfe Carlo distributions for the point estimates of 7 are Reported
in table 6.1.
Results for 7
We
start
=
1.3
and
6
=
.97
by discussing the results obtained using the more "'moderate" values of the
preference parameters TSl which were used by Tauchen (1986). Figure 6.1 includes a repli-
it
is
known from Tauchen
phenomenon can be seen
is
below the 45 degree
is
always
less
converges),
which
that the over-identifying restrictions test '"under rejects."
This
in
line.
the upper
left
conditions Ml.
plot in figure 6.1
It is
an example
by noting that the dot-dash
line
Since the minimized value of the continuous-updating estimator
than or equal to that of the iterative estimator (when the iterative estimator
we expect the under
restrictions tests
is
moment
in
cation of findings in Tauchen (1986) using
is
more substantial
rejection to be
more pronounced when the
over-identifying
based on the continuous-updating estimator. Indeed, the under rejection
for
both the two-step and continuous-updating estimators. Interestingly,
the underlying central limit approximation for the continuous-updating estimator looks quite
good
as depicted
by the upper right plot
dence sets are evaluated
in
the lower
left
in figure 6.1.
The
plot in figure 6.1.
criterion-function based confi-
The confidence
sets
based on
the continuous-updating estimator do not contain 7 as often as predicted by the asymptotic
theory as might be anticipated from the
downward
distortion of the minimized criterion
functions. In contrast, the limit theory works well for the confidence intervals based on the
Wald
statistic
when the continuous-updating estimator
Consider next the set of eight
ment conditions
relative to
approximations are much
sample
less
moment
size,
is
used.
conditions M2. Given the large
it is
number
mo-
of
not surprising that the underlying central limit
While the
accurate (see the upper right plot in figure 6.2).
asymptotic approximations are better with the continuous-updating estimator, at least
smaller probability values, the criteria evaluated at the true parameters
when compared
to the
magnitudes predicted by the asymptotic theory.
much
the minimized criteria %^cti<3iis are
updating estimator with probability values
still
On
are too large
the other hand,
better behaved, especially for the continut»»asless
than .15 (see the upper
left plot in figure
6.2).
Confidence intervals based on criteria function behavior performed poorly
ting,
although they performed better
other two estimators.
for
in this set-
continuous-updating estimator than
for the
Moreover, the criterion-function based confidence intervals
for the
for the
continuous-updating estimator proved to be more reliable that the confidence intervals based
on the Wald
statistic.
Figures 6.3 and 6.4 report plots for the sets of four
moment
reduction in
conditions (relative to
M2)
moment
leads to an
conditions
M3
improvement
in
and M4. The
the underly-
ing central limit approximations depicted in the upper right portions of the figures.
is
especially true for the continuous-updating estimator used in conjunction with
conditions M4.
restrictions
The minimized
behave
ment conditions
as predicted
M3
are used.
estimation methods when
is
below the
moment
criterion function values used to test the over-identifying
by the asymptotic theory
In contrast there
moment
solid line in the
This
conditions
upper
left
M4
is
for all three estimators
when mo-
substantial under rejection for
all
three
are used. Interestingly, the dot-dashed line
plot in figure 6.4.
This means that the minimized
values of the criteria for the continuous-updating estimator are not always below those of
26
'
The reason
the iterative estimator.
for this
apparent anomaly
is
that the iterations on the
weighting matrix often did not converge for this experiment. The criterion-function-based
confidence intervals work quite well for the continuous-updating estimator
conditions
M3
are used and both
the two-step estimator
methods
for constructing confidence intervals
when moment conditions
summary, with the possible exception
In
moment
when moment
M4
work
well for
are used.
of the over-identifying restriction test using
conditions Ml, the asymptotic approximations for inferences based on the iterative
estimator perform worse than the corresponding approximations for the other two estimators.
In
comparing the continuous-updating estimator to the two-step estimator, we found the
following. First the asymptotic distribution for the over-identifying restrictions tests
when based on the Ifcntinuous-updating
reliable
^sstimator.
Second,
for
both
is
more
estiniafegrS'l*'
confidence intervals constructed based on the original criterion functions, are often distorted,
but with the exception of
Ml
they are no
less reliable
and often more
than confidence
reliable
intervals constructed via quadratic approximations.
Of
course, assessing the reliability of the asymptotic theory as applied to the different
parameter estimators
is
a different question than assessing the performance of the parameter
estimators themselves. In regards to this latter question, the results in the
table 6.1
show that the continuous-updating estimator tends
considerably
less bias in
to
first
portion of
have either comparable or
the medians than the other two estimators.
On
the other hand, the
dispersion in the estimators as measured by the width between the .10 and .90 quantiles
always
less
and often much
less for
is
the iterative estimator than for the continuous-updating
estimator.
Results for
-
=
13.7
Next we consider
(1990b) (TS2
using
moment
in
and
=
S
1.139
results using the preference specification considered
table 3.1).
conditions
We
M3
only consider the performance of
and M4. Our
6'
by Kocherlakota
MM estimators obtained
results are displayed in figures 6.5
and
6.6
and
With
table 6.1.
this
change
parameter configuration, the results
in
there
is
M3 moment
M4 moment
However under the
conditions are similar to those in figure 6.3.
for the
no longer evidence of under rejection of the over-identifying
conditions,
As
restrictions.
the weighting matrices for the iterative method often failed to converge for the
In contrast to our earlier findings, the limit theory no longer provides a
the coverage probabilities for the criterion-function-based confidence sets
estimator
is
used in conjunction with
In regards to the
median
moment
conditions
M4
before,
M4
runs.
good guide
when the two-step
(compare figures
6.4
and
parameter estimates, the continuous-updating estimator again has
bias than the other
two estimators but more dispersion
for
6.6).
less
measured by the distance
(as
between quantiles.)
Time SeparaBle
6.2
To
e.xplore the extent to
for larger
sample
Preferences, Monthly Data
which the limiting distribution provides a better guide
sizes (with less
extreme data points), we redid some our calculations using
simulations calibrated to monthly data as described in subsection
sively
on the more "moderate" preference configuration adjusting
we only looked
at estimators constructed using
ticularly interested in
post war data.
all
Our
moment
for inference
set
M2
moment
because of
its
results are reported in figures 6.7
and
We
M2
and M3.
use in practice
6.8
focused exclu-
8 accordingly. In this case
conditions
common
3.2.
and table
We
are par-
when analyzing
6.2.
Notice that
of the asymptotic approximations are consistently reliable for the continuous-updating
estimation method. In sharp contrast, large sample inferences for the two-step estimator are
of particularly
M3.
.\lso. of
poor quality with the exception of the over- identifying restrictions
note
is
that the iterative estimates
very close to one another
6.2 as well as in the
reason for this
is
when M3
is
test using
and the continuous-updating estimates are
used. This
is
"Minimized" and "Constrained
reflected in quantiles reported in table
-
Minimized" graphs. Presumably, the
that the weighting matrix tends to be a relatively "flat" function of the
parameters.
28
parameter estimates of
In regards to the
both the continuous-updating estimator and
7,
the iterative estimators have distributions that are
parameter value than the distributions
for
much more concentrated around
the two-step estimator (again see table 6.2).'^
These particular Monte Carlo experiments are ones
restrictions provide
much more
the true
which the unconditional moment
in
identifying information about the
power parameter 7 than
any of the other experiments we report on. Not only are estimates more accurate than the
estimates obtained from the
Monte Carlo experiments
the estimates from the log-normal
the
same sample
6.3
Monte Carlo experiments reported
in section 5
which have
size.^'^
Time Nonseparable
As we discussed
calibrated to annual data, but also
Preferences, Annual Data
in subsection 6.1, the
reliable inference in the case of
continuous-updating estimator generally provides more
time separable preferences when data are generated from the
annual Markov Chain model. However even
for that estimator,
ditions M.3 are used that the distributions of the criteria
TJj
it is
only
when moment
("Minimized"),
TJj
con-
("True")
and T{Jj — Jj) ("Constrained-Minimized") accord well with the corresponding chi-square
distributions. For these reasons
only
moment
conditions
M3
we consider
and only
results with
for the
time nonseparable preferences using
continuous-updating estimator. To construct
our Monte Carlo data sets we used the three time nonseparable settings of the parameters
listing in table 3.2 as
TNSl, TNS2 and TNS3
data generated with ^
separable preferences
=
0,
but
when the
still
{9
=
1/3,^
= -1/3,^ =
estimated the parameter
restriction ^
=
is
imposed
in
9.
-2/3).
We
also used
Unlike the case of time
the estimation, the estimator
'-The performance of the two-step estimator could potentially be improved by using a different weighting
For example, the residuals from nonlinear two-stage least squares applied to each
Euler equation could be used to estimate the asymptotic covariance matrix of the moment conditions.
Another possibility is to use the covariance matrix of the prices of the "synthetic" securities implicit in the
matri.K in the first-step.
use of instrumental variables. See Hansen and Jagannathan (1993) for a discussion of this weighting matrix.
'^Presumably, the main reason for this disparity is that the Markov chain models are calibrated to div-
idend behavior rather than return behavior.
As
is
well
known from
the empirical asset pricing literature,
the dividend calibrations imply returns that are less volatile than historical time series because of
fundamental model misspecification.
29
some
accommodates an MA(1) structure
Vrib) of the asymptotic covariance matrix
equation errors. As
in section 5
was not positive definite
in
we used the Vrib) estimator given
in (5.1)
in
the Euler
except when
it
which case we shifted to Durbin's (1960) estimator with a 4th
order autoregression.
Figure 6.9 presents the p- value plots for the criterion functions for the different settings
0.^'*
for
of the
time separable case (the
In contrast to the
minimized
imply small sample over-rejection of the moment conditions
criteria
each of the settings for
solid line in figure 6.3), the distribution
Further, even
$.
when evaluated
at the true parameters, the criterion
functions are not distributed as a chi-square. This occurs for
^
=
moment cc^itidns, which assumes an MA(1)
small sample distortion of the
GMM criterion function.
VVald statistics are very far from being chi-square
in section 5
and subsections
6.1
much
criterion function perform
all
four settings of 6 including
Evidently the estimator of the asymptotic covariance
(time separable preferences).
matrix of the
and
6.2,
.
structure for the errors, caiSses
Notice that the distributions of the
Consistent with the results reported
confidence intervals for 7 constructed using the
better than those based on the
Wald
statistic.
Table 6.3 reports statistics summarizing the properties of the estimators of 7 and
estimator of 7 does not
in
the estimators of 7
at least in part
\1.\(1)
terms
estimators of
and
9
The median
= —1/3 and
=
1/3,
The
77
is
substantially below the true value
=
for 9
when time
1/3.
Further, the dispersion of
separability
is
correctly imposed,
due to having to estimate an additional parameter and to accommodate
estimator of the asymptotic covariance matrix
there
is
V'(6).
Regarding the
substantial dispersion for each case as evidenced by the
909c quantiles. Notice further that there
cases of ^
for
above
considerably larger than
is
in the
9,
=
0.
general perform as well as in the time separable preference case
(see table 6.1 for the comparison).
of 7 in the case of ^
for
is
some median
and -1/3 (panels A. B and C of table
10% and
bias in the estimators of 9 for the
6.3).
In
summary, the annual data
^^The Monte Carlo e.xperiments used to construct these results are independent across the different values
6. The p- value plots of Figures 5.1, 5.2 and 6.1-6.8 considered results for fixed preference parameters and
the three different estimators. The same Monte Carlo data was used for the experiments for each estimator
of
in these plots.
30
do not permit simultaneous estimation of 9 and 7 with any reasonable precision,
moment
conditions M3.
Time Nonseparable
6.4
at least for
Preferences, Monthly Data
Using the Monthly Markov chain model we also examined the time nonseparable model
—
for 9
and -1/3 and
1/3,
M2
conditions
for
moment
conditions M3.
We
further considered
moment
and since the continuous-
since these conditions are often used in practice
updating estimators demonstrated reasonable small sample properties under time-separable
We
preferences.
did not consider the case of ^
Markov chain model
with the monthly model because the
implies that the covariance matrix of the Euler equation errors
to being singular at this
When
= —2/3
valu^f 9^ Once again we used the estimator
of V{b) given
is
by
close
(5.1).
Durbin's (1960) estimator was necessary we used a r2th order autoregression.
Figure
6.
10 reports the results for criterion functions using
M2, the minimized
moment
conditions M2. Under
criterion performs reasonably well for all three settings of 9.
Tests of
the overidentifying restrictions of the model have the correct small sample size in this case.
Notice further that the criterion T{Jj
—
Jj) ("Constrained-Minimized")
chi-square distributed, but that the distribution of the
square. Hence,
much more
Finally,
we
we continue
table 6.4, panels
estimator of
10%
-,
summary
.\.
to be
B and
statistic
to find that inferences based directly
reliable than those based
report
Wald
is
is
close to being
very far from
chi-
on criterion functions are
on quadratic approximations to the criterion functions.
statistics of the central
tendency of the estimators of 7 and 9
in
C. Lack of prior knowledge of the parameter 9 again causes the
much
less precise as
quantiles. (For instance,
Figure 6.11 presents the
compare the
measured by the distance between the 90% and
first
p- value plots for
column
moment
of table 6.2 to panel
conditions M3.
B
of table 6.4.)
In this case the dis-
tribution of the minimized criterion functions and the "Constrained-Minimized" criteria are
not chi-square.
The modefs over
identifying conditions are rejected too often for
parameter settings and confidence intervals
for
31
the parameter
-/
ail
of the
have the wrong coverage
probabilities.
substantial
With moment conditions M3, allowing
number
for
time nonseparability results
of very large estimates of 7 as reflected by the size of the
(see table 6.4, panels C,
D and
(the difference between
moment
E).
It
90%
in
a
quantiles
appears that the addition of returns as instruments
conditions
M2
and M3) improves the performance
of the
estimator and the quality of the central limit approximations.
Concluding Remarks
7
In this
paper we examined the
that differ in the
way
in
finite
sample properties of three alternative
which the moment conditions are weighted. Particular attention
was paid to both the performance of
alternative
we used
tests of over-identifying restrictions
ways of consti^ctmg apnfidence
sets.
In
documenting
several different specifications of the consumption-based
differed substantially in the
interest.
GMM estimators
amount sample information
there
is
finite
and
to
comparing
sample properties,
CAPM. The
experiments
about the parameters of
While the experiments do not uniformly support the conclusion that one estimator
dominates the others, some interesting patterns emerged.
•
Continuous-updating
in
conjunction with criterion-function based inference often per-
formed better than other methods
mations are
•
In
still
for
annual data, however the large sample approxi-
not very reliable.
monthly data the central
mation method applied
in
limit approximations for the continuous-updating esti-
conjunction with the criterion function-based method of
inference performed well in most of our experiments, including ones in which the pa-
rameters are estimated very accurately and ones
in
which there
is
a substantial
amount
of dispersion in the estimates.
•
Confidence intervals constructed using quadratic approximations to the criterion function
performed very poorly
in
many
of our experiments.
32
•
The continuous-updating estimator
timators, but the
much
•
The
had
typically
Monte Carlo sample
less
median
bias than the other es-
distributions for this estimator sometimes had
fatter tails.
tests for over identifying restrictions are,
the weighting matrix
reliable test statistic.
is
by construction, more conservative when
continuously updated, and in
many
cases this led to a
more
(However, we made no attempts at comparing power even
for
size corrected tests.)
Our Monte Carlo experiments
us to reexamine
tests of the
[see, for
is
some
monthly data were
for
sufficiently successful to convince
of the empirical evidence for the consumption-based
consumption-based
CAPM,
CAPM.
In
most
the model's over identifying conditions ar€;¥eje?ted
example, Hansen and Singleton (1982)]. Since the two-step or iterative estimator
typically used in practice, one potential explanation for these rejections could be the
tendency of these estimators to result
assess this possibility
in
over rejection of the model in small samples. To
we estimated the time separable and time nonseparable models
the continuous-updating estimator.
moment
subsection 3.1 along with
We
used the consumption and return data described
M2
conditions
given in table
Estimation of the time separable model resulted
720.65 respectively. This
estimated value of
0.43 hence
it
7.
is
an example where the
The minimized
GMM
in
tail
-3.3.
behavior of the criterion results
criterion
we found
in several of
far
little
near the true parameter values so
at plausible values of the
not at
our Monte Carlo experiments, with the continuous-
deterioration in the criterion function
in practice
parameters.
is
of
from being economically
updating estimator, extreme point estimates of the parameters are possible.
those cases there typically was
in large
was 5.94 with an implied p-value
appears that the continuous-updating estimator implies that the model
.\s
in
point estimates of 6 and 7 of 0.25 and
odds with the data. However the estimate parameters are very
plausible.
using
it
is
in
when evaluated
important to evaluate the criterion tunction
In this case
33
However
we
restricted 7 to the range
[0
20]
and
estimated 8 for each hypothetical value of
of 7
is
7.
The
resulting
minimized criterion
as a function
plotted in the top panel of figure 7.1. Notice that for this range of 7 the minimized
criterion function
once a plausible
is
well
set of
above 30 where the implied
parameters
is
p- value is essentially zero.
considered, the model
is still
rejected
As a
result,
when using
the
continuous-updating estimator.
In estimation of the time nonseparable
model the point estimates of the parameters were
also quite implausible with estimates of 6,
The bottom panel
of 13.55 with an implied p-value of .035.
In
is still
of 1.20, 267.96
and 0.32
respectively.
of figure 7.1 presents the criterion function for the continuous-updating
estimator with 7 restricted to the range
model
7 and
[0 20].
As a
At 7 = 20 the
result,
even at
criterion reaches a
this
extreme value
substantialip'at odds with the data.
minimum
for
7 the
i^
''
summary, although the continuous-updating estimator does not save the consumption-
based
CAPM,
use in
manv
the experiments that
we have presented provide evidence
GMM estimation environments.
34
that
it
should be of
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Sawa, T. (1969): "The Exact Sampling Distribution of Ordinary Least Squares and TwoStage Least Squares Estimator." Journal of American Statistical Association, 64, 923937.
S. (1989): "Intertemporally Dependent Preferences and the
sumption and Wealth."' Review of Financial Studies, 2, 73-89.
Sundaresan,
Volatility of
Con-
Tauchen. G. (1986): "Statistical Properties of Generalized Method-of-Moments Estimators
of Structural Parameters Obtained from Financial Market Data," Journal of Business
and Economic Statistics, 4: 397-425.
Tauchen, G. and R. Hussey (1991): "Quadrature- Based Methods for Obtaining Approximate Solutions to Nonlinear Asset Pricing Models," Econometrica, 59: 371-396.
"Estimation of Functions of Population Means and Regression CoefA Minimum Expected Loss (MELO) Approach," Journal of Econometrics, 8, 127-158.
Zellner, A. (1978):
ficients Including Structural Coefficients;
•
-(
Jt
-^
Table
Order of C(L)
3.1:
MLE
of
Monthly Law
of
Motion
Table
5.1:
Properties of Estimators of 7, Log-Normal Model
Continuous- Updating
A.
Median
No Time
Averaging, True 7
3.72
1.64
7171.17
1.81
4.47
1.81
10% Quantile
-3.67
-0.23
90%
18.75
4.00
Mean
Truncated Mean"
Quantile
B.
Median
Mean
Truncated Mean''
10% Quantile
90%
Quantile
Time
Two-Step
Iterative
.Averaging, True 7
3.48
1.75
-518.14
1.88
3.18
1.88
-10.48
-0.29
11.52
4.22
=
4.55
1.73
=
4.99
Table
6.1:
Properties of Estimators of 7,
Markov Chain Model, Annual Data
Continuous- Updating
Mean
Truncated Mean"
Median
10% Quantile
90% Quantile
Mean
Truncated Mean"
Median
10% Quantile
90% Quantile
Mean
Truncated Mean"
Median
10% Quantile
90% Quantile
Mean
Truncated Mean"
Median
10% Quantile
90%
Quantile
Mean
Truncated .Mean"
Median
10% Quantile
90% Quantile
Mean
Truncated Mean"
Median
10% Quantile
90% Quantile
Iterative
Two-Step
Table
6.2:
Properties of Estimators of 7,
True 7
=
1.3, 6
Markov Chain Model, Monthly Data
=
Continuous-Updating
0.97i/'2
Table
6.3:
Properties of Estimators of 7 and
6,
Markov Chain Model, Annual Data
Continuous-Updating Estimator, True 7
Mean
Truncated Mean"
Median
10% Quantile ^-
90%
Quantile
Mean
Truncated Mean"
Median
10% Quantile
90%
Quantile
Mean
Truncated Mean"
Median
10% Quantile
90% Quantile
Mean
Truncated Mean"
Median
10% Quantile
90% Quantile
=
1.3,6
=
.97
Table
6.4:
Properties of Estimators of 7 and
9,
Markov Chain Model, Monthly Data
Continuous-Updating Estimator, True 7
Mean
Truncated Mean"
Median
10% Quantile
90%
Quantile
Mean
Truncated Mean"
Median
10% Quantile
90%
Quantile
Mean
Truncated Mean"
Median
10% Quantile
90% Quantile
Mean
Truncated Mean"
Median
10% Quantile
90%' Quantile
Mean
Truncated Mean"
.Median
10% Quantile
90% Quantile
.Mean
Truncated Mean"
Median
10% Quantile
90% Quantile
=
1.3,6
=
.97*'''^
Figure
5.1
Criterion Functions
Monthly Log-Normal Model, No Time Averaging
....Iterative,
Minimized
0.5
0.4
0.3
0.2
0.1
Two-Step,
Continuous-Updating
Figure 5.2
Criterion Functions
Monthly Log-Normal Model, Time Averaging
.Iterative,
Minimized
0.5
0.4
0.3
0.2
0.1
Two-Step,
Continuous-Updating
Figure 5.3
Smoothed Distribution of Estimated Angle implied by (1, 7r]
Monthly Log-Normal Model, No Tim^ Averaging
Continuous-Updating
Two
Iterative
Step
I
1.5
46
Figure 5.4
Smoothed Distribution of Estimated Angle implied by
Monthly Log-Normal Model, Time Averaging
Continuous-Updating
(1,
7t]
Iterative
1
Two
Step
I
1.5
47
1.5
Figure 5.5
Criterion Function for Continuous-Updating Estimator
Monthly Log-Normal Model, No Time Averaging
(D
_3
CO
>
c
g
O
Figure
6.1
Criterion Functions
Annual Markov Chain Model
=
.97,
7
.Iterative,
=
1.3.
^
=
0.
Moment
Two-Step,
Conditions Ml.
Continuous-Updating
Figure 6.2
Criterion Functions
Annual Markov Chain Model
=
7
.97,
=
1.3, ^
=
0.
Moment Conditions M2.
Two-Step,
.Iterative,
Continuous-Updating
Minimized
0.1
0.2
Constrained
0.1
0.2
0.3
-
True
0.4
0.5
0.1
0.4
0.3
0.4
0.5
0.4
0.5
Wald
Minimized
0.3
0.2
0.5
0.1
50
0.2
0.3
Figure 6.3
Criterion Functions
Annual Markov Chain Model
/^
=
.97,
7
=
1.3.
^
=
0.
Moment
Two-Step,
.Iterative,
Conditions- M3.
Continuous-Updating
Minimized
0.1
0.2
Constrained
0.1
0.2
0.3
-
True
0.4
0.5
0.1
0.4
0.3
0.4
0.5
0.4
0.5
Wald
Minimized
0.3
0.2
0.5
0.1
51
0.2
0.3
Figure 6.4
Criterion Functions
Annual Markov Chain Model
=
.97,
7
=
1.3, ^
=
0.
Moment
Two-Step,
.Iterative,
Conditions M4.
Continuous- Updating
Minimized
0.1
0.2
Constrained
0.1
0.2
0.3
-
True
0.4
0.5
0.1
0.4
0.3
0.4
0.
Wald
Minimized
0.3
0.2
0.5
0.1
02
0.2
0.3
0.4
0.5
Figure 6.5
Criterion Functions
Annual Markov Chain Model
=
1.139, 7
=
13.7, ^
=
0.
Moment
Two-Step,
.Iterative,
Conditions M3.
Continuous-Updating
Minimized
0.1
0.2
Constrained
0.1
0.2
0.3
-
True
0.4
0.5
0.1
0.4
0.3
0.4
0.
0.4
0.5
Wald
Minimized
0.3
0.2
0.5
0.1
53
0.2
0.3
Figure 6.6
Criterion Functions
Annual Markov Chain Model
l3
=
1.139, 7
=
13.7, ^
=
0.
Two-Step,
.Iterative,
Moment
Conditions M4.
Continuous-Updating
Minimized
0.1
0.2
Constrained
0.5
/
0.4
0.3
0.2
0.1
0.3
-
True
0.4
0.5
Minimized
y^ y^ '.'
0.1
0.2
0.3
Wald
0.4
0.5
Figure 6.7
Criterion Functions
Monthly Markov Chain Model
l3
=
.97^/>2
7
=
1.3,
^
=
0.
Two-Step,
.Iterative,
Moment
Conditions M2.
Continuous-Updating
Minimized
0.1
0.2
Constrained
0.1
0.2
0.3
-
True
0.4
0.5
0.1
0.4
0.3
0.4
0.5
Wald
Minimized
0.3
0.2
0.5
0.1
00
0.2
0.3
0.4
0.5
Figure 6.8
Criterion Functions
Monthly Markov Chain Model
/?
=
.97^/^2^
7
=
1.3, ^
=
0.
Two-Step,
.Iterative,
Moment
Conditioas M3.
Continuous-Updating
Minimized
0.1
0.2
Constrained
u.o
0.3
-
True
0.4
0.5
Minimized
0.1
0.2
0.3
Wald
0.4
0.5
Figure 6.9
Criterion Functions
Annual Markov Chain Model, Continuous- Updating Estimator
= .97, 7 = 1.3, Time Nonseparable. Moment Conditions M3.
13
e
=
^
1/3
=
0,
_ ^ = -1/3, •
Minimized
0.2
Constrained
0.5
0.4
0.3^
0.2
0.1
0.3
-
= -2/3
True
«J-
0.1
e
0.4
0.5
Minimized
,!»
0.1
0.2
0.3
Wald
0.4
0.
Figure 6.10
Criterion Functions
Monthly Markov Chain Model, Continuous- Updating Estimator
.97^/*^, 7 = 1.3, Time Nonseparable. Moment Conditions M2.
l3 =
......e
=
1/3
e
= o,_0 ^ -1/3
Minimized
0.1
0.2
Constrained
0.1
0.2
0.3
True
0.4
0.5
0.1
0.3
0.4
0.3
0.4
0.5
0.4
0.5
Wald
Minimized
-
0.2
0.5
0.1
58
0.2
0.3
MIT LIBRARIES
3 9080 00897 4716
Figure 6.11
Criterion Functions
Monthly Markov Chain Model, Continuous-Updating Estimator
/J = .97*/*^, 7 = 1.3, Time Nonseparable. Moment Conditions M3.
e=
1/3
^
=
0,
Minimized
0.4
0.3
0.2
0.1
0.2
0.3
-1/3
True
0.5
0.1
_^ =
0.4
0.5
y/r'y.
tl76
ni^i4
Date Due
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