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Working Paper Series
Hypothesis Testing and Finite Sample
Properties of Generalized Method of
Moments Estimators: A Monte Carlo
Study
WP 90-12
This paper can be downloaded without charge from:
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Ching-Sheng Mao
Federal Reserve Bank of Richmond
Working Paper 90-12
Hypothesis Testing and Finite Sample Properties
Generalized Method of Moments Estimators:
A Monte Carlo Study
Ching-Sheng Mao
Federal Reserve Bank of Richmond
November 1990
of
1.
Introduction
Econometric methods based on the first-order conditions of intertemporal
optimization models have gained increasing popularity in recent years.
To a
large extent, this development stems from the celebrated Lucas critique,
which argued forcibly that traditional econometric models are not structural
with respect to changes in the economic environment caused by policy regime
shifts.
The generalized method of moments
(GMM) estimation procedure
developed by Hansen (19821 is a leading example of a large research program
in estimating parameters of taste and technology that are arguably invariant
This estimation procedure has been used by many
to shifts in policy rules.
researchers to estimate nonlinear rational expectations models and has had a
major impact on the practice of macroeconometrics.
In this paper I set out to examine the finite sample properties of GMM
estimators using conventional Monte Carlo simulation techniques.
This study
is motivated by the fact that little is known about the performance of the
GMM estimation in a small sample setting.
Most of the desirable properties
of GMM estimators are based on large sample approximations. There does exist
some work on similar problems (Tauchen (1986) and Kocherlakota (1988)).
The
current study differs from these previous studies in several important
aspects.
First, the model I use to assess the performance of GMM estimators
involves not only the saving decision of a representative agent but the
leisure decision as well.
decision.
Previous studies have abstracted from the leisure
The second distinct feature of this paper concerns the way that
random data are generated.
Here, an equilibrium business cycle model was
utilized to simulate an artificial economy in which the production technology
and the forcing process are explicitly specified. This model has been widely
used in the real business cycle literature to calibrate the U.S. economy.
Our approach is different from the previous studies in which random data were
1
generated from an endowment (barter) economy.
It is well known that Monte Carlo experiments have serious limitations.
In particular, the sampling results they generate can only be applied to the
parameter values that were considered by the experiments because the sampling
distribution of the estimates may depend in a nontrivial way on the parameter
values.
With this caution in mind, our experiments were carried out along a
number of dimensions.in order to make the experimental results as robust as
possible.
First, the parameter governing intertemporal substitution or
relative risk aversion was varied over a relatively wide range of values.
This parameter has been very difficult to pinpoint in empirical studies (see
Eichenbaum, Hansen, and Singleton
(19831, Hall
Singleton
Second,
(1982,
1983,
1988)1.
the
(19891, and
parameter
Hansen
and
governing
the
persistence of the random shock was also varied in order to understand the
sensitivity of the sampling distribution of the estimates to shifts in the
forcing process.
Thirdly, the estimation was performed using various sample
sizes, ranging from SO to SO0 observations.
The case of 500 observations
allows us to see how the asymptotic properties of GMM estimators hold up in
this seemingly large sample environment. As it turned out, a sample of this
size may not be large enough for making correct inferences.
Finally, the
estimation was performed using different numbers of lags in forming the
instrumental vector.
As will be seen, the performance of the GMM estimation
and the associated specification test is fairly sensitive to this parameter.
These experiments produce a number of results that are quite different
from those of previous studies.
Perhaps the most striking finding is that
the GMM specification test tends to over-reject the true model even for large
samples. This is particularly so when relative risk aversion is high and the
lag lengths used to form instruments are relatively large.
For moderate
sample sizes (say, 300 observations) the rejection rates of the model can be
2
as high as 30% at the significance level of 5%.
The poor performance of this
specification test is mainly due to the asymptotic sampling bias.
In fact,
our experiments show that the disparities between the sampling distribution
of the test statistic and the asymptotic distribution are fairly substantial.
The tails of these sampling distributions are much thicker than those of the
asymptotic distribution. In many instances, the objective function which the
GMM estimation procedure tries to minimize is also ill behaved.
This result
explains why the specification test tends to over-reject the model.
Our experiments also indicate that the test results are very sensitive
to the lag lengths used to form instruments. Specifically, as the number of
lags used to form instruments decreases, the approximations of the objective
function and the test statistic are much more accurate, and the performance
of the test improves by a significant margin.
The rejection rates in these
cases are close to the corresponding significance levels, particularly, for
large sample sizes.
This last result is consistent with that of Tauchen
(1986) and strongly suggests that whenever possible shorter lag lengths
1
should be used to form instruments.
The rest of the paper is organized as follows. The next section briefly
describes a representative agent model and derives the Euler equations.
GMM estimation procedure is reviewed in section 3.
The
In section 4 I discuss
the data generating process and perform a statistical test on the accuracy of
the simulated data.
The experiment results are discussed in section 5.
The
final section contains a brief summary and conclusions.
2.
The Model
This section lays out a prototype representative agent model which has
'My results are somewhat different from those of Tauchen who found that the
test tends to under-reject the model.
3
received considerable attention in macroeconomics.
This model provides the
basic framework in which the GMM estimator will be assessed in this paper.
The economy is assumed to be populated by a large number of identical
and infinitely lived consumers.
At each date t the representative consumer
values service from consumption of a single commodity ct and leisure 1
tPreferences are assumed to be
represented by
a
constant relative risk
aversion (CRRA) utility function:
u(ct,lt) =
1
' 1 (l-e)]l-l'c- l},
t
l-l/crI[Ct
8 In ct + (l-0) In It,
if (r > o and (r# 1,
if (r= 1,
where 8 E (O,ll. The parameter Q has the interpretation of the intertemporal
elasticity of substitution with respect to a "composite good" defined as a
2
Cobb-Douglas function of ct and It.
At each date t the consumer is endowed with a pre-determined capital kt
and earns capital income r k
where r is a stochastic one period return (or
t t'
t
rental rate) in consumption units.
The consumer also receives in period t a
wage income wtnt, where nt represents hours worked and w
market-determined wage rate.
The total income is divided between consumption
and investment so that the budget constraint at time t is c
5 wtnt + rtkt.
is the stochastic
t
t + kt+l
- (l-6)kt
It is assumed that capital depreciates at the rate 0 5 6 5 1
and the agent is endowed with one unit of time each period.
The consumer chooses a sequence of consumption and labor supply, taking
prices as given, so as to maximize expected lifetime utility subject to a set
of intertemporal budget constraints. Formally, the consumer's problem is
2
The CRRA utility.function has been widely used in the real business cycle
literature (see, for example, King, Plosser, and Rebel0 (19881 and Kydland
and Prescott (198311. Because of its popularity, many empirical studies have
attempted to estimate the parameters of this utility function. Note that the
inverse of (r is a measure of relative risk aversion.
4
max
E.
{ct,kt+l,ItBy)
Bt u(ct.it) ]s
O<@<L
subject to
Ct
+ kt+l
- (l-6)kt 5 rtkt + wtnt,
It + nt = 1,
where p is the time
for all t,
for all t,
preference discount factor.
The information set at time
t over which expectations are taken is assumed to contain current and past
values of all decision variables and prices.
Consumers, however, do not know
future wage rates and rental rates.
The Euler equations that characterize the consumer's equilibrium are
(1)
ul(ct'lt)/uc(ct.lt) = Wt,
Uc(Ctplt) = P Et
where u
C
and u
1
~~(c~+~,l~+~l (1 - 6 + rt+l)
1
,
(21
represent the marginal utility of consumption and leisure,
respectively. These two equations have the usual interpretations. Equation
(1) states that the rate of substitution between consumption and leisure in a
given period must equal the cost of leisure, which is the real wage rate.
Equation (2) implies that in equilibrium the consumer is indifferent between
consuming one extra unit of goods today and investing it in the form of
capital and consuming tomorrow.
Given the assumed CRRA utility function, (1) and (2) imply the following
relation:
(1
- 6 + rt+l)
(w~+~/w~)
Cl-e)(lh-1)
1
_ 1
= o
(3)
This expression is obtained by solving leisure from (1) and substituting into
(2).
It states that the expectation of the Euler equation residual ct+l
(i.e., the term defined in the bracket) is zero, conditional on information
at time t.
That is, any variables contained in the information set It should
5
be uncorrelated with E~+~.
These restrictions are commonly referred to by
economists as the orthogonality conditions.
The moments restrictions implied by equations like (31 are the building
blocks for constructing a large class of instrumental variables estimators of
the underlying parameters of the utility function.
Many of these estimation
procedures, such as the maximum likelihood or two stage least squares (2SLS),
3
require additional distributional assumptions which may not be true.
The
generalized method of moments procedure proposed by Hansen (1982) does not
require such ad hoc restrictions and has gained increasing popularity for
estimating nonlinear rational expectations models.
3.
The GMM Estimation Procedure
In this section I describe the GMM estimation using the above model as
an example.
The discussion follows closely that of Hansen and Singleton
(1982). The main purpose here is to fix notation, so the discussion will be
brief. A rigorous treatment of this subject can be found in Hansen (1982).
Suppose an econometrician who observes time series data {x,;t = l;..,T)
on consumption c
t'
the rate of return on capital (1 - 6 + rt), and the real
wage rate w t, wishes to estimate the parameter vector ;r = [/3c 9l'.4
assumed that the joint process of (5,) is stationary.
It is
Let ct+l = h(xt+l,Z)
be the residual defined in (31 and Z+ a (qxl) vector of variables contained
in the information set I
with q being greater than or equal to the number
t.'
3
For example, Hansen and Singleton (1983) estimated an asset pricing model
using a maximum likelihood procedure. The same model was estimated by Hall
(1989) using a 2SLS procedure. These studies assumed that the logarithm of
consumption growth and asset returns follow a joint Gaussian process. As
pointed out by Hansen and Singleton (19821, the maximum likelihood estimator
is biased and inconsistent if this assumption is false. The finite sample
properties of 2SLS estimators were studied by Mao (19891.
4
For simplicity, both consumers and the econometrician are assumed to observe
the gross rate of return on investment so that the depreciation rate 6 needs
not be estimated.
6
of parameters to be estimated.
As discussed before, the Euler equation (3)
implies a set of q population orthogonality conditions:
E[g(xt+l,“t,Zg)l
= E[h(xt+po)
ztl
= 0.
(4)
where E is the unconditional expectation operator and z, is a vector of the
true parameter values.
The Q
used to form the product in equation (4) are
In the context of our example, the
the instruments for the estimation.
-t usually includes a constant term and current and lagged values of
vector z
the rate of growth in consumption, asset returns, and the rate of change in
the real wage rate.
Condition (4) is the basis for constructing the GMM estimator of 1,.
It
is obtained by choosing the value of z that makes the sample counterpart of
the population orthogonality conditions "close" to zero.
Specifically, let
the sample average of the function g be given by
i,(z)
= (l/T)
; g(~t+l,ztsgl.
t=1
where the subscript T indicates that the value of the function depends on the
sample size.
Zy
zll.
Ek("t+l+n
Note that g,(z) is a method of moments estimator of E[g(xt+l,
Since ET(z) +
E[g(xt+l,Zt,~)l almost surely in z, and from (4)
x0) 1 = 0, the value of the function gT(z), evaluated at ;r = z,,
should be close to zero for a sufficiently large sample size T. Using this
A
fact, the GMM estimator z of ;ro can be obtained by minimizing the quadratic
form JT given by
J.&g)
= gT(pWTiT(LL
(51
where WT is a q x q symmetric, positive definite matrix that satisfies UT +
W almost surely, and W is a symmetric and nonsingular constant matrix.
choice of the weighting matrix W
T'
The
which can depend on sample information,
7
defines a metric that makes ET close to zero.
Hansen (19821 showed that under regularity conditions the estimator ;r
constructed in this way is a consistent estimator of z. with an asymptotic
variance-covariance matrix that depends on the choice of the weighting matrix
wT'
Hansen also showed that it is possible to select "optimally" a weighting
matrix that minimizes (in the matrix sense) the limiting covariance matrix of
A
;r.This smallest asymptotic variance-covariance matrix is given by5
&+l'Zo)
-1
1 L
zt 'W* E $~t+l,~ol
-1
where W* = (E[g(x
-t+1,Zt,Lo)g(Xt+l’Zt.Zo)‘l)
gt
,
,
(6)
which is the inverse of the
variance-covariance matrix of the random variable g("t+l,zt,To).
Since both
W* and .X*depend on the unknown parameters, they must be estimated.
In order to obtain consistent estimates of the weighting matrix W* and
the covariance matrix X*, Hansen and Singleton (1982) implemented a two-step
procedure which will be followed in this paper.
Initially, a 2SLS weighting
-1
matrix (X~;z,l is employed to obtain the first step estimates of z,.
estimates are used to construct a consistent estimate of W*,
6
These
which is then
used in the second step to obtain final estimates of z, and Z*.
The GMM estimation provides a convenient test of the overidentifying
restrictions implied by the model.
In particular, Hansen (1982) showed that
5
The expression (6) is the smallest asymptotic variance-covariance matrix of
the GMM estimators among all possible choices of weighting matrices, holding
constant the sequence of instruments. This expression, therefore, depends on
the choice of instruments. Hansen (1985) developed a method to calculate the
greatest lower bound of this covariance matrix as the instruments vary over
an admissible set. This method can be used to select instruments that yield
the (asymptotically) smallest covariance matrix. Tauchen (1986) studied the
properties of the GMM estimator using this optimal procedure. This technique
is not considered in this paper.
6
The weighting matrix can be estimated by the inverse of the sample variance
covariance matrix of the random variable g evaluated at the initial estimates
of the parameter vector. This estimate may be modified to account for the
autocorrelation of the disturbance term (see Hansen and Singleton (1982)). My
estimation procedure sets the order of the moving average term to zero.
8
the statistic TJT(i)' which is the sample size times the minimized value of
the objective function, is distributed asymptotically as a chi-squared random
variable with degrees of freedom equal to the dimension of g(x
-t+lsz,,;r) less
the number of parameters being estimated.
This statistic has been used in
many studies to test the overall specification of the underlying economic
model.
One of the objectives of this paper is to understand the behavior of
this statistic and to evaluate the consequences of this specification test in
a finite sample environment.
4.
The Data Generating Process
The model I used to generate artificial data is a standard real business
cycle model which has received considerable attention in the literature.
In
this section I briefly outline this model and describe a numerical method to
obtain equilibrium solutions.
Consider the following optimization problem of a central planner:
max
E.
{ctsk t+l.lt,nt)
,"
=
Pt u(ct.ItI ]s
o<p<1,
subject to
Ct + kt+l
- il-6)kt s AtF(kt,nt), for all t,
It + nt = 1,
for all t,
where F(kt,nt) is a constant returns to scale technology and ht is a positive
random shock.
This model is identical to the consumer's problem except that
income is generated from an endogenous production process.
Solutions of this
optimization problem will be used below to generate time series data for the
sampling experiments.
Note that in equilibrium the rental rate and the wage
rate equal the marginal product of capital and labor, which can be used to
generate data on prices once the model is solved.
As in King, Plosser, and Rebel0 (19881, I assume that the technology is
9
a l-a
,
given by a Cobb-Douglas production function, i.e., AtF(kt,nt) = Atkt nt
where a e (0, 1).
Also, the technology shock is assumed to follow a discrete
stationary Markov process with a transition matrix that is structured in such
a way to yield a first-order autoregressive representation of the process,
i.e., In $+l
= p In ht + u~+~, where p E (0, 1) and u~+~ is an i.i.d. random
disturbance. Using a technique proposed by Rebel0 and Rouwenhorst (1989). I
7
employed a five-state Markov chain to approximate this AR(l) process.
A discrete dynamic programming algorithm will be used to solve the above
maximization problem. This method has recently been applied to solve a large
class of dynamic models where equilibria are optimal (e.g., Christian0 (1989)
and Rebel0 and Rouwenhorst (198911. Details of this method can be found in
Bertsekas (1976) and will not be presented here.
Basically, this numerical
method approximates the policy functions for capital and labor on a finite
number of grid points over the state space.
Starting from an initial guess,
the numerical procedure iterates on the value function of the problem using
the conventional successive approximation algorithm (see Bertsekas (1976)).
The equilibrium solutions for capital and labor are obtained when the value
function converges to a fixed point.
Once the capital stock and labor hours
are solved, other quantities and prices can be derived.
These solutions can
then be used to generate pseudodata for the sampling experiments.
The values of parameters used in solving the model are a = 0.3, 8 = 0.1,
s = 0.96 and 8 = 0.3.
experiment.
These parameters will be held constant throughout the
Two parameters that are of special interest are p and (r, which
will be varied in solving the model.
The benchmark value for p is set to 0.9
7
It should be pointed out that the number of states used for the shock is not
important for the results of this paper. I assume that the technology shock
lies on five distinct points over a bounded interval. The mean and variance
of the log of the shock are 0 and 0.001, respectively. The benchmark value
for p is set to 0.9. These figures are well within the values used in the
real business cycle literature.
10
and later changed to 0.0.
The parameter d will take four different values,
i.e., (r= 0.1, 0.5, 1.0, and 2.5.
It is essential that the artificial data constructed in this fashion are
reasonably accurate. To ensure this accuracy, I adopted 2500 grid points for
the capital stock.
These grids were defined over the ergodic set of capital
8
in order to improve accuracy.
In addition, a statistical test suggested by
Den Haan and Marcet (1989) was performed to test whether the simulated data
satisfy the orthogonality conditions implied by the Euler equation (31.' For
each case under consideration a sample path containing 3000 observations were
generated from the model and the statistic calculated.
The results of this
experiment are given in Table 1 where the probability value (i.e., the tail
area) of the test statistic is indicated in the parenthesis.
It is clear
form this table that the values of the chi-squared statistic are small and
statistically insignificant, indicating that the orthogonality conditions are
satisfied by the solutions. These results justify use of the simulated data
in the GMM estimation, which is based on the same set of restrictions tested
by the above statistical procedure.
8
Intuitively, the ergodic set of capital is a set of numbers {k 1 k 5 k 5 k>
such that its complement has probability zero. This means that once capital
falls into this set, it stays there forever and never moves out. Restricting
capital over this smaller set avoids wasting grid points and thereby improves
accuracy. In my numerical procedure the ergodic set is approximated by first
solving the problem over the feasible set using coarse grids. The implied
ergodic set is then used in the second run to define a new range for capital.
The process continues until the number of grids contained in the ergodic set
exceeds 90 percent of grids being used. For more discussion on the concept
of ergodicity, see Brock and Mirman (19721 and Sargent (1980).
9
The Den Haan-Marcet statistic for testing the orthogonality condition (4) is
l-‘[Tgt~zl~,
where i is a vector of OLS estimates in a
[~zi~ztct~l
.
regression of E~+~ on the instruments z
This statistic has an asymptotic
-t*
chi-squared distribution with degrees of freedom equal to the dimension of
m = i’
IXz;gzl
The value of this statistic should be "small" if the orthogonality
ft.
conditions are satisfied.
11
5.
Simulation Results
The pseudodata generated from the artificial economy are used in this
section to estimate the parameter 1 of the utility function, using the GMM
procedure described in section 3.
10
These experiments were carried out along
several dimensions in order to assess the robustness of the sampling results.
The following are some pertinent features of the experimental design.
The experiments were organized along two types of perturbations.
The
first perturbation concerns variations of those parameters that are important
for generating pseudodata that have different stochastic properties.
parameters include p,
c,
These
which controls the persistence of the random shock, and
which controls relative risk aversion or intertemporal substitution of
consumption and leisure. For these two parameters the following setups were
considered: p = 0, 0.9, and Q = 0.1, 0.5, 1.0, and 2.5.
The second perturbation concerns two aspects of estimation that directly
affect the finite sample properties of GMM estimators.
These parameters are
the sample size, T, and the number of lags used to form instruments, NLAG.
Throughout the experiment the instrumental vector zt is selected to include a
constant and the lagged values of consumption growth c
investment (l-6+r
t++
t+l'ct'
the return on
and the rate of change in the real wage rate w
t+l'wt.
The following cases were considered: NLAG = 1 and 2. and T = 50, 150. 300,
and 500.
For each of these experiments, 400 repetitions were carried out and
sampling statistics calculated. Within each experiment the same set of data
were used for different sample sizes and different lag lengths in order to
reduce variability among experiments.
This variance reduction technique was
frequently used in Monte Carlo study to control intra-experiment variations
10
The numerical routine I used to carry out the estimation is a GAUSS program
written by David Runkel and Gregory Leonard. In order to verify its accuracy
I checked this program against a GMM subroutine provided in the RATS package.
These two numerical routines yield virtually identical results.
12
(see Hendry (1984)).
General Characteristics of the Sampling Distribution of 7
Table 2 displays the estimated mean, the standard deviation (SD), and
the median of the GMM estimates of 8, (r,and 8.
As previously mentioned, the
values for the two parameters p and 8 are fixed throughout the experiments,
which are 0.96 and 0.'3,respectively. The four control parameters are p,
(r,
T, and NLAG.
In panel A I report the results of the first set of experiments which
involve a highly persistent shock (p = 0.9).
The number of lags used to form
instruments in these experiments was 2 so that there are 7 variables included
in the instrumental vector gt (i.e., a constant plus two lagged values of
consumption growth, asset returns, and wage growth). It is clear from this
A
table that the estimate 8 performs extremely well regardless of the Q values
or the sample sizes used.
The estimated mean and median are almost identical
to the pseudo value of (3. The standard deviation is very small, indicating
,.
that the sampling distribution of 8 is tightly concentrated around the true
value.
This result, which is also true for the other experiments displayed
in panels B and C, indicates that the parameter 8 can be reliably estimated
using the GMM procedure.
A
A
The performance of (r and 8 is less clear.
Except for cases where (r is
(r = 0.11, the GMM procedure tends to underestimate (r. Both the
A
mean and median of v are below the pseudo value that was used to generate the
small
data.
(i.e.,
However, as the sample size increases these central measures converge
to the true value, which is to be expected. For most cases
(r is within one standard deviation of o‘about its mean.
the true value of
This result suggests
that the magnitudes of the bias might not be quantitatively important. This,
however, is not true for the estimate of 8.
13
As the Table shows, the estimate
0 is severely biased, particularly, for cases where 0‘ is close to one.
This
is not surprising because when (r= 1 the parameter e is not identifiable and,
therefore, cannot be estimated with any precision."
The Table shows that the
dispersion of 8 when (r= 1 is very large, and both mean and median are skewed
To some extent, this result
A
As the Table shows, the estimate 8
toward negative values, which is meaningless.
applies also to the case of (r = 0.5.
is small
perform relatively better as c moves away from one. In fact, when r.r
8%
(i.e., (r= 0.1) the estimate 8 converges from below to the true value as the
sample size increases, and when (r is large (i.e., C = 2.5) it converges from
above to the true value as the sample size increases. Note that for small
A
samples 8 is still severely biased regardless of the (rvalues.
Panel B displays the sampling results for cases where the random shocks
are purely temporary (p
the same as before.
were considered.
=
0).
The number of lags used to form instruments is
For these experiments, only the cases of (r= 0.5 and 2.5
As can be seen, the chief difference here is that the
estimate of the curvature parameter Q is upward biased, which is in contrast
to the results displayed in panel A.
Although it seems apparent that this
difference is due to the forcing process that was used to generate the data,
it is not easy to identify the specific sources that cause these biases.
For
example, the estimated correlation coefficient between consumption growth and
asset returns when (r= 0.5 is higher in the case of p = 0 than in the case of
P
=
0.9.
However, when c = 2.5 this correlation becomes smaller for p = 0.
There does not appear to exist a clear pattern in the correlation structure
of the simulated data that helps explain or identify the bias.
It is
,.
,.
interesting to note that the sampling distribution of (r (and f3)gets tighter
11
Note that the utility function becomes additively separable when (r= 1. In
this case, the marginal utility of consumption does not depend on the leisure
decision, which implies that the wage term will not appear in equation (3).
As a result, the parameter 8 cannot be identified.
14
as the shock becomes less persistent.
Panel C gives the sampling results where the number of lags used to form
The value of the parameter p is still
,.
A
Examination of the results indicate that both (rand 8 appear to perform
instruments was decreased from 2 to 1.
0.9.
better than the first set of experiments in terms of the central measures.
The mean as well as the median of these two estimates are closer to the true
values.
Although the dispersion (i.e., standard deviation) of the sampling
distribution tends to rise as NLAG decreases from 2 to 1, the magnitudes do
not seem unusually large.
This last result, which will be made more clear
below, is somewhat different from that of Tauchen (1986) who found that there
is a strong bias/variance trade-off as NLAG increases.
h
n
Bias and Root Mean Squared Error (RMSE) of (rand 8
Table 3 contains some specific statistics regarding the performance of cr
A
and 8. Two conventional measures were computed. The first measure, bias, is
the sample average of the estimates less the true parameter value, and the
second measure, RMSE, is the root mean squared error about the true parameter
value.
For the purpose of comparison, these two measures were divided by the
estimated standard deviation of the estimates, and the results were given in
12
the brackets in the Table.
Notice that the standardized RMSE should have a
value that is close to but greater than one.
Several conclusions regarding the accuracy of i seem apparent from Table
3.
First, the estimate i is in general biased, but the magnitudes are not
very large.
2.
This result is consistent with the material presented in Table
Except for a few cases the bias is about half of the estimated standard
deviation.
The normalized RMSE is more or less around the anticipated value
12
Because the estimated standard deviation is itself a random variable, this
division introduces some noises in the standardized measures.
15
of one.
Note thatCor
shorter lag lengths (see panel Cl the bias as well as
the normalized RMSE are smaller than those displayed in panel A.
This result
indicates that the slightly higher standard deviation that is associated with
NLAG = 1 is dominated by the improvement in performance in terms of the bias.
The above findings, which are also true for the 8 estimates, suggest that the
smaller standard deviation that might be obtained by using longer lag lengths
may be outweighed by the larger bias.
Later we will see that statistical
inferences based on longer lag lengths usually yield misleading conclusions.
A
The results of Table 3 also show that the performance of (rand 8 depends
on the values of u and p in a nontrivial way (comparing panels A and B).
For
example, when d‘ is small (i.e., Q = 0.51, the accuracy of these two estimates
deteriorates as p gets smaller.
But when (r is large (i.e., (r = 2.51, their
performance improves as p becomes smaller.
This conclusion, however, is not
1
,.
unambiguous because, as mentioned before, the standard deviations of c and 8
decrease with the value of p.
Thus, in terms of the normalized measures of
bias and RMSE, the performance of
these two estimates (especially, for
smaller sample sizes) becomes worse as the value of p gets smaller.
Because
of the inherent nonlinearity of the model and the GMM estimation procedure,
it is difficult to identify the sources that cause these disparities.
Testing the Overidentifying Restrictions
As discussed in section 3, the GMM estimation provides a general test of
the specification of the model.
In particular, the restrictions that the
Euler equation residual should be uncorrelated with variables contained in
the information set constitute a set of overidentifying restrictions that can
be tested. The statistic used to perform this test is the sample size times
the minimized value of the objective function. This statistic is distributed
asymptotically as a chi-squared random variable with degrees of freedom equal
16
13
to the number of overidentifying restrictions.
This subsection summarizes
the results of this specification test.
Table 4 reports the proportion of time that the model was rejected out
of the 400 repetitions.
The rejection rates were calculated at the nominal
significance levels of 5% and 10%. The degrees of freedom of these tests are
also indicated in the Table.
Since the model is correctly specified in all
experiments, the rejection rates (i.e., type I errors) should be close to the
corresponding significance level, in particular, for large sample sizes.
As
the Table clearly demonstrates, the model restrictions were rejected much
more frequently than expected, particularly, for cases where p = 0.9 and NLAG
= 2.
Looking at panel A, it is striking that even for a relatively large
sample (i.e., T = 5001, the rejection rates are more than 10% in most cases
and can be as high as 29% for CT= 0.1.
14
As should be expected, the rejection
rates increase as the sample size decreases. For a small sample such as T =
50, these rejection rates are in the range of 30% to 55%, depending on the
values of (r. Note that the model is rejected more frequently as (r becomes
smaller.
These results are sharply different from those of Tauchen (1986)
who found that, in a somewhat different context, the rejection rates of the
model were more or less in line with the significance levels.
Table 4 also shows that the specification test is very sensitive to the
number of lags used to form instruments. As shown in panel C (NLAG = 11, the
rejections rates are much lower than those associated with NLAG = 2.
As the
sample size increases, these rejection rates converge to the nominal rates.
13
In our example, the number of overidentifying restrictions is equal to the
dimension of the instrumental vector less the number of parameters being
estimated (i.e., q-3).
14
An experiment using 1000 observations was performed for the case of Q = 0.1.
The performance of the test did not improve very much. The rejection rate
dropped from 29% to 26% at 5% significance level. This finding suggests that
the sampling bias is quite substantial for small 6 values. This point will
be addressed in more detail later on.
17
These findings clearly suggest that the risk of making incorrect inferences
increases with the lag lengths.
To check this conclusion more carefully, I
increased NLAG to 4 for some experiments and found that the performance of
the test worsen dramatically and the rejection rates appear to depend more on
15
NLAG than on the sample size.
In addition to the results reported here, some further experiments were
conducted in order to see the sensitivity of the test with respect to certain
16
aspects of the estimation procedure.
Specifically, instead of using the
two-step procedure suggested by Hansen and Singleton (19821, the number of
iterations was increased until the minimized values of the objective function
differ by less than 1 percent over consecutive iterations. These experiments
indicate that the estimation usually converges in 4 to 6 steps
and
the
final
value of the objective function is not very different from that using the two
steps procedure.
Consequently, the test results are virtually identical to
those reported in Table 4. 17
Behavior of the Test Statistic and the Objective Function
The results reported so far strongly suggest that rejection of the model
is likely due to the asymptotic sampling bias.
This subsection provides some
pertinent information concerning the behavior of the test statistic as well
as the objective function.
A useful way to pose our problem is as follows: In order to make correct
inferences (say, to reject the model about 5% of time at the corresponding
nominal rate), what is the correct critical value that should be used for the
15
For example, the rejection rates under p = 0.9 and (r= 0.5 were 46% and 40%
for T = 50 and 500, respectively. These results are not reported and can be
obtained from the,author upon request.
16
The following experiments were suggested to me by Martin Eichenbaum.
17
It should be cautioned that for these experiments only a limited number of
cases were considered.
18
test statistic? The last two columns of Table 4 contain relevant statistics
The first column lists the values of a chi-squared
to answer this question.
random variable at 5% significance level.
These numbers, which were used to
perform the test, represent the theoretical values that should be used if the
sampling distribution of the test statistic is "close" to a true chi-squared
distribution. The last column displays the critical values that should have
been used for making correct inferences. These figures were calculated from
the approximate distribution of the test statistic.
It is clear that the
sampling values are much higher than the theoretical values, particularly,
The two values become closer for NLAG = 1 although
for longer lag lengths.
some minor disparities do exist.
This simple exercise suggests that the
tails of the sampling distribution of the test statistic might be thicker
than the theoretical distribution.
Figures la and lb plot the "inverted" sampling distribution of the test
statistic for c = 0.5 and 2.5, respectively. For comparisons, these figures
were plotted against a theoretical chi-squared distribution.
The heights of
these inverted distributions represent the marginal probabilities of the test
statistic at the corresponding values shown on the horizontal axis.
line is used to mark the 5% significance level.
A dotted
These figures show that the
asymptotic sampling biases are quite substantial for NLAG = 2.
As suggested
above, the rear ends of the sampling distribution are thicker than those of
the true distribution.
This is the reason why the specification test tends
to over-reject the model.
The sampling distribution appears to converge to
the true distribution as the sample size increases.
exist some significant disparities for T = 500.
of NLAG = 1 performs much better.
However, there still
On the other hand, the case
The two distributions are almost identical
for T = 500, a result which is consistent with the results of Table 4.
The reliability of the GMM estimator and the associated test is directly
19
related to the behavior of the objective function.
As Hansen (19821 pointed
out, the values of the objective function evaluated at the true parameter
values are distributed asymptotically as chi-square statistics with degrees
of freedom equal to the number of orthogonality conditions.
If this minimum
chi-squared property is violated, one expects a poor performance of the GMM
estimator and
the associated specification test.
Figure
2
plots
the
distribution of the objective function for (r = 0.1 and 2.5 using NLAG = 2.
For comparisons, the true chi-squared distribution is also plotted.
As can
be seen, these figures are very similar to those in figure la and lb.
In
particular, the sampling distribution of the objective function is far away
from the asymptotic distribution when (r is small.
These results explain why
GMM estimators perform poorly for small o‘ values.
Again, the sampling
distribution converges to the true distribution as the sample size increases.
6.
Conclusions
In this paper I have examined the performance of the GMM estimation in a
simple neoclassical model which has received considerable scrutiny in recent
years.
Most empirical studies have found that this model or a similar one
did not seem consistent with the actual data.
The findings of this article
suggest that rejection of the model might have been caused by an inadequacy
of the asymptotic approximations. For cases where the curvature parameter of
the utility function is small or risk aversion is high, these sampling biases
are fairly substantial and could lead to inappropriately high rejections of
the true model.
In order to make correct inferences, it may require a large
number of observations which are simply not feasible in practice.
One way to
alleviate this problem is to use shorter lag lengths in forming instruments.
To some extent, the better performance with shorter lag lengths is due to a
reduction in the number of restrictions that are being tested.
20
But, more
important, it is because the sampling errors are smaller and the estimators
perform better.
Another implication suggested by our experiments is that the curvature
parameter of the utility function can be reasonably estimated using the GMM
procedure.
This result is contrary to the existing belief that it might be
very difficult to pin down this parameter. This perception is perhaps due to
the wide range of estimates that exist in the literature.
Our findings
suggest that, if the model is correctly specified, failure to uncover this
curvature parameter accurately may result from measurement errors that plague
the data.
Such errors are particularly characteristic of consumption data.
To keep the cost of simulation to its minimum, several important issues
have not been addressed in this paper.
In particular, problems concerning
the power of the specification test were ignored. .Consequently, very little
can be said about the power of the test against false specifications. Recent
studies by Singleton (19851 have shown that it is feasible to discriminate
among competing economic models within the GMM environment.
Such endeavors
will require much more careful and finer calibrations of the model and are
left for future studies.
21'
References
Bertsekas, D. P.,Dynamic Programming and Stochastic Control.
Academic Press, 1976.
New
York:
Brock, W. A., and L. Mirman. "Optimal Economic Growth and Uncertainty: The
Discounted Case," Journal of Economic Theory, 4(1972), 479-513.
Christiano, L. J. "Solving a Particular Growth Model by Linear Quadratic
Approximation and by Value Function Iteration," Discussion Paper #9,
Institute for Empirical Macroeconomics, Federal Reserve Bank of
Minneapolis, January 1989.
Den Haan, W. J., and A. Marcet. "Accuracy in Simulations," Manuscript,
Carnegie-Mellon University, 1989.
Eichenbaum, M., L. P. Hansen, and K. J. Singleton. "A Time Series Analysis of
Representative Agent Models of Consumption and Leisure Choices Under
Uncertainty," Manuscript, Carnegie-Mellon University, 1983.
Hall, R. E. "Intertemporal Substitution in Consumption," Journal of Political
Economy, 96(19891, 339-357.
Hansen, L. P. "Large Sample Properties of Generalized Method of Moments
Estimators," Econometrica, 50(19821, 1029-1054.
"A Method for Calculating Bounds on the Asymptotic Covariance
Matrices of Generalized Method of Moments Estimators," Journal of
Econometrics, 30(1985), 293-238.
and K. J. Singleton. "Generalized Instrumental Variables
Estimation of Nonlinear Rational Expectations Models," Econometrica,
50 (1982). 1269-1286.
, and K. J. Singleton. "Stochastic Consumption, Risk Aversion,
and the Temporal Behavior of Asset Returns," Journal of Political
Economy, 91(1983), 249-265.
and K. J. Singleton. "Efficient Estimation of Linear Asset
Pricing Models with Moving Average Errors," Manuscript, University of
Chicago, April 1988.
Hendry, D. F. "Monte Carlo Experimentation in Econometrics," Chapter 16,
Handbook of Econometrics, Vol. II, A. Griliches and M. D. Intriligator,
eds. Amsterdam: North-Holland, 1984.
King, R. G., C. Plosser, and S. Rebelo. "Production, Growth and Business
Cycles I. The Basic Neoclassical Model," Journal of Monetary Economics,
May 1988.
Kocherlakota, N. R. "In Defense of the Time and State Separable Utility-Based
Asset Pricing Model," Working paper 63, Department of Finance,
Northwestern University, October 1988.
Kydland, F., and E. Prescott. "Time to Build and Aggregate Fluctuations,"
Econometrica, SO(19821, 1345-1370.
22
Mao, C. S. "Estimating Intertemporal Elasticity of Substitution: The Case of
Log-Linear Restrictions," Economic Review, Federal Reserve Bank of
Richmond, 75(19891, 3-14.
Rebelo, S., and G. Rouwenhorst. "Linear Quadratic Approximations versus
Discrete State Space Methods: A Numerical Evaluation," Manuscript,
University of Rochester, March 1989.
Sargent, T. J. "Tobin's q and the Rate of Investment in General Equilibrium.
Carnegie-Rochester Conference Series on Public Policy, 12, K. Brunner
and A. H. Meltzer, eds. Amsterdam: North- Holland, 1980.
Singleton, K. J. "Testing Specifications of Economic Agents' Inter-temporal
Optimum Problems in the Presence of Alternative Models," Journal of
Econometrics, 30(1985), 391-413.
Tauchen, G. "Statistical Properties of Generalized Method-of-Moments
Estimators of Structural Parameters Obtained from Financial Market
Data," Journal of Business and Economic Statistics, 4tOctober 19861,
397-416.
23
Table 1
The Den Haan-Marcet Statistic for Testing the Orthogonality Conditions
(Fixed parameters: a = 0.3, 6 = 0.1, /3= 0.96, 6 = 0.3, p = 0.91
6
NLAG = 1
NLAG = 2
0.1
1.20
(0.88)
4.91
(0.671
0.5
2.14
(0.71)
2.73
(0.91)
1.0
2.49
(0.65)
3.51
(0.83)
2.5
2.50
(0.65)
8.44
(0.30)
Note: 1. The statistic is computed based on a sample of 3000 observations.
2. The instruments are chosen to include a constant and the lagged ~
values of consumption growth, asset returns and wage growth.
3. NLAG = number of lags used to form instruments.
4. Test results are similar for smaller sample size and larger
degrees of freedom, which are not reported.
24
,.
A
A
Table 2: Moments of the Sampling Distributions of 8, (rand 8
,.
8
,.
Parameters
QO
T
/i (8, =
Mean
0‘
0.96)
Median
SD
Mean
SD
Median
Mean
@O
= 0.3)
SD
Median
0.078
0.113
0.172
0.097
0.086
0.150
0.204
0.228
Panel A: p = 0.9, NLAG = 2
0.084
0.170
0.241
0.251
0.1
50
150
300
500
0.960
0.960
0.960
0.960
0.003
0.002
0.001
0.001
0.960
0.960
0.960
0.960
0.125
0.120
0.124
0.117
0.060
0.048
0.058
0.038
0.110
0.110
0.109
0.110
0.5
50
150
300
500
0.960
0.960
0.960
0.960
0.001
0.001
0.000
0.000
0.960
0.960
0.960
0.960
0.406
0.449
0.474
0.478
0.089
0.119
0.118
0.092
0.387 -0.150 0.420 -0.107
0.422 -0.006 0.176 -0.024
0.094 0.237 0.076
0.449
0.461
0.121 0.745 0.133
1.0
50
150
300
500
0.960
0.960
0.960
0.960
0.001
0.001
0.000
0.000
0.960
0.960
0.960
0.960
0.719
0.850
0.937
0.934
0.212
0.355
0.443
0.216
0.686
0.773
0.841
0.887
-0.887
-0.416
-0.737
-0.523
2.003
4.125
4.212
5.567
-0.692
-0.562
-0.438
-0.337
2.5
50
150
300
500
0.960
0.960
0.960
0.960
0.000
0.000
0.000
0.000
0.960
0.960
0.960
0,960
1.433
1.789
2.129
2.233
0.518 1.323
0.736 1.614
0.913 1.858
0.788 2.028
1.996
0.985
0.617
0.502
1.892
0.529
0.297
0.210
1.757
0.685
0.574
0.494
Panel B: p = 0.0, NLAG = 2
0.5
50
150
300
500
0.960
0.960
0.960
0.960
0.001 0.960
0.000 0.960
0.000 0.960
0.000 0.960
0.724
0.601
0.554
0.536
0.082
0.073
0.053
0.042
0.732
0.602
0.554
0.535
-0.321
0.116
0.213
0.245
0.442
0.154
0.085
0.060
-0.246
0.138
0.224
0.253
2.5
50
150
300
500
0.960 0.001 0.960
0.960 0.000 0.960
0.960 0.000 0.960
0.960 0.000 0.960
3.317
2.837
2.682
2.604
0.469
0.330
0.231
0.190
3.327
2.837
2.688
2.602
0.468
0.382
0.348
0.329
0.062
0.074
0.061
0.049
0.483
0.392
0.356
0.335
0.104
0.197
0.283
0.261
0.168
0.233
0.788
0.235
0.091
0.152
0.192
0.207
Panel C: p = 0.9, NLAG = 1
0.1
50
150
300
500
0.960
0.960
0.960
0.960
0.004
0.002
0.001
0.001
0.960
0.960
0.960
0.960
0.093
0.098
0.099
0.098
0.052
0.061
0.066
0.057
0.086
0.083
0.084
0.084
0.5
50
150
300
500
0.960
0.960
0.960
0.960
0.002 0.960
0.001 0.960
0.000 0.960
0.000 0.960
0.413
0.468
0.498
0.480
0.121
0.183
0.211
0.092
0.385 -0.063 0.386 -0.072
0.423
0.079 0.730 0.061
0.451
0.242 0.666 0.148
0.456
0.242 0.228 0.190
1.0
50
150
300
500
0.960
0.960
0.960
0.960
0.001 0.960
0.001 0.960
0.001 0.960
0.000 0.960
0.783
0.906
0.977
0.979
0.337
0.368
0.342
0.273
0.702
0.806
0.881
0.920
-0.654
-0.640
-0.446
-0.536
2.5
50
150
300
500
0.960
0.960
0.960
0.960
0.001 0.960
0.000 0.960
0.000 0.960
1.607
2.028
2.353
2.375
0.673 1.439
0.970 1.747
0.977 2.003
0.843 2.140
1.381
0.709
0.468
0.422
on
Iterations
Note:
Statistics
are
0.000
computed
0.960
based
400
25
of
each
2.479
5.088
5.587
5.404
-0.574
-0.301
-0.268
-0.184
2.842 1.164
0.512 0.655
0.285 0.432
0.208 0.407
experiment.
A
A
Table 3: Bias and Root ,Mean Squared Error (RMSE) of crand 8
"0
Bias
T
e
6
Parameters
Norm.
Bias
RMSE
Norm.
RMSE
Bias
Norm.
Bias
RMSE
Norm.
RMSE
Panel A: p = 0.9, NLAG = 2
0.1
50
150
300
500
0.025
0.020
0.024
0.017
0.5
50
150
300
500
1.0
2.5
1
[
[
t
0.421
0.421
0.421
0.461
0.065
0.052
0.062
0.041
Il.081
[l.081
[l.081
t1.091
-0.216
-0.130
-0.059
-0.049
l-2.781
i-1.151
r-o.341
[-0.501
0.230
0.172
0.181
0.109
12.961
Il.521
12.311
El.121
-0.094
-0.051
-0.026
-0.022
I-1.061
t-o.431
i-0.221
I-O.231
0.129
0.129
0.121
0.094
Il.461
Il.091
Il.021
il.031
-0.450
-0.306
-0.206
-0.179
i-1.071
r-l.731
1-O.871
t-0.241
0.615
0.353
0.314
0.765
Il.461
i2.001
l1.321
11.031
50
150
300
500
-0.280
-0.150
-0.063
-0.066
i-1.331
i-O.421
L-O.141
f-O.301
0.351
0.385
0.447
0.225
il.661
il.091
L1.011
L1.041
-1.187
-0.716
-1.037
-0.823
1-O.591
t-0.171
I-0.241
L-O.151
2.326
4.182
4.368
5.621
Il.161
[l.Oll
Il.031
(1.011
50
150
300
500
-1.067
-0.711
-0.371
-0.267
I-2.061
I-O.971
i-o.411
r-O.341
1.185
1.022
0.984
0.831
t2.291
(1.391
il.081
[1.051
1.696
0.685
0.317
0.202
[
[
[
[
0.901
1.301
1.071
0.961
2.539
0.865
0.434
0.291
Il.341
il.641
(1.461
t1.391
I-1.411
I-1.201
i-l.021
l-O.911
0.762
0.240
0.122
0.081
f1.721
Il.561
t1.431
il.351
t
[
[
[
2.701
1.091
0.781
0.571
0.179
0.110
0.077
0.057
[2.881
Il.481
(1.271
Il.151
Panel B: p = 0.0, NLAG = 2
0.5
50
150
300
500
0.224
0.101
0.054
0.036
t
[
[
[
2.741
1.381
1.021
0.871
0.238
0.124
0.076
0.055
Il.911
Il.701
Il.431
Il.321
-0.621
-0.184
-0.087
-0.055
2.5
50
150
300
500
0.817
0.337
0.182
0.104
[
r
[
[
1.741
1.021
0.791
0.551
0.942
0.472
0.293
0.216
12.011
[l.431
[l.271
[l.141
0.168
0.081
0.048
0.029
Panel C: D = 0.9. NLAG = 1
0.1
50
150
300
500
-0.007
-0.002
-0.001
-0.002
I-O.141
L-O.041
I-O.021
1-o.041
0.053
0.061
0.066
0.057
f1.011
[l.OOl
Il.001
r1.001
-0.196
-0.103
-0.017
-0.039
I-l.171
1-o.441
[-0.021
f-0.171
0.258
0.254
0.787
0.238
11.541
(1.091
[l.OOl
[l.Ol]
0.5
50
150
300
500
-0.087
-0.032
-0.002
-0.021
L-0.721
l-O.181
r-o.011
E-O.221
0.149
0.186
0.211
0.094
il.231
l1.011
[l.001
Il.021
-0.363
-0.221
-0.058
-0.058
i-0.941
l-O.301
I-0.091
L-O.251
0.529
0.762
0.668
0.234
Il.371
(1.041
[I.001
il.031
1.0
50
150
300
500
-0.217
-0.094
-0.023
-0.021
f-0.641
1-O.261
1-O.071
t-0.081
0.400
0.389
0.342
0.273
r1.191
[l.031
[1.001
t1.001
-0.954
-0.940
-0.746
-0.836
l-O.381
1-O.181
i-O.131
I-0.151
2.653
5.168
5.629
5.462
il.071
Il.021
il.011
Il.011
2.5
50
150
300
500
-0.893
-0.472
-0.147
-0.125
I-1.331
t-O.491
1-o.151
I-O.151
1.117
1.077
0.987
0.851
El.661
il.111
Il.011
t1.011
1.081
0.409
0.168
0.122
I
[
[
I
3.038
0.655
0.331
0.241
[1.071
Il.281
[l.161
Il.161
Note:
1.
2.
Statisitics
are
computed
based
on 400 iterations
pormalized
Fl
ures
re
orted
in the
brackets
are
mated
standard
deviation)
.
BIi!S
/ (es!1
26
0.381
0.801
1.361
0.591
of
Bias
each
and
ex er ment.
RUEE te.g.,
Table 4: Rejection Rate (%I of the Overidentifying Restrictions
Parameters
Degrees
of
Freedom
Rejection Rate at
Significance Level of
10 %
5%
Critical Value at
5% Significance Level
True
Sampling
Panel A: p = 0.9, NLAG = 2
0.1
50
150
300
500
4
4
4
4
55
43
31
29
65
50
39
38
9.49
9.49
9.49
9.49
21.09
31.42
28.84
23.68
0.5
50
150
300
500
4
4
4
4
40
31
20
12
52
40
28
19
9.49
9.49
9.49
9.49
20.39
24.04
18.83
15.18
1.0
50
150
300
500
4
4
4
4
36
24
18
12
45
32
24
18
9.49
9.49
9.49
9.49
18.99
21.23
17.87
13.94
2.5
50
150
300
500
4
4
4
4
29
25
19
13
41
33
21
26
9.49
9.49
9.49
9.49
20.15
18.96
16.58
.14.20
Panel B: p = 0.0, NLAG = 2
0.5
50
150
300
500
4
4
4
4
25
17
13
11
33
25
17
16
9.49
9.49
9.49
9.49
15.41
14.16
13.85
12.91
2.5
50
150
300
500
4
4
4
4
17
13
9
7
26
17
13
13
9.49
9.49
9.49
9.49
13.45
13.13
10.56
10.46
Panel C: p = 0.9, NLAG = 1
0.1
50
150
300
500
1
1
1
1
12
12
7
9
18
19
13
15
3.84
3.84
3.84
3.84
6.55
6.26
4.52
5.10
0.5
50
150
300
500
1
1
1
1
16
8
6
6
22
12
11
11
3.84
3.84
3.84
3.84
9.72
5.18
4.47
3.99
1.0
50
150
300
500
1
1
1
1
14
7
6
6
20
12
12
11
3.84
3.84
3.84
3.84
8.47
4.78
4.47
3.86
2.5
50
150
300
500
1
1
1
1
12
7
7
5
18
12
12
11
3.84
3.84
3.84
3.84
7.36
4.66
4.36
3.76
Note:
Statistics
are
computed
based
on
400
27
iterations
of
each
experiment.
Figure
1 a: Test
I -lLdF(y):
Statistic
Asymptotic
vs.
Sampling
(Parameters:0 = 0.5,p = 0.9)
NLAG
NLAG
= 2
T=50
=
1
T=50
0,
03
Ad
I
.-
4
..-co
I
I
I
I
5
10
15
20
k0
0.
h
“0
5
10
15
20
25
‘0
X
25
X
T=500
T=500
0-l
I
I
I
5
10
15
I
he
..-co
g”
4
On
k d
0 .
OO
5
10
15
20
25
“0
X
Asymptotic:
X
Sampling:
- - -
20
25
I b: Test
Figure
1 -sidF(y):
Asymptotic
vs.
0 = 2.5,
(P arameters:
NLAG
Statistic
p
Sampling
=
0.9)
NLAG
= 2
=
1
T=50
T=50
a!
..--co
$6
so
.-
Oo
5
10
15
20
25
15
20
25
X
T=500
"0
5
10
T=500
15
20
25
“0
5
10
X
Asymptotic:
X
Sampling:
- - -
Figure
1 -JidF(y):
2:
Objective
Asymptotic
Function
vs.
Sampling
(parameters:p = 0.9, NLAG = 2)
CT =
T=50
T=50
OO
5
10
15
20
25
30
OO
5
10
15
X
5
10
15
20
25
30
20
25
30
x
T=500
OO
2.5
CT =
0.1
T=500
20
25
30
“0
5
10
X
Asymptotic:
15
X
Sampling:
- - -
@FedFRASER