BOOTSTRAP METHODS IN LIMITED
DEPENDENT VARIABLE MODELS
By
JINYONG CHEN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1993
I dedicate this dissertation to the glory of my Lord Jesus Christ. Because of
His blessings, my life as a graduate student has been an exciting and rewarding
experience.
ACKNOWLEDGEMENTS
I truly appreciate the patient encouragement,
guidance and support of Dr.
G.S. Maddala. He helped me to organize this dissertation. I would like to thank
Drs. Alan Agresti, David Denslow, Steven Donald, Larry Kenny and Mark Rush
who
provided
me with
valuable
comments
Drs.
Frank
Martin,
Dennis
Wackerley and Thomas Santner who were willing to spend hours brainstorming
and challenging the statistical methodology employed. I appreciate many helpful
comments from Dr. Jinook Jeong.
I thank my colleagues Mr.
Yikang Li and Mr.
Li Zhu for their many helpful discussions. I would also like to thank the many
other people who helped me in this work.
In particular, those in the computer
laboratory who provided me with easy access to many computers, dramatically
reducing my computational time.
I also thank those who have been praying for m
my dear brother in Christ Dr. Nehemiah Cherng,
I owe a special thanks to
who gave me extraordinary
support in many ways. I owe deep thanks to my wife, Xiaohui, for her continuous
support, encouragement, patience, understanding and prayers.
TABLE OF CONTENTS
ACKNOW LEDGEM ENTS . . . . . . . . . iii
ABSTRACT. . . . . . . . . . . . . v
INTRODUCTION. .
A QUASIPIVOTAL METHOD FOR GENERATING
BOOTSTRAP CONFIDENCE INTERVALS
Introduction . . .
The QuasiPivotal Method .........
Usual Bootstrap Confidence Intervals. .
Comparisons of the Confidence Intervals
Comparisons of the Bootstrap Methods
Summary
. . . . . . 8
. .. f f f 9 f f f f f 14
. .ft f f S f f f 15
** *** *f23^
BOOTSTRAP METHODS IN BINAI
RESPONSE VARIABLE MOD]
3.1 Introduction . .
3.2 Bootstrap Methods in Binary
3.3 Generalised Residuals . .
3.4 Monte Carlo Experiments for
3.5 Summary . . .
ELS
Response Variable Mod'
Bootstrap Methods
* ft f
* ft f
. . . 3 2
. . . 3 2
els. . . 33
. . . 3 7
. . . 4 1
. . . 4 6
IV BOOTSTRAP METHODS IN THE TOBIT
4.1 Introduction . . . . .
4.2 Applications of Bootstrap Methods
4.3 Monte Carlo Experiments for Boots
4.4 Summary . . . . .
t
;t
MODEL . . .. . 51
. . . . . . 5 1
.o the Tobit Model. .. . 52
rap Methods . . . 56
. . . . . . 59
TESTS OF HYPOTHESES IN
DEPENDENT VARIABLE
5.1 Introduction . . .
5.2 Wald, Likelihood Ratio a
5.3 Monte Carlo Experiment
5.4 Summary . . .
LIMITED
MODELS.
nd Lagrang
s for Hypoti
* . . a 64
* S 9 f f f U. f f f 64
e Multiplier Tests. . . 67
hesis Testing . . . 69
. . . . . . . 76
VI CONCLUSIONS .
BIBLIOffGRAPHY. . f . . .
lTrf'flfi A "YT TTrf AT t '" TrrT T n
<
J
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
BOOTSTRAP METHODS IN LIMITED
DEPENDENT VARIABLE MODELS
By
Jinyong Chen
December 1993
Chairman: Dr. G.
S. Maddala
Major Department: Economics
Since the bootstrap method was introduced by Efron in 1979,
there have been
many applications in estimating models, generating bootstrap confidence intervals,
and testing of hypotheses using bootstrap based critical values.
The dissertation's
bootstrap methods: b
main emphasis is on comparing the performance of two
bootstrapping the residuals and bootstrapping the data,
method suggested by Efron. The comparison is made with special reference to the
logit, the probit and the tobit models and also in obtaining the correct significance
levels for the Wald, likelihood ratio and Lagrange multiplier tests. Several methods
of bootstrapping the residuals have been investigated. An extension is also made
on the method of generating short bootstrap confidence intervals.
The general
conclusion that emerges from the several Monte Carlo experiments conducted is
that the method of bootstrapping the residuals is to be preferred to the method of
CHAPTER 1
INTRODUCTION
The bootstrap method, introduced by Efron (1979), is a resampling method
whereby information in the sample data is "recycled" for the purpose of inference.
Resampling methods are not new. Thejackknife, introduced by Quenouille (1956
is one of the resampling methods used to reduce bias and provide more reliable
standard errors. Unlike thejackknife, the bootstrap resamples at random. In other
words, while the jackknife systematically deletes a fixed number of observations
in order (without replacement),
the bootstrap randomly picks a fixed number of
observations from
the original sample with replacement.
It serves not only of
reducing bias and providing more reliable standard errors, but also giving interval
estimators and tests of hypotheses. Though thejackknife is shown in Efron (1979)
to be a linear approximation to the bootstrap, the bootstrap method is more widely
applicable than the jackknife,
major
applications
bootstrap
method
are point estimation,
interval estimation, and tests of hypotheses. These are not three individual parts,
but an interwoven system in statistics. There is an enormous statistical literature
on bootstrap
confidence
intervals.
repeated
sampling,
bootstrap
can
approximate
unknown
true distribution
a statistic with
empirical
bootstrap cumulative distribution. Instead of getting only the point estimator of
a parameter as with a regular estimation procedure,
we can obtain the interval
point estimator,
but also
have
a distribution
of this
point
estimator
confidence interval. To get a better bootstrap point estimator, we might also trim
the distribution by discarding observations that lie beyond a given confidence
interval. Obviously the confidence interval must be one with good features, such
as one with a relatively good coverage or short interval. The point estimator, called
the bootstrap
trimmed mean,
is then calculated
using only the
"untrimmed"
observations.
Tests of hypotheses are another major application of the bootstrap method.
With
the bootstrap approximation of the unknown
true distribution
of a
statistic,
we might have an accurate true significance level with respect to the
hypothesized nominal level. To improve bootstrap hypothesis testing, Hall (1991)
suggested that the resampling be done in a way that reflects the null hypothesis,
even when the true hypothesis is distant from the null. The second suggestion was
that bootstrap hypothesis tests should employ methods that are already known
to have
good
features
in the
closely
related
problem
confidence
interval
construction.
From
these
recommendations
we can
see that
bootstrap
hypothesis
testing is
very
closely
related
to the
construction
a bootstrap
confidence interval.
There are two ways to classify bootstrap methods. The first classification is
that we can either bootstrap residuals of a model or bootstrap the data itself. To
bootstrap
residuals of a model,
we may first estimate the model and get the
residuals, then randomly draw residuals with replacement to form a new set of
bootstrap
residuals
same
sample
size.
Another
way
of bootstrapping
residuals is
to randomly
generate a
new set
of bootstrap
residuals
from
form
a new
bootstrap
data
set {y:ix}
of the same sample size.
The second
classification
scheme
to divide
bootstrapping
three
categories:
parametric, semiparametric, and nonparametric.
does not depend on assumption.
The nonparametric bootstrap
If we try to estimate a tobit model using the
parametric bootstrap method, we first estimate the model with the original sample
(y,x) to get an estimate i
we generate an error term
of the true parameter Pf. Second, for each bootstrapping,
* from the normal distribution with mean zero and
estimated variance. Third, we can get a new
* from the model using x and
Then we obtain, from the y" and x, the bootstrap point estimate
This bootstrap
procedure is called the parametric bootstrap. The semiparametric bootstrap is any
method in between these two methods. For an efficient estimation method with a
correctly specified model,
the best bootstrap method is
the parametric.
While
perhaps
not quite
as good
as the
parametric
bootstrap,
nonparametric
bootstrap method is relatively better when we use an efficient, nonparametric
estimation method.
As Efron (1990) and many other authors pointed out, the computational
burden of the bootstrap simulations for reliable bootstrap confidence intervals is
a problem even with today's
fast machines. Several resampling methods to reduce
the computational burden have been
devised.
One group of methods
tries
reduce the required number of bootstrap replications through more sophisticated
resampling
schemes
such
as balanced
sampling,
importance
sampling,
antithetic sampling. All of these methods were originally developed for Monte Carlo
analysis in 1960s. Efron (1990) proposed a post hoc correction method to reduce
the firstorder bias in bootstrap estimates. Another line of research has focused
4
For large samples, we have well developed asymptotic theory. But for small
samples,
it is
very
difficult
to find
approximate
cumulative
distribution
function of a statistic. However, with the bootstrap method, we can approximate
an exact cumulative distribution function of a statistic for any sample size.
Since the bootstrap is only a technique used in estimation, the goodness of
result from
a bootstrap
estimation
depends
not only
on the
method
bootstrapping but also on the estimation method.
With an efficient estimation
method, we will likely have an efficient bootstrap estimation method. With a robust
estimation method,
we will probably get a robust bootstrap estimation method.
With a sensitive estimation method, such as the maximum likelihood estimation
method for the tobit model, the bootstrap based on this estimation method would
still be sensitive.
While, this may seem a disadvantage of the bootstrap method,
it is not a major drawback, because we can easily change to another estimation
method to improve the result. Therefore with bootstrapping, to obtain satisfactory
results,
we need
to apply not only the
best estimation method,
but also
bootstrapping method that works the best with the selected method of estimation.
There
other
disadvantages
bootstrap
method.
Since
bootstrapping is very computer intensive, it may take too much computer time,
especially
getting
bootstrap
confidence
intervals.
Second,
since
bootstraps depend on the original sample, the reliability of the original sample
becomes very important.
Since the first paper by Efron (1981) on bootstrapping censored data, there
have been few papers applying bootstrap methods in limited dependent variable
models, that is, regression models for which the range of the dependent variable
5
estimation uses the nonparametric method of KaplanMeier (1958) and concluded
through simulation that the procedure performs reasonably well in finite samples.
Teebagy and Chatterjee (1989) applied Efron's nonparametric bootstrap
method with MLE to the logistic regression model. They got satisfactory results in
their Monte Carlo study. Manski and Thomson (1986) studied the use of bootstrap
in the maximum score estimation of binary response variable models.
Using the
same bootstrap procedure as Teebagy and Chatterjee (1989), they found that the
bootstrap standard errors are very close to the true ones. However, Adkins
1990)
applied one of the parametric bootstrap methods with MLE to the probit model
and got different results from those of TeebagyChatterjee in that the bootstrap
method tends to be quite unreliable in small samples. The conflict between these
papers leads us to further investigate why different bootstrap methods provide
conflicting
results
similar
models
what
effect
from
bootstrapping
data
bootstrapping
residuals.
Indeed,
major
motivation of this dissertation.
Flood (1985) applied the augmented bootstrap method to the tobit model
and found,
through simulation,
that
the augmented bootstrap gives standard
errors that are close to the true values.
These three studies of TeebagyChatterjee, Adkins, and Flood concentrate
on estimating standard errors of estimated parameters of the logit,
probit, and
tobit models. The standard error is then used to form confidence intervals, which
have already been well provided by bootstrap methods. The major advantage of the
bootstrap method is to form reliable confidence intervals,
but the standard interval
generated
from
bootstrap
standard
errors
does
perform
as well
as the
6
standard error to the asymptotic standard error is not appropriate because even
if the two of them agree,
there can be a large difference in the corresponding
confidence intervals if the bootstrap distribution is sufficiently skewed.
Wald,
likelihood ratio,
and Lagrange multiplier tests are the most
commonly used hypothesis tests in limited dependent variable models. There are
two major problems with these tests.
The first one is the conflict between tests;
one test may reject the null hypothesis, but another may fail to reject the null. The
second problem is that the true significance level is frequently larger than the
nominal level. This means that the tests overreject the null hypothesis. These two
problems
arise
from
using
asymptotic
chisquare
critical
regions
as an
approximation.
If we
can
approximately
true
distributions
those
statistics,
which is the advantage of the bootstrap method,
then the problems
might be eliminated.
Because of the importance of bootstrap confidence intervals, the accuracy
of bootstrap hypothesis testing, and the goodness of bootstrap trimmed estimates
in limited dependent variable models, it is hoped that this work will motivate
practicing econometricians to consider the bootstrap methods in their analyses.
It is also hoped that this dissertation will provide an idea about how to choose a
suitable bootstrap method for making inferences in limited dependent variable
models. Finally, this dissertation provides comparisons between the parametric
bootstrap method, the semiparametric bootstrap methods, and the nonparametric
bootstrap method.
In this dissertation, an extension is made to the method of generating the
short
bootstrap
confidence interval. Also
methods
to create a
bootstrap
7
bootstrapping residuals, as applied to estimate the logit, probit, and tobit models,
some guidance is provided on the relative merits of the two approaches. As an
application
bootstrap
hypothesis
testing,
approximations
exact
distributions
of the
Wald,
likelihood ratio,
and Lagrange multiplier tests
provided through bootstrapping.
The structure of the dissertation is as follows: In chapter 2,
a quasipivotal
method
generating
bootstrap
confidence
intervals
is discussed
application
method
generating
bootstrap
confidence
intervals
presented.
With bootstrap confidence intervals, we suggest a new method to find
the bootstrap trimmed mean.
In chapter 3, a comparison between the parametric bootstrapping residuals
method and the nonparametric bootstrapping data method is studied by Monte
Carlo experiments for correctly specified logit and probit models.
chapter
investigation
conducted
through
Monte
Carlo
experiments to see the differences between the parametric bootstrap method, the
semiparametric
bootstrap
method,
nonparametric
bootstrap
method
under correctly specified tobit models.
In chapter 5,
an application of bootstrap methods to approximate the exact
distribution of the Wald, likelihood ratio, and Lagrange multiplier test statistics
for the logit,
probit,
and tobit models is discussed in detail, once again using
Monte
Carlo
experiments.
At the
same
time,
comparison
between
parametric bootstrapping residuals method and nonparametric bootstrapping data
method is studied.
The final chapter presents the summary and conclusions.
CHAPTER 2
A QUASIPIVOTAL METHOD FOR GENERATING
BOOTSTRAP CONFIDENCE INTERVALS
Introduction
Instead of getting a point estimator, the bootstrap method can be used to
give us an interval estimator. In addition to this,
we can also get an empirical
cumulative distribution function of the estimator. So by bootstrapping,
we can
gain
much
more
information
beyond
point
estimator
its standard
deviation.
Therefore a major goal in refining bootstrap methods is to generate
better bootstrap confidence intervals. The percentile method, which takes 5% off
each
when
we try to
form a
90% confidence interval,
introduced and
developed by Efron (1981, 1982, 1985,
1987),
gives bootstrap confidence intervals
for parameters of interest. Beran's B method
1987) gives more accurate bootstrap
confidence
takes
too much
computing
time,
requiring
1000x1000
bootstraps
one
confidence
interval.
more
convenient
method,
which
automatically corrects for bias, is the bootstrapt.
Hall (1992) has applied the bootstrap method to generate a short confidence
interval. He provided the asymptotic theory and concluded that the accuracy of the
coverage for the short bootstrap confidence interval is O(n2) and the accuracy of
its length is O,(n7t/2). In this chapter, we extend the method of generating the short
bootstrap confidence interval, called the quasipivotal method, to be a correction
confidence interval,
which will be presented in section 2.3.
We present Monte
Carlo experiments showing that the performance of the quasipivotal method is
better than that of the bootstrap confidence intervals generated by the other
methods.
The idea
behind
the quasipivotal method
comes
directly from
original idea of the pivotal quantity method,
which minimizes the length of the
bootstrap confidence interval given its confidence level.
Leger and Romano (1990) consider the problem of using the bootstrap to
adaptively choose the trimming proportion in an adaptive trimmed mean.
suggest a new method to find a bootstrap trimmed mean by trimming the sum of
squares of errors of a regression model.
The pivotal and the quasipivotal methods will be presented in section 2.2.
A real bootstrap confidence interval for a parameter will be given in section 2.3,
and comparisons among the different confidence intervals,
which are generated
different bootstrap
methods,
are discussed in section 2.4.
Finally,
we will
compare
the methods of generating bootstrap confidence intervals
to see the
differences between bootstrapping data and bootstrapping residuals in section 2.5.
2.2 The QuasiPivotal Method
Let Q(X,e) be some function of the random variable X and the parameter e
such that the distribution of Q does not depend on 8.
Then Q(X,0) is called a
pivotal quantity. To construct a confidence interval for z(0) at level 1
2a using the
pivotal quantity method, we need to find a pair of numbers q1 and q2, such that
P{q,
12ac
(2.1)
Thpe mlntitr OfY A1 rCan he nivntPd in thep snn that n.
10
Here, we have infinitely many choices of q, and q2, but we should select that pair
and q2
that will make T,(X) and T2(X) close together in some sense.
instance,
if T2(X)T,(X),
which is
the length
of the confidence interval,
is not
random, then we might select that pair of q, and q2 that minimizes T2(X)T1(X).
This is what we use in our Monte Carlo experiments.
Alternatively, if T2(X)T,(X)
is random, then we might select the pair of q1 and q, that minimizes the average
length of the interval. If Q(X,6) has a symmetric distribution, like the standard
normal distribution, then q2=q1, and the resulting interval is the same as Efron's
percentile interval.
Consider the random variable X having cumulative distribution function H
with mean p. Let
 13 P
SE(j3)
T(Z, )
(2.2)
which is assumed to be distributed with cumulative distribution function F and
is a function of the data and an unknown parameter.
is an estimator of p,
z{X 2;
 Xn}.
Let T(Z*,B)
be an estimator ofT(Z,[p). Then, by using the
bootstrap method, the empirical cumulative distribution function G of
an estimate of the cumulative distribution function F
of T(Z,[), and
T(Z*,3)
T(Z*,6)
actually a statistic.
Therefore, to give a short bootstrap confidence interval,
suggest the following method,
which we call a quasipivotal method. And this
method also could be used as a "correction method" for the percentile method,
Beran's B, the bootstrapt, and the real bootstrap confidence interval wherever the
percentile method is
applied.
To get a bootstrap confidence interval of T(Z,) at level 12a, assume that
of q,
P{qT(Zt,)q2} =
12a
(2.3)
We suggest finding q, and q2 by minimizing (q2 q) subject to G(q2)G(qj)=12a and
q2>ql.
Step 1
The detailed procedure is as follows:
: Draw a bootstrap sample zi" from
z of size n with replacement.
Step 2:
Use the bootstrap sample
(b) to get p"b)
, the standard error of j*(bJ
T(z* 3).
Step
Repeat
steps
to 2 B times
to get
an empirical
cumulative
distribution function G of the statistic
T(Z*,l)
for sufficiently large
Step 4:
Given q,, minimize qo, subject to
G(q2)
0(q1)
> 12a
to get
as a function of q,.
Step 5:
Minimize
(2(q1) q1)
with respect to q, to get
then
Step 6:
Step 7:
The short bootstrap confidence interval of T(Z, ) is
Report the short bootstrap confidence interval of (3 at level of 12a
to be
 q2SE().,
I q1SE(1)}
Note
that,
if T(z,3)
where
=
a linear
increasing
function,
bootstrapt as in equation (2.2),
then the length of the short bootstrap confidence
interval of p is also minimized.
Actually,
quasipivotal
method
always
gives
best
confidence
12
each tail. Because the empirical cumulative bootstrap distribution is usually not
symmetric, the percentile method does not perform well. Efron's biascorrected
percentile method (known as the BC method) corrects the bias of the percentile
method, and Efron's accelerated biascorrected percentile (BCa) method (Efron
1987) corrects its skewness.
We could show that, under the assumption of a single peak for the density
function
empirical
cumulative
distribution
function,
the quasipivotal
method contains the central densest part of the empirical cumulative bootstrap
distribution.
This follows from the first order condition, as will be shown below.
To construct a confidence interval at level 12a, we want to minimize q2ql subject
to
0(q2)
 G(q,)
= 12a
(2.5)
Let G and g be the cumulative distribution function and its density function (i.e.
g is the derivative function of the cumulative distribution function G). Let L be the
Lagrange function
L = q2q, 1+ 2aIG(q2)G{q,)l}
then
= l+g(q,) = 0
= 1g (q2)
thus
S Ilr
In r\
13
Many methods partially apply the percentile method to generate a bootstrap
confidence interval, for instance, the bootstrapt, Beran's B and the real bootstrap
confidence interval (which will be presented in next section).
Both the quasipivotal
method and the percentile method directly find two end points of a confidence
interval. As we have shown above the quasipivotal method is the best one among
those methods of generating bootstrap confidence intervals.
The quasipivotal method is also our suggested correction method. To take
one example,
we can correct the bootstrapt by using the quasipivotal method
instead of using the percentile method when generating a bootstrap confidence
interval. To get a bootstrap confidence interval ofT(Z,p) at level 12a, by using the
quasipivotal method, the short bootstrap confidence interval is given as (qq2)
where q, and q2 are found by following step
to step 4 described earlier.
When
using
percentile
method
, the
bootstrap
confidence
interval
is given
(G (a),G
1a ),
where
G is the
empirical
cumulative
bootstrap
distribution
function. Unless the empirical cumulative distribution function is symmetric, the
corrected (by the quasipivotal method) bootstrapt method is always better than
the bootstrapt method because it automatically corrects for skewness whereas the
bootstrapt does not. In the case of symmetry, these two methods are equivalent,
because of the equivalency of the quasipivotal method and the percentile method,
and so no correction is needed.
In this way, Beran's B method, the real bootstrap confidence interval and
even the percentile method itself could be corrected by the quasipivotal method
as long as the true distribution of T(Z,p) is asymmetric. In practice, especially for
small samples, bootstrap distributions are skewed.
Therefore the quasipivotal
2.3 Real Bootstrap Confidence Intervals
In inference theory, if we build a confidence interval, say (a,b), for the true
parameter p at level a,
we interpret it as meaning that with repeated sampling,
100(1a)% of
our confidence intervals would contain
the true parameter p.
would be very costly, and sometimes impossible,
to do the repeated sampling.
Therefore
analysts usually get a confidence interval
(a,b)
and say that with
100(1a)% confidence, the true parameter p fall in this interval. But things are
different if we
bootstrap,
since
we are
actually
doing repeated
sampling
resampling). So the bootstrap method may give us a closer to the exact (or true)
confidence interval than we usually have.
It can be argued that the percentile interval given by Efron (1981) is not
really a confidence interval, since it gives central 90% populations of the empirical
cumulative distribution function of bootstrap estimates of the true parameter p,
which also makes it hard to draw reasonable inferences about P based on just the
bootstrap distribution.
To have a more meaningful confidence interval, let us consider the following
bootstrap procedure:
Step 1:
From
bootstrap
sample
we can
estimate
quantity
T(X*r,p))
which is
(2.7)
SE([ *(b))
Step 2:
From
bootstrapping,
we can
obtain
empirical
cumulative
distribution
function,
called
T(X*(b),).
Then
bth
bootstrap sample, the real confidence interval for [ at level 12a is
~(X*cb,,B)
Step
Repeat steps 1 to 2 B times. Then the averaged bootstrap confidence
interval,
which is called the real bootstrap confidence interval, at
level 12a is
B bi
(2.9)
GIB ) SEbp*)
B b=1
corrected
quasipivotal
method)
bootstrap
confidence interval at level 12a would be
B b
(2.10)
b=1
The method we describe above, with the advantage of the bootstrap method,
gives us a real confidence interval1 for the true parameter p.
Comparisons of the Confidence Intervals
Consider
random
variables
* *, Xn,
generated
from
cumulative distribution function F with mean P3.
We are interested in getting a
confidence
interval
P by
using
bootstrap
methods.
course,
estimator
of B,
would
We can
use
this simple model,
to compare
methods of generating bootstrap confidence intervals.
To detect
bias and skewness of the different
bootstrap confidence
intervals, we need to generate data sets from skewed distributions with different
skewness.
We generate a random variable X to
form four data sets from
Gamma distribution with
P(X_) =0.560
Xr(r=5,1
=20);
from the ChiSquare
distribution with degree of freedom 1,
P(Xj3) =0.542
XT(r= 10,X= 10);
from
the exponential distribution with mean 2,
P(Xi) =0.530
Xr(r=20,X= 10);
from
standard
normal
distribution
with
PX3) =0.500
XN(O,
0.05).
For the Gamma distribution, the skewness depends inversely on
the parameter r (figure 2.1). These distributions have different shapes; we want to
examine if the quasipivotal method always gives the confidence interval closest
to the true.
Let (QL, QR) be a confidence interval.
We define two criteria
(2.11
QR X
XQ,L
RC/LC =
P(X Ts
(2.12)
P(QL
sTsX)
With respect to the sample mean, RL/LL is the ratio of the right length to the left
length.
The right (left) length means the distance between the sample mean and
the right (left) end of the bootstrap confidence interval. This criterion reflects the
shape due
to the bias of the empirical
cumulative distribution.
Thus,
if this
quantity is very
close
to the exact
value
then
method
corrects
bias
satisfactorily. Efron and Tibshirani (1986) used this RL/LL criterion to compare
bootstrap confidence intervals. RC/LC is the ratio of the right coverage to the left
coverage. The right (left) coverage means the probability that the true parameter
rP 11
I..1 .
4
I .1 
,. \a
... ... tin na~~mr rf ta an tmr nvr a 1, fi flI In~ n+ an n fl flf 4 ,* fl .4I I l**fr.f t4r.a t n
I 1
exact value
then
the method
corrects
the shape and skewness satisfactorily.
These two criteria would give us a good description of the shape of the empirical
cumulative
distribution.
third
criterion
use
in these
Monte
Carlo
experiments is the length of the bootstrap confidence interval.
Along with decreasing the true skewness of the underlying distribution of
to zero, we construct four tables by generating bootstrap confidence intervals
using the different methods.
Each
table gives
us the average results of 500
replications with 1000 bootstraps of sample size 20 for each replication. We have
nine different methods of constructing bootstrap confidence intervals. The exact
confidence interval (EXACT) is generated by the pivotal quantity method directly
from the true distribution of
The percentile method (PC),
the quasipivotal
method (QP),
the biascorrected
percentile method
(BC),
the accelerated bias
corrected percentile method (BCJ),
the real bootstrap confidence interval (RCI),
corrected (by the quasipivotal method) real bootstrap confidence interval (CRCI),
the bootstrapt method (BST) and the corrected (by the quasipivotal method)
bootstrapt
method
(CBST)
generate
bootstrap
confidence
intervals
from
empirical
cumulative
distribution
estimator
mean
With
replications,
we give the averaged bootstrap confidence intervals in tables 2.1
2.4. Of the three criterion columns in tables 2.1 through 2.4, we use the results
from the exact confidence interval as an index for comparisons.
As we can see from tables 2.1
through 2.4,
the quasipivotal corrected
methods, of PC, RCI and BST, which are labeled QP, CRCI and CBST in the tables,
provide better estimates of the exact confidence intervals of the mean Ip from the
true distribution than do the uncorrected methods. And the quasipivotal method
CBST in tables 2.1
to 2.3,
have small negative effects on RC/LC,
but greatly
improve RL/LL to yield an overall improvement over the uncorrected methods
shown in table 2.5. The corrected real bootstrap confidence interval becomes the
closest to the true when X is generated from a symmetric normal distribution
shown
table
(biascorrected
percentile
method)
(accelerated biascorrected percentile method), reported in tables 2.1 through 2.5,
do not perform as well as expected.
Adding three criterion columns for each method then dividing by three for
each sample distribution, we get the first four columns of table 2.5 corresponding
to tables 2.1
to 2.4.
The average values of these first four columns in table 2.5
form the last column in the table.
With decreasing skewness from column 1
column 4, all of the entries for each method are decreasing, which means that the
less the skewness, the closer are the results to the exact confidence intervals for
each
method.
When
underlying
distribution
is symmetric,
maximum
underestimate or overestimate is only about 2.4%. It is shown in the last column
of table 2.5 that the quasipivotal method gets the best results, and all of the
corrected methods perform better than the uncorrected methods.
Comparisons of the Bootstrap Methods
To compare the methods of generating bootstrap confidence intervals by
investigating the difference in effects from bootstrapping data and bootstrapping
residuals, we consider the following linear regression
, I P x1+ P2 u,
(2.13)

19
with sample size 40. The true parameter values are Po=0.4, Pji= 1.0, and p2= 1.0. We
estimate the model (2.13) with the nonparametric estimation method of ordinary
least square (OLS). Three different bootstrap methods are applied: one is Efron's
nonparametric
bootstrapping
data method;
the second
is the semiparametric
bootstrapping residuals method; and the third is the parametric bootstrapping
residuals method.
We first
generate
exogenous variables
from
U(2,2),
then
generate
errors. The procedure for generating the errors and estimating the parameters of
this linear model by Efron's nonparametric bootstrap method, which bootstraps
the data, is as follows:
Step 1:
Generate two exogenous variables from U(2,2) and errors (u,
from
the standard normal distribution, then get (y,
according to equation
(2.13) to create the sample
(Y,X)={(y1,x1)
Then get the
..,(yn)}
estimate
Step
of the true B by ordinary least square (OLS).
Bootstrap the sample (Y,X) in pairs by repeatedly randomly picking
n pairs of ((y1,x)) with replacement to form a new bootstrap sample
(Y,X*)
= {(yx)
S.. (yi~4J}.
Step
Estimate this linear regression model by OLS with this bootstrap
sample (Y,X') to get the bootstrap estimate I'.
Step 4:
Step 5:
Repeat step 2 to step 3 B=500 times.
Find the mean and bootstrap trimmed mean of estimates, mean sum
of squared residuals ofY (MSSR9),
bootstrap confidence interval, and
other criteria.
Step 6:
Repeat step 1 through step 5 M=500 times (this is the super loop) to
20
The procedure for generating y, and estimating the parameters of the linear
regression
model
semiparametric
bootstrapping
residuals
method
different only for step 2 to step 3 from the previous method:
Step 2:
Find the residuals
according to
(2.14)
= y,(Ij o +Xu+2x2,)
In each bootstrap replication, bootstrap
to get
Then get
according
to equation
(2.13)
to have
a new
bootstrap
sample
.~w~,.(J.3~
I f
111~
[r.x)=I
IY1
Step 3:
Estimate the model by OLS with this bootstrap sample (YX) to get
the bootstrap estimate Pi.
The procedure for generating the errors and estimating the parameters of
linear
regression
model
by the
parametric
bootstrap
method,
(this
bootstrapping residuals method),
is different only for step 2 to step 3 from the
method of bootstrapping data:
Step
Generate errors
from the normal distribution with estimated
variance, then get
according to equation (2.13) to have a new
bootstrap sample (Y,X).
Step
Estimate the model by OLS with this bootstrap sample (Y,X) to get
the bootstrap estimate I.
The purpose of this Monte Carlo study is to study the differences between
the two bootstrap methods, bootstrapping data and bootstrapping residuals.
serve
purpose,
we first
discuss
the bootstrap
trimmed
mean and
other
criteria.
symmetric trimmed mean by using the percentile confidence interval. i.e.
found mean of X's
they
only for those inside the percentile confidence interval of p,
which is the true mean of X.
Their atrimmed mean is defined as follows:
nInal
2[nal)
(2.15)
i=Inctl+1
where aE [0, ),
'] is the greatest integer function, and Xm), Xf2,...
statistics. This method will be applied to
(b)
{po(
Xn) are the order
separately as
i ;(b)}
'"Trim II" in our Monte Carlo experiment presented in table 2.6.
We also use a new method to find a bootstrap trimmed mean by trimming
the sum of squares of errors of a regression model.
For 500 bootstraps,
we have
500 estimates {fP'")}.
We put them back in one model with the data (Y,X) that is
formed from all 500 bootstrap samples. Then,
we apply the above trim II method
to trim
chosen.
(SSE*(b)}.
If a SSE'"m has been chosen, then the corresponding p'" will be
The last step is to find the means of those chosen P"'s.
This method is
called '"Trim I" in table 2.6 and table 2.7
The advantage of this trimming method
is that it keeps P"'s
in pairs, trims them off in pairs,
and averages them in pairs.
Since
errors are
generated from
normal
distribution
true
confidence interval for P should be symmetric.
That is to say
a correction for bias
or skewness is not needed.
Thus we consider only the percentile method for
generating bootstrap confidence intervals in this Monte Carlo experiment.
The fact that the true confidence interval for P is symmetric. Means that,
the true RC/LC= 1 and the true RL/LL= 1
We calculate the TRIM I RC/LC (RL/LL)
using the trim I mean (ref eqs.
(2.11) and (2.12)),
by taking the sum of squared
differences between the bootstrap RC/LC (RL/LL) and 1 over all 500 replications.
(n
22
Note that the TRIM I (or BOOT) RC/LC (or RL/LL) is different from RC/LC (or
RL/LL) in tables 2.1
to 2.5.
The coverage (that is,
how many times does
calculated confidence interval include the true coefficient) will be our first priority
criterion for a bootstrap confidence interval. The TRIM I
the second
or BOOT) RC/LC will be
, and the TRIM I (or BOOT) RL/LL will be the third. The smaller of the
last two criteria gives the better bootstrap method.
In table 2.6
, the "OLS" is the averaged estimates of IB over 500 replications.
Also in the table
the "BOOTSTRAP"
'TRIM I"
and "TRIM II"
are the bootstrap
mean, trimming I mean, and trimming II mean respectively over 500 replications.
When
comes
to point
estimates
, comparing
these
three
bootstrap
methods, we can see from table 2.6 that Efron's
nonparametric bootstrapping data
method
, the semiparametric bootstrapping residuals method,
and the parametric
bootstrap method are almost equivalent,
MSSR, to the true value.
with the first one having the closest
The bootstrap mean, trimming I mean, and trimming II
mean are all equivalent within each bootstrap method.
For obtaining the 90% confidence interval, comparing these three bootstrap
methods
we can
see from
table
2.7 that
the semiparametric
bootstrapping
residuals
method
does
not give
good
coverage
on fo.
Efron's
nonparametric
bootstrapping
data
method
gives
coverage
a large
sum
squared
differences.
The parametric bootstrap method gives a slightly low coverage, and
almost
smallest
sum
squared
differences.
Therefore
parametric
bootstrap method performs better than Efron's
nonparametric bootstrap method
in this Monte Carlo experiment.
Summary
The quasipivotal method suggested in
this paper gives the best result
theoretically
empirically
true
distribution
is skewed.
true
distribution is symmetric, theoretically, the quasipivotal method is equivalent to
the percentile method. So this bootstrap method performs satisfactorily for any
underlying true distribution.
Using this method as a correction method yields
results that are much better than those provided by the uncorrected method.
In the case of symmetry, the corrected real bootstrap confidence interval
gives the closest result to the true in our Monte Carlo experiments. In reality,
however,
most
distributions
are asymmetric,
especially
small
samples.
Therefore, bootstrap confidence intervals, which are most useful in small samples,
should be generated by either the quasipivotal method or a method corrected by
the quasipivotal method.
Under the correct specification,
the parametric bootstrapping residuals
method
generating
bootstrap
confidence
intervals
model
estimation
performs
satisfactorily
our
Monte
Carlo
experiments.
model
misspecified,
we would expect that Efron's nonparametric bootstrap method to
perform better and the parametric bootstrap method to perform worse. In the case
of symmetry, the bootstrap mean, the proposed trimmed I mean, and trimmed II
mean are all equivalent within each bootstrap method. Finally, for asymmetric
distributions, the trimmed means are expected to perform better.
Table 2.1:
X generated from r(r=0.25,.=1), P(Xj3)=0.560, sample size is 20
with 1000 times bootstrapping for 500 replications.
METHOD INTERVAL LENGTH RC/LC RL/LL
EXACT
0.075, 0.417
0.140, 0.457
0.128, 0.432
0.150, 0.476
0.161, 0.515
0.161
CRCI
BST
CBST
1.000
0.927
0.888
0.956
1.035
1.395
1.257
1.535
1.383
0.638
0.133, 0.563
0.150, 0.675
0.120, 0.593
1.000
1.331
1.232
1.414
1.544
1.340
1.509
1.331
1.498
1.000
1.238
1.014
1.458
1.876
2.640
1.731
2.643
1.732
Table
X, P(X 3)= 0.542,
degree of freedom,
generated from Chisquare distribution with
sample
is 20 with 1000 times bootstrapping
for 500 replications.
METHOD INTERVAL LENGTH RC/LC RL/LL
EXACT 0.488, 1.493 1.000 1.000 1.000
PC 0.526, 1.492 0.925 1.184 1.195
QP 0.529, 1.433 0.900 1.103 0.996
BC 0.590, 1.541 0.946 1.256 1.400
BCa 0.622, 1.628 1.001 1.357 1.747
RCI 0.614, 1.832 1.211 1.183 2.220
CRCI 0.539, 1.656 1.111 1.327 1.468
BST 0.590, 1.900 1.304 1.184 2.229
CBST 0.509, 1.710 1.195 1.328 1.475
Table 2
X, P(X3p)=0.530, generated fr
mean=2, sample size is 20 with
replications.
"om
exponential
distribution
with
1000 times bootstrapping for 500
METHOD INTERVAL LENGTH RC/LC RL/LL
EXACT 1.270, 2.716 1.000 1.000 1.000
PC 1.378, 2.771 0.963 1.123 1.136
QP 1.344, 2.710 0.945 1.073 1.004
BC 1.412, 2.824 0.977 1.170 1.276
BCa 1.452, 2.914 1.011 1.241 1.513
RCI 1.436, 3.063 1.125 1.126 1.708
CRCI 1.358, 2.904 1.069 1.226 1.292
BST 1.404, 3.125 1.190 1.123 1.709
CBST 1.321, 2.956 1.131 1.222 1.293
Table 2.4:
X generated from N(0,1), P(X<)=0,500, sample size is 20 with
1000 times bootstrapping for 500 replications.
METHOD INTERVAL LENGTH RC/LC RL/LL
EXACT
0.368, 0.368
0.357, 0.
QP
BC
BCa
RCI
CRCI
BST
CBST
0.351, 0.349
0.360, 0.350
0.362, 0.350
0.376, 0.367
0.369, 0.363
0.391, 0.381
0.383, 0.377
1.000
0.963
0.951
0.965
0.967
1.010
0.995
1.049
1.034
1.000
1.010
1.014
1.002
0.999
1.015
1.011
1.010
1.008
1.000
0.998
1.008
0.989
0.988
0.996
1.001
0.997
1.001
Table 2.5:
The average of the absolute differences from the exact results from
the columns of LENGTH, RC/LC and RL/LL in previous tables.
METHOD TABLE 1 TABLE 2 TABLE 3 TABLE 4 TOTAL
EXACT
PC
QP
BC
BCa
RCI
CRCI
BST
CBST
1.000
1.214
1.119
1.305
1.485
1.792
1.499
1.836
1.538
1.000
1.151
1.069
1.237
1.368
1.538
1.302
1.572
1.333
1.000
1.099
1.044
1.156
1.255
1.320
1.196
1.341
1.215
1.000
1.013
1.024
1.016
1.015
1.010
1.006
1.021
1.014
1.000
1.119
1.064
1.179
1.281
1.415
1.251
1.443
1.275
Table 2.6:
Estimates of the parameters and mean squared errors of the linear
regression model with B=500, N=40 and M=500.
METHOD Po P1 t2 MSSR,
TRUE 0.400 1.000 1.000 1.000
OLS 0.401 0.989 1.006 0.987
EFRON'S NONPARAMETRIC BOOTSTRAPPING OF DATA
BOOTSTRAP 0.401 0.989 1.005 0.988
TRIM I 0.401 0.989 1.005 0.988
TRIM II 0.401 0.989 1.005 0.988
SEMIPARAMETRIC BOOTSTRAPPING OF RESIDUALS
BOOTSTRAP 0.401 0.989 1.006 1.043
TRIM I 0.401 0.989 1.006 1.043
TRIM II 0.401 0.989 1.006 1.043
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
BOOTSTRAP 0.402 0.989 1.005 1.119
TRIM I 0.402 0.989 1.005 1.119
TRIM II 0.402 0.989 1.005 1.119
Table
Estimates of the parameters and mean squared errors of the linear
regression model with B=500, N=40 and M=500.
CRITERIA Po PI P2
EFRON'S NONPARAMETRIC BOOTSTRAPPING OF DATA
INTERVAL 0.141, 0.661 0.765, 1.214 0.776, 1.234
COVERAGE 0.892 0.876 0.876
TRIM I RC/LC 2.194 2.099 2.198
TRIM I RL/LL 3.664 5.594 5.417
BOOT RC/LC 1.949 2.472 2.838
BOOT RL/LL 2.217 3.270 3.089
SEMIPARAMETRIC BOOTSTRAPPING OF RESIDUALS
INTERVAL 0.143, 0.660 0.764, 1.214 0.779, 1.232
COVERAGE 0.604 0.900 0.896
TRIM I RC/LC 2.165 1.887 1.873
TRIM I RL/LL 3.432 2.795 2.661
BOOT RC/LC 2.118 1.933 1.879
BOOT RL/LL 2.163 2.036 1.971
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
INTERVAL 0.140, 0.664 0.760, 1.218 0.776, 1.236
COVERAGE 0.876 0.874 0.878
TRIM I RC/LC 1.901 1.966 2.296
TRIM I RL/LL 2.688 2.678 2.740
BOOT RC/LC 1.712 1.777 2.168
BOOT RL/LL 2.139 2.056 2.166
distl:
xr(r=5,X=20)
and
P(Xp1) =o
.560
with mean=0.25.
dist2:
dist3:
xr(r=io,x
xP (r=20o,A
=10)
=10)
and
and
P(X B) =0
) =0
.542
.530
with mean=1.
with mean=2.
Figure 2.1
Three gamma distributions of ~.
CHAPTER 3
BOOTSTRAP METHODS IN BINARY
RESPONSE VARIABLE MODELS
Introduction
During recent
years
bootstrap
method
been
applied
many
econometric applications. A survey paper by Jeong and Maddala (1992), however,
points out that there are several questions left in the area that need to be studied
further.
This chapter discusses the differences between bootstrapping data and
bootstrapping residuals in binary response variable models oflogit and probit. For
models with limited dependent variables, simple bootstrap methods fail to keep the
censoring properties of the model. There are different ways to modify the bootstrap
method to avoid the flaw. For example, Efron (1981) proposed a bootstrap method
for censored data which keeps the properties of censoring. For binary response
variable models, we compare four modifications in this chapter.
The first one is bootstrapping data,
which is a nonparametric bootstrap
method.
Teebagy
Chatterj ee
1989)
apply
Efron's
bootstrapping
data
procedure (Efron 1981) to the logistic regression model.
The Monte Carlo study
they conducted shows that the results are satisfactory. The rest of the methods
depend on bootstrapping residuals. Adkins (1990) estimated bootstrap standard
errors using a parametric bootstrapping residuals method in a probit model, got
unstable results, and argued that the bootstrap method is not superior to MLE for
33
method by generating errors from the underlying distribution for each bootstrap
iteration. The last method we consider, which turns out to be unsuccessful, is to
bootstrap generalized residuals to estimate the binary response variable models.
The concept of generalized residuals deserves some explanation. Since the
first definition of the generalized residual by Cox and Snell (1968), the generalized
residual has been applied to several limited dependent variable models. Lancaster
(1985ab) defined generalized residuals and featured them in diagnostic statistics
to detect omitted
covariates
neglected
heterogeneity in
duration models.
Chesher and Irish (1987) applied graphical and numerical analysis of residuals to
censored data. Gourieroux et al. (1987a) proposed a new definition of generalized
residuals
that
can
used
a fairly
general
context,
especially
in limited
dependent
variable
models.
In this
chapter,
we apply
Gourieroux
et al.
definition of the generalized residual
to binary response variable models.
generalized residual bootstrap method is a semiparametric method, and the model
is estimated by solving a nonlinear equation system instead of using the maximum
likelihood estimation method.
Bootstrap Methods in Binary Response Variable Models
As we know,
logit and
probit models are approximately equivalent
(Maddala 1983,
p23), and estimates of one model can be transformed into the
estimates of the other. This means that if we use two different methods to estimate
these two types of models, we should have comparable results. This implies that
conclusions
models
conflict,
then
at least
one
method
questionable.
This
kind of
difference between Teebagy
& Chatterjee's method
34
We should note that the bootstrap method is useful only in small samples.
In large samples,
the bootstrap method does not give more accurate estimates
while taking a lot more computer time than the asymptotic method.
To bootstrap
data,
that
, to apply
nonparametric
bootstrap
method
, we first generate the data from a binary response variable model:
(3.1)
otherwise
We will
then have the data
cy,xI
Next,
from
these data
randomly
draw
with
replacement
create
new
bootstrap
sample
(Y*,X*)= {(y ,x),
Finally, we apply the maximum likelihood estimation
method to the bootstrap sample (YV,X) to get an estimate of the parameter. This
procedure is repeated B times (number of bootstraps) to either form a bootstrap
confidence interval of the parameter or to get a bootstrap point estimate of the
parameter by averaging these B estimates.
Teebagy and Chatterjee (1989) apply
this modified bootstrap method to a logit model in a Monte Carlo experiment. They
concluded that the bootstrap estimator consistently overestimates the true value
of the standard errors while the asymptotic estimator using the Fisher information
matrix consistently underestimates them. They also argued that in small samples
the bootstrap standard errors are substantially closer to the true values than are
the asymptotic standard errors.
To bootstrap
residuals,
we have
three
methods
to discuss:
parametric
bootstrap
method; Adkins
bootstrap
method;
generalized
residual
method
(this method will be discussed in
the next section).
We first
x:P+ u,
r(Yn'~n))
error
' from the underlying cumulative distribution function, call it F, to get the
new dependent variable
I,3P u'$1
(3.2)
otherwise
Then
we can
apply
maximum
likelihood
estimation
method
to the
new
bootstrap
sample
cy*(b)x)
to get
an estimate
parameter
This
procedure is repeated B times.
Adkins'
bootstrap method,
we make a
different modification of the
bootstrap method. Instead of generating errors from the cumulative distribution
function F, we generate
maximum likelihood estima
of (Y,X).
from the uniform distribution of (0, 1). Also we use the
te ( of the true parameter [3 from the original sample
Then for each bootstrap replication, a binary dependent variable
given as
[0,F(x )]
(3.3)
e*Eb) ({F(X1 ), 11
where F is the underlying cumulative distribution function of error u in model
(3.1).
Thus we have a bootstrap sample (Y",X), from which we can estimate the
parameter of interest. Adkins (1990) applied this method to a probit model and
concluded that this bootstrap method is not superior to MLE and gives unstable
estimates, making it inappropriate for the probit model.
In model (3.2),
the errors {u)} are directly generated from the underlying
function of u1 is the standard normal,
we have the probit model. Actually,
parametric
bootstrap method
and Adkins'
bootstrap
method
are theoretically
equivalent. Let us consider the probability of Y,=1 from model (3.1)
P(Y= 1)=P(u, > ~P)
= 1F(x: p)
= F^B )
which
leads
to the
generation
* from
model
(3.3)
Adkins'
bootstrap
method.
Even though the two methods are equivalent, there is an important reason
to consider them as two different methods. In particular, while they may be almost
equivalent numerically, the parametric bootstrap method is more widely applicable
than Adkins'
bootstrap method, since the latter is restricted to binary response
variable models. The parametric bootstrap method, though, can be applied to tobit
models as well.
For simplicity,
the following discussion about the parametric
bootstrap method also applies to Adkins' bootstrap method.
Returning to the differences between bootstrapping data and bootstrapping
residuals, if we have a sample (Y,X) from the binary response variable model of
equation (3.1
then we can choose between two methods to estimate the model.
The first one, bootstrapping data, a nonparametric method, is to directly estimate
the model by using the maximum likelihood estimation method with a bootstrap
sample (Y*,X'). The second method, bootstrapping residuals, a parametric method,
is to generate the error term from the underlying cumulative distribution function,
and get a new
* according to equation (3.2) by using the error term
uL. the
I I
Suppose
model
is correctly
specified,
in the
sense
that
error
distribution is what we assume. In the first nonparametric method, we only have
a partial observation from the true distribution, which may not be representative
of the population, especially in small samples. This sample distribution might be
skewed
, and the estimates of the parameters might be biased because we are
resampling
generate y
from
sample
according to
data.
For the second
a correctly specified
error
parametric
method,
distribution,
if we
unusual
features of the sample will be somewhat mitigated. The parametric method should,
therefore, outperform the nonparametric method when the model specification is
correct.
When the model is misspecified, however, the nonparametric method might
work
better,
because
parametric
method
is based
on a
wrong
error
distribution.
That is,
the parametric method is not only inappropriate for the
model, but also uses a possibly misspecified model to change the observed data
from Y to Y'
Since a new data set for the variable Y* is generated according to the
misspecified model for every bootstrap replication,
the estimates of the model
would be pulled away from
the true parameter in
the same direction in each
replication. Hence, the biases would accumulate throughout the replications.
We should note that the maximum likelihood estimation method we use is
a parametric estimation method, which is sensitive to the assumption of the model
specification. Hence,
only the correctly specified model will be studied in this
chapter.
3.3 Generalized Residuals
38
hand, nonetheless it remains instructive to explore this approach. The generalized
residual method is used to estimate a regression model. Think of the residuals as
the estimates of the errors of a model. Consider a linear regression model
I3P'x,
i=1,2,
(3.4)
where E, is the error of the model. Assume
then
is an estimate of the parameter (,
x,
is the residual for the ith observation. In the case of a nonlinear model,
i=1,2,
S.., nj
(3.6)
where
errors
are independently
identically
distributed,
equation for the ith observation has a unique solution for s,,
h,(y,,x,43).
(3.7)
This defines
replace f3 by a
the generalized error for the ith observation of the model.
n
estimate
If we
we have the generalized residual defined in the sense
of Cox and Snell (1968),
=hy1x;3
i=1,2,
S.., (3.8)
If the data are censored, as in the logit model, we can not use this definition
of the
generalized
residual for
every
observation
since
it depends
on the
unobservable variable
7 In other words, in a censored regression model,
difficult to find the error terms directly. It seems natural to replace the errors {es,()}
by their best prediction
{Ep[e,(p) yIl}
This leads us to the following Gourieroux
"'t n
Si (Xr) P1EI)
E~B[c(P
(3.9)
and the generalized residual for the ith observation is
(3.10
where
is the ML or any other consistent estimate of p.
To use the generalized residual bootstrap method to estimate a model, we
assume that the dependent variables follow an exponential distribution, i.e. the
p.d.f. (probability density function) can be written as
(3.11)
Then the loglikelihood function of the latent model is
3.12)
L (; y x)=
1=(, )~l A~,P+ xy)
and the normal equations, as proved by Gourieroux et al. (1987a), can be written
as
dLcBtYI4x
where
Ti,
d13 VS
is a generalized residual for the ith observation and
(3.13)
is an estimate
of the parameter p.
The p.d.f. of a dichotomous logit model is
exp(f3'x,)
i+ exp(3' x,
1x 1
1 exp(p'x)
eXPto IxY
 log[1 + exp(3'x )l1
(3.14)
s,(=
yy, I x,;p)= exp[Q' (x,;p) Tty,) +A(r,;P)+B(x,, y,))
I(ylx,;P)
(3.15)
1+exp(j' x,)
which is the difference between
the observed y, and its expectation.
With this
bilinear exponential density function (3.13),
the normal equations from (3.10) and
(3.12) are
(3.16)
While solving this normal equation, we will have exactly the same solution as the
maximum likelihood estimate for both the logit and probit models. If we bootstrap
generalized
residuals
we obtain
bootstrap
generalized
residuals
Then the new bootstrap normal equations are
,qtt
Since we need to solve
(3.17)
the actual nonlinear equation system is
1 + exp(P'x,.)
= 0 (3.18)
where the i'
only represents the position, and after bootstrapping, the
is no
longer the same as x,
When we used Newton's iterative methods to solve this nonlinear equation
system to obtain the estimate of p,
the estimates did not get convergence.
major reason for this might be that the bootstrapping causes xj not to be matched
with
So that the
is not necessarily matched with
As a result, we
were unable to use the generalized residuals method.
It;)
3.4 Monte Carlo Experiments for Bootstrap Methods
We generate the data from the following binary response variable model
(3.19)
othenvise
to compare the different bootstrap methods. Four methods are discussed: the logit
maximum likelihood estimation method, Efron's nonparametric bootstrap method,
the parametric bootstrap method, and Adkins' parametric bootstrap method. Since
the bootstrap generalized residual method did not converge, we can not present
the results of this method. The model we describe has two continuous exogenous
variables and an intercept. The true parameter values are Po=0.4, Pi= 1, and ,2= 1.
About 40 percent of the observations of Y are censored in the experiment. Both of
the continuous variables are drawn from the uniform distribution over the range
(2,2).
The sample size is 40.
We first generate two exogenous variables from U(2,2).
We then generate
the errors. The procedure for generating the errors and estimating the parameters
logit
model
Efron's
nonparametric
bootstrap
method,
which is
bootstrapping data method, is as follows:
Step 1:
Generate two exogenous variables from U(2,2) and errors
from
the logistic distribution, then get {yj} according to equation
3.19) to
have the sample
Step 2:
(Y,X)={(y ,x,),
., (yx)}.
Bootstrap the sample (Y,X) in pairs by repeatedly randomly picking
n pairs of {(y,xj)} with replacement to form a new bootstrap sample
(Y*, X'={ (y;,x~)
**
9~*
(Yn,'4d}
Po +P1X1I+P2XL,+ UL> 0
estimate pl
Step 4:
Repeat step 2 to step 3 B=100 times.
Step 5:
Find the mean of estimates, E(Y),
and sum of squared differences
between y, and its prediction.
Step 6:
Repeat step 1 through step 5 M= 1000 times (this is the super loop)
to obtain the averages of the bootstrap estimates, biases, and their
mean squared errors.
The procedure for generating the errors and estimating the parameters of
the logit model by the parametric bootstrap method,
residuals method, is different only for step 2 to step
which is the bootstrapping
3 from the previous method:
Step
Generate errors
from the logistic distribution, then get
according to equation (3.19) to have a new bootstrap sample (Y',X).
Step 3:
Estimate
logit model
by the maximum likelihood
estimation
method
with
bootstrap
sample
(Ytx)
to get
bootstrap
estimate p'.
The procedure for generating y, and estimating the parameters of the logit
model by Adkins' parametric bootstrap residuals method is different only for step
2 to step 3 from the method of bootstrapping data:
Step
Generate
from uniform (0,1), then get
have a new bootstrap sample
Step 3:
Estimate
logit model
(Y,X)={ (y;,x).
by the maximum
according to (3.3) to
". (y,xn) .
likelihood
estimation
method
with
bootstrap
sample
(r,X)
to get
bootstrap
estimate p'.
The purpose of this Monte Carlo experiment is to study the differences
43
To compare the point estimates, we will first check their biases and their
mean squared errors of J, using the fact that
1A1~
Al I
1M 
13P*FU3*(P
(3.20)
MS~EI5
RSSI ]
BIsvrJ
for each individual W*
. Since the three estimates of j's may behave differently, we
might need two kinds of overall criteria for the estimation.
The first set is the
expectation of Y and its bias.
E(Y)J
= P(Y,= 1)
= 1F[(P + Px,, +2 Ix)
For the logit model it is given by
E(11)zN
(3.21)
1 +exp[(p +px
+ P2,)l
For the probit model it is given by
iN (+
N fr
where a is
(3.22)
the cumulative distribution function of the standard normal.
approximately true value of E(Y) is estimated by using true values of p's instead
of P"'s
in equations (3.21) and (3.22).
The second comparison uses the sum of squared differences between y, and
its prediction, denoted as sum of squared residuals of Y (SSRy
We also consider
 a. a'. ww I fl fl Ii
= PIu,
(Po + Plx,,+ Pzxz,)]
E( Y)
1
In n rr r
11
N
Z=: (u, 9;12
i=I
(3.23)
1 + exp[ 3p +Px,+P2)U1
(logit model)
(3.24)
SSGR,
y0( + 3;x+p )]
(probit
model)
(3.25)
where the prediction of Y is
*~+P~,, ~
(3.26)
otherwise
As we know,
the parametric
bootstrap method is equivalent to Adkins'
bootstrap
method
theoretically,
so they should
be equivalent in Monte Carlo
experiment.
through 3
This turns out to be true, as can be seen by examining tables 3.1
There is no significant difference in any criterion between these two
methods. Keep in mind, however, that the parametric bootstrap method is more
general.
Comparing the differences between bootstrapping data and bootstrapping
residuals,
we can see from table 3.1 and table 3.2 that the results from Efron's
nonparametric bootstrapping data method are very close to the results from the
method of parametric bootstrapping residuals. For the logit model, they are very
close on biases, and the MSEp of the nonparametric bootstrap method is only
about 5% less than on MSEB of the parametric bootstrap method. For the probit
model, the nonparametric bootstrap method gives lower values for both biases and
SSR,
SSGR,
significantly
higher
Efron's
nonparametric
bootstrap
method
than
parametric bootstrap method for both the logit and probit models. This suggests
that the parametric bootstrap method is more reliable than the method of directly
bootstrapping data. Again, we should note that we are dealing with the parametric
estimation method
i.e. the maximum likelihood estimation method.
Comparing the maximum likelihood
estimation (MLE)
method with
parametric bootstrap method, we can see that the estimates of the parameters are
significantly
biased
have a
smaller
MSE,
for the
MLE
than for
parametric bootstrap method in both the logit and probit models (table 3.1 and
table 3.2). But the latter gives lower SSRy
about 10% (table 3.3).
s for both the logit and probit models by
Since the overall expectations E(Y) are both the same, the
parametric bootstrap method might be better because of the smaller confidence
bounds and lower variance when the specification is correct.
Comparing the estimation between the logit model and the probit model, we
can see from tables 3.1 through 3.3 that there are greater biases, greater RSSp, as
well as greater MSE; for the estimates of the probit model than for those of the
logit model,
which has greater error variance at the time of generation. For the
logit model,
the RSS0's
for the three different B's are close.
But for the probit
model, the RSSB of Po is smaller than that from the logit model. However the RSSp's
of pi
and P2 are greater than those from the logit model. Overall, the estimates of
the probit model have greater variance about the true parameters, but have a
lower
SSR,
than
logit
model
(table
3.3).
In addition,
iterations
estimating the probit model converged much slower than those of the logit model.
Regardless of the models or the methods, all have excellent estimations of
From table 3.3,
we can see that the sum of squared generalized residuals
(SSGRy) and sum of squared residuals (SSRy) seem to have the same power as
criteria. Even though they have different values, they have the same pattern of
variation for both the logit and probit models.
This implies that the generalized
residuals represent the residuals well.
In the case of misspecification,
a parametric estimation method,
such as
the maximum
likelihood
estimation
method,
would
be sensitive.
Therefore
nonparametric
semiparametric)
bootstrap
method
with
a nonparametric
estimation method might outperform the parametric bootstrap method.
Summary
In a
correctly
specified
model,
with
an efficient
parametric
estimation
method, the parametric bootstrap estimation method gives better results than the
nonparametric bootstrapping data method.
The parametric bootstrap method,
which is more general, is equivalent to Adkins' bootstrap method in these binary
response variable models. The parametric bootstrap method gives smaller variance
of the prediction and
greater mean squared
errors of the estimates
than
maximum likelihood estimation method.
In a
misspecified
model,
we need first
to find
an efficient and robust
estimation method, then according to the parametric property of this estimation
method to choose an appropriate bootstrap method. Probably the nonparametric
bootstrap method would be an appropriate method.
For a correctly specified logit model, the parametric bootstrap method with
logit
maximum
likelihood
estimation
method
provides
most
reliable
47
estimation method gives the most reliable estimates among the other bootstrap
estimates.
Using the maximum likelihood estimation method, the estimates of the logit
model are more reliable,
have less variance,
and faster convergence
than
estimates of the probit model.
Because of bootstrapping the nonlinear equation system, we were not able
to apply the bootstrap method to the generalized residual estimation method.
Table 3.1: Correctly specified logit model as B= 100, N=40 and M=1000.
BETA TRUE MEAN BIAS RSS, MSE,
LOGIT MAXIMUM LIKELIHOOD ESTIMATION
Po 0.400 0.501 0.010 0.386 0.396
Pi 1.000 1.188 0.035 0.348 0.383
P2 1.000 1.204 0.042 0.383 0.424
E(Y) 0.544 0.544 0.000  
EFRON'S NONPARAMETRIC BOOTSTRAPPING OF DATA
Po 0.400 0.635 0.055 0.658 0.713
iP 1.000 1.473 0.223 0.642 0.866
P2 1.000 1.495 0.245 0.697 0.942
E(Y)' 0.544 0.544 0.000  
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
Po 0.400 0.631 0.053 0.687 0.740
i 1.000 1.473 0.224 0.702 0.926
P2 1.000 1.500 0.250 0.749 0.999
E(Y)' 0.544 0.544 0.000  
ADKINS' PARAMETRIC BOOTSTRAPPING OF RESIDUALS
Po 0.400 0.635 0.055 0.686 0.741
P1 1.000 1.473 0.224 0.681 0.905
P2 1.000 1.499 0.249 0.769 1.018
E(Y)' 0.544 0.544 0.000  
* the value for E(Y) is approximate.
Table
Correctly specified probit model
as B=100, N=40 and M=500.
BETA TRUE MEAN BIAS RSS, MSE,
PROBIT MAXIMUM LIKELIHOOD ESTIMATION
Po 0.400 0.484 0.007 0.153 0.160
P1 1.000 1.268 0.072 0.510 0.582
P2 1.000 1.254 0.064 0.535 0.599
E(Y)' 0.636 0.638 0.000  
EFRON'S NONPARAMETRIC BOOTSTRAPPING OF DATA
Po 0.400 0.613 0.045 0.310 0.356
P1 1.000 1.665 0.442 0.856 1.298
P2 1.000 1.631 0.398 0.885 1.284
E(Y)' 0.636 0.637 0.000  
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
Po 0.400 0.644 0.059 0.353 0.413
P1 1.000 1.763 0.583 1.188 1.771
P2 1.000 1.736 0.542 1.197 1.738
E(Y)" 0.636 0.637 0.000  
ADKINS' PARAMETRIC BOOTSTRAPPING OF RESIDUALS
Po 0.400 0.631 0.054 0.341 0.395
P, 1.000 1.767 0.588 1.255 1.843
P2 1.000 1.738 0.544 1.232 1.776
E(Y)' 0.636 0.637 0.000  
* the value for E(Y) is approximate.
Table
Comparison of criteria among different methods.
METHOD SSR, SSGRy BIAS OF E(Y)
LOGIT MODEL ESTIMATION (M= 1000)
LOGIT MLE 8.691 5.877 0.000
EFRON 19.038 13.909 0.000
PARAMETRIC 7.834 5.337 0.000
ADKINS 7.846 5.338 0.000
PROBIT MODEL ESTIMATION (M=500)
PROBIT MLE 6.322 4.296 0.000
EFRON 17.642 14.195 0.000
PARAMETRIC 5.598 3.787 0.000
ADKINS 5.613 3.800 0.000
CHAPTER 4
BOOTSTRAP METHODS IN THE TOBIT MODEL
Introduction
Many of the recent developments in econometric methods have been in the
area of limited dependent variable models, that is, regression models where the
range of the dependent variable is restricted to some subset of the real line. The
regression model with a nonnegative constraint on the dependent variable,
called tobit model, was proposed by Tobin (1958).
the so
The strong consistency and the
asymptotic normality of the maximum likelihood estimator of the tobit model were
proved by Amemiya (1973). And it was shown by Olsen (1978) that if the iterative
process of the maximum likelihood estimation (MLE) yields a solution, it will be
the global maximum of the likelihood function; i.e., with the tobit MLE method,
given
initial
value,
converges,
then
estimator will
only
consistent and asymptotically normal maximum likelihood estimator.
However, it is well known that the tobit ML estimator is sensitive to the
assumptions
normality
homoskedasticity.
presence
either
nonnormality or heteroskedasticity can result in inconsistency of the maximum
likelihood
estimator.
There
are several
papers
discussing
sensitivity
nonnormality (Arabmazar & Schmidt 1982, Goldberger 1983) and the sensitivity
to heteroskedasticity (Arabmazar
& Schmidt
1981,
Hurd
1979) of the model.
Pnwell fl 9841 nrnnnped a n alternative tn the mnxiriim likelihnndl estimatnr which
52
is a generalization of the least absolute deviations estimation for a standard linear
model, that is robust to heteroskedasticity. Later on, he (Powell 1986) proposed
a symmetrically censored least squares estimator. Both estimators have certain
robustness properties, but can be very inefficient under the correct specification
for they
disregard
the information
contained in
parametric assumptions.
Peracchi
1990) introduced a class of boundedinfluence estimators for the tobit
model. These estimators provide a compromise between efficiency and robustness,
thereby attaining high efficiency in the tobit model and being robust in probability
distribution.
Efron (1981) applied a nonparametric bootstrap method to censored data
to keep the property of censoring by bootstrapping data directly.
Flood (1985)
introduced an augmented semiparametric bootstrap method to obtain standard
errors of system tobit coefficients,
but this method does not retain the property of
censoring of the data.
In this chapter, we investigate the differences between bootstrapping data
and bootstrapping residuals in the tobit model.
To this end,
we also propose a
mixed, semiparametric, bootstrap method based on the tobit MLE; and we apply
the balanced resampling technique to Efron's nonparametric bootstrap method to
estimate the tobit model.
4.2 Applications of Bootstrap Methods to the Tobit Model
There are
many ways
to get
estimators
tobit model
under
assumptions of the model. For instance, the probit maximum likelihood estimator
(Amemiya 1978) is consistent; Heckman's twostep estimation (Heckman 1976) is
53
converges. In addition, weighted least squares, nonlinear least squares, nonlinear
weighted least squares,
EM algorithm (Hartley
1958),
censored least absolute
deviations
estimation
(Powell
1984),
symmetrically
censored
least
squares
estimation (Powell 1986) and many more techniques can be used.
With
correct
specification,
because
its strong
consistency
uniqueness,
the tobit maximum likelihood estimator is more efficient and also
easy to obtain.
Combining the tobit MLE with a bootstrap method,
we can get
several bootstrap estimators. The purpose of this chapter is to compare bootstrap
methods
comparing
bias
reduction
mean
squares
error
(MSE,)
reduction of the different bootstrap estimates, and also the relative sum of squared
residuals of Y (SSRy).
The tobit model is
P x +P2 XI +I2X2t+
RHS,
(4.1)
otherwise
where the u, are independently and identically distributed normal with mean zero
and variance o2. Let xj=(1 x,, x,,) and P=(Po 1 Pt2) be column vectors. We can derive
the augmented bootstrap procedure as the following: First, estimate the model by
tobit MLE
compute
where
is the
vector
residuals
for the
observations
for which
the y,'s
are positive.
Second,
an augmented
residual
vector
is constructed where
. fi4
a4J].
If the total sample size is N and the
are positive
r observations,
then
vector
order
Third,
u is resampled with replacement to create a bootstrap sample u
of size N.
Fourth,
is constructed
using
according to
Ip
y1 =max(RHSt,0)
from
Through this semiparametric bootstrapping procedure,
we can see that, first,
if we want the bootstrap sample to maintain the censoring property, then (y,x)
should
paired
as in Efron
(1981).
This,
however,
is not
case
in the
augmented bootstrap method proposed by Flood (1985).
errors have been forced to be symmetric,
Therefore,
Second, the augmented
which may not be the case in reality.
we will propose two more bootstrap methods besides the augmented
semiparametric bootstrap method,
Efron's nonparametric bootstrap method, and
the parametric bootstrap method.
The first method, which is a nonparametric bootstrap method, applies the
balanced resampling technique to
Efron's nonparametric bootstrap method to
reduce the bias in Efron's bootstrap estimates. To bootstrap B times, we copy the
original sample
B times to make a group with BxN
pairs
of (y,x),
then randomly
draw without replacement
to form B
bootstrap
samples of size N. For each bootstrap sample, we can use the tobit MLE to get the
balanced bootstrap estimates.
second
method,
which
is semiparametric,
mixes
augmented
bootstrap with Efron's nonparametric bootstrap method. For residuals
have noncensored positive {y,}, thus we keep corresponding pairs of
{774*},
(x0ti;).
for residuals
{4
we do not know if the corresponding observation is censored
or not.
We may
choose
the same kinds of xt
to pair with
them
by randomly
drawing x; with replacement from the entire set of x.
Then we get the new pair
Finally we can form the augmented sample as
(xA,uA) = (,x )
(4.2)
55
So with Efron's bootstrap or balanced bootstrap methods, we avoid forcing
the errors to be symmetric and we maintain the censoring property.
mixed augmented bootstrap method,
With the
we force the errors to be symmetric and
partially maintain the censoring property.
To estimate the model by the tobit MLE, we use Fair's iteration method (Fair
1977). Let
api/a
le /2dt
(4.3)
1 le 'xWl1
(1/21t
(4.4)
For model (4.1),
we have the loglikelihood function
logL=~ log(1 0c + log(
0 1
where
2i ra2
E(y 1 3x,)
1 20o p
the summation Eo is over the No observations
for which y,=0, and
summation E, is over the N,
observations for which y>0. From the firstorder
condition for a maximum, we have (see Maddala 1983, p152153)
N1
(4.6)
 a(x,'x,j Xo o
(4.7)
where 1is is
the least squares
estimator for (3 obtained from
nonzero
observations on y.
matrix of values of xj for y1=0.
/ is a 3xN1 matrix of values of x, for nonzero y,.
To'(7 N, 1
/ is a 3xNo
is a 1xNo vector of values of y, for
(4.5)
0(2 =
fXLS
(4.8)
1D,.
Then Fair's iteration method for obtaining the maximum likelihood estimates of
B and
from equations (4.6) and (4.7) can be processed with X=0.4 (see Maddala
1983, p154).
For the bootstrap estimation of the model, the mean of the squared errors
of pi (MSE,) can be partitioned into two terms:
M
M(m Bi
M I
+ (IB3)2
(4.9)
where
is the average bootstrap estimate of the true parameter P in the mth
replication, and
is the average of
{53*(W)}
over M replications. The first term in
right hand side of equation (4.9) is the residual sum of squares of P3 (RSSp), and
the second term is the bias. For a given data set, the better method of estimation
should have a lower level of bias and/or lower level of MSEp.
Monte Carlo Experiments for Bootstrap Methods
We generate the data from the tobit model of equation (4.1) to compare the
different
bootstrap
methods
their
estimates.
Five
bootstrap
methods
applied: Efron's nonparametric bootstrapping data method; Efron's nonparametric
bootstrap
method
modified
balanced
resampling
technique;
Flood's
augmented
semiparametric
bootstrap
method;
mixed
augmented
semiparametric bootstrap method; and the parametric bootstrapping residuals
method.
We estimate three models.
The first model has two continuous exogenous variables and an intercept.
are
57
drawn from the uniform distribution over the range (0,5). The sample size is N=40.
The purpose of the first Monte Carlo experiment is
to see how the proposed
bootstrap methods work.
The detailed procedure for estimating the first tobit model by the first four
bootstrap methods mentioned above is as follows:
Step 1:
Generate a random sample of data according to
(4.1) with error
terms distributed from the standard normal.
Step 2:
Obtain estimates of the parameters of the tobit model using the tobit
MLE.
Step 3a:
Bootstrap the sample using Efron's nonparametric bootstrap method
to get B=100 bootstrap samples, then estimate the model with the
tobit MLE for each bootstrap sample. Finally, find the mean of the
bootstrap estimates.
Step 3b:
Bootstrap the sample with the balanced bootstrap method to get
B= 100 bootstrap samples. Estimate the model by the tobit MLE with
each bootstrap sample, and then find the mean of the bootstrap
estimates.
Step 3c:
Bootstrap the sample with the augmented bootstrap method to get
B=100 bootstrap samples. Next estimate the model using the tobit
MLE for each
bootstrap sample,
then find
the mean
of the
bootstrap estimates.
Step 3d:
Bootstrap the sample with the mixedaugmented bootstrap method
to get B=100 bootstrap samples; next estimate the model using the
tobit MLE for each bootstrap samples,
then find the mean of the
Step 4:
Repeat step 1 through step 3 M= 100 times (this is the super loop) to
obtain the averages of the estimates and the residual sum of squares
(RSSp) for each bootstrap method and the tobit maximum likelihood
estimates.
Comparing
those
four
bootstrap
methods
in table
balanced
resampling method
is almost
equivalent
to Efron's
nonparametric
bootstrap
method, but with a significant increase in computer time. The mixed augmented
bootstrap method greatly reduces RSS,'s, but at the same time it enlarges the
biases significantly,
which causes the method to be inefficient.
For the other three bootstrap methods,
we repeat the same Monte Carlo
study with 500 replications. Those results are presented in table 4.2 and table 4.4.
The third model we choose has two continuous exogenous variables generated
from the uniform distribution over the range (2,2), and an intercept.
parameter values are Po3=0.4, P= 1.0, and (=1.0.
use these Monte Carlo experiments to see the dif
The true
The sample size is still 40.
ferences between bootstrapping
data and bootstrapping residuals.
Comparing the differences between bootstrapping data and bootstrapping
residuals, we can see from table 4.2 and table 4.3 that the results from Efron's
nonparametric bootstrapping data method are very close to the results from the
parametric bootstrapping residuals method, except for the large bias for E(Y) from
the first method. But when we look at table 4.4 for the tobit II and tobit III models,
the SSRy's
(sum of squared residuals of Y) are significantly higher for Efron's
nonparametric bootstrap method than for the parametric bootstrap method. This
suggests that the parametric bootstrap method is reliable and has less variance
59
Comparing Flood's augmented semiparametric bootstrap method with the
parametric bootstrap method in models II and III (table 4.2 & table 4.3), we can
see that the results from these two methods are nearly equivalent. And, as we look
at table 4.4
, the SSRy's are a little smaller for the parametric bootstrap method,
but equivalent for the SSGRy
Comparing the maximum likelihood estimation
method
with
augmented
bootstrap
method
or the
parametric
bootstrap
method, we can see from tables 4.2 to 4.4 that the maximum likelihood estimation
method
smaller
MSEp's,
larger
SSR,'s.
So the
augmented
bootstrap
method or the parametric bootstrap method should provide reliable estimates.
From table 4.4, we can see that the sum of squared generalized residuals
(SSGRy) and the sum of squared residuals of Y (SSRy) seem to have the same
power as criteria. Even though they have different values,
they have a similar
pattern of variation for both models. This implies that the generalized residuals
represent the residuals well in the tobit model.
4.4 Summary
In a correctly specified tobit model, with an efficient parametric estimation
method,
the augmented semiparametric bootstrap method and
the parametric
bootstrap
estimation method
give
better results
than
nonparametric
bootstrapping data method and the tobit maximum likelihood estimation method.
The parametric bootstrap method, which is widely applicable, is almost equivalent
to Flood's augmented bootstrap method in the tobit model.
Table 4.1: Correctly specified tobit model (I) as B=100, N=40 and M=100.
BETA TRUE MEAN BIAS RSS, MSE,
TOBIT MAXIMUM LIKELIHOOD ESTIMATION
Po 3.000 3.132 0.017 0.724 0.741
P1 0.500 0.506 0.000 0.021 0.021
P2 0.200 0.224 0.001 0.017 0.018
E(Y) 0.033 0.075 0.002  
EFRON'S NONPARAMETRIC BOOTSTRAPPING OF DATA
Po 3.000 3.211 0.045 0.788 0.833
P1 0.500 0.518 0.000 0.022 0.022
P2 0.200 0.229 0.001 0.017 0.018
E(Y) 0.033 0.082 0.002  
BALANCED NONPARAMETRIC BOOTSTRAPPING OF DATA
Po 3.000 3.202 0.041 0.787 0.828
P1 0.500 0.518 0.000 0.022 0.022
P2 0.200 0.229 0.001 0.017 0.018
E(Y) 0.033 0.073 0.002  
FLOOD'S SEMIPARAMETRIC BOOTSTRAPPING OF RESIDUALS
Po 3.000 3.109 0.012 0.723 0.735
P1 0.500 0.505 0.000 0.020 0.020
P2 0.200 0.224 0.001 0.016 0.017
E(Y) 0.033 0.057 0.001  
MIXEDAUGMENTED SEMIPARAMETRIC BOOTSTRAP
Po 3.000 2.201 0.638 0.470 1.108
P1 0.500 0.391 0.012 0.014 0.026
P2 0.200 0.180 0.000 0.012 0.012
E(Y) 0.033 0.179 0.045  
Table 4.2: Correctly specified tobit model (II) as B=100, N=40 and M=500.
BETA TRUE MEAN BIAS RSS, MSE,
TOBIT MAXIMUM LIKELIHOOD ESTIMATION
Po 3.000 3.077 0.006 0.808 0.814
Pi 0.500 0.504 0.000 0.022 0.022
p2 0.200 0.213 0.000 0.017 0.017
E(Y)' 0.361 0.369 0.000  
EFRON'S NONPARAMETRIC BOOTSTRAPPING OF DATA
Po 3.000 3.133 0.018 0.873 0.891
pi 0.500 0.511 0.000 0.023 0.023
P2 0.200 0.216 0.000 0.017 0.017
E(Y)' 0.361 0.257 0.011  
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
Po 3.000 3.141 0.020 0.878 0.898
pI 0.500 0.511 0.000 0.023 0.023
P2 0.200 0.217 0.000 0.017 0.018
E(Y)' 0.361 0.377 0.000  
FLOOD'S SEMIPARAMETRIC BOOTSTRAPPING OF RESIDUALS
Po 3.000 3.091 0.008 0.895 0.903
P1 0.500 0.505 0.000 0.024 0.024
P2 0.200 0.214 0.000 0.017 0.017
E(Y)' 0.361 0.380 0.000  
* the value for E(Y) is approximate.
Table 4.3: Correctly specified tobit model (III) as B=100, N=40 and M=500.
BETA TRUE MEAN BIAS RSS, MSE,
TOBIT MAXIMUM LIKELIHOOD ESTIMATION
Po 0.400 0.392 0.000 0.045 0.045
P 1.000 1.012 0.000 0.031 0.031
P2 1.000 1.000 0.000 0.033 0.033
E(Y) 0.892 0.898 0.000  
EFRON'S NONPARAMETRIC BOOTSTRAPPING OF DATA
Po 0.400 0.380 0.000 0.047 0.047
Pi 1.000 1.019 0.000 0.031 0.031
P2 1.000 1.003 0.000 0.034 0.034
E(Y) 0.892 0.564 0.108  
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
Po 0.400 0.380 0.000 0.047 0.047
Pi 1.000 1.019 0.000 0.032 0.032
P2 1.000 1.004 0.000 0.034 0.034
E(Y)' 0.892 0.903 0.000  
FLOOD'S SEMIPARAMETRIC BOOTSTRAPPING OF RESIDUALS
Po 0.400 0.387 0.000 0.047 0.047
Pi 1.000 1.014 0.000 0.032 0.032
P2 1.000 0.999 0.000 0.033 0.033
E(Y)' 0.892 0.904 0.000  
* the value for E(Y) is approximate.
Table 4.4:
Comparison of criteria among different methods for tobit models II
and III.
METHOD SSRy SSGR,, BIAS OF E(Y)
TOBIT MODEL (II) ESTIMATION RESULTS (M=500)
TOBIT MLE 15.455 2.857 0.000
EFRON 32.795 11.188 0.011
PARAMETRIC 14.201 2.599 0.000
FLOOD 14.288 2.582 0.000
TOBIT MODEL (III) ESTIMATION RESULTS (M=500)
TOBIT MLE 18.669 1.782 0.000
EFRON 115.210 33.665 0.108
PARAMETRIC 16.917 1.578 0.000
FLOOD 17.227 1.565 0.000
CHAPTER 5
TESTS OF HYPOTHESES IN LIMITED
DEPENDENT VARIABLE MODELS
Introduction
Three general principles employed for hypothesis testing in econometrics
are the Wald (W),
likelihood ratio (LR),
and Lagrange multiplier (LM) criteria.
W test was introduced by Wald
1943).
Aitchison and Silvey (1958),
and Silvey
(1959) first developed the LM test. The LM test is also the same as the score test,
(1947).
Although
those
hypothesis
tests
consider
general
issue
hypothesis testing from different perspectives and have different critical regions
for small samples,
asymptotically the three procedures perform identically.
For testing linear restrictions on the coefficients of certain linear models,
Savin (1976),
Berndt and Savin (1977),
and Breusch
1979) showed that there
exists a systematic numerical inequality1 between the test statistics. Specifically,
this is W
> LR LM. Because of this inequality, in use there may be conflicts
among these tests,
i.e. sometimes one rejects a null hypothesis using one test but
another test fails to reject the null.
Two problems arise from using the asymptotic
chisquare distribution as an approximation. Evans and Savin
1982) reported
The inequality
disturbances provided
relation
that the
only
holds
unknown
for a general
linear
model
with
normal
elements of the covariance matrix can

65
that the probability of conflict can be substantial when the three tests are based
on the asymptotic chisquare critical
value.
They also concluded
that in
classical linear regression model the conflict between the W
due to the tests not having the correct significance level. T
LR, and LM tests is
his is another major
problem of these three tests. Note that there is no conflict between the three tests
when they are based on exact distributions (Evans and Savin, 1982).
Breusch and Pagan
1979),
Godfrey (1978),
Griffiths and
Surekha
(1986) found in their Monte Carlo experiments that the LM test rejects the null
hypotheses less frequently than indicated by its nominal size. In other words, the
nominal size of the test tends to overestimate the true probability of type I error
in finite samples.
There are two kinds of correction methods that can be used to solve the
significance level problem of these three tests in general linear regression models.
One is to adjust the critical value of the tests. Harris (1985) proposed a general
sizecorrected
LM test
procedure with
a rigorous
theoretical grounding.
With
tedious algebra, Honda (1988) applied Harris'
method to provide the formula for
the size correction
to the LM
test for
heteroskedasticity.
The second
kind
correction method is to modify the test statistic. Evans and Savin (1982) compared
two correction methods, one from Gallant (1975) and the other from Rothenberg
(1977), and concluded that the three Edgeworth sizecorrected tests have almost
right significance
levels
that
probability
conflict
between
the size
corrected tests is of no consequence under commonly satisfied conditions.
For nonlinear regression models, the inequality relation between values of
statistics is no longer available.
Thus, it will be interesting to see if there is any
Recent studies show that,
for the logit model,
we do not have to
rely on the
asymptotic distribution of a test statistic.
Because we can get exact inference
conditional on the sufficient statistic.
For details
, see the survey of exact inference
for contingency tables by Agresti (1992).
There
been a
effort devoted
to solving the significance level
problem in nonlinear models.
Gallant
(1975)
suggested
using degrees of
freedom corrections in nonlinear models, and Rothenberg (1977) has suggested
using Edgeworth sizecorrections in the multivariate regression model.
Rocke
1989)
applied
bootstrap
Bartlett
adjustment
to the
likelihood
ratio
statistic
for the
seemingly
unrelated regression model.
Rayner
(1990),
using
Edgeworth expansions, showed that a bootstrap Bartlett adjustment to the LR test
statistic may be used to estimate p values with error of order improved to n
for the W and LM tests there is not any improvement.
Davidson and MacKinnon (1984) proposed several LM tests and a LR test
for the logit and probit models.
They found one of the LM tests outperforms the
other tests by having more accurate type I error with respect to the chisquare
distribution.
none
tests
clearly
larger
power.
Taylor
(1991
compared two kinds of LM tests for the tobit model. Instead of the asymptotic chi
square critical values,
he used
empirical
finite sample critical values from a
simulated exact distribution by generating ten thousand replications for each
sample size. He concluded that the Hessian LM test would be more powerful than
the outerproduct of the gradient variant of the LM test. His choice of LM test
coincides with that of Davidson and MacKinnon.
Horowitz
(1991)
applied
a bootstrap
method
a set of Monte
Carlo
we have seen,
there
are two
major problem
with
these
three
tests
because of using the asymptotic chisquare distribution as an approximation: The
first is the significance level problem, and the second is the conflict among these
three
tests.
purpose
chapter
is to apply
bootstrap
methods
approximate the exact distribution of these three test statistics for the logit, probit,
tobit
models,
to investigate
effects
differences
between
bootstrapping data and bootstrapping residuals on these hypothesis tests. We use
Hessian
, Hessian LM
LR tests,
as well
as the bootstrap
Bartlett
adjusted LR test.
Wald, Likelihood Ratio and Lagrange Multiplier tests
The Wald approach starts at the alternative and asks whether movement
toward the null would be an improvement.
This involves estimation under the
alternative and the value hypothesized under the null,
where the metric is the
expected value of the Hessian matrix evaluated under the alternative. In contrast,
the Lagrange multiplier approach starts at the null and
considers
movement
toward
the alternative.
This requires evaluating the slope of the loglikelihood
function (the score) when the parameters are constrained to the space of the null,
where the metric is the inverse of the expected value of the Hessian evaluated
under the null. Finally, the likelihood ratio approach compares the two hypotheses
directly on an equal basis. This involves estimating the model under both the null
alternative
then
comparing
difference
in the loglikelihood
functions.
Which one to use usually depends on such factors as small sample
behavior or computational convenience.
68
of parameters, s(J) be the gradient of the loglikelihood function, and I(p) be the
information matrix of a model.
Then for the null hypothesis Ho: p=Po, the Wald,
likelihood ratio, and Lagrange multiplier test statistics are
(5.1)
LR = 2(lnL(3) lnL(B))
(5.2)
(5.3)
(13111U3)L's(13)
For the logit model
(5.4)
otherwise.
Let x,=(l
x1l X21)
be a
column vector and p be a
column vector.
The log
likelihood function for the model is
lnL=
ID 11Y
I1
(5.5)
 ln[Il+exp(p'x,)]
1=1
We consider the null hypothesis
P21P20'
leaving Io and f,
to be nuisance
parameters. Hence, the gradient of the loglikelihood function for f2 is
s~(I3)
N
= Ex2yt
(=1Xl
1l
i=
exp(p'x)
(5.6)
1 + exp(p' x)
from
the second
derivative
loglikelihood
function
we derive
information matrix
N exp(Q'x)
(rc:71
,n\
= (P P,)' rcB] ((i P,1
Po +Px,,+ Pzxz, Y:
I
I2 33) 2
M o2)
(5.8)
LR = 2 (lnL(f) lnL($))
(5.9)
LM=
() tI()b] s,()1
(5.10)
For the probit and tobit models, there will be similar equations from (5.4)
to (5.10).
See Maddala (1983) for more details.
Three bootstrap methods will be used: the first is Efron's nonparametric
bootstrapping data method; the second is the parametric bootstrapping residuals
method, and the third is Flood's augmented bootstrap method. Details of the first
two bootstrap methods have been presented in chapter 3
details for the third were
discussed in chapter four.
bootstrap test statistics W"
Using equations (5.8) to (5.10),
we can compute the
and LM'.
Monte Carlo Experiments for Hypothesis Testing
We generate the data from the logit model of the equation (5.4).
Our goal is
to improve the accuracy of significance levels of the Wald, likelihood ratio, and
Lagrange multiplier tests by finding bootstrap critical regions instead of asymptotic
chisquare critical regions. Three different bootstrap methods are applied, Efron's
nonparametric bootstrapping data method, the parametric bootstrap method, and
Flood's
augmented
semiparametric
bootstrap
method.
model
continuous exogenous variables and an intercept. The true parameter values are
Po=0.4, P,=1.0, P2=1.0. Both continuous variables x1 and x2 are randomly drawn
from the uniform distribution over the range (2,2). Approximate 40 percent of the
(iB2
70
Using this same setting, we do Monte Carlo experiments for the probit and tobit
models with the error term of the tobit model generated from the standard normal
distribution.
The detailed procedure for the logit model is as follows:
Step 1:
Generate a
random sample of
data
CY,X~
N according to
equation (5.4) with error generated from the logistic distribution.
Obtain restricted ML estimates2 and unrestricted ML estimates of
Step
the model from sample (Y,X).
Then compute the W
LR, and LM test
statistics according to equations from (5.8) to (5. 10).
Call their values
Wo, LRo, and LMo.
Step 3a:
Bootstrap a
sample (Y,X) using Efron's nonparametric
bootstrap
method
to get
a new
bootstrap
sample
(YtXl.
Then
estimate
restricted and unrestricted ML estimates.
Step 3b:
Generate errors from the normal distribution with the restricted ML
estimated variance. Then, by using both the hypothesized parameter
values of the null and restricted ML estimates
from the original
sample (Y,X),
we can obtain
to get a new parametric bootstrap
sample (Yt,X) with which to estimate restricted and unrestricted ML
estimates.
Step 3c:
For the augmented error method,
we can obtain a new bootstrap
sample to estimate restricted and unrestricted ML estimates. (Details
of the procedure are presented in chapter 4).
Step 4:
Compute the three test statistics for each bootstrap method. Call
their values W("b)
LR('b)
and LM'(b)
Step 5:
For each bootstrap method, estimate the alevel critical values of
these three tests from the empirical distribution of
{w(b)},
{LR*(b)
{LM'(b)}
that are obtained by repeating step 3 and step 4 B= 100
times.
Let C,(ac
CR(a),
CL(a)
denote
the estimated
critical
values.
Step 6:
For each bootstrap
method,
reject the model being tested at the
nominal alevel based on the bootstrap critical values ifWo>C,(a) for
the Wald test,
LRo>CLR(a) for the LR test, and LMo>CLM(a) for the LM
test. Reject the model at the nominal alevel based on the asymptotic
chisquare critical value if Wo>X2(1a) for the Wald test,
LRo>2(1
for the LR
test,
LMo>X2(1a)
for the
LM test
with
degree of
freedom one.
In addition to the test statistics of the Wald
, likelihood ratio, and Lagrange
multiplier, we also consider the bootstrap Bartlett adjusted LR test. This procedure
is as follows: first we get LR% from the original sample as in step 2.
have the bootstrap LR test statistic LR'(b) in the step
Second
, and whose average
over the 100 bootstraps estimates the true average value of the LR statistic under
the null hypothesis.
Finally
the bootstrap Bartlett adjusted LR statistic is
LR,
L% / L9
(5.11)
which is tested against a chisquare distribution with one degree of freedom.
From tables 5.1 to 5.6
true significance levels, for s
, we give the actual percentages of rejection,
seven different nominal levels.
i.e. the
We also provide the
atln i la 1.. an nC Ih a rai +4 ra an eta at l.ta +r. *< 1^ ns raw l an rnwr frntn +1tn nn a1 i~ nTa1 7&1
the nominal level is a=0. 10 (or
8.8%),
10%), and the true significance level is 0.088 (or
then the absolute value of the relative change of the true level away from
the nominal level is 0.12 (or 12% in the tables) because the absolute value of the
difference between 0.10 and 0.088 divided by 0.10 is 0.12.
In table 5.7
we add all
the relative changes of the seven columns.
from this averaging because of its bad resu
Efron's
We exclude the asymptotic Wald test
Its. We also exclude the discussion of
nonparametric bootstrap method because of its extremely unsatisfactory
results for all the Monte Carlo experiments.
The purpose of these Monte Carlo experiments is to find the best hypothesis
test with the correct true significance levels for each model; to see if applying
bootstrap methods improves hypothesis testing over the Wald, likelihood ratio, and
Lagrange multiplier tests;
to investigate the problem of conflict between these
three tests in these three limited dependent variable models; and finally, to
see the
differences
between
bootstrapping
data
method
the bootstrapping
residuals method for these hypothesis tests.
For the logit model
table 5.7),
Flood's
augmented bootstrap method does
not yield accurate
true significance levels.
It has an average of about
41.6%
(66.2%
for the relative changes of the true levels away from the nominal levels
with sample size N=50
100)
This method consistently underestimates the
nominal levels by a large margin.
The reason for this might be that the augmented
bootstrap
method forces
the error terms
to be symmetric in
each
bootstrap
sample.
The probability of conflict between the tests based on Flood's
augmented
bootstrap critical values is small, an average of 0.019 for both sample sizes (table
5.10).
73
5.2 and 5.7). All the tests with sample size N=100 generated from the parametric
bootstrap method have satisfactory performance with little overestimation. We can
see from table 5.10 that the probability of conflict between the tests based on
parametric bootstrap critical values is small, an average of 0.011 for both sample
sizes.
The probability of conflict between these three tests based on asymptotic
chisquare critical values is moderate, an average about 0.05 for both sample
sizes.
Overall for the logit model,
the LM
test with the asymptotic chisquare
critical values gives the most accurate true significance levels.
And the Wald test
with the parametric bootstrap critical values, as well as the parametric bootstrap
Bartlett adjusted LR test with the chisquare critical values,
gives satisfactory true
significance levels.
For the probit model (table 5.7),
all four tests with the small sample N=50
generated from both the parametric bootstrap method and Flood's
bootstrap method give similar results,
augmented
with an average of 35% relative change. At
a sample size of 100, the tests generated from the parametric bootstrap method
give
satisfactory
results,
specially
Wald,
bootstrap
Bartlett
adjusted LR tests. For both sample sizes, however, the LM test with asymptotic
chisquare critical values gives the most accurate true significance levels, and the
LM test generated from the parametric bootstrap method gives satisfactory results.
The average probability of conflict (table 5.10) between these three tests generated
from the parametric bootstrap method is about 0.039 with a maximum of 0.068
for sample size of 50 and only about 0.013 with a maximum of 0.024 for sample
size of 100.
For the tobit model
, the tests based on asymptotic chisquare critical values
change of 73%
62%) for sample size N=50 (N=100).
For sample sizes of both 50
and 100, the Wald,
LR, and LM tests generated from both bootstrap methods,
augmented
semiparametric
bootstrap
method
parametric
bootstrap
method
, give satisfactory true significance levels,
with the performance of the
augmented bootstrap method better on average (tables 5.5 to 5.7).
The bootstrap
Bartlett adjusted LR test performs well in the large sample. For the tobit model,
the Wald, LR, and LM tests are almost equivalent whichever sample size we choose
and whichever bootstrap method we apply.
The maximum probability of conflict
between these three tests in table 5.10 is 0.01 with an average of 0.004 over five
hundred replications.
Even for the badly behaved chisquare approximations,
average probability of conflict is only 0.021 out of five hundred replications for
both sample sizes.
Therefore
, for testing in the tobit model,
these Monte Carlo
experiments suggest
use of the
Wald
,LR,
and LM
tests
with
the augmented
bootstrap based critical values.
For the Wald
, LR, and LM tests based on asymptotic chisquare critical
values
there
are large
probabilities
conflict
(table
5.10).
average
probability of conflict is about 0.05 for the logit model and 0.145 for the probit
model.
From table 5.7
we can
see that the Wald test with the chisquare critical
region substantially overrejects the null hypotheses for all three models with
average relative change of 110%. All three tests perform poorly in the tobit model.
The LR test performs well in both the logit and probit models with sample size of
100 of average relative change about 9.6%. And it overrejects the null hypothesis
in both the logit and probit models,
where with a sample size of 50,
the average
relative change about 33.7%.
The LM test performs excellently in both the logit
75
The bootstrap Bartlett adjusted LR test performs well only in large samples,
which is not the advantage of bootstrap methods (table 5.7).
For two bootstrap methods,
we can see from table 5.7 that overall tests
using the parametric bootstrap based critical values perform better than tests
using augmented bootstrap based critical values. The latter performs satisfactorily
only in the tobit model. For the augmented bootstrap method, it is strange that the
tests perform better in small samples than in large samples. As the sample size
increases
to 100 in
both
logit and
probit models,
two methods
opposite results: the parametric bootstrap method generates better tests with a
small average relative change, and the augmented bootstrap method generates
worse tests with a large average relative change.
From tables 5.1 through 5.6, we get table 5.9 by summing each percentage
column of absolute values of relative changes of the true levels to the nominal
levels for each table. We can see that the 1% column is very sensitive and difficult
to match, while the 50% column is stable and the easiest to match. Since there are
a lot of significantly large relative changes in an absolute sense in the 1% column,
we can create a new table 5.8 by removing the 1% column from table 5.7. For the
parametric bootstrap method in table 5.8, more than 30% of the total sums of
relative changes have been reduced for each of these three tests with respect to
table 5.7. The same results hold for the asymptotic chisquare distribution in table
This means
that it is really difficult for these two methods
to accurately
estimate the 0.01 nominal level, and relatively much easier to estimate the other
nominal levels.
Summary
There are many hypothesis tests that can be used for econometric models,
but for each hypothesis there should be only one best test.
Monte Carlo experiments,
With these restricted
we can suggest that for the logit and probit models,
LM test based on chisquare critical values provides accurate true significance
levels.
For the tobit model
the Wald
, LR, and LM tests using Flood's
augmented
bootstrap
based
critical
values
are all
equivalent and
provide
accurate
true
significance levels.
For these three models, the Wald test using chisquare based
critical values always substantially overrejects the null hypothesis.
Since we are testing parametric models with parametric estimation methods
parametric
hypothesis
tests
, not surprisingly
, the tests
generated
using
Efron's
nonparametric bootstrapping of data give us useless results in our Monte
Carlo experiments.
The bootstrap Bartlett adjusted likelihood ratio test does not perform as
well as expected in small samples.
With
parametric
bootstrap
method
augmented
bootstrap
method
, the probabilities of conflict between the Wald, LR, and LM tests are of no
consequence for the logit and probit models,
and especially for the tobit model.
However,
when these three tests use chisquare based critical values,
there are
some substantial conflicts among them in the logit and probit models.
Table 5.1:
True
significance
differences
between
levels
Lthe
with
true
absolute
levels
values
nomina
percentage
l levels in
parentheses for the logit model of sample
N=50 and M=500.
NOMINAL 1% 5% 10% 15% 20% 25% 50%
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
WALD 1.6 6.0 10.4 16.6 22.8 27.6 54.8
(60) (20) (4) (11) (14) (10) (10)
LR 2.2 6.2 12.0 17.6 22.2 26.6 55.0
(120) (24) (20) (17) (11) (6) (10)
LRAD 1.0 5.6 11.0 15.2 21.4 26.6 54.2
(0) (12) (10) (1) (7) (6) (8)
LM 2.4 7.2 13.0 18.0 24.4 28.8 54.6
(140) (44) (30) (20) (22) (15) (9)
FLOOD'S AUGMENTED BOOTSTRAPPING OF RESIDUALS
WALD 0.2 2.4 5.4 7.8 11.2 13.2 36.2
(80) (52) (46) (48) (44) (47) (28)
LR 0.4 3.4 5.2 9.2 12.2 14.2 37.6
(60) (32) (48) (39) (39) (43) (25)
LRAD 0.0 2.0 4.8 6.8 12.8 14.0 41.6
(100) (60) (52) (55) (36) (44) (17)
LM 1.0 3.8 7.8 11.0 13.6 16.6 35.2
(0) (24) (22) (27) (32) (34) (30)
ASYMPTOTIC CHISQUARE DISTRIBUTION
WALD 2.6 10.8 16.2 20.2 29.4 33.6 58.6
(160) (116) (62) (35) (47) (34) (17)
LR 1.6 7.8 12.6 17.6 25.0 28.8 56.2
(60) (56) (22) (17) (25) (15) (12)
LM 1.0 5.0 11.2 16.0 24.0 27.6 55.0
(0) (0) (12) (7) (20) (10) (10)
Table
True
significance
levels
wfth
absolute
values
percentage
differences
between
the true levels and
the nominal levels in
parentheses for the logit model of sample size N=100 and M=500.
NOMINAL 1% 5% 10% 15% 20% 25% 50%
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
WALD 1.0 5.8 11.2 15.6 20.8 26.8 51.0
(0) (16) (12) (4) (4) (7) (2)
LR 1.2 5.8 11.4 15.8 20.6 27.2 51.6
(20) (16) (14) (5) (3) (9) (3)
LRAD 0.2 4.6 10.8 14.2 20.4 26.4 51.0
(80) (8) (8) (5) (2) (6) (2)
LM 1.4 4.6 11.0 14.8 20.4 27.8 51.6
(40) (8) (10) (1) (2) (11) (3)
FLOOD'S AUGMENTED BOOTSTRAPPING OF RESIDUALS
WALD 0.0 0.6 2.2 4.8 7.8 9.2 23.8
(100) (88) (78) (68) (61) (63) (52)
LR 0.2 1.0 2.6 6.0 8.2 10.0 25.2
(80) (80) (74) (60) (59) (60) (50)
LRAD 0.0 1.0 2.4 4.8 7.8 10.0 32.8
(100) (80) (76) (68) (61) (60) (34)
LM 0.4 1.8 3.4 7.0 9.6 10.6 26.2
(60) (64) (66) (53) (52) (58) (48)
ASYMPTOTIC CHISQUARE DISTRIBUTION
WALD 1.3 9.4 16.2 21.4 28.6 34.2 56.2
(30) (88) (62) (43) (43) (37) (12)
LR 0.4 4.8 11.4 15.2 21.4 27.0 51.8
(60) (4) (14) (1) (7) (8) (4)
LM 0.6 3.8 10.6 14.4 21.2 26.4 52.0
(40) (24) (6) (4) (6) (6) (4)
Table
True
significance
differences
between
levels
with
absolute
the true levels and
parentheses for the probit model of sample
values
percentage
the nominal levels in
N=50 and M=500.
NOMINAL 1% 5% 10% 15% 20% 25% 50%
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
WALD 0.0 0.4 4.4 10.8 18.6 26.4 50.6
(100) (92) (56) (28) (7) (6) (1)
LR 0.0 1.4 5.2 13.2 22.0 28.0 52.4
(100) (72) (48) (12) (10) (12) (5)
LRAD 0.0 0.8 2.8 9.6 19.4 26.8 52.2
(100) (84) (72) (36) (3) (7) (4)
LM 1.4 4.0 9.4 17.6 23.8 29.6 52.6
(40) (20) (6) (17) (19) (18) (5)
FLOOD'S AUGMENTED BOOTSTRAPPING OF RESIDUALS
WALD 0.0 1.2 7.8 17.0 24.6 30.2 47.4
(100) (76) (22) (13) (23) (21) (5)
LR 0.2 1.6 7.2 18.0 24.6 29.4 47.2
(80) (68) (28) (20) (23) (18) (6)
LRAD 0.0 1.2 6.4 11.4 22.4 28.6 48.8
(100) (76) (36) (24) (12) (14) (2)
LM 0.4 2.0 7.8 17.8 23.4 27.0 47.2
(60) (60) (22) (19) (17) (8) (6)
ASYMPTOTIC CHISQUARE DISTRIBUTION
WALD 12.8 22.8 29.8 34.8 41.2 43.6 63.6
(1180) (356) (198) (132) (102) (74) (27)
LR 1.4 8.4 14.4 20.8 27.0 32.0 55.4
(40) (68) (44) (39) (35) (28) (11)
LM 0.4 4.4 7.8 13.4 20.8 26.8 54.2
(60) (12) (22) (11) (4) (7) (8)
Table 5.4:
True
significance
levels
with
absolute
values
percentage
differences
between
the true levels and
the nominal levels in
parentheses for the probit model of sample size N= 100 and M=500.
NOMINAL 1% 5% 10% 15% 20% 25% 50%
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
WALD 1.4 4.4 8.6 14.0 18.6 24.6 49.8
(40) (12) (14) (7) (7) (2) (0)
LR 1.4 4.6 8.8 15.0 19.2 25.0 50.4
(40) (8) (12) (0) (4) (0) (1)
LRAD 0.6 4.8 8.6 13.2 19.0 23.0 50.6
(40) (4) (14) (12) (5) (8) (1)
LM 2.2 6.0 11.0 15.4 20.0 25.0 49.6
(120) (20) (10) (3) (0) (0) (1)
FLOOD'S AUGMENTED BOOTSTRAPPING OF RESIDUALS
WALD 3.8 6.4 9.0 11.8 16.4 19.4 34.4
(280) (28) (10) (21) (18) (22) (31)
LR 2.4 6.2 7.2 11.0 14.0 17.8 33.6
(140) (24) (28) (27) (30) (29) (33)
LRAD 1.8 5.0 9.0 10.4 14.4 18.0 37.0
(80) (0) (10) (31) (28) (28) (26)
LM 1.6 4.4 6.4 9.0 11.8 15.2 31.2
(60) (12) (36) (40) (41) (39) (38)
ASYMPTOTIC CHISQUARE DISTRIBUTION
WALD 5.0 14.4 22.0 26.2 32.8 36.8 62.8
(400) (188) (210) (75) (64) (47) (26)
LR 1.0 4.4 9.0 15.0 20.8 26.4 52.4
(0) (12) (10) (0) (4) (6) (5)
LM 0.8 2.8 8.2 11.8 20.0 23.4 51.2
(20) (44) (18) (21) (0) (6) (2)
Table
True
significance
differences
between
levels
Lthe
with
true
absolute
levels
parentheses for the tobit model of sample
values
nomina
percentage
1 levels in
N= 50 and M=500.
NOMINAL 1% 5% 10% 15% 20% 25% 50%
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
WALD 0.8 3.0 7.6 15.6 19.6 24.8 52.0
(20) (40) (24) (4) (2) (1) (4)
LR 0.8 3.2 8.0 15.6 19.4 25.0 51.8
(20) (36) (20) (4) (3) (0) (4)
LRAD 0.8 2.6 6.6 11.6 16.2 20.4 48.4
(20) (48) (34) (23) (19) (18) (3)
LM 0.8 2.6 7.8 15.0 19.6 25.0 51.6
(20) (48) (22) (0) (2) (0) (3)
FLOOD'S AUGMENTED BOOTSTRAPPING OF RESIDUALS
WALD 1.0 4.4 9.2 15.0 19.6 24.6 50.6
(0) (12) (8) (0) (2) (2) (1)
LR 1.4 4.8 9.0 15.0 19.6 24.6 50.6
(40) (4) (10) (0) (2) (2) (1)
LRAD 1.6 4.6 6.8 11.2 15.8 19.6 46.6
(60) (8) (32) (25) (21) (22) (7)
LM 1.0 4.4 8.8 14.4 19.0 24.4 50.8
(0) (12) (12) (4) (5) (2) (2)
ASYMPTOTIC CHISQUARE DISTRIBUTION
WALD 3.0 7.4 14.6 19.2 25.0 28.8 56.2
(200) (48) (46) (28) (25) (15) (13)
LR 3.4 9.2 16.6 21.6 27.8 31.2 58.0
(240) (84) (66) (44) (39) (25) (16)
LM 4.2 10.0 18.2 22.4 28.0 33.0 59.8
(320) (100) (82) (49) (40) (32) (20)
Table 5.6:
True
significance
differences
between
levels
with
absolute
the true levels and
values
percentage
the nominal levels in
parentheses for the tobit model of sample size N= 100 and M=500.
NOMINAL 1% 5% 10% 15% 20% 25% 50%
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
WALD 0.6 4.8 11.4 16.8 21.8 26.8 52.6
(40) (4) (14) (12) (9) (7) (5)
LR 0.8 5.4 11.0 17.0 21.8 26.2 53.0
(20) (8) (10) (13) (9) (5) (6)
LRAD 1.0 4.4 10.0 14.2 21.6 25.4 50.8
(0) (12) (0) (5) (8) (2) (2)
LM 0.6 5.0 11.0 17.2 22.0 27.0 52.8
(40) (0) (10) (15) (10) (8) (6)
FLOOD'S AUGMENTED BOOTSTRAPPING OF RESIDUALS
WALD 1.0 4.6 9.6 15.8 21.2 26.4 52.0
(0) (8) (4) (5) (6) (6) (4)
LR 1.0 4.8 10.4 15.6 22.2 26.6 52.0
(0) (4) (4) (4) (11) (6) (4)
LRAD 1.0 4.4 9.6 13.2 19.8 25.6 51.8
(0) (12) (4) (12) (1) (2) (4)
LM 1.0 4.4 10.0 15.8 21.6 26.4 52.2
(0) (12) (0) (5) (8) (6) (4)
ASYMPTOTIC CHISQUARE DISTRIBUTION
WALD 1.8 9.4 15.8 22.4 30.0 34.4 59.6
(80) (88) (58) (49) (50) (38) (20)
LR 2.0 9.4 16.0 22.2 29.6 33.8 59.6
(100) (88) (60) (48) (48) (35) (20)
LM 2.8 10.2 17.6 23.6 30.4 34.6 60.4
(180) (104) (76) (57) (52) (38) (21)
Table
Total of absolute values of percentage differences between the true
levels and the nominal levels with each test for different methods as
M=500.
MODEL LOGIT PROBIT TOBIT
N 50 100 50 100 50 100
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
WALD 129(0) 45(0) 290(2) 82(1) 95(2) 91(2)
LR 208(0) 70(0) 259(3) 65(3) 87(2) 71(1)
LRAD 44(0) 111(4) 306(2) 84(1) 165(0) 29(3)
LM 280(0) 75(5) 125(4) 154(1) 95(1) 89(1)
AVERAGE 23.6 10.8 35.0 13.8 15.8 10.0
FLOOD'S AUGMENTED BOOTSTRAPPING OF RESIDUALS
WALD 345(0) 510(0) 260(3) 410(5) 25(1) 33(2)
LR 286(0) 463(0) 243(3) 311(5) 59(4) 33(1)
LRAD 364(0) 479(0) 264(2) 203(1) 175(1) 35(2)
LM 169(0) 401(0) 192(3) 266(1) 37(1) 35(1)
AVERAGE 41.6 66.2 34.3 42.5 10.6 4.9
ASYMPTOTIC CHISQUARE DISTRIBUTION
WALD 471(0) 315(0) 2069(0) 1010(0) 375(0) 383(0)
LR 207(0) 98(5) 265(0) 37(3) 514(0) 399(0)
LM 59(0) 90(4) 124(3) 111(1) 643(0) 528(0)
AVERAGE' 19.0 13.4 27.8 10.6 73.0 62.4
* average absolute values of percentage differences of LR and LM tests only.
Table
Total of absolute values of percentage differences between the true
levels and the nominal levels with each test for different methods
without 1% column as M=500.
MODEL LOGIT PROBIT TOBIT
N 50 100 50 100 50 100
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
WALD 69(0) 45(0) 190(2) 42(0) 75(2) 51(1)
LR 88(0) 50(0) 159(3) 25(1) 67(2) 51(1)
LRAD 44(0) 31(2) 206(2) 44(1) 145(0) 29(3)
LM 140(0) 35(1) 85(2) 34(1) 75(1) 49(1)
AVERAGE 14.2 6.7 26.7 6.0 15.0 7.5
FLOOD'S AUGMENTED BOOTSTRAPPING OF RESIDUALS
WALD 265(0) 410(0) 160(3) 130(1) 25(1) 33(2)
LR 226(0) 383(0) 163(3) 171(1) 19(1) 33(1)
LRAD 264(0) 379(0) 164(2) 123(0) 115(0) 35(2)
LM 169(0) 341(0) 132(3) 206(1) 37(1) 35(2)
AVERAGE 38.5 63.0 25.8 26.3 8.2 5.7
ASYMPTOTIC CHISQUARE DISTRIBUTION
WALD 311(0) 285(0) 889(0) 610(0) 175(0) 303(0)
LR 147(0) 38(1) 225(0) 37(3) 274(0) 299(0)
LM 59(0) 50(4) 64(3) 91(1) 323(0) 348(0)
AVERAGE' 17.2 7.3 24.1 10.7 42.9 52.8
* average absolute values of percentage differences of LR and LM tests only.
Table 5.9:
Total of each percentage column of absolute values of relative change
of true level to nominal level for each table from 5.1 to 5.6.
PERCENT LOGIT PROBIT TOBIT
N 50 100 50 100 50 100
1% 780 610 1960 1220 940 460
5% 440 486 984 352 440 340
10% 328 420 554 372 354 240
15% 267 312 351 237 181 225
20% 297 300 255 201 160 212
25% 264 265 213 188 119 153
50% 176 214 80 164 74 96
Table 5.10:
The maximum probability of conflict for each different method
among three tests in three models. All entries are in percentage.
NOMINAL 1 % 5% 10% 15% 20% 25% 50%
PARAMETRIC BOOTSTRAPPING OF RESIDUALS
50 0.8 1.2 2.6 1.4 2.2 2.2 0.4
LOGIT
100 0.4 1.2 0.4 1.0 0.4 1.0 0.6
50 1.4 3.6 5.0 6.8 5.2 3.2 2.0
PROFIT
100 0.8 1.6 2.4 1.4 1.4 0.4 0.8
50 0.0 0.6 0.4 0.6 0.2 0.2 0.4
TOBIT
100 0.2 0.6 0.4 0.4 0.2 0.8 0.4
FLOOD'S AUGMENTED BOOTSTRAPPING OF RESIDUALS
50 0.8 1.4 2.6 3.2 2.4 3.4 2.4
LOGIT
100 0.4 1.2 1.2 2.2 1.6 1.4 2.4
50 0.4 0.8 0.6 1.0 1.2 3.2 0.2
PROBIT
100 2.2 2.0 2.6 2.8 4.6 4.2 3.2
50 0.4 0.4 0.4 0.6 0.6 0.2 0.2
TOBIT
100 0.0 0.4 0.8 0.2 1.0 0.2 0.2
ASYMPTOTIC CHISQUARE DISTRIBUTION
50 1.6 5.8 5.0 4.2 5.4 6.0 3.6
LOGIT
100 0.7 5.6 5.6 7.0 7.4 7.8 4.4
50 12.4 18.4 22.0 21.4 20.4 16.8 9.4
PROBIT
100 4.2 11.6 13.8 14.4 12.8 13.4 11.6
50 1.2 2.6 3.6 3.2 3.0 4.2 3.6
TOBIT
100 1.0 0.8 1.8 1.4 0.8 0.8 0.8
CHAPTER 6
CONCLUSIONS
This dissertation extends Hall's (1992) short bootstrap confidence intervals
to the quasipivotal
method
which
provides
a further
correction
method
generating bootstrap confidence intervals. It is shown theoretically and empirically
through Monte Carlo experiments that, among all methods generating bootstrap
confidence intervals, the bootstrap quasipivotal method is the best. This method
would also
be a
useful
correction
method
as long as
the confidence interval
generating method uses the percentile method. The results based on the correction
method are better than those based on uncorrected methods.
Also the generation of a real bootstrap confidence interval is proposed. This
performs
very
satisfactorily
when
underlying
distribution
is symmetric.
However,
in reality,
most
underlying
distributions are asymmetric.
Therefore,
bootstrap confidence intervals, which are most useful in small samples, should be
generated by the quasipivotal method. A method of finding a bootstrap trimmed
mean is proposed, which yields satisfactory results for a linear regression model
with a symmetric error distribution.
logit
model,
parametric
bootstrap
method
with
the logit
maximum likelihood estimation method provides the best reliable estimates among
other bootstrap estimates, and gives satisfactory significance levels for tests of
hypotheses.
For the probit model,
the parametric bootstrap method with the probit
maximum likelihood estimation method gives the most reliable estimates among
other bootstrap estimates, and also yields satisfactory significance levels for tests
of hypotheses,
though only at a sample size 100 or larger.
For the tobit model, the parametric bootstrap method and the augmented
bootstrap
method
are equivalent
in estimation,
both
satisfactory
significance levels for tests of hypotheses.
hypothesis
testing
in the logit
probit
models,
Lagrange
multiplier
based
on chisquare
critical
values
provides
accurate
true
significance levels.
For testing in the tobit model, the Wald, likelihood ratio,
Lagrange multiplier tests using the augmented semiparametric bootstrap based
critical values are equivalent and provide the best true significance levels.
Wald test using chisquare based critical values always substantially overestimates
the nominal levels.
The bootstrap Bartlett adjusted likelihood ratio test does not
perform as well as expected in small samples.
With
parametric
bootstrap
method
augmented
bootstrap
method
probabilities
conflict
between
Wald
likelihood
ratio,
Lagrange multiplier tests are of almost no consequence for either the logit, probit,
or tobit models.
In the logit model, the nonparametric bootstrap method gives a small mean
squared errors of P3'
(MSEB) because of a small residual sum of squares of 13's
(RSSJ),
but gives a significantly large sum of squared residuals of Y (SSRy),
causes large variation of the prediction.
That is,
which
the nonparametric bootstrap
method does not produce reliable estimates for the logit model.
89
SSRy, which causes large variation of the prediction. That is, the nonparametric
bootstrap method does not produce reliable estimates for the probit model.
In both
tobit
II and
tobit
III models,
nonparametric
bootstrap
method has a large bias for E(Y), and a significantly large SSRy. This causes large
variation of the prediction from the model. That is, the nonparametric bootstrap
method does
not produce reliable estimates for both
the tobit II and
tobit III
models.
In the linear regression model, the nonparametric bootstrap method gives
a better1 mean SSRy, and the coverage of its confidence interval is almost the
same as
the coverage of the confidence interval generated
by the
parametric
bootstrap method.
But with the other two criteria for confidence interval,
nonparametric bootstrap method does not give an interval estimate as good as the
one generated from the parametric bootstrap method.
In hypothesis testing, the nonparametric bootstrap method yields useless
test statistics for the Wald, likelihood ratio, and Lagrange multiplier in the logit,
probit, and tobit models.
We use the efficient parametric estimation method, maximum likelihood
estimation
probit,
(MLE) method,
tobit.
to estimate
use
the parametric models of the logit,
the efficient nonparametric estimation method,
ordinary least squares (OLS) estimation method, to estimate the linear regression
model. By using the nonparametric bootstrap method with the MLE,
we do no
have a reliable point estimate because of the significantly large SSRy. By using the
nonparametric bootstrap method with OLS,
we obtain a reliable point estimate
because of the better mean SSRy, but we still do not have a good interval estimate.
90
That is to say, the nonparametric bootstrap method performs better if the efficient
estimation method is nonparametric.
However,
using
parametric
bootstrap
method
with
efficient
estimation methods,
we obtain reliable point estimates in the estimation of the
logit, probit, tobit, and linear regression models; we have reliable interval estimate
in the estimation of the linear regression model; we have reliable estimates of true
significance levels; and we have no more trouble with the conflict problem.
Therefore,
with an efficient estimation method
, the parametric bootstrap
method is highly recommended, but the nonparametric bootstrap method does not
perform well.
The performance of a semiparametric bootstrap method with an
efficient estimation method depends on different models and the detailed structure
bootstrap
method.
instances,
Flood's
augmented
semiparametric
bootstrap method performs satisfactorily for the tobit model, but not for the logit
and probit models. The mixedaugmented semiparametric bootstrap method does
not perform satisfactorily at all.
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