Bootstrap methods in limited dependent variable models

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Bootstrap methods in limited dependent variable models
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Thesis (Ph. D.)--University of Florida, 1993.
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Includes bibliographical references (leaves 91-96).
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by Jinyong Chen.

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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 QUASI-PIVOTAL METHOD FOR GENERATING


BOOTSTRAP CONFIDENCE INTERVALS
Introduction . . .
The Quasi-Pivotal 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 . . .


lTr-f'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


"un-trimmed"


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 first-order 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 Kaplan-Meier (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 Teebagy-Chatterjee 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 Teebagy-Chatterjee, 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 over-reject the null hypothesis. These two


problems


arise


from


using


asymptotic


chi-square


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 quasi-pivotal


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 QUASI-PIVOTAL 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 bootstrap-t.

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(n-2) 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 quasi-pivotal 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 quasi-pivotal method is

better than that of the bootstrap confidence intervals generated by the other


methods.


The idea


behind


the quasi-pivotal 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 quasi-pivotal 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 Quasi-Pivotal 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,

1-2ac


(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 quasi-pivotal method. And this


method also could be used as a "correction method" for the percentile method,

Beran's B, the bootstrap-t, and the real bootstrap confidence interval wherever the


percentile method is


applied.


To get a bootstrap confidence interval of T(Z,) at level 1-2a, assume that


of q,










P{qT(Zt,)q2} =


1-2a


(2.3)


We suggest finding q, and q2 by minimizing (q2 -q) subject to G(q2)-G(qj)=1-2a 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)


> 1-2a


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 1-2a


to be


- q2SE().,


I- -q1-SE(1)}


Note


that,


if T(z,3)


where


-=


a linear


increasing


function,


bootstrap-t as in equation (2.2),


then the length of the short bootstrap confidence


interval of p is also minimized.


Actually,


quasi-pivotal


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 bias-corrected

percentile method (known as the BC method) corrects the bias of the percentile

method, and Efron's accelerated bias-corrected 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 quasi-pivotal


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 1-2a, we want to minimize q2-ql subject

to


0(q2)


- G(q,)


= 1-2a


(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 = q2-q, 1+ -2a-IG(q2)-G{q,)l}


then


= -l+g(q,) = 0


= 1-g (q2)


thus


S Ilr


In r\







13

Many methods partially apply the percentile method to generate a bootstrap

confidence interval, for instance, the bootstrap-t, Beran's B and the real bootstrap


confidence interval (which will be presented in next section).


Both the quasi-pivotal


method and the percentile method directly find two end points of a confidence

interval. As we have shown above the quasi-pivotal method is the best one among

those methods of generating bootstrap confidence intervals.

The quasi-pivotal method is also our suggested correction method. To take


one example,


we can correct the bootstrap-t by using the quasi-pivotal method


instead of using the percentile method when generating a bootstrap confidence

interval. To get a bootstrap confidence interval ofT(Z,p) at level 1-2a, by using the

quasi-pivotal 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-


1-a ),


where


G is the


empirical


cumulative


bootstrap


distribution


function. Unless the empirical cumulative distribution function is symmetric, the

corrected (by the quasi-pivotal method) bootstrap-t method is always better than

the bootstrap-t method because it automatically corrects for skewness whereas the

bootstrap-t does not. In the case of symmetry, these two methods are equivalent,

because of the equivalency of the quasi-pivotal 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 quasi-pivotal 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 quasi-pivotal









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(1-a)% 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(1-a)% 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


b-th


bootstrap sample, the real confidence interval for [ at level 1-2a 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 1-2a is


B b-i


(2.9)


G-IB ) SEbp*)
B b=-1


corrected


quasi-pivotal


method)


bootstrap


confidence interval at level 1-2a 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


X-r(r=5,1


=20);


from the Chi-Square


distribution with degree of freedom 1,


P(Xj3) =0.542


X-T(r= 10,X= 10);


from


the exponential distribution with mean 2,


P(Xi) =0.530


X-r(r=20,X= 10);


from


standard


normal


distribution


with


PX3) =0.500


X-N(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 quasi-pivotal 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
X-Q,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 t-mr 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 quasi-pivotal


method (QP),


the bias-corrected


percentile method


(BC),


the accelerated bias-


corrected percentile method (BCJ),


the real bootstrap confidence interval (RCI),


corrected (by the quasi-pivotal method) real bootstrap confidence interval (CRCI),

the bootstrap-t method (BST) and the corrected (by the quasi-pivotal method)


bootstrap-t


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 quasi-pivotal 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 quasi-pivotal 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


(bias-corrected


percentile


method)


(accelerated bias-corrected 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 quasi-pivotal 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 a-trimmed mean is defined as follows:


n-Inal


-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 quasi-pivotal method suggested in


this paper gives the best result


theoretically


empirically


true


distribution


is skewed.


true


distribution is symmetric, theoretically, the quasi-pivotal 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 quasi-pivotal method or a method corrected by

the quasi-pivotal 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 Chi-square 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:


x-r(r=5,X=20)


and


P(Xp1) =o


.560


with mean=0.25.


dist2:

dist3:


x-r(r=io,x
x-P (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)

= 1-F(-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 i-th observation. In the case of a nonlinear model,


i=1,2,


S.., nj


(3.6)


where


errors


are independently


identically


distributed,


equation for the i-th observation has a unique solution for s,,


-h,(y,,x,43).


(3.7)


This defines

replace f3 by a


the generalized error for the i-th 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 i-th 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 log-likelihood 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 i-th 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)


= 1-F[-(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,


y--0( + 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 bounded-influence 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 two-step 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 'x-Wl1
(1/21t


(4.4)


For model (4.1),


we have the log-likelihood 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 first-order


condition for a maximum, we have (see Maddala 1983, p152-153)


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)


1-D,.


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 B-i


M I-


+ (IB-3)2


(4.9)


where


is the average bootstrap estimate of the true parameter P in the m-th


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 mixed-augmented 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 -- --
MIXED-AUGMENTED 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


chi-square 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 chi-square 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


size-corrected


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 size-corrected 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 size-corrections 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 chi-square


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 outer-product 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 chi-square 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 log-likelihood


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 log-likelihood


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 log-likelihood 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
I-1


(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 log-likelihood function for f2 is


s~(I3)


N
= Ex2yt
(=1Xl


-1l
i=


exp(p'x)


(5.6)


1 + exp(p' x)


from


the second


derivative


log-likelihood


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

chi-square 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 a-level 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 a-level 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 a-level based on the asymptotic


chi-square critical value if Wo>X2(1-a) for the Wald test,


LRo>2(1


for the LR


test,


LMo>X2(1-a)


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 chi-square 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 n-C -Ih a rai +4 ra an eta at l.-ta +r. *< 1^ ns raw l an rnwr frnt-n +1-tn 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


chi-square critical values is moderate, an average about 0.05 for both sample


sizes.


Overall for the logit model,


the LM


test with the asymptotic chi-square


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 chi-square 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

chi-square 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 chi-square 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 chi-square 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 chi-square 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 chi-square critical


region substantially over-rejects 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 over-rejects 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 chi-square 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 chi-square 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 chi-square based


critical values always substantially over-rejects 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 chi-square 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)
LR-AD 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)
LR-AD 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 CHI-SQUARE 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)
LR-AD 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)
LR-AD 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 CHI-SQUARE 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)
LR-AD 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)
LR-AD 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 CHI-SQUARE 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)
LR-AD 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)
LR-AD 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 CHI-SQUARE 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)
LR-AD 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)
LR-AD 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 CHI-SQUARE 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)
LR-AD 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)
LR-AD 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 CHI-SQUARE 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)
LR-AD 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)
LR-AD -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 CHI-SQUARE 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)
LR-AD 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)
LR-AD -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 CHI-SQUARE 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 CHI-SQUARE 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 quasi-pivotal


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 quasi-pivotal 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 quasi-pivotal 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 chi-square


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 chi-square 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 mixed-augmented semiparametric bootstrap method does

not perform satisfactorily at all.













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