A MONTE CARLO ANALYSIS OF ALTERNATIVE ESTIMATORS OF … · A MONTE CARLO ANALYSIS OF ALTERNATIVE ESTIMATORS OF THE TOBIT MODEL Getachew Asgedom Tessema No. 73 - April, 1994 ISSN ISBN

A MONTE CARLO ANALYSIS OF

ALTERNATIVE ESTIMATORS OF THE TOBIT MODEL

Getachew Asgedom Tessema

No. 73 - April, 1994

ISSN

ISBN

0 157 0188

1 86389 182 X

A Monte Carlo Analysis of Alternative Estimators of the TobitModel1

by

Getachew Asgedom Tessema2

Department of EconometricsUniversity of New England

Armidale, NSW, 2351.

February 1994

1This is a revised version of the paper presented at the Ph.D. Conference in Economicsand Business, University of Western Australia, Perth, November 3-5,1993.

~I would like to thank Prof. William E. Griffiths and Associate Prof. Howard E. Doranfor their guidance and encouragement. I am also grateful to Tim Coelli who answeredmany of my programming problems.

ABSTRACT

In recent years, many types of Tobit models have been suggested andvarious estimation methods proposed [see the survey by Amemiya (1981,1984)]. However, little attention is given to the finite sample properties ofthe various estimators.

This paper examines the small sample properties of some of the estimatorsof the Tobit model. These estimators include, among others, the Heckman’stwo-step estimator (H2S), the maximum likelihood estimator (MLE) and twononlinear least squares estimators. Further, a three-step estimator (35E)and its weighted version, a weighted three-step estimator (W3SE), are alsosuggested and investigated. The paper investigates the effects of sample size,degree (level) of censoring and error distribution on the properties of theestimators.

Under normally distributed error terms, the MLE estimator performedbetter than all estimators, followed by the 3SE estimator. However, boththe MLE and 3SE estimators appear to be .sensitive for skewed @hi-square)distributed error terms. On the other hand, given a low level of censoring,the MLE estimator performs well under the student’s-t distribution. If thedegree of censoring is high, the MLE estimates under the student’s-t dis-tribution can be less efficient than the 3SE estimator. The H2S estimator,although less efficient compared to 3SE or MLE estimators in all cases, seemsto perform well, given low levels of censoring. However, the H2S estimatorcan be misleading if used for hypothesis testing and confidence intervals evenfor large sample sizes and low degrees of censoring.

ii

1 INTRODUCTION

1 Introduction

Tobit models refer to regression models involving dependent variables forwhich observations are limited to a certain range. The Tobit model wasfirst suggested in the pioneering work of Tobin (1958) in which he analysedhousehold expenditure on durable goods by considering the fact that thedependent variable cannot take negative values.

In recent years numerous applications of tobit models have appearedover a wide range of areas in economics. Examples of applications includelabour supply models [Heckman (1976), Keely, Robins, Spiengelman andWest (1978) and Wales and Woodland (1980)], demand for housing [Lee andTrost (1978)], modelling for public decision [Foot and Poirier (1980)], house-hold expenditure models [Jarque (1987)], demand for imports [Wu (1992)].Theoretical and empirical surveys on tobit models were given by Amemiya(1981, 1984).

On the other hand, many types of tobit models have been suggestedand various estimation methods proposed. Examples of papers which arerelated to the theoretical aspects of various estimators and their propertiesinclude those of Amemiya (1973, 1978, 1981), Ooldberger (1981), Greene(1981, 1983, 1991), Heckman (1979), Olsen (1978), White (1980), Powell(1984, 1986), Peracchi (1990) and Chib (1992). However, almost all thetheoretical studies are concerned with the asymptotic or large sample prop-erties and/or computational ease of alternative estimators. In other wordslittle attention is given to the finite (small) sample properties of the variousestimators [Paarsch (1984), Flood (1985)]. Thus the purpose of this study isto make a contribution towards this end. Specifically, we use Monte Carlotechniques to assess the relative performance of the various estimators of themodel.

The paper is organized as follows. Section 2 presents the specification ofthe model and briefly reviews the various estimators of the model. Section3 presents the design of the Monte Carlo experiment followed by the dis-cussion of the Monte Carlo results in Section 4. Finally, Section 5 presentsconclusions.

2 THE MODEL AND ITS ESTIMATORS 2

2 The Model and its Estimators

The standard Tobit model is defined as follows:3

y~ = x~#+ui, i= 1,...,N, (i)

~ = ~’ if~’ > O,= 0 if y~ _< 0, (2)

where u~ are assumed to be independently and identically distributeddrawings from N(0,~2), # is a (kxl) vector of unknown parameters and it isalso assumed that y~ and x~, i=l,...,N, are observed but y~’ are unobserved ify~_<0.

Since there are many generalizations of the Tobit model, the model speci-fied in (1)-(2) above is usually referred as the Standard Tobit Model, hereafterthe Tobit model. Note that the model (1)-(2) is also referred as the censoredregression model.

Given this model, a number of estimators have been suggested to estimatethe parameters (fl and ~r2) of the Tobit model. Some of these estimators willbe considered below.

2.1 Properties of Least Squares Estimators of the To-bit Model

Given the model in (1)-(2), it is important to examine the properties of theordinary least squares estimators of the model. To do this we first define thefollowing: Let No be the number of observations for which yi = 0, and N1 bethe number of observations for which y~ > 0 such that N = No + N1. Further,we define f~ and Fi to be the density function and the distribution functionof the standard normal variable, respectively, evaluated at z~ =

Suppose we consider the observations in which the dependent variable ispositive (i.e. y~ > 0). That is

3Amemiya (1984) used five types of classifications for Tobit models, for convenience,and defined the Standard Tobit model as Type I.

2 THE MODEL AND ITS ESTIMATORS

Further, if we assume normality as in the Tobit model, equation(3) canbe shown by straightforward integration to be

(4)where ~(z~) = f(zi)/F(z~) which is known as the hazard rate in reliability

theory and its inverse is known as Mill’s ratio.As can be seen from equation (4) applying least squares omits the term

~r~(zi) which is not independent of x~. This leads to a biased and inconsistentestimator of 9. However, the magnitude and direction of the bias is unknownwithout making further assumptions. Goldberger (1981) assumed that thexi are normally distributed and showed that ordinary least squares based onpositive values of y~ (OLSP) is biased towards zero.

Similarly, consider the least squares estimators when applied to the modelusing all the limit and non-limit observations. That is, in this case we considerthe unconditional expectation which yields the equation

E[y~] = F(z~)(@3) + o’f(z~)

Following the same assumpt ions as Goldberger (I 981 ), Greene ( 1981 ) showedthat the ordinary least squares (OLS) estimators based on all observationsare biased and he proposed a consistent estimator which is referred to as thecorrected ordinary least squares (COLS) estimator. However, a Monte Carlostudy by Flood (1985) indicates that, under normality, the COLS estimatoris biased even when the sample size increases.

In general we conclude that the least squares estimators using both thepositive observations of y~ or al! the N observations do not provide good es-timators for the Tobit model. Thus alternative estimators have been devisedwhich are discussed in the following sections.

2.2 Maximum Likelihood Estimation of the Tobit Model

The maximum-likelihood procedure can be applied to obtain consistent esti-mators of the parameters, fl and cr~. For convenience we assume that, withoutloss of generality, the first NI observations contain the non-zero observationsand the remaining No observations contain zero observations.

Thus the likelihood function of the model can be written as!

L = I-i011 -/2/]H1 (2~rcr~)l/2 exp{-(yi- x’ifl)/2o-~} (6)


where the first product is evaluated over the No observations for which y~ = 0and the second product is evaluated over N1 observations for which y~ > 0.

The log-likelihood function is

1logL = ~o log(1- Fi)+ ~ logtz )"’’7r°’~’1’~"

where E0 is the summation over the No observations for which y~ = 0 andis the summation over the N~ observations for which y~ > 0.The first derivatives of log L used to obtain the maximum are4

The maximum-likelihood estimators of the parameters of the Tobit modelare defined as a solution to the partial derivative equations (8) and (9) whenthey are equated to zero. These equations are nonlinear in the parametersand hence must be solved using iterative methods. Amerniya (1973) provedthat the Tobit ML estimators are strongly consistent and asymptoticallynormal with the asymptotic variance-covarince given by the inverse of theinformation matrix which is defined by

Where

4The second derivatives of log L are given in Amemiya (1973, pp. 1005-1006).

(11)


Amemiya (1973) showed that the Tobit likelihood function is not globallyconcave with respect to the original parameters, ~3 and cr2. However, Olsen(1978) proved the global concavity of log L based on the reparametrizationa = fl/o and h = 1/a. This implies that a standard iterative method such asNewthon-Raphson or the method of scoring always converges to the globalmaximum of log L.~ Further, because of the invariance property of maximumlikelihood estimators one can obtain a unique solution in terms of the originalparameters of the tobit model. Tobin (1958) suggested the same procedureand in order to speed up convergence of the estimation process, he proposedan initial estimator based on a linear approximation of the reciprocal of theMill’s ratio. However, Amemiya (1973) showed that Tobin’s (1958) initial es-timator is inconsistent, and he proposed an alternative consistent estimator.He also showed that the second-round estimator based on his initial estima-tor is consistent and asymptotically normal. However, based on a simulationstudy of a single replication of sample sizes of 1000 and 5000, Wales andWoodland (1980) indicated that Amemiya’s initial estimator is inefficient.

Given the assumptions of normality and homoscedasticity, as assumedin the Tobit model, the maximum likelihood estimators are consistent andasymptotically efficient. However, they are sensitive to departures from theseassumptions. If the assumption of normality of the disturbances is violated,Goldberger (1980) and Arabmazar and Schmidt (1982) have shown that themaximum likelihood estimators lead to inconsistent estimates. Further, asshown by Hurd (1979), Maddala and Nelson (1975) and Arabmazar andSchmidt (1981), heteroscedasticity of the disturbances can cause inconsis-tency of the parameter estimates, even when the shape of the error density iscorrectly specified. Thus the maximum likelihood estimators are not robustto the assumptions of the model. Further, except for the simplest cases, thederivatives of the loglikelihood functions are very complicated and hence thecomputational cost of the maximum likelihood estimators can be very high.

5Greene (1990) investigated the possibilty of multiple roots of the Tobit !oglikelihoodfunctions based on the original parameters and showed that the problem of multiple rootsin the Tobit model is less obvious than suggested in earlier literature.


2.3 Heckman’s two-step Estimator

Heckman (1976) proposed an alternative estimator which yields consistentestimates of the parameters based on a two-step procedure6. The two-stepestimator uses the observations for which y~ > 0 and proceeds as follows.

Consider the conditional expectation for which y~ > 0. That is equation

(4) can be written as

(12)

where e~ = y~- E(y;ly~ > 0) so that E(e~) = 0 and the variance of ~ isgiven by

= - _ (la)Note that as we have discussed earlier, a and hence ~(.) is unkown in

equation (12). Thus ordinary least squares estimates of y on x gives biasedestimates. Heckman (1976) treats the bias as a result of omitted variablesand he suggested a two-step procedure which involves the estimation of theomitted variables using the probit maximum likelihood estimator in the firststep of his procedure and then ordinary least squares can be applied in thesecond step after replacing the omitted variables by their consistent esti-mates. Specifically, Heckman’s two-step procedure is given by

Step 1. Estimate a by probit MLE, say &, to obtain ~(x~&). These areconsistent since the probit maximum likelihood estimator is consistent.

Step 2. Replace ~(x~a) by ;~(x~&) and then regress Yi on x~ and ;~(x~&) toobtain consistent estimates of fl and ~r based on the observations forwhich yl > 0.

To discuss Heckman’s two-step estimator further, equation (12) can bewritten as~

y~ = x~/3 + cri(x~a) + e~ + ,~ (14)where r]~ = cr[,k(x~a)- .~(x~&)]. And using vector notation, equation (14) canbe presented as

6Heckman’s two-step estimator was originally suggested for a system of two equations,but can be used in a single equation with some adjustments.

7See Amemiya (1984, 1985).


(15)where Y is an Nlxl vector of the non-zero observations on the dependentvariable, X is a Nixk matrix of explanatory variables corresponding to Y,/~is an Nix1 vector whose elements are ~(x~a), and ~ and , are vectors of N,elements of ~ and r]~, respectively.

Further, equation (15) can be written as

Y = 27 + (e + (16)

where 2 = (X,/~) and 7 =Thus, Heckman’s 2-step estimator of "~ is defined by

~/ = (2,2)-12,Y. (17)

where ~ is asymptotically normal with mean 7 and the asymptotic variance-covariance matrix is given by

V.~ = o2(Z’Z)-Iz’[Y]~ q-(1~- ~)X(X’D1X)-Ix’(I- ~)]Z(Z’Z)-1 (18)

where cr~P~ = E(ee’) is an NlxN1 diagonal matrix whose diagonal elementsare var(e~) as defined in(13), X is Nlxk matrix of observations and Di isNlxN1 diagonal matrix whose diagonal elements are given by F(x~e~)-l[1 -F(x:a)]-l f(x:o~)2.

Note that the H2S estimates are likely to be imprecise because of themulticollinearity problem, which is often very strong, between the explana-tory variables; i.e., the x’s and ),(.) in (12). Further, the H2S estimator doesnot guarantee that the estimate for ~r, which is the coefficient of ~(.) in theright hand side of equation (12), will be positive. However, Heckman (1976)suggested that ~ may also be estimated as follows:

Equation (12) can be rewritten as

y~ = ~[x’~ + a(.)] + ~ (19)

It can be shown that both the left hand side, yi, and the right hand side,x~cz + ,~(.), variables in equation (19) are positive. This implies that leastsquares estimates of y~ on x~ + .~(.), where c~ and hence .~(.) is estimatedusing probit MLE in the first stage, are guaranteed to produce a positiveestimate of ~r, a feature which is not guaranteed directly from equation (12).


Given this, we can estimate the coefficients of the model, fl’s, by addingone more step as follows.

Let ~0 be an estimate of ~r from equation (19). Then substituting a0 inequation (12) and rearranging the model gives

(20)

which can be written as(21)

where #~ = y~ - cr0A(.) and other components of the model are as definedin (12).

Note that equation (21) is a simple linear model and hence one can es-timate the parameters of the model by regressing ~)~ on the x’s. This proce-dure can be referred to as The 3-step estimator (3SE). Interestingly, itcan be shown that the H2S and 3SE estimators have the same asymptoticproperties.S

2.4 Variations of Heckman’s two-step Estimator

Below, we consider alternative techniques of estimating equation (12). Specif-ically, we discuss estimation of the model using weighted least squares, non-linear least squares and nonlinear weighted least squares based on the obser-ations for which y~ > 0.

2.4.1 Weighted Heckman~s two-step (WH2S) Estimator

It should be noted that equation (12) represents a heteroscedastic regressionmodel which implies that the least squares estimates are not efficient. Heck-man (1976, 1979) suggested that more efficient estimtes can be obtained usingwieghted least squares in the second step of the procedure with the weightsgiven by (13).

Let the resulting WH2S estimator based on the N1 observations be de-noted by ~/w. It can be shown that it is consistent and asymptotically nor-mal with the asymptotic variance-covariance matrix given by [see Amemiya(1984), pp. 12-16]

SThe asymptotic results are not included in this paper but will be available in anotherrelated paper soon.


V~w = ~2{Z’[E ÷ (I- E)X(X’DiX)-IX’(I- E)]-IZ}-~ (22)

Note that using the same analogy we can obtain a weghted version of the3SE estimates given in equations (19)-(21) above, with the weights given by(13). This estimation procedure is reffered as the weighted 3-step estimator(W3SE).

2.4.2 Nonlinear Least Squares (NLS) and Nonlinear WeightedLeast Squares (NWLS) Estimators

The expression given by (12) can be viewed as a nonlinear problem. Thereforeone can apply nonlinear least squares with/3 and ~r estimated simultaneouslyby minimizing

~iN=~l[Yi -- X,~ -- O’/~(X’i~3/O’)]2 (23)

It should be noted that even if x}/3 is linear in /3 it involves a nonlinearestimation problem in view of the dependence of ,~(.) on/3 and

Let ~N be the nonlinear least squares estimator obtained by minimizingequation (23) based on the N1 observations. Then it can be shown that ~Nis asymptotically normal with mean 3’ and asymptotic variance-covariancematrix given by

= (24)

wheres = (ZX, D2a), (25)

and D~ is the N~xN~ diagonal matrix whose ith element is [1 ÷ (x~a)~ ÷

Alternatively, equation (12) can be estimated using nonlinear weightedleast squares. That is to say, the parameters /3 and a can be estimatedsimultaneously by minimizing the weighted sum of squares of the residualswhich is given by

[Y~ - ~/3 - ~(~)]~ (~6)

If we let ~NW to be ~he nonlinear weighted least squares estimator basedon the N~ observations, i~ can be shown ~hat i~ is asymptotically normal with


mean 7 and the asymptotic variance-covariance matrix is given as:

Y~Nw = ~2(s,~-1 s)-i (27)

2.5 Two-step Estimators based on Unconditional Ex-pectation

It should be noted that Heckman’s two-step estimator uses only those ob-servations for which y~ > 0. Wales and Woodland (1980) suggested that asimilar procedure may be applied using al! the observations. That is, includ-ing the observations for which y~ -- 0. A further discussion of the estimatorsbased on all observations and their asymptotic distributions was presentedin Amemiya (1984, 1985). The estimation procedure using all observationsproceeds as follows:

The unconditional expectation of y~ given by (5) can be written as

(28)

where ,~ = ~ - E[~] such that E[~d = 0 and

Further, equation (28) can be written as

y~ = F(x’~&)[x~fl + cr.~(x~&)] + 5~ + ~ (30)

where(~ = [F(x~a) - F(x~&)]x~fl + ~r[f(x’~a) -

In matrix notation (30) can be expressed by

Y=b2~+5+~ (31)

where b is an NxN diagonal matrix whose elements are F(x~&), ~ and ~ arevectors of order N whose elements are 5i and ~i, respectively, and 2 and ~are as defined in (16).

Note that the models given by (28) and (31) have the same form asthe previous models given by (12) and (16), respectively. One noticeabledifference between the two sets of models is that while models (12) and (16)


were derived using the N1 observations, the other two are obtained basedon all obseravations, N. Next, we consider the models (28) and/or (31) andapply estimators which are analogous to those discussed in the preceedingsection, namely, ordinary least squares, weighted least squares, nonlinearleast squares and nonlinear weighted least squares estimators.

2.5.1 Heckman’s two-step Estimator based on Unconditional Ex-pectation (H2SU) of the Model

One way of estimating model (31) is to apply ordinary least squares in thesecond step of the procedure. It should be clear that the first step stillinvolves the estimation of a using probit MLE estimation.

Let ~ be the H2SU estimator of 7 based on all observations. Then it isdefined by

It can be easily shown that 9 is consistent and asymptotically normallydistributed with mean 3’ and the asymptotic variance-covariance matrix isgiven by [see amemiya (1984), pp. 14-15].

(33)

where ~r2f~ is an NxN diagonal matrix whose i*h elements are Var(Si) givenby (29).

2.5.2 Weighted Heckman’s two-step Estimator based on the Un-conditional Expectation (WH2SU) of the Model

The model given by (28) is a heterscedastic model implying that more efficientestimates can be obtained using weighted least squares in the second step ofthe procedure, with the weights given by (29). Let the WH2SU estimator,based on all observations, be denoted by ~/w. Then ~w is consistent andasymptotically normal with the asymptotic variance-covariance matrix given

byV~,W = o’2(Z’D2a-IZ)-~ (34)


2.5.3 Nonlinear Least Squares (NLS) and Nonlinear WeightedLeast Squares (NWLS) Estimators

Nonlinear least squares and nonlinear weighted least squares estimators canbe applied to estimate model (28). These estimators are consistent and theirasymptotic distributions can be obtained by noting that the results of alinear regression model hold for a nonlinear regression model asymptoticallyby treating the derivative of the nonlinear regression function with respectto the parameter vector as the regression matrix.9 Following this procedureand using all observations it can be shown that the NLS estimator, ~/N, hasthe same asymptotic distribution as the H2SU estimator, "~. Similarly, theNWLS estimator, ;’/NW, has the same asymptotic distribution as the WH2SUestimator, ~/w.

Finally, given the above estimators of the Tobit model, the followingpoints are worth noting:

The estimators, "~ and "~ cannot be ranked on the basis of their asymptoticcovariance matrices. This is because the difference of the matrices given by(18) and (33) is generally neither positive nor negative definite. This impliesthat the preference of any of the two estimators depends on parameter val-ues. Similarly, one cannot make definite comparison between the covariancematrices given by (18) and (24) or between the covariance matrices given by(22) and (24). Therefore the choice between the corresponding estimators"~ and ~N and ~iw and "~NW, respectively, depends on the empirical values.Further, since the estimators ~/N and ~NW have the same asymptotic distri-butions as ;~ and ;~w, respectively, their relative performance can only bedetermined based on empirical results. Thus the objective of this study is toanswer some or part of these questions.

In a standard linear regression model, the least squares estimators areunbiased and consistent for a wide class of distributions of the disturbances.However, the situation is quite different for Tobit models. That is, the as-sumption of normality of the disturbances, which is a common feature forboth the MLE and Heckman’s two-step estimators, is essential for the proofsof consistency. In general the properties of these estimators are very sensitiveto violations of the assumptions of the model. This has led to the develop-ment of estimators which are robust (or less sensitive) to the functional form

9See Amemiya (1983) in Handbook of Econometrics, pp. 337-341, Edited by ZviGriliches.

3 DESIGN OF THE EXPERIMENT !3

of the distribution of the disturbances. These types of estimators include thesemi-parametric estimators; Powell’s (1984) least absolute deviations (LAD)estimator and Powe!l’s (1986) symmetrically censored least squares (SCLS)estimator. Other estimators of the model, which have been proposed morerecently, include the bounded influence estimators [Peracchi (1990)] and theBayessian estimator by Chib (1992). Again, the small sample properties ofmost of these estimators need to be investigated.

3 Design of the Experiment

The specific model to be investigated in this study is of the form:

i= 1-,...,N (35)

(36)

where X=(1, xl, x2) is an Nx3 matrix of observations containing a columnvector of l’s corresponding to the constant term and observations on theexplanatory variables xl and x2,

y*, the latent variable, is an Nxl vector which is assumed to be observedonly if it is positive,

y is an Nxl vector of observations on the dependent variable consistingof N~ positive (non limit) observations corresponding to the positive valuesof y* and No = N - N1 zero (limit) observations,

/3=(/30, fl~, fl2)’ is a 3xl vector of unknown parameters to be estimated,and

u is an Nxl vector of identically and independently distributed randomerrors with mean/*=0 and variance a2. Note that if ui’s are normally dis-tributed model (35)-(36) becomes the standard Tobit model defined by (1)-(2).

Given this model, the objectives of the Monte Carlo experiment are toinvestigate the effects of the following on the estimators of the Tobit modelwhich we have discussed in Section 2.

1. To investigate the effects of changes in distributional assumptionsfor the disturbance term,

3 DESIGN OF THE EXPERIMENT 14

2. To investigate the effects of the degree of censoring, and

3. To investigate the effects of sample size.

To investigate the effects of distributional assumptions we consideredthree distributions. These are:

The standard normal distribution which represents the usual assump-tions of the model,

The students’-t distribution with three degrees of freedom that repre-sents a symmetric but fat tailed distribution (i.e., symmetric departuresof the model), and

The chi-square distribution with four degrees of freedom that representsa skewed distribution.

The three distributions are generated in such a way that the mean andthe variance will be zero and one, respectively.

The effects of the degree of censoring on the performance of the estimatorsof the model are examined at three levels, namely, 25%, 50% and 75% degreesof censoring. These levels of censoring are chosen to represent a wide rangeof economic or other data to which the model may be applied.

Three levels of sample size are considered in order to examine the effectsof sample size on the performance of the estimators of the model. That is,the sample sizes of 100, 200 and 400 which correspond to small, medium andlarge sizes, respectively.

The various components of the model (35) are determined as follows:The observations on the explanatory variable xi~, i=l,...,N, are generated

from the interval [0,4] equidistantly where the distance depends on the samplesize. Similarly, the observations on the second explanatory variable,i=l,...,N, are generated uniformly from the interval [-1,1]. Once the valuesof the explanatory variables are determined from their respective intervalsthey remain the same throughout the experiment.

The parameters fll and f12 are set to be equal to one, (i.e., fll = /3~ =1). However, /30 is used to determine the degree of censoring and hencetakes different values depending on the particular level of degree of censoringand type of distribution. For example, given the above data generationprocess, if the disturbances are normally distributed with mean zero and

4 DISCUSSION OF RESULTS 15

variance equal to one, i.e., the standard normal distribution, then a value offl0 = -0.75 yields approximately 25 percent degree of censoring. Similarly,the approximate levels of degree of censoring of 50 percent and 75 percentcan be obtained by setting fl0 equal to -2.00 and -3.25, respectively. Somepreliminary experiments were conducted to determine the value of fl0 for thethree types of distribution and degree of censoring.

Finally, since the explanatory variables are fixed, the coefficient of deter-mination, R~’, is controlled by setting the variance of the disturbance termto a certain constant. In this case cr~ = 1. This also implies that the signalto noise ratio be constant; the signal being the variation of the dependentvariable due to variations of the systematic component and the noise repre-sents the variance of the error term of the model. In general, the design ofour experiment is similar to that of Wales and Woodland (1980) and Paarsch(1984).

Given the data generation process, a run length of M=3000, (i.e., thenumber of replications for each sample), is used throughout the experiment.

4 Discussion of Results

In this Section we discuss the results obtained for the various estimators ofthe tobit model. A total number of 11 estimators are included in this analysis.For convenience, the estimators of the tobit model can be divided into twomain categories: those estimators using only non-limit, N1, observations;and those which use all N observations. This division is made purely fordiscussion purposes because of the large number of estimators involved inthe study. Section 4.! presents the results obtained for those estimatorsusing only N1 observations. Results of the estimators of the tobit modelwhich are obtained using all N observations are discussed in Section 4.2. Afew estimators are selected based on the analysis made in Sections 4.1 and4.2 and further analysis and comparison of these estimators is presented inSection 4.3.

4.1 Estimators using only N1 observations

Below, we discuss the Monte Carlo results obtained for the estimators usingonly the positive observations on y~. These estimators include:


(i) The ordinary least squares estimator using only positive (non-limit)observations on y~ which is denoted by OLSP,(ii) The Heckman’s two-step estimator (H2S),(iii) The weighted Heckman’s two-step esimator (WH2S),(iv) The three-step estimator (3SE),(v) The weighted three-step estimator (W3SE), and(vi) The nonlinear least squares estimator using only positive observationson y~ (NLSP).Summary statistics of the Monte Carlo results for these estimators are

provided in Tables 1-4. As a guide to interpreting the tables, consider Table1 which presents estimated results of estimators using N1 observations, givena sample size of 100, 25% degree of censoring and for the three distributions,namely, normal, students’-t and chi-square distributions. The first row ofTable 1 presents the list of distributions and the parameters (/~1 andto be estimated under each distribution. The corresponding true values ofthe parameters are listed in row 3 of Table 1 where fll=fl2=l in all cases.The numbers in brackets indicate the column number in the table. Thelist of estimators are given in column (1). In column (2) are the summarystatistics for each estimator, i.e., the Average Monte Carlo Estimate (AME),the Standard Error (SE), the Bias (BIAS) and the Root Mean Square Error(RMSE). Finally, the corresponding estimates are presented in columns (3)-(8). For example, given a sample size of 100, 25% degree of censoring andnormally distributed error terms, the AME of ~1 using OLSP is equal to

0.760 ( see column(3) of Table 1). And this is obtained as follows:

1 y~3ooo~AME(~,) = fl~ - 3000~=, ~,,i -- 0.760 (37)

Similarly, continuing downwards in the same column is its standard error(SE) which is calculated as

3000 ^sEO1) = ~~)~/3oooand its bias is given by

--0.093 (38)

BIAS(~I) = fl~ - fl~ (39)

= 0.760- 1.000 = -0.240 (40)


Finally, its RMSE is calculated by

3000 ^RMSE(~I) = ~/~i=1 (flli -- /~1)2/3000 : 0.257 (41)

Other tables may be interpreted in a similar way. As discussed in Section2, it is a well known fact that the Ordinary Least Squares estimator using onlypositive values of y~ (OLSP) provides estimates which are biased. However,what may be of interest in this case is the degree and probably the direction ofbias which may result due to changes in sample size, degree of censoring anddistributional assumptions of the error term, and the relative performancecompared with other estimators.

Table 1 shows that, given a sample size of 100 and 25% degree of censor-ing, /31 and/32 are estimated, respectively, with biases of 24.0% and 21.2%under normally distributed error terms, 17.5% and 16.8% under student’s-tdistribution and 24.7% and 23.2% under chi-square distribution using theOLSP estimator. These biases are substantially large compared, say, to theH2S estimates in which the coefficients are estimated with at most 2% biasunder similar conditions. The bias of the OLSP estimates remains high evenfor large sample sizes. Further, the biases of the OLSP estimates increasealmost linearly with the degree of censoring.

Note that our results show that the OLSP provides estimates which are’biased towards zero’ in all cases, i.e., for all sample sizes, degrees of cen-soring and distributional assumptions. In general, one cannot determine thedirection of bias of the OLSP estimator without making further assumptions.Goldberger (1981) has shown that, if the explanatory variables are normallydistributed, OLSP estimators are biased towards zero.

The H2S estimator, compared to the OLSP, provides consistent estimates.As shown in Table 1, given a small sample size and low degrees of censoring,the biases of the estimates of/31 and f12 using H2S estimator, respectively,are 1.1% and 0.9% under a normal distribution, 2.1% and 1.6% under t-distribution and 1.3% and 0.5% for the chi-square distribution. These resultsare in contrast to the biases of the OLSP estimator which are in the rangeof about 16 to 25 percent. Bias appears to be relatively small for H2S esti-mates for all sample sizes and distributions provided that the number of limitobservations remains low (i.e., 25% degree of censoring). However, as the de-gree of censoring gets higher, bias becomes a problem for the non-normaldistributions.


Further, our results indicate that the efficiency of the H2S estimates de-clines dramatically with higher levels of degree of censoring. For example,given a large sample size of 400 and normally distributed error terms, theestimates of the SE’s of fll and f~2 increased by almost 50% when the propor-tion of limit observations in the sample increased from 25 to 50 percent, andare six to seven times higher when the degree of censoring is 75%. In general,the effects of the degree of censoring on the performance of the H2S estima-tor is very severe. The H2S estimates deteriorate further under non-normaldistributions coupled with higher levels of censoring.

It is important to note that the weighted Heckman’s two-step estimator(WH2S), compared to the H2S estimator, provides more efficient estimatesof f!l and f12 only when the errors are normally distributed. As can beseen from Tables 1-3, the WH2S estimates are relatively more efficient thantheir H2S counterparts under normally distributed error terms. Otherwise,the WH2S estimator gives results which are relatively inferior to the H2Sestimates. Similar results can be observed for all sample sizes and degreesof censoring. In general, the WH2S estimator is sensitive to changes indistributional assumptions of the error structure.

On the other hand, the 3SE estimator provides results which are superiorto the H2S estimator or its weighted version, the WH2S, in all cases. Given asmall sample size (i.e., N=100), the RMSE for the 3SE estimator are alwayslower for the three levels of degree of censoring as shown in Tables 1-3.Similarly, the results for the medium and large sample sizes depict that the3SE estimator gives more efficient estimates under al! levels of degrees ofcensoring. However, what is more important about these results is that the3SE estimator is not only more efficient but also less sensitive to changes inthe degree of censoring in a given sample. For example, Table 3 depicts thatwhen the degree of censoring increases from 25 to 75 percent the RMSE for flland f12 increased by just over two times, i.e., from 0.114 and 0.186 to 0.285and 0.402, respectively. This result is incomparable to the H2S estimateswhere the increase in RMSE of the coefficients is six to eight times for thesame changes in degrees of censoring. This is also true for the medium andlarge samples which suggests that the 3SE estimator is much less sensitiveto changes in the degree of censoring for a given sample.

As to the effects of distributional assumptions of the error term, the 3SEestimator seems to perform better under normality conditions and gets worseunder the chi-square distribution. This is particularly significant for higher


levels of degrees of censoring. However, it is important to note that underall conditions the 3SE estimator actually outperforms the H2S estimates interms of reliability (efficiency). Furthermore, the 3SE estimator is muchless sensitive to increases in the propotion of limit observations in a sample,compared to the H2S estimator.

In general, given the estimators which use only N1 observations, the 3SEestimates have the lowest RMSE, except when the errors are normally dis-tributed. In such cases it’s weighted version, the W3SE, yields more efficientestimates. Similar to the WH2S estimator, the W3SE estimator is sensitiveto changes in distributional assumptions of the error structure. One possibleexplanation for the sensitiveness of the weighted estimators to non-normalityof error terms could be that the estimation of the weights involves resultswhich are obtained on the assumption of normality. That is, the expressionsfor the weights themselves depend on normality. Specifically, as shown inSection 2, the weights involve results such as the probability and cumulativedensity functions of the standard normal distribution. Hence, any departurefrom normality of the error terms may result in inefficiency and/or inconsis-tency of estimates. Note that, as discussed in Section 2, the 3SE estimatorguarantees that the estimated value of ~r, which is given in the right handside of equation (12), is positive in all cases, which is not the case for theH2S estimator.

Considering the NLSP estimator, the complex nature of the function hasaffected the speed and convergence of the nonlinear estimates in our exper-iment. In SHAZAM the nonlinear regressions are estimated by maximumlikelihood procedure, assuming that the errors are additive and normallydistributed. The estimation procedure uses the algorithm known as a Quasi-Newton method. However, although not difficult, the nonlinear estimationin SHAZAM is rather slow and depends on the complexities of the nonlinearfunction. In this case the NLSP estimator takes significantly longer computertime to converge than do other estimators.1°

Note that convergence is not always guaranteed in nonlinear least squares,even after a large number of iterations. In this experiment we considered amaximum of 100 iterations, and to help speed up convergence, the true pa-

1°For example, it takes approximately 4 minutes CPU time to obtain results from 100samples (replications) using NLSP compared to about 20 seconds CPU time for H2S or3SE estimators.


rameter values were used as starting values for the NLSP estimates. However,some did not converge after 100 iterations. For example, given a sample sizeof 200 observations and 75% degree of censoring, of the 3000 replications(samples), 4, 2, and 5 percent of the samples did not convege, respectively,for normal, student’s-t and chi-square distributions. Convergence is more dif-ficult in small samples and higher degrees of censoring. Results for sampleswhich did not converge are excluded from the experiment.

Given this, our results in general show that the NLSP estimator providesrelatively very inefficient estimates. As shown in Table 1, given normallydistributed error terms and 25% degree of censoring, the SE’s of the estimatesof fll and f12 are 0.215 and 0.291, respectively, for a sample size of 100. Theseresults are over 10 percent larger than the corresponding H2S estimates, anda further 50 to 90 percent larger than the 3SE estimates. By increasingthe sample size to 200 the NLSP estimates come relatively close to the H2Sestimates, but are still much less efficient than the 3SE estimates. This isalso true for the large sample size.

Further, as the proportion of limit observations in the sample increases,the quality of the NLSP estimates deteriorates quickly. Table 3 depicts thatthe RMSE of the NLSP estimates almost doubled when the degree of cen-soring increased from 25% to 50% and further increased to over eight timesfor 75% degree of censoring.

As to the effects of distributional assumptions of the error term, the NLSPestimator performs relatively better under normal conditions. Especiallyfor degrees of censoring of 50% and above, the NLSP provides relativelybetter results under normality conditions and gets worse under the chi-square(skewed) distribution (see Table 2 and 3). In other words the NLSP estimatoris not robust to changes in error structure of the model.

Finally, the following points can be concluded from the preceeding dis-cussions:

The 3SE estimator provides estimates which are relatively efficient andless sensitive to changes in degrees of censoring. Bias is not a problem forlower levels of censoring. However, bias becomes a problem for higher degreesof censoring and non-normal error terms. It also seems to perform slightlybetter under normality conditions. The 3SE estimator appears to be thebest, given the estimators which use only the positive observations on

The H2S estimator is generally less efficient compared to the 3SE esti-mator and can be very sensitive to changes in degree of censoring. The H2S


estimator seems to be robust to changes in distributional assumptions, pro-vided that the degree of censoring is low. However, bias becomes a seriousproblem for higher degrees of censoring and non-normal distributions.

It is important to emphasize that the weighted versions of the Heckman’stwo-step (WH2S) and the three-step (W3SE) estimators may improve theefficiency of their conterparts; i.e., the H2S and 3SE estimators, respectively,if and only if the errors are normally distributed. Otherwise, the WH2S andW3SE estimators provide estimates which are inferior to their respective un-weighted estimates. Thus one should take proper caution before applying theweighted versions of the estimators to obtain more efficient estimates as theassumption of normality may not actually hold. In other words, preliminarysteps such as pre-testing for normality of the error structure may be nec-cessary. On the other hand, it should be noted that, even under normalityconditions of the error term, the improvement of the weighted estimates overthe unweighted estimates seems to be marginal.

The NLSP estimator is not one of the best estimators for the followingreasons. It provides inefficient estimates relative to that of the H2S and3SE estimates. Further, the NLSP is not robust to changes in distributionalassumptions of the error term. Bias seems to be a problem and gets worstfor skewed distributions and higher levels of censoring. It is computationallymore difficult and may not always converge. Specifically, convergence isdifficult for small sample sizes and higher degrees of censoring.

Not surprisingly, the OLSP estimator provides poor estimates in all cases.

4.2 Estimators using all N observations

In this Section we discuss the results for estimators which use all limit andnon-limit observations on y~. These estimators may include the following:

(i) The simple ordinary least squares estimator (OLS) using all observa-tions,

(ii) The maximum likelihood estimator (MLE),(iii) The Heckman’s two-step estimator based on the unconditional ex-

pectation of the tobit model (H2SU),(iv) The weighted Heckman’s two-step estimator based on the uncondi-

tional expectation of the model (WH2SU). That is, the weighted version of(iii), and


(v) The nonlinear least squares estimator applied to the unconditionalexpectation of the model (NLSU).

Given these estimators, summary statistics of the Monte Carlo results areprovided in Tables 5-8. The interpretation of the results is similar to those inthe previous tables, except that these estimators utilize all the observationson y~.

As discussed in Section 2, the traditional OLS estimator provides incon-

sistent estimates and hence is not considered appropriate. The interest inthis case is to see its relative performance compared to that of the consistentestimators. Results for a sample of 100 and 25% degree of censoring aretabulated in Table 5. These results suggest that bias is a serious problemfor OLS estimates, it averages around 20 percent for all distributions com-pared to the MLE estimates which range from a minimum of 0.2 percentunder normally distributed errors to a maximum of 4.3 percent for the chi-square distribution. The bias of the OLS estimates remains high and doesnot seem to decline even when the sample size is large (see Table 6). Further,the severity of bias of the OLS estimates increases proportionately with thenumber of limit observations in the sample, irrespective of the total samplesize or distributional assumptions of the error structure of the model. Forexample, given a sample size of 200 and normally distributed error terms,as shown in Table 8, the estimated biases of fll and f12 are given by, respec-tively, -0.197 and -0.211 for a 25% degree of censoring, and -0.505 and -0.550for a 50% degree of censoring. The biases of the OLS estimates of fll andf12 increased further to -0.797 and -0.757, respectively, for a 75% degree ofcensoring. Similar results are observed for low and large sample sizes.

It is not, however, surprising for the OLS estimator to perform badly.What is surprising, from our results, is the poor performance of the estima-tors suggested by Wales and Woodland (1980), the H2SU and the WH2SUestimators. Table 5 depicts that the biases of these estimates ranges from12 to 22 percent for H2SU estimates and deteriorates further to 21 to 47percent for WH2SU estimates. More surprisingly, the bias of the estimatesdoes not seem to decline with the increase in sample size or with changesin the distribution of the error structure of the model (see Tables 6 and 7).Also, as shown in Table 8, the estimates get worse for higher levels of censor-ing. To sum up, despite their large sample properties (i.e., consistency andasymptotic normality), the H2SU and the WH2SU estimators provide verypoor results in all cases.


The nonlinear least squares estimator using all observations (NLSU) seemsto perform well compared to others, with the exception of the MLE estima-tor. As shown in Table 5, given a sample size of 100 and 25% degree ofcensoring, the biases of the NLSU estimates ranges between 4 to 9 percentcompared to about 20% or more for the H2SU or WH2SU estimators over thethree distributions. The biases of the NLSU estimates decline further withincreases in sample size (see Tables 6-7). However, bias becomes a problemfor higher levels of degree of censoring (see Table 8). Furthermore, the NLSUestimates are very inefficient compared to the MLE estimates. For example,Table 5 depicts that the standard errors of the NLSU estimates of fll and f12,given normally distributed error terms, are about 290 and 158 percent higherthan their respective MLE estimates. The results are similar for medium andlarge sample sizes as welt as for the non-normal distributions.11

In general, our results indicate that the MLE estimator provides relativelybetter estimates under all circumstances, given the estimators which use allobservations. Note that under normality conditions, it is widely claimedthat the MLE provides consistent and more efficient estimates. In Table 5,the bias for the MLE of fll and f12 is about 0.2 percent under normally dis-tributed error terms. This is compared to about 2.5 and 4 percent bias understudent’s-t and chi-square distributions, respectively. The bias of the MLEestimates under a normal distribution disappears when sample size increases,but remains almost at the same level for the non-normal distributions (seeTable 6). These results may suggest that, although the bias is relativelysmall, the MLE is not robust to the distributional assumption of the errorterms.

With regard to the efficiency of the MLE estimates, the MLE estimatesappear to be most efficient under normal and student’s-t distributions com-pared to the skewed (chi-square) distribution. Table 5 depicts that, givensmall sample size and low degrees of censoring, the standard errors of theMLE estimates when the errors have chi-square distribution are about 10percent higher than the standard errors of the estimates for normal andstudent’s-t distributions. Similar observations can be made for medium andlarge sample sizes. However, the quality of the MLE esimates decline for

lZNote that the nonlinear estimation is very slow compared to the MLE estimation.For example, given a sample size of 400, normally distributed errors and 25% degree ofcensoring, it takes about 5 minutes CPU time for NLSU, compared to 58 seconds for MLEestimates, to obtain results from 100 replications.


higher levels of censoring and chi-square distributed error terms. On theother hand, it is important to note that the MLE seems to be robust tosymmetric but wide tailed distributions, as can be seen from the resultsfor t-distributed error terms. That is, although there exists some bias un-der t-distributed error terms, the MLE estimates for normal and student’s-tdistribution are almost the same in most cases. That is, using the RMSE

criteria, the results for MLE estimates under t-distributed error terms areas good as those from normally distributed error terms. As to the effectsof censoring, Table 8 indicates that doubling the degree of censoring from25% to 50% percent may cause a 25 to 30 percent increase in stadard errorsof estimates under normally distributed error terms, and gets worse for chi-square distributed error terms. A further increase of the degree of censoringresults in more inefficient estimates.

Finally, given that the estimators using all observations, one can makethe following conclusions:

Overall, the MLE performs better in all circumstances; i.e., for all samplesizes, distributions and degrees of censoring. In particular, the MLE performsmuch better under normal and student’s-t distributions compared to the chi-square distribution.

The H2SU and WH2SU estimators are no better than the simple OLSestimator and provide very poor results in all cases. Bias is a serious problemand gets worse with higher degrees of censoring.

The NLSU estimator performs well, given that the degree of censoring islow. However, bias becomes a problem as the degree of censoring increases.Moreover, the NLSU estimator is very inefficient compared to the MLE esti-mator in all cases.

4.3 Further Analysis of Selected Estimators

This Section presents further comparisons of estimators which are selected

from Sections 4.1 and 4.2 above. Estimators which are either biased and/orrelatively highly inefficient (or in general terms too poor to be candidates)are excluded from further discussion on the basis of the preceding discus-sions. Specifically, the ordinary least squares (OLSP) and the nonlinearleast squares (NLSP) estimators are excluded, from the estimators usingonly positive observations on y~ (i.e., from Section 4.1). Further, given theestimators using all observations (i.e., from Section 4.2), the simple ordi-


nary least squares (OLS), the Heckman’s two-step estimator based on theunconditional expectation of the model (H2SU) and its weighted version, theWH2SU estimator, are excluded due to their relative poor performances.

In total, six out of eleven estimators are discussed below. Given theseestimators, a summary of relative root mean square errors (RMSE) of theestimators are provided in Tables 9 and 10 for 25% and 50% degrees ofcensoring, respectively, and for all sample sizes and distributions. Theserelative RMSE are obtained by dividing the RMSE of each estimator bythe corresponding RMSE of the MLE estimator, for a given sample size,distribution and degree of censoring. For example, given a sample size of100, normally distributed error terms and 25% degrees of censoring, theRMSE of t31 for H2S estimator is equal to 0.190 (see Table 1), and thecorresponding RMSE for MLE estimator is equal to 0.099 (see Table 5).Thus the relative RMSE for t31 using the H2S estimator is given by the ratio0.190/0.099=1.919, which is shown at the top of Column 4 of Table 9. Othersfollow similar procedures.

As can be seen from Table 9, the NLSU estimator has the largest rela-tive RMSE value for all sample sizes and distributions which implies thatthe NLSU estimator is relatively poor (inei~cient) compared to others. Forexample, given a sample size of 100 and normally distributed error terms,the RMSE of the NLSU estimates of/3, and/32 are, respectively, 2.394 and1.605 times greater than their respective MLE estimates. The relative RM-SEs of the NLSU estimates of/3, and/32 further increase to 3.104 and 1.667,respectively, when the sample size becomes large (i.e., 400). This is becausethe relative ei~ciency of the MLE estimates increases at a higher speed withincreases in sample size compared to that of the NLSU estimates. Further,the NLSU estimates deteriorate for higher levels of censoring as shown inTable 10. Note that the relative RMSE values of the NLSU estimates seemto decline for the non-normal distributions, however, they still remain highcompared to other estimators.

The NLSU estimator is followed by the H2S estimator and its weightedversion, the WH2S estimator. As shown in Table 9 and 10, both estimatorsexhibit very high ratios of relative RMSE’s compared to, say, the 3SE orMLE estimators. Further, Table 10 depicts that the relative performanceof the H2S and WH2S estimators deteriorates for higher levels of censoringbased on the relative t~MSEs. In general, the H2S and WH2S estimatorsare no closer to the MLE estimator or to the 3SE estimator. Note that, as


discussed earlier in this Section, the WH2S estimator provides marginallymore efficient estimates than the unweighted H2S estimator only when theerrors are normal. This is also true for the W3SE and 3SE estimators.

On the other hand, it is important to note that the relative RMSE valuesindicate that the 3SE estimator provides results which are comparable tothe MLE estimates. Given a low level of degree of censoring, the relativeRMSE for 3SE estimates remain very close to the MLE estimates for normaland students’-t distributions. The relative performance of the 3SE estimatorbecomes better under t-distributed errors for all samples. More interestingly,or perhaps surprisingly, the evidence in Table 10 indicates that, as the de-gree of censoring increases, the relative performance of the 3SE estimatorunder student’s-t distribution compared to the MLE estimator improves sig-nificantly. For example, given a 50% degree of censoring, a sample size of400 and t-distributed error terms, the relative RMSE values for the 3SE es-timates of fll and/32, are, respectively, about 26 and 10 percent less than thecorresponding relative RMSEs of MLE estimates. Results for 75% degree ofcensoring and large sample size also exhibited that the 3SE estimator per-forms relatively better than the MLE estimator, if the errors have students’-tdistribution. This is, however, not the case for the H2S estimator, which de-teriorates for higher degrees of censoring, and its relative RMSEs remainsignificantly larger than both the MLE and 3SE estimators in all cases.

Next we compare the asymptotic variances and the Monte Carlo (true)variances of the estimators for all samples and distributions. These are listedin Tables 11 and 12 for 25% and 50% degrees of censoring, respectively. Notethat the asymptotic variances of the estimators are obtained by substitut-ing actual values into the respective analytical (asymptotic) formulas of theestimators which are provided in Section 2 of this study. For example, theasymptotic variances for /31 and /3~. for the MLE estimator are calculatedusing the diagonal elements of the variance-covariance matrix given by equa-tion (10) in Section 2 and so on. These are then compared with the MonteCarlo (true) variances of the estimators which are obtained based on the 3000replications (samples). Given these, the main points are discussed be!ow.

As shown in Tables 11 and 12, just by considering the asymptotic vari-ances of the estimators one can observe the vast difference in relative effi-ciency of the various estimators. Specifically, the large values of the asymp-totic variances, as well as the true variances, of the NLSU estimates revealits relative inefficiency in all cases, i.e., for all samples, distributions and


degrees of censoring and it becomes increasingly less efficient with higherdegrees of censoring. For example, Table 11 depicts that, given a samplesize of 100 and normally distributed error terms, the asymptotic variancesof the MLE estimates of fll and/32 are, respectively, 19.3 and 45.2 percentof the corresponding NLSU estimates. These values drop further to 5.6 and16.9 percent, respectively, if the degree of censoring becomes 50% (see Table12). It is also important to note that the asymptotic variances of the NLSUestimates are not good approximations of the corresponding true variances.

Regarding the MLE estimates, the asymptotic variances more or less pro-vide very close (sometimes accurate) approximations of the true variances,given that the errors have a normal distribution. This is particularly true forlarger samples. Further, it is also intersting to note that the MLE estimatorseems to perform fairly well under students’-t dstribution, given that the de-gree of censoring is low. Whereas, for the chi-square distributed error termsthe true variances of the MLE estimates overestimate the asymptotic vari-ances in all cases. In other words the MLE estimates are inaccurate under achi-square distribution compared to the normal and students’-t distributions.

Tables 11-12 also reveal interesting results with regard to the variancesof the H2S and 3Stg estimators. Note that the H2S and 3SE estimators havethe same asymptotic variances. However, the true variances of the H2S es-timates are signifcantly higher than their respective asymptotic variances inall cases. For example, given a 25% degree of censoring, a sample size of 400and normally distributed error terms, the true variances of the H2S estimatesof fll and f12 are, respectively, about 30! and 161 percent larger than the re-spective asymptotic variances. Results for the non-normal distributions arealso similar. The gap between the asymptotic variances and the true vari-ances of the H2S estimates widens for higher levels of censoring. On the otherhand, the situation is different for the 3SE estimator in which the asymptoticvariances yield very close approximations of the true variances and performbetter under normal and students’-t dstributions. In general, one can con-clude the following important points based on the variances of the H2S and3SE estimators. One, the results indicate that under no circumstances canthe asymptotic variances of the H2S estimates be used to approximate thetrue variances of the estimates. In other words, the asymptotic variancesunderestimate the true variances of the estimates. Two, the 3Stg estimatesare much more efficient than their corresponding H2S estimates. Further,the asymptotic and the true variances of the 3SE estimates become closer


under normal and students’-t dsitributions.In the discussions so far, we have considered the relative performance of

the various estimators, mainly focussing on the unbiasedness and efficiencyof the point estimates with respect to changes in sample size, distributionof the error term and degree of censoring. However, equally important inapplied research is to see the performance of the estimators viz-a-viz sta-tistical inference. That is, to examine the performance of the estimators inhypotheses testing and/or confidence intervals of the population parameters.Specifically, we test the hypotheses:

H0: ]3~ = 1 (42)H1:/3~ ~ 1, k=1,2. (43)

To test the hypotheses we use the test statistic:

(44)

where under the null hypothesis the statistic t is asymptotically distributedas a standard normal random variable, ~)k is the sample estimate of/3k ands.e.(A) is the standard error of A. a nounal level of significance isconsidered so that the expected percentage of rejections whenever the nullhypothesis is true is equal to 5%.

Or equivalently, a 95% confidence interval can be constructed such that:

P[~ - z × s.e.(~) </~ < ~ + z × s.e.(¢)~)] = 0.95 (45)

which is equivalent to

P(-1.96 < t < 1.96) = 0.95 (46)

where t is defined by (44) and the standard z value at a 5% significant levelis 1.96. Thus, we obtained the percent of coefficients contained in the 95%confidence interval for the H2S, 3SE and MLE estimators. The results arelisted in Table 13 for all sample sizes and distributions, given a 25% degreeof censoring. These results reveal that the percentage of confidence intervalsfor the H2S estimator, which actually contain the true parameters, are faraway from the expected 95~ closure rate. For example, given a sample size of


Table 13: 95% Confidence Intervals of Estimators for all Sample Sizes andDistributions, given 25% Degree of Censoring.

Sample Size Estimator Para--meter

% of Coefficients Contained in 95~ C.I.*Normal Student’s-t Chi-Square

--* C.I. stands for Confidence Interval.

100 H2S fll 69.80 74.!0 83.17fl~ 84.50 86.43 90.37

3SE fll 89.60 89.13 88.37¢72 91.87 92.00 92.33

MLE fll 94.10 94.30 93.40f12 94.17 94.47 93.40

200 H2S fll 66.93 66.23 78.03fl~ 81.93 82.22 89.33

3SE ~1 90.30 88.20 88.00fl~ 92.47 91.20 92.20

MLE fl~ 94.70 95.13 91.37f12 93.87 94.67 93.63

400 H2S fll 68.00 66.73 77.63f12 83.17 82.07 89.92

3SE fl~ 88.37 87.90 83.87fl~ 90.60 90.43 89.67

MLE fl~ 94.93 94.43 87.13fl~ 94.83 95.20 93.40

5 SUMMARY AND CONCLUSIONS 36

100 and normally distributed error terms, only 69.8 % and 84.5% of the 3000confidence intervals contain the true parameters,/31 and/32, respectively. Theresults do not improve even for large sample sizes. Thus any inference basedon the H2S estimates is likely to be misleading.

On the other hand, the percent confidence intervals for the 3SE estimator,compared to the H2S estimator, are a lot closer to the expected 95% closurerate; however, they are not as good as for the MLE estimator. The results forthe 3SE estimator are similar for the three distributions for small and mediumsample sizes. If the sample size becomes large, the 3SE estimator seemsto perform better under the symmetric distributions. The MLE estimatorperforms quite well, specifically under normal and students’-t distributions.

5 Summary and Conclusions

In this paper we examined the small sample properties of some of the es-timators of the Tobit model. These estimators include, two ordinary leastsquares estimators: one using only the positive (non-limit) observations onthe dependent variable, y~, (OLSP) and the other using all limit and non-limit observations on yi (OLS), the Heckman’s two-step estimator (H2S) andits weighted version, the weighted Heckman’s two-step estimator (WH2S),Heckman’s two-step estimator based on the unconditional expectation of themodel (H2SU) and its weighted version, the WH2SU, the maximum likeli-hood estimator (MLE) and two nonlinear least squares estimators. Further, athree-step estimation procedure which is reffered to as the three-step estima-tor (3SE) and the weighted three-step estimator (W3SE) are also suggestedand investigated. The effects of sample size, degree (level) of censoring anddistributional assumptions of the error structure of the model are investi-gated.

The least squares estimators are seriously affected by the degree of cen-soring and provide biased estimates; the bias being an increasing functionof the degree of censoring. Similarly, the H2SU and the WH2SU estimatorsare no better than the least squares estimators. In other words, the H2SUand the WH2SU estimators are biased and inefficient and get worse withincreases in the degree of censoring.

The nonlinear least squares estimators are generally less et~icient andcomputationally very slow compared to the MLE or 3SE estimators. Most

5 SUMMARY AND CONCLUSIONS 37

importantly, the nonlinear least squares are sensitive to the degree of censor-ing.

Under normality conditions, the MLE estimator gives the best resultsfollowed by the 3SE estimator. The loss in efficiency of the 3SE estimatorcompared to the MLE estimator is marginal. However, both the MLE and3SE estimators appear to be sensitive for the skewed (chi-square) distributederror terms. On the other hand, given low levels of censoring, the MLE esti-mator performs well under student’s-t distribution. If the degree of censoringis high, the MLE estimates under student’s-t distribution can be less efficientthan the 3SE estimator. It is also important to note that the t-tests and con-fidence intervals based upon the MLE estimates seem to perform quite wellfor both normal and student’s-t distributions.

The H2S estimator, although less efficient compared to 3SE or MLE es-timators in all cases, seems to perferm well, given !ow levels of censoring.However, it can be highly inefficient for degrees of censoring as high as 50%.Further, the H2S estimator can be misleading if used for hypothesis testingand confidence intervals even for large sample sizes and low degrees of censor-ing. This result is in contrast to that of the 3SE estimator which yields resultswhich are very close to the MLE estimates. Further research on the aboveand on semi-parametric and Bayesian estimators of the model is underway.

References 38

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Re[ere~ ce~ 41

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A MONTE CARLO ANALYSIS OF ALTERNATIVE ESTIMATORS OF … · A MONTE CARLO ANALYSIS OF ALTERNATIVE ESTIMATORS OF THE TOBIT MODEL Getachew Asgedom Tessema No. 73 - April, 1994 ISSN ISBN