Generalizing Full Rank Conditions in Heteroscedastic Censored Regression Models Songnian Chen Hong Kong University of Science and Technology Shakeeb Khan¤ University of Rochester June 1998 Abstract Powell's(1984) Censored Least Absolute Deviations (CLAD) estimator for the censored linear regression model has been regarded as a desirable alternative to maximum likelihood estimation methods due to its robustness to conditional heteroscedasticity and distributional misspeci¯cation of the error term. However, the CLAD estimation procedure has failed in certain empirical applications (e.g. Honor¶e et.al.(1997) and Chay(1995)) due to the restrictive nature of the \full rank" condition it requires. This condition can be especially problematic when the data is heavily censored. In this paper we introduce estimation procedures for heteroscedastic censored linear regression models with a much weaker identi¯cation restriction than that required for the CLAD, and which are °exible enough to allow for various degrees of censoring. The new estimators are shown to have desirable asymptotic properties and perform well in small scale simulation studies, and can thus be considered as viable alternatives for estimating censored regression models, especially for applications in which the CLAD fails.
JEL Classi¯cation: C14,C23,C24 Key Words: censored regression, full rank condition, heavy censoring, heteroscedasticity.
¤
Corresponding author. Department of Economics, University of Rochester, Rochester, NY 14627; e-mail:
[email protected]. We are grateful to J.L. Powell and B.E. Honor¶e for their helpful comments.
1
Introduction and Motivation
The censored regression model, sometimes referred to by economists as the \Tobit" model, has been the focus of much attention in both the applied and theoretical econometrics literature since the seminal work of Tobin(1958). In its simplest form the model is written as: yi = max(x0i ¯0 + ²i ; 0)
(1)
where yi is an observable response variable, xi is a d¡dimensional vector of observable covariates, ²i is an unobservable error term, and ¯0 is the d¡dimensional \parameter of interest". As many economic data sets are subject to various forms of censoring, the development of consistent estimation procedures for this model and variations thereof has become increasingly important. Traditionally this model has been estimated by maximum likelihood methods after imposing homoscedasticity and parametric restrictions on the underlying error terms. In the past ¯fteen years a number of consistent estimators have been proposed which allow for much weaker restrictions on the error terms, such as constant conditional quantiles (Powell(1984,1986a) Nawata(1990), Khan and Powell(1997), Buchinsky and Hahn(1998)), conditional symmetry (Powell(1986b)) and independence between the errors and regressors (Horowitz(1986,1988), Moon(1989), Honor¶e and Powell(1994)). The weakest such restriction is the constant conditional quantile restriction. Powell(1984) exploited a conditional median restriction on the error term, and proposed the Censored Least Absolute Deviations (CLAD) estimator, de¯ned as the minimizer of : Sn (¯) =
n 1X jyi ¡ max(x0i ¯; 0)j n i=1
(2)
This median restriction was generalized to any speci¯c quantile, ® 2 (0; 1) in Powell(1986a), generalizing the objective function to : Qn (¯) =
n 1X ½®(yi ¡ max(0; x0i ¯)) n i=1
(3)
where ½® (¢) ´ j ¢ j + (2® ¡ 1)¢ denotes the \check" function introduced in Koenker and Bassett(1978). These estimators are very attractive due to weak assumptions they require,
making them \robust" to conditional heteroscedasticity and non-normality of the error distribution. Furthermore, Powell(1984,1986a) proved these estimators to have desirable asympp totic properties, notably their parametric ( n) rate of convergence, and limiting normal distribution. 1
However, there is one serious drawback to the use of the CLAD estimator which has been encountered in certain practical applications. In the empirical work of Chay(1995) and Honor¶e et.al.(1997) the CLAD estimator has failed. The reason for this is that the identi¯cation condition for the CLAD estimator requires the matrix: E[I[x0i ¯0 > 0]xi x0i ]
(4)
to be of full rank1 , which in the empirical examples mentioned, could not be satis¯ed for estimates of ¯0 which minimized (??). In general, this full rank condition creates stability problems for the CLAD estimation procedure in data sets where the index x0i ¯0 is negative with a high probability, as would usually be the case when the data is heavily censored. This full rank condition becomes more \°exible" if we minimize (??) instead of (??) using a quantile ® > 0:5. In this case we can write the full rank condition as E[I[p® (xi ) > 0]xi x0i ]
(5)
where p® (xi ) denotes the ®th conditional quantile function. Since p® (xi ) ¸ p0:5(xi ) when ® > 0:5 this condition is more likely to be satis¯ed in practical applications.
However, estimating ¯0 through the minimization of (??) using various quantiles necessarily rules out the possibility of conditional heteroscedasticity, since it requires that all conditional quantiles are constant. In other words, if conditional heteroscedasticity is to be allowed for, ® must be ¯xed a priori, and not determined by the degree of censoring in the data set. For any statistical model in which conditional heteroscedasticity is present, the most sensible location restriction to impose on the error terms is a conditional mean or median restriction. It is well known that the censored regression model is not identi¯ed under a conditional mean restriction, leaving the conditional median restriction, and hence the identi¯cation restriction in (??), as necessary for estimating ¯0. In this paper we aim to address this problem by proposing estimators for the censored regression model which permit conditional heteroscedasticity, yet allow for much less stringent identi¯cation conditions than that required by the CLAD. We do so by restricting the structural form of conditional heteroscedasticity to be multiplicative, modelling the error 1
It should be noted that this full rank condition is necessary given the conditional quantile restriction
(see Powell(1984,1986a)). Thus it is also needed for estimators in the literature based on this restriction, such as Nawata(1990), Khan and Powell(1997), Buchinsky and Hahn(1998).
2
term as the product of a nonparametrically speci¯ed \scale" function of the regressors, and a homoscedastic error term: ²i = Á(xi )ui
P (ui · ¸jxi ) ´ P (ui · ¸) 8¸ 2 R; xi a.s.
(6)
Note that this structure still allows for conditional heteroscedasticity of very general forms, as Á(¢) is left unspeci¯ed. We propose two estimation procedures, based on two restrictions on the homoscedastic error term ui . The ¯rst estimator is based on the assumption that ui has a standard normal distribution.2 While this restricts the error term behavior a great deal further, it may not be as serious of an assumption. For example, it has been concluded in Powell(1986a), Donald(1995), and Horowitz(1993) that heteroscedasticity is a far more serious problem than departures from normality when estimation of ¯0 is concerned. Their conclusions are consistent with our simulation results discussed later in this paper. The second estimation procedure we introduce allows us to do away with the normality distribution, only requiring that ui have a smooth, positive density function on the real line. As detailed in the next section, both estimators involve two stages, and are very similar in spirit to the estimators introduced in Chen and Khan(1998a,b). The ¯rst stage involves nonparametric quantile regression, and the second stage adopts a simple least-squares type ¯tting device. The asymptotic properties of these estimators will thus follow from the same arguments used in those two papers. The paper is organized as follows. The following section motivates the estimation procedures we propose and discusses each of the two stages involved in greater detail. Section 3 outlines regularity conditions used in proving the asymptotic properties of the estimators, and then outlines the steps involved for the proof. Section 4 explores the ¯nite sample properties of these estimators through a small scale simulation study, and compares their performance to the CLAD estimator. Finally, section 5 concludes by summarizing our results and examining possible extensions and areas for future research. 2
Note that setting the variance of ui to one is just a normalization that is required by leaving Á(¢)
unspeci¯ed.
3
2
Models and Estimation Procedures
We consider censored regression models of the form: yi = max(x0i ¯0 + ²i ; 0)
(7)
²i = Á(xi )ui
(8)
The only restriction we impose on the \scale" function Á(¢) is that it satisfy certain \smoothness" properties, as detailed in the next section. We ¯rst consider an estimator based on a normality assumption on ui : P (ui · ¸jxi ) = ©(¸)
(9)
where ©(¢) denotes the c.d.f. of the standard normal distribution. To construct an estimator for ¯0 based on this restriction, we note that for any quantile ® 2 (0; 1), the equivariance property of quantiles implies that:
p® (xi ) = max(x0i ¯0 + cZ® Á(xi ); 0) where p®(¢) denotes the ®th conditional quantile function and cZ® denotes the ®th quantile of the standard normal distribution. Thus if we have for some point xi such that p®i (xi ) > 0 for two distinct quantiles ®1 ; ®2, we can combine the relations: p®1 (xi ) = x0i ¯0 + cZ®1 Á(xi )
(10)
p®2 (xi ) = x0i ¯0 + cZ®2 Á(xi )
(11)
to yield the relationship: cZ®2 p®1 (xi ) ¡ cZ®1 p®2 (xi ) = x0i ¯0 cZ®2 ¡ cZ®1
(12)
This suggests that if the values of the quantile functions were known for d observations, the parameter of interest ¯0 could be recovered by solving the above system of equations. The condition necessary for a unique solution to this system is the full rank of the random variable: I[p®1 (xi ) > 0]xi 4
assuming without loss of generality that ®1 < ®2 . This generalizes the full rank condition of the CLAD estimator as long as ®1 > 0:5. Of course, the above system of equations cannot immediately translate into an estimator of ¯0 since the values of the conditional quantile functions (p® (¢)) are unknown. However, they can be estimated nonparametrically in a preliminary stage, and these estimated values (^ p®(¢)) can be \plugged" into the system of equations, de¯ning a two-step estimator of ¯0 as "
n 1X ^ I[^ p®1 (xi ) > 0]xi x0i ¯= n i=1
where y^i ´
#¡1
n 1X y^i xi n i=1
(13)
^®2 (xi ) ^®1 (xi )¡cZ cZ ®1 p ®2 p . Z cZ ®2 ¡c®1
We next consider relaxing the normality assumption on ui , only requiring that it have a continuous distribution with a positive density function. In contrast to the estimator we introduced, which treats the two quantile function values as pseudo-dependent variables, we will now treat the average of the quantile functions as a pseudo-dependent variable and their di®erence as a pseudo-regressor. Speci¯cally, letting c®i denote the (unknown) quantile values of ui , for a value of xi where both quantile functions are positive, we have the following relationship: p¹(xi ) = x0i ¯0 +
c¹ ¢p(xi ) ¢c
(14)
where p¹(¢) ´ (p®2 (¢) + p®1 (¢))=2, ¢p(¢) ´ p®2 (¢) ¡ p®1 (¢), c¹ ´ (c®2 + c®1 )=2, ¢c ´ c®2 ¡ c®1 . This suggests an alternative second stage least squares ¯tting device, regressing p^¹(¢) on xi and ¢^ p(¢). Speci¯cally, let ¯^ 2 Rd and the \nuisance parameter" °^1 2 R minimize the least squares function:
n 1X I[^ p®1 (xi ) > 0](^p¹(xi ) ¡ x0i ¯ ¡ °1¢^ p(xi ))2 n i=1
We note that both estimators are de¯ned as functions of nonparametric estimators, which p are known to converge at a rate slower than n. The next section shows how the second stage estimator can still achieve the parametric rate of convergence under regularity conditions which are common in the literature. Before proceeding however, we need to discuss each of the stages of our two procedures in greater detail. Speci¯cally, we discuss the nonparametric estimation procedure adopted in the ¯rst stage, and some technical complications which require the modi¯cation of the second stage objective functions. 5
2.1
First Stage of the Estimators
The ¯rst stage involves nonparametrically estimating the conditional ®1 ®2 quantiles of the observed dependent variable yi given the regressors xi . While several conditional quantile estimators have been recently proposed in the statistics and econometrics literature, we use the local polynomial estimator introduced in Chaudhuri(1991a,b). This estimation procedure is computationally simple (it involves minimization of a globally convex objective function which can be handled using the simplex method) and it allows for simple control of the order of the bias by selecting the appropriate order of the polynomial. Its description is facilitated by introducing new notation, and the notation adopted has been chosen deliberately to be as close as possible to that introduced in Chaudhuri(1991a,b). For a vector a let [a] denote the sum of its components. For any two vectors of the same b
dimension a; b, let ab denote ¦a(i)(i) where subscripts denote components. For any vector a, and any non-negative integer k, let ak denote the vector of elements fabi g where bi is a vector
of non-negative integers such that [bi ] = k.
Now assuming for ®1 ; ®2, the conditional quantile function p® (¢) has order of di®erentiability k, let s(A) denote the dimension of the set of all vectors of non-negative integers fbi g where [bi ] · k. Let I(¢) be an indicator function, and let ±n denote a \bandwidth" sequence used to smooth the data. The local polynomial estimator of the conditional ®th
quantile function at a point x involves ®-quantile regression(see Koenker and Bassett (1978)) on observations which are \close" to x. Speci¯cally, let the vectors (µ^0; µ^1 ; :::µ^k ) minimize the kernel weighted objective function3 : n X i=1
I[xi 2 Cn (x)]½®(yit ¡ µ0 ¡ µ10 (xi ¡ x) ¡ :::µk0 (xi ¡ x)k )
(15)
where Cn is a sequence of cubes centered at x whose sides are of length 2±n , so (j)
xi 2 Cn (x) , jxi ¡ x(j) j · ±n j = 1; 2:::d The conditional quantile estimator which will be used in the ¯rst stage will be the value µ^0. 3
For technical reasons used in proving asymptotic properties of the second stage estimator, we actually
require that this objective function be minimized over a compact subset of Rs(A) .
6
The motivation for including a higher order polynomial in the check function and estimating the nuisance parameters (µ^1; :::µ^k ) is to achieve bias reduction of the nonparametric estimator, analogous to using a \higher order" kernel with kernel estimation. One of the disadvantages of working with nonparametric conditional quantile estimators as opposed to nonparametric conditional mean estimators, is that the former is not linear in the dependent variable. This problem can be overcome by using the local Bahadur representation developed in Chaudhuri(1991a) and Chaudhuri et.al.(1997). The exact form of this representation requires the introduction of some additional notation. For a sample of n observations, we will let Nn (x) denote the number of observations whose regressors values lie in Cn (x). Let ºi® denote the \residuals", yi ¡ p® (xi ), and letting fºi® jXi (¢) and fXi (¢) denote
conditional and marginal density functions (where de¯ned) respectively. We de¯ne the s(A) dimensional vector b(±n ; xj ¡ xi ) as : b(±n ; xj ¡ xi ) = ±n¡[u] (xj ¡ xi )u ; [u] · k and the s(A) £ s(A) matrix G®n by the components: R
[¡1;1]d
¿ u ¿ v fºi® jXi (0jx + ±n ¿ )fXi (x + ±n ¿ )d¿ R ; [u] · k; [v] · k [¡1;1]d fXi (x + ±n ¿ )d¿
Also, let es(A) denote the s(A) dimensional vector whose ¯rst component is 1 and whose remaining components are 0. For an observation xi let pn;® (xi ; x) denote the kth order Taylor polynomial of p® (xi ) around x. Then we can work with the following linear representation: Lemma 1 (From Lemma 4.1 in Chaudhuri et.al.(1997)) Under assumptions on the density functions, bandwidth sequence and smoothness of pt (¢) discussed in the next section, we have
p^® (x) ¡ p® (x) = Nn (xi )¡1 G®n (xi )¡1 + R®n (x)
n X i=1
b(±n ; xi ¡ x)(® ¡ I[yit · pn;®(xi ; x)]I[xi 2 Cn (x)] (16)
where the remainder R®n (¢) converges to 0 as the sample size n tends to in¯nity uniformly over x in a compact set.
7
2.2
Second Stage of each Estimator
The second stage of our estimators treat the reduced form values estimated in the ¯rst stage as \raw" data, and adopts weighted least squares type ¯tting devices to estimate ¯0. As mentioned above, positive weight will only be given to observations whose estimated quantile function values exceed the censoring point. We thus propose minimizing second stage objective functions of the form: Ã
n cZ p^® (xi ) ¡ cZ®1 p^®1 (xi ) 1X ¿(xi )!(^ p®1 (xi )) ®1 2 Z ¡ x0i ¯ n i=1 c®2 ¡ cZ®1
!2
for the estimator under the normality assumption, and n ³ ´2 1X p(xi )°1 ¿(xi )!(^ p®1 (xi )) p^¹(xi ) ¡ x0i ¯ ¡ ¢^ n i=1
without the normality assumption. Here !(¢) is a \smooth" weighting function which only keeps observations for which the \reduced form" values exceed the censoring value, and ¿ (¢) is a trimming function which serves to bound the density of the regressors from 0. This yields closed form, least squares type estimators of the form: ¯^ =
Ã
n 1X ¿ (xi )!(^ p®1 (xi ))xi x0i n i=1
!¡1
£
n cZ p^® (xi ) ¡ cZ®1 p^®2 (xi ) 1X ¿(xi )!(^ p®1 (xi ))xi ®2 1 Z n i=1 c®2 ¡ cZ®1
and, letting °^ denote our second estimator for °0 ´ (¯00 ; °1 )0 , °^ =
Ã
n 1X ¿ (xi )!(^ p®1 (xi ))^ zi z^i0 n i=1
!¡1
£
n 1X ¿(xi )!(^ p®1 (xi ))^ zi p^¹(xi ) n i=1
where z^i denotes the vector (x0i ; ¢^ p(xi ))0 . We note that the proposed estimators fall into the class of \semiparametric two-step" estimators, for which many general asymptotic results have been developed (see Andrews(1994), Newey and McFadden(1994), Sherman(1994)). As p proven in the next section, under appropriate regularity conditions the parametric ( n) rate of convergence can be obtained for the second stage estimators despite the nonparametric rate of convergence of the ¯rst stage estimator. 8
3
Asymptotic Properties of the Estimators
The necessary regularity conditions will ¯rst be outlined in detail before proceeding with the proofs of consistency and asymptotic normality; speci¯c assumptions are imposed on the parameter space, the error terms and the regressors, the order of smoothness of the scale function Á(¢), and the kernel regularity conditions needed for the ¯rst stage:
3.1 3.1.1
Assumptions Assumptions on the Parameter Space
A1 The true parameter value ¯0 lies in parameter space B, a subset of Rd , where d = dim(X).
3.1.2
Assumptions on Error Terms and Regressors
B1 The sequence of d + 1 dimensional vectors (²i ; Xi ) are independent and identically distributed. B2 The regressor vector Xi has support which is a convex subset of Rd with nonempty interior, and has a continuous density with respect to Lebesgue measure, denoted by fXi (¢) which is uniformly bounded away from in¯nity, and bounded away from zero except on the boundary of its support. B3 The error terms ²i are of the form ²i = Á(xi )ui where Á(¢) is a deterministic function of the regressors, and ui is a random variable, distributed independently of the regressors, with the following properties: B3.1 If the ¯rst proposed estimator is used, ui is assumed to have a standard normal distribution. B3.1' If the second proposed estimator is used, it is only required that ui has a continuous distribution with density function that is bounded, positive, and continuously di®erentiable on the real line. B4 Let Vx be a compact subset with non-empty interior of the support of Xi . Vx will be assumed to have the following properties: B4.1 fX (x) ¸ ²0 > 0 8x 2 Vx , for some constant ²0 . 9
B4.2 (Identi¯cation) Letting J denote the d £ d matrix: E[¿(xi )!(p®1 (xi ))xi x0i ] and letting J ¤ denote the (d + 1) £ (d + 1) matrix: E[¿(xi )!(p®1 (xi ))zi zi0 ] where zi ´ (x0i ; ¢p(xi ))0 , then we assume that J and J ¤ are of full rank. 3.1.3
Assumptions on the Multiplicative Scale Function
C1 The multiplicative heteroscedasticity function Á(xi ) is assumed to be bounded away from in¯nity and 0, and to be di®erentiable up to order k on Vx with derivatives of order k being uniformly continuous with exponent ° 2 (0; 1]. Letting p = k + ° denote the order of smoothness of these functions, it will be assumed that p > 3=2d.
3.1.4
Assumptions on the Weighting and Trimming Functions
D1 The weighting function, !(¢) : R ! R+ has the following properties: D1.1 ! ´ 0 if its argument is less than c, an arbitrarily small positive constant. D1.2 !(¢) is twice continuously di®erentiable. D1.3 !(¢) is strictly increasing on the interval (c; 1). D2 The trimming function ¿ (¢) : Rd ! R+ is bounded and takes the value 0 i® its argument lies outside Vx .
3.1.5
First Step Kernel Regularity Conditions
E1 The kernel function in the ¯rst step estimators is the product of d Uniform kernels. Speci¯cally, letting Cn (Xi ) denote the cube in Rd , centered at Xi , with side length 2±n . Then the other observations in the sample which are use to estimate the quantile function at Xi are indexed by the set Sn (Xi ) = fj : 1 · j · n; j 6= i; Xj 2 Cn (Xi )g
10
E2 The ¯rst stage bandwidth sequence, denoted ±n is of the form: ±n = c1n¡» where c1 is some positive constant, and » 2 (1=2[p]; 1=3d) where [ ] denotes the integer
argument. 3.1.6
Remarks on the Assumptions
1. Assumption B2 restricts attention to continuous explanatory variables only. The asymptotic results derived in the next section can easily be extended to include categorical regressors by following the approach taken in Khan(1997a,b). 2. Assumption D2 will allow for the exogenous trimming of the data, incorporated into the weighting function, essentially only keeping observations for which the density of the regressors is bounded away from 0 via assumption B3.1. This will ensure avoiding the \denominator" problem which arises in two-step semiparametric estimators, and help with other technical issues. 3. Assumption B3.1', required for the second estimator, imposes no location restriction on the homoscedastic error term. For the second estimation procedure this restriction is not necessary to identify an intercept term, as long as conditional heteroscedasticity is present (see following remark). 4. Assumption B4.2 characterizes the \full rank" conditions which illustrate the advantages of these estimators over Powell's CLAD estimator. The ¯rst such condition can be roughly interpreted as a full rank requirement on the matrix: E[I[x0i ¯0 + cZ®1 > 0]xi x0i ] which, by appropriate selection of the quantile ®1 , is much less stringent than Powell's condition: E[I[x0i ¯0 > 0]xi x0i ] 5. The full rank condition imposed on J ¤ will not be satis¯ed if ²i is homoscedastic. While homoscedastic models are not the concern of this paper, as the full rank condition of the CLAD can be easily generalized in this case, it is still worth noting that the slope 11
coe±cients of our second estimator will still be estimable. This can best be understood by verifying the estimatibility condition discussed on page 58 of Amemiya(1985). A necessary and su±cient condition for the estimability of linear combination of regression coe±cients ¯, expressed as F 0 ¯, is that there exist a matrix A such that F = X 0 A, where X is the regression matrix. In the context of our second estimator, for observations for which the quantile function is positive, the (infeasible) regression matrix can be expressed as : X = [1n X ¤ ¾1n ] where 1n denotes an n-dimensional vector of 1's, ¾ denotes the standard deviation of the ²i in the case of homoscedasticity, and X ¤ denotes the n£d ¡ 1 matrix of regressors
exlcuding the intercept. Thus if °0 denotes the d+1 dimensional vector of coe±cients in
our second estimator, extracting the subset of slope coe±cients corresponds to setting F 0 = [0d¡1 Id¡1 0d¡1], where 0d¡1 is a d ¡ 1-dimensional vector of 0's and Id¡1 is a d ¡ 1 £ d ¡ 1 identity matrix. To ¯nd a n £ d ¡ 1 matrix A such that F = X 0 A, it will
su±ce that the n £ d matrix [1n X ¤] be of full column rank, which follows from the full rank condition in the ¯rst part of Assumption B4.2.
6. Assumption D1 imposes additional restrictions on the weighting function. It ensures that estimation is based only on observations for which the conditional quantile functions are greater than the censoring values. It is essentially a smooth approximation to an indicator function, that will help avoid certain technical di±culties. Note that assumption D1.1 implies that the support of the weighting function is bounded away from the censoring value; this is necessary to avoid the \boundary problem" that arises when nonparametric estimation procedures are used with censored data. 7. Assumption E2 allows for a range of bandwidth sequences, but rules out the sequence which yields the optimal rate of convergence of the ¯rst step estimator as discussed in Chaudhuri(1991a). It imposes \undersmoothing" of the ¯rst stage nonparametric estimator.
3.2
Proof of Asymptotic Normality of the Estimators
In this section it will be proven that under the assumptions detailed in the previous section, the proposed (second stage) estimators for the slope coe±cients converges at the parametric 12
p ( n) rate despite the nonparametric rate of convergence of the ¯rst stage estimator. Our proof involves virtually identical steps to those taken in the proof of Theorems 1 and 3 in Chen and Khan(1998b), so we just provide a sketch of the main steps involved, referring the reader to that paper for the details involved. Note we can write: ¯^ ¡ ¯0 = Ã
Ã
n 1X ¿ (xi )!(^ p®1 (xi ))xi x0i n i=1
!¡1
£ !
n 1X c® p^1(xi ) ¡ c®1 p^2 (xi ) ¿ (xi )!(^ p®1 (xi ))xi ( 2 ¡ x0i ¯0 ) n i=1 c®2 ¡ c®1
and °^ ¡ °0 = Ã
Ã
n 1X ¿ (xi )!(^ p®1 (xi ))^ zi z^i0 n i=1
!¡1
£
!
n 1X ¿ (xi )!(^ p®1 (xi ))^ zi (p^¹(xi ) ¡ z^i0 °0 ) n i=1
We aim to establish a linear representation for the proposed estimators; following Chen and Khan(1998) we establish the asymptotic properties of the denominator and numerator in the above expressions separately. The ¯rst lemma evaluates the probability limits of the denominator terms: Lemma 2 Letting J denote the d £ d matrix: E[¿(xi )!(p®1 (xi ))xi x0i ] and J ¤ the (d + 1) £ (d + 1) matrix: E[¿(xi )!(p®1 (xi ))zi zi0 ] then Ã
!
(17)
!
(18)
n 1X p ¿ (xi )!(^ p®1 (xi ))xi x0i ! J n i=1
and Ã
n 1X p ¿ (xi )!(^ p®1 (xi ))^ zi z^i0 ! J ¤ n i=1
13
We next turn attention to the numerator terms, focusing initially on estimator requiring normality; a mean value expansion of the weighting function around the true values of the quantile functions yields : n 1X c® p^1 (xi ) ¡ c®1 p^2 (xi ) ¡ x0i ¯0 ) ¿ (xi )!(p®1 (xi ))xi ( 2 n i=1 c®2 ¡ c®1
+
n 1X c® p^1 (xi ) ¡ c®1 p^2 (xi ) ¡ x0i ¯0 )(^ ¿ (xi )! 0 (~ p®1 (xi ))xi ( 2 p®1 (xi ) ¡ p®1 (xi )) n i=1 c®2 ¡ c®1
+Rn
(19)
(20)
(21)
Note that equation (??) can be decomposed as follows: n c®2 1X ¿(xi )!(p®1 (xi ))xi (^ p® (xi ) ¡ p®1 (xi ))¡ n i=1 c®2 ¡ c®1 1
(22)
n 1X c®1 ¿(xi )!(p®1 (xi ))xi (^ p® (xi ) ¡ p®2 (xi )) n i=1 c®2 ¡ c®1 2
(23)
We can linearize the nonparametric component in equation (??)(the same argument can be used with the nonparametric part of equation (??)). Following Lemma 3 in Chen and Khan(1998), we can express (??) as a second order U-statistic with a kernel that depends on the sample size:
Lemma 3 n 1X c®2 ¿(xi )!(p®1 (xi ))xi (^ p® (xi ) ¡ p®1 (xi )) = n i=1 c®2 ¡ c®1 1
X 1 c®2 ¡1 ¿ (xi )!(p®1 (xi ))xi es(A)G¡1 n®1 (xi )pn®1 (xi )£ n(n ¡ 1) i6=j c®2 ¡ c®1
b(±n ; xj ¡ xi )I[yj · p®1 (xj )] ¡ ®1)I[xj 2 Cn (xi )] + op (n¡1=2 )
We can express this U-statistic as an average of mean zero i.i.d random variables: 14
(24)
Lemma 4 Let ±®1 (yi ; xi ) be de¯ned as the mean zero random variable: ¿ (xi )!(p®1 (xi ))(®1 ¡ I[yi · p®1 (xi )])fº¡1 ®1 (0jxi )xi jXi i
then X 1 ¡1 ¿ (xi )!(p®1 (xi ))xi es(A)G¡1 n®1 (xi )pn®1 (xi )£ n(n ¡ 1) i6=j
b(±n ; xj ¡ xi )(I[yj · p®1 (xj )] ¡ ®1 )I[xj 2 Cn (xi )] can be represented as n 1X ±® (yi ; xi ) + op (n¡1=2 ) n i=1 1
(25)
The linear term in (??) can be shown to be asymptotically negligible. Lemma 5 = op (n¡1=2 ).
n c® p^1 (xi ) ¡ c®1 p^2 (xi ) 1X ¿(xi )! 0 (~ p®1 (xi ))xi ( 2 ¡ x0i ¯0 )(^ p®1 (xi ) ¡ p®1 (xi )) n i=1 c®2 ¡ c®1
This concludes the linear representation for the numerator term of the estimator requiring normality. For the estimator without normality, we proceed by decomposing the numerator term as follows: n c® 1X ¿(xi )!(^ p®1 (xi )) 2 (^ p® (xi ) ¡ p®1 (xi ))zi + n i=1 ¢c 1
(26)
n 1X c® ¿(xi )!(^ p®1 (xi )) 2 (^ p® (xi ) ¡ p®1 (xi ))(^ zi ¡ zi )¡ n i=1 ¢c 1
(27)
n 1X c® ¿(xi )!(^ p®1 (xi )) 1 (^ p® (xi ) ¡ p®2 (xi ))zi ¡ n i=1 ¢c 2
(28)
n 1X c® ¿(xi )!(^ p®1 (xi )) 1 °2(^ p®2 (xi ) ¡ p®2 (xi ))(^ zi ¡ zi )+ n i=1 ¢c
(29)
n 1X c¹ ¿(xi )!(^ p®1 (xi ))(¹ p(xi ) ¡ x0i ¯0 ¡ ¢p(xi ))zi n i=1 ¢c
(30)
15
Following a mean value expansion around !(^ p®1 (xi )) for each of the above terms, the same arguments in the previous lemmas can be used, following the arguments used in the proof of Theorem 3 in Chen and Khan(1998b). Speci¯cally, it follows that the terms in equations (??), (??), and (??) are op (n¡1=2 ), and the terms in equations (??) and (??) can be represented as, respectively : n 1X ±¤ (yi ; xi ; zi ) + op (n¡1=2 ) n i=1 ®1
and n 1X ±¤ (yi ; xi ; zi ) + op (n¡1=2 ) n i=1 ®2
where ±®¤ 2 (yi ; xi ; zi ) ´ ¿ (xi )!(p®1 (xi ))fº¡1 ®2 jX (0jxi ) i
c®1 (®2 ¡ I[yi · p®2 (xi )])zi ¢c
±®¤ 1 (yi ; xi ; zi ) ´ ¿ (xi )!(p®1 (xi ))fº¡1 ®1 jX (0jxi ) i
c®2 (®1 ¡ I[yi · p®1 (xi )])zi ¢c
and
thus completing the linear representation of the second estimator's numerator term. We are now ready to prove the main theorem, which establishes the limiting distributions of the two proposed estimators: Theorem 1 Let - denote the d £ d matrix: E[(±®2 (yi ; xi ) ¡ ±®1 (yi ; xi ))(±®2 (yi ; xi ) ¡ ±®1 (yi ; xi ))0 ] and let -a st denote the (d + 1) £ (d + 1) matrix: E[(±®¤ 2 (yi ; xi ; zi ) ¡ ±®¤ 1 (yi ; xi ; zi ))(±®¤ 2 (yi ; xi ; zi ) ¡ ±®¤ 1 (yi ; xi ; zi ))0 ] then p ^ n(¯ ¡ ¯0 ) ) N (0; J ¡1 -J ¡1 )
(31)
p n(^ ° ¡ °0) ) N(0; (J ¤ )¡1 -¤ (J ¤ )¡1 )
(32)
and
16
Proof: The preceding arguments enable us to write: n X
1 ¯^ ¡ ¯0 = (J ¡1 + op (1)) ±® (yi ; xi ) ¡ ±®1 (yi ; xi ) + op (n¡1=2 ) n i=1 2 and °^ ¡ °0 = ((J ¤ )¡1 + op (1))
n 1X ± ¤ (yi ; xi ; zi ) ¡ ±®¤ 1 (yi ; xi ; zi ) + op (n¡1=2 ) n i=1 ®2
so the desired results follows by the central limit and Slutsky theorems.
Q.E.D.
For purposes of inference, we propose a consistent estimators for the limiting covariance matrices above. As discussed in Chen and Khan(1998a,b), the \outer score" term requires nonparametric density estimation. For this we propose a Nadaraya-Watson kernel estimator ®
using the estimated residuals º^i j = yi ¡ p^®j (xi ) i = 1; 2; ::n j = 1; 2: f^º ®j jXi (0jxi ) =
P
j6=i
(1)
(2)
j6=i
Kh1 (xj ¡ xi )
Kh1 (xj ¡ xi )Kh2 (^ º ®j )
P
(1)
where K (1) ; K (2) , are continuously di®erentiable kernel functions on compact subsets of Rd and R respectively, that are positive, symmetric about 0, and integrate to 1. Kh(1) (¢); Kh(2) (¢) 1 2 (1) ¢ (2) ¢ denote, respectively, h¡d ( h1 ) and h¡1 ( h2 ), where h1 and h2 are bandwidth sequences 1 K 2 K
which satisfy : 1. h1 = o(1); h2 = o(1) 2. nhd1 ! 1; n1=8 h2 ! 1 Using this kernel estimator for the conditional density, the following theorem proposes an estimator for the limiting covariance matrix and establishes its consistency. A proof can be found in Theorem 2 in Chen and Khan(1998b). Theorem 2 De¯ne J^ and J^¤ by the matrices: n 1X ¿(xi )!(^ p®1 (xi ))xi x0i n i=1
and n 1X ¿(xi )!(^ p®1 (xi ))^ zi z^i0 n i=1
17
respectively. Let ±^®j (xi ) j = 1; 2 be de¯ned by the vectors q
®j (1 ¡ ®j )¿ (xi )!(^ p®1 (xi ))f^º¡1 ®j (0jxi )xi jXi
and let ±^®0 j (xi ; z^i ) ´ = 1; 2 be de¯ned by: q
®j (1 ¡ ®j )¿ (xi )!(^ p®1 (xi ))f^º¡1 ®j (0jxi )^ zi jXi
^ and ^ ¤ can be de¯ned as : so that n cZ cZ cZ cZ 1X ( Z ®1 Z ±^®2 (xi ) ¡ Z ®2 Z ±^®1 (xi ))( Z ®1 Z ±^®2 (xi ) ¡ Z ®2 Z ±^®1 (xi ))0 n i=1 c®1 ¡ c®2 c®2 ¡ c®1 c®1 ¡ c®2 c®2 ¡ c®1
and n 1X ((1=2 ¡ °^1)±^®¤ 2 (xi ; z^i ) ¡ (1=2 + °^1)±^®¤ 1 (xi ; z^i ))((1=2 ¡ °^1)±^®¤ 2 (xi ; z^i ) ¡ (1=2 + °^1)±^®¤ 1 (xi ; z^i ))0 n i=1
respectively; then p ^ J^¡1 ! J ¡1 -J ¡1 J^¡1 -
and p ^ ¤(J^¤ )¡1 ! (J ¤)¡1 -¤(J ¤ )¡1 (J^¤ )¡1-
4
Monte Carlo Results
In this section, the ¯nite sample properties of the proposed estimators are examined through the results of a small scale simulation study. In the study we consider various designs, with varying degrees of censoring, and compute basic summary statistics for the two estimators we introduce in this paper, referred to in this section as WNPQN (weighted non-parametric quantile regression with normal errors) and WNPQ (weighted non-parametric quantile regression), as well as Powell's CLAD estimator. These results are reported in Tables I-VI. We simulated from models of the form yi = max(® + xi ¯0 + Á(xi )ui ; 0) where xi was a random variable distributed uniformly between -2 and 2, ¯0 was set to 0.5, and the error Á(xi )ui varied to allow for three di®erent designs: 18
1. homoscedastic normal (Tables I,II): Á(xi ) ´ 2:5, ui » standard normal. 2. homoscedastic Cauchy (Tables III,IV): Á(xi ) ´ 1, ui » standard Cauchy. 3. heteroscedastic (Tables V,VI): Á(xi ) = exp(0:65x2i ); ui » standard normal. For each of these restrictions on the error terms, we simulated for four di®erent values of the intercept term ®: 0, -0.25, -0.5, -0.75 , to allow for di®erent degrees of censoring. Fraction of observations uncensored are stated in parentheses for each design. Each design was replicated 801 times, for sample sizes of 100, 200, 400 and 800. The tables report mean bias, median bias, mean squared error, and median absolute deviation for the three estimators. Our simulation study was performed in GAUSS. To compute the CLAD estimator, we adopted the iterative linear programming method introduced in Buchinsky(1994)4 The ¯rst stage of our two estimators were computed using the linear programming method discussed in Buchinsky(1994), which is guaranteed to converge to the global minimum. This method was used to ¯t a local linear model for the conditional quantile functions at ®1 = 0:75; ®2 = 0:85. We set the bandwidth to 2 ¢ n¡2=7 for the purpose of computational simplicity, well aware that a data driven approach would probably be more suitable.
As the results on the tables indicate, there are signi¯cant advantages to adopting either of our two suggested procedures over the CLAD estimator when the degree of censoring exceeds 50%. For the homoscedastic normal design, the CLAD begins to break down when the intercept is decreased to ¡0:5 (corresponding to 42.2% of the observations uncensored)
as indicated by its MSE not decreasing when the sample size is increased from 400 to 800 observations. As ® is decreased to -0.75, the CLAD completely collapses, exhibiting median biases and MAD's which remain constant at -0.5 and 0.5 respectively. In contrast, the WNPQN and WNPQ estimators perform well for all degrees of censoring5. Even when the intercept is reduced to ¡0:75. Though both estimators exhibit somewhat large bias for
observations less than 400, the estimators clearly exhibit consistency, with mean squared errors constantly reducing by a factor of a half as the sample size doubles. Also, even for the 4
This algorithm only converges to a local minimum of the objective function. For some of the designs
considered, our results varied with the starting value selected. For such designs, we used three starting values: the LAD estimated values, the ordinary least squares estimated values, and the true parameter values. We reported the results corresponding to the starting value for the which the CLAD performed the most favorably. 5 As alluded to earlier, the limiting regression matrix of the WNPQ estimator will be singular in the ¯rst two designs. By the explanation discussed earlier, estimability of the slope parameter is not an issue, as con¯rmed by the results in Tables I-IV.
19
lower degrees of censoring where the CLAD performs well, it is still outperformed in terms of mean squared error and MAD by the other two estimators. One result we ¯nd somewhat surprising is that WNPQ estimator outperforms the WNPQN estimator in this design, even though the errors are in fact normal. For the design with Cauchy errors, the CLAD again breaks down as the intercept is set to -0.5, with mean squared errors not shrinking with the sample size. Again the other two estimators perform well for all levels of censoring when the sample size exceeds 200. An interesting point is that the WNPQN performs very well despite the fact that the errors are non-normal. This is consistent with ¯ndings in Powell(1986a) and Horowitz(1993) who both conclude heteroscedasticity is far mor serious a concern than non-normality when estimating censored regression models. In the context of our estimator, this conclusion follows from the fact that the distance between the 0:75th and 0:85th quantiles is very similar for the Cauchy and normal distributions. Also, as was the case for the normal error design, the WNPQ outperforms the CLAD in terms of mean squared error at all levels of censoring. Our results are most dramatic for the heteroscedastic design considered. Here, the CLAD performs poorly for all levels of censoring and begins to break down at ® = ¡0:25, where
45% of the observations are uncensored. In contrast, both our estimators perform as well as under homoscedasticity, exhibiting biases and measures of dispersion which are far smaller than those of the CLAD. In summary, the results of our simulation study regarding the CLAD follow the predictions of the theory behind it, and agree with certain empirical problems which have been encountered in practice. Furthermore the positive results for the two estimators we introduce here suggest they are viable alternatives to using the CLAD. We conclude this section with two additional remarks regarding our simulation results. First, it should be noted that for all designs we considered, there is a fraction of observations for which the index x0i ¯0 is positive. Thus it is not necessary for the full rank condition to be literally violated for the CLAD estimator to break down. Second, it should be noted that the designs considered only involved one regressor. It is expected that the results for the WNPQN and WNPQ would be worse if the number of regressors were increased, as the ¯rst stage estimator would su®er from the usual curse of dimensionality, and possibly have a (second order) e®ect on the second stage estimator.
20
5
Summary and Concluding Remarks
This paper introduces two new estimators for censored regression models with conditional heteroscedasticity of very general forms. One of the estimators requires no parametric speci¯cation of the homoscedastic error term which is multiplied by a scale function. The advantage of these estimators is that they allow for more general rank conditions than estimators based on a conditional quantile restriction, notably Powell's CLAD estimator. This generalization proves especially useful in the presence of heavily censored data. The proposed estimators are shown to have desirable asymptotic properties, and a simulation study indicates that the estimators have favorable ¯nite sample properties at various degrees of censoring. There are two shortcomings with our proposed procedures. The ¯rst is that they involve nonparametric estimation in the ¯rst stage which introduces the usual problems of bandwidth selection, as well as a curse of dimensionality, which in this context will have a second order e®ect on the second step of the estimators. The second shortcoming is that our procedures are not asymptotically e±cient. Since both procedures are based on selecting only two quantiles, one can easily conceive of constructing more e±cient estimators based on several quantiles, analogous to the e±ciency gain discussed in Powell(1986a). Thus any skepticism regarding our simulation results being based on only one regressor should be weighed against the fact that they were also only based on two quantiles, suggesting that dimensionality may not be as serious a problem in this case. The procedures introduced in this paper can be extended to other models, suggesting areas for future research. For example, Powell's(1986b) symmetrically trimmed estimators also require a positive index for identi¯cation, which could be relaxed in a similar fashion. Also, estimators which are bivariate extensions of these symmetrically trimmed estimators (e.g. Honor¶e(1993), Honor¶e and Powell(1994), Honor¶e et.al.(1997)) have analogous rank conditions which could possibly be relaxed by adopting similar procedures.
References [1] Amemiya, T. (1985), Advanced Econometrics, Cambridge, MA: Harvard University Press [2] Andrews, D.W.K. (1994), \Empirical Process Methods in Econometrics", in Engle, R.F. and D.McFadden (eds.), Handbook of Econometrics, Vol. 4, Amsterdam: North21
Holland. [3] Buchinsky, M. (1994), \Changes in the U.S. Wage Structure 1963-1987: Application of Quantile Regression", Econometrica, 62, 405-458 [4] Buchinsky, M. and J. Hahn (1998), \An Alternative Estimator for the Censored Quantile Regression Model", Econometrica, 66, 653-672 [5] Chaudhuri, P. (1991a), \Nonparametric Quantile Regression", Annals of Statistics, 19, 760-777 [6] Chaudhuri, P. (1991b), \Global Nonparametric Estimation of Conditional Quantiles and their Derivatives", Journal of Multivariate Analysis, 39, 246-269 [7] Chaudhuri, P., K. Doksum, and A. Samarov (1997), \On Average Derivative Quantile Regression", Annals of Statistics, 25, 715-744 [8] Chay, K.Y. (1995), \Evaluating the Impact of the 1964 Civil Rights Act on the Economic Status of Black Men using Censored Longitudinal Earnings Data", Unpublished manuscript, Princeton University [9] Chen, S. and S. Khan (1998a), \Estimation of a Semilinear Censored Regression Model", manuscript, University of Virginia [10] Chen, S. and S. Khan (1998b), \Quantile Estimation of Non-stationary Panel Data Censored Regression Models", manuscript, University of Virginia [11] Fan, J. and I. Gijbels (1996), Local Polynomial Modelling and its Applications', New York: Chapman and Hall. [12] Honor¶e, B.E. (1992) \Trimmed LAD and Least Squares Estimation of Truncated and Censored Regression Models with Fixed E®ects", Econometrica, 60, 533-565 [13] Honor¶e, B.E., Kyriazidou, E. and C. Udry(1997) \Estimation of Type 3 Tobit Models Using Symmetric Trimming and Pairwise Comparisons", Journal of Econometrics, 76, 107-128 [14] Honor¶e, B.E. and J.L. Powell (1994), \Pairwise Di®erence Estimators of Censored and Truncated Regression Models", Journal of Econometrics, 64, 241-278 [15] Horowitz, J.L. (1986), \A Distribution-Free Least Squares Estimator for Censored Linear Regression Models", Journal of Econometrics, 32, 59-84 22
[16] Horowitz, J.L. (1988), \Semiparametric M-Estimation of Censored Linear Regression Models", Advances in Econometrics, 7, 45-83 [17] Horowitz, J.L. (1993), \Semi-Parameric and Non-Parametric Estimation of Quantal Response Models", in Maddala, G.S., Rao, C.R. and H.D. Vinod(eds.), Handbook of Statistics Vol 11, Amsterdam: Elsevier Science [18] Khan, S. and J.L. Powell (1996), \2-Step Quantile Estimation of the Censored Regression Model ", manuscript, Princeton University [19] Koenker, R. and G.S. Bassett Jr. (1978), \Regression Quantiles", Econometrica, 46, 33-50 [20] Nawata, K. (1990), \Robust Estimation Based on Group-Adjusted Data in Censored Regression Models", Journal of Econometrics, 43, 337-362 [21] Newey, W.K. and D. McFadden (1994) \Estimation and Hypothesis Testing in Large Samples", in Engle, R.F. and D. McFadden (eds.) , Handbook of Econometrics, Vol. 4, Amsterdam: North-Holland. [22] Powell, J.L. (1984) \Least Absolute Deviations Estimation for the Censored Regression Model", Journal of Econometrics, 25, 303-325 [23] Powell, J.L. (1986a) \Censored Regression Quantiles", Journal of Econometrics, 32, 143-155 [24] Powell, J.L. (1986b) \Symmetrically Trimmed Least Squares Estimation of Tobit Models", Econometrica, 54, 1435-1460 [25] Powell, J.L., J.H. Stock, and T.M. Stoker (1989) \Semiparametric Estimation of Index Coe±cients", Econometrica, 57, 1404-1430. [26] Sherman, R.P. (1994), \U-Processes in the Analysis of a Generalized Semiparametric Regression Estimator", Econometric Theory, 10, 372-395
23
Table I Monte Carlo Simulation Homoscedastic Design- Normal Errors ® = 0 (50%)
® = ¡0:25 (46.4%)
CLAD
WNPQN
WNPQ
CLAD
WNPQ
WNPQN
Mean Bias
-0.0447
-0.0908
-0.0522
-0.1075
-0.0787
-0.0661
Med. Bias
-0.1692
-0.0655
-0.0623
-0.2750
-0.0747
-0.0871
MSE
0.3243
0.2656
0.1494
0.4755
0.2337
0.1836
MAD
0.2768
0.3279
0.2360
0.3595
0.3108
0.2405
Mean Bias
-0.0519
-0.0632
-0.0309
-0.1090
-0.0873
-0.0485
Med. Bias
-0.1321
-0.0650
-0.0264
-0.2279
-0.0828
-0.0586
MSE
0.1698
0.1198
0.0398
0.2278
0.1178
0.0388
MAD
0.2218
0.2255
0.1359
0.3077
0.2336
0.1406
Mean Bias
-0.0278
-0.0333
-0.0211
-0.0599
-0.0541
-0.0370
Med. Bias
-0.0938
-0.0378
-0.0271
-0.1817
-0.0498
-0.0352
MSE
0.1168
0.0635
0.0190
0.2616
0.0690
0.0207
MAD
0.1897
0.1669
0.0911
0.2595
0.1619
0.0978
Mean Bias
-0.0206
-0.0228
-0.0100
-0.0460
-0.0407
-0.0149
Med. Bias
-0.0741
-0.0277
-0.0031
-0.1298
-0.0439
-0.0147
MSE
0.0657
0.0315
0.0100
0.1680
0.0358
0.0094
MAD
0.1518
0.1135
0.0618
0.2203
0.1299
0.0644
n = 100:
n = 200:
n = 400:
n = 800:
24
Table II Monte Carlo Simulation Homoscedastic Design- Normal Errors ® = ¡0:5 (42.2%)
® = ¡0:75 (38.4%)
CLAD
WNPQN
WNPQ
CLAD
WNPQN
WNPQ
Mean Bias
-0.1916
-0.1088
-0.1191
-0.2974
-0.1905
-0.1031
Med. Bias
-0.3974
-0.1046
-0.1304
-0.5000
-0.1786
-0.1316
MSE
0.5144
0.2609
0.2527
0.3953
0.3325
0.4487
MAD
0.4704
0.3135
0.2866
0.5000
0.3561
0.3329
Mean Bias
-0.2214
-0.1319
-0.0806
-0.3414
-0.2045
-0.0984
Med. Bias
-0.4226
-0.1252
-0.0830
-0.5000
-0.2018
-0.1086
MSE
0.2897
0.1295
0.0527
0.2834
0.1816
0.0609
MAD
0.4956
0.2454
0.1521
0.5000
0.2846
0.1649
Mean Bias
-0.2043
-0.0757
-0.0598
-0.3404
-0.1364
-0.0784
Med. Bias
-0.3494
-0.0767
-0.0634
-0.5000
-0.1448
-0.0843
MSE
0.2165
0.0848
0.0247
0.3631
0.0982
0.0283
MAD
0.4430
0.1909
0.1060
0.5000
0.2054
0.1238
Mean Bias
-0.1709
-0.0762
-0.0351
-0.4095
-0.1161
-0.0617
Med. Bias
-0.2774
-0.0716
-0.0372
-0.5000
-0.1152
-0.0663
MSE
0.2111
0.0398
0.0103
0.2572
0.0518
0.0140
MAD
0.3568
0.1350
0.0661
0.5000
0.1476
0.0887
n = 100:
n = 200:
n = 400:
n = 800:
25
Table III Monte Carlo Simulation Homoscedastic Design- Cauchy Errors ® = 0 (50.0%)
® = ¡0:25 (43.8%)
CLAD
WNPQN
WNPQ
CLAD
WNPQN
WNPQ
Mean Bias
-0.0447
-0.1415
-0.0697
-0.1075
-0.1265
-0.0841
Med. Bias
-0.1692
-0.0325
-0.1204
-0.2750
-0.0353
-0.1766
MSE
0.3243
2.8677
0.4947
0.4755
3.1428
0.6639
MAD
0.2768
0.5932
0.2766
0.3595
0.6288
0.3611
Mean Bias
-0.0519
-0.0544
-0.0618
-0.1090
-0.0544
-0.0961
Med. Bias
-0.1321
-0.0228
-0.0820
-0.2279
-0.0228
-0.1276
MSE
0.1698
0.5526
0.0623
0.2278
0.5526
0.0719
MAD
0.2218
0.3539
0.1461
0.3077
0.3539
0.1747
Mean Bias
-0.0278
-0.0419
-0.0561
-0.0599
-0.0657
-0.0860
Med. Bias
-0.0938
-0.2285
-0.0581
-0.1817
-0.0253
-0.0982
MSE
0.1168
0.2407
0.0221
0.2616
0.3143
0.0312
MAD
0.1897
0.2602
0.0960
0.2595
0.2732
0.1266
Mean Bias
-0.0206
-0.0198
-0.0387
-0.0460
-0.0527
-0.0622
Med. Bias
-0.0741
-0.0107
-0.0417
-0.1298
-0.0299
-0.0705
MSE
0.0657
0.0776
0.0091
0.1680
0.1103
0.0147
MAD
0.1518
0.1703
0.0658
0.2203
0.1879
0.0915
n = 100:
n = 200:
n = 400:
n = 800:
26
Table IV Monte Carlo Simulation Homoscedastic Design- Cauchy Errors ® = ¡0:5 (37.8%)
® = ¡0:75 (32.3%)
CLAD
WNPQN
WNPQ
CLAD
WNPQN
WNPQ
Mean Bias
-0.2116
-0.1780
-0.1213
-0.3578
-0.2961
-0.1797
Med. Bias
-0.3786
-0.0273
-0.2271
-0.5000
-0.1317
-0.2975
MSE
0.2942
4.3889
1.3639
0.3465
6.6233
1.4293
MAD
0.4226
0.7296
0.4292
0.5000
0.7325
0.4374
Mean Bias
-0.2002
-0.1135
-0.1235
-0.3827
-0.1605
-0.1526
Med. Bias
-0.2940
-0.0785
-0.1593
-0.5000
-0.0771
-0.2335
MSE
0.1733
0.9828
0.2718
0.2835
1.8000
0.7680
MAD
0.3380
0.4624
0.2739
0.5000
0.5374
0.3160
Mean Bias
-0.1866
-0.0346
-0.0915
-0.4505
-0.1620
-0.1314
Med. Bias
-0.2458
-0.0275
-0.1119
-0.5000
-0.1250
-0.1677
MSE
0.1722
0.5492
0.0573
0.2374
0.6267
0.0911
MAD
0.3020
0.3365
0.1666
0.5000
0.3775
0.2167
Mean Bias
-0.1229
-0.0365
-0.0814
-0.4554
-0.0624
-0.1139
Med. Bias
-0.1649
-0.0266
-0.0950
-0.5000
-0.0485
-0.1396
MSE
0.1275
0.1745
0.0251
0.2466
0.2962
0.0448
MAD
0.2323
0.2433
0.1200
0.5000
0.3022
0.1657
n = 100:
n = 200:
n = 400:
n = 800:
27
Table V Monte Carlo Simulation Heteroscedastic Design ® = 0 (50%)
® = ¡0:25 (45%)
CLAD
WNPQN
WNPQ
CLAD
WNPQN
WNPQ
Mean Bias
0.2957
-0.1036
0.0209
0.4215
-0.0439
0.0460
Med. Bias
-0.2560
-0.0524
0.0448
-0.4715
-0.0137
0.0356
MSE
6.9389
1.8026
0.9811
17.479
1.5908
0.9911
MAD
0.4629
0.8251
0.6054
0.5000
0.7801
0.6340
Mean Bias
0.3648
-0.0596
0.0357
0.2102
-0.0703
-0.0124
Med. Bias
-0.1487
-0.0590
0.0247
-0.3764
-0.0878
-0.0073
MSE
12.958
1.0425
0.5717
3.0417
1.0185
0.6027
MAD
0.3667
0.6245
0.5166
0.5000
0.6568
0.5196
Mean Bias
-0.1571
-0.0449
-0.0491
0.2987
-0.0042
-0.0748
Med. Bias
-0.0719
-0.0394
-0.0724
-0.3166
-0.0172
-0.1049
MSE
0.9476
0.4751
0.3231
20.5311
0.4621
0.3352
MAD
0.2929
0.4385
0.3921
0.5000
0.4492
0.4014
Mean Bias
-0.1056
-0.0008
-0.0125
0.0918
-0.0323
-0.0292
Med. Bias
-0.0361
-0.0131
-0.0059
-0.2690
0.0026
-0.0237
MSE
0.4964
0.2989
0.2053
1.8531
0.3044
0.2085
MAD
0.2014
0.3605
0.3179
0.5000
0.3623
0.3194
n = 100:
n = 200:
n = 400:
n = 800:
28
Table VI Monte Carlo Simulation Heteroscedastic Design ® = ¡0:5 (39.8%)
® = ¡0:75 (35%)
CLAD
WNPQN
WNPQ
CLAD
WNPQN
WNPQ
Mean Bias
0.5856
-0.0660
0.0554
0.0599
-0.1809
0.1038
Med. Bias
-0.5000
-0.0338
-0.0125
-0.5000
-0.1758
-0.1203
MSE
11.199
1.9428
1.3164
4.0091
2.8080
2.2658
MAD
0.5000
0.8254
0.6716
0.5000
0.9967
0.7415
Mean Bias
0.4312
-0.1269
-0.0466
-0.1152
-0.1377
0.0357
Med. Bias
-0.5000
-0.1288
-0.0558
-0.5000
-0.1575
0.0247
MSE
15.156
1.1285
0.7433
4.6252
1.2553
0.5717
MAD
0.5000
0.7236
0.5098
0.5000
0.6950
0.5166
Mean Bias
0.1299
-0.0816
-0.0864
-0.2644
-0.0784
-0.0491
Med. Bias
-0.5000
-0.0353
-0.0594
-0.5000
-0.1190
-0.0724
MSE
1.3755
0.6092
0.3730
1.3755
0.6856
0.3231
MAD
0.5000
0.4852
0.4038
0.5000
0.5423
0.3921
Mean Bias
0.1021
-0.1003
-0.0548
-0.4245
-0.0937
-0.0125
Med. Bias
-0.5000
0.0006
-0.0565
-0.5000
-0.0974
-0.0059
MSE
1.5946
0.3032
0.2525
0.4012
0.3567
0.2053
MAD
0.5000
0.3480
0.3433
0.5000
0.3920
0.3179
n = 100:
n = 200:
n = 400:
n = 800:
29
