Download Extending Powell's Semiparametric Censored Estimator to Include Non-Linear Functional Forms and Extending Buchinsky's Estimation Technique

DISCUSSION PAPERS IN ECONOMICS
Working Paper No. 98-27
Extending Powell's Semiparametric Censored Estimator to
Include Non-Linear Functional Forms and
Extending Buchinsky's Estimation Technique
Gregory D. Berg
Department of Economics, University of Colorado at Boulder
Boulder, Colorado
October 1998
Center for Economic Analysis
Department of Economics
University of Colorado at Boulder
Boulder, Colorado 80309
© 1998 Gregory D. Berg
Advantages of Censored Regressions with LAD
The “Tobit", from Tobin (1958) for censored regression models, has received much
attention with parametric and semiparametric techniques suggested in the literature. The
advantage to semiparametric techniques is that no assumption on the error terms is needed.
As such, it is robust to non-normality and heteroskedasticity. One such estimator is by
Powell (1984).
The Powell's censored least absolute deviation estimator for censored regressions has
two main advantages over the censored maximum likelihood estimation. The first main
advantage is that it is robust to non-normality of the error terms. Secondly it is robust to
heteroskedasticity as is common in most cross-sectional data sets.
The performance of Powell's censored LAD is shown in Paarsch(1984). Paarsch
compared the censored LAD and the censored ML when the assumption of normality did not
hold and the censored LAD performed better shown by closer estimates to the true values of
the parameters. The censored maximum likelihood performed better with closer estimates to
the truth only when the errors were normally distributed. It would appear that the censored
LAD is a preferred estimator for censored regressions.
However, Powell’s censored LAD appears in the economic literature only rarely.
The reason for this is threefold. First the censored LAD does not appear in any statistical
package to date. As such, it is much more difficult for the applied econometrician to
implement the censored LAD. Secondly, even if the applied econometrician were able to
program a computer to calculate the censored LAD, it has been computationally difficult.
Thirdly, the functional form must be linear. So the merits of Powell’s censored LAD
estimator as being semi-parametric and robust to non-normality and heteroskedasticity are
2
diminished by the computational complexity, the lack of the estimator in standard
econometric software and the limitation that the functional form must be linear.
This paper extends Powell's (1984) censored LAD to include non-linear functional
forms. Although the non-linear censored least absolute deviations (NLC-LAD) extensions
are not proven theoretically, a Monte Carlo study shows that this is a good topic for future
research. The Monte Carlo study shows that when the assumptions of the censored ML are
not met the NLC-LAD performs better. This is important in econometrics because there is
now an estimator that outperforms the Tobit in cross-sectional data sets with over 1000
observations when heteroskedasticity is not correctly specified. This many observations are
common in today's cross-sectional surveys and there are infinite specifications that may be
the correct specification of heteroskedasticity.
Review of Literature
The first group of literature is about estimators. There are three estimators discussed
here: the Tobit from Tobin (1958), the censored LAD from Powell (1984), and one by
Buchinsky and Hahn (1998). The “Tobit” from Tobin (1958) is a maximum likelihood
estimator for a censored regression. Sometimes the literature refers to all censored
regressions as Tobits. This approach is not taken here. The Tobit in this paper is meant to be
the maximum likelihood estimator. The maximum likelihood estimator is a parametric
estimator because it specifies a functional form for both the regression equation and for the
distribution of the error process. A semiparametric estimator assumes a functional form for
the regression but assumes no functional form for the error process. This is an advantage to
semiparametric techniques because of no assumption on the error terms is needed. As such it
is robust to non-normality and heteroskedasticity. One such estimator is by Powell (1984).
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Powell's (1984) censored least absolute deviation (LAD) estimator for censored
regressions has two main advantages over the censored maximum likelihood estimation.
First, it is robust to non-normality of the error terms. Second, it is robust to
heteroskedasticity as is common in most cross-sectional data sets. Unlike the parametric
Tobit procedures that specify a functional form for demand and a functional form for the
disturbances, the LAD estimator does not specify any distribution for the errors. As such, it
is a fortiori robust to heteroskedasticity (Powell 1984, 305).
“The assumptions on the distribution of the error term ut required for
consistency are much weaker than those for maximum likelihood or least squares
estimators for the censored regression model. It is the fact that the median of the
censored variable yt does not depend upon the functional form of the density of the
errors that makes censored LAD a ‘distribution-free’ estimator, a property not shared
by the mean (if it exists) of yt. Hence least absolute deviations estimation is a natural
approach for the censored data when the assumption of normality of the errors is
suspect”(Powell 1984, 307).
Paarsch (1984) shows the superior performance of the censored LAD compared to the
censored maximum likelihood to departures from non-normality in a Monte Carlo study. The
censored maximum likelihood performed well only when the errors were normally
distributed. It would appear that the censored LAD is a preferred estimator for censored
regressions.
Paarsch had a linear censored model where yi = max{0,a+bxi}. The censored LAD
performed better than the Tobit (censored ML) when the errors were distributed as Cauchy.
Both estimators appeared to be similar when the errors were distributed as Laplace whereas
when the errors were distributed as Normal, the censored ML was a much better estimator.
Unfortunately in practice the exact nature of the errors is not known which gives reason to
use the censored LAD. Paarsch (1984) has three objectives: (1) to show the effect of
different distributions; (2) to show the effect of sample size for small to medium size
samples; and (3) to show the effect of the degree of censoring. Some of Paarsch's results are
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in Table 1 comparing the censored maximum likelihood estimator and the censored LAD of
Powell. Cauchy, laplace ,and normal distributions were used in the comarison.
TABLE 1
Cauchy
Tobit
a
b
Powell
a
b
Truth
-10.00
1.00
-10.00
1.00
Mean
-108.33
7.30
-16.73
1.42
Median
-57.66
3.50
-12.83
1.23
S.D.
423.46
37.21
13.52
0.86
Laplace
Tobit
a
b
Powell
a
b
Truth
-10.00
1.00
-10.00
1.00
Mean
-12.09
1.14
-12.78
1.17
Median
-11.70
1.10
-10.99
1.12
S.D.
3.59
0.27
7.02
0.44
Normal
Tobit
a
b
Powell
a
b
Truth
-10.00
1.00
-10.00
1.00
Mean
-10.23
1.02
-13.69
1.30
Median
-9.86
1.01
-11.65
1.19
S.D.
3.08
0.24
9.24
0.70
As can be seen the censored LAD performed better than the Tobit (censored ML)
when the errors were distributed as Cauchy. Both estimators appeared to be similar when the
errors were distributed as Laplace whereas when the errors were distributed as Normal, the
censored LAD was a much better estimator. Unfortunately, in practice the exact nature of the
errors is not known which gives reason to use the censored LAD. Paarsch (1984) has three
objectives: (1) to show the effect of different distributions; (2) to show the effect of sample
size for small t medium size samples; and (3) to show the effect of the degree of censoring.
Paarsch concludes that:
"The most striking feature of these tables is the relatively poor performance
of the Tobit ML estimator and Heckman's two-step method and the stable
performance of Powell's LAD estimator when the errors are Cauchy. Of interest too
is the relative performance of the three estimators when errors are normal. Although
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Heckman's two-step method and Powell's LAD estimator do appear biased in small
samples, this bias appears to shrink quickly [with increased sample size]. Of more
concern is the relative inefficiency of both when compared to Tobit ML. As is to be
expected the Tobit ML estimator performed well under normality, but the fact that its
relative bias is low when the errors are Laplace is indeed surprising" (Paarsch, 1984,
p. 210-211).
Two results of Paarsch's (1984) study show that Powell's (1984) LAD performs better
as the degree of censoring is reduced and as the sample size is increased.
A recent estimator that is similar to Powell is by Buchinsky and Hahn (1998) who
estimate censored quantile regressions (censored LAD is the 50th quantile) by first estimating
nonparametric quantiles and conditional distributions and then using linear programming to
estimate the parameters of interest. The benefits are (1) estimation is possible when the
censoring point is not known and (2) the objective function is globally convex so it
guarantees a global minimum in a finite number of simplex iterations. However,
nonparametric estimation is still required first, which may be a complicated process.
Deaton (1997) shows that for heteroskedasticity and a sample size of 100, Powell's
estimator is closer to the truth in only 55 out of 100 replications with a larger standard
deviation. So for small samples the Tobit may be better since, although it is biased in small
samples, it has a smaller standard deviation. However the bias to variance tradeoff turns out
to Powell's censored LAD favor when the sample size increases. With a sample size of 1000
instead of 100, the censored LAD was closer to the truth than the censored ML 96 percent of
the time. Since most household surveys have sample sizes at least this large, Deaton suggests
using the censored LAD and at the very least to compare it to the censored ML as a guide to
failure of homoskedasticity (Deaton 1997, p. 90).
However, Powell’s censored LAD appears in the economic literature only rarely.
The reason for this is twofold. First, the censored LAD does not appear in any statistical
package to date, so that it is difficult for the applied econometrician to implement the
6
censored LAD. Second, even if the applied econometrician were able to program a computer
to calculate the censored LAD, some methods of computation are difficult.
The problem stems from the objective function having several local minimums along
with the global minimum. The objective function is not globally convex. See Nawata (1994)
for a level surface of the objective function with two estimated parameters. Nawata suggests
a random search procedure to obtain a global minimum.
The censored regression comes from the model where Yi* = a+bxi + ui. Yi* is the
latent demand and if positive, Yi* is equal to the actual demand Yi. However when Yi* is
negative, the actual demand is equal to zero from censoring, the minimum allowed. The
actual demand Yi therefore equals max{0, a+bxi + ui}.
The censored regression model is
y i = max{0, x i' b + u i }
i = 1,2,..., N
where the error term ui and the parameter vector b are unobserved.
Powell's original objective function with only two parameters to be estimated, a and
b, with two explanatory variables, a constant and xi is:
N
N
Min ∑ y i − m(a , b, x i ) ⇔ Min ∑ y i − max( 0, a + bx i ) .
< a ,b >
< a ,b >
i =1
i =1
Powell's (1984) estimator is restricted to linear functional forms. Powell shows that
the median function m(a,b,xi) is equal to the function max(0,a+bxi). The censored LAD
objective then is to minimize the following by choosing a vector of parameters b:
(
{
})(
{
})
1 N
'
'
∑ sign y i − max 0, x i b y i − max 0, x i b
N i =1
7
which has first order conditions,
(
{
}) ( )
1 N
'
'
∑ sign y i − max 0, x i b 1 x i b = 0
N i =1
The above first order condition cannot hold exactly. The indicator function, 1(xi'b)
shows that when estimating b, the only observations for which xi'b > 0 are used. Dropping
the observations for which xi'b < 0 is of no consequence because yi - max{0, xi'b} = yi , which
does not depend on the parameters b.
The second group of literature is about estimation techniques to obtain Powell's
(1984) estimator. There are four approaches to estimate the censored least absolute deviation
(LAD) estimator of Powell. The first is by Butler et. al (1990) which uses a gradient
approach as to calculate a LAD that is not censored and this approach can be extended to the
censored case as well. This approach calculates a gradient of the objective function and then
decides where to move from there. A criticism of this approach is that the censored LAD
estimator objective function is not differentiable and thus a gradient method that assumes
differentiability will not work very well. However, since the objective function is continuous,
numerical gradients can be calculated and implemented in a censored framework.
The second approach is by Buchinsky (1994). Since max(0,a+bxi) is not linear in the
parameters a and b, linear programming cannot be used. Still an LP algorithm can be used.
Buchinsky(1994) suggests the Iterative Linear Programming Algorithm (ILPA) to obtain
Powell's(1984) estimator. The ILPA starts with the whole data set and estimates the median
regression which is a standard LP problem. From the estimated parameters, truncate the data
set by keeping only the observations where the predicted demands are positive. From this
truncated data set, estimate the median regression again. Go back to the whole data set and
8
keep the predicted demands using the updated parameters where the predicted demands are
positive. On the new truncated sample repeat the process. Stop the iteration procedure when
two sets of consecutive iterations are the same. If the iteration number is finite then a local
minimum is guaranteed (Buchinsky 1994, 412). The proof of the ILPA is in Buchinsky
(1991).
Deaton (1997) suggests using the Buchinsky (1994, 412) method to obtain Powell's
(1984) estimator. Furthermore Deaton suggests that if the ILPA results in a set of parameter
estimates that are not the same in consecutive iterations but rather the same in say every fifth
iteration, then cycling is said to occur through a finite set of estimates. If cycling occurs,
Deaton suggests using the estimates that fit the criterion function best.
Buchinsky's iterative method starts with the whole data set with the objective
N
Min
< a , b ,e i >
∑e
i
i =1
s.t.
(a )e i ≥ 0
(b)(a + bx i ) − y i ≤ e i
(c) y i − (a + bx i ) ≤ e i
which is a linear program. This linear program is equivalent to a standard LAD
N
Min ∑ y i − (a + bx i )
< a ,b >
i =1
which is the beginning of the iteration process. The estimated parameters are used to
calculate predicted demands.
The third method to calculate Powell's (1984) censored LAD is by Nawata (1994).
This approach uses a random search procedure to obtain the global minimum of the objective
9
function. This is a time consuming process which does not guarantee the global minimum is
reached.
The fourth method is by Paarsch (1984). Paarsch uses nonlinear programming and a
spline method to estimate the censored LAD. The max{0,a} function is splined. Paarsch
uses nonlinear programming with a splined function and 3N constraints where N is the
number of observations. One of the problems with the Paarsch approach is the nonconvexity of one of the constraints. As such a global minimum is not guaranteed and a grid
search must be used. Paarsch's estimation technique is still preferred by Rao and Zhao
(1995).
Standard errors can be calculated using the asymptotic variance matrix that Powell
develops for his estimator (Powell 1984, 312) or the bootstrap estimates of the standard
errors. Paarsch shows that a technique to estimate a, b is:
N
Min
< a , b ,e i >
∑e
i =1
i
s.t.
y i − max{0, a + bx i } ≤ e i
⇔
N
Min
< a , b ,e i >
∑e
i =1
i
s.t.
(1) y i − max{0, a + bx i } ≤ e i
(2) max{0, a + bx i } − y i ≤ e i
(3)e i ≥ 0
for
i = 1,2,..., N
10
So there are 3N constraints. Paarsch says that max{0,a+bxi} can be splined and the resulting
non-linear program solved. The non-convexity problem is in the first constraint when the
parameter "b" is positive.
Extensions
Powell (1984) derives an estimator and asymptotic covariance matrix of the
parameters. Powell's model is robust to non-normality and heteroskedasticity. It is a semiparametric model that assumes a functional form for the model but no specification for the
errors is required.
Powell's model is:
y i = max{0, x i' b + u i }, i = 1,2,..., N
where yi is the censored observable dependent variable, xi is a vector of k explanatory
variables, b is a vector of k parameters, and ui is the unobservable disturbance term. Powell
develops the censored LAD estimation for b and the estimated asymptotic variance
^ −1 ^
^ −1
C MC
where
^
^
^
^
^
C = 2(c i N ) −1 ∑ 1( x 'i b > 0)1(0 ≤ u i ≤ c N ) x i x i'
i
^
M=
^
1
1( x 'i b > 0)x i x 'i
∑
N i
covariance matrix:
1(A) = A if A is true and 0 if A is false
^
^
u i = y i − x i' b
^
^
^
^
c N = c 0 N −g median{u i : u i > 0, x i' b > 0}
c0 > 0
1
g ∈ (0, )
2
11
Powell's model is restricted to linear censored models where the linearity in
parameters comes from x'ib. The suggestion here is to extend the functional form to include
forms that are nonlinear in the parameters. Powell's estimator can be generalized to include
models nonlinear in the parameters such as:
y i = max{0, f ( x i , b) + u i }
where i = 1,2,…N and f(xi,b) is a function nonlinear in the parameters.
This is useful in demand estimation with demand equations resulting from a constant
elasticity of substitution (CES) model as a simple case or a nonlinear consumption function
in the time domain.
It is suggested here, not proven, that Powell's proofs of consistency and asymptotic
normality might hold in a general model by replacing xi'b in Powell's work with f(xi, b). The
asymptotic variance covariance matrix changes slightly with the replacement of
x i x 'i
with
^
^



 ∂f (x i , b)  ∂f ( x i , b) 
^
^



 ∂ b  ∂ b 
'
Now since a suggestion of a censored non-linear least absolute deviations estimator is
at hand, an estimation technique is needed. Buchinsky's (1996) Iterative Linear
Programming Algorithm is extended here to a similar Iterative Non-Linear Programming
Algorithm.
Buchinsky's (1994) method shows that since the function max{0,xi'b} is not linear in
the parameters, linear programming (LP) cannot be used. Still an LP algorithm can be used.
12
Buchinsky (1994) suggests the Iterative Linear Programming Algorithm (ILPA) to obtain
Powell's (1984) estimator. The ILPA starts with the whole data set and estimates the median
regression which is a standard LP problem. From the estimated parameters, truncate the data
set by keeping only the observations where the predicted demands are positive. From this
truncated data set, estimate the median regression again. Go back to the whole data set and
keep the predicted demands using the updated parameters where the predicted demands are
positive. On the new truncated sample repeat the process. Stop the iteration procedure when
two sets of consecutive iterations are the same. If the iteration number is finite then a local
minimum is guaranteed (Buchinsky 1994, 412). The proof of the ILPA is in Buchinsky
(1991).
The censored LAD objective is to minimize the following by choosing b:
(
{
})(
{
})
1 N
'
'
∑ sign y i − max 0, x i b y i − max 0, x i b
N i =1
which has first order conditions,
(
{
}) ( )
1 N
'
'
∑ sign y i − max 0, x i b 1 x i b = 0
N i =1
The above first order condition cannot hold exactly. The indicator function, 1(xi'b)
shows that when estimating b, the only observations for which xi'b > 0 are used. Dropping
the observations for which xi'b < 0 is of no consequence because yi - max{0, xi'b} = yi , does
not depend on b.
The extension here to the Iterative Non-linear Programming Algorithm is to replace
the objective function in the ILPA, with a similar objective function except replacing the
linear latent equation with a nonlinear latent equation. The underlying latent variable is yi*
13
which is observed only when positive. The observed variable is yi which may be censored to
be equal to zero if yi* is non positive. That is:
yi = yi* if yi* >0 or yi = 0 if yi* <=0.
The equation that determines yi* is:
yi* = xi'b + ei
where: xi is an (1xk) vector of k exogenous variables for i = 1, 2, …, N observations,
b is a (kx1) vector of parameters to be estimated, and ei is a (1x1) vector of unobserved
random errors for i = 1, 2, …, N.
The extension suggested here is to make the above linear specification into a more
general specification of:
yi* = f(xi,b)+ei
where f(xi,b)is a linear or nonlinear function of the exogenous variables and parameters.
The objective function of the ILPA is to minimize:
(
{
})(
{
})
1 N
sign y i − max 0, x 'i b y i − max 0, x 'i b .
∑
N i=1
The objective of the extended model is to minimize:
(
{
})(
{
})
1 N
∑ sign yi − max 0, f ( x i , b) yi − max 0, f ( x i , b) .
N i=1
Now since the objective function has changed so do the first order conditions. The
FOC change from:
(
{
}) ( )
(
{
}) (
1 N
∑ sign yi − max 0, x i' b 1 x 'i b = 0
N i=1
to
)
1 N
∑ sign yi − max 0, f ( x i , b) 1 f ( x i , b) = 0.
N i=1
14
Again, the above first order condition cannot hold exactly. The indicator function, 1(f(xi,b)),
shows that when estimating b, the only observations for which f(xi,b) > 0 are used. Dropping
the observations for which f(xi,b) < 0 is of no consequence because yi - max{0, f(xi,b) } = yi ,
which does not depend on the parameters to be estimated, b.
So the suggested extension of Powell's (1984) semiparametric estimator for censored
regressions with an extension of the method of computation suggested by Buchinsky (1994)
are now at hand. Even though this particular paper does not prove the extensions, it does
give indication that further study is warranted with the results of the below Monte Carlo
study.
Monte Carlo Study
Before theoretical work is done to prove the asymptotic results of the non-linear
extended Powell estimator, a Monte Carlo study was undertaken as a first step. The
nonlinear censored least absolute deviation estimator (NLC-LAD) was compared to the
censored maximum likelihood (ML) estimator. The class of non-linear functional forms is
large so there might be some restrictions on functional forms that are allowed in ongoing
work to prove the NLC-LAD has the desired properties of consistency and an asymptotically
normal covariance matrix.
The nonlinear functional form chosen is most similar to a consumption function in
economics, Consumption = a + (Income)b, where ‘a’ and ‘b’ are parameters to be estimated.
The model used for generation of data is:
C = a +Yb + e,
15
where: C is consumption, Y is income, a = -1, b = 0.5, and the random component e is
distributed in test 1 as independently normal with variance of 25, and in test 2 as
independently normal with a variance of 9*Y. The exogenous income, Y, is distributed
equidistant in the interval [0,20]. Test 2 is an introduction of heteroskedasticity. This tests
the NLC-LAD to a censored ML with a misspecification of heteroskedasticity. Often in
applied work the exact nature of the heteroskedasticity is unknown. The previous results in
the Chapter V Demand Estimation and Analysis show that the exact nature of the
heteroskedasticity is difficult specify so specifications that have been used in the past are
used. The Hausman test showed that for the 14 demand equations that 4 were not consistent
when estimated with the censored ML. This arises from the unknown exact nature of the
heteroskedasticity. Table 29 shows the results of the two Monte Carlo tests.
TABLE 2
Test 1
Censored ML
NLC-LAD
Test 2
Censored ML
NLC-LAD
Parameter
a
b
a
b
Truth
-1
0.5
-1
0.5
Mean
-0.93
0.45
-0.73
0.42
Median
-0.91
0.49
-0.70
0.46
S.D.
1.11
0.29
1.15
0.23
Parameter
a
b
a
b
Truth
-1
0.5
-1
0.5
Mean
-3.22
0.67
-0.92
0.37
Median
-3.26
0.70
-0.65
0.39
S.D.
1.68
0.14
1.47
0.25
For test 1 (Homoskedasticity) the percent censoring was 35%, the number of
observations = 100, and the number of estimation trials = 100. NLC-LAD is closer to the
truth 48.9 percent of the time for parameter ‘a’. NLC-LAD is closer to the truth 43.2 percent
of the time for parameter ‘b’. These results show the relative power of censored ML when all
of the assumptions in favor of censored ML are met. Censored ML performed better based
on the percent of time the censored ML estimates were closer to the truth compared to the
16
number of times NLC-LAD was closer to the truth. The relative efficiency of the two
estimators was similar with the censored ML having a smaller standard deviation of the
estimated parameter 'a' but a larger standard deviation of the estimated parameter 'b'.
For test 2 (Heteroskedasticity) the percent censoring was 43 %, the number of
observations = 100, and the number of estimation trials = 100. NLC-LAD is closer to the
truth 78.0 percent of the time for parameter ‘a’. NLC-LAD is closer to the truth 57.0 percent
of the time for parameter ‘b’. These results show that when heteroskedasticity is not
correctly specified in the censored ML that the parameter estimates are not consistent. The
inconsistency can be proven analytically as in Deaton (1997) and supported here with the
Monte Carlo results. Since it is asserted that the NLC-LAD is consistent although not proven
analytically, the results of the Monte Carlo study show that the NLC-LAD was closer to the
truth more often than the inconsistent censored ML. The relative efficiency was different for
the different parameters. The censored ML had a smaller standard deviation for parameter 'b'
while having a larger standard deviation for the parameter 'a'.
It is suggested here that the NLC-LAD is consistent. To give more support to this
claim the suggested estimator should be estimated with larger sample sizes. Deaton (1997)
suggests sample sizes of at least 1000 observations which are commonly met in household
surveys. Unfortunately even a sample size of 200 was not possible given the current state of
technological progress in the computer lab at the University of Colorado Economics
Department. The workspace of Gauss is limited by the amount of computer RAM. Even so,
the results of this preliminary Monte Carlo study give great encouragement to the suggestion
of extending Powell's (1984) censored least absolute deviations estimator to include
nonlinear functional forms. This would be an important improvement in the estimation of
censored regressions. First, the conditions of a large sample size are easily met with today's
17
household surveys and secondly, the nature of the NLC-LAD to be a distribution free
estimator make the non-linear censored least absolute deviations estimator a viable
alternative to the censored ML estimator when the normality of the errors is suspect or when
there is heteroskedasticity in the data.
18
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Buchinsky, Moshe, “Methodological Issues in Quantile Regressions.” Chapter 1 of The
Theory and Practice of Quantile Regression. Ph.D. dissertation, Harvard University
(1991).
Buchinsky, Moshe and Hahn, Jinyong. “An Alternative Estimator for The Censored
Quantile Regression Model.” Econometrica. 66 (1998) 653-671.
Butler, Richard J.; McDonald, James B.; Nelson, Ray D. “Robust and Partially Adaptive
Estimation of Regression Models.” The Review of Economics and Statistics. 72
(1990): 321-327.
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