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Semiparametric median estimation of the Type 3 Tobit model
Economics Letters 016%1765/94/$07.00
44 (1994) 349-352 0 1994 Elsevier
349 Science
B.V. All rights
reserved
Semiparametric median estimation the Type 3 Tobit model Songnian
Chen”
Department of Economics, Received Accepted
of
19 July 1993 18 September
Princeton University, Princeton,
NJ 08.544, USA
1993
Abstract This paper considers estimation of the Type 3 Tobit model. The estimator is a two-step LAD estimator when a bivariate version of the median restriction is imposed on the error terms. A consistent estimator for the asymptotic covariance matrix is proposed. JEL
classification: C13, Cl4
1. Introduction This paper considers estimation of the Type 3 Tobit model under a weak restriction imposed on the distribution of the unobservable error terms. The model is obtained by adding more information to the censored sample selection model about the selection variable. It therefore makes the identification and estimation of parameters of the main equation easier. To date, various distribution-free estimators have been proposed for the censored sample selection model, all of which involve specific non-parametric functional estimation with some undesirable finite sample and asymptotic properties. The only estimators to date designed specifically for the Type 3 Tobit model are suggested by Honor6 et al. (1992) under symmetry and independence restrictions, and Chen (1993) under the independence restriction, all of which sidestep specific non-parametric estimation. We propose another estimator here for the Type 3 model also with only a weak restriction imposed on the error terms. Like those of Honor6 et al. and Chen, the estimator presented here is a two-step estimator where the selection equation is estimated first, and the estimator obtained from that equation is used to eliminate the effect of the selection on the main equation. The estimator considered here is a median regression estimator; the first-step estimator of the selection equation is Powell’s censored least absolute deviations estimator. In the second step, the first-step estimator is used to perform one-sided trimming in order to restore the constant conditional median restriction on the error term of the second equation. The estimator of the parameters of the main equation is then obtained through another median regression. Unlike the first estimator in Honore et al. based on the symmetry assumption, the estimator considered here does not impose a strong shape restriction on the part of the error distribution. The next section describes the model and discusses the estimators and their motivation. Large sample properties of the estimators are investigated in section 3. * I am grateful SSDI
to Professor
0165-1765(93)00353-P
James
Powell
for his constant
advice
and helpful
comments.
350
S. Chen
I Economics
Letters
44 (1994) 349-352
2. The model and estimator We consider Y* =
estimation
of the parameter
the latent
y0 of a Type 3 Tobit
max(y,0) ,
variable
model
defined
by
(1)
w*=(Z’Y,+ E&>o) where
vector
(2)
7
y is of the form
Y = x’P0 + El >
(3)
and 1 (A) represents the indicator function of event A. Here y* and w* are the dependent variables, x and z are vectors of exogenous variables, PO and ‘yOare unknown parameter vectors, and e1 and Ed are error terms. In the case considered here, only a weak restriction is imposed on the error distribution. The constant conditional median restriction is assumed and the proposed estimator is a generalization of Powell’s LAD estimator for the censored regression model. Up to now, the conditional median restriction has been exploited exclusively to estimate univariate semiparametric models in the literature. In the bivariate model here, l1 and E* are each assumed to have constant median 0 conditional on the regressors; we also assume some mild restriction on the joint distribution of (Ed, E*) in order to accommodate the bivariate setting. Honor6 et al. consider the estimation of the same model under the joint symmetry restriction with strong shape implication, while here it is replaced by something like sign independence between l1 and Q, which can also be utilized to eliminate the sample selection bias. As mentioned above, one-sided trimming is performed to restore the constant conditional median restriction. The idea behind the trimming is to restrict estimation of -yOto observations with asymptotic constant conditional median for the error term. Here censoring occurs when e1 > -x’&. If we only keep the observations for which x’p,, > 0 and y > x’&, it yields a new sample with observations with l1 > 0 and x’/?,, > 0. With some auxiliary conditions, the ‘new arrow term’ of the main equation, E: = Ed, conditional on the event y > 0 and 1 ~rl,o,x~Po,,,), would have a constant conditional median. As a result, a two-step LAD estimator is suggested. Specifically, the estimator of 7, is defined as
where
b is the first-step
LAD
b=argy&lini:
estimator: (yi-max{xip,O}I,
(5)
i=l
where
B and G denote
the parameter
3. Large sample properties
spaces
for PO and yO, respectively.
of the estimator
In this section we discuss the asymptotic properties of the estimator suggested above under the constant conditional median restriction. The way to establish the consistency and asymptotic normality will be similar to that of Honor6 et al. (1992). We make the following assumptions.
S. Chen I Economics Letters 44 (1994) 349-352
Assumption
3.1.
B is a compact
Assumption
3.2.
The random
and PO E int B and G is bounded.
subset
sample
3.51
consists
of n i.i.d.
draws of (y, W, x, z) from Eqs.
(l)-(3).
Assumption 3.3. Assume that conditional on (x, z), l1, e2 each has median 0, respectively, and P(E~ > 0, e1 > 0 1X, z) = P(E* < 0, or > 0 lx, z) almost surely. Also, assume that (e,, Q) is continuously distributed conditional on (x, z) and the conditional density of E, given x is bounded from above uniformly in x; assume that P,{f,+(O) > 0} > 0 and P,{f,,~,,,lso,~~po,o(0) > 0} > 0 almost surely. Assumption
3.4.
EXX’
and Ezz’ exist and El{x’&
> O}XX’ and El{x’P,
> 0, cl >O)ZZ’
are of full
rank. Assumption 3.5. x’p borhood of PO.
is continuously
distributed
in a neighborhood
of 0 for all p in a neigh-
Here the parameter space G of y0 is not assumed to be compact because of the convexity of the objective function in the second step - a feature of the convex objective function as illustrated by Newey and Powell (1987). The assumption P(E~ > 0 1l1 > 0, x, z) = P(E~ < 0 1lr > 0, x, z) is made to guarantee the constant conditional median of the error term after trimming, which is much weaker than the symmetry assumption imposed by Honor6 et al. In the one-dimensional case the zero median of an error term implies that it takes on positive and negative values with equal probability. One sufficient condition for P(E* > 0 ( l1 > 0, x, z) = P(E~ < 0 1or > 0, X, z) is that the probability (Ed, e2) in each of the four quadrants in l/4. Other assumptions are more or less standard in an LAD framework. Theorem
1. Under Assumptions
3.1-3.5,
(8, 9) is a consistent estimator of (&, y,,) and
(6) where
with
(7) and
r=
El~x~~o>~~Ll,x(0)~~’ EJ&(O)(P(E,
< 0) 1or = 0, x, z)z~‘l~,~~,>,,~
0 EI{x’P,
> O)J&(O)P(E,
> 0 I EZ = 07x7 Z) > ’ (8)
Proof. The proof is similar to that of Honor6 et al. and hence it is omitted here. Consistency of fl is proved in Powell (1984). The part for the consistency of 3 follows from a standard procedure by
352
S. Chen
I Economics
Letters 44 (1994) 349-352
a version of the uniform law of large numbers. Because of the nature of the non-differentiability of the objective function, the standard Taylor series approach does not apply here. Instead, the result of Pakes and Pollard (1989) is used to prove the asymptotic normality. In order for large sample inference on y0 to be carried out using the estimator 9, a consistent estimator of the asymptotic covariance matrix of 9 needs to be provided. It is straightforward to establish that the following v is a consistent estimator of V: (9) Also it follows directly from Pakes and Pollard (1989, pp. 1043-1044) that one consistent estimator of r is the numerical derivative estimator f. Let 0i = (p’, y ‘)‘, and (10) Then, the jth column of f is 1
”
where qlj is the unit vector with 1 in its jth place and e,, converges to zero at an appropriate rate with the sample size. f converges in probability to r if n-1’2ein* = o,(l), since {~$(8,x, z)} is a Euclidean class [see Pakes and Pollard (1989)]. Therefore
consistently estimates Zi. In order to evaluate and improve upon the current estimator, the next natural step is to compute the semiparametric efficiency bound for the model under the weak semiparametric restriction imposed here and find the corresponding efficient estimator.
References Chen, S., 1993, Semiparametric estimation of Type 3 Tobit model, unpublished manuscript. Honor&, B.E., and J.L. Powell, 1992, Pairwise differences of linear, censored and truncated regression models, unpublished manuscript. Honori, B.E., E. Kyriazidou and C. Udry, 1992, Estimation of Type 3 Tobit models using symmetric trimming and pairwise comparisons, draft. Newey, N.K. and J.L. Powell, 1987, Asymmetric least squares estimates and testing, Econometrica 55, 819-847. Pakes, A. and D. Pollard, 1989, Simulation and the asymptotics of optimization estimators, Econometrica 57, 1027-1057. Powell, J.L., 1984, Least absolute deviations estimation for the censored regression model, Journal of Econometrics 25, 303-325. Powell, J.L., 1986, Symmetrically trimmed least squares estimation for Tobit models, Econometrica 54, 1435-1460. Powell, J.L., 1989, Semiparametric estimation of censored selection models, unpublished manuscript. eds., Handbook of Powell, J.L., 1992, Estimation of semi-parametric models, in: R. Engle and D. McFadden, Econometrics (North Holland, Amsterdam).
44 (1994) 349-352 0 1994 Elsevier
349 Science
B.V. All rights
reserved
Semiparametric median estimation the Type 3 Tobit model Songnian
Chen”
Department of Economics, Received Accepted
of
19 July 1993 18 September
Princeton University, Princeton,
NJ 08.544, USA
1993
Abstract This paper considers estimation of the Type 3 Tobit model. The estimator is a two-step LAD estimator when a bivariate version of the median restriction is imposed on the error terms. A consistent estimator for the asymptotic covariance matrix is proposed. JEL
classification: C13, Cl4
1. Introduction This paper considers estimation of the Type 3 Tobit model under a weak restriction imposed on the distribution of the unobservable error terms. The model is obtained by adding more information to the censored sample selection model about the selection variable. It therefore makes the identification and estimation of parameters of the main equation easier. To date, various distribution-free estimators have been proposed for the censored sample selection model, all of which involve specific non-parametric functional estimation with some undesirable finite sample and asymptotic properties. The only estimators to date designed specifically for the Type 3 Tobit model are suggested by Honor6 et al. (1992) under symmetry and independence restrictions, and Chen (1993) under the independence restriction, all of which sidestep specific non-parametric estimation. We propose another estimator here for the Type 3 model also with only a weak restriction imposed on the error terms. Like those of Honor6 et al. and Chen, the estimator presented here is a two-step estimator where the selection equation is estimated first, and the estimator obtained from that equation is used to eliminate the effect of the selection on the main equation. The estimator considered here is a median regression estimator; the first-step estimator of the selection equation is Powell’s censored least absolute deviations estimator. In the second step, the first-step estimator is used to perform one-sided trimming in order to restore the constant conditional median restriction on the error term of the second equation. The estimator of the parameters of the main equation is then obtained through another median regression. Unlike the first estimator in Honore et al. based on the symmetry assumption, the estimator considered here does not impose a strong shape restriction on the part of the error distribution. The next section describes the model and discusses the estimators and their motivation. Large sample properties of the estimators are investigated in section 3. * I am grateful SSDI
to Professor
0165-1765(93)00353-P
James
Powell
for his constant
advice
and helpful
comments.
350
S. Chen
I Economics
Letters
44 (1994) 349-352
2. The model and estimator We consider Y* =
estimation
of the parameter
the latent
y0 of a Type 3 Tobit
max(y,0) ,
variable
model
defined
by
(1)
w*=(Z’Y,+ E&>o) where
vector
(2)
7
y is of the form
Y = x’P0 + El >
(3)
and 1 (A) represents the indicator function of event A. Here y* and w* are the dependent variables, x and z are vectors of exogenous variables, PO and ‘yOare unknown parameter vectors, and e1 and Ed are error terms. In the case considered here, only a weak restriction is imposed on the error distribution. The constant conditional median restriction is assumed and the proposed estimator is a generalization of Powell’s LAD estimator for the censored regression model. Up to now, the conditional median restriction has been exploited exclusively to estimate univariate semiparametric models in the literature. In the bivariate model here, l1 and E* are each assumed to have constant median 0 conditional on the regressors; we also assume some mild restriction on the joint distribution of (Ed, E*) in order to accommodate the bivariate setting. Honor6 et al. consider the estimation of the same model under the joint symmetry restriction with strong shape implication, while here it is replaced by something like sign independence between l1 and Q, which can also be utilized to eliminate the sample selection bias. As mentioned above, one-sided trimming is performed to restore the constant conditional median restriction. The idea behind the trimming is to restrict estimation of -yOto observations with asymptotic constant conditional median for the error term. Here censoring occurs when e1 > -x’&. If we only keep the observations for which x’p,, > 0 and y > x’&, it yields a new sample with observations with l1 > 0 and x’/?,, > 0. With some auxiliary conditions, the ‘new arrow term’ of the main equation, E: = Ed, conditional on the event y > 0 and 1 ~rl,o,x~Po,,,), would have a constant conditional median. As a result, a two-step LAD estimator is suggested. Specifically, the estimator of 7, is defined as
where
b is the first-step
LAD
b=argy&lini:
estimator: (yi-max{xip,O}I,
(5)
i=l
where
B and G denote
the parameter
3. Large sample properties
spaces
for PO and yO, respectively.
of the estimator
In this section we discuss the asymptotic properties of the estimator suggested above under the constant conditional median restriction. The way to establish the consistency and asymptotic normality will be similar to that of Honor6 et al. (1992). We make the following assumptions.
S. Chen I Economics Letters 44 (1994) 349-352
Assumption
3.1.
B is a compact
Assumption
3.2.
The random
and PO E int B and G is bounded.
subset
sample
3.51
consists
of n i.i.d.
draws of (y, W, x, z) from Eqs.
(l)-(3).
Assumption 3.3. Assume that conditional on (x, z), l1, e2 each has median 0, respectively, and P(E~ > 0, e1 > 0 1X, z) = P(E* < 0, or > 0 lx, z) almost surely. Also, assume that (e,, Q) is continuously distributed conditional on (x, z) and the conditional density of E, given x is bounded from above uniformly in x; assume that P,{f,+(O) > 0} > 0 and P,{f,,~,,,lso,~~po,o(0) > 0} > 0 almost surely. Assumption
3.4.
EXX’
and Ezz’ exist and El{x’&
> O}XX’ and El{x’P,
> 0, cl >O)ZZ’
are of full
rank. Assumption 3.5. x’p borhood of PO.
is continuously
distributed
in a neighborhood
of 0 for all p in a neigh-
Here the parameter space G of y0 is not assumed to be compact because of the convexity of the objective function in the second step - a feature of the convex objective function as illustrated by Newey and Powell (1987). The assumption P(E~ > 0 1l1 > 0, x, z) = P(E~ < 0 1lr > 0, x, z) is made to guarantee the constant conditional median of the error term after trimming, which is much weaker than the symmetry assumption imposed by Honor6 et al. In the one-dimensional case the zero median of an error term implies that it takes on positive and negative values with equal probability. One sufficient condition for P(E* > 0 ( l1 > 0, x, z) = P(E~ < 0 1or > 0, X, z) is that the probability (Ed, e2) in each of the four quadrants in l/4. Other assumptions are more or less standard in an LAD framework. Theorem
1. Under Assumptions
3.1-3.5,
(8, 9) is a consistent estimator of (&, y,,) and
(6) where
with
(7) and
r=
El~x~~o>~~Ll,x(0)~~’ EJ&(O)(P(E,
< 0) 1or = 0, x, z)z~‘l~,~~,>,,~
0 EI{x’P,
> O)J&(O)P(E,
> 0 I EZ = 07x7 Z) > ’ (8)
Proof. The proof is similar to that of Honor6 et al. and hence it is omitted here. Consistency of fl is proved in Powell (1984). The part for the consistency of 3 follows from a standard procedure by
352
S. Chen
I Economics
Letters 44 (1994) 349-352
a version of the uniform law of large numbers. Because of the nature of the non-differentiability of the objective function, the standard Taylor series approach does not apply here. Instead, the result of Pakes and Pollard (1989) is used to prove the asymptotic normality. In order for large sample inference on y0 to be carried out using the estimator 9, a consistent estimator of the asymptotic covariance matrix of 9 needs to be provided. It is straightforward to establish that the following v is a consistent estimator of V: (9) Also it follows directly from Pakes and Pollard (1989, pp. 1043-1044) that one consistent estimator of r is the numerical derivative estimator f. Let 0i = (p’, y ‘)‘, and (10) Then, the jth column of f is 1
”
where qlj is the unit vector with 1 in its jth place and e,, converges to zero at an appropriate rate with the sample size. f converges in probability to r if n-1’2ein* = o,(l), since {~$(8,x, z)} is a Euclidean class [see Pakes and Pollard (1989)]. Therefore
consistently estimates Zi. In order to evaluate and improve upon the current estimator, the next natural step is to compute the semiparametric efficiency bound for the model under the weak semiparametric restriction imposed here and find the corresponding efficient estimator.
References Chen, S., 1993, Semiparametric estimation of Type 3 Tobit model, unpublished manuscript. Honor&, B.E., and J.L. Powell, 1992, Pairwise differences of linear, censored and truncated regression models, unpublished manuscript. Honori, B.E., E. Kyriazidou and C. Udry, 1992, Estimation of Type 3 Tobit models using symmetric trimming and pairwise comparisons, draft. Newey, N.K. and J.L. Powell, 1987, Asymmetric least squares estimates and testing, Econometrica 55, 819-847. Pakes, A. and D. Pollard, 1989, Simulation and the asymptotics of optimization estimators, Econometrica 57, 1027-1057. Powell, J.L., 1984, Least absolute deviations estimation for the censored regression model, Journal of Econometrics 25, 303-325. Powell, J.L., 1986, Symmetrically trimmed least squares estimation for Tobit models, Econometrica 54, 1435-1460. Powell, J.L., 1989, Semiparametric estimation of censored selection models, unpublished manuscript. eds., Handbook of Powell, J.L., 1992, Estimation of semi-parametric models, in: R. Engle and D. McFadden, Econometrics (North Holland, Amsterdam).