ESTIMATION OF SOME NON-NORMAL LIMITED 
DEPENDENT VARIABLE MODELS 
by 
Lung-Fei Lee 
Discussion Paper No. 81 - 148, June 1981 
Center for Economic Research 
Department of Economics 
University of Minnesota 
Minneapolis, Minnesota 55455 
ESTIMATION OF SOME NON-NORMAL LIMITED DEPENDENT 
VARIABLE MODELS 
Lung-Fei Lee{*) 
Department of Economics, University of Minnesota, Minneapolis, 
and 
Center for Econometrics and Decision Sciences, 
University of Florida, Gainesville 
Abstract 
Estimation of the Tobit models where the distribution of the 
disturbances belongs to the Pearson family of distribution is considered. 
Some simple recursive relations between the moments of the truncated 
distribution are derived. Those recursive relations provide some 
structural equations which are linear in variables but nonlinear in 
coefficients. The estimation method proposed is a nonlinear two stage 
least squares method. 
Address: (valid after July 1, 1981) 
Department of Economics 
University of Minnesota 
1035 Business Administration 
271 19th Avenue South 
Minneapolis, MN 55455 
(valid until June 30, 1981) 
Department of Economics 
University of Florida 
Gainesville, Florida 32611 
ESTIMATION OF SOME NON-NORMAL LIMITED DEPENDENT 
VARIABLE MODELS 
by 
Lung-Fei Lee 
1. Introduction 
May 1981 
It is known that classical least squares method does not provide 
consistent estimates for the Tobit regression models when the dependent 
variables are truncated or censored. To overcome this problem, specific 
distribution for the disturbances is assumed. The commonly specified 
distribution is the normal distribution. Based on the normal distribution, 
the maximum likelihood estimator has been shown to be consistent, 
asymptotically normal and asymptotically efficient in Amemiya [1973]. 
Consistent instrumental variable method has also been proposed in Amemiya 
[1973]. Unfortunately, these estimates are not robust. The misspecification 
of normality may create severe asymptotic bias in the estimates. Other 
misspecifications, such as the assumption of homoscedaatic errors when 
there are heteroscedasticity, will also create such problem. The consequences 
of such misspecifications have been investigated in Goldberger [1980], 
Hurd [1979] and Nelson [1981]. Since misspecification in the Tobit model 
can cause problems, it is desirable to have testing procedures to test the 
assumptions or to consider models which will relax the strong assumptions. 
For the censored-normal Tobit model, a specification test is suggested in 
Nelson [1981]. To test the normality assumption, some Lagrangean multiplier 
tests are provided in Lee [1981] for both the censored and truncated Tobit 
models. 
-2-
Since normality is a rather strong assumption, it is desirable to 
relax this assumption and consider models with more flexible distributions. 
In this article, we consider the estimation of the Tobit models where the 
distribution of the disturbances belong to the Pearson family of 
distributions. The Pearson family of distributions is attractive since it 
contains distributions of various widely different shapes and contains 
the normal, student t, and gamma distributions as special cases. 
This article is organized as follows. In Section 2, the Tobit 
model with the Pearson family of distributions is specified. In Section 3, 
we analyze the moments of the distribution of the Tobit models when there 
is a single truncation. In Section 4, recursive relations between the 
moments of the doubly truncated distributions are derived. In Section 5, 
consistent estimation methods are suggested. Finally, we draw our 
conclusions. 
-3-
2. The Tobit Models and the Pearson Family of Distributions 
A typical censored Tobit model is specified as 
i=1,2, ... ,N 
where x. is a 1 x k row vector of exogenous variables which contains a 
1. 
(2.1) 
constant term; E(u.)=O and the disturbances u. are independent and identically 
1. 1. 
distributed; the dependent variable Yli is an unobservable variable but is 
related to the observed dependent variable Yi as follows:!/ 
= 0 otherwise 
This model was originated in the pioneer paper in Tobin [1959] with the 
additional normality assumption imposed on the distribution of the disturbance 
u. To relax the normality assumption, we assume that the distribution of 
u belongs to the Pearson family of distributions. This system of distributions 
was originated by K. Pearson, see, e.g. Elderton and Johnson [1969] and 
Johnson and Kotz [1970]. For each number of the system, the probability 
density function g(u) satisfies the following differential equation, 
dQ,ng(u) = 
du 
a+u 
2 b
o
+bl u+b 2u 
2/ As the disturbance u has zero mean, it implies that a=-bl .- After some 
(2.2) 
reparameterizations, the density function g(u) with zero mean is characterized 
by the differential equation, 
(2.3) 
-4-
Let f(ylix) be the conditional density function·of Yl conditional on the 
exogenous variable vector x. It follows that 
= dQ.ng(u) 
du 
u = y -xS 1 
(2.4) 
The formal solution of the density function g(u) from the differential 
equation (Z.3) depends on the characteristics of the solutions of the 
. Z 
quadratic equation c
o
-cl u+c2u =0. The general density function, however, 
can be expressed informally as 
g(u) = exp 
Define a dichotomous indicator I., 
1. 
I. = J 1 
1. l 0 
if Yli > 0, 
otherwise. 
dt (Z.5) 
g(u)du be the probability 
-00 
that I. = O. The log likelihood function for this model is 
1. 
where 
L 
c 
q(t) = 
c , c l ' c Z) + I. q(y.-x.S) o 1. 1. 1. 
_ Ii'n JOOexp q(t)dt) 
_00 
(Z.6) 
(2.7) 
-5-
The normal density function corresponds to the case that c1=O and cZ=O. 
The (central) t distribution belongs to the family with c1=0. 
The Tobit model described above is a censored regression model in 
which the exogenous variables vector x. is always observable and the number 
1. 
of observations corresponding to 1.=0 is also known. There are situations 
1. 
in which the samples are drawn only from the populations with Y1>0. These 
samples are known as truncated samples and the corresponding Tobit model 
is a regression model with truncated dependent variable. For the truncated 
Tobit model with the Pearson family of distributions, the general density 
function is 
g(ulx) 
where, for given x, u is in the interval [-xS, ooJ. For a given samples 
{y.} of size N, the log likelihood function for the truncated Tobit model is 
1. 
LT = Li~l{q(Yi-xiS) - ~n f exp q(t)dt} 
-x.S 
1. 
(2.9) 
(Z.8) 
The Pearson family of distributions of u uses four parameters and is characterized 
by the first four moments of u (see Elderton and Johnson [1969J, pp. 38-39). 
It is possible to expand the system to the family of distributions 
characterized by the differential equation, 
d£ng(u) = 
du 
a+u 
with higher order polynomials in the denominator. However, as argued in 
the statistical literatures, such complications are unnecessary for most 
practical purposes. 
3. Incomplete Moments and Singly Truncated Distribution 
While the maximum likelihood method for the Tobit model with normal 
distribution is attractive, it seems computationally extremely complicated 
for the estimation of the Tobit model with the Pearson family of distributions. 
As pointed out in Johnson and Kotz ([1970], p. 12), fitting the Type IV 
distribution by maximum likelihood is difficult and it is practically never 
3/ 
attempted.- As a practice alternative, the method of moments are used for 
the general distribution fitting purpose, see Elderton and Johnson ([1969], 
Chapters 4 and 5). The method of moments have been generalized to the 
estimation of the truncated Pearson distributions in Cohen [1951, 1953]. 
The models considered by Cohen are not regression-type models. For the 
estimation of the Tobit models, the method of moments, however, can be 
modified to the method of instrumental variables estimation. Instrumental 
variables estimation of the Tobit models with normal distribution was first 
proposed in Amemiya [1973]. 
The estimation method that we will consider utilizes only the sample 
information on the observed values of Yli and are applicable to both the 
censored and truncated models. For both models, it follows from (2.4) that 
the density function of Yl conditional on x satisfies the following relation 
where a(x), b.(x), i=0,1,2 are functions of x; 
1 
(3.1) 
-7-
a(x) = c l + xS (3.2) 
b (x) 2 (3.3) = c + clxS + c2 (xS) 0 0 
bl (x) = cl + 2c 2xS (3.4) 
b 2(x) c 2 (3.5) 
For the sake of notational simplicity, the vector x in the density function 
(3.1) and the functions (3.2)-(3.5) will be subpressed. The differential 
equation "in (3.1) can be used to derive very simple recursive relations for 
the moments of the truncated Pearson distribution. Such relations can be found 
in Cohn [1952].i/ Equation (3.1) implies that 
(3.6) 
Multiplying both sides by y~, n£{O,1,2, .•. ,} and integrating over the range 
from ° up to the upper limit of Yl' denoted as "00", we have 
By the integration by parts, it follows that 
00 
f y~(a-Yl)f(Yl)dYl 
° 
+ (n+2)b2y~+lJdYl • a f~Y~f(Yl)dYl 
° 
(3.7) 
(3.8) 
Suppose that lim y~f(Yl) = ° for all n, n=O,1,2, ••• i.e., at the upper end 
Yf~ 
point, y~f(Yl) vanish, the equation (3.8) can then be further simplified to 
and 
Let 
-8-
-bof(O) - foo[-bl+2b2Yllf(Yl)dYl = 
o 
~n = IooY~f(Yl)dYl / Ioof(Yl)dYl 
o 0 
n=1,2, •.. 
be the nth moment of the truncated density function f(yl ) / Joof(Yl)dYl . 
o 
Equation (3.9) implies 
(1-2b2)~1 = a-b + b f(o) 1 0 F 
where F Joof(Yl)dYl , and equation (3.10) implies 
o 
(3.9) 
(3.10) 
(3.11) 
(3.12) 
(3.13) 
Substituting the expressions (3.2)-(3.5) into the equations (3.12) and (3.l3~, 
we have explicitly the following set of equations: 
1 [co+cl xS + 
2 f(olx) 
Y = xS + l-2c c2 (xS) 1 + nl 2 F(x) 
(3.14) 
where F(x) = (f(Yllx)dYl; nl = Y-~ and 1 
0 
n+l n-l n n n-l Y = ny c* - ny c* + Y x(l-nc* )6 + ny xc* 6 
on In 2n In 
(3.15) 
n=l ,2, ... 
-9-
where 
c* = c
o
/(1-(n+2)c2) (3.16) on 
c* = c/(1-(n+2)c2) (3.17) In 
c* 2n c2/(1-(n+2)c 2) (3.18) 
and 
X(yn_~ )(l-nc* )6 
n 2n 
n-l (y -~ l)nxcl* 6 n- n 
(3.19) 
n-l 5/ 
- nCy -~ 1)Cx~x)c*2 CS~S).-
n- n 
Obviously, conditional on Yl being observed, all the disturbances have zero 
mean, i.e., ECnniyl>O) = 0 for all n, ne{O,1,2, ... }. 
-10-
4. Incomplete Homents and Doubly Truncated Distribution 
The derivations of the moments and the recursive formulae in the 
n previous section are based on the assumption that ylf(yl ) evaluated at the 
upper end point of the support or ~ are zero for all n, ns{O,l, •.. }. This 
assumption is quite restrictive and rules out many distributions in the 
Pearson family. This problem can be avoided if we artifically truncate 
the distribution of Yl from above with a nonstochastic threshold k, k>O, 
which eliminates the observations y. with value greater than k. Thus the 
1. 
model considered becomes a two limits Tobit model with the Pearson family 
of distributions which generalizes the model in Rossett and Nelson [1975].i/ 
From equation (3.6), it implies 
and hence 
for all n, ns{0,1,2, ••• }. 
Let 
Vn = JkY~f(Yl)dYl / 
o 
dYl = Jky~(a-Yl)f(Yl)dYl 
o 
(4.1) 
(4.3) 
be the nth moment of 
and F = Jkf(Yl)dYl . 
o 
the doubly truncated density function f(yl)/Jkf(Yl)d Yl , 
o 
(4.2) 
-11-
The equations in (4.2) become 
(b -b k + b kZ) f(k) - b f(O) - [-bl + Zb Z v ] = a v 01 Z F of 1 - 1 (4.4) 
and 
= av - v 
n n+l n = 1,2, ... 
Equations in (4.5) imply, in turn, the following recursive relation, 
(4.6) 
-k(n-l)b v 
o n-2' n = 2,3, .•. 
Substituting the functions in (3.2)-(3.5) into (4.4) and (4.6), we have 
explicitly the following equations, 
and 
y = xS + l-~C [co+clxS + C2 (XS)2]f(0Ix)/F(X) 2 (4;4)' 
n+l y 
1 2 2 f (k I x) 
1-2c [co-clk + cZk + (cl -2c Zk)xS + c 2 (xS) ] F(x) + nl Z 
n n n-l y (-nc* + k(l+c* » + y xS(l-nc* ) + y (nc* +(n-l)kc* ) In 2n 2n on In 
n-l n-l 
+ y xS(ncIn - k(1-(n-2)c~n» + ny (x~x)(S~S)c~n 
n-2 n-2 n-2 
- y k(n-l)c* - k(n-l)y xSc* - k(n-l)y (x~x)(S0S)c* +n 
on 1 2n n 
n = 2,3, ... (4.5) , 
where E(nniO < Yl < k) = 0 for all n; 
and 
C* 
on 
c* In 
c* 2n 
(4.7) 
(4.8) 
(4.9) 
-12-
5. Instrumental Variables Esimations 
For the Tobit models with normal distribution, Aroemiya [1973] 
has derived the relations between the first and the second moments of a 
singly truncated normal distribution and proposed a simple consistent 
instrumental variables estimation method. This kind of method can be 
extended to the estimation of our models. 
To estimate the models, the relations in (3.15) or the ones in 
(4.5)' can be used when the dependent variables is singly truncated. For 
the doubly truncated cases, only the relations in (4.5)' can be used. The 
basic vector of parameters in our model is 9 = (S', co' c l ' c2)'. The 
equations (3.15) and (4.5)' are linear in variables but nonlinear in 
parameters. A consistent estimation method that can be used is the nonlinear 
two stage least squ~res procedure (see Aroemiya [1974]). With loss of 
generality, let us briefly describe the estimation procedure for the doubly 
truncated models. For each n, nE{2,3, ••• ,}, it is 'convenient to rewrite the 
equation (4.5)' implicitly as 
(5.1) 
where the functions Y. (9), j=l, ••• ,8, are nonlinear vector-value functions In 
of the basic parameter vector 9 and are defined from the equation (4.5)', 
(4.7)-(4.9) . Let z 
n 
n n n-l n-l n-l n-2 n-2 n-2 
= ( y , y x, y , y x, y ( x~x), y ,Y x, y (x~x) ) 
be the vector of explanatory variables on the right hand side of the equation 
(5.1). Hence, equation (5.1) can be rewritten as 
-13-
n+l = z y (9) + n y n n n (5.2) 
Without loss of 
generality, assume that the sample size of the observed samples on y with 
o < y < k is N. Let Y 1 and Z be the data matrices of the dependent 
n+ n 
variable yn+l and the explanatory variable vector z. In matrix notations, 
n 
equation (5.2) is 
y +1 = Z Y (9) + E (5.3) 
n n n n 
where E is the corresponding vector of the disturbances n " To estimate 
n n1 
this equation, valid instrumental variables need to be constructed for z . 
n 
Since, in general, the vector of exogenous variables, x, contains a constant 
term, some of the columns in Z are identical. Of course, one can eliminate 
n 
the duplicated columns and combine the corresponding coefficients. But this 
is rather tedious to do in practice. Let x = (1, w). The distinct explanatory 
n n n-l 
variables in z are the variables, y , y w, y 
n 
n-l n-l n-2 n-2 y w, y (wiw), Y , Y w 
n-2 
and y (~.naw) . To construct a set of valid instruments for these variables, 
we can regress the dependent variable y on the exogenou8variables 1, wand 
some low order polynomials of wand then construct the least square predictor 
y for y. The vector of instrumental variables can be constructed as 
~n ~n ~n-l ~n-l ~n-l ~n-2 ~n-2 ~n-2 
q = (y , y w, y ,y w, y (wiw), Y , Y w, Y (wQw» (5.4) 
n 
Let Q denote such constructed instrumental variables matrix. For each 
n 
n, n=2,3, •.. , a nonlinear two stage least squares estimator e of e 
n 
can be derived by 
min (Yn+1 8 
-14-
Z y (8))'Q (0'0 )-lQ,(y +1 - Z y (8)). 
n n n -n'n n n n n 
For this estimation, iterative methods such as the Newton's method or its 
variants need be used. Let e(m) be the estimator derived in the mth step 
n 
iteration. th The m+1 step iteration will give a modified estimate as 
e(m+1) 
n 
ay' (e (m)) 
= a(m) + [ n n Z'Q (Q'Q )-lQ,Z 
n a8 n n n n n n 
ay' (B(m)) 
[ n n Z'Q (QIQ )-lQ, (y -Z y (B(m)))] 
a8 n n n n n n+1 n n n 
The final estimate is derived when the iteration converges. As in Amemiya 
[1973, 1974], the estimator can be easily shown to be consistent and 
asymptotically normal under very general regular conditions. The asymptotic 
covariance matrix can be computed as 
ay 1(8) 
[ n Z'Q (QIQ )-10,Z ao n n n n 'n n 
ay (8) -1 ay'(8) 
n ] [n Z' 0 (0 I Q ) -10 I rl (8) a8' a8 n -n -n n -n n . 
Q (QIQ )-lQ,z 
ay (8) ay'(8) ay (8) -1 a~' ] [a~ Z'Q (QIQ )-lQ'Z n ] 
n n n n n n n n n n n a8' 
where rl (8) is the covariance matrix of E. The analytical expression for 
n n 
rl (8) involves the evaluation of higher order truncated moments of y, the 
n 
density function f(yllx) at Yl = 0 and k, and the probability ~kf(Ylx)dY as 
in the equations (4.4) and (4.5). A relatively computational simple 
method to estimate the asumptotic variance matrix is to use the matrix V 
n,n 
(5.4) 
-15-
V 
ay' (8 ) 
[ n n 2'Q (Q'Q )-lQ'2 
a8 n n n n n n 
ay (8 ) -1 ay , (8 ) 
n n][ n n 2'0 (O'Q )-1 
a8' a8 n 'n 'n n n,n 
ay (8 ) 
n n 
a8' ] 
ay' (8 ) 
[ n n 2'Q (Q'Q )-lQ'2 
a8 n n n n n n (5.5) 
where nni 
n+l 
= y. -z.y (8 ), i=l, ... ,N, are the estimated residuals. 1. n1. n n Under 
the regular conditions as in Amemiya [1973J, this is a consistent estimate of 
h . . 7/ t e covar1.ance matr1.x.-
The above nonlinear two stage method gives a separate estimate of 
8 for each n, nE{2,3, .•• }, for the doubly truncated Tobit model. In practice, 
it is enough to use several of the low order moments for the estimation purpose 
without much information lost on the higher moments. Suppose we have used 
the equations in (5.1) for n=2,3, ••• ,m, and derived the corresponding estimates 
8 for each n, it is desirable to pool the estimators so as to derive a more 
n 
efficient one. A relatively simple pooling procedure is the following mixed 
estimation procedure. 
8 
m 
= 
I 
I 
The set of estimators 8 can be rearranged as 
n 
8 + (5.6) 
where ~ = 8 - 8. The asymptotic covariance matrix of ~n can be estimated 
n n 
bv V in (5.5). Since the estimators 8 , n=2,3, ... ,m, are derived from the 
. ~ n 
same samples, the disturbances ~n' ~~, n#~, are correlated. The asymptotic 
covariance of ~n and ~~ can be estimated by Vn~ where 
-16-
ay' (e ) 
Vnn = [ n n Z'O (Q'Q )-lQ'Z 
~ ae n'n n n n n 
ay' (8 ) [ n n 
ae 
ay (e ) -1 
n n J 
ae' 
The equation (5.6) can then be estimated by the generalized least 
(5.7) 
squares 
method. Let V = [V
ni ] be the (m~l)x(m-l) block matrix consisting of the 
matrices Vni,and Vni be the submatrix in the (n,i)th position of the inverse 
-1 -
matrix V . The pooled estimator e from the equation (5.6) is 
(5.8) 
m m ji-l 
and its asymptotic covariance matrix can be estimated by (Lj=2Li=2V ) . 
The above estimation method is, of course, not the most efficient 
method. A more efficient estimation method is to estimate e from a set of 
equations in (5.3) with n=2, ••• ,m, by the generalized nonlinear three stage 
least squares. This method, however, is rather complicated and,will not be 
8/ 
recommended for our general model.-
The above estimation method can be greatly simplified if the 
distribution is assumed to be normal. The normal distribution is a member of 
the Pearson family of distributions and corresponds to the case that cl =c 2=0. 
For the normal distribution, the recursive relation (3.15) for the singly 
truncated case will become 
n+l y n n-l y xS + ny Co + nn+l n=1,2, .•. (5.9) 
-17-
and the recursive relation (4.5)' for the doubly truncated case will become 
n+l k n y - y n n-l n-l n-Z (y -ky )x8 + (ny -k(n-l)y )c + n , 
o n 
(5.10) 
n=Z,3, •.• 
Since the equations in (5.10) are linear in coefficients, the nonlinear two 
stage least squares method will become the two stage least square~ method and 
the computations will be simplified. This two stage method generalizes the 
instrumental variable method in Amemiya [1973J. Since the pooled estimate 
uses more information, it will be asymptotically more efficient than his one. 
Since the instrumental variable estimate can be easily derived for the normal 
distribution, it may also be useful as an initial estimate to start the 
iteration for the estimation of the general model. Similar to the Amemiya 
procedure, our procedure does not impose inequality constraint on the 
estimation of the variance c in (5.10). Hence it is possible to have negative 
o 
estimate of the variance c when the sample size is not sufficiently large.1! 
o 
Similar problem may occur in the estimation of the general model with the 
Pearson family of distribution. The variance is c
o
!(1-3cZ) for the general 
model. Since in many empirical studies, the main interest is to estimate the 
vector of coefficients 8, the unconstrained estimation method will suffice 
for this purpose. 
-18-
6. Conclusions 
In this article, we have considered the estimation of the Tobit 
models when the distribution of the disturbances belongs to the Pearson 
family of distributions. Model with single truncation and model with double 
truncations are considered. The estimation method we have proposed in this 
article is a nonlinear two stage least squares method. We have derived from 
the differential equation, which characterizes the Pearson family of 
distributions, some simple recursive relations between the moments of the 
truncated distributions. The recursive relations provide some structural 
equations which are linear in variables but nonlinear in coefficients. The 
nonlinear two stage least squares methods are applied to estimate those 
equations. 
-19-
Footnotes 
(*) The author is an associate professor of economics, University of 
Minnesota, and a visiting assQciate professor for the Center for 
Econometrics and Decision Sciences, University of Florida. Financial 
support from the National Science Foundation under grant SES 8006481 
to the University of Minnesota, is gratefully acknowledged. A 
preliminary version of this article was presented at the CEME Conference 
on Qualitative Decision Theory and Discrete Data Analysis at Harvard 
University on April 2-4, 1981. I appreciate having valuable comments 
from Dale Poirier and the conference participants. Any errors are of 
my own. 
1. The specified truncation threshold is specified as zero. More general 
case is that Y1i is observed if Y1i > c where c is a known constant; 
otherwise Y1i = c. This case can be transformed into our specification 
by rewriting the equation (2.1) into Y1i - c = XiS - c + u i and 
regarding Y1i - c as the dependent variable. If the truncation threshold 
Ci is varying for different i, the regression equation Y1i - c i = x.S - c. + u. 1 1 1 
will have a known coefficient in the exogenous variable c. on the right 
1 
hand side. For this case, our estimation methods proposed below need 
to be modified slightly to take into account this constraint. 
2. See, e.g., Elderton and Johnson [1969], p. 38. 
3. Type IV distribution is one of the main type distribution and corresponds 
2 to the case that the solutions of c
o
-c1u + c 2u = 0 are complex. 
4. Some of such related references were pointed out to me by Dale Poirier. 
- 20-
5. The operator ~ denotes the Kronecker product. It should note that the 
relation a~'(S~S) = (IK~S) + (S~IK)' where IK is is a k x k identity 
matrix, is useful for our .estimation procedure proposed below. 
6. The distribution of the disturbances in the model of Rosett and Nelson 
7. 
is assumed to be normal. They call it the two limit probit model. 
More regular statement is that NV is a consistent estimate 
n,n 
covariance matrix for the limiting distribution of IN (~ - e). 
n 
of the 
8. The computation burden in this procedure is on the evaluation of the density 
function and the probability function. For some specific members of the 
Pearson family such as normal distribution, the computations of the 
density and probability functions are rather simple and the generalized 
nonlinear three stage least squares are attractive. 
9. Another reason may be due to the misspecification of the model. 
-21-
REFERENCES 
Amemiya, T. (1973), "Regression Analysis When the Dependent Variable is 
Truncated Normal," Econometrica 41, pp. 997-1016. 
Amemiya, T. (1974), "The Nonlinear Two-Stage Least-Squares Estimator," 
Journal of Econometrics 2, pp. 105-110. 
Cohen, A. C., Jr. (1951), "Estimation of Parameters in Truncated Pearson 
Frequency Distributions," Annals of Mathematical Statistics 22, pp. 256-65.-
Cohen, A. C., Jr. (1953), "Estimating Parameters in Truncated Pearson 
Frequency Distributions Without Resort to Higher Moments," Biometrika 
40, pp. 50-57. 
Elderton, W. P. and N. J. Johnson (1969), Systems of Frequency Curves, 
Cambridge University Press. 
Goldberger, A. S. (1980), "Abnormal Selection Bias," Discussion paper 
no. 8006, Social Systems Research Institute, University of 
Wisconsin, Madison. 
Hurd, M. (1979), "Estimation in Truncated Samples when There is 
Heteroscedasticity," Journal of Econometrics 11, pp. 247-258. 
Johnson, N. L. and S. Kotz (1970), Continuous Univariate Distributions - 1, 
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Lee, 1. F. (1981), "A Specification Test for Normality Assumption for 
the Truncated and Censored Tobit Models," Manuscript, U. of Florida. 
Nelson, F. D. (1979), "The Effect of and A Test of Misspecification in 
the Censored-Normal Model," Social Science Working Paper 291, 
California Institute of Technology, forthcoming in Eco~~~~trica. 
Rosett, R. N. and F. D. Nelson (1975), "Estimation of the '1"~.o Limit 
Probit Regression Model," Econometrica 43, pp. 141-146. 
Tobin, J. (1958), "Estimation of Relationships for Limited Dependent 
Variables," Econometrica 26, pp. 24-26.