Truncated and Censored
Regression Models
(excerpts taken directly from Franses and Paap book)
1. Truncated Regression : Case where ( yi , xi ) is observed only when yi  a (left
truncation) or when yi  b (right truncation) or when c  yi  d (double truncation).
Example: Let yi = the profit of the i-th firm as a percentage of assets and xi = the
four firm concentration ratio of industry the firm is in. Suppose only firms with
positive profit rates are observed and firms with negative profit rates are not
observed. In this case a = 0 and we have a problem where the dependent variable is
left truncated.
yi   0  1 xi   i where yi  0 for i  1,, N and  i  N (0,  2 )
Neglecting the truncation can lead to biased estimates of  0 and  1 .
E[ yi | yi  0, xi ]   0 1 xi  
 ((  0  1 xi ) /  )
1  ((  0  1 xi ) /  )
The last term on the right-hand-side of the above equation is called the Mills ratio.
2. Censored Regression model Case where all ( yi , xi ) are observed it is just that when
yi "passes" the truncation point, yi is recorded as the truncation point. As in the
truncation model you can have left-censoring, right-censoring, or upper and lowercensoring. For discussion purposes let's consider the case where yi  0 as in the
Franses and Paap book. They consider a contribution to charity example. Some
people give to the designated charity and some people do not.
Type-1 Tobit Model
Latent model: y i*  X i    i
yi  0 if y i*  X i    i  0
yi  X i    i if y i*  X i    i  0
E ( yi | yi  0, X i ) is given above in the truncated regression model discussion
(where X i is a scalar explanatory variable).
The unconditional expectation of y given X is
E ( yi | X ) = (1  ( X i  /  )) X i    ( X i  /  )
The total effect of a change in the k-th explanatory variable x k ,i on the expectation of
yi is
E[ yi | X i ]
= (1  ( X i  /  ))  k
x k ,i
Note: The Type-I Tobit model presented above can be written as a combination of
two familiar models. The first model is a Probit model, which determines whether
the yi variable is zero or positive and the second model is a Truncated Regression
model for the positive values of yi . The Type-I Tobit model assumes that the
parameters for the effect of the explanatory variables on the probability that an
observation is censored and the effect on the conditional mean of the non-censored
observations are the same. This distinguishes the Type-I model from the Type-II
model presented below.
Type-II Tobit Model
yi  0 if yi*  X   1,i  0
y i  X i    2,i if yi*  X i   2,i  0
where   ( 0 ,, K ) , where  1,i  N (0,1) because it concerns the Probit part, and
where  2,i  N (0,  22 ) . Both errors may be correlated and hence E[ 1,i  2,i ]   12 . This
model contains a Probit model for yi being zero or positive and a standard Regression
model for the positive values of yi . The Probit model may, for example describe the
influence of explanatory variables X i on the decision whether or not to donate to charity,
while the Regression model measures the effect of the explanatory variables on the size
of the amount for donating individuals.
The Type-II Tobit model is more flexible than the Type-I model. Owing to potentially
different  and  parameters, it can for example describe situations where older
individuals, but, given a positive donation, younger individuals perhaps donate more than
older individuals. The explanatory variable age then has a positive effect on the donation
decision but a negative effect on the amount donated given a positive donation. This
phenomenon cannot be described by the Type-1 Tobit model.
The expected contribution of an individual who decides to contribute is given by
E[ yi | yi*  0, X i ]  X i    12
 (  X i )
1   (  X i )
A special case is called the two-part model where the covariance between the Probit and
the Regression equation  12 is 0. In that case the expectation simplifies to X i  . The
advantage of a two-part model over a standard regression model for only those
observations with non-zero value concerns the possibility of computing the unconditional
expectation of yi as shown below.
The effect of a change in the k-th explanatory variable x k ,i on the expectation of noncensored yi for the Type-II Tobit model is given by
E[ y i | y i*  0, X i ]
  k   12 (2i  ( X i )i ) k
x k ,i
where i   ( X i ) . The above expectation is the expected donated amount given a
positive donation.
If one wants to analyze the effect of x k ,i on the expected donated amount without
conditioning on the decision to donate to charity, one must calculate the following
unconditional expectation.
E[ yi | X i ]  X i  (1  ( X i ))   12 ( X i ) .
The effect of the k-th explanatory variable x k ,i on the unconditional expectation is
E[ yi | X i ]
 (1  ( X i ))  k  X i ( X i ) k   12 ( X i ) ( X i ) k .
xk ,i
This partial derivative captures both the changes in probability that an observation is not
censored and the changes in the conditional mean of positive yi observations.