In this note I show that a typo in Stata’s help file may have contributed in wrongly estimating marginal changes after truncated and censored regression models. This has significant implications for empirical practice and results from published studies may need to be reconsidered.Stata; mfx; tobit; truncreg; marginal effects; truncated regression; censored regression

Drichoutis, Andreas

English

Research Papers in Economics

Page 1 of 7 
(Re)estimating marginal changes after “truncreg” and “tobit” in Stata
1 
Andreas C. Drichoutis 
 Dept. of Economics, University of Ioannina, Greece 
 
1. Introduction 
In censored and truncated regression models, coefficient estimates represent the effect of an 
independent variable on a latent variable.  This effect is generally not useful, because the latent 
variable is unobserved.  Thus we are mostly interested in estimating marginal changes which 
represent the effect of an independent variable on the conditional mean function. A typo in 
Stata’s help file has contributed to the incorrect estimation of marginal changes in such cases.  
This has significant implications for empirical practice, and results from some published studies 
may need to be reconsidered. 
Popular software programs, such as Stata and Limdep, include automatic routines that ease the 
computational burden of estimating marginal changes and corresponding standard errors. 
Unfortunately,  a typo that appears in the help files of some of Stata’s commands could 
potentially lead some users in choosing the wrong prediction option when estimating marginal 
changes after the associated commands (i.e., the truncreg and tobit commands). I should 
note that this is not a problem with the code but a problem with the help files. Stata has already 
acknowledged this error in personal communication. The same problem is pervasive in other 
commands of the censored regression model like xttobit (the random effects variant of tobit) 
and ivtobit (tobit model with continuous endogenous regressors). Stata has corrected the help 
files for ivtobit and published an erratum for the reference manuals on their website while an 
erratum was still pending at time of writing for the truncreg, tobit and xttobit commands. 
However, this typo has been fixed in the final free update of Stata 11 (as of September 1, 2011) 
and does not appear in Stata 12. 
                                                 
1 I would like to thank, without implicating, William Greene for providing a preview version of Limdep ver. 10 and 
for sorting out several of my queries. All errors and omissions are my own. Page 2 of 7 
The purpose of this note is to alert researchers about the potential flaws of past empirical studies 
and to avoid similar flaws in future empirical research.  Relying on a single program for doing 
empirical economic analysis is potentially dangerous.  It is unsettling to think of how many 
papers may have misreported the real effects of their models, with significant implications for 
empirical practice and policy making.  I first discuss the formulas for estimating marginal 
changes in censored and truncated regression models and then pinpoint the error in Stata’s help 
files.  Then I provide an empirical illustration in which I compare Stata’s output with Limdep’s 
output and my own calculations in Stata. 
 
2. The censored regression model 
In the censored regression model, the latent underlying regression is: 
*
ii i Yu   bX   ,            ( 1 )  
while the observed dependent variable is: 
ii YL     if  
*
ii YL   (lower tail censoring), 
ii YU     if  
*
ii YU   (upper tail censoring), and 
*
ii ii YY u    bX    if  
*
ii i L YU  . 
For the latent variable, the conditional mean function is simply   
* | ii i EY   Xb X .  For an 
observation randomly drawn from the population, which may or may not be censored, the 
conditional mean function is (Greene, 2003, 2007): 
          
 
|1
ii
ii i i i i i i i
ii
EY L U
 
   

 
              
Xb X  ,  (2) 
where 
ii
i
L


 

bX
 and 
ii
i
U


 

bX
 .  The most familiar censored regression model is the 
Tobit model, which arises by setting  i U     and  0 i L   in (2).   Page 3 of 7 
The marginal effect of a continuous variable  j X  is given by differentiating (2) with respect to it: 
      
* |P r o b
j Xi i j i j i i j i i i ME E Y X b b Y             X   .    (3) 
If  j X  is a dummy, then marginal effects are only approximately correct, and discrete changes 
should be estimated.  From (2) we get: 
  |, 1 |, 0
j Xi j i j DC E Y X E Y X   XX .                    (4) 
      
3. The truncated regression model 
The truncated regression model can be seen as a special case of the censored regression model 
where only data from the third group are observed. The latent regression is still given by (1), and 
the observed dependent variable is: 
*
ii ii YY u    bX    if  
*
ii i L YU  . 
The conditional mean function in this case is: 
   
 
|,
ii
ii ii i i
ii
EY L Y U
  

 

   
 
Xb X   ,        ( 5 )      
where 
ii
i
L


 

bX
 and 
ii
i
U


 

bX
 . 
If  j X  is continuous, then marginal effects can be derived by differentiating (5): 
 
 
 
  
2
1,
j
ii i i i i
Xj j
ii ii
ME b b h b
     

 
    
                  
  ,    (6) 
where   , hb  is the scale factor. 
If  j X  is a dummy, then discrete changes should be estimated as: 
    |, 1 , |, 0 ,
j Xi j i i ii j i i i DC E Y X L Y U E Y X L Y U       XX   .    (7) Page 4 of 7 
4. Estimating marginal changes in Stata 
To estimate marginal changes in Stata, one needs to use the mfx module (the mfx module has 
been superseded by margins in Stata 11, although the error in the help file remained at time of 
writing).  The help file of Stata indicates that the specified options in mfx, predict(options) 
can be, among others: 
e(a,b) which calculates   | E x bu ax bub    , the expected value of  | y x conditional on 
| y x being in the interval (a,b), meaning that  | y x is censored,  
or 
ystar(a,b) which calculates  
* Ey , where 
* y a   if  xbua   , 
* y b   if  xbub , and 
* y xb u   otherwise, meaning that 
* y  is truncated. 
However, if one tries to use e(a,b) after a censored regression model or ystar(a,b) after a  
truncated regression model, one gets incorrect results.  Stata’s technical support has 
acknowledged in personal communication that “the prediction option for 'ystar(a,b)' 
corresponds to the case when 
* y  is censored (not truncated), whereas the prediction option for 
e(a,b) corresponds to the case when  | y x is truncated (not censored).” 
To illustrate these errors with an empirical application, I use one of Stata’s example datasets and 
similar specifications that appear in the help files for tobit and truncreg. The dataset is a 
1978 automobile dataset which contains information on different car models, their selling price, 
fuel efficiency (miles per gallon), weight, gear ratio as well as the origin of the car (foreign vs. 
domestic)  etc.  
I regress miles per gallon (mpg)  on a continuous variable (weight) and a dummy variable 
indicating origin of the car (foreign). This specification is similar to the one that appears in the 
example of the viewer window of Stata if one types: help tobit 
postestimation##predict. The only addition is the foreign variable. The lower limit is 17 
and the upper limit is 24.  I used the nlcom module in Stata to calculate standard errors for 
marginal changes of own calculations.  Page 5 of 7 
It is obvious from Table 1 that, in the censored regression model, Limdep’s output, the 
ystar(a,b) option, and my own calculations coincide.  According to Stata’s official help file, 
the e(a,b) option should have been used, but it provides a very different set of results.  Note 
that Limdep results were obtained from the preview version of Limdep ver. 10. The effect of the 
dummy variable foreign in Limdep, is slightly different than the corrected output of Stata and my 
own calculations in Stata. This is because the routine that requests estimation of marginal 
changes at the means, uses the scaled coefficient, i.e., it treats the dummy as a continuous 
variable. In essence, the program uses formula (3) instead of (4) and formula (6) instead of  (7) 
which, however, produces only approximately correct results. 
For the truncated regression model I use the same dataset and a specification similar to the one 
that appears in the example of the viewer window of Stata if one types: help truncreg 
postestimation##predict. I regress price of cars on miles per gallon (mpg) and origin of 
the car (foreign). The lower limit is 4,000 and the upper limit is 9,500.  Similarly, in the 
truncated regression model, the e(a,b) option, Limdep’s output
2, and own calculations 
coincide.  However, according to Stata’s help file, the ystar(a,b) option rather than the 
e(a,b) option should have been used
3.  The former evidently produces a different set of results. 
 
5. Conclusions 
Estimating marginal changes in Stata after a censored or a truncated regression model has been a 
black box for many researchers.  Unfortunately, a simple typo in the help file of Stata may have 
caused researchers to use the wrong formulas for calculating marginal changes.  This could have 
had significant impact on policy recommendations if Stata’s users ignored the mathematics 
(which appear correct) but not the labels in the help file. If this is the case, many studies may 
                                                 
2 Users of Limdep ver. 9 may also notice that standard errors in the case of the truncated regression are different 
than the ones reported in Table 1 that were obtained with ver. 10. This is due to a simplification of an extremely 
complicated derivative that was used by the program in version 9. Limdep ver. 10 now uses the full correct 
derivative and standard errors will coincide with output of Stata, as evident in Table 1. 
3 In the case of a truncated regression the results obtained with the wrong option i.e., the ystar(a,b)option, are 
meaningless. In contrast, in a censored regression model the results obtained with the wrong option, i.e., the 
e(a,b) option, correspond to the marginal effects on the expected value of the dependent variable conditional on 
being uncensored. Page 6 of 7 
need to publish errata of their empirical findings after re-estimating marginal changes for 
truncated or censored regression models. 
 
6. References 
Greene, W.H. 2003. Econometric Analysis (5th ed.). Prentice Hall: New Jersey. 
Greene, W.H. 2007. Limdep version 9.0, Econometric modeling guide (Vol. 1). Econometric 
Software Inc.: Plainview, New York. 
 
 Page 7 of 7 
Table 1. Estimation results using Stata and Limdep 
  Stata’s output  Limdep’s output  Own calculations in 
Stata 
 
Variables 
e(a,b) ystar(a,b)    
 
Coef. 
Std. 
Error 
Coef. 
Std. 
Error 
Coef. 
Std. 
Error 
Coef. 
Std. 
Error 
Censored regression 
(lower limit=17, upper 
limit=24, Dependent 
variable is mpg) 
weight  -0.328 0.059 -0.582 0.066 -0.582 0.066 -0.582 0.066 
foreign  -1.541 0.492 -2.706 0.780 -2.846 0.914 -2.706 0.780 
Truncated regression 
(lower limit=4000, 
upper limit=9500, 
Dependent variable is 
price) 
mpg  -78.901  31.836  -89.264  60.430  -78.905   31.836    -78.901  31.836 
foreign  1119.988 439.570 1541.508 676.680 876.649 283.876    1119.978 439.570 
 

(Re)estimating marginal changes after “truncreg” and “tobit” in Stata

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