This page shows an example of tobit regression analysis with footnotes
explaining the output. The data in this example were gathered on undergraduates
applying to graduate school and includes undergraduate GPAs, the reputation of
the school of the undergraduate (a topnotch indicator), the students’ GRE score, and whether or not the
student was admitted to graduate school.

The range of
possible GRE scores is 200 to 800. This means that our outcome variable is both left censored
and right-censored.  In other words, if two students score an 800, they
are equal according to our scale but might not truly be equal in aptitude.
(In other words, we have a ceiling effect.)  The same is true of two students scoring 200
(a floor effect).  Tobit regression generates a model that
predicts the outcome variable to be within the specified range.

If we are interested in predicting a student’s GRE score using their
undergraduate GPA and the reputation of their undergraduate institution, we
should first consider GRE as an outcome variable.

use https://stats.idre.ucla.edu/stat/stata/dae/logit.dta, clear

summarize(gre)

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
         gre |       400       587.7    115.5165        220        800

histogram gre, bin(10) freq

To generate a tobit model in Stata, list the outcome variable followed by the
predictors and then specify the lower limit and/or upper limit of the outcome
variable.  The lower limit is specified in parentheses after
ll and the upper limit is
specified in parentheses after ul.
A tobit model can be used to predict an outcome that is censored
from above, from below, or both.

tobit gre gpa topnotch, ll(200) ul(800)

Refining starting values:

Grid node 0:   log likelihood = -2332.8456

Fitting full model:

Iteration 0:   log likelihood = -2332.8456  
Iteration 1:   log likelihood = -2331.4413  
Iteration 2:   log likelihood = -2331.4314  
Iteration 3:   log likelihood = -2331.4314  

Tobit regression                                    Number of obs     =    400
                                                           Uncensored =    375
Limits: Lower = 200                                     Left-censored =      0
        Upper = 800                                    Right-censored =     25

                                                    LR chi2(2)        =  70.93
                                                    Prob > chi2       = 0.0000
Log likelihood = -2331.4314                         Pseudo R2         = 0.0150

------------------------------------------------------------------------------
         gre | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         gpa |   111.3085   15.19669     7.32   0.000     81.43266    141.1843
    topnotch |   46.65774   15.75359     2.96   0.003     15.68709     77.6284
       _cons |   205.8515   51.24085     4.02   0.000      105.115    306.5881
-------------+----------------------------------------------------------------
   var(e.gre)|   12429.62   923.9586                      10739.66     14385.5
------------------------------------------------------------------------------

Tobit Regression Output

Tobit regression                                  Number of obsb   =        400
                                                  LR chi2(2)c      =      70.93
                                                  Prob > chi2d     =     0.0000
Log likelihooda = -2331.4314                      Pseudo R2e       =     0.0150

------------------------------------------------------------------------------
         gref|      Coef.g  Std. Err.h     ti   P>|t|j    [95% Conf. Interval]k
-------------+----------------------------------------------------------------
         gpa |   111.3085   15.19665     7.32   0.000     81.43273    141.1842
    topnotch |   46.65774   15.75356     2.96   0.003     15.68716    77.62833
       _cons |   205.8515   51.24073     4.02   0.000     105.1152    306.5879
-------------+----------------------------------------------------------------
  var(e.gre)l|   12429.62   923.9586                      10739.66     14385.5
------------------------------------------------------------------------------

a. Log likelihood – This is the log likelihood of the fitted model. It
is used in the Likelihood Ratio Chi-Square test of whether all predictors’
regression coefficients in the model are simultaneously zero.

b. Number of obs – This is the number of observations in the dataset
for which all of the response and predictor variables are non-missing.

c. LR chi2(2) – This is the Likelihood Ratio (LR) Chi-Square test that at least one of the predictors’ regression
coefficient is not equal to zero.  The number in the parentheses indicates the
degrees of freedom of the Chi-Square distribution used to test the LR Chi-Square
statistic and is defined by the number of predictors in the model (2).

d. Prob > chi2 – This is the probability of getting a LR test
statistic as extreme as, or more so, than the observed statistic under the null
hypothesis; the null hypothesis is that all of the regression coefficients
are simultaneously equal to zero.  In other words, this is the
probability of obtaining this chi-square statistic (70.93) or one more extreme if there is in fact
no effect of the predictor variables.  This p-value is compared to a specified
alpha level, our willingness to accept a type I error, which is typically set at
0.05 or 0.01.  The small p-value from the LR test,  <0.0001, would lead us to
conclude that at least one of the regression coefficients in the model is not
equal to zero. The parameter of the chi-square distribution used to test the
null hypothesis is defined by the degrees of freedom in the prior line,
chi2(2)

e. Pseudo R2 – This is McFadden’s pseudo R-squared.  Tobit
regression does not have an equivalent to the R-squared that is found in OLS
regression; however, many people have tried to come up with one.  There are a
wide variety of pseudo-R-square statistics.  Because this statistic does not
mean what R-square means in OLS regression (the proportion of variance of the
response variable explained by the predictors), we suggest interpreting this
statistic with great caution. For more information on pseudo R-squareds, see
What are Pseudo R-Squareds?.

f. gre – This is the response variable predicted by the model.
We are using a tobit model because this response variable is censored: the GRE
scores are scaled from 200 to 800 and cannot fall outside of this range.

g. Coef. – These are the regression coefficients. Tobit regression coefficients are
interpreted in the similiar manner to OLS regression coefficients; however, the linear effect
is on the uncensored latent variable, not the observed outcome. The expected
GRE score changes by Coef. for each unit increase in the
corresponding predictor.

    gpa – If a subject
were to increase his gpa by one point, his expected GRE score would
increase by 111.3085 points while holding all other variables in the model constant.
Thus, the higher a student’s gpa, the higher the predicted GRE score.

topnotch – If a subject attended a topnotch
institution for her undergraduate education, her expected GRE score would be 46.65774
points higher than a subject with the same grade point average who attended
a non-topnotch institution.  Thus, subjects from topnotch undergraduate
institutions have higher predicted GRE scores than subjects from
non-topnotch undergraduate institutions if grade point averages are held
constant.

_cons – If all of the predictor variables in
the model are evaluated at zero, the predicted GRE score would be _cons = 205.8515.  For subjects from non-topnotch undergraduate institutions (topnotch
evaluated at zero) with
zero gpa, the predicted GRE score would be 205.8515.  This may seem very
low, considering the mean GRE score is 587.7, but note that evaluating
gpa at zero is out of the range of plausible values for gpa.

h. Std. Err. – These are the standard errors of the individual
regression coefficients.  They are used in both the calculation of the t test statistic, superscript
i, and the
confidence interval of the regression coefficient, superscript k.

i. t –  The test statistic t is the ratio of the Coef. to the Std. Err. of the respective predictor.  The
t value is used to test against a two-sided alternative hypothesis that the
Coef. is not equal to zero.

j. P>|t| – This is the probability the t test statistic (or a more extreme test statistic) would be observed under the null hypothesis
that a particular predictor’s regression coefficient is zero, given that the
rest of the predictors are in the model.  For a given alpha level, P>|t| determines whether or not the null hypothesis
can be rejected. If P>|t|
is less than alpha, then the null hypothesis can be rejected and the parameter
estimate is considered statistically significant at that alpha level.

    gpa – The t test
statistic for the predictor gpa is (111.3085/15.19665) = 7.32 with an
associated p-value of <0.001.  If we set our alpha level to 0.05, we would reject the null hypothesis and conclude that the regression coefficient for
gpa has been
found to be statistically different from zero given topnotch is in the model.

topnotch -The t test
statistic for the predictor topnotch is (46.65774/15.75356) = 2.96 with an
associated p-value of 0.003.  If we set our alpha level to 0.05, we would reject the null hypothesis and conclude that the regression coefficient for
topnotch has been
found to be statistically different from zero given gpa is in the model.

_cons – The t test
statistic for the intercept, _cons, is (205.8515/51.24073) = 4.02 with
an associated p-value of < 0.001.  If we set our alpha level at 0.05, we would
reject the null hypothesis and conclude that _cons has been found to be
statistically different from zero given gpa and topnotch are in the model and evaluated at zero.

k. [95% Conf. Interval] – This is the Confidence Interval (CI) for an
individual coefficient given that the other predictors are in the model.  For a
given predictor with a level of 95% confidence, we’d say that we are 95%
confident that the “true” coefficient lies between the lower and upper limit of
the interval.  The CI is equivalent to the t test statistic: if the CI includes zero,
we’d fail to reject the null hypothesis that a particular regression coefficient
is zero given the other predictors are in the model with alpha level of zero.  An advantage of a CI is
that it is illustrative; it provides a range where  the “true” parameter may
lie.

l. var(e.gre) – This is the estimated variance of the regression.
In earlier versions of Stata, sigma was given in the output.  Sigma is the square root of the variance that is given the in current output.