Interpreting marginal effects after applying a Tobit model

[Jonas Heinrich]

Hi,

recently asked a question on how to interpret outputs generated with marginal effects after estimating a Tobit model. I am using Stata 13, so I figured I'd use the command margins - which I find very helpful. However, I am still a little bit lost when it comes to interpreting the results. I am researching firm level data in regards to innovation output and would like to understand wether certain companies use innovation input more efficiently than others.

I have two issues:

(1) Overcoming the short come of the Tobit output: namely that it shows the impact of changes in the regressors on the latent variable y instead of on the observed variable y*.
(2) Making sure that I actually estimate efficiency. I am just not sure if I am doing it correctly.


To make things a little more tangible (especially compared to my earlier post that failed to gain any attraction due to my stupid asking style; see the link below), I figured a simulate some data and analyze them accordingly:

***Generate simulated Data
*Input
gen x=rnormal()^2
*Company Type
gen t=round(runiform())
*Market environment
gen m=round(runiform())

***Functional forms
gen y=2*x-3*t+4*x*t+5*t*m+rnormal()

***Creating Tobit conditions (left censored at 0)
replace y=0 if _n<1000

*Tobit: Let yi be the observed var bounded from below and yi* the latent var
tobit y c.x##i.t i.m##i.t, ll(0)

*Marginal effect on the censored expected value E(yi)
margins, dydx(x) predict(ystar(0,.)) atmeans over(t)
margins, dydx(x) predict(ystar(0,.)) at(m=(0 1)) over(t)
marginsplot

I generate output bounded from below via the function y=2*x-3*t+4*x*t+5*t*m+rnormal(), where x depicts a metric input variable, t indicates the firm type (coded 0;1), m acts as an indicator for market uncertainty (coded 0;1 for simplicity reasons). The function thus simulates data, where y grows with the input of x, declines when company type 1 is in question, but where company type 1 generates more y per unit x invested, and does so even more, when the market is certain (m=1).

Running the Tobit model does yield the results expected.

Tobit regression results:

Tobit regression                                  Number of obs   =      10000
                                                  LR chi2(5)      =   15604.32
                                                  Prob > chi2     =     0.0000
Log likelihood = -19054.298                       Pseudo R2       =     0.2905

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   1.963289   .0294802    66.60   0.000     1.905502    2.021076
         1.t |  -3.409549   .1061183   -32.13   0.000    -3.617562   -3.201536
             |
       t#c.x |
          1  |   3.769483   .0401508    93.88   0.000     3.690779    3.848186
             |
         1.m |   .0056122   .0880544     0.06   0.949    -.1669922    .1782166
             |
         m#t |
        1 1  |   5.580141   .1144427    48.76   0.000      5.35581    5.804471
             |
       _cons |  -.7479231   .0703441   -10.63   0.000    -.8858117   -.6100346
-------------+----------------------------------------------------------------
      /sigma |   2.926497   .0126382                      2.901723     2.95127
------------------------------------------------------------------------------
  Obs. summary:       3129  left-censored observations at y<=0
                      6871     uncensored observations
                         0 right-censored observations

However, I have some issues interpreting the marginal effects.

Results marginal effects:

. margins, dydx(x) predict(ystar(0,.)) atmeans over(t)

Conditional marginal effects                      Number of obs   =      10000
Model VCE    : OIM

Expression   : E(y*|y>0), predict(ystar(0,.))
------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
x            |
           t |
          0  |   1.310562    .021878    59.90   0.000     1.267682    1.353442
          1  |   5.351779   .0214539   249.46   0.000      5.30973    5.393828
------------------------------------------------------------------------------

. margins, dydx(x) predict(ystar(0,.)) at(m=(0 1)) over(t)
------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
x            |
       _at#t |
        1 0  |    1.17447   .0196007    59.92   0.000     1.136054    1.212887
        1 1  |   2.626861    .024471   107.35   0.000     2.578899    2.674823
        2 0  |   1.175705   .0195122    60.25   0.000     1.137462    1.213949
        2 1  |   5.033166   .0211266   238.24   0.000     4.991759    5.074574
------------------------------------------------------------------------------

I interpret them as follows: Holding all covariates at their respective mean, company type 2 generates 5.35 units of y per unit x invested, whilst company type 1 only amounts to 1.31 units y per unit x invested. As such, company type 2 are on average more efficient. Moreover, this relationship increases as m increases. Does this make sense? And how can I be sure, that I am acctually measure "efficiency". Any help is gladly appreciated.

Thank you loads,
Jonas

PS: Earlier related question: http://www.statalist.org/forums/foru...ng-the-results