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I have estimated the following model on Stata 13:

tobit y x1 x2 x1x2

where:

  • $y$ is aid spending by a charity in a developing country
  • $x_1$ is GDP per capita of developing country
  • $x_2$ is proportion of official funding that charity receives (so is a fraction between 0 and 1)
  • $x_1x_2$ is an interaction term between GDP per capita and proportion of official funding (both of which are continuous variables)

Edit: I am interested in the expected value of $y$ (censored and uncensored), i.e. ($E(y|x)$). Therefore I calculated the marginal effect of the interaction term (evaluated at the mean of $x_1$ and $x_2$). Specifically I found ($∂E(y│x)$)/($∂x_1 ∂x_2$).

Suppose I find that ($∂E(y│x)$)/($∂x_1 ∂x_2$) is negative.

I am inclined to interpret this as: As the proportion of official funding that the charity receives rises, the charity sends less aid to countries with a higher GDP per capita.

Would this be correct?

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  • $\begingroup$ The statement above seems to equate the marginal effects with the conditional expectation, which is not the case. The marginal effect is the derivative of that expectation. $\endgroup$ – Dimitriy V. Masterov Feb 20 '14 at 21:35
  • $\begingroup$ @DimitriyV.Masterov please see my edit which I hope makes more clear what I actually have done. Sorry for the earlier confusion. Would really appreciate your advice on my interpretation in light of this edit. $\endgroup$ – user40632 Feb 21 '14 at 0:19
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With the Tobit, there are at least four types of marginal effects that may be interesting.

They are marginal effects on the conditional expected value of

  1. the latent dependent variable (unobserved)
  2. the variable conditional on being uncensored (folks away from the zero boundary)
  3. the censored variable (observed)
  4. the probability of being uncensored

You can think of latent as desired and censored as actual or observed. If negative charity spending does not quite make sense to you, it's just a consequence of normalization. If you observe $y$ when $\alpha + \beta'x+\varepsilon>L$, this can be rewritten as $(\alpha-L) + \beta'x+\varepsilon>0$. This is probably why the intercept below is negative.

It sounds like you're interested in (1), (2) and (3), which correspond to

  1. $\frac{\partial E[y^* \vert x]}{\partial x} = \beta$
  2. $\frac{\partial E[y \vert x,y>0]}{\partial x} = \beta \cdot \Phi(x'\beta/\sigma)$
  3. $\frac{\partial E[y \vert x]}{\partial x} = \beta \cdot \{1-\frac{\phi(x'\beta/\sigma)}{\Phi(x'\beta/\sigma)}- \left( \frac{\phi(x'\beta/\sigma)}{ \Phi(x'\beta/\sigma)} \right)^2 \}$

Yours would be the right interpretation for the marginal effect on the latent or desired spending $y^*$, but not for the censored versions of $y$. The marginal effects of just the interactions in the latter cases are much more difficult since you have a non-linear expectation, and for a non-linear model, the magnitude, sign, and significance of the interaction coefficient may not yield reliable information about the true marginal effect. This is explained nicely in the context of a probit in the Ai, Norton, and Wang paper. Which marginal effect you care about depends on your research question.

Perhaps something could be hacked together like inteff for probit/logit, but I have not seen anyone actually work it out. You can find the expectations in either one of Cameron and Trivedi's Microeconometrics books. The derivatives are fairly elementary (if messy), but the hard part will be the standard errors.

But there are ways to achieving something close to what you want. Here I model expenditure on ambulance trips as a function of age and education. Age is measured in decades, so 3 corresponds to 30. I would probably model this outcome in logs, but that will complicate things.

First I fit your model with interactions (using factor variable notation):

. use http://cameron.econ.ucdavis.edu/musbook/mus16data.dta, clear
. tobit ambexp c.age##c.educ, ll(0)

Tobit regression                                  Number of obs   =       3328
                                                  LR chi2(3)      =     169.24
                                                  Prob > chi2     =     0.0000
Log likelihood = -26621.837                       Pseudo R2       =     0.0032

------------------------------------------------------------------------------
      ambexp |      Coef.   Std. Err.      t    P>|t|     [95% Conf. nterval]
-------------+----------------------------------------------------------------
         age |   558.9861   233.4216     2.39   0.017     101.3215    1016.651
        educ |   105.0704   75.83215     1.39   0.166    -43.61203    253.7528
             |
c.age#c.educ |  -2.147978   17.28228    -0.12   0.901    -36.03295    31.73699
             |
       _cons |  -2483.576   1022.405    -2.43   0.015    -4488.182   -478.9699
-------------+----------------------------------------------------------------
      /sigma |   2781.473   37.78852                      2707.382    2855.564
------------------------------------------------------------------------------
  Obs. summary:        526  left-censored observations at ambexp<=0
                      2802     uncensored observations
                         0 right-censored observations

Both education and age raise expenditures on ambulance trips, though the education effect is insignificant. You can just read off the insignificant coefficient on the interaction (-2.147978) to get the effect on the latent outcome (desired expenditure).

If you are willing to give up the interaction, you can still have a non-linear effect on the non-latent outcomes in (2) and (3) since $\Phi()$ and $\phi()$ are changing with education and age. In our case, the effect of education will be bigger for an older person even though $\beta$ is the same for both in this model.

Here's the Tobit without an interaction:

. tobit ambexp age educ, ll(0) // no interaction

Tobit regression                                  Number of obs   =       3328
                                                  LR chi2(2)      =     169.23
                                                  Prob > chi2     =     0.0000
Log likelihood = -26621.845                       Pseudo R2       =     0.0032

------------------------------------------------------------------------------
      ambexp |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   530.5004   44.21754    12.00   0.000     443.8041    617.1967
        educ |   95.95908   19.38974     4.95   0.000     57.94206    133.9761
       _cons |  -2362.819   318.0821    -7.43   0.000    -2986.476   -1739.163
-------------+----------------------------------------------------------------
      /sigma |   2781.431    37.7829                      2707.351    2855.511
------------------------------------------------------------------------------
  Obs. summary:        526  left-censored observations at ambexp<=0
                      2802     uncensored observations
                         0 right-censored observations

Now we calculate the marginal effects of one more year of education at ages 30, 40, and 50. Not surprisingly, they will be the same as the Tobit coefficients above and redundant:

. /* ME on latent variable ystar */
. margins, dydx(educ) atmeans at(age = (3(1)5))

Conditional marginal effects                      Number of obs   =       3328
Model VCE    : OIM

Expression   : Linear prediction, predict()
dy/dx w.r.t. : educ

1._at        : age             =           3
               educ            =    13.40565 (mean)

2._at        : age             =           4
               educ            =    13.40565 (mean)

3._at        : age             =           5
               educ            =    13.40565 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
educ         |
         _at |
          1  |   95.95908   19.38974     4.95   0.000      57.9559    133.9623
          2  |   95.95908   19.38974     4.95   0.000      57.9559    133.9623
          3  |   95.95908   19.38974     4.95   0.000      57.9559    133.9623
------------------------------------------------------------------------------

But since the $\Phi$ and $\phi$ functions are non-linear, the censored effects will vary by education (they will be smaller for younger people, though not significantly so). Note that the effect on the censored outcome below is much smaller and rises with age:

. /* ME on censored E[y|x,y>0] */
. margins, dydx(educ) atmeans predict(e(0,.)) at(age = (3(1)5))

Conditional marginal effects                      Number of obs   =       3328
Model VCE    : OIM

Expression   : E(ambexp|ambexp>0), predict(e(0,.))
dy/dx w.r.t. : educ

1._at        : age             =           3
               educ            =    13.40565 (mean)

2._at        : age             =           4
               educ            =    13.40565 (mean)

3._at        : age             =           5
               educ            =    13.40565 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
educ         |
         _at |
          1  |   38.93137   7.884372     4.94   0.000     23.47829    54.38446
          2  |   43.49449   8.786361     4.95   0.000     26.27354    60.71544
          3  |   48.40415   9.774295     4.95   0.000     29.24689    67.56142
------------------------------------------------------------------------------

Now we will add the zeros, which raises the effect, but not all the way to the latent magnitudes near $100:

. /* ME for censored E[y|x] */
. margins, dydx(educ) atmeans predict(ystar(0,.)) at(age = (3(1)5))

Conditional marginal effects                      Number of obs   =       3328
Model VCE    : OIM

Expression   : E(ambexp*|ambexp>0), predict(ystar(0,.))
dy/dx w.r.t. : educ

1._at        : age             =           3
               educ            =    13.40565 (mean)

2._at        : age             =           4
               educ            =    13.40565 (mean)

3._at        : age             =           5
               educ            =    13.40565 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
educ         |
         _at |
          1  |   55.02846   11.15398     4.93   0.000     33.16707    76.88985
          2  |   62.03843   12.53252     4.95   0.000     37.47515    86.60172
          3  |   68.56482   13.84297     4.95   0.000     41.43309    95.69654
------------------------------------------------------------------------------

Finally, some caveats and advice. The Tobit is extremely sensitive to departures from normality and heteroskedasticity of the errors. You can and should test these. It also makes a strong assumption that the zeros are created by the same process at the positives, which is often false. A nice example of this is expenditures on family vacation and number of children. More children probably makes a stay-cation more likely, but conditional on going, more children lead to higher bill. A Tobit cannot handle this case. Finally, the Tobit assumes that it's possible for donations to take on a value arbitrarily close to the limit. That is somewhat unrealistic for your scenario since no one would bother to donate a penny, for instance. For example, if the cheapest car is \$6,000, and I only want to spend $5,999, then 6K should be the limit rather than zero. In the durable goods literature, it is common to use the lowest price option as the limit. Maybe something like that makes sense here as well, at least as a robustness check.

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