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
- the latent dependent variable (unobserved)
- the variable conditional on being uncensored (folks away from the zero boundary)
- the censored variable (observed)
- 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
- $\frac{\partial E[y^* \vert x]}{\partial x} = \beta$
- $\frac{\partial E[y \vert x,y>0]}{\partial x} = \beta \cdot \Phi(x'\beta/\sigma)$
- $\frac{\partial E[y \vert x]}{\partial x} = \beta \cdot
\{1-\frac{\phi(x'\beta/\sigma)}{\Phi(x'\beta/\sigma)}- \left(
\frac{\phi(x'\beta/\sigma)}{ \Phi(x'\beta/\sigma)} \right)^2 \}$
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.
