Tobit predicts values above lower bound

[Marlene Fruehling]

Good morning,

I have a censored dataset, where individuals reported a willigness to pay (WTP) in a field experiment for a product, but where told that this value could not be below a certain value (let's say 5 USD). Individuals with a WTP below those 5 USD hence did not Report a WTP. All Independent variables are available for those indiviudals.

I estimate the following:

tobit WTP `controls' , ll (4.999999) vce(cluster Village)
predict WTP_predicted /*for censored value prediction*/
quietly margins, dydx(*) atmeans post
est store rWTP /*for Regression Output*/

When I look at the predicted values (WTP_predicted), some households whose WTP was censored have predicted values that are above the lower bound of 5 USD, which of course makes no sense.

Is the Tobit model the wrong model? Or am I doing something wrong here?
Help would be very much appreciated!

Thanks a lot in advance,
Marlene

[Clyde Schechter]

The default statistic for -predict- after -tobit- is xb, which is just the uncensored linear combination. To get censored predictions you need to specify the -ystar()- option. See -help tobit postestimation##predict- for more information. The same applies to -margins- after -tobit-.

[Marlene Fruehling]

Dear Clyde,

I appreciate your help a lot, Thanks, that makes sense!
I have a small two follow-up questions - to be on the safe side -, I hope that is okay.


tobit Log(WTP) treatment `controls', ll(4.999999) vce(cluster Village)
local pr2 `e(r2_p)'
predict LogWTP_Tobit, ystar(4.99999,.)
margins, dydx(*) atmeans post predict(ystar(4.99999,.))
est store rWTP

outreg2 [rWTP] using "`file'.doc", append ///
label ///
nodepvar ///
bdec(3) ///
stats(coef pval) ///
drop(`controls1) ///
addstat(Pseudo R2, `pr2') ///
asterisk(pval)

}
}
xxx

1. Question: Am I understanding correctly that the variable LogWTP_Tobit now gives my predicted log(WTP) values taking into account the censored sample?
2. Question: Am I understanding correctly that I can interpret an coefficient of, lets say, 0.07 as "receiving the treatment is associated with a mean increase by 7 percent"? This 7 percent increase would then mean WTP is 7 higher than the mean WTP without treatment? Note that the treatment variable is a dummy variable taking 0 for no treatment and 1 for having treatment, and the WTP is in logarithm and in monetary terms (continuous).

I am very sorry for these beginner-questions, unfortunately I have nobody else to ask.
Thanks again,
Marlene

[Clyde Schechter]

Question 1. Correct.

Question 2. More or less correct. For the moment, let's disregard the tobit and pretend this were OLS. Then a unit difference in the treatment variable (1 vs 0) is associated with a 0.07 difference in log(WTP). This in turn correspond to a WTP ratio of exp(0.07) = approx. 1.073, or a 7.3% relative increase in WTP. And to avoid overstating our precision, it would be reasonable to call this approximately 7%.

Now, the -tobit- makes it more complicated. The coefficient relates the difference in the treatment variable to the difference in the linear predictor, which is that latent outcome that gets censored at 5. Suppose you have some subgroup of people whose covariate values are such that under the no-treatment condition the expected value of this latent outcome (latent log WTP) is 3. Then the observed outcome, due to censoring, is 5. Now suppose you have an otherwise identical group that is treated. Then the latent outcome will be approximately 3 + 0.07 = 3.07, which, after censoring, is still 5--i.e. no change in the observed outcome. So it is not possible to give a simple, clean characterization of the coefficient as an elasticity. It works like an elasticity when outcomes are far away from the censoring boundaries, but fails with outcomes that are censored or nearly so.

[Marlene Fruehling]

Thank you very much, Clyde!