Estimating a multivariate Tobit with endogenous covariates

[Martin Trombetta]

Hey everyone!

I'm trying to estimate an unrestricted system of two equations where both endogenous variables are left-censored (therefore a multivariate Tobit with endogenous covariates) using gsem. It seems pretty straightforward but I get the following error:

invalid path specification;
a nonrecursive system among the variables 'y1' and 'y2' is not allowed

I have consulted the theoretical literature (specially Amemiya (1974), which seems to be the seminal paper) and haven't found a reason why such a model can't be estimated. Let me show you what I'm talking about by a simple simulation, try running the following code:

clear
set obs 200
gen x=runiform()
gen u1=rnormal(0,1)
gen u2=rnormal(0,1)
gen y1=x+u1+u2
gen y2=x+0.5*u1+1.5*u2
replace y1=0 if y1<0
replace y2=0 if y2<0
gsem (y1 <- y2 x, family(gaussian,lcensored(0))) (y2 <- y1 x, family(gaussian,lcensored(0)))

All comments and suggestions will be appreciated.

Regards

[Phil Bromiley]

It may not apply to your model, but in traditional simultaneous equation models, you cannot have identical exogenous rhs variables in both equations.

[Martin Trombetta]

Oh well that is true, identification conditions must be satisfied for estimation to be possible. That problems exists in the code I provided (though not in my original data, I think) so let me try again:

clear
set obs 200
gen x1=uniform()
gen x2=uniform()
gen u1=rnormal(0,1)
gen u2=rnormal(0,1)
gen y1=x1+u1+u2
gen y2=x2+0.5*u1+1.5*u2
replace y1=0 if y1<0
replace y2=0 if y2<0
gsem (y1 <- y2 x1 x2, family(gaussian,lcensored(0))) (y2 <- y1 x1 x2, family(gaussian,lcensored(0)))

The problem persists despite the fact that identification conditions hold (or at least they would hold in the linear case). Are identification conditions different under Tobit specification?

[Jeff Wooldridge]

A simultaneous Tobit model where each variable is a function of the other is very tricky — conceptually and computationally. The reduced form doesn’t even always exist. There’s an old literature on the similar problem with binary responses; I wouldn’t be surprised if papers don’t exist for the Tobit case. There are weird restrictions in the parameters for the model to make sense. I’m not surprised gsem doesn’t allow it.

You’re trying to estimate

y1 = max(0,a1*y2 + b1*x1 + u1)
y2 = max(0,a2*y1 + b2*x2 + u2)

This is very nonlinear. Frankly, I don’t know how I’d summarize the results even if I could estimate the model.

If you say more about the application you might get more suggestions.

[Martin Trombetta]

Thank you very much for your answer, Jeff. Let me tell you a little bit about the specific application.

I'm investigating the determinants of hours worked both in the household and in the labor market. I consider both variables to be jointly determined, which is why I'm estimating a system. Of course, I could do it using linear specification (which I've done and results don't look bad) but both variables have a relevant proportion of zeros and I'd like my model to deal with that information in a sensible way. That's why I'm considering Tobit, the specification is exactly the one you wrote. My covariates include traditional control variables such as gender, age, income and the like. I'm not concerned with causal identification right now, as I understand that would add too much complication to my current research stage.

In the end, what I'd most like to see are average marginal effects, particularly that of one endogenous on the other. Do you think that can be done in this context? Thanks in advance for any suggestions.

[Jeff Wooldridge]

Your example fails what I call the "autonomy requirement." In my MIT Press book I discuss this. Just because two variables are jointly determined doesn't mean it makes sense to write one as a function of the other, or vice versa. In fact, if you solved any economic model, you would obtain each as a function of exogenous variables only. SEMs were meant to apply to equilibrium relationships from two different sides of a market (supply and demand). That is why we take hours and wages in an SEM. One side is the individual, the other is the firm. But do not use them when two variables are chosen by the same entity. I don't put quantity of chicken as an explanatory variable for quantity of beef and call that a demand function. Each quantity is a function of all prices, but not other quantities.

[Martin Trombetta]

I see your point Jeff, but do you agree the marginal effects I'm interested in exist? Say labor hours were raised exogenously (because of new regulations), that would impact hours worked in the household, wouldn't it? And the same would be true if hours worked in the household rose, say, because washing machines became more expensive, right? How would you estimate such effects?