Simultaneous Probit Model - Binary dependent and endogenous

[Michael Zuze]

I am trying to avoid the Forbidden regression. I have two simultaneous equations one for poverty and the other for informal employment specified as follows

Poor = B₀ + B₁ Informal employment + B₂X +B₃Z1+ U ..........(1)

Informal Employment = B₀ + B₁ Poor + B₂X +B₃Z2+ U...........(2)

where both dependent and endogenous variables are binary for the two equations and vector X has same exogenous variables for eqn 1 &2 , Z1 and Z2 are instruments

I was following Maddala(1983), who suggested estimating probit ML in the first and second stages; however, after reading Angrist, I discovered this is impossible and leads to forbidden regression. rather i should use LPM. Kindly assist me in working this out for my two simultaneous equations..

Thanks

[Jeff Wooldridge]

It's not a bad idea to use linear models and estimate each equation by 2SLS. It's true estimating first-stage probits doesn't get you very far.

I believe the -cmp- command can estimate the joint model by MLE if you want to account for the binary nature, although logical consistency and convergence can be issues.

I do wonder what kind of causality you're hoping to establish. That informal employment "causes" poverty? I'm not sure the two equation passes the so-called "autonomy" requirement. Does each equation live on its own, apart from the other? Good SEM applications have this feature. A demand function is one side of the market, supply is the other. Cities choose a police force size, criminal commit crimes. And so on. It's a gray area in your case, but worth thinking about. I've said in my writing before: just because two variables are jointly determined doesn't mean an SEM is appropriate.

[Michael Zuze]

It's not a bad idea to use linear models and estimate each equation by 2SLS. It's true estimating first-stage probits doesn't get you very far.

I believe the -cmp- command can estimate the joint model by MLE if you want to account for the binary nature, although logical consistency and convergence can be issues.

I do wonder what kind of causality you're hoping to establish. That informal employment "causes" poverty? I'm not sure the two equation passes the so-called "autonomy" requirement. Does each equation live on its own, apart from the other? Good SEM applications have this feature. A demand function is one side of the market, supply is the other. Cities choose a police force size, criminal commit crimes. And so on. It's a gray area in your case, but worth thinking about. I've said in my writing before: just because two variables are jointly determined doesn't mean an SEM is appropriate.

Thank you for the cmp suggestion.

So, from the literature, there is a bidirectional relationship between informality and poverty, with various factors influencing both. Low earnings from informal employment significantly contribute to household poverty. On the other hand, household poverty can compel the head of the household to accept informal work, as they cannot afford to remain unemployed. so I wanted to analyse this bidirectional relationship. Given the simultaneity of informal employment and household poverty, estimating the equations by ordinary least squares would result in inconsistent and biased estimates of the determinants of household heads’ informal employment and of the implications on household poverty as suggested by https://www.journals.uchicago.edu/do...urnalCode=edcc. So that's why I wanted to employ a simultaneous maximum likelihood probit model to account for the existing reverse causality. I have tried the cmp however the system is not converging maybe since the two variables appear in both equations either as dependent or endogenous simultaneously.

[Jeff Wooldridge]

For an individual, quantity of chicken and quantity of broccoli are also simultaneously determined. Would you use an SEM for them? No, because demand functions are quantities as functions of prices and income and tastes. Modeling one quantity as a function of other quantities doesn’t answer an interesting question. This is the sort of example I’m thinking about.