Endogenous Categorical Variable (non-binary, possibly ordered)

[Vincent Sullivan]

Dear All,

I would be very grateful if anyone could help me. Maybe other people have the same issue.

I have cross sectional micro-data, 2000 obs. I want to look at the effect of a categorical variable (three categories) on a continuous variable. I have strong reasons to think that this categorical variable is endogenous.
So I have:
Y = B0 + B1*CAT_2 + B2*CAT_3 + exogenous variables

It could be argued that the categorical variable is ordered in the sense that there is a latent variable increasing over the categories (although I could put the case that the variable is not ordered). So, I need an ordered or multinomial first stage.I have two instruments (but I could stretch it to three instruments, although my third isn’t as plausible).

So my question is how to estimate the system to infer causality. If I just had one binary endogenous regressor I could use treatreg, or biprobit or just estimate the whole thing as LPMs.

Could I just bootstrap the following?
. oprobit CAT Z1 Z2 X
. predict PCAT_1 PCAT_2 PCAT_3, p
. regress Y CAT_2 CAT_3 X

or run the following series of LPMs:
. ivregress 2sls Y X (CAT_2 CAT_3 = Z1 Z2)
but I don’t think ivregress allows for the fact that the categories are mutually exclusive and ordered

I have looked at cmp but I am not sure how to execute this
I have run:
. cmp (y = X CAT_2 CAT_3) (CAT_2 = X Z1 Z2) (CAT_3 = X Z1 Z2), ind($cmp_cont $cmp_probit $cmp_probit)
but again I don’t think this allows for the fact that the categories are mutually exclusive and ordered

So I think I need an ordered (or multinomial first stage) but I don’t know what to do..
Something like this?
cmp (y = X ?) (CAT = X Z1 Z2), ind($cmp_cont $cmp_oprobit)
What should I put in the first equation? How would I interpet it.

Any help would be greatly appreciated. It's my first time posting here so apologies if I have not been clear enough.

Best wishes,

Vincent

[Phil Bromiley]

Please read the FAQs on asking questions. Treatreg will handle multi valued treatments although at the cost of assuming the errors in the treatment equation are uncorrelated with errors in the outcome equation. If cmp doesn't do it, you might look at gsem.
Phil