Probit with panel data - assumptions violated with endogenous covariate?

Question

I am trying to understand how does the fact that a regressor is endogenous (as a result of reverse causality) in a dynamic probit model violate any assumptions that I may need to make for this model. E.g. for an OLS model, it is pretty straightforward - the orthogonality condition for the regressor being uncorrelated with the error term will not hold. Can similar logic be extended to probit models with panel data? I would appreciate simple explanations and/or references that explain this.

Answer 1

Yes it does. According to the Verbeeks guide to modern econometrics (pp418) standard panel fixed effects binary model assumes that error has “a symmetric distribution with distribution function $F(.)$, i.i.d. across individuals and time and independent of all $x_{is}$” [with the model being $y_{it}^*=x_{it}’ \beta+ \alpha_i+u_{it}$].

Just a two pages later Verbeek also writes about Random effects probit model $y_{it}^*=x_{it}’ \beta +\epsilon_{it}$ that $\epsilon_{it}$ should also besides other things be independent of $x$.

Now both of the above panel models are technically static since no lags are involved but I don’t see any reason why this assumption could be relaxed just by making model dynamic. In fact there is a such thing as panel IV logit/probit, probably people would not invest time into development of those if this assumption could be relaxed by just adding lags.