Probit with panel data - assumptions violated with endogenous covariate? 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.
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.