I would like to estimate a logit model in the presence of endogeneity. The dependent variable is binary (actually, it is non-binary with multiple ordinal categories, but from what I've read dealing with endogeneity in a multinomial logit model is even more complicated, so I plan on jointly estimating multiple binomial logit models) and most of the regressors, some of which are endogenous, are discrete as opposed to continuous.
Now, according to the answer to this question I cannot use instrumental variables in a non-linear model such as logit. Since my endogenous regressors are categorical, I cannot use the control-function approach either (see this paper, p. 1: "The control-function approach, however, does not work if any of the endogenous regressors are non-continuous."), nor can I use the special regressor approach (see this, slide 24).
What can I use in this setting to deal with potential confounders? Maximum likelihood estimation (see slide 19)?