Should we use IV to estimate dynamic linear probability models with fixed effects?

I give a set of pros and cons of this procedure and conclude that this procedure should be treated with caution, especially in fixed-T settings. Even if we ignore the possibility that average marginal effects may not be point-identified, directly applying IV/GMM estimators to this dynamic LPM identifies incorrectly-weighted average marginal effects, which may differ from the true average marginal effect, under large-n, fixed-T or large-n, large-T asymptotics. I also show that there exist certain DGPs that can push the large-n, fixed-T limits of these IV estimators outside the identified set for the true average marginal effect. The only good news is that nonparametrically testing the point null of zero first-order state dependence is possible with default routines. Unfortunately, this nonparametric test can have low power. In relation to this, I demonstrate through an empirical example that the resulting IV/GMM estimates of the average treatment effect of fertility on female labor force participation are outside the nonparametric bounds under monotonicity.