Zero-inflated models (e.g., ZI poisson, ZI negative binomial, hurdle) assume two processes for the generation of the observed outcome variable: a process for deciding whether the outcome is zero or not, and a process, for those for whom the outcome is not definitely zero, for assigning a value to the outcome. In these models, each process has its own regression model and (implied) disturbance. Presumably, in most cases, the disturbances are correlated (i.e., the unmeasured causes of one process also cause the other process).
In the nonzero part of the model, we condition on not receiving a zero. For example, in the hurdle model, the nonzero part can be estimated directly by simply excluding those who received a zero and estimating a zero-truncated model for everyone else. Those who did receive a zero do not contribute to the likelihood for the nonzero process model estimation.
Conditioning on receiving a zero would seem to me to be conditioning on a collider, opening up a noncausal path from the predictors to the nonzero outcome through the shared disturbance, which to me would seem to bias the relationship between the predictors and the observed nonzero outcome.
Does this bias indeed occur, or is my logic or understanding faulty? Are there ways to mitigate this bias other than trying to eliminate the shared disturbance through control variables or instrumental variables?