I have a categorical treatment variable, MessageType, that has 12 different values. The outcome variable, Crash, sometimes determines these values. So more crashes lead to certain types of messages, and fewer crashes lead to other kinds of messages. I am interested in estimating the causal effect of each message type on the number of crashes, however, since the message itself is determined by the outcome, I have an endogeneity problem. In my regression, I am controlling for different factors (traffic, weather, road conditions etc.) that can affect the number of crashes to achieve some sort of conditional independence. However, I think even after controlling for all the covariates the endogeneity is still there especially for a particular type of message, i.e., "Crash Ahead". This message is almost always determined by the outcome and therefore it gives me a large causal effect if included in the regression like other message types.
The question is what is the best way to handle this particular treatment which is determined by the outcome to get consistent treatment effects for "CrashAhead" and other message types? I was thinking of using a lag of outcome as an instrument but it doesn't seem like a good instrument as it fails to satisfy the exclusion restriction. My baseline model is Poisson fixed effects and I have a long panel (large T, small N).