# How to deal with treatment variable which is determined by the outcome variable

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).

• Can anyone please help with this? Any ideas or thoughts you may have would help a lot Jun 1, 2020 at 18:31
• If the message is determined by the outcome, $O\to M,$ then you don't have an endogeneity problem; you have a violation of the fundamental law of causality: causes must precede effects! Jun 3, 2020 at 21:32
• @AdrianKeister The causality runs both ways. Crashes cause the message, and the message may also have effect on crashes in the current time period. It's a recursive setting $O\rightarrow M \rightarrow O$ Jun 3, 2020 at 21:35
• I would strongly recommend that you think about a causal diagram. For example, you might or might not need to be controlling for some of the things you're controlling for. It depends on whether any of them set up a backdoor path - something much easier to determine with a causal diagram. Jun 3, 2020 at 21:38

Here's a partial answer: for any moment in time $$t$$, think about the outcomes at $$t$$, the messages at $$t+1,$$ and the outcomes at $$t+2,$$ like so: What this diagram is saying (and we can alter the diagram easily to suit your mental model) is that the outcome at $$t$$ influences the messages at $$t+1$$ and the outcomes at $$t+2,$$ and the messages at $$t+1$$ influence only the outcome at $$t+2.$$
You could then take this model to set up your regression. Note that I'm not considering the lag as an instrumental variable. The diagram above is more of a straight-forward confounding diagram, where, perhaps, the backdoor adjustment formula might be useful: $$P(O(t+2)|\operatorname{do}(M(t+1)))=\sum_{O(t)}P(O(t+2)|M(t+1),O(t))\,P(O(t)).$$