I've been struggling with how to think about the causal estimate of a program on two outcomes, when one of the two outcomes affects the other outcome. It seems sort of like simultaneous equations, but reading up on SEM, I don't see immediately how it could apply. Would appreciate any ideas for how to approach this problem. Here is the setup:
Two outcomes: $Y_1$ and $Y_2$
One treatment: $T$
Data is a panel, and we believe that $E[\epsilon|\alpha, T] = 0$
The true model is something like: $$ Y_{1,it} = \alpha_i + \delta T_{it} + \gamma Y_{2,it} + \beta (Y_{2,it}\times T_{it})+ \epsilon_{it}\\ Y_{2,it} = \alpha_i + \delta T_{it} + \eta_{it} $$ So, $T$ has some effect on $Y_2$, and both $T$ and $Y_2$ have an effect on $Y_1$. Importantly, $Y_2$ happens before $Y_1$.
If I were to estimate the first equation by OLS, $\delta$ would be the average effect of $T$ not accounting for $Y_2$. But that's not really what I'm after -- I want the average effect of $T$ on $Y_1$, accounting for both channels through which $T$ works.
Estimating the second equation via OLS is valid (right?)
Would it be valid to plug in fitted values of $\hat Y_2$ into the first equation? Probably not, because the standard errors wouldn't account for uncertainty in the second equation... One could sort of justify it algebraically, but I wouldn't know how to interpret the fitted model.
This question falls under the heading of "bad controls" -- but perhaps one aspect of it that makes it a bit more difficult than typical "bad control" problems is that I really think that the interaction there is important, and because $Y_2$ isn't determined entirely by $T$.
How would I approach this problem? Is this something that can be solved by some well-known technique? If so, which one?
Thanks in advance for any advice!
EDIT: Thinking about it more, I'd get a causal estimate of the effect of $T$ alone by running $Y_{1,it} = \alpha_i + \delta T_{it} + \epsilon_{it}\\$. But I also want an estimate of $\gamma$ and particularly $\beta$. Given that the $\delta$ from the second equation is nonzero, is there some problem with estimating the first equation by OLS? Do the parameters have causal interpretation? If not, how do I get to causal interpretation?