# causal inference with correlated multivariate outcomes

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?

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?

• Are the $\alpha$s the same in both equations or do they need a subscript? – Dimitriy V. Masterov Mar 1 '14 at 23:23
• The alphas are fixed-effects, or individual-specific intercepts. They'll have different values in each regression, but represent the same variable. – generic_user Mar 1 '14 at 23:26
• @generic_user if $Y_{2, it}$ happens before $Y_{1, it}$ why have you given them the same subscript $t$? – AdamO Jul 20 '17 at 20:26

If the model is as you write it, then you can estimate both models with OLS using the exogenous $Y_{2, it}$ values as a covariate in the model for $Y_{1, it}$ provided the error terms are IID with constant variance.
However, if there's measurement error, 2 stage least squares may be better suited to estimating the problem. 2SLS is a special type of SEM which considers the possibility that $Y_{2, it}$ has measurement error.