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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?

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

As an example, suppose I follow patient measures of blood pressure over repeated hospitalizations and their risk for ensuing stroke. The blood pressure I observe is a snapshot of a continuous, cumulatively accruing risk factor for stroke (higher blood pressure leads to hypertrophy of the heart, which increases the risk of throwing clots or impairing blood flow to the brain). I will never know if my snapshot represents a peak or a trough of such a process, and so there is substantial measurement error. There are other variables however which help me predict this error, such as the time of day, the age of the patient, and variability in stroke outcomes after conditioning on such things.

The problem with fitting two separate regression models here is that I use nothing of the parent model (the model for stroke) to improve my prediction of the blood pressure in the patient during his or her hospitalization. 2SLS uses that parent model, or more specifically the conditional errors in the outcome, to better estimate the effects in the instrumental variable model.

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