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I am wondering about the effect of an event $Z$ on an outcome $Y$. I assume $Z$ does not affect $Y$ except through the channel of an intermediate outcome $D$, that is endogenous to $Y$.

To understand the effect of $Z$ on $D$, I run the following regression:

$$ D = X'\beta + \sum_{\tau \neq -1} \delta_\tau Z_\tau + \epsilon $$ Where $X$ is a vector of controls, and $Z_\tau$ is a set of dummy variables equal to 1 if the time period is $\tau$ periods since the event.

How can I use the event, $Z$, as an instrument for the intermediate outcome $D$, and show a period-by-period effect of $D$ on $Y$?

Are there papers with examples using something similar?

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Without modeling the influence of $D$ on $Y$ you won't be able to identify the influence of $Z$ on $Y$.

For example, assuming linear relation $Y=\gamma_0+\gamma_1 D + \eta$ and that $\eta$ and $\epsilon$ are independent, then the effect of $Z$ on $Y$ is given by $\gamma_1 \cdot \delta_\tau$. The general rules of causal inference are provided in Pearl (2009). The example above is a simple instance of a linear structural equation model with a causal structure of chain.

Pearl, J. (2009). Causality: models, reasoning and inference. Second edition. Cambridge: MIT press.

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  • $\begingroup$ So, given the linear relation you suggested, I could get an effect for each period ($\gamma_1 \cdot \delta_1$,$\gamma_1 \cdot \delta_2$,$\gamma_1 \cdot \delta_3$, $\gamma_1 \cdot \delta_4$, ...)? $\endgroup$ – gannawag Apr 12 '17 at 20:51
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    $\begingroup$ Yes, that's the idea. Your comment makes me however wonder, what the meaning of the state in which $\sum_{\tau \neq -1}Z_t =0$ is? Because the effect describes the change in $Y$ due to change from$Z_\tau=0$ to $Z_\tau=1$. $\endgroup$ – matus Apr 12 '17 at 21:21

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