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I want to find a data generating process implying homogeneous individual treatment effects.

Specifically, consider two potential outcomes $y_i^1$ and $y_i^0$.

The first one is the individual $i$'s (potential) outcome if she took a treatment.

and the second one is the individual $i$'s (potential) outcome if she did not take any treatment.

In this situation, I want to consider a data generating process under which the individual treatment effect, $y_i^1 - y_i^0$ is identical across all individual, that is $y_i^1 - y_i^0=y^1 - y^0$.

What data generating process yield the homogeneous individual treatment effects?

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Your question sounds a bit vague to me. But this is what I understood:

$y_{i} \sim N(\mu_{i}, \sigma^2)$

$\mu_{i} = \alpha_{0} + \alpha_{1} * treatment$

where $y_{i}$ is the outcome for individual i which is normally distributed and we model its mean using $\alpha_{0}$ the general intercept and $\alpha_{1}$ is the effect of treatment where treatment takes 1 and 0. Note that $\alpha_{1}$ is fixed and doesn't change depending on the individual. Is that what you want?

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