# A Data Generating Process Implying Homogeneous Individual Treatment Effects

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?

$$y_{i} \sim N(\mu_{i}, \sigma^2)$$
$$\mu_{i} = \alpha_{0} + \alpha_{1} * treatment$$
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?