I would like to see whether black and white participants are treated (0/1 treatment indicator) differently in my participant level data. However, black and white participants have different underlying characteristics that I capture in a vector of many indicator/binary variables I'll call $C$.
Can I run a logistic model to first find the the predicted probability of treatment based on the vector $C$:
$$ \hat Y = C\beta+\epsilon$$
and then plug this predicted value into a similar logistic regression with the same dependent treatment variable regressed on an indicator/dummy variable for black participants $black$ and single column of my probability of treatment, $\hat Y$ :
$$ Y = \beta_1 black+\beta_2 \hat Y+ u$$
and check if $\beta_1$ is statistically significant? I would like to properly adjust for these underlying participant characteristics, but I would like to not have to simply run one regression including the whole vector $C$ along with $black$:
$$ Y = \gamma_1 black+C\beta+ v$$
for other methodological reasons, one of which is that I would also like to include the interaction term $ (black)x(\hat Y)$ in my second equation to allow differential treatment along the distribution of $\hat Y$ (and the whole point of my project was to see if I can improve upon this).
I have run some simulations and found that the $\beta_1$ above is slightly different than $\gamma_1$ even if $black$ and $C$ are correlated, but I want to know if there is something very wrong with this approach.