Does Propensity Score theorem invalidate linear regression? The propensity score theorem in Causal Inference implies that $E[T|X]=E[T|p(X)]$, i.e. it is sufficient to control for the propensity score to retrieve the average causal effect. Then, we can use a linear regression which only controls for $p(X)$, and not all confounders $X$ to retrieve the Average Causal Effect.
This implies a regression equation of the form $$Y=\beta_0+\beta_1T+\beta_2p(X)+\varepsilon$$ However, since $p(X)$ is almost always a non-linear function of $X$ (clearly in a logistic regression, even in an LPM we have to restrict probabilities to between 0 and 1 which introduces non-linearities). If we used regression adjustment instead, our model would be of the form
$$Y=\delta_0+\delta_1T+\sum_{k=1}^K \gamma_k x_k+\eta $$
This is clearly linear in $X$ and otherwise has the same form has the propensity score regression. As I understand this, this has two consequences:

*

*Linear Regression can never retrieve the ACE

*To retrieve the ACE in a regression without the propensity score, we need a partially linear model, whose non-linear term includes all confounders $X$ (and, coincidentally, which approximates the propensity score $p(X)$).

Are these consequences correct? Does this pose a problem for regression adjustment, or do we simply (as usual in OLS) argue that linear regression retrieves the best linear approximation to the (partially) non-linear model involving the propensity score?
 A: I disagree with your assertion that
$$Y=\beta_0+\beta_1T+\beta_2p(X)+\varepsilon$$
can be used generally to estimate the ACE simply by the PS theorem. The PS theorem is nonparametric; it doesn't refer any models and says nothing about the way the ACE should be estimated from the PS. The PS theorem says that if your estimand is $E[Y|T,X]$, then any estimator of $E[Y|T,p(X)]$ is also an estimator of $E[Y|T,X]$. The above model may be an estimator of $E[Y|T,p(X)]$, but that doesn't mean that it is consistent or a good estimator. That requires knowledge of the functional form of the relationship between the PS and the outcomes. If that relationship is not linear, then the model above is not in general consistent for the ACE.
So, the consequences are not correct:

*

*Linear regression on $X$ can recover the ACE if the true outcome model is linear in $X$ (there are a few other conditions; see Chattopadhyay and Zubizarreta (2022, Theorem 5) for a list). Linear regression on $p(X)$ can recover the ACE if the true outcome model is linear in the PS. I think your intuition is that at most one of these can be true, but it's not the case that at least one of these must be true.

*If the true outcome model is linear in $p(X)$, then it is likely true that the outcome model is not linear in $X$. That doesn't mean regression on $X$ will not be sufficient to remove confounding; again, see the reference above for all the conditions in which linear regression is sufficient to remove bias. But nothing guarantees that the outcome model is linear in $p(X)$, so there is no inherent conflict.

