In an RCT, does running OLS on $Y_i = \beta_0 + \tau D_i + \varepsilon_i$ and recovering $\tau$ recover ATE or ATT Let's say I run an RCT and then run OLS on $Y_i = \beta_0 + \tau D_i + \varepsilon_i$ where $D_i$ is a dummy variable indicating whether an individual $i$ received the treatment. If I were to take the expectation of $y_i$, would $\tau$ be the ATE or ATT?
I understand taking the expectation works as follows:
$\mathbb{E}[y_i | D_i = 1]=\beta_0 + \tau + \mathbb{E}[\varepsilon_i | D_i = 1]$
$\mathbb{E}[y_i | D_i = 0]=\beta_0 + \mathbb{E}[\varepsilon_i | D_i = 0]$
Subtract the two to arrive at:
$\mathbb{E}[y_i | D_i = 1] - \mathbb{E}[y_i | D_i = 0] = \tau + \mathbb{E}[\varepsilon_i | D_i = 1] - \mathbb{E}[\varepsilon_i | D_i = 0]$
where $\mathbb{E}[\varepsilon_i | D_i = 1] - \mathbb{E}[\varepsilon_i | D_i = 0] = 0$ if there is no selection bias.
Thus, I understand the right-hand side of the equation, but am confused as to the interpretation of the left-hand side.
Is the left-hand side ATE or ATT or am I missing something entirely? Thank you.

In an RCT, the ATT and ATE are equal, so the OLS estimator is consistent for both.
Define $\tau_{ATE} = E[Y^1] - E[Y^0]$ and $\tau_{ATT} = E[Y^1|D=1] - E[Y^0|D=1]$.
In an RCT, $Y^d \perp D, \forall d$.
Therefore, $E[Y^d|D=1] = E[Y^d]$, so $\tau_{ATE} = \tau_{ATT}$.
