# Linear Regression: Calculating a treatment effect directly in regression vs. averaging potential outcomes

Suppose I have the following true model, where an individual $$i$$ at a particular point in time $$t$$ is either treated ($$W=1)$$ or untreated ($$W=0$$). The outcome for individual $$i$$ at time $$t$$ under treatment status $$W$$ is random variable $$Y(W)_{it}$$.

The true model is:

$$y_{it}(0)=a_{i}+\gamma_{t}+\epsilon_{it}$$ $$y_{it}(1)=a_{i}+\gamma_{t}+\eta+\epsilon_{it}$$ $$\epsilon_{it}\sim\cal{{N}}(0,\sigma^2)$$

Where $$\eta$$, then, is the treatment effect, or parameter, of interest.

Suppose we have the following two ways of estimating $$\eta$$:

1. Run the regression from: $$y_{it}(W)=a_{i}+\gamma_{t}+\eta\cdot W_{it}+\epsilon_{it}$$ and obtain the coefficient estimate of $$\hat{\eta}_{OLS}$$

2. Exclude all observations for which $$W=1$$. Estimate every $$y(0)$$ with: $$y_{it}(0)=a_{i}+\gamma_{t}+\epsilon_{it}$$ Then estimate $$\eta$$ with $$\hat{\eta}_{ATT}=\dfrac{1}{N}\sum\left(y_{it}(1)-\widehat{y_{it}(0)}\right)$$

I know the distribution of $$\hat{\eta}_{OLS}$$. I am wondering if the distribution of $$\hat{\eta}_{ATT}$$ is different and how I can see this.

My intuition is that the distributions are not the same, and that the first estimator is more efficient than the second.