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.


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