Estimate of SATE and ATE

Question

I am trying to understand the connection between sample average treatment effect (SATE) and average treatment effect (Population)-PATE or ATE

$SATE=\frac{1}{n} \sum^n_i (Y_i(1)-Y_i(0))$

$ATE=\frac{1}{N} \sum^N_i (Y_i(1)-Y_i(0))=E(Y_i(1)-Y_i(0))$

The estimator:

$\hat{\Delta}=\frac{1}{n} \sum^n_i (Y_i*(T_i=1)-Y_i(T_i=0))$, $T_i$ is the treatment status and we assume treatment is randomly assigned.

$\hat{\Delta}$ is sample mean difference between treated and untreated subjects.

So $\hat{\Delta}$ is the estimate for both $SATE$ and $ATE$, right?

One more question: Why should we be interested in this $SATE$? this $SATE$ may vary from sample to sample. Why should we care whether treatment works or not in one particular sample?

Answer 1

The main difference can be seen in the sum indices. The sample ATE uses a sample $1\dots n$, whereas the population ATE includes the entire population $1\dots N$. The estimator $\hat{\Delta}$ is still based on a sample. You'd be correct in saying that it gives you an estimate of the sample ATE, which is also an estimate of the population ATE if the data is from an RCT, but it is not the same as the population ATE.