Stata teffects ATET How does stata estimates ATET (Average Treatment Effect on the Treated) using teffects psmatch.
I understand  the average treatment effect (ATE) is computed by taking the average of the difference between the observed and potential outcomes for each subject done by  teffects psmatch.
What about ATET?
 A: The ATET is estimated by finding a control unit to pair with each treated unit and then computing the difference between the mean outcome in the treated units with the mean of the matched control units. It's actually a bit simpler than estimating the ATE, which requires doing two rounds of matching, one for the treated group and one for the control group.
A: Considering PSM 1:X (1 treated matched with X non-treated), there is just one round of matching in which the program looks for one (X) observation (non-treated) per observation treated. The difference between ATE and ATT (also called ATET) is related to the sample (the rows of the data) you used to estimate the effect.
Calculating ATT, we can say that you only use the rows of the treated population and compare their outcome measured (after treatment) to the potential outcome that the counterfactual observation had. This last value is located in the same row as the treated observation. Actually, you use the whole data because you link information from other rows (non-treated) to the row of treated observations; however, your impact measure is only focused on the treated population (treated rows).
On the other hand, when you calculate the ATE, you calculate the impact using all the rows of the observations matched. It means, in addition to the rows used in ATT (previous paragraph), you add the rows of the non-treated population, which also have on the right (in the same row) the potential outcome obtained by their counterfactual observation (in this case, treated, that were matched in the unique and previous round of matching).
In summary, with ATE, you are using the two counterfactual sides of treatment status, comparing their potential outcomes. It means what would happen if both treatment conditions were opposite, while ATT is focused on only one counterfactual side, what would happen to suppose the treatment people were non-treated.
