Potential Outcomes Framework
For each unit $i$, define two potential outcomes:
\[Y_i(1)=\text{outcome if unit }i\text{ receives treatment},\] \[Y_i(0)=\text{outcome if unit }i\text{ receives control}.\]Individual and average effects
The individual treatment effect is
\[\tau_i=Y_i(1)-Y_i(0).\]The population average treatment effect is
\[\boxed{ATE=E[Y(1)-Y(0)]}.\]For a finite sample of $N$ units, the sample average treatment effect is
\[\tau_S=\frac{1}{N}\sum_{i=1}^N[Y_i(1)-Y_i(0)].\]In a finite-population randomized experiment this fixed-sample quantity is often written $\tau$ or $\tau_N$.
Treatment assignment and observed outcome
Let
\[W_i= \begin{cases} 1,&\text{treated},\\ 0,&\text{control}. \end{cases}\]The observed outcome is
\[\boxed{ Y_i^{obs}=W_iY_i(1)+(1-W_i)Y_i(0) }.\]If $W_i=1$, this reduces to $Y_i^{obs}=Y_i(1)$; if $W_i=0$, it reduces to $Y_i^{obs}=Y_i(0)$.
Fundamental problem of causal inference
Only one potential outcome is observed for each unit. Therefore $\tau_i$ cannot generally be computed directly. Research design replaces the missing counterfactual with a credible comparison group and targets an average effect.
Stable treatment definition
The notation normally assumes that a unit’s potential outcome depends on its own assigned treatment and that treatment versions are well defined. Interference between units or multiple hidden versions of treatment requires richer potential-outcome notation.
Identification versus estimation
- The estimand is the target, such as $ATE$ or the finite-sample $\tau_S$.
- Identification states when observable quantities determine that target.
- The estimator is the sample rule used to estimate it, such as $\bar Y_1-\bar Y_0$.