35.3 Framework for Generalization

Let:

• $P_t$, $P_c$: treated and control populations
• $N_t$, $N_c$: random samples drawn from $P_t$, $P_c$
• $\mu_i$, $\Sigma_i$: means and covariance matrices of the $p$ covariates in group $i \in \{t, c\}$
• $X_j$: vector of covariates for individual $j$
• $T_j \in \{0, 1\}$: treatment indicator (1 = treated, 0 = control)
• $Y_j$: observed outcome
• Assume $N_t < N_c$ (i.e., more controls than treated)

The conditional treatment effect is:

$$\tau(x) = R_1(x) - R_0(x), \quad \text{where } R_1(x) = E[Y(1) \mid X = x], \quad R_0(x) = E[Y(0) \mid X = x]$$

If we assume constant treatment effects (parallel trends), then $\tau(x) = \tau$ for all $x$. If this assumption is relaxed, we can still estimate an average effect over the distribution of $X$.

Common Estimands

• Average Treatment Effect (ATE): Average causal effect across all units.
• Average Treatment Effect on the Treated (ATT): Causal effect for treated units only.

---