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