Statistical models of the generalized synthetic control method and difference-in-differences estimates.

<p>i = hospitals. T = time period (T0, before the new health policy (financial incentive) was implemented; T1, after the new health policy was implemented); A = treatment variable (A = 0 if the unit did not receive the policy, A = 1 if the unit received the policy); A*T = the average treatment effect on the treated (interaction between the treatment indicator and the time indicator); C = set of unit-time-varying covariates that affect (and not affected by) the outcome and which represents the set of covariates sufficient for confounding control; μ = time-invariant unobserved confounders; λ = time-varying effects that are assumed to be the same for the treated and control units. X<sub>1i</sub> = intervention at hospitals (0 = no incentive [ineligible], 1 = incentive [eligible]); X<sub>2i</sub> = time indicator (0 = pre-intervention, 1 = post-intervention); <i>X<sub>it</sub></i> = a covariate that can vary across units i and time t; β0 = constant; β1 = treatment group-specific effect; β2 = time trend common to the eligible and ineligible hospitals; β3 = difference-in-differences estimates.</p> <p>(DOCX)</p>