Regression Discontinuity Designs

by Charlie Zhang

Sharp RD (SRD)

Assumptions:

1. \(Y_{i}(0), Y_{i}(1) \perp W_{i} | X_{i}\) (Unconfoundness)
2. Continuity of Conditional Regression Functions (Imben and Lee, 2008) \(E[Y(0)|X]\) and \(E[Y(1)|X]\) are continuous in \(x\) as a result of the violation of the RCM;
3. Continuity of Conditional Distribution Functions

Treatment status is a deterministic and discontinuous function of a covariate (MHE, p.189). It is formally expressed as \(P(T = 1 | x < x^∗) = 1\) and \(P(T = 1 | x ≥ x^∗) = 0\) in the case the forcing variable is x and the threshold is \(x^*\).

Then, the Average causal effect of the treatment at the discontinuity point is: $$\tau_{SRD} = E[Y_{1} − Y_{0} | X_{i} = c] = \lim_{x \downarrow c} E[Y_{i} | X_{i} = x] − \lim_{x \uparrow c} E[Y_{i} | X_{i} = x]$$

Fuzzy RD (FRD)

Assumptions:

Treatment status is a deterministic and discontinuous function of a covariate (MHE, p.189). It is formally expressed as \(P(T = 1 | x < x^∗) = 1\) and \(P(T = 1 | x ≥ x^∗) = 0\) in the case the forcing variable is x and the threshold is \(x^*\).

Then, the Average causal effect of the treatment at the discontinuity point is: $$\tau_{FRD} = \frac{\lim_{x \downarrow c} E[Y|X=x] - \lim_{x \uparrow c} E[Y|X=x]}{\lim_{x \downarrow c} E[W|X=x] - \lim_{x \uparrow c} E[W|X=x]} \\ = E[Y_{i}(1) - Y_{i}(0) | \text{unit i is a complier and } X_{i} = c]$$

Others

The problem of external validity

1. Both SRD and FRD can, at best, provide estimates of the average treatment effect for a subpopulation, namely the subpopulation with covariate value equal to \(X_{i} = c\);
2. RD designs might have a relatively high degree of internal validity based on unconfoundness.