For a definition of data centricity that we use here, see [datacentricity.org](https://datacentricity.org/). If staying true to the data is making the correct assumptions about the data, then the [[Design Pattern II - Regression Discontinuity (RD)|RD pattern]] makes its central assumption about a single quantity: the value of the running variable $X$ at the cutoff $c$. Everything that gives RD its causal interpretation flows from one piece of structure: at the cutoff, treatment changes discontinuously while potential outcomes are assumed to evolve smoothly. As we covered in [[Local identification by a cutoff]], that minimalism is also the source of the design's fragilities. For a unit $i$ with running variable $X_i$, treatment $D_i$, and potential outcomes $Y_i(0), Y_i(1)$, define the cutoff $c$ such that $D_i = \mathbb{1}[X_i \ge c] \quad \text{(sharp RD)}$ The observed outcome is $Y_i = D_i Y_i(1) + (1 - D_i) Y_i(0)$, and the [[Local Average Treatment Effect|treatment effect at the cutoff]] is $\tau_{\text{RD}} = \mathbb{E}[Y_i(1) - Y_i(0) \mid X_i = c]$ Identifying $\tau_{\text{RD}}$ from observed data requires the following assumptions to hold: /1. **Continuity of potential outcomes at the cutoff** requires that the conditional expectations of both potential outcomes are continuous functions of $X$ at $c$: $\lim_{x \uparrow c} \mathbb{E}[Y_i(d) \mid X_i = x] = \lim_{x \downarrow c} \mathbb{E}[Y_i(d) \mid X_i = x] \quad \text{for } d \in \{0, 1\}$ This is the central identifying assumption of RD. It cannot be tested empirically because we never observe $Y_i(0)$ for units with $X_i \ge c$ or $Y_i(1)$ for units with $X_i < c$. Indirect evidence comes from balance checks on observed pre-treatment covariates, which should *also* be continuous at $c$ if the assumption holds. /2. **No precise manipulation of the running variable** has a key diagnostic implication: the density of $X$ should be continuous at the cutoff: $f_X(x) \text{ is continuous at } x = c$ Unlike continuity of potential outcomes, evidence against this assumption can be probed via the [[McCrary density test|McCrary density test]]. A statistically significant discontinuity in the density at $c$ suggests that units sorted themselves into or out of treatment by manipulating $X$, which would violate the as-good-as-random-near-the-cutoff logic. /3. **No compound treatment at the cutoff** requires that no other intervention $W$ also changes discontinuously at $c$: $\lim_{x \uparrow c} \mathbb{E}[W_i \mid X_i = x] = \lim_{x \downarrow c} \mathbb{E}[W_i \mid X_i = x]$ This assumption cannot be ruled out empirically unless the relevant co-moving variables are observed. In addition, the **sharp/fuzzy distinction** is part of the design specification, not an assumption to be tested *per se*. A *fuzzy* RD relaxes the deterministic relationship between $X$ and $D$ to a discontinuity in the probability of treatment: $\lim_{x \uparrow c} \Pr(D_i = 1 \mid X_i = x) \neq \lim_{x \downarrow c} \Pr(D_i = 1 \mid X_i = x)$ Identification under fuzzy RD requires the additional assumption of **monotonicity** — crossing the cutoff cannot reduce anyone's probability of treatment (no defiers) — and the resulting estimand is a local Wald ratio mathematically equivalent to the [[Design Pattern I - Instrumental Variable (IV)|IV]] estimator with the cutoff indicator as the instrument. What does it mean to violate these assumptions? Violating continuity is a complete failure of causal identification: there is no credible causal effect to recover from the observed jump. Violating the no-manipulation assumption is the same: sorting unwinds the local randomness that the design relies on. A compound-treatment violation does not destroy identification but changes the interpretation: $\tau_{\text{RD}}$ becomes the bundled effect of the treatment rather than the focal one. None of these violations are remedied by changing the estimator; they are remedied only by changing the design or the question. > [!info]- Last updated: May 14, 2026