所属模块：
M1 · 数学与统计基础 (Mathematics & Statistics Fundamentals)| 专题分类：因果推断 (Causal Inference)| 难度等级：Medium
一、核心一句话结论 (One-Sentence Summary)
利用某个连续变量在阈值处的’跳跃’识别因果；关键假设是阈值附近个体除处理外无系统性差异。
Regression Discontinuity Design (RDD) exploits exogenous threshold cutoffs on continuous running variables (e.g. Credit score $ge 700$ gets loan), comparing units immediately above and below the threshold as a quasi-randomized trial.
二、核心考点要义 (Key Insights)
- 📌 阈值 c 处处理状态发生跳变
- 📌 只识别阈值附近的局部效应（LATE at cutoff）
- 📌 需检验协变量在阈值处是否连续
English Insights:
– Sharp RDD: Treatment assignment is a deterministic step function of running variable $X$: $T_i = mathbb{I}(X_i ge c)$.
– Fuzzy RDD: Cutoff increases the probability of treatment discontinuously: $P(T_i=1mid X_i)$ jumps at $c$ (analyzed via Instrumental Variables).
– Core Continuity Assumption: Potential outcomes $E[Y(0)mid X]$ and $E[Y(1)mid X]$ must be smooth and continuous at cutoff $c$; units cannot precisely manipulate their running variable around the threshold.
三、核心数学原理与机理推导 (Mathematical Principles & Derivation)
$$tau=lim_{xdownarrow c}mathbb E[Ymid X=x]-lim_{xuparrow c}mathbb E[Ymid X=x]$$
RDD 的核心思想：当处理由某连续变量 X 是否越过阈值 c 决定时（如’考试分数 ≥60 分才录取’、’年龄 ≥65 岁才领养老金’），比较阈值两侧个体的结果差异即可识别因果效应——因为在阈值附近，个体除’是否越过阈值’外几乎相同（近似随机化）。统计量：τ=lim_{x↓c}E[Y|X=x]−lim_{x↑c}E[Y|X=x]，即阈值处的跳跃幅度。关键假设：① 连续性——若无处理，E[Y|X] 在阈值处连续（即除处理外没有其他因素在阈值处跳变）；② 无操纵（no manipulation）——个体不能精确控制 X 使其恰好落在阈值一侧（如不能精确改分数）；③ 阈值附近的个体是可比的。识别的是局部效应——只对’阈值附近的依从者’有效（LATE at cutoff），不能外推到远离阈值的个体（如不能从’60 分录取’推断’90 分录取’的效果）。
📖 查看英文严格数学推导 (English Mathematical Derivation)
Identification theorem: The treatment effect at cutoff $c$ is identified as the discontinuous jump in conditional expectation: $tau_{text{RDD}}(c) = lim_{x downarrow c} E[Ymid X=x] – lim_{x uparrow c} E[Ymid X=x]$. Expanding potential outcomes: $lim_{x downarrow c} E[Ymid X=x] = lim_{x downarrow c} E[Y(1)mid X=x] = E[Y(1)mid X=c]$ (by continuity). Similarly, $lim_{x uparrow c} E[Ymid X=x] = lim_{x uparrow c} E[Y(0)mid X=x] = E[Y(0)mid X=c]$. Subtracting gives: $tau_{text{RDD}}(c) = E[Y(1) – Y(0)mid X=c]$, which is the exact causal Local Average Treatment Effect at the threshold. Local linear regression fits separate slopes on each side of $c$ within bandwidth $h$: $min_{alpha, beta, tau, gamma} sum_{i: |X_i – c| le h} left(Y_i – alpha – beta(X_i – c) – tau T_i – gamma(X_i – c)T_iright)^2 Kleft(frac{X_i – c}{h}right)$.
四、工业级落地权衡与工程考量 (Industrial Trade-offs)
实践要点：① 检验假设——(a) 协变量连续性：检验其他协变量（性别、年龄、背景）在阈值处是否连续（若不连续说明存在操纵或混杂）；(b) 操纵检验（McCrary 检验）：检查 X 的密度在阈值处是否有跳跃（个体聚集在阈值一侧说明可能操纵）；(c) 带宽敏感性：用不同带宽（bandwidth）估计，结果应稳健。② 估计方法——用局部线性回归（在阈值两侧分别拟合，取阈值处的截距差），而非全局多项式（高次多项式在边界外推不稳定，Gelman & Imbens 建议避免）；带宽可用 Imbens-Kalyanaraman 或 Calonico 等最优带宽选择器。③ 模糊 RDD（Fuzzy RDD）——若越过阈值只是提高处理概率而非完全决定（如’60 分以上才可申请’），则用阈值作为工具变量（2SLS）估计，此时识别的是依从者的效应——这是 RDD 与 IV 的结合。④ 应用实例——教育（班级规模、奖学金）、公共政策（养老金、医保资格）、平台规则（信用分阈值、推荐位门槛）；实践中电商的’满减门槛’、’免运费门槛’都是天然 RDD 场景。⑤ 局限——只能评估阈值附近的效应、需要大样本（阈值附近的样本量受限）、对带宽选择敏感。
⚙️ 查看英文落地权衡分析 (English Systems & Trade-offs)
Validating RDD assumptions: (1) McCrary Density Test: Checks whether the density of the running variable $f(X)$ exhibits a sudden spike/cliff at cutoff $c$. A spike indicates that users strategically manipulated their score (e.g. Submitting fake receipts to cross a bonus threshold), violating local randomization. (2) Covariate Continuity: Pre-treatment covariates (age, gender, prior activity) must be smooth and continuous across the cutoff.
五、常见面试避坑陷阱 (Common Pitfalls & Traps)
- ⚠️ 用全局高次多项式拟合 RDD（边界不稳定）
- ⚠️ 把阈值附近的局部效应外推到全体
English Pitfalls:
– Extrapolating RDD treatment effects far away from the cutoff (RDD estimates are strictly local to $X=c$).
– Using high-degree global polynomials (e.g. 5th-order polynomial regression), which Gelman & Imbens (2019) proved produces noisy, biased boundary artifacts; local linear or local quadratic regression with triangular kernels must be used.
六、高频深度面试追问与预测 (Follow-Up Questions)
- RDD 与 IV 的关系？
- Why did Gelman & Imbens prove that high-order global polynomials should never be used in RDD?
- 如何检验操纵（manipulation）？
- How does the McCrary Density Test detect manipulation of the running variable?
七、知识图谱对齐 (Knowledge Graph Anchor)
- 🔗 关联底层卡片：
因果推断框架：潜在结果模型、倾向评分匹配与双重差分(Causal Inference: Potential Outcomes, PSM & DiD) - 🗺️ 知识图谱模块：
数据科学与因果实验导图
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