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Fuzzy Regression Discontinuity Design

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Instrumental Variables Regression Discontinuity Design +1 more→

Core Idea

In fuzzy RDD, the probability of treatment jumps discontinuously at the threshold c*, but not from 0 to 1. The running variable serves as an instrument for treatment. The estimand is the LATE (Local Average Treatment Effect) for units near the cutoff whose treatment status is affected by the discontinuity.

Explainer

From your study of sharp RDD, you know the basic idea: if assignment to treatment is determined by whether a running variable crosses a threshold, units just below and just above the cutoff are nearly identical, making the discontinuity a natural experiment. The clean assumption in sharp RDD is that every unit above the threshold receives treatment and every unit below does not — the assignment rule is perfectly enforced. But in many real applications, the threshold only *changes the probability* of treatment — some units above it don't receive treatment, and some below it do. This is the fuzzy RDD setting.

The classic example is a scholarship program that automatically sends an eligibility letter to students scoring above a test cutoff. Most students who receive the letter take up the scholarship, but some don't bother, and a few below the cutoff receive it through discretionary decisions by administrators. The running variable (test score) no longer perfectly determines treatment — it only shifts the probability. Near the cutoff, you observe a jump in the fraction treated, but not from 0 to 1. Graphically, if you plot treatment takeup against the running variable, you see a discontinuous *jump* at the threshold, but the treatment probability remains strictly between 0 and 1 on both sides.

Here is where your knowledge of instrumental variables becomes essential. Being just above versus just below the cutoff serves as an instrument for actual treatment receipt. Think through the IV conditions: (1) Relevance: crossing the threshold increases the probability of treatment — this is the first stage, and it is directly visible as the jump in treatment rate. (2) Exclusion: being just above the cutoff only affects outcomes through its effect on treatment take-up, not through any direct channel. The local nature of RDD — comparing only units very close to the cutoff — makes this exclusion assumption far more credible than in a typical IV setup, because units just above and just below are nearly identical in all other respects.

The fuzzy RDD estimand is the Local Average Treatment Effect: the effect of treatment for "compliers" near the cutoff — those whose treatment status would change depending on which side of the threshold they fall. This is the same LATE you encountered in IV: compliers are the units who take treatment when nudged by the instrument but would not otherwise. The estimator is the ratio of the reduced-form discontinuity (the jump in outcomes at the threshold) to the first-stage discontinuity (the jump in treatment probability). This is exactly the IV ratio estimator, implemented locally. When the first-stage jump is 1 — when everyone above takes up and no one below does — fuzzy and sharp RDD coincide, and the LATE equals the ATE for compliers at the cutoff.

Practice Questions 5 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → Counting to 20 → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Number Bonds to 10 → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Indefinite Integrals → Basic Integration Rules → Riemann Sums → Definite Integral Definition → Probability Density Functions and Continuous Distributions → Cumulative Distribution Functions → Continuous Random Variables → Probability Density Functions → Expected Value → Weak Law of Large Numbers → Probability Axioms and Rules → Conditional Probability → Independence of Events → Sampling Distributions → Standard Error of Estimators → Hypothesis Testing: Framework and Logic → P-values and Statistical Significance → Effect Size and Practical Significance → Hypothesis Testing: Framework and Logic → Z-Tests and T-Tests for Means → One-Sample Z-Test for Means → One-Sample and Two-Sample T-Tests → Inference in Linear Regression → Prediction Intervals in Regression → Linear Regression Basics → Residuals and Goodness of Fit (R²) → Simple (Bivariate) OLS Regression → Classical OLS Assumptions (Gauss-Markov) → Multiple Regression → Interpreting Regression Coefficients → Hypothesis Testing in Regression → F-Test and Joint Significance → R-Squared and Model Fit → Omitted Variable Bias → Causal Inference and the Identification Problem → Potential Outcomes and the Rubin Causal Model → Regression Discontinuity Design → Sharp Regression Discontinuity Design → Fuzzy Regression Discontinuity Design

Longest path: 120 steps · 593 total prerequisite topics

Prerequisites (3)

Regression Discontinuity Designhard Instrumental Variableshard Sharp Regression Discontinuity Designhard

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