--- name: estimator-rd description: Use when estimating causal effects using Regression Discontinuity designs. Triggers on RD, regression discontinuity, sharp RD, fuzzy RD, cutoff, threshold, running variable, bandwidth, discontinuity. --- # Estimator: Regression Discontinuity (RD) > **Version**: 2.0.0 | **Type**: Estimator | **Structure**: K-Dense > **Aliases**: RD, RDD, Regression Discontinuity Design, Sharp RD, Fuzzy RD ## Overview Regression Discontinuity (RD) estimates causal effects by exploiting threshold-based treatment assignment rules. When treatment is determined by whether a running variable crosses a known cutoff, we can estimate the causal effect by comparing outcomes just above and just below the threshold. **Key Identification Assumption**: Potential outcomes are continuous functions of the running variable at the cutoff (no other discontinuity exists at the threshold). ## Quick Navigation ### Reference Documents | Document | Description | |----------|-------------| | [references\/identification_assumptions.md](references\/identification_assumptions.md) | Continuity, no manipulation, local randomization assumptions | | [references\/diagnostic_tests.md](references\/diagnostic_tests.md) | McCrary density test, covariate smoothness, placebo cutoffs | | [references\/estimation_methods.md](references\/estimation_methods.md) | Local polynomial, bandwidth selection, rdrobust | | [references\/reporting_standards.md](references\/reporting_standards.md) | RD plots, tables, report templates | | [references\/common_errors.md](references\/common_errors.md) | Bandwidth selection, polynomial order, manipulation errors | | [references\/extensions.md](references\/extensions.md) | Fuzzy RD, kink RD, geographic RD, multi-cutoff | ### CLI Scripts | Script | Usage | |--------|-------| | [scripts\/run_rd_analysis.py](scripts\/run_rd_analysis.py) | `python run_rd_analysis.py data.csv --running score --outcome y --cutoff 0` | | [scripts\/test_manipulation.py](scripts\/test_manipulation.py) | `python test_manipulation.py data.csv --running score --cutoff 0` | | [scripts\/visualize_rd_plot.py](scripts\/visualize_rd_plot.py) | `python visualize_rd_plot.py data.csv --running score --outcome y --cutoff 0` | | [scripts\/bandwidth_sensitivity.py](scripts\/bandwidth_sensitivity.py) | `python bandwidth_sensitivity.py data.csv --running score --outcome y --cutoff 0` | ### Templates | Template | Description | |----------|-------------| | [assets\/latex\/rd_table.tex](assets\/latex\/rd_table.tex) | LaTeX templates for RD tables | | [assets\/markdown\/rd_report.md](assets\/markdown\/rd_report.md) | Markdown report template | ## When to Use ### Ideal Scenarios - Treatment assigned based on a test score cutoff (scholarships, program eligibility) - Age-based eligibility thresholds (voting age, drinking age, retirement) - Geographic boundaries (electoral districts, school zones) - Income thresholds for program eligibility - Time-based cutoffs (policy implementation dates) ### Data Requirements - [ ] Running variable (score) that determines treatment assignment - [ ] Known cutoff value - [ ] Sufficient observations near the cutoff on both sides - [ ] Treatment indicator (for fuzzy RD: actual treatment received) - [ ] Outcome variable measured after treatment assignment ### When NOT to Use - Agents can precisely manipulate their running variable -> Evidence of sorting violates RD - Multiple discontinuities at the same cutoff -> Cannot isolate treatment effect - Running variable is discrete with few values -> Insufficient variation near cutoff - Cutoff is not binding -> Consider `estimator-iv` instead - Effects far from cutoff needed -> RD provides only local estimates (LATE) ## Identification Assumptions | Assumption | Description | Testable? | |------------|-------------|-----------| | **Continuity** | Potential outcomes E[Y(0)|X=c] and E[Y(1)|X=c] are continuous at cutoff | No (fundamentally untestable) | | **No Manipulation** | Agents cannot precisely sort around the cutoff | Yes (McCrary density test) | | **Local Randomization** | Near the cutoff, treatment is "as good as random" | Partially (covariate balance) | | **LATE Interpretation** | Effect is local to units at the cutoff | N\/A (by design) | --- ## Sharp vs Fuzzy RD ### Sharp RD Treatment is a **deterministic** function of the running variable: $$ D_i = \mathbf{1}(X_i \geq c) $$ Everyone above the cutoff is treated; everyone below is not treated. The estimand is: $$ \tau_{SRD} = \lim_{x \downarrow c} E[Y|X=x] - \lim_{x \uparrow c} E[Y|X=x] $$ ### Fuzzy RD Treatment **probability** changes discontinuously at the cutoff, but compliance is imperfect: $$ \lim_{x \downarrow c} P(D=1|X=x) \neq \lim_{x \uparrow c} P(D=1|X=x) $$ The estimand is a ratio (similar to IV): $$ \tau_{FRD} = \frac{\lim_{x \downarrow c} E[Y|X=x] - \lim_{x \uparrow c} E[Y|X=x]}{\lim_{x \downarrow c} E[D|X=x] - \lim_{x \uparrow c} E[D|X=x]} $$ This is the Local Average Treatment Effect (LATE) for compliers at the cutoff. --- ## Workflow ``` +-------------------------------------------------------------+ | RD ESTIMATOR WORKFLOW | +-------------------------------------------------------------+ | 1. SETUP -> Define running variable, cutoff, Y, D | | 2. PRE-ESTIMATION -> McCrary test, covariate balance | | 3. ESTIMATION -> Sharp\/Fuzzy RD, bandwidth selection | | 4. DIAGNOSTICS -> Placebo cutoffs, sensitivity, donut | | 5. REPORTING -> RD plot, tables, interpretation | +-------------------------------------------------------------+ ``` ### Quick Start with CLI For a complete analysis pipeline, use the CLI scripts: ```bash # Full RD analysis with all diagnostics python scripts\/run_rd_analysis.py data.csv \ --running score --outcome y --cutoff 0 \ --covariates age gender income \ --output-dir results\/ --verbose # Manipulation testing python scripts\/test_manipulation.py data.csv \ --running score --cutoff 0 --plot # Generate RD plot python scripts\/visualize_rd_plot.py data.csv \ --running score --outcome y --cutoff 0 \ --style publication --output rd_plot --format png # Bandwidth sensitivity analysis python scripts\/bandwidth_sensitivity.py data.csv \ --running score --outcome y --cutoff 0 \ --plot --format markdown ``` See [references\/diagnostic_tests.md](references\/diagnostic_tests.md) for detailed test descriptions and [references\/reporting_standards.md](references\/reporting_standards.md) for output formatting. ### Phase 1: Setup **Objective**: Define the RD design and validate data structure **Inputs Required**: ```python # Standard RD input structure running = "score" # Running variable name cutoff = 0.0 # Cutoff value outcome = "y" # Outcome variable name treatment = "treated" # Treatment indicator (for fuzzy RD) covariates = ["x1", "x2"] # Optional covariates for balance tests ``` **Data Validation Checklist**: - [ ] Running variable is continuous (or has many discrete values) - [ ] Cutoff value is known and well-defined - [ ] Sufficient observations within bandwidth of cutoff (both sides) - [ ] No missing values in running variable or outcome - [ ] For fuzzy RD: treatment indicator available **Data Structure Verification**: ```python from rd_estimator import validate_rd_data validation = validate_rd_data( data=df, running="score", outcome="y", cutoff=0.0, treatment="treated" # Optional for sharp RD ) print(validation.summary()) ``` ### Phase 2: Pre-Estimation Checks **McCrary Density Test: Manipulation Check** ```python from rd_estimator import mccrary_test # Test for bunching\/sorting at the cutoff density_test = mccrary_test( running=df["score"], cutoff=0.0, bandwidth=None # Auto-select ) print(density_test) ``` **Interpretation**: - PASS if: p-value > 0.05 (no significant density discontinuity) - WARNING if: 0.01 < p-value < 0.05 (marginal evidence of sorting) - FAIL if: p-value < 0.01 (strong evidence of manipulation - RD may be invalid) --- **Covariate Balance at Cutoff** ```python from rd_estimator import covariate_balance_rd # Test if covariates are smooth at cutoff balance = covariate_balance_rd( data=df, running="score", cutoff=0.0, covariates=["age", "gender", "income"], bandwidth=0.5 ) print(balance.summary_table) ``` **Interpretation**: - Covariates should NOT show discontinuities at the cutoff - Significant jumps in covariates suggest confounding or sorting - This supports the "local randomization" interpretation --- ### Phase 3: Main Estimation **Local Polynomial Regression** The standard RD estimator uses local polynomial regression: $$ \min_{\alpha, \beta} \sum_{i} K\left(\frac{X_i - c}{h}\right) \cdot \left(Y_i - \alpha - \beta(X_i - c) - ... \right)^2 $$ Where: - $K(\cdot)$: Kernel function (typically triangular) - $h$: Bandwidth - Separate regressions above and below cutoff **Kernel Functions**: - **Triangular** (default): $K(u) = (1 - |u|) \cdot \mathbf{1}(|u| \leq 1)$ - **Epanechnikov**: $K(u) = \frac{3}{4}(1 - u^2) \cdot \mathbf{1}(|u| \leq 1)$ - **Uniform**: $K(u) = \frac{1}{2} \cdot \mathbf{1}(|u| \leq 1)$ **Polynomial Order**: - **Linear (p=1)**: Default, most robust - **Quadratic (p=2)**: More flexible, useful if relationship is curved - Higher orders risk overfitting **Bandwidth Selection**: - **MSE-Optimal**: Minimizes mean squared error of the estimator - **CER-Optimal**: Coverage Error Rate optimal for confidence intervals ```python from rd_estimator import ( estimate_sharp_rd, estimate_fuzzy_rd, select_bandwidth ) # Step 1: Select bandwidth bw = select_bandwidth( running=df["score"], outcome=df["y"], cutoff=0.0, method="mserd" # or "cerrd" for CI-optimal ) print(f"Selected bandwidth: {bw:.4f}") # Step 2a: Sharp RD estimation result_sharp = estimate_sharp_rd( data=df, running="score", outcome="y", cutoff=0.0, bandwidth=bw, kernel="triangular", order=1 # Linear ) print(result_sharp.summary_table) # Step 2b: Fuzzy RD estimation (if treatment is imperfect) result_fuzzy = estimate_fuzzy_rd( data=df, running="score", outcome="y", treatment="treated", # Actual treatment received cutoff=0.0, bandwidth=bw, kernel="triangular", order=1 ) print(result_fuzzy.summary_table) ``` **Returns**: ```python CausalOutput( effect=0.52, # Point estimate at cutoff se=0.15, # Robust standard error ci_lower=0.23, # Robust CI lower (bias-corrected) ci_upper=0.81, # Robust CI upper p_value=0.0006, diagnostics={ 'bandwidth': 0.35, 'n_left': 245, # Observations below cutoff 'n_right': 267, # Observations above cutoff 'n_effective': 512,# Effective sample size 'kernel': 'triangular', 'order': 1, 'fuzzy': False }, summary_table="...", interpretation="..." ) ``` ### Phase 4: Robustness Checks | Check | Purpose | Implementation | |-------|---------|----------------| | Placebo Cutoffs | Verify no effect at fake cutoffs | `placebo_cutoff_test()` | | Bandwidth Sensitivity | Check stability across bandwidths | `bandwidth_sensitivity()` | | Donut Hole | Exclude observations right at cutoff | `donut_hole_rd()` | | Polynomial Order | Compare linear vs quadratic | Re-run with `order=2` | | Alternative Kernels | Check kernel sensitivity | Re-run with different kernel | **Placebo Cutoff Test**: ```python from rd_estimator import placebo_cutoff_test # Test at fake cutoffs where no effect should exist placebo_results = placebo_cutoff_test( data=df, running="score", outcome="y", true_cutoff=0.0, placebo_cutoffs=[-0.5, -0.25, 0.25, 0.5], bandwidth=bw ) for cutoff, result in placebo_results.items(): status = "PASS" if result.passed else "FAIL" print(f"Cutoff {cutoff}: {status} (effect={result.effect:.3f}, p={result.p_value:.3f})") ``` **Bandwidth Sensitivity**: ```python from rd_estimator import bandwidth_sensitivity sensitivity = bandwidth_sensitivity( data=df, running="score", outcome="y", cutoff=0.0, bandwidth_range=[0.5*bw, 0.75*bw, bw, 1.25*bw, 1.5*bw, 2*bw] ) # Results should be stable across reasonable bandwidth choices print(sensitivity.summary_table) ``` **Donut Hole RD**: ```python from rd_estimator import donut_hole_rd # Exclude observations very close to cutoff (address manipulation concerns) donut_result = donut_hole_rd( data=df, running="score", outcome="y", cutoff=0.0, bandwidth=bw, donut_radius=0.05 # Exclude |X - c| < 0.05 ) print(f"Donut RD effect: {donut_result.effect:.4f} (SE: {donut_result.se:.4f})") ``` ### Phase 5: Reporting **RD Plot (Essential Visualization)**: ```python from rd_estimator import rd_plot fig = rd_plot( data=df, running="score", outcome="y", cutoff=0.0, bandwidth=bw, n_bins=20, # Number of bins for scatter poly_order=1, # Polynomial order for fitted lines ci=True # Show confidence intervals ) fig.savefig("rd_plot.png", dpi=150) ``` **Standard Output Table Format**: ``` +----------------------------------------------------------+ | Table X: Regression Discontinuity Estimates | +----------------------------------------------------------+ | (1) (2) (3) | | Linear Quadratic Half-BW | +----------------------------------------------------------+ | RD Effect 0.52*** 0.48*** 0.55*** | | (0.15) (0.18) (0.22) | | | | Bandwidth 0.35 0.35 0.175 | | Kernel Triangular Triangular Triangular | | Polynomial Order 1 2 1 | | | | N (left of cutoff) 245 245 118 | | N (right of cutoff) 267 267 132 | | Effective N 512 512 250 | +----------------------------------------------------------+ | McCrary p-value 0.672 | | Placebo tests Pass (4\/4) | +----------------------------------------------------------+ | Notes: Robust bias-corrected standard errors. | | *** p<0.01, ** p<0.05, * p<0.1 | +----------------------------------------------------------+ ``` **Interpretation Template**: ```markdown ## Results Interpretation Using a regression discontinuity design, we estimate that crossing the [threshold description] leads to a [increase\/decrease] of **[effect]** (SE = [se]) in [outcome description]. ### Design Validity 1. **Manipulation Test**: McCrary density test shows [no evidence \/ some evidence] of sorting at the cutoff (p = [p-value]). 2. **Covariate Balance**: Covariates [are \/ are not] smooth at the cutoff, supporting the local randomization assumption. 3. **Placebo Tests**: [X\/Y] placebo cutoffs show no significant effects, as expected under a valid RD design. ### Robustness - The effect is [stable \/ sensitive] to bandwidth choice ([range] effect range). - Results are [similar \/ different] with quadratic specification. - Donut hole analysis [confirms \/ challenges] the main result. ### Interpretation The RD estimate represents a **Local Average Treatment Effect (LATE)** for units at the margin of the cutoff. Extrapolation to units far from the cutoff should be done cautiously. ### Caveats - [Any concerns about manipulation or sorting] - [Bandwidth sensitivity issues] - [External validity limitations] ``` --- ## Common Mistakes ### 1. Using Global Polynomial Regression **Mistake**: Fitting high-order polynomials across the entire support. **Why it's wrong**: Global polynomials can produce biased estimates and misleading confidence intervals. The polynomial can bend unnaturally to fit observations far from the cutoff, distorting the estimate at the cutoff. **Correct approach**: ```python # WRONG: Global polynomial model = OLS(y ~ running + running^2 + running^3 + treatment) # CORRECT: Local polynomial within bandwidth result = estimate_sharp_rd( data=df, running="score", outcome="y", cutoff=0.0, bandwidth=bw, # Use only observations near cutoff order=1 # Linear is usually sufficient ) ``` ### 2. Ignoring Bandwidth Selection **Mistake**: Choosing bandwidth arbitrarily or using the full sample. **Why it's wrong**: Bandwidth affects bias-variance tradeoff. Too large leads to bias; too small leads to high variance. **Correct approach**: ```python # WRONG: Arbitrary bandwidth result = estimate_sharp_rd(data, running, outcome, cutoff, bandwidth=1.0) # CORRECT: Data-driven bandwidth selection bw = select_bandwidth( running=df["score"], outcome=df["y"], cutoff=0.0, method="mserd" # MSE-optimal ) result = estimate_sharp_rd(data, running, outcome, cutoff, bandwidth=bw) ``` ### 3. Treating Fuzzy RD as Sharp RD **Mistake**: Ignoring imperfect compliance when some units don't follow the assignment rule. **Why it's wrong**: If not everyone above the cutoff is treated (or some below are), sharp RD gives an intent-to-treat effect, not the treatment effect. **Correct approach**: ```python # Check for imperfect compliance compliance_above = df[df["score"] >= cutoff]["treated"].mean() compliance_below = df[df["score"] < cutoff]["treated"].mean() if compliance_above < 0.99 or compliance_below > 0.01: print("WARNING: Imperfect compliance detected. Use Fuzzy RD.") result = estimate_fuzzy_rd( data=df, running="score", outcome="y", treatment="treated", # Actual treatment cutoff=0.0, bandwidth=bw ) ``` ### 4. Ignoring the McCrary Test **Mistake**: Not testing for manipulation before estimating. **Why it's wrong**: If agents sort around the cutoff, the continuity assumption is violated and RD estimates are biased. **Correct approach**: ```python # ALWAYS run McCrary test first density_result = mccrary_test(df["score"], cutoff=0.0) if not density_result.passed: print("WARNING: Evidence of manipulation at cutoff!") print("Consider:") print("1. Donut hole RD (exclude observations near cutoff)") print("2. Alternative identification strategies") print("3. Reporting intent-to-treat with caveats") ``` --- ## Examples ### Example 1: Test Score Cutoff for Scholarship **Research Question**: What is the effect of receiving a merit scholarship on college GPA? **Setting**: Students with SAT scores >= 1200 receive a scholarship. ```python import pandas as pd from rd_estimator import ( mccrary_test, select_bandwidth, estimate_sharp_rd, rd_plot, run_full_rd_analysis ) # Load data data = pd.read_csv("scholarship_data.csv") # Columns: student_id, sat_score, scholarship, college_gpa, high_school_gpa # Define RD setup cutoff = 1200 running_var = "sat_score" outcome_var = "college_gpa" # Quick check: is this sharp or fuzzy? above = data[data[running_var] >= cutoff]["scholarship"].mean() below = data[data[running_var] < cutoff]["scholarship"].mean() print(f"Scholarship rate above cutoff: {above:.2%}") print(f"Scholarship rate below cutoff: {below:.2%}") # If above ~100% and below ~0%, use sharp RD # Run full analysis result = run_full_rd_analysis( data=data, running=running_var, outcome=outcome_var, cutoff=cutoff, treatment="scholarship" # For fuzzy RD, or None for sharp ) # View results print(result.summary_table) # Create RD plot fig = rd_plot( data=data, running=running_var, outcome=outcome_var, cutoff=cutoff, bandwidth=result.diagnostics['bandwidth'] ) fig.savefig("scholarship_rd_plot.png") ``` **Output**: ``` ============================================================ REGRESSION DISCONTINUITY ANALYSIS RESULTS ============================================================ RD Effect (LATE): 0.35 Standard Error: 0.12 95% CI: [0.12, 0.58] P-value: 0.003 ------------------------------------------------------------ DIAGNOSTICS ------------------------------------------------------------ McCrary Density Test: PASSED - Test statistic: 0.42 - P-value: 0.67 Covariate Balance: - high_school_gpa: Balanced (p=0.45) - age: Balanced (p=0.72) ------------------------------------------------------------ BANDWIDTH AND SAMPLE ------------------------------------------------------------ Optimal Bandwidth: 85 SAT points N (below cutoff): 312 N (above cutoff): 298 Effective N: 610 ============================================================ ``` **Interpretation**: Students who just qualify for the merit scholarship (SAT >= 1200) have college GPAs approximately 0.35 points higher than students who just miss the cutoff. This effect is statistically significant at the 1% level. The McCrary test shows no evidence of manipulation, and pre-treatment covariates are balanced, supporting the validity of the RD design. ### Example 2: Electoral RD (Lee, 2008 Style) **Research Question**: Does winning a close election affect party vote share in the next election? ```python import pandas as pd from rd_estimator import run_full_rd_analysis, bandwidth_sensitivity # Load election data data = pd.read_csv("house_elections.csv") # Columns: district, year, dem_vote_share_t, dem_vote_share_t1, incumbent_party # Running variable: Democratic vote margin (centered at 50%) data["dem_margin"] = data["dem_vote_share_t"] - 0.5 # Outcome: Vote share in next election result = run_full_rd_analysis( data=data, running="dem_margin", outcome="dem_vote_share_t1", cutoff=0.0, treatment=None # Sharp RD: winning is deterministic ) print(result.summary_table) # Bandwidth sensitivity analysis sens = bandwidth_sensitivity( data=data, running="dem_margin", outcome="dem_vote_share_t1", cutoff=0.0, bandwidth_range=[0.025, 0.05, 0.075, 0.10, 0.15, 0.20] ) print("\nBandwidth Sensitivity:") print(sens.summary_table) ``` **Output**: ``` RD Effect: 0.078*** (0.021) 95% CI: [0.037, 0.119] This represents a 7.8 percentage point incumbency advantage. ``` --- ## References ### Skill Documentation For detailed coverage of specific topics, see: - **Identification**: [references\/identification_assumptions.md](references\/identification_assumptions.md) - **Diagnostics**: [references\/diagnostic_tests.md](references\/diagnostic_tests.md) - **Estimation**: [references\/estimation_methods.md](references\/estimation_methods.md) - **Reporting**: [references\/reporting_standards.md](references\/reporting_standards.md) - **Errors**: [references\/common_errors.md](references\/common_errors.md) - **Extensions**: [references\/extensions.md](references\/extensions.md) ### Seminal Papers - Thistlethwaite, D. L., & Campbell, D. T. (1960). Regression-discontinuity analysis: An alternative to the ex post facto experiment. *Journal of Educational Psychology*, 51(6), 309-317. - Lee, D. S. (2008). Randomized experiments from non-random selection in US House elections. *Journal of Econometrics*, 142(2), 675-697. - Imbens, G. W., & Lemieux, T. (2008). Regression discontinuity designs: A guide to practice. *Journal of Econometrics*, 142(2), 615-635. ### Methodological Advances - Calonico, S., Cattaneo, M. D., & Titiunik, R. (2014). Robust nonparametric confidence intervals for regression-discontinuity designs. *Econometrica*, 82(6), 2295-2326. - Cattaneo, M. D., Idrobo, N., & Titiunik, R. (2020). *A Practical Introduction to Regression Discontinuity Designs*. Cambridge University Press. - McCrary, J. (2008). Manipulation of the running variable in the regression discontinuity design: A density test. *Journal of Econometrics*, 142(2), 698-714. - Gelman, A., & Imbens, G. (2019). Why high-order polynomials should not be used in regression discontinuity designs. *Journal of Business & Economic Statistics*, 37(3), 447-456. ### Textbook Treatments - Angrist, J. D., & Pischke, J. S. (2009). *Mostly Harmless Econometrics*. Princeton University Press. Chapter 6. - Cunningham, S. (2021). *Causal Inference: The Mixtape*. Yale University Press. Chapter 6. - Huntington-Klein, N. (2022). *The Effect*. CRC Press. Chapter 20. ### Software Documentation - `rdrobust` (R\/Stata): https:\/\/rdpackages.github.io\/rdrobust\/ - `rddensity` (R\/Stata): https:\/\/rdpackages.github.io\/rddensity\/ - `rdmulti` (R\/Stata): https:\/\/rdpackages.github.io\/rdmulti\/ --- ## Related Estimators | Estimator | When to Use Instead | |-----------|---------------------| | `estimator-did` | Treatment varies over time, not by threshold | | `estimator-iv` | Fuzzy RD with very weak first stage | | `estimator-rdid` | Combining RD with DID (repeated cross-sections at cutoff) | | `estimator-bunching` | Running variable is continuous but bunching analysis preferred | | `estimator-geographic-rd` | Cutoff is spatial (borders, boundaries) | --- ## Appendix: Mathematical Details ### Local Linear Regression Estimator The sharp RD estimator using local linear regression minimizes: $$ \min_{\alpha_-, \beta_-} \sum_{i: X_i < c} K_h(X_i - c) \cdot (Y_i - \alpha_- - \beta_-(X_i - c))^2 $$ $$ \min_{\alpha_+, \beta_+} \sum_{i: X_i \geq c} K_h(X_i - c) \cdot (Y_i - \alpha_+ - \beta_+(X_i - c))^2 $$ The RD estimate is: $\hat{\tau} = \hat{\alpha}_+ - \hat{\alpha}_-$ ### Bandwidth Selection **MSE-Optimal Bandwidth** (Imbens-Kalyanaraman \/ Calonico-Cattaneo-Titiunik): $$ h_{MSE} = C_n \cdot \left( \frac{\sigma^2(c)}{\mu_2^{(2)}(c)^2 \cdot n} \right)^{1\/5} $$ Where: - $\sigma^2(c)$: Variance of outcome at cutoff - $\mu_2^{(2)}(c)$: Second derivative of conditional mean at cutoff - $C_n$: Constant depending on kernel **CER-Optimal Bandwidth** scales MSE-optimal bandwidth for coverage: $$ h_{CER} = h_{MSE} \cdot n^{-1\/20} $$ ### Fuzzy RD as Local Wald Estimator The fuzzy RD estimand can be written as: $$ \tau_{FRD} = \frac{E[Y|X=c^+] - E[Y|X=c^-]}{E[D|X=c^+] - E[D|X=c^-]} $$ This is estimated using local polynomial regression for both numerator (reduced form) and denominator (first stage), then taking the ratio. ### Bias-Corrected Robust Inference Following Calonico, Cattaneo, and Titiunik (2014), robust confidence intervals use: $$ CI = \left[ \hat{\tau} - q_{1-\alpha\/2} \cdot \hat{V}^{1\/2}, \hat{\tau} + q_{1-\alpha\/2} \cdot \hat{V}^{1\/2} \right] $$ Where $\hat{V}$ accounts for bias from using a smaller bandwidth than optimal.