27.8 Specification Checks

To validate the credibility of an RD design, researchers perform several specification checks:

27.8.1 Balance Checks

Also known as checking for discontinuities in average covariates, this test examines whether covariates that should not be affected by treatment exhibit a discontinuity at the cutoff.

Null Hypothesis ( $H_0$ ): The average effect of covariates on pseudo-outcomes (i.e., those that should not be influenced by treatment) is zero.
If rejected, this raises serious doubts about the RD design, necessitating a strong justification.

27.8.2 Sorting, Bunching, and Manipulation

This test, also known as checking for discontinuities in the distribution of the forcing variable, detects whether subjects manipulate the running variable to sort into or out of treatment.

If individuals can manipulate the running variable—especially when the cutoff is known in advance—this can lead to bunching behavior (i.e., clustering just above or below the cutoff).

If treatment is desirable, individuals will try to sort into treatment, leading to a gap just below the cutoff.
If treatment is undesirable, individuals will try to avoid it, leading to a gap just above the cutoff.

Under RD, we assume that there is no manipulation in the running variable. However, bunching behavior, where firms or individuals strategically manipulate their position, violates this assumption.

To address this issue, the bunching approach estimates the counterfactual distribution—what the density of individuals would have been in the absence of manipulation.
The fraction of individuals who engaged in manipulation is then calculated by comparing the observed distribution to this counterfactual distribution.
In a standard RD framework, this step is unnecessary because it assumes that the observed distribution and the counterfactual (manipulation-free) distribution are the same, implying no manipulation

27.8.2.1 McCrary Sorting Test

A widely used formal test is the McCrary density test (McCrary 2008), later refined by (Cattaneo, Idrobo, and Titiunik 2019).

Null Hypothesis ( $H_0$ ): The density of the running variable is continuous at the cutoff.
Alternative Hypothesis ( $H_a$ ): A discontinuity (jump) in the density function at the cutoff, suggesting manipulation.
Interpretation:
- A significant discontinuity suggests manipulation, violating RD assumptions.
- A failure to reject $H_0$ does not necessarily confirm validity, as some forms of manipulation may remain undetected.
- If there is two-sided manipulation, this test will fail to detect it.

27.8.2.2 Guidelines for Assessing Manipulation

J. L. Zhang and Rubin (2003), Lee (2009), and Aronow, Baron, and Pinson (2019) provide criteria for evaluating manipulation risks.
Knowing your research design inside out is crucial to anticipating possible manipulation attempts.
Manipulation is often one-sided, meaning subjects shift only in one direction relative to the cutoff. In rare cases, two-sided manipulation may occur but often cancels out.
We could also observe partial manipulation in reality (e.g., when subjects can only imperfectly manipulate). However, since we typically treat it like a fuzzy RD, we would not encounter identification problems. In contrast, complete manipulation would lead to serious identification issues.

Bunching Methodology

Bunching occurs when individuals self-select into specific values of the running variable (e.g., policy thresholds). See Kleven (2016) for a review.
The method helps estimate the counterfactual distribution—what the density would have been without manipulation.
The fraction of individuals who manipulated can be estimated by comparing observed densities to the counterfactual.

If the running variable and outcome are simultaneously determined, a modified RD estimator can be used for consistent estimation (Bajari et al. 2011):

One-sided manipulation: Individuals shift only in one direction relative to the cutoff (similar to the monotonicity assumption in instrumental variables).
Bounded manipulation (regularity assumption): The density of individuals far from the threshold remains unaffected (Blomquist et al. 2021; Bertanha, McCallum, and Seegert 2021).

27.8.2.3 Steps for Bunching Analysis

Identify the window where bunching occurs (based on Bosch, Dekker, and Strohmaier (2020)). Perform robustness checks by varying the manipulation window.
Estimate the manipulation-free counterfactual distribution.
Standard errors for inference can be calculated (following Chetty, Hendren, and Katz (2016)), where bootstrap resampling of residuals is used in estimating the counts of individuals within bins. However, this step may be unnecessary for large datasets.

If the bunching test fails to detect manipulation, we proceed to the Placebo Test.

McCrary Density Test (Discontinuity in Forcing Variable)

library(rdd)

set.seed(1)

# Simulated data without discontinuity
x <- runif(100, -1, 1)
DCdensity(x, 0)  # No discontinuity

#> [1] 0.06355195

# Simulated data with discontinuity
x <- runif(1000, -1, 1)
x <- x + 2 * (runif(1000, -1, 1) > 0 & x < 0)
DCdensity(x, 0)  # Discontinuity detected

#> [1] 0.001936782

Cattaneo Density Test (Improved Version)

library(rddensity)

# Simulated continuous density
set.seed(1)
x   <- rnorm(100, mean = -0.5)
rdd <- rddensity(X = x, vce = "jackknife")
summary(rdd)
#> 
#> Manipulation testing using local polynomial density estimation.
#> 
#> Number of obs =       100
#> Model =               unrestricted
#> Kernel =              triangular
#> BW method =           estimated
#> VCE method =          jackknife
#> 
#> c = 0                 Left of c           Right of c          
#> Number of obs         66                  34                  
#> Eff. Number of obs    38                  25                  
#> Order est. (p)        2                   2                   
#> Order bias (q)        3                   3                   
#> BW est. (h)           0.922               0.713               
#> 
#> Method                T                   P > |T|             
#> Robust                0.6715              0.5019              
#> 
#> 
#> P-values of binomial tests (H0: p=0.5).
#> 
#> Window Length              <c     >=c    P>|T|
#> 0.563     + 0.563          26      20    0.4614
#> 0.603     + 0.580          27      20    0.3817
#> 0.643     + 0.596          30      20    0.2026
#> 0.683     + 0.613          32      21    0.1690
#> 0.722     + 0.630          32      22    0.2203
#> 0.762     + 0.646          33      22    0.1770
#> 0.802     + 0.663          33      23    0.2288
#> 0.842     + 0.680          35      24    0.1925
#> 0.882     + 0.696          36      24    0.1550
#> 0.922     + 0.713          38      25    0.1299

# Plot requires customization (refer to package documentation)

27.8.3 Placebo Tests

Placebo tests, also known as falsification checks, assess whether discontinuities appear at points other than the treatment cutoff. This helps verify that observed effects are causal rather than artifacts of the method or data.

There should be no jumps in the outcome at values other than the cutoff ( $X_i < c$ or $X_i \geq c$ ).
The test involves shifting the cutoff along the running variable while using the same bandwidth to check for discontinuities in the conditional mean of the outcome.
This approach is similar to balance checks in experimental design, ensuring no pre-existing differences. Remember, we can only test on observables, not unobservables.

Under a valid RD design, matching methods are unnecessary. Just as with randomized experiments, balance should naturally occur across the threshold. If adjustments are required, it suggests the RD assumptions may be invalid.

27.8.3.1 Applications of Placebo Tests

Testing No Discontinuity in Predetermined Covariates: Covariates that should not be affected by treatment should not exhibit a jump at the cutoff.
Testing Other Discontinuities: Checking for discontinuities at other arbitrary points along the running variable.
Using Placebo Outcomes: If an outcome variable that should not be affected by treatment shows a significant discontinuity, this raises concerns about RD validity.
Assessing Sensitivity to Covariates: RD estimates should not be highly sensitive to the inclusion or exclusion of covariates.

27.8.3.2 Mathematical Specification

The balance of observable characteristics on both sides of the threshold can be tested using:

$Z_i = \alpha_0 + \alpha_1 f(X_i) + I(X_i \geq c) \alpha_2 + [f(X_i) \times I(X_i \geq c)]\alpha_3 + u_i$

where:

$X_i$ = running variable
$Z_i$ = predetermined characteristics (e.g., age, education, etc.)
$\alpha_2$ should be zero if $Z_i$ is unaffected by treatment.

If multiple covariates $Z_i$ are tested simultaneously, simulating their joint distribution avoids false positives due to multiple comparisons. This step is unnecessary if covariates are independent, but such independence is unlikely in practice.

27.8.4 Sensitivity to Bandwidth Choice

The choice of bandwidth is crucial in RD estimation. Different bandwidth selection methods exist:

Ad-hoc or Substantively Driven: Based on theoretical or empirical reasoning.
Data-Driven Selection (Cross-Validation): Optimizes bandwidth to minimize prediction error.
Conservative Approach: Uses robust optimal bandwidth selection methods (e.g., (Calonico, Cattaneo, and Farrell 2020)).

The objective is to minimize mean squared error (MSE) between estimated and actual treatment effects.

27.8.5 Assessing Sensitivity

Results should be consistent across reasonable bandwidth choices.
The optimal bandwidth for estimating treatment effects may differ from the optimal bandwidth for testing covariates but should be fairly close.

# Load required package
library(rdd)

# Simulate some data
set.seed(123)
n <- 100  # Sample size
running_var <- runif(n, -1, 1)  # Running variable centered around 0
treatment   <- ifelse(running_var >= 0, 1, 0)  # Treatment assigned at cutpoint 0
outcome_var <- 2 * running_var + treatment * 1.5 + rnorm(n)  # Outcome variable

# Compute the optimal Imbens-Kalyanaraman (IK) bandwidth
bandwidth <-
    IKbandwidth(running_var,
                outcome_var,
                cutpoint = 0,
                kernel = "triangular")

# Print the bandwidth
print(bandwidth)
#> [1] 0.4374142

27.8.6 Manipulation Robust Regression Discontinuity Bounds

McCrary (2008) linked density jumps at cutoffs in RD studies to potential manipulation.
- If no jump is detected, researchers proceed with RD analysis; if detected, they halt using the cutoff for inference.
- Some studies use the “doughnut-hole” method, excluding near-cutoff observations and extrapolating, which contradicts RD principles.
  - False negative could be due to a small sample size and can lead to biased estimates, as units near the cutoff may still differ in unobserved ways.
  - Even correct rejections of no manipulation may overlook that the data can still be informative despite modest manipulation.
  - Gerard, Rokkanen, and Rothe (2020) introduces a systematic approach to handle potentially manipulated variables in RD designs, addressing both concerns.
The model introduces two types of unobservable units in RD designs:
- always-assigned units, which are always on one side of the cutoff,
- potentially-assigned units, which fit traditional RD assumptions.
  - The standard RD model is a subset of this broader model, which assumes no always-assigned units.
Identifying assumption: manipulation occurs through one-sided selection.
The approach does not make a binary decision on manipulation in RD designs but assesses its extent and worst-case impact.

Two steps are used:

Determining the proportion of always-assigned units using the discontinuity at the cutoff
Bounding treatment effects based on the most extreme feasible outcomes for these units.

For sharp RD designs, bounds are established by trimming extreme outcomes near the cutoff; for fuzzy designs, the process involves more complex adjustments due to additional model constraints.
Extensions of the study use covariate information and economic behavior assumptions to refine these bounds and identify covariate distributions among unit types at the cutoff.

Setup

Independent data points $(X_i, Y_i, D_i)$ , where $X_i$ is the running variable, $Y_i$ is the outcome, and $D_i$ indicates treatment status (1 if treated, 0 otherwise). Treatment is assigned based on $X_i \geq c$ .

The design is sharp if $D_i = I(X_i \geq c)$ and fuzzy otherwise.

The population is divided into:

Potentially-assigned units ( $M_i = 0$ ): Follow the standard RD framework, with potential outcomes $Y_i(d)$ and potential treatment states $D_i(x)$ .
Always-assigned units ( $M_i = 1$ ): These units do not require potential outcomes or states, and always have $X_i$ values beyond the cutoff.

Assumptions

Local Independence and Continuity:
- $P(D = 1|X = c^+, M = 0) > P(D = 1|X = c^-, M = 0)$
- No defiers: $P(D^+ \geq D^-|X = c, M = 0) = 1$
- Continuity in potential outcomes and states at $c$ .
- $F_{X|M=0}(x)$ is differentiable at $c$ , with a positive derivative.
Smoothness of the Running Variable among Potentially-Assigned Units:
- The derivative of $F_{X|M=0}(x)$ is continuous at $c$ .
Restrictions on Always-Assigned Units:
- $P(X \geq c|M = 1) = 1$ and $F_{X|M=1}(x)$ is right-differentiable (or left-differentiable) at $c$ .
- This (local) one-sided manipulation assumption allows identification of the proportion of always-assigned units among all units close to the cutoff.

When always-assigned unit exist, the RD design is fuzzy because we have

Treated and untreated units among the potentially-assigned (below and above the cutoff)
Always-assigned units (above the cutoff).

Causal Effects of Interest

causal effects among potentially-assigned units:

$\Gamma = E[Y(1) - Y(0) | X = c, D^+ > D^-, M = 0]$

This parameter represents the local average treatment effect (LATE) for the subgroup of “compliers”—units that receive treatment if and only if their running variable $X_i$ exceeds a certain cutoff.

The parameter $\Gamma$ captures the causal effect of changes in the cutoff level on treatment status among potentially-assigned compliers.

RD designs with a manipulated running variable	“Doughnut-Hole” RD Designs:
Focuses on actual observations at the cutoff, not hypothetical true values. Provides a direct and observable estimate of causal effects, without reliance on hypothetical constructs.	Exclude observations around the cutoff and use extrapolation from the trends outside this excluded range to infer causal effects at the cutoff Assumes a hypothetical population existing in a counterfactual scenario without manipulation. Requires strong assumptions about the nature of manipulation and the minimal impact of extrapolation biases.

Identification of $\tau$ in RD Designs

Identification challenges arise due to the inability to distinguish always-assigned from potentially-assigned units, thus Γ is not point identified. We establish sharp bounds on Γ
These bounds are supported by the stochastic dominance of the potential outcome CDFs over observed distributions.

Unit Types and Notation:

$C_0$ : Potentially-assigned compliers.
$A_0$ : Potentially-assigned always-takers.
$N_0$ : Potentially-assigned never-takers.
$T_1$ : Always-assigned treated units.
$U_1$ : Always-assigned untreated units.

The measure $\tau$ , representing the proportion of always-assigned units near the cutoff, is point identified by the discontinuity in the observed running variable density $f_X$ at the cutoff

Sharp RD:

Units to the left of the cutoff are potentially assigned units. The distribution of their observed outcomes ( $Y$ ) are the outcomes $Y(0)$ of potentially-assigned compliers ( $C_0$ ) at the cutoff.
To determine the bounds on the treatment effect ( $\Gamma$ ), we need to assess the distribution of treated outcomes ( $Y(1)$ ) for the same potentially-assigned compliers at the cutoff.
Information regarding the treated outcomes ( $Y(1)$ ) comes exclusively from the subpopulation of treated units, which includes both potentially-assigned compliers ( $C_0$ ) and those always assigned units ( $T_1$ ).
With $\tau$ point identified, we can estimate sharp bounds on $\Gamma$ .

Fuzzy RD:

Note: Table on page 848 (Gerard, Rokkanen, and Rothe 2020)
Subpopulation	Types of units
$X = c^+, D = 1$	$C_0, A_0, T_1$
$X = c^-, D = 1$	$A_0$
$X= c^+, D = 0$	$N_0, U_1$
$X = c^-, D = 0$	$C_0, N_0$

Unit Types and Combinations: There are five distinct unit types and four combinations of treatment assignments and decisions relevant to the analysis. These distinctions are important because they affect how potential outcomes are analyzed and bounded.
Outcome Distributions: The analysis involves estimating the distribution of potential outcomes (both treated and untreated) among potentially-assigned compliers at the cutoff.
Three-Step Process:
1. Potential Outcomes Under Treatment: Bounds on the distribution of treated outcomes are determined using data from treated units.
2. Potential Outcomes Under Non-Treatment: Bounds on the distribution of untreated outcomes are derived using data from untreated units.
3. Bounds on Parameters of Interest: Using the bounds from the first two steps, sharp upper and lower bounds on the local average treatment effect are derived.
Extreme Value Consideration: The bounds for treatment effects are based on “extreme” scenarios under worst-case assumptions about the distribution of potential outcomes, making them sharp but empirically relevant within the data constraints.

Extensions:

Quantile Treatment Effects: alternative to average effects by focusing on different quantiles of the outcome distribution, which are less affected by extreme values.
Applicability to Discrete Outcomes
Behavioral Assumptions Impact: Assuming a high likelihood of treatment among always-assigned units can narrow the bounds of treatment effects by refining the analysis of potential outcomes.
Utilization of Covariates: Incorporating covariates measured prior to treatment can refine the bounds on treatment effects and help target policies by identifying covariate distributions among different unit types.

Notes:

Quantile Treatment Effects (QTEs): QTE bounds are less sensitive to the tails of the outcome distribution, making them tighter than ATE bounds.
- Inference on ATEs is sensitive to the extent of manipulation, with confidence intervals widening significantly with small degrees of assumed manipulation.
- Inference on QTEs is less affected by manipulation, remaining meaningful even with larger degrees of manipulation.
Alternative Inference Strategy when manipulation is believed to be unlikely. Try different hypothetical values of $\tau$

devtools::install_github("francoisgerard/rdbounds/R")

library(formattable)
library(data.table)
library(rdbounds)
set.seed(123)
df <- rdbounds_sampledata(1000, covs = FALSE)
#> [1] "True tau: 0.117999815082062"
#> [1] "True treatment effect on potentially-assigned: 2"
#> [1] "True treatment effect on right side of cutoff: 2.35399944524618"
head(df)
#>            x         y treatment
#> 1 -1.2532616  2.684827         0
#> 2 -0.5146925  5.845219         0
#> 3  3.4853777  6.166070         0
#> 4  0.1576616  3.227139         0
#> 5  0.2890962  7.031685         1
#> 6  3.8350019 10.238570         1

rdbounds_est <-
    rdbounds(
        y = df$y,
        x = df$x,
        # covs = as.factor(df$cov),
        treatment = df$treatment,
        c = 0,
        discrete_x = FALSE,
        discrete_y = FALSE,
        bwsx = c(.2, .5),
        bwy = 1,
        
        # for median effect use 
        # type = "qte", 
        # percentiles = .5, 
        
        kernel = "epanechnikov",
        orders = 1,
        evaluation_ys = seq(from = 0, to = 15, by = 1),
        refinement_A = TRUE,
        refinement_B = TRUE,
        right_effects = TRUE,
        yextremes = c(0, 15),
        num_bootstraps = 5
    )
#> [1] "The proportion of always-assigned units just to the right of the cutoff is estimated to be 0.38047"
#> [1] "2025-02-08 18:14:33.753569 Estimating CDFs for point estimates"
#> [1] "2025-02-08 18:14:34.145708 .....Estimating CDFs for units just to the right of the cutoff"
#> [1] "2025-02-08 18:14:35.938183 Estimating CDFs with nudged tau (tau_star)"
#> [1] "2025-02-08 18:14:35.993495 .....Estimating CDFs for units just to the right of the cutoff"
#> [1] "2025-02-08 18:14:38.747617 Beginning parallelized output by bootstrap.."
#> [1] "2025-02-08 18:14:43.191747 Computing Confidence Intervals"
#> [1] "2025-02-08 18:14:54.779613 Time taken:0.35 minutes"

rdbounds_summary(rdbounds_est, title_prefix = "Sample Data Results")
#> [1] "Time taken: 0.35 minutes"
#> [1] "Sample size: 1000"
#> [1] "Local Average Treatment Effect:"
#> $tau_hat
#> [1] 0.3804665
#> 
#> $tau_hat_CI
#> [1] 0.4180576 1.4333618
#> 
#> $takeup_increase
#> [1] 0.6106973
#> 
#> $takeup_increase_CI
#> [1] 0.4332132 0.7881814
#> 
#> $TE_SRD_naive
#> [1] 1.589962
#> 
#> $TE_SRD_naive_CI
#> [1] 1.083977 2.095948
#> 
#> $TE_SRD_bounds
#> [1] 0.7165801 2.3454868
#> 
#> $TE_SRD_CI
#> [1] -1.200992  7.733214
#> 
#> $TE_SRD_covs_bounds
#> [1] NA NA
#> 
#> $TE_SRD_covs_CI
#> [1] NA NA
#> 
#> $TE_FRD_naive
#> [1] 2.583271
#> 
#> $TE_FRD_naive_CI
#> [1] 1.506573 3.659969
#> 
#> $TE_FRD_bounds
#> [1] 1.273896 3.684860
#> 
#> $TE_FRD_CI
#> [1] -1.266341 10.306116
#> 
#> $TE_FRD_bounds_refinementA
#> [1] 1.273896 3.684860
#> 
#> $TE_FRD_refinementA_CI
#> [1] -1.266341 10.306116
#> 
#> $TE_FRD_bounds_refinementB
#> [1] 1.420594 3.677963
#> 
#> $TE_FRD_refinementB_CI
#> [1] NA NA
#> 
#> $TE_FRD_covs_bounds
#> [1] NA NA
#> 
#> $TE_FRD_covs_CI
#> [1] NA NA
#> 
#> $TE_SRD_CIs_manipulation
#> [1] NA NA
#> 
#> $TE_FRD_CIs_manipulation
#> [1] NA NA
#> 
#> $TE_SRD_right_bounds
#> [1] -2.056178  3.650820
#> 
#> $TE_SRD_right_CI
#> [1] -11.044937   9.282217
#> 
#> $TE_FRD_right_bounds
#> [1] -2.900703  5.272370
#> 
#> $TE_FRD_right_CI
#> [1] -14.75421  15.58640

rdbounds_est_tau <-
    rdbounds(
        y = df$y,
        x = df$x,
        # covs = as.factor(df$cov),
        treatment = df$treatment,
        c = 0,
        discrete_x = FALSE,
        discrete_y = FALSE,
        bwsx = c(.2, .5),
        bwy = 1,
        kernel = "epanechnikov",
        orders = 1,
        evaluation_ys = seq(from = 0, to = 15, by = 1),
        refinement_A = TRUE,
        refinement_B = TRUE,
        right_effects = TRUE,
        potential_taus = c(.025, .05, .1, .2),
        yextremes = c(0, 15),
        num_bootstraps = 5
    )
#> [1] "The proportion of always-assigned units just to the right of the cutoff is estimated to be 0.38047"
#> [1] "2025-02-08 18:14:56.245033 Estimating CDFs for point estimates"
#> [1] "2025-02-08 18:14:56.449024 .....Estimating CDFs for units just to the right of the cutoff"
#> [1] "2025-02-08 18:14:58.23866 Estimating CDFs with nudged tau (tau_star)"
#> [1] "2025-02-08 18:14:58.276159 .....Estimating CDFs for units just to the right of the cutoff"
#> [1] "2025-02-08 18:15:01.104187 Beginning parallelized output by bootstrap.."
#> [1] "2025-02-08 18:15:05.392548 Estimating CDFs with fixed tau value of: 0.025"
#> [1] "2025-02-08 18:15:05.468534 Estimating CDFs with fixed tau value of: 0.05"
#> [1] "2025-02-08 18:15:05.527238 Estimating CDFs with fixed tau value of: 0.1"
#> [1] "2025-02-08 18:15:05.578757 Estimating CDFs with fixed tau value of: 0.2"
#> [1] "2025-02-08 18:15:06.69184 Beginning parallelized output by bootstrap x fixed tau.."
#> [1] "2025-02-08 18:15:09.681396 Computing Confidence Intervals"
#> [1] "2025-02-08 18:15:22.407704 Time taken:0.44 minutes"

causalverse::plot_rd_aa_share(rdbounds_est_tau) # For SRD (default)

# causalverse::plot_rd_aa_share(rdbounds_est_tau, rd_type = "FRD")  # For FRD

References

Aronow, Peter M, Jonathon Baron, and Lauren Pinson. 2019. “A Note on Dropping Experimental Subjects Who Fail a Manipulation Check.” Political Analysis 27 (4): 572–89.

Bajari, Patrick, Han Hong, Minjung Park, and Robert Town. 2011. “Regression Discontinuity Designs with an Endogenous Forcing Variable and an Application to Contracting in Health Care.” National Bureau of Economic Research.

Bertanha, Marinho, Andrew H McCallum, and Nathan Seegert. 2021. “Better Bunching, Nicer Notching.” arXiv Preprint arXiv:2101.01170.

Blomquist, Sören, Whitney K Newey, Anil Kumar, and Che-Yuan Liang. 2021. “On Bunching and Identification of the Taxable Income Elasticity.” Journal of Political Economy 129 (8): 2320–43.

Bosch, Nicole, Vincent Dekker, and Kristina Strohmaier. 2020. “A Data-Driven Procedure to Determine the Bunching Window: An Application to the Netherlands.” International Tax and Public Finance 27: 951–79.

Calonico, Sebastian, Matias D Cattaneo, and Max H Farrell. 2020. “Optimal Bandwidth Choice for Robust Bias-Corrected Inference in Regression Discontinuity Designs.” The Econometrics Journal 23 (2): 192–210.

Cattaneo, Matias D, Nicolás Idrobo, and Rocı́o Titiunik. 2019. A Practical Introduction to Regression Discontinuity Designs: Foundations. Cambridge University Press.

Chetty, Raj, Nathaniel Hendren, and Lawrence F Katz. 2016. “The Effects of Exposure to Better Neighborhoods on Children: New Evidence from the Moving to Opportunity Experiment.” American Economic Review 106 (4): 855–902.

Gerard, François, Miikka Rokkanen, and Christoph Rothe. 2020. “Bounds on Treatment Effects in Regression Discontinuity Designs with a Manipulated Running Variable.” Quantitative Economics 11 (3): 839–70.

Kleven, Henrik Jacobsen. 2016. “Bunching.” Annual Review of Economics 8: 435–64.

Lee, David S. 2009. “Training, Wages, and Sample Selection: Estimating Sharp Bounds on Treatment Effects.” The Review of Economic Studies, 1071–1102.

McCrary, Justin. 2008. “Manipulation of the Running Variable in the Regression Discontinuity Design: A Density Test.” Journal of Econometrics 142 (2): 698–714.

Zhang, Junni L, and Donald B Rubin. 2003. “Estimation of Causal Effects via Principal Stratification When Some Outcomes Are Truncated by ‘Death’.” Journal of Educational and Behavioral Statistics 28 (4): 353–68.