Bunching Estimation

Motivating Example#

An economist wants to estimate how much taxpayers adjust their reported income in response to marginal tax rate changes. The US income tax schedule has kink points -- thresholds where the marginal tax rate jumps discretely. For example, at the boundary of the 10% and 12% brackets, the marginal rate increases by 2 percentage points. If taxpayers are responsive, they should cluster their reported income just below the kink to avoid the higher rate.

Here is the problem: the economist cannot run an experiment. She cannot randomly assign taxpayers to different kink points. She cannot compare taxpayers at different income levels, because high-income and low-income taxpayers differ in countless unobservable ways. A simple regression of income on tax rates would be hopelessly confounded by ability, education, occupation, and preferences.

The bunching estimation approach, developed by Saez (2010), solves this problem by exploiting the shape of the income distribution around the kink. If taxpayers are unresponsive to taxes, the income distribution should be smooth through the kink point -- there is no reason for a discontinuity in density. But if taxpayers respond by reducing their reported income to stay below the kink, we should see an excess mass of taxpayers bunching at the kink relative to a smooth counterfactual density.

The key insight is that the amount of excess bunching is proportional to the behavioral elasticity. More bunching means taxpayers are more responsive. The researcher estimates the counterfactual density by fitting a polynomial to the observed income histogram, excluding the bins near the kink, and then interpolating through the excluded region. The difference between the observed density and the counterfactual in the bunching region measures the excess mass, from which the elasticity is recovered.

This approach has been extended from kinks (where the marginal rate changes) to notches (where the average tax rate jumps discretely), where the incentive to bunch is much stronger and the implied elasticities can be identified more precisely (Kleven & Waseem, 2013).

A. Overview#

What Bunching Estimation Does#

Bunching estimation identifies behavioral responses to policy thresholds by measuring the excess mass of observations clustered at a kink or notch point relative to a smooth counterfactual density. The method proceeds in four steps:

1. Bin the data: create a histogram of the running variable (e.g., reported income) with fine bins
2. Exclude the bunching region: remove bins near the threshold where bunching occurs
3. Fit a counterfactual: estimate a polynomial through the remaining bins to approximate the density that would have prevailed without the threshold
4. Compute excess mass: integrate the difference between the observed density and the counterfactual over the bunching region

The excess mass $B$ -- or the normalized excess mass $b = B / h_0(z^*)$, where $h_0(z^*)$ is the counterfactual density height at the threshold -- is the key estimand. The implied elasticity is then:

where $t_0$ and $t_1$ are the marginal tax rates below and above the kink.

Kinks vs. Notches#

Different types of thresholds produce different bunching patterns:

• Kink point: the marginal tax rate changes (the slope of the tax schedule changes). Bunching is typically modest because the incentive to adjust is proportional to the tax rate change. The income distribution shows a spike but no "missing mass" above the kink.

• Notch point: the average tax rate jumps discontinuously (the level of the tax schedule jumps). The incentive to bunch is much stronger -- there is a region above the notch where no rational agent should locate (the "dominated region"). The income distribution shows both a spike at the notch and a "hole" (missing mass) above it. Kleven and Waseem (2013) developed the formal framework for notch bunching.

The Counterfactual Density#

The counterfactual density is the distribution that would have prevailed in the absence of the threshold. It is estimated by fitting a polynomial of order $p$ (typically $p = 5$ to $9$) to the binned frequency counts, excluding bins within the bunching window $[z^* - \delta_L, z^* + \delta_R]$. The polynomial is then used to interpolate through the excluded region, and an integration constraint ensures that the total mass is preserved -- the excess mass at the threshold must equal the "missing mass" above it.

How It Differs from RDD#

Bunching estimation and regression kink design (RKD) both exploit kink points, but they answer different questions with different identifying assumptions:

• RKD estimates the causal effect of a treatment on an outcome using the kink in the treatment assignment function. It requires that the conditional expectation of the outcome is smooth through the kink.
• Bunching estimates the elasticity of the running variable itself (e.g., reported income) with respect to the tax rate change at the kink. It requires that the density of the running variable would be smooth in the absence of the kink.

Common Confusions#

When to Use#

1. Agents face a clear threshold in their budget set. Tax kinks, notch points, regulatory cutoffs, benefit eligibility thresholds -- any setting where the incentive structure changes discretely at a known point.

2. The running variable is continuous and finely measured. Income, revenue, asset holdings, or other variables measured with sufficient precision to construct a histogram with fine bins.

3. You want to estimate a behavioral elasticity. Bunching directly identifies how responsive agents are to changes in incentive structures, which is the key policy parameter for welfare analysis and optimal policy design.

4. You have a large sample. Bunching requires enough observations in each bin to construct a reliable density estimate. Administrative data (tax records, social insurance registers) are ideal.

Do NOT Use Bunching When:#

1. The threshold is not known to agents. If agents are unaware of the kink or notch, they cannot respond to it, and bunching tells you about information, not preferences.

2. The running variable is coarsely measured or discrete. If income is reported in broad brackets or round numbers only, the histogram is too coarse to estimate a smooth counterfactual density.

3. Round-number bunching confounds the threshold. If the kink point coincides with a salient round number, behavioral bunching at the kink and round-number bunching are confounded.

4. You want to estimate effects on a different outcome. Bunching estimates the elasticity of the running variable. If you want the causal effect on, say, labor hours or firm investment, you need RDD or RKD instead.

Connection to Other Methods#

Bunching estimation is related to several other causal inference methods:

• Regression Kink Design (RKD): Both exploit kink points, but RKD estimates the causal effect of a kink-assigned treatment on an outcome (e.g., the effect of unemployment insurance generosity on unemployment duration), while bunching estimates the elasticity of the running variable itself (e.g., how much reported income responds to the tax rate change). RKD requires a smooth conditional outcome expectation; bunching requires a smooth counterfactual density.

• Regression Discontinuity (RDD): RDD exploits a cutoff where treatment status jumps. In bunching, the "treatment" is the change in incentives, and the response is the distortion of the density. The McCrary density test in RDD checks for manipulation (bunching) as a threat; in bunching estimation, the density distortion is the estimand.

• Difference-in-Differences (DiD): Some bunching applications use a DiD-like comparison, exploiting tax reforms that change the location or size of kink points. By comparing bunching before and after a reform, researchers can difference out round-number effects and other non-behavioral density features.

• Instrumental Variables (IV): Bunching can be viewed as a distributional version of IV. The kink or notch is the "instrument" -- an exogenous change in the budget set -- and the excess mass is the "first stage" response in the density of the running variable.

B. Identification#

For bunching estimation to provide valid elasticity estimates, three key assumptions must hold (Saez, 2010).

Assumption 1: Smooth Counterfactual Density#

Plain language: In the absence of the threshold, the density of the running variable would be smooth through the kink or notch point. There would be no "natural" pile-up or gap at the threshold -- any excess mass or missing mass is caused by behavioral responses to the threshold.

Formally: The counterfactual density $h_0(z)$ is continuously differentiable in a neighborhood of $z^*$, the threshold. This rules out round-number bunching, institutional features, or other non-behavioral reasons for density discontinuities at the threshold.

This assumption is tested by checking for bunching at placebo thresholds -- points in the income distribution where no tax rate change occurs. If similar bunching appears at placebo kinks, the identifying assumption is likely violated.

Assumption 2: No Other Behavioral Responses at the Threshold#

Plain language: The threshold affects behavior only through the mechanism of interest (e.g., the tax rate change). There is no other policy, regulation, or incentive that kicks in at the same threshold and could independently cause bunching.

For example, if a tax kink coincides with an eligibility threshold for a government transfer program, bunching could reflect responses to the transfer, the tax change, or both. The elasticity estimate would be confounded.

Assumption 3: Known Functional Form of Response#

Plain language: The standard bunching estimator assumes a specific model of individual behavior -- typically isoelastic preferences -- to map from the observed excess mass to the structural elasticity. Blomquist and Newey (2017) showed that the bunching estimator identifies the elasticity only under these functional form restrictions. Without them, the same amount of bunching is consistent with a range of elasticities.

This assumption is the most debated in the bunching literature. Sensitivity analysis should explore how the implied elasticity changes under alternative preference specifications.

C. Visual Intuition#

Adjust the elasticity and tax rate change to see how bunching emerges at the kink point. The counterfactual polynomial estimates what the density would look like without the threshold; the excess mass between the observed and counterfactual densities identifies the behavioral response.

D. Mathematical Derivation#

The Bunching Estimator#

E. Implementation#

Bunching Estimation: Step by Step#

1library(bunching)
2
3# ---- Step 1: Set parameters ----
4zstar    <- 10000   # kink point (threshold)
5binwidth <- 500     # bin width
6poly     <- 7       # polynomial order
7
8# ---- Step 2: Run bunching estimator ----
9result <- bunchit(
10z_vector  = df$income,
11zstar     = zstar,
12binwidth  = binwidth,
13bins_l    = 20,    # bins left of excluded region
14bins_r    = 20,    # bins right of excluded region
15poly      = poly,
16t0        = 0,     # kink (not notch)
17notch     = FALSE
18)
19
20# ---- Step 3: Extract key results ----
21summary(result)
22# B     = excess mass
23# b     = normalized excess mass
24# e     = implied elasticity
25# B_se  = bootstrap SE

F. Diagnostics#

F.1 Placebo Tests at Non-Kink Points#

Apply the same bunching estimation procedure at income levels where no kink or notch exists. If you find significant "bunching" at placebo points, the smooth counterfactual assumption may be violated -- round-number effects, institutional features, or other non-behavioral factors are producing density irregularities that the polynomial cannot capture.

F.2 Integration Constraint (Notch Designs)#

For notch bunching, the excess mass at the threshold must equal the missing mass above it. This constraint is an internal consistency check: agents who bunch at the notch must come from somewhere (the dominated region above the notch). If excess mass exceeds missing mass, the model is misspecified -- perhaps the dominated region boundaries are wrong, or there is bunching from agents who were initially below the threshold.

F.3 Polynomial Order Sensitivity#

Plot the estimated excess mass $\hat{B}$ as a function of the polynomial order $p$ for p = 3, 4, 5, 6, 7, 8, 9, 10. The estimate should stabilize for moderate p values. If the excess mass changes substantially as p increases, the counterfactual is sensitive to functional form and the estimate is fragile. Low-order polynomials (p below 5) may be too inflexible; high-order polynomials (p above 9) may overfit.

F.4 Bunching Window Sensitivity#

The choice of the excluded region $[z^* - \delta_L, z^* + \delta_R]$ affects the estimate. Too narrow: the polynomial is fit using bins that are part of the bunching response, biasing the counterfactual downward and understating excess mass. Too wide: the polynomial has fewer data points to fit and the interpolation is less precise. Report results for a range of window widths.

F.5 Visual Inspection#

Always produce two plots:

1. Histogram with counterfactual overlay: the observed binned density as bars, the polynomial counterfactual as a dashed line, and the bunching region shaded. The excess mass should be visually apparent.
2. Sensitivity plots: excess mass and implied elasticity as functions of polynomial order and bunching window width.

1# Sensitivity to polynomial order
2poly_range <- 5:9
3B_by_poly <- sapply(poly_range, function(p) {
4res <- bunchit(z_vector = df$income, zstar = zstar,
5                binwidth = binwidth, bins_l = 20,
6                bins_r = 20, poly = p, notch = FALSE)
7res$B
8})
9
10plot(poly_range, B_by_poly, type = "b", pch = 19,
11   xlab = "Polynomial Order", ylab = "Excess Mass (B)",
12   main = "Sensitivity to Polynomial Order")
13
14# Placebo test at a non-kink point
15placebo_zstar <- 15000
16placebo <- bunchit(z_vector = df$income,
17                  zstar = placebo_zstar,
18                  binwidth = binwidth, bins_l = 20,
19                  bins_r = 20, poly = 7, notch = FALSE)
20cat("Placebo B:", placebo$B, "\n")

Reading the Output#

The key outputs from a bunching estimation are:

What to Report#

A well-reported bunching analysis should include:

1. The histogram with counterfactual overlay and bunching region shaded
2. Excess mass $\hat{B}$ (or normalized $\hat{b}$) with bootstrap confidence interval
3. Implied elasticity $\hat{e}$ with standard error
4. Polynomial order used and sensitivity across orders 5-9
5. Bunching window boundaries and sensitivity to window width
6. Bin width used and sensitivity
7. Placebo tests at non-kink points
8. Discussion of frictions -- are the estimates naive or friction-corrected?

G. What Can Go Wrong#

H. Practice#

H.1 Concept Checks#

H.2 Guided Exercise#

H.3 Error Detective#

H.4 You Are the Referee#

I. Swap-In: When to Use Something Else#

• Regression Kink Design (RKD): when you want to estimate the causal effect of a kink-determined treatment on a downstream outcome (e.g., the effect of unemployment insurance on job search duration), rather than the elasticity of the running variable itself.

• Regression Discontinuity (RDD): when treatment status (not just the incentive slope) changes at the threshold. RDD estimates the local treatment effect; bunching estimates the behavioral elasticity. If you have a notch, you can use RDD for the outcome effect and bunching for the elasticity -- they complement each other.

• Instrumental Variables (IV): when you want to estimate a behavioral parameter but do not have a density-based approach -- for example, using variation in tax rates across jurisdictions as instruments for reported income.

• Structural estimation: when frictions are severe and you need a fully parametric model of the decision problem to identify the elasticity. The Kleven and Waseem (2013) model is a hybrid: it combines reduced-form bunching with a structural friction distribution.

J. Reviewer Checklist#