Dynamic Bunching Estimation with Panel Data

2.1 Failure of Identification with Cross-Sectional Data Analysis

Within this simple two-period model, I construct several examples in which the static bunching estimator using cross-sectional data is not consistent. Mathematical details for each example are provided in Supplementary Material A. Each example starts with a base year in which the traditional method identifies ϵ. Researchers estimate the distribution F z t z t over some range of income surrounding the notch at z _t = ρ _t and the reduced-mass range z t ∈ ρ t , ρ t + △ z t from which agents bunch.^[6] The width of this reduced range identifies a homogeneous elasticity, as Bertanha, McCallum, and Nathan (2023) and Blomquist et al. (2021) have shown in static models.

In practice, researchers have relaxed the assumption of a homogeneous elasticity but have relied on stronger empirical assumptions. This is because real-world data rarely exhibit the range of zero mass that would be predicted if the elasticity is homogeneous and there are no optimization frictions. However, an average elasticity can be derived from the excess mass or reduced mass (Kleven and Waseem 2013). In the simpler model, one can estimate any of the excess mass, reduced mass, or bunching range.

Here, I will formalize the parametric assumption in the bunching literature. Doing so will allow for a precise comparison of this assumption with the version for dynamic bunching estimation. Denote by f z t z t the PDF of z _t .

Assumption S (in static estimation): Choose a continuous function of finite parameters C z t over a range Z which is such that ∀ z t ∉ ρ t , ρ t + △ z t , C z t = f z t z t . Assume that ∀z _t ∈ Z, f z t * z t * = C z t .

The bunching literature most often assumes that f z t * z t * is a polynomial of a particular order over the range Z. It is inherently parametric because two functions that take the same values for z t ∉ ρ t , ρ t + △ z t need not take the same values for the omitted range z t ∈ ρ t , ρ t + △ z t . This lack of unique identification has been the subject of recent econometric critiques of bunching estimation. This paper provides concrete examples in which Assumption S fails and demonstrates that even in these cases a panel-data version of the design can provide consistent estimates.

When is Assumption S likely to fail? I provide three simple examples two-period examples. In the base year of each example that follows, the distribution of potential income is uniform, i.e. the density f z 0 * z 0 * is constant except in the omitted range around the notch. The excess mass below the notch ρ ₀ is F z t ρ 0 − F z t * ρ 0 , and in these examples with a uniform distribution, the counterfactual mass F z t * ρ 0 can be accurately quantified by extrapolating the constant density that is observed away from the notch. In the very next year, however, Assumption S fails to hold.

(1) Fixed costs of adjustment. Suppose that some agents face a large fixed cost of adjusting their next-year income, such as the cost of hiring a tax expert or the psychic cost of avoiding or evading. Such a setting can be modeled as a share of agents for whom z 1 * = z 0 . If some of these agents bunch in the base year, then in the next year they will have z 1 = z 1 * = ρ 0 , implying excess mass at this income level even if there is no notch when t = 1. Gelber, Jones, and Sacks (2020) show a version of this result for kinks (rather than notches) and use it to estimate fixed costs of adjustment using data spanning the removal of the kink. In contrast, I highlight the implication for identification. Even if the notch remains in place, the static approach does not generally provide a consistent estimator because what appears as bunching includes both the active bunching that the researcher wishes to quantify and some lingering effects of past bunching. Fixed costs of adjustment are a form of serial dependence, and the appendix derives results for examples with either discrete or continuous distributions of income.

(2) Continuous serial dependence. Income may be serially dependent if an agent’s salary or hours are a function of those in the previous period or if business opportunities include those of the previous period and a random shock. Serial dependence creates issues even if the transition distribution F z 1 * | z 0 * z 1 * | z 0 * , z 0 does not exhibit discrete mass points. In the next example, this distribution is continuous but gives greater probability to values of z 1 * that are close to z ₀. In other words, income growth from one year to the next is most likely to take on small absolute values, and larger values of z ₀ increase the likelihood of larger values of z ₁. If agents experience large-absolute-value changes in income with positive probability, then bunching in the base year even affects the density at next-year incomes that are far from the notch. Supplementary Material B provides an additional illustration of this bias using the first 10 years of earnings from each household in the PSID. In this example, one can see that serial dependence can distort the entire distribution of income, yet it may not be apparent from the observed distribution that such distortion has occurred.

Note that allowing for serial dependence in income does not imply determinism; agents make a choice in each period over whether or not to bunch. However, these examples demonstrate that when income is serially dependent, the income distribution reflects accumulated effects, and the standard methodology does not recover the behavioral parameters governing these effects. This accumulation of effects may explain findings that bunching at kinks in individual income taxes has increased over time since 1996 (Chetty, Friedman, and Saez (2013) and Mortenson and Whitten (2020)).

(3) Notch-related attrition/extensive-margin responses. The bunching literature has examined notch-related attrition in the form of extensive-margin responses, i.e. those who choose income of zero over either the potential income above the notch or the bunching income just below it. A point made by Kleven and Waseem (2013) and generalized by Best and Kleven (2018) is that such responses should not occur just above the notch because agents can simply bunch by giving up very little income relative to their preferred income level. Kopczuk and Munroe (2015) note that even if this holds in the limit as potential income approaches the notch, extensive-margin responses may still occur close enough to the notch to affect estimation, and that it is not possible (with the standard bunching design) to measure extensive-margin responses without making strong assumptions. Moreover, a notch may induce other forms of systematic attrition, such as if crossing the notch increases the probability of being audited and removed from the sample. If this attrition is not determined by agent maximization, then it need not approach zero near the notch.

These examples may be thought of as mechanisms for the identification concerns documented in existing papers. These concerns can be mitigated with panel data. The approach allows for agents to make static or dynamic choices, as it is like the static approach in that it estimates reduced-form parameters rather than uniquely identifying structural parameters. The estimation is dynamic in the sense that it exploits the distribution of income growth rather than the distribution of cross-sectional income. It requires no changes in the notch policy, though these can be useful illustrating identification and for identifying additional parameters.

2.2 Identification with Panel Data

The ideal setting for estimating responses to a notch would be to observe agents’ choices with and without the notch in the budget set. In practice, this will rarely be possible without a within-subjects experiment. Observational studies must therefore make an assumption about what is a credible counterfactual when all agents face a notch. With panel data, one can make identifying assumptions about the distribution of income growth that are likely to hold even when the corresponding assumptions for a cross-section are violated. This can be seen in the preceding examples, where F z 1 z 1 fails to identify F z 1 * z 1 * , but for all observations not bunching in the base year (i.e. z 0 * = z 0 ), F z 1 | z 0 * z 1 | z 0 identifies F z 1 * | z 0 * z 1 * | z 0 * , z 0 .

Figure 2 provides intuition for dynamic designs using panel data. Income levels and growth are drawn from the smoothed PSID distribution as in the simulations presented in Section 3.1, and the real-world bunching of charities produces a very similar figure (Marx 2018). Panel A of the figure shows the distribution of next-year log income (relative to the notch) for two illustrative levels of base-year income. For each group, the distribution of next-year income is distorted around the notch as one would expect, with excess mass just to the left and reduced mass just to the right. Panel B shows the distribution of growth rates for each group, defined as the difference between next-year log income and base-year log income.

Panel B of Figure 2 shows that each base-year-income group can be used as a counterfactual for the other. The two groups have similarly-shaped growth distributions, except that the two groups require different income growth rates to reach the notch, and hence the bunching distortions lie in different parts of the two groups’ growth distributions. The extent of the distortions can therefore be estimated by comparing the shape of one group’s growth distribution around its notch to the corresponding, undistorted section of the growth distribution among households in the other group.

To formalize the intuition, consider again an observed distribution of base-year income F z 0 z 0 and a latent distribution of potential next-year income F z 1 * z 1 * = ∫ F z 1 * | z 0 z 1 * | z 0 d F z 0 z 0 . To generalize from the homogeneous elasticity above, allow △z ₁ to have distribution F △ | z 0 △ z 1 | z 0 with domain 0 , △ z 1 ̄ . Note that the possibility of △z ₁ = 0 allows for optimization frictions. The resulting empirical elasticity is the appropriate subject for a paper studying estimation, and this estimate can be translated to different structural elasticities depending on the additional assumptions that the researcher makes about the setting (Einav, Finkelstein, and Schrimpf 2017). To simplify the exposition in the text, I assume away attrition and extensive-margin responses, which I incorporate into a generalization in Supplementary Material A. However, I allow for heterogeneous elasticities ϵ _i .

Consider a monotonic income-growth function g z 0 mapping a base-year income to a value that income could possibly take in the next year. For example, g z 0 = 1.1 z 0 would represent income growth of 10 percent. I will first show that for any such growth function there is an identifying assumption similar that is similar to Assumption S but that holds in the preceding examples where Assumption S fails. While this assumption remains parametric, I then note that comparing distributions for multiple growth functions g z 0 offers an identification opportunity more akin to comparing distributions over multiple budget sets.

Consider a range Z = z 0 ̲ , z 0 ̄ such that g z 0 ̲ < ρ and g z 0 ̄ > ρ + △ z 1 ̄ . Denote by Z ̃ ≔ z 0 ∈ Z : g z 0 ∉ ρ , ρ + △ z 1 ̄ the set of base-year incomes in Z for which income growth g z 0 does not lead to next-year income in the reduced-mass range. Thus, for z 0 ∈ Z ̃ , F z 1 | z 0 g z 0 | z 0 = F z 1 * | z 0 g z 0 | z 0 . Denote by f z 1 * | z 0 g z 0 | z 0 the PDF of z 1 * conditional on z ₀.

Assumption D (in dynamic estimation): Choose a continuous function of finite parameters C z 0 over a range Z which is such that ∀ z 0 ∈ Z ̃ , C z 0 = f z 1 * | z 0 g z 0 | z 0 , the conditional PDF. Assume that ∀z ₀ ∈ Z, f z 1 * | z 0 g z 0 | z 0 = C z 0 .

The proof of the theorem appears in Supplementary Material A. To interpret the result, note that the numerator is observed and is equal to f z 1 | z 0 z 1 | z 0 if F △ △ z 1 | z 0 is differentiable over △ z 1 ∈ 0 , △ z 1 ̄ . Consider a small elasticity and corresponding small △ z 1 ̲ . Conditional on z ₀, the share of agents with small enough elasticities that △ z 1 ≤ △ z 1 ̲ is equal to F △ | z 0 △ z 1 ̲ | z 0 = d d z 1 F z 1 | z 0 ρ + △ z 1 ̲ | z 0 C z 0 = f z 1 | z 0 ρ + △ z 1 ̲ | z 0 f z 1 * | z 0 ρ + △ z 1 ̲ | z 0 , which is the share of the agents with potential income of ρ + △ z 1 ̲ who do not bunch.

For intuition about the identification, consider a 50 percent growth rate ( g z 0 = 1.5 z 0 ) and the special case in which the distribution of growth rates does not depend on base-year income. If ρ + △ z 1 ̲ is a value in the next-year reduced range, then 50 percent growth would take an observation with base-year income of z 0 = ρ + △ z 1 ̲ / 1.5 to the reduced range in the next year. Thus, the probability of growing by 50 percent from this level of base-year income should be reduced because some agents whose income would have grown at this rate will instead only grow income enough to reach the notch. To quantify this distortion, one can simply compare Pr ρ + △ z 1 ̲ | z 0 = ρ + △ z 1 ̲ / 1.5 with, say, Pr ρ + △ z 1 ̲ | z 0 = ρ + △ z 1 ̲ / 2 . The undistorted latter probabilities provide a simple counterfactual in the special case of a growth distribution that does not vary with base-year income, in which case these probabilities should be equal if not for bunching. More generally, Assumption D maintains that the growth probability for a particular level of base-year income can be predicted using surrounding values of base-year income for which such growth should not be distorted. Such an assumption can be tested by creating placebo notches and checking whether observations surrounding any income level in the base year predict the probability of growing at any particular rate.

Assumption D differs from Assumption S in that it restricts the distribution of within-agent changes in income. It can therefore be implemented with a generalized difference-in-differences design rather than simple differences in a cross section. Moreover, while the static approach imposes an assumption about the density near the threshold, dynamic estimates can be obtained from observations that are far from the threshold in the base year. This difference is likely to be particularly important for notches that are in place for multiple periods, causing distortions around the notch to accumulate. Researchers can estimate bunching separately for multiple combinations of base-year-income and growth, or these estimates can be combined, as noted in the following corollary to the theorem.

This corollary exploits an assumption in the literature that in the small range of incomes z 1 * from which agents bunch, z 1 * may be uncorrelated with ϵ _i and hence △z ₁, such as if the distribution of ϵ _i is the same for all agents. In this case, there are multiple potential sources of identification for the distribution of △z ₁ and hence ϵ _i . In essence, different levels of base-year income provide budget set variation because the cost of exceeding the notch is imposed at different amounts of income growth.

3.1 Static Bunching Estimation

Consider a notch that imposes a discrete cost on agents i if choice variable z _i is greater than a threshold ρ. Let z _i be collapsed into bins of width ω. Let bin _b denote the maximum value of z _i in a bin. The bin count is c b = ∑ i 1 z i ∈ bin b − ω , bin b . The standard estimating equation for bunching at a notch is

where n _E and n _R are the numbers of bins in the excess and reduced ranges, respectively. The counterfactual is a polynomial of order K with parameters α _k . This equation provides two measures of bunching that should be equal if there are no extensive-margin responses or notch-related attrition: ∑ h = 0 n E − 1 β E , h gives the excess mass of bunchers just below the notch, and ∑ j = 1 n R β R , j gives the reduction of mass just above the notch.

Visual examples of the standard approach are provided in Figure 1. These use data from the charity application in Section 2 and the PSID. Relative to the counterfactual polynomial in each figure, mass is elevated in the bunching range just below the notch and reduced in the reduced range above the notch. Often this mass is divided by the counterfactual level of the density at the notch, and this bunching ratio gives an estimate of the amount by which the average buncher has reduced the choice variable (Kleven and Waseem 2013).

The standard methodology has also been augmented to address some further complications. Recognizing agents may reduce the choice variable from all levels above a notch or especially a kink, researchers have estimated the counterfactual separately on either side of the threshold or adjusted the counterfactual above the threshold so as to equalize the excess- and reduced-mass estimates (e.g. Chetty et al. 2011). As another example, the mass in a strictly dominated region just above the notch can be used to estimate the number of agents facing frictions that prevent bunching (Kleven and Waseem 2013). These adjustments are straightforward to incorporate into the dynamic designs proposed in this paper.

3.2 Dynamic OLS Estimation

As shown in Section 2, a dynamic design can identify bunching parameters using the panel nature of the data. Such dynamic designs can be implemented in a variety of ways. In theory, one could estimate the entire multivariate density of the choice variable in all available years, but allowing for such generality would be computationally intensive. I propose two methods of implementation that focus on pairs of the base-year logged choice variable ( r i t = ln z i t ) and growth to its next-year value (g _it = r _it+1 − r _it ).

3.2.1 Concept

To develop intuition before considering the full OLS implementation, consider restricting a dataset to agents for whom g _it falls within a particular range, i.e. a growth rate bin. Figure 3 provides an illustration using the data described in Section 5. I restrict the sample to charity-year observations with g i t ∈ 0.1 , 0.2 , then plot the (conditional) mean growth rate by bins of r ̃ i t = ln z i t 100,000 , which is base-year income recentered relative to the notch at $100,000. For example, among these charities with g i t ∈ 0.1 , 0.2 , the ones that are around 0.5 log points below the notch in the base year r ̃ i t ≈ − 0.5 have a mean of g _it is close to 0.147. This conditional mean should not be distorted by the notch because even if g _it = 0.2, which is the maximum value of growth for all depicted observations, they will have r ̃ i t + 1 ≈ − 0.3 , which is well below the notch. Similarly, there should be no distortion for charities well above the notch. All such bins are depicted with dark circles, and the curve with standard error bands illustrates a quadratic fit to their outcomes.

Figure 3:

Growth of treated charities versus counterfactual for an illustrative bin of growth rates. Notes:The figure shows growth of income from the current year to the next year as a function of current income (recentered around the reporting notch at $100,000). The figure sample consists of organizations in an illustrative growth bin that includes are organizations with growth between 0.1 and 0.2 log points. The marker with a 95-percent confidence interval represents the bin (defining the “near notch _it ” dummy described in the text) for which growth of 0.1–0.2 implies that future receipts lie in the “omitted range” straddling the notch. The conditional average growth rate of these charities is just below 0.145, which is significantly less than the counterfactual growth rate interpolated from charities with higher and lower current incomes. The difference is interpreted as a measure bunching; some charities that approach the notch reduce their income to stay below it, and therefore conditional average growth is less than predicted. N = 152,191. Omitted range is −0.08–0.07. Bin width = 0.05.

The target of estimation is the outcome of the “treatment” bin that appears in the figure as the filled circle with a confidence interval. For this bin, g i t ∈ 0.1 , 0.2 translates to a range of future income that straddles the notch, thereby including those that cross the notch and also those that bunch below it. This bin is therefore not selected based on whether an agent is a buncher, but it includes agents “treated” with the opportunity to bunch, and therefore offers an instrumental variable for bunching. The instrument’s relevance can be seen from the bin’s mean growth rate, which is significantly reduced relative to the quadratic counterfactual because some charities in the bin reduce their growth in order to bunch. To avoid bias of this counterfactual, the researcher may wish to drop nearby bins, depicted with lighter gray circles, for g i t ∈ 0.1 , 0.2 translates to some values of r ̃ i t + 1 that are close enough to the notch on one side or the other to overlap with the bunching or reduced range.

3.3 Maximum Likelihood Estimation

Dynamic OLS estimation is straightforward and provides a number of potential advantages over static OLS estimation. However, OLS estimation will not account for extensive-margin responses without additional assumptions and will not be efficient because binning the data treats observations within a bin as equivalent. A maximum likelihood bunching estimator can address these limitations.

4.1 Static Method Simulations

With the theory having demonstrated some threats to identification, I use simulations to assess the magnitude of biases. To conduct simulations, I use a panel of household taxable income from the Panel Study of Income Dynamics. Use of real-world data from the PSID shows that results are not driven by a particular distribution of known functional form. In addition, these data are publicly available, which will facilitate replication and extension of the proposed methods. I restrict attention to the nationally-representative sample within the PSID and exclude the over-sample of low-income families. The sample covers years from 1967 through 2013, and I inflate all amounts to 2013 dollars using the Consumer Price Index. The variable of interest is the household’s taxable income.^[9] In order to evaluate static and dynamic estimates on the same set of observations, I use each pair of consecutive years in which household income is observed, which I label the “base year” and “next year.”

To control the true amount of bunching in this naturally occurring data, I estimate the smoothed joint distribution of income and one-year growth. I define r _it as the log of realized base-year income and g _it = r _it+1 − r _it as the growth rate from base-year income to next-year income. I first bin the inverse CDF of g _it and estimate it as a flexible function with range 0,1 so that I can generate random numbers u _i from a uniform distribution and convert them to values of g _it .^[10] I then bin the joint distribution of r _it and g _it to estimate the inverse CDF F − 1 r i t | g i t as a flexible function with range 0,1 so that I can generate random numbers v _i from a uniform distribution and convert them to values of r _it .^[11] This allows me to generate a sample of any size according to the smoothed joint distribution of income and growth.

I generate bunching at a simulated notch with a heterogeneous distribution of abilities and elasticities. I employ the quasilinear utility function u c t , z t , a t = c t − a t 1 + 1 / ϵ z t a t 1 + 1 / ϵ as in Section 2. I assume a linear budget constraint c t ≤ 1 − τ z t − T t ⋅ 1 z t > ρ t for marginal tax rate τ and fixed cost T _t imposed when income exceeds the notch ρ _t . I set τ = 0.2 and create a hypothetical notch for certain years t, as described for each exercise, with ρ _t = $40, 000 and T _t = $1000. I assume that ϵ = 0 for half of all households (e.g. due to optimization frictions), and for the other half of households, ϵ ∼ Unif 0,1 . Note that larger elasticities require a wider reduced range for both the static and dynamic methods. Larger elasticities or fewer optimization frictions also imply more bunching, which will result in greater bias for the static method. Households bunch in the base year, and in the next year the notch is removed. A notch need not be removed for biases to arise, but I remove the notch in the simulation because this means that there should be exactly zero bunching after the reform, and therefore the mean estimate gives the bias of the estimator.

To examine the severity of the problem posed by notch-related attrition or extensive-margin responses, I take 10,000 draws of 100,000 observations from the smoothed distribution of incomes in the PSID and generate attrition from a fraction of those whose incomes should lie above the notch. Across simulations, I vary the extent of this notch-related attrition response between 0 and 10 percent of households, the largest value motivated by an estimated notch-related attrition rate for charities of 9.3 percent in Marx (2018) and below. No households bunch. I report static estimates of the average income foregone by bunchers using each of the excess mass below the threshold and the reduction in mass above the threshold. Because the true amount of bunching is zero, the average estimate gives the bias, and the percentage of statistically significant results gives the coverage, or simulated rejection rate. I also report the root mean squared error (root-MSE).

Table 1 shows the effect of notch-related attrition on static bunching estimates. Estimates using the reduced mass above the notch show small biases, as well as coverage rates close to 0.05, when the attrition rate is 0 or 1 percent. When the attrition rate is as small as 2 percent, the reduced-mass estimates falsely attribute them to bunching and reject too frequently. If 5 percent of agents respond on the extensive margin, the reduced-mass estimate rejects nearly three times as often as it should. The pattern is similar for the excess-mass estimates, though these show inflated coverage rates even for small rates of attrition. This may be surprising, given that the attrition response occurs entirely above the notch and not in the bunching range. Even so, these estimates are biased because the reduced mass above the omitted range lowers the counterfactual mass within the omitted range, including below the notch.

I next quantify the bias arising in static estimates when income is serially dependent. I impose bunching in the base year but then estimate bunching in the next year when there is no notch and hence no active bunching. If income is not serially dependent, then bunchers’ next-year incomes are determined by their base-year potential income. If income is serially dependent, then bunchers’ next-year incomes are determined by their base-year actual income or a combination of actual and potential income. I vary the degree of serial dependence over the simulations by choosing different weights η and determining next-year income by adding the randomly drawn growth rate g _it to a weighted average of base-year potential income (with weight η) and base-year actual income (with weight 1 − η). I consider possibilities ranging from full weight on actual income, as suggested by estimated one-for-one causal effects of base-year income on future-year income in Marx (2018), to a case of double weight on potential income as in the case of pure retiming of the income that is reduced in order to bunch.

Results appear in Table 2. When income exhibits no serial dependence (η = 1), static estimates have coverage rates close to 0.05 and relatively small bias. With positive serial dependence (η < 1, i.e. bunching reduces next-year income), both estimates show a positive bias, and coverage rates rise. Indeed, for the one-for-one case found of η = 0 found by Marx (2018), the reduced-mass estimator is off by nearly $1000 on average and rejects in more than one third of samples. With negative serial dependence (η > 1, i.e. bunching increases next-year income), the bias in the excess-mass estimates becomes negative, while coverage rates are again too high for both estimates.

4.2 Dynamic OLS Simulation

Dynamic OLS estimation should be consistent under serial dependence of income. I evaluate this design in the same way as the static estimator, with 10,000 random samples of 100,000 observations, base-year bunching at $40,000, and no bunching in next-year income.^[12]

Table 3 presents the results of the dynamic OLS simulation. The outcome in the left half of the table is income, which would be reduced in the treated bins approaching the notch if households in these bins chose to bunch, as would the outcome in the right half, which is the probability of crossing. Across the different types of serial dependence shown in each row, the coverage rate for the effect on income is close to 0.05. Coverage rates for the second outcome are closer to 0.06, but they do not change much with serial dependence, and they remain lower than the rates for the static estimator in all but ideal conditions.

Since the bunching ratio is the static estimate of the effect on income, results for this outcome can also be compared to those of the static estimates in the middle row of Table 2, in which there is no serial dependence. The root-MSE for the dynamic estimate is 55.6, compared to 549 and 222 for the static reduced-mass and excess-mass estimates, respectively. This suggests that the dynamic method is more precise, but one can also see that root-MSE is closely related to the bias, which is nonzero for both of these parametric methods. In general, the dynamic OLS method will lose some precision due to the dropped bins and to not estimating any bunching that is occurring in the base year. This will not be a problem if there is a base year prior to the existence of the notch that wouldn’t be used at all in the static design. If this is not the case, then the dynamic estimator would have relatively less precision if the number of years in the panel is small.

The dynamic OLS design estimates conditional growth probabilities as polynomials of base-year income. A potential downside of this approach is that it may not obtain an adequate fit if the distribution of conditional growth rates changes rapidly across values of base-year income r _i . To test this possibility, I run additional simulations in which I alter the growth rates of the draws from the smoothed PSID distribution. For context, the growth rate of log income in that distribution increases monotonically from −0.14 to −0.01 across deciles of base-year income, and the standard deviation varies non-monotonically between 0.45 and 0.53. To increase the variation in these conditional moments, I define a normalized base-year variable r ̃ i = r i − min r i max r i − min r i ∈ 0,1 and then consider transformations of g _i using the linear, quadratic, and exponential functions of r ̃ i . In the first three simulations, these functions are added to g _i , thereby changing the conditional mean but not the variance. Three more simulations change both moments by multiplying g _i by the same functions of r ̃ i .

Results appear in Table 4. For all three of the additive transformations, results are similar to those with the untransformed distribution, with coverage rates close to 0.05. The same is true for the first and third multiplicative transformations. However, multiplication by r ̃ i 2 produces coverage rates in excess of 0.16 for both outcomes. This finding shows that the proposed method can fail if the conditional distribution of growth differs in particular ways across levels of base-year income. In light of this, researchers can plot the binned conditional growth rate probabilities for undistorted bins and visually examine how well these are approximated by polynomials of base-year income. They can also test for “effects” on growth to placebo thresholds and assess robustness to varying the order of the polynomial.

4.3 Dynamic MLE Simulation

The simulation of MLE dynamic bunching estimation follows the earlier simulations. Here, the 10,000 random samples each contain 10,000 observations (reduced from the sample size of 100,000 used for OLS estimation due to the longer estimation time). I again impose bunching in the base year but no bunching or notch-related attrition responses in the next year. To allow for notch-related attrition responses, I randomly select observations to attrit at the rate that I observe among high-income households in the raw data, which is roughly four percent.^[13] I treat all attritors as having r i t + 1 = log 10,000 , with corresponding g _it , and then estimate a probability mass for this level of g _it that can vary with r _it and g _it . I capture notch-related attrition responses among households whose next-year income should exceed the notch by incorporating a parameter that, if ρ _it is the growth rate that would bring a household to the notch, shifts mass from ρ _it to log 10,000 .

Table 5 presents results from the simulation. I report estimates describing bunching and attrition responses, both of which have a true value of zero. The top row describes estimates of the share of bunchers among the total number of households that should move to the reduced region. The second row reflects the estimated effect on the income of these households, and the third row represents estimates of the share, of those that should cross the notch, that instead exit the sample. All of these estimates exhibit a bias close to zero and a coverage rate close to 5 percent.

The results in Table 5 also provide some information about precision. While MLE is efficient when the distribution is correctly specified, the estimator in the simulation does not use the functional form of the actual generating distribution, but rather a distribution function meant to provide enough flexibility for researchers to reasonably approximate a variety of empirical distributions. The estimated income reduction is the same quantity that is estimated using the bunching ratio in the static design, and therefore results from the middle row of Table 4 can therefore be compared to those in Tables 1 and 2. The root-MSE of the dynamic estimate is about 10 percent large than that of the static estimate of excess mass and less than half that of the static estimate of the reduced mass.^[14] Thus, the off-the-shelf MLE estimator performs nearly as well as the better of the static estimators, and precision can be further improved by tailoring the MLE to reduce unnecessary parameters and to better fit the empirical distribution outside of the omitted range.