Finance and Economics Discussion Series
Divisions of Research & Statistics and Monetary Affairs
Federal Reserve Board, Washington, D.C.
Better Bunching, Nicer Notching
Marinho Bertanha, Andrew H. McCallum, and Nathan Seegert
2021-002
Please cite this paper as:
Bertanha, Marinho, Andrew H. McCallum, and Nathan Seegert (2021). “Better Bunching,
Nicer Notching,” Finance and Economics Discussion Series 2021-002. Washington: Board
of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2021.002.
NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary
materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth
are those of the authors and do not indicate concurrence by other members of the research staff or the
Board of Governors. References in publications to the Finance and Economics Discussion Series (other than
acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.
BETTER BUNCHING, NICER NOTCHING
Marinho Bertanha∗
Andrew H. McCallum†
Nathan Seegert
‡
First draft: August 5, 2017
This draft: August 14, 2020
Abstract
We study the bunching identification strategy for an elasticity parameter that
summarizes agents’ response to changes in slope (kink) or intercept (notch) of a
schedule of incentives. A notch identifies the elasticity but a kink does not, when the
distribution of agents is fully flexible. We propose new non-parametric and
semi-parametric identification assumptions on the distribution of agents that are
weaker than assumptions currently made in the literature. We revisit the original
empirical application of the bunching estimator and find that our weaker identification
assumptions result in meaningfully different estimates. We provide the Stata package
bunching to implement our procedures.
JEL: C14, H24, J20
Keywords: partial identification, censored regression, bunching, notching
∗
Corresponding author. Department of Economics, University of Notre Dame, 3060 Jenkins Nanovic
Halls, Notre Dame IN 46556. Email: mbertanha@nd.edu. Website: www.nd.edu/∼mbertanh.
†
Trade and Quantitative Studies Section, International Finance, Board of Governors of the Federal
Reserve System. Email: andrew.h.mccallum@frb.gov. Website: www.andrewhmccallum.com.
‡
Eccles School of Business, University of Utah. Email: nathan.seegert@eccles.utah.edu. Website:
www.nathanseegert.com.
1
Introduction
Estimating agents’ responses to incentives is a central objective in economics and many
other social sciences. A continuous distribution of agents that face a piecewise-linear schedule
of incentives results in a distribution of responses with mass points located where the slope
or intercept of the schedule changes. For example, a progressive schedule of marginal income
tax rates induces a mass of heterogeneous individuals to report the same income at the level
where marginal rates increase. Many studies in economics use mass points in the response
distribution to recover primitive parameters that govern agents’ responses to incentives.
Pioneering work by Saez (2010), Chetty, Friedman, Olsen, and Pistaferri (2011), and
Kleven and Waseem (2013) develop bunching estimators to use mass points in response
distributions to recover primitive parameters. These estimators are widely applied in
economics and rely on the idea that a mass point is larger, the more responsive agents are to
incentives. The size of the mass point, however, also depends on the unobserved distribution
of agents’ heterogeneity. Current methods are only able to map the size of mass points to
primitive parameters because they make specific assumptions about the unobserved
distribution.
This paper places bunching estimators on a statistical foundation and makes three
contributions on the identification of a primitive parameter that summarizes agents’
responses to incentives. First, we clarify how the mapping of observed variables to an
elasticity parameter depends on assumptions about the unobserved distribution of
heterogeneity. The elasticity parameter captures the log percentage change of a response to
a log percentage change in an incentive. A change in the intercept of the incentive schedule
admits non-parametric point identification of the elasticity but a change in slope does not.
Second, we examine the assumptions made by current bunching methods and propose
weaker assumptions for partial and point identification of the elasticity. Third, we revisit the
original empirical application of the bunching estimator, which is in the economics literature
2
that examines the largest means-tested cash transfer program in the United States —the
Earned Income Tax Credit (EITC). Our weaker assumptions about the unobserved
distribution of heterogeneity result in meaningful changes in estimates of individual
responses to taxes.
Our first contribution is to clarify the importance of assumptions about unobserved
heterogeneity for the identification of the elasticity. Many existing estimates are based on an
agent optimization problem with a piece-wise linear constraint that has one change in slope
or intercept. Slope changes in the constraint are often referred to as ‘‘kinks’’ while intercept
changes are often called ‘‘notches.’’ We generalize the constraint of the agent’s problem to a
schedule with multiple changes in intercepts and slopes because agents typically encounter a
combination of both kinks and notches.
We highlight three insights about identification with kinks and notches assuming a
non-parametric family of distributions for unobserved heterogeneity that have continuous
probability density functions (PDFs). First, if the constraint has at least one notch, it is
possible to point identify the elasticity. Identification comes from using the empty interval
in the support of the observed distribution that is created by agents’ responses to a notch.
Second, point identification is impossible if the incentive schedule only contains kinks.
Identification is impossible because there always exists an unobserved distribution that
reconciles any elasticity with the observed distribution of responses. Third, inference
methods designed for one kink can be applied in cases with multiple kinks at each kink
separately, as long as there are no notches preceding the kink under study. This is because
the range of heterogeneous agents that bunch at a kink is the same regardless of whether it
is the first kink in the schedule or if it is followed by another kink. In contrast, the range of
agents that bunch at a kink changes if that kink is preceded by a notch. Thus, methods
designed for one kink could be invalidated by a preceding notch, but our new identification
strategies can handle both kinks and notches simultaneously.
Our second contribution is to propose three novel identification strategies for the
3
elasticity if the incentive schedule has kinks but no notches. Each of these strategies relies
on weaker assumptions than those implicit in current implementations of the bunching
estimator. Our first strategy identifies upper and lower bounds on the elasticity —partially
identifies the elasticity —by making a mild shape restriction on the non-parametric family of
heterogeneity distributions. The other two strategies point identify the elasticity using
covariates and semi-parametric restrictions on the distribution of heterogeneity.
The first strategy partially identifies the elasticity by assuming a bound on the slope
magnitude of the heterogeneity PDF, that is, Lipschitz continuity. Intuition for
identification of the elasticity in this setting is as follows. We observe the mass of agents
who bunch, which equals the area under the heterogeneity PDF inside an interval. The
length of this bunching interval depends on the unknown elasticity. The maximum slope
magnitude of the PDF implies upper and lower bounds for all possible PDF values inside the
bunching interval that are consistent with the observed bunching mass. This translates into
lower and upper bounds, respectively, on the size of the bunching interval, which
corresponds to lower and upper bounds on the elasticity. These bounds allow researchers to
examine the magnitude of the impossibility result in their empirical context. Depending on
the data, it might take an unreasonably high slope magnitude on the heterogeneity PDF to
produce bounds that include all possible elasticity values. In other settings, the difference
between upper and lower bounds may be economically large even for small slope magnitudes.
The next two strategies rely on the fact that bunching can be rewritten as a censored
regression model with a middle censoring point. We stress that while these strategies
necessarily add structure to point identify the elasticity, they do not require fully parametric
assumptions, such as normality, on the unconditional distribution of heterogeneity.
The second strategy identifies the elasticity by estimating a maximum likelihood
mid-censored model, using data truncated to a window local to the kink. The likelihood
function assumes that the unobserved distribution conditional on covariates is parametric,
but we demonstrate that correct specification of the conditional distribution is not necessary
4
for consistency, as long as the unconditional distribution is correctly specified. For example,
conditional normality yields a mid-censored Tobit model, which has a globally concave
likelihood and is easy to implement. Nevertheless, consistency only requires that the
unobserved distribution is a semi-parametric mixture of normals, and conditional normality
is not necessary. Truncating the sample around the kink point improves the fit of the model
and further weakens these distribution assumptions.
The third strategy restricts a quantile of the unobserved distribution, conditional on
covariates, and point identification follows existing theory for censored quantile regressions
(Powell, 1986; Chernozhukov and Hong, 2002; Chernozhukov, Fernández-Val, and Kowalski,
2015).
Both of the two semi-parametric methods are censored regression models that
incorporate covariates. These approaches extend bunching estimators to control for
observable heterogeneity for the first time. Observable individual characteristics generally
account for substantial variation across agents and leave less heterogeneity unobserved. This
fact suggests that identification strategies that utilize covariates should be preferred over
identifying assumptions that only restrict the shape of the unobserved distribution without
covariates.
Our third contribution is to illustrate the empirical relevance of our methods by
revisiting Saez (2010)’s original influential application of bunching in the distribution of U.S.
income caused by kinks in the EITC schedule. That approach implicitly assumes the
unobserved PDF of agents that bunch is linear and uses a trapezoidal approximation to
compute the bunching mass. This assumption fits poorly when the true density is non-linear
or the interval of agents that bunch is large. We compare elasticity estimates based on our
identification assumptions with estimates based on the trapezoidal approximation using
annual samples of U.S. federal tax returns from the Internal Revenue Service (IRS).
Our partial identification method indicates that households adjust their reported income
in response to marginal tax rates by a considerable amount. Placing a conservative limit on
5
the slope magnitude, the lower bound for the elasticity is 0.34 —that is, a one percent
increase in the marginal tax rate results in a reduction in reported income of at least 0.34
percent. This estimate contrasts with the estimate of 0.43 using the trapezoidal
approximation. The difference in these estimates matters. For example, Saez (2001) shows
that the optimal top marginal tax rate for an economy with an elasticity of 0.34 is 13
percentage points higher than when the elasticity is 0.43.
The truncated Tobit model with covariates fits well the observed distribution of income
making our semi-parametric consistency result operative. Elasticity estimates from this
model differ substantially from estimates based on the trapezoidal approximation for some
categories of U.S. taxpayers. For example, we estimate an an elasticity of 0.72 versus a
trapezoidal estimate of 1.10 for married and self-employed individuals. This large difference
highlights the sensitivity of estimates to functional form assumptions, as well as the need for
methods that rely on weaker assumptions.
Our three new methods provide a suite of ways to recover elasticities from bunching
behavior. Each method differs in the assumptions they make about the unobserved
distribution to achieve identification. There is no way to determine which assumption is
correct because the unobserved distribution is not fully identified. Nevertheless, estimates
that are stable across many methods indicate that different identifying assumptions do not
play a major role in the construction of those estimates. On the contrary, estimates that are
sensitive to different assumptions are dependent on the validity of those assumptions.
Therefore, we recommend that researchers examine the sensitivity of elasticity estimates
across all available methods as a matter of routine.
Bunching estimators are widely applied in settings including fuel economy regulations
(Sallee and Slemrod, 2012), electricity demand (Ito, 2014), real estate taxes (Kopczuk and
Munroe, 2015), labor regulations (Garicano, Lelarge, and Van Reenan, 2016), prescription
drug insurance (Einav, Finkelstein, and Schrimpf, 2017), marathon finishing times (Allen,
Dechow, Pope, and Wu, 2017), attribute-based regulations (Ito and Sallee, 2018), education
6
(Dee, Dobbie, Jacob, and Rockoff, 2019; Caetano, Caetano, and Nielsen, 2020b), minimum
wage (Jales, 2018; Cengiz, Dube, Lindner, and Zipperer, 2019), and air-pollution data
manipulation (Ghanem, Shen, and Zhang, 2019), among others. Variation in the size of the
mass point across groups of individuals has also been used as a first stage in a two stage
approach to control for endogeneity (Chetty, Friedman, and Saez, 2013; Caetano, 2015;
Grossman and Khalil, 2019).1 An additional complication in many applications arises when
the bunching mass is spread over a range instead of being a mass point. Blomquist, Kumar,
Liang, and Newey (2019) provide a discussion about the potential sources for this
complication and Cattaneo, Jansson, Ma, and Slemrod (2018) propose a filtering method to
resolve it. Kleven (2016) reviews the many applications and branches of the bunching
literature and Jales and Yu (2017) relates bunching to regression discontinuity design
(RDD).
In the context of kinks, Blomquist and Newey (2017) were the first to prove the
impossibility of point identification and the possibility of partial identification —and earlier
Blomquist, Kumar, Liang, and Newey (2015) provide the outline for those proofs. We derive
partial identification bounds by assuming the PDF has a bounded slope, whereas Blomquist
and Newey (2017) assume the PDF of heterogeneity is monotone. We developed our
impossibility and partial identification results independently of theirs. Our partial
identification approach has three valuable features: closed-form solutions, observed bunching
always implies a positive elasticity, and nesting of the original bunching estimator.
Blomquist and Newey (2018) explain that a notch can identify the elasticity and a formal
proof of identification appears contemporaneously in an earlier version of our paper
(Bertanha, McCallum, and Seegert, 2018). To the best of our knowledge, ours is the first
paper to demonstrate point identification using censored regression models, covariates, and
semi-parametric assumptions on the distribution of heterogeneity. More generally, the
1
Econometric approaches using bunching for causal identification include Khalil and Yildiz (2017), Caetano
and Maheshri (2018), Caetano, Kinsler, and Teng (2019), and Caetano, Caetano, and Nielsen (2020a).
7
theory demonstrating that a kink fails to point identify the elasticity relates to the literature
on impossible inference reviewed by Bertanha and Moreira (2020).
The paper proceeds with an utility maximization model subject to a piecewise-linear
budget constraint in Section 2. Section 3 investigates the identification of the elasticity in
the case of kinks and notches. We propose the three identification strategies for the
elasticity in Section 4 and illustrate these methods empirically in the context of the EITC in
Section 5. Section 6 concludes. Appendix A contains all proofs, and supplemental Appendix
B collects auxiliary results and examples. Finally, we developed the Stata command
bunching that implements our procedures.2
2
Utility Maximization Subject to Piecewise-Linear Constraints
Firms’ and individuals’ optimization problems often face piecewise-linear constraints.
The nature of constraints is dictated by differential tax rates, insurance reimbursement rates,
or contract bonuses. A budget set is fully characterized by a sequence of intercepts and
slopes that change at known points. A change in the intercept is referred to as a notch, and
a change in the slope is referred to as a kink.
2.1
Model Setup
We start with the labor supply characterization employed by the vast majority of the
literature, which follows the seminal work of Saez (2010) and Kleven and Waseem (2013).
Agents maximize an iso-elastic quasi-linear utility function and choose consumption and
labor subject to a piecewise-linear budget set. Well-known models that fit into this category
include those of Burtless and Hausman (1978), Best and Kleven (2018), Einav, Finkelstein,
and Schrimpf (2017), among others. For ease of exposition, we focus on budget sets with one
2
The Stata package is available at the Statistical Software Components (SSC) online repository. Type ssc
install bunching in Stata to install the package. The package is also available for download from the
website of the authors.
8
kink or one notch in the main text. In supplemental Appendices B.1 and B.2, we generalize
the literature to any combination of kinks and notches. Section 3 below briefly discusses new
insights for the identification of the elasticity that arise in the problem with multiple kinks
and notches.
Consider a population of agents that are heterogeneous with respect to a scalar variable
N ∗ , referred to as ability. Ability is distributed according to a continuous probability density
function (PDF) fN ∗ , with support (0, ∞), and a cumulative distribution function (CDF)
FN ∗ . Agents know their N ∗ , but the econometrician does not observe the distribution of N ∗ .
Agents maximize utility by jointly choosing a composite consumption good C and labor
supply L. Utility is increasing in C and decreasing in L. These variables are constrained by
a budget set, where the agent may consume all of its labor income net of taxes plus an
exogenous endowment I0 . For simplicity, we assume the price of labor and consumption are
equal to one, such that taxable labor income Y is equal to L.
In the budget constraint with a kink, the tax rate increases from t0 to t1 as income
increases above the kink value K. The budget constraint has a notch when the agent is
charged a lump-sum tax of ∆ > 0 as income crosses K. Agent type N ∗ maximizes utility
U (C, Y ; N ∗ ) as follows,
max
C,Y
N∗
C−
1 + 1/ε
Y
N∗
1+ 1ε
(1)
s.t.
C = I{Y ≤ K}[I0 + (1 − t0 )Y ] + I{Y > K} [I1 + (1 − t1 ) (Y − K)] ,
(2)
where I{·} is the indicator function; the budget line has intercept I0 and slope 1 − t0 if
Y ≤ K, but intercept I1 = I0 + K(1 − t0 ) − ∆ with slope 1 − t1 if Y > K; and ε is the
elasticity of income Y with respect to one minus the tax rate when the solution is interior.
In the case of a kink, ∆ = 0, and the budget frontier is continuous; otherwise, in the case of
a notch, it has a jump discontinuity of size ∆ at Y = K. The solution is always on the
9
budget frontier in Equation 2.
2.2
Model Solution
The solution for Y in Problem 1 is well known in the literature, when K is a kink (Saez,
2010) and when K is a notch (Kleven and Waseem, 2013):
N ∗ (1 − t0 )ε , if 0 < N ∗ < N
Y =
K
, if N ≤ N ∗ ≤ N
N ∗ (1 − t1 )ε , if N < N ∗ ,
(3)
where the expressions for the thresholds N and N are given below.
In the case of a kink, N = K(1 − t0 )−ε , and N = K(1 − t1 )−ε . The budget frontier is
continuous, but its slope suddenly decreases at Y = K. For values of N ∗ inside the bunching
interval [N , N ], the agent’s indifference curve is never tangent to the budget frontier, and we
have the non-interior solution Y = K. For values of N ∗ outside of the bunching interval, the
indifference curve is always tangent to some point on the budget frontier.
In the case of a notch, the solution is interior for N ∗ < N = K(1 − t0 )−ε , but there are
no tangent indifference curves for N ∗ ∈ [K(1 − t0 )−ε , K(1 − t1 )−ε ], just as in the case of a
kink. Although tangency occurs for N ∗ > K(1 − t1 )−ε , some of the resulting utility levels are
lower than the utility at the notch point. The budget frontier with a jump-down
discontinuity at Y = K has an interval of income values (K, Y I ] that no agent ever chooses.
The value Y I > K corresponds to the interior solution of the agent with N ∗ = N I ; that is,
the smallest N ∗ such that the agent’s utility is equal to the utility of the agent choosing
Y = K. Thus N = N I , and the solution is at Y = K for N ∗ ∈ [N , N ]. As the ability N ∗
increases above N I , the utility gets larger than the utility at K, and again there is an
interior solution. Supplemental Appendix B.2 has a formal definition of N I in Equation B.3.
To make the solution more tractable, we take the natural logarithm of all variables.
10
Define y = log(Y ), n∗ = log(N ∗ ), k = log(K), s0 = log(1 − t0 ), and s1 = log(1 − t1 ).
n∗ + εs0 , if n∗ < n
y=
k
, if n ≤ n∗ ≤ n
n∗ + εs1 , if n < n∗ .
(4)
As ability n∗ increases, the optimal choice of y increases, except when n∗ falls inside the
bunching interval [n, n], in which y remains constant and equal to k.
2.3
Bunching and the Counterfactual Distribution of Income
The solution in the previous section expresses income as a function of the model
parameters and n∗ . For given values of (t0 , t1 , k, ε), the continuously distributed n∗ maps
into a mixed continuous-discrete distribution for y. The model predicts bunching in the
distribution of y at a kink or notch point (i.e. P(y = k) > 0), but a continuous distribution
of y otherwise. The amount of bunching depends on the elasticity ε and the unobserved
distribution n∗ ,
∗
B ≡ P (y = k) = P (n ≤ n ≤ n) =
Z
n
n
fn∗ (u) du = Fn∗ (n) − Fn∗ (n) ,
(5)
where the length of the interval [n, n] varies with ε.
The literature typically defines B in terms of the counterfactual distribution of income in
the scenario without any kinks or notches. Let counterfactual income be y0 in such case.
The solution to Problem 1 is simply y0 = n∗ + εs0 for every value of n∗ . The variable y0 has
continuous PDF fy0 and CDF Fy0 . The bunching mass is derived as
B=
Z
k+∆y
k
fy0 (u) du = Fy0 (k + ∆y) − Fy0 (k) ,
(6)
where ∆y = ε(s0 − s1 ). Figure 1, Panels a and b, illustrates the distributions of y and y0 ,
and how they relate to each other, to B, and to fn∗ .
11
Saez (2010)’s insight is that the mass of agents bunching B is increasing in the elasticity
ε for a given distribution of y0 . Stated another way, the more agents shift income to the
kink-point k, the more sensitive they are to changes in tax rates. All current bunching and
notching estimators use this insight to identify the elasticity. First, the researcher obtains an
estimate of the counterfactual distribution of y0 and the bunching mass B. Plugging these
into Equation 6 allows us to solve for an estimate of the elasticity.
The treatment of the problem thus far abstracts from the existence of optimization and
friction errors in the solution of Problem 1. In reality, instead of y, researchers typically
observe the distribution of ye = y + e, where e is a random variable accounting for
optimization frictions. In this case, the distribution of ye has the bunching mass distributed
over a range around the kink point, as opposed to being right at the kink.
In a recent survey article, Kleven (2016) summarizes an identification strategy commonly
used in the literature to estimate the distribution of y0 . The ‘‘polynomial strategy’’ was first
proposed by Chetty et al. (2011) (Equations 14-15 and Figures 3-4), and it consists of fitting
a flexible polynomial to an estimate of the PDF of y. The polynomial regression excludes
observations that lie in a range around the kink point. The researcher chooses the range
based on the support of the distribution of friction errors. The polynomial fit is then
extrapolated to this excluded region as a way of predicting fy0 . The procedure is widely used
in the bunching literature; see, for example, Figure 6 by Bastani and Selin (2014) , Figure 1
by Devereux, Liu, and Loretz (2014), and Figure 4 by Best and Kleven (2018). In
supplemental Appendix B.3, we give more details in the context of a simple example, where
n∗ is uniformly distributed. The example shows that such an identification strategy fails to
recover both B and fy0 , even when the proposed polynomial fit is perfect.
The strategy fails for two reasons. First, the distribution y is observed with error, and a
proper deconvolution method must be used to retrieve the distribution of y, given the
distribution of ye. Second, even when the distribution of y is known, it is not possible to
obtain the distribution of y0 inside the integration domain of Equation 6. Although y = y0
12
when n∗ < n, we have that y = k, while y0 = n∗ + εs0 when n∗ ∈ [n, n] (Figures 1a and 1b).
The shape of the distribution of y0 is unidentified when n∗ falls in the bunching interval.
The rest of this paper focuses on the second problem of the identification strategy,
namely the problem of identifying the elasticity, ε, using the distribution of y instead of the
distribution of ye = y + e. In fact, our methods apply to the many examples of bunching that
do not have friction errors, for example, Figure 4 by Glogowsky (2018) and Figure 1 by
Goncalves and Mello (2018). The study of identification in the presence of optimization
frictions is deferred to future research. In work in progress, Cattaneo et al. (2018) study
identification of the distribution of y given the distribution of ye plus minimal assumptions on
the distribution of e.
3
Identification
The general solution to Problem 1 with multiple kinks and notches in supplemental
Appendix B.2 brings new insights to the identification of the elasticity, when compared to
the particular solution in the case of one kink or notch. First, in a budget set with multiple
kinks but no notches, the general solution is simply a combination of solutions local to each
kink. The bunching intervals of consecutive kinks do not overlap (Equation B.4). As a
result, inference methods for the elasticity that are valid in the case of one kink may still be
used locally to each kink.
Second, a notch at k creates an empty interval in the support of the distribution of y
right after k. Such an empty interval may or may not contain the next tax change point
k ′ > k, depending on the value of ε. For example, eligibility for Medicaid benefits in the
United States creates a sizeable notch that may overshadow the next tax change in the
budget set of some individuals. In this case, inference methods that focus on kinks without
accounting for other notches may produce misleading conclusions about the elasticity.
The rest of this section investigates identification with one notch or one kink. We show
that identification is possible with one notch without any restriction on the distribution of
13
n∗ . On the other hand, the identification in case of a kink is impossible, unless the
researcher imposes restrictions on the distribution of n∗ .
3.1
Identification with at Least One Notch
In the problem without optimization error, the existence of one notch produces
additional identifying information in the observed distribution of income. However, the
identification strategy is different from previous studies, which solely rely on Equation 6.
Even when N ∗ has full support (0, ∞), there exists an empty interval in the distribution
of Y , to right of the notch.3 Following the solution in Equation 3, the empty interval is
(K, Y I ], where Y I = N I (1 − t1 )ε , and N I is defined above. Once Y I is identified from the
support of the distribution of Y , we numerically solve for ε that satisfies the indifference
condition Equation 7 below.
Theorem 1. Suppose the support of N ∗ is equal to (0, ∞), that K is a notch, and that the
upper limit of the empty interval in the support of Y to the right of K is equal to Y I . Then
the indifference condition that defines Y I is equivalent to
I
Y + εK
K
YI
1ε
= (1 + ε)
C + I1 + K(1 − t1 )
1 − t1
,
(7)
where C is the consumption value on the budget frontier at the notch point. Moreover, there
exists an unique ε that solves Equation 7 as a function of Y I , K, C, I1 , t1 . Therefore the
elasticity is identified.
A proof for this theorem is in Appendix A.1 and all our other proofs are in Appendix A.
3.2
Lack of Identification With One Kink
Although bunching is increasing in the elasticity for a fixed distribution of y0 or n∗ , it is
also true that, for a fixed elasticity, bunching increases as fn∗ becomes more concentrated
3
In this subsection, it is analytically simpler to work with the solution in levels rather than in logs.
14
between n and n. If all we know about fn∗ is that it is continuous with full support and that
its integral over [n, n] equals B, then there is no way to identify both the elasticity and fn∗
using only Equation 5; equivalently, there is no way to identify both the elasticity and the
distribution of y0 using only Equation 6. Intuitively, identification using only (5) or (6) is
impossible because each uses one equation to solve for two unknowns. This is shown by
Blomquist et al. (2015) and Blomquist and Newey (2017). We present the impossibility result
in this section as a building block to our novel identification strategies in the next sections.
Formally, the data and model comprise five objects: 1) the CDF of earnings Fy , 2) the
kink point k, 3) the slopes of the piecewise-linear constraint s0 and s1 ; 4) the CDF of the
latent variable Fn∗ , and 5) the elasticity ε. Equation 4 is a mapping T that takes objects
(2)–(5) and maps them into the CDF of optimal incomes across agents:
Fy = T (k, s0 , s1 , Fn∗ , ε).
The researcher observes objects (1)–(3), but does not observe the last two, Fn∗ and ε.
The problem of identification consists of inverting the mapping T such that the unobserved
ε is a function that only depends on the first three objects (Fy , k, s0 , s1 ), regardless of what
Fn∗ may be. We denote the class of admissible distributions of n∗ as Fn∗ . If the class Fn∗
contains all possible continuous distributions of n∗ , then identification of ε is impossible.
Lemma 1. Let Fn∗ be the class of all CDFs Fn∗ that have continuous PDFs fn∗ with
support (−∞, ∞). Let Fy be the class of all CDFs Fy that are mixed continuous-discrete
with one mass point at k, and continuous PDF fy otherwise. Take Fy , k, s0 , and s1 as
givens. For every elasticity ε ∈ (0, ∞), there exists Fn∗ ,ε ∈ Fn∗ such that
Fy = T (k, s0 , s1 , Fn∗ ,ε , ε). Therefore it is impossible to point-identify ε.
Figure 1 provides intuition for the proof of Lemma 1. It illustrates that the observable
PDF fy in Figure 1a is generated by applying Equation 4 to two different combinations of
latent variable distributions and elasticities, fn∗ ,ε and fn∗ ,ε′ in Figures 1c and 1d,
respectively. Lemma 1 clarifies that current bunching methods are either implicitly
restricting Fn∗ or simply inconsistent for the true elasticity.
15
A direct consequence of Lemma 1 is that it is impossible to test restrictions on Fn∗ .
Below we consider a couple of examples of identifying restrictions from the literature.
Example 1. Saez (2010) implicitly restricts Fn∗ when using a trapezoidal approximation to
solve the integral in Equation 6, in levels rather than in logs (Saez’s Equation 4 on page
186). That is,
B=
Z
K+∆Y
K
fY0 (u) du ∼
=
fY0 (K + ∆Y ) + fY0 (K)
2
∆Y,
(8)
where ∆Y = K [((1 − t0 )/(1 − t1 ))ε − 1]. A sufficient condition for the approximation to be
true is to assume fY0 (u) is an affine function of u for values of u ∈ [K, K + ∆Y ]. Given
that Y0 = N ∗ (1 − t0 )ε , the PDF fN ∗ (u) = fY0 (u(1 − t0 )ε )(1 − t0 )ε is restricted to be an affine
function of u inside the interval [K(1 − t0 )−ε , K(1 − t1 )−ε ]. This is equivalent to restricting
fn∗ to have an exponential shape within [k − εs0 , k − εs1 ].
The rest of Saez’s identification strategy uses the fact that fY0 (K) = fY (K − ), and
fY0 (K + ∆Y ) = fY (K + )((1 − t1 )/(1 − t0 ))ε , where fY is the PDF of the continuous portion
of the distribution of Y , and fY (K ± ) denotes side limits limY →K ± fY (K ± ). Substituting
these into Equation 8,
1
B∼
=
2
+
fY (K )
1 − t1
1 − t0
ε
−
+ fY (K ) K
1 − t0
1 − t1
ε
−1 ,
(9)
which is Equation 5 by Saez (2010). It is then possible to solve implicitly for ε as a function
of the side limits of fY , the tax rates, the kink point, and the bunching mass.
One may argue that the affine assumption is a good approximation to any potentially
non-linear density fN ∗ , if the bunching interval [K(1 − t0 )−ε , K(1 − t1 )−ε ] is small. The
problem with this argument is that the size of the interval is itself a function of the elasticity.
It is impossible to state that the interval is small and the linear approximation is a good one
without a priori knowledge of the elasticity.
16
Example 2. The derivation by Chetty et al. (2011) of Equation 6 on page 761 assumes that
the PDF fY0 is constant inside the bunching interval [K, K + ∆Y ]. This is equivalent to
assuming that N ∗ is uniformly distributed in that region and thus restricts the class Fn∗ .
For some scalar a, assume FY0 (u) = a + fY0 (K)u for u ∈ [K, K + ∆Y ], so that the PDF of
Y0 is constant and equal to fY0 (K) in the bunching interval. Then,
Z
K+∆Y
fY0 (u) du = FY0 (K + ∆Y ) − FY0 (K)
ε
1 − t0
−1
=fY0 (K)∆Y = fY0 (K)K
1 − t1
1
−
t
0
∼
=fY0 (K)Kε ln
1 − t1
B/fY0 (K)
,
ε∼
=
1−t0
K ln 1−t1
B=
K
(10)
where the second to last approximate equality uses [(1 − t0 )/(1 − t1 )]ε − 1
∼
= ln[(1 − t0 )/(1 − t1 )]ε for small tax changes; and the last approximate equality is Equation 6
by Chetty et al. (2011). The rest of their identification procedure relies on the polynomial
strategy to obtain B and fY0 (K), as described in the supplemental Appendix B.3.
The constant PDF assumption on fY0 is more restrictive than the affine PDF assumption
that justifies Saez’s trapezoidal approximation. The trapezoidal approximation allows for fY0
to have a non-zero slope in the bunching interval, whereas the constant PDF assumption
does not.
There are more flexible restrictions one could impose on Fn∗ . For example, one could say
n∗ follows a distribution inside a parametric family of distributions.
Example 3. In general, let Fn∗ = {Gn∗ (n; θ) , θ ∈ Θ}, where Gn∗ are CDFs indexed by a
p × 1 vector of parameters θ in a parameter space Θ. Identification of the elasticity requires
that the bunching mass and the shape of the distribution of y around the kink point are
sufficient to identify θ and ε. That is, the family of distributions Fn∗ is such that, for any
feasible choice of (k, s0 , s1 , ε, θ), there exists an unique solution (ε̄, θ̄) = (ε, θ) to the following
17
system of equations:
Gn∗ (k − εs1 ; θ) − Gn∗ (k − εs0 ; θ) = Gn∗ k − ε̄s1 ; θ̄ − Gn∗ k − ε̄s0 ; θ̄
(11)
Gn∗ (u − εs0 ; θ) = Gn∗ (u − ε̄s0 ; θ̄) for ∀u < k
(12)
Gn∗ (u − εs1 ; θ) = Gn∗ (u − ε̄s1 ; θ̄) for ∀u > k.
(13)
For example, the family of normal distributions with unknown mean and variance satisfies
these conditions (see supplemental Appendix B.4). Identification is also possible in families
with more than just two parameters. The objects on the left-hand side (LHS) of the three
equations above, evaluated at the true (ε, θ), are identified from the data. Thus, if Fn∗
satisfies (11)-(13), then the elasticity and Fn∗ are identified.
4
Solutions
The rest of the paper focuses on methods that identify the elasticity in the kink case. We
present three types of identification assumptions on the distribution of ability, from less
restrictive to more restrictive. We start with a non-parametric shape restriction that bounds
the slope magnitude of fn∗ , which leads to partial identification of ε. Next, we connect
bunching to the literature on censored regressions, where n∗ is the regression error. It
becomes natural to use covariates to explain n∗ , and we propose two types of
semi-parametric restrictions on the distribution of n∗ that point-identify the elasticity. The
first restricts the distribution of n∗ , conditional on covariates; and the second restricts a
quantile of the distribution of n∗ , conditional on covariates. In general, more data variation
and structure are needed to provide any information about the elasticity.
4.1
Non-parametric Bounds
Our partial identification approach relies on restricting the class Fn∗ to PDFs, fn∗ , that
are Lipschitz continuous with constant M ∈ (0, ∞). In other words, the slope magnitude of
18
any fn∗ ∈ Fn∗ is bounded by M . The following theorem gives the partially identified set for
ε as a function of identified quantities and the maximum slope magnitude M .
Theorem 2. Assume Fn∗ contains all distributions with PDF fn∗ that are Lipschitz
continuous with constant M ∈ (0, ∞). Then the elasticity ε ∈ Υ, where
|fy (k+ )−fy (k− )| [fy (k+ )+fy (k− )]
∅
,
if
B
<
2M
fy (k+ )−fy (k− )| [fy (k+ )+fy (k− )]
|
Υ=
≤B<
[ε, ε] , if
2M
[ε, ∞) , if fy (k+ )2 +fy (k− )2 ≤ B
2M
fy (k+ )2 +fy (k− )2
2M
,
where ∅ is the empty set, and
1/2
2 [fy (k + )2 /2 + fy (k − )2 /2 + M B]
ε=
M (s0 − s1 )
−2 [fy (k + )2 /2 + fy (k − )2 /2 − M B]
ε=
M (s0 − s1 )
− (fy (k + ) + fy (k − ))
1/2
+ (fy (k + ) + fy (k − ))
.
Figures 1c and 1d provide the intuition behind the derivation of the bounds in Υ. For a
fixed value of ε, the length of the interval [n, n] is fixed. If the magnitude of the derivative
of fn∗ is bounded by M , we obtain maximum and minimum areas under fn∗ over [n, n]. We
repeat this exercise for every value of ε to get a range of possible areas associated with each
ε. Given the probability of bunching B is the area under the true fn∗ over [n, n], the
partially identified set has all values of ε whose range of possible areas contains B. The
partially identified set is empty if M is not big enough to allow for the existence of a
continuous function fn∗ which connects fy (k − ) = fn∗ (k − εs0 ) to fy (k + ) = fn∗ (k − εs1 ). The
partially identified set is unbounded if M is large enough to allow fn∗ to be zero inside the
interval [n, n].
The expression for the partially identified set depends on the value of M and the
researcher must specify this value to compute the bounds. The uniform approximation in
Example 2 says that fn∗ has zero slope inside the bunching interval, that is, M = 0. The
trapezoidal approximation in Example 1 implicitly chooses M = m0 such that m0 is the
19
smallest value of M for which we have bounds that are well defined. Formally, m0 solves
B = |fy (k + ) − fy (k − )| [fy (k + ) + fy (k − )] /2m0 , which makes ε = ε and point-identifies ε.
Thus the exercise of computing bounds necessarily involves assumptions weaker than the
uniform and trapezoidal approximations.
Lemma 1 makes clear that it is impossible to identify, and thus estimate, the value of M .
A useful starting point for the magnitude of M comes from the maximum slope magnitude
of the continuous part of fy , say m1 . The PDF fy is identified and is the shifted PDF of n∗ .
Thus, the maximum slope of fn∗ outside of the bunching interval is identified and equal to
m1 . If we assume that that the slope of fn∗ inside the bunching interval is never bigger than
outside, then M = m1 .
As a rule of thumb, we recommend researchers to plot the bounds in Theorem 2 as a
function of M for a range of values that includes m0 , m1 , and possibly bigger values, e.g., up
to 2m1 . Theorem 2 is important to quantify the magnitude of the impossibility problem
presented in Lemma 1. If the bounds plotted for a range of M values admit elasticities that
are too different in economic terms, then the identifying assumptions play a critical role in
determining the elasticity. We give full details and implement this sensitivity analysis in the
empirical section using our bunching Stata package (Section 5).4
While we assume the PDF has bounded slope, Blomquist and Newey (2017) partially
identify the elasticity by assuming the PDF of heterogeneity is monotone. Our approach has
three valuable properties. The first is that the bounds of our partially-identified set have
closed form solutions. Second, an observed mass point implies a positive elasticity even for
large values of the slope M , which is in line with the theoretical prediction that agents
respond to a change in incentives. Third, it nests and is easily comparable to the original
bunching estimator based on the trapezoidal approximation.
4
It is important to clarify that the problem of choosing M is different than the typical problem of choosing
a tuning parameter, e.g., a bandwidth or polynomial order in non-parametric estimation. The value of M
represents a choice of functional form assumption, while in non-parametric estimation, you typically choose
the tuning parameter to achieve desirable properties of the estimator for a given functional form assumption.
20
We end this subsection with the case of a budget set with several kinks kj , j = 1, . . . , J,
but no notches. One may ask whether the existence of several kinks helps identify the
elasticity. As noted above, the bunching intervals do not overlap across kinks, that is,
N j = Kj (1 − tj−1 )−ε < Kj (1 − tj )−ε = N j . Lemma 1 applies to each kink, and multiple
kinks do not necessarily point-identify ε, because the distribution of n∗ may be very different
across different bunching intervals.
Multiple kinks do help with the identification of ε, as long as the researcher restricts the
slope of fn∗ and believes the model in Equation 1 applies to all individuals. This arises from
the fact that every individual is assumed to have the same elasticity parameter ε, and that
the bounds of Theorem 2 vary in length as Bj , fy (kj± ), sj vary across cutoffs j = 1, . . . , J.
The partially identified set is narrowed down by the intersection of bounds specific to each
one of the multiple kinks.
Corollary 1. Assume the conditions of Theorem 2 for each kink kj , j = 1, . . . , J. Then the
T
elasticity ε ∈ Jj=1 Υj , where Υj is the partially identified set of Theorem 2 applied to kink
kj .
4.2
Semi-parametric Identification with Covariates
Identification with kinks is impossible when the distribution of ability n∗ belongs to the
non-parametric class of all continuous distributions. Parametric functional form assumptions
identify the elasticity, but identification relies on fitting such functional form to
non-bunching individuals and extrapolating the functional form to bunching individuals.
This section considers alternative identification assumptions that rely on the existence of
additional covariates in the dataset. There is strong empirical evidence suggesting that
ability is well explained by individual characteristics, such as age, demographics, filing status,
etc. For example, the ability distribution of young workers may have a very different mean
and variance, compared to that of older workers. Extrapolations based on covariates that
predict n∗ are much more reasonable than extrapolations solely based on the shape of the
21
PDF of n∗ . The key assumption is that covariates that help explain the distribution of n∗ for
non-bunching individuals also help explain the distribution of n∗ for bunching individuals.
We start by connecting bunching to censored regression models. This allows us to relate
to the vast econometrics literature in this area. Consider again the data generating process
given by Equation 4. The model for y is a mid-censored model, where the error term is n∗ ,
the intercept to the left of the kink is εs0 , the intercept to the right of the kink is εs1 , and
the censoring point is k. The main difference between (4) and a typical censored regression
model is that the latter has the censoring point at either the minimum or maximum of the
distribution of y (see Equation 15 in the next subsection). Identification, estimation, and
inference in these models have been widely studied in econometrics since Tobin (1958).
There are many advantages of framing the estimation of ε as estimation of a censored
model. Surveys of censoring models and their applications are provided by Maddala (1983),
Amemiya (1984), Dhrymes (1986), Long (1997), DeMaris (2005), and Greene (2005). There
are straightforward extensions that account for optimizing frictions. Moreover, censored
models are easily estimated with a number of different techniques that are available in many
computer packages. Most importantly, it becomes extremely practical to add covariates as
explanatory factors for the distribution of n∗ .
Assume the researcher has access to a vector of covariates X ∈ R1×(d+1) , where X
contains an intercept variable and the distribution of X is unrestricted. We build on
censoring models with covariates to identify the elasticity by imposing two types of
semi-parametric assumptions on the distribution of n∗ .
The first type of assumption states that the distribution of n∗ is a mixture of normal
distributions averaged over the distribution of covariates. This assumption does not imply
conditional normality of n∗ given X but it is implied by conditional normality of n∗ .
Although the Tobit likelihood assumes normality of the unobserved distribution conditional
on covariates, we demonstrate that the Tobit estimator remains consistent under the
semi-parametric class of normal mixtures. In addition, the researcher may estimate a
22
truncated Tobit model on data in a small neighborhood of the kink point, which requires
even weaker distribution assumptions for consistency. This robustness property remains true
if we replace the normal distribution by another parametric distribution to form the
semi-parametric mixture. For example, the maximum likelihood estimator for the elasticity
that assumes that n∗ conditional on X is exponential remains consistent when the
unconditional distribution of n∗ is a mixture of exponentials averaged over X, whether or
not the distribution of n∗ conditional on X is exponential. In the rest of this section, we keep
this first type of assumption in terms of normals for ease of exposition and practical reasons:
the Tobit likelihood is globally concave and software to estimate Tobit models is ubiquitous.
The second type of assumption imposes a parametric functional form on a quantile of the
conditional distribution of n∗ , given X. Sufficient variation in covariates yields
point-identification of the elasticity, which is consistently estimated by mid-censored
quantile regressions.
4.2.1
Tobit Regression
The first type of semi-parametric assumption is formally stated in Lemma 2 below. In
the meantime, we construct the Tobit estimator by assuming that there exists unique
(β, σ) ∈ R1×(d+1) × R+ , such that
Fn∗ |X (n, x) = Φ
n − xβ
σ
,
(14)
where Fn∗ |X denotes the CDF of n∗ conditional on X, and Φ(·) is the CDF of a standard
normal distribution. Assumption 14 does not restrict the distribution of X; thus the
unconditional CDF of n∗ lives in a semi-parametric class and needs not to be normal. The
more variation in covariates one has, the richer is this class of distributions.
The elasticity parameter ε is consistently estimated using a mid-censored Tobit
regression. Define the error term U = n∗ − Xβ, the latent variables y0∗ = εs0 + Xβ + U , and
23
y1∗ = εs1 + Xβ + U , where y1∗ < y0∗ , since ε > 0 and s0 > s1 . Then y follows a mid-censored
Tobit model
y ∗ , if y1∗ < y0∗ < k
0
∗
∗
y=
k , if y1∗ ≤ k ≤ y0∗ = min{y0 ; max{k; y1 }}.
y ∗ , if k < y ∗ < y ∗
1
1
0
(15)
This is different from the classic Tobit model, where the censoring point is either at the
minimum or at the maximum of the distribution of y. A possible estimation strategy is to
adapt the two-step Heckit estimator to our setting (Heckman, 1976, 1979). In the first step,
estimate a binary outcome for bunching and not bunching individuals including covariates.
In the second step, regress income of not bunching individuals on covariates and the
equivalents of the inverse Mills ratio. Another extremely practical way of estimating this
mid-censored Tobit model is to estimate two classic Tobit models. To see that, construct the
variables y0 = min{y, k} and y1 = max{k, y}. It turns out that y0 follows a right-censored
Tobit with intercept εs0 + β0 , slope coefficients β1 , . . . , βd , where β = (β0 , β1 , . . . , βd ).
Similarly, y1 follows a left-censored Tobit with intercept εs1 + β0 , and slope coefficients
β1 , . . . , βd . Thus, the elasticity is consistently estimated by the difference of both intercepts
(εs1 + β0 ) − (εs0 + β0 ) divided by (s1 − s0 ). Despite its practicality, this estimation strategy
does not constrain the slope coefficients and variances to be equal on both sides of the cutoff,
which translates into loss of efficiency. The mid-censored Tobit likelihood naturally takes
these constraints into account and provides the most efficient estimates. It is therefore our
preferred implementation.
Let (yi , Xi ), i = 1, . . . , n be an iid sample of observations. The maximum likelihood
estimator (MLE) for (ε, β, σ) is constructed by maximizing the log-likelihood function of the
24
sample of yi s, conditional on Xi s.
L(y1 , . . . , yn |X1 , . . . , Xn ; ε, β, σ)
n
1
yi − εs0 − Xi β
1X
I {yi < k} log
φ
=
n i=1
σ
σ
k − εs1 − Xi β
k − εs0 − Xi β
+ I {yi = k} log Φ
−Φ
σ
σ
yi − εs1 − Xi β
1
+ I {yi > k} log
φ
σ
σ
n
1X
≡
ℓi (ε, β, σ).
(16)
n i=1
Regardless of the true distribution Fn∗ |X , the MLE is consistent for the parameter that
maximizes the population average of the log-likelihood function, that is,
(ε̄, β̄, σ̄) ∈ arg max E[li (ε, β, σ)]. Standard textbook analyses of Tobit models demonstrate
uniqueness of (ε̄, β̄, σ̄) as solution to the maximization problem.5 We say the elasticity is
identified by a mid-censored Tobit when the true parameter ε coincides with ε̄. We show
that the normality Assumption 14 is not necessary for identification of ε.
Lemma 2. Let Gn∗ (n; β, σ, FX ) = E Φ
n−Xβ
σ
, where the expectation is taken over the
distribution of X with CDF FX . Assume the true distribution of n∗ belongs to the
semi-parametric family
Fn∗ = {Gn∗ (n; β, σ, FX ), (β, σ, FX ) ∈ R1×(d+1) × R+ × FX },
(17)
where FX is the class of all CDFs of X. Suppose Fn∗ satisfies (11)–(13) for the true FX .
Define Gy (y; ε, β, σ, FX ) to be the unconditional CDF of y obtained by transforming
Gn∗ (n; β, σ, FX ), according to Equation 4 and a given value of ε. Let Fy (y) be the true CDF
of y. If Gy (y; ε̄, β̄, σ̄, FX ) = Fy (y), then ε̄ equals the true elasticity, regardless of Assumption
14. This remains true if we replace the normal distribution by another parametric
5
For example, see Hayashi (2000), Section 8.3.
25
distribution to form the semi-parametric mixture in (17).
If the Tobit best-fit distribution of y matches the true distribution of y, Lemma 2
guarantees that the elasticity estimated by the Tobit is consistent for the true elasticity,
regardless of whether Fn∗ |X is normal. Essentially, Lemma 2 requires the unconditional
distribution of n∗ to be a mixture of normal distributions, where the average is taken across
the distribution of covariates. Standard quasi-MLE asymptotic inference procedures apply
here. Namely, the MLE (ε̂, β̂, σ̂) obtained from (16) and centered at (ε̄, β̄, σ̄) is
asymptotically normal, with zero mean and the usual variance-covariance matrix in the
‘‘sandwich form.’’
One of the features of bunching estimators is the reliance on data local to the kink point.
With the mid-censored Tobit model, the researcher may also restrict the sample to
observations of y lying in a small neighborhood of k and estimate a truncated Tobit.6 The
truncated Tobit is an attractive estimation strategy, because consistency of ε̂ relies on a
much weaker version of Assumption 14. Moreover, the smaller the truncation window, the
easier it is to fit the unconditional distribution of y with a Tobit, and the stronger is the
robustness result of Lemma 2.
As a matter of routine, we recommend researchers estimate a truncated Tobit model for
various window sizes around the kink point and examine two things: first, the plot of the
estimated elasticity as a function of the size of the truncation window; second, the plot of
the best-fit Tobit distribution of y compared to the histogram of y for various sizes of
truncation windows. The distribution fit tends to improve as the size of the window
decreases. The better the fit, the more likely the conditions of Lemma 2 are met, and the
closer is the elasticity to the truth. We illustrate this exercise with simulated data below and
with real data in Section 5.
6
The truncated Tobit model has a log-likelihood that is slightly different than (16). Instead of the
log-likelihood of y|X, it maximizes the log-likelihood of y|X, k − δ < y < k + δ for δ > 0, which has a
truncated normal distribution.
26
Consider the following simulation experiment. Let U1 and U2 be Bernoulli with
√
probability of success 2/2, and U3 be normal with mean 1.59 and variance 0.72 , where all
three variables are independent. Let the covariates be X1 = U1 and X2 = U1 U2 , and ability
√
be n∗ = 2X1 + 2X2 + U3 . This model was chosen to match moments of the real data in
Section 5. Generate an iid sample with 500,000 observations of X1 , X2 , and y, according to
Equation 4 with ε = 1. As in the EITC example in Section 5, the kink point is k = 2.1494
(i.e., log of 8.580), with t0 = −0.34 and t1 = 0.
The first exercise estimates a mid-censored Tobit that is correctly specified with both
covariates X1 and X2 . We start with the full sample of simulated data and produce
estimates for truncation windows that are symmetric around the kink point and shrink in
size. For example, Figures 2a and 2b show the histogram of simulated data for y, and the
best-fit Tobit distributions for two truncation sizes, 100% and 40%. Although fn∗ |X1 ,X2 is
normal, it is clear from the figures that fy is not a censored normal and therefore fn∗ is not
normal. Figure 2c displays the elasticity estimate as a function of the percentage of data used
in each truncated estimation. The elasticity estimate is stable over all truncation windows,
because the model is correctly specified. The Tobit fits the distribution of y perfectly for all
truncation windows, and the estimated elasticity is approximately equal to the truth.
The second exercise estimates a misspecified model using the same simulated data.
Specifically, we drop X2 out of the model. In this case, fn∗ |X1 does not have a normal
distribution, and Assumption 14 is not satisfied. Estimation using all of the data does not fit
the distribution of y (Figure 2d). On the other hand, Figure 2e demonstrates that the
truncated Tobit matches the distribution of y perfectly for windows that use 40% of the data
or less. In line with Lemma 2, elasticity estimates converge to the truth, as the truncation
window decreases below 40%.
4.2.2
Censored Quantile Regressions
Another type of semi-parametric assumption on the ability distribution consists of
27
restricting a quantile of the distribution of n∗ , conditional on X. Namely, for τ ∈ (0, 1), we
assume that there exists an unique β(τ ) ∈ R1×(d+1) such that
Qτ (n∗ | X) = Xβ(τ ),
(18)
where Qτ denotes the τ -th quantile of a distribution. A common choice in applied work is
τ = 1/2 or the median regression. The restriction in (18) may be a flexible one if one includes
transformations of X on the right-hand side, e.g., polynomials and interaction terms.
Equation 15 leads to y = min{εs0 + n∗ ; max{k; εs1 + n∗ }}, which is an increasing and
continuous function of n∗ . The quantile of an increasing and continuous function of n∗ is
equal to that same function evaluated at the quantile of n∗ . Using Assumption 18,
Qτ (y | X) = min{εs0 + Xβ(τ ); max{k; εs1 + Xβ(τ )}}.
(19)
For those observations such that X ′ β(τ ) < k − εs0 or X ′ β(τ ) > k − εs1 , the quantile
Qτ (y | X) varies linearly with X; otherwise, it is constant and equal to k. Intuitively, if
there is enough variation in X for uncensored observations, then the slope coefficients and
the intercepts are identified. This leads to identification of ε.
Lemma 3. Define X̃ = [X, I {Qτ (y | X) > k}], a random vector in R1×(d+2) . Assume
h
i
E I {Qτ (y | X) 6= k} X̃ ′ X̃ has full rank and that Assumption 18 holds. Then ε is
identified.
In the absence of covariates or restrictions on Qτ (y | X), the rank condition is never
satisfied. This confirms the impossibility demonstrated in Lemma 1. For example, suppose
the researcher has two dummy variables, W1 and W2 . An unrestricted Qτ (y | W1 , W2 )
contains four parameters, because the conditional quantile takes at most four possible values.
In the best case scenario for identification, these four values are all different from k. In
terms of Lemma 3, d = 3, and X = [1, W1 , W2 , W1 W2 ] is 1 × 4. The matrix
28
h
i
E I {Qτ (y | X) 6= k} X̃ ′ X̃ is 5 × 5 but has rank equal to 4 at most. Thus Qτ (y | X) must
be restricted to fewer parameters for identification to be possible.
Theoretical work on estimation and inference of parameters in censored quantile
regression (CQR) models dates back to the 1980s (Powell (1984, 1986)). Recent advances
include the computationally attractive three-step estimator by Chernozhukov and Hong
(2002), and CQR with endogeneity by Chernozhukov et al. (2015). In the simpler case of
Qτ (y | X) = Xβ(τ ), Koenker and Bassett (1978) show that a consistent estimator for β(τ )
is obtained by the solution to the problem
min
b∈Rd+1
n
X
i=1
[ρτ (yi − Xi b)] ,
(20)
where (yi , Xi ) i = 1, . . . , n is an iid sample and ρτ (u) = (τ − 1 (u ≤ 0)) u is the so-called
‘‘check function.’’ In our case, the parametric conditional quantile function Qτ (y | X) is
given in Equation 19. The slope and intercept coefficients are estimated by
(b̂(τ ), δ̂(τ )) = arg min
b∈Rd ,δ∈R
n
X
i=1
[ρτ (yi − min{Xi′ b; max{k; Xi′ b + δ}})] ,
(21)
where b̂(τ ) is consistent for β(τ ) + [εs0 , 0, . . . , 0]′ , and δ̂(τ ) is consistent for ε(s1 − s0 ).
Therefore the elasticity is consistently estimated by ε̂ = δ̂/(s1 − s0 ), and it is asymptotically
normal.
The optimization problem in Equation 21 is computationally difficult. For the left (or
right) censored case, Chernozhukov and Hong (2002) proposed a fast and practical estimator
that consists of three steps. Our case of middle censoring requires a straightforward
modification of their method. We delineate practical steps to obtain ε̂ and its standard error
using CQR in Section B.5 of the supplemental appendix.
29
5
Application to EITC
We demonstrate and compare our new methods using bunching behavior created by
kinks in the earned income tax credit (EITC). Each method differs in the assumptions they
make about the unobserved distribution to achieve identification. There is no way to
determine which assumption is correct because the unobserved distribution is not fully
identified. Nevertheless, estimates that are stable across many methods indicate that
different identifying assumptions do not play a major role in the construction of those
estimates. On the contrary, estimates that are sensitive to different assumptions are
dependent on the validity of those assumptions. Patel, Seegert, and Smith (2016) provide an
empirical illustration of this sensitivity.
First, we use our non-parametric bounds to provide initial information about how
sensitive the elasticity estimate is to different shapes of the underlying ability distribution.
When the bounds are tight, then the shape of the underlying distribution is not critical. But
when the bounds are wide, then the shape is critical. In this case, reducing the range of
possible elasticities requires either stronger restrictions on the shape of the ability
distribution or additional data on determinants of ability.
Second, we combine observed determinants of ability with our semi-parametric approach
to point identify the elasticity. We compare the resulting best-fit Tobit income distribution
to the observed distribution for alternative samples that range from using all observations to
using only data local to the kink. When the best-fit Tobit distribution coincides with the
observed distribution, the estimated elasticity is consistent (Lemma 2). Furthermore, if the
Tobit elasticity is within narrow non-parametric bounds, then the identifying assumptions
are inconsequential; if within wide bounds, then the identifying assumptions are not
contradictory and the covariates provide point identification. In contrast, if the Tobit
elasticity is outside of the bounds, then the elasticity estimate is not robust to the two
alternative identifying assumptions. Finally, when the best-fit Tobit distribution does not
30
coincide with the observed distribution, the determinants of ability used for estimation are
uninformative or the semi-parametric assumption is inappropriate.
We recommend that researchers examine the sensitivity of elasticity estimates across all
available methods as a matter of routine. We illustrate these steps in the context of the
EITC in the rest of this section.
5.1
Data
We use data from the Individual Public Use Tax Files, constructed by the IRS. The
annual cross-section for each year 1995 to 2004 includes sampling weights which allow
interpretation of any estimates as being based on the population of U.S. income tax returns.
This data was initially used by Saez (2010) to demonstrate how to use bunching to estimate
an elasticity.7
The income distribution for individuals with one child demonstrates clear bunching
around the $8,580 kink (year 2008 dollars) in the EITC schedule. Because the marginal tax
rate increases from −34 percent to 0 percent at $8,580, individuals have strong incentives to
report less income than if the tax rate had remained −34 percent above the kink point.
Observed bunching in the distribution of income suggests that people do respond to
changes in tax rates. To effectively set tax rates, however, it is imperative to quantify this
response precisely. Small variation in elasticity estimates imply large differences in optimal
tax rates. For example, variation in the elasticity of taxable income between 0.1 and 0.2
implies an optimal top marginal tax rate between 82% or 69%.8 As demonstrated above,
7
We replicate some of the estimates in Saez (2010) using publicly available code from the website of the
American Economic Journal: Economic Policy and report them in supplemental Appendix B.6.
8
This example comes from Saez (2001). In particular, Equation 9 states τ̄ = (1 − g)/(1 − g + εu εc (a − 1)),
where g is defined as the value the government has for the marginal consumption of high income earners
(often set to 0), a is the Pareto parameter (with baseline value of 2), and εc and εu are the compensated and
uncompensated elasticities of taxable income. For the calculation in the text, we utilize εu = εc , a Pareto
parameter of 2, and a g value of 0.1.
31
identifying the elasticity requires information on the amount of bunching and the income
distribution. The following sections show how different methods leverage different types of
variation to identify the elasticity.
The methods of this paper are designed for data without friction errors or sharp
bunching. For examples of sharp bunching data, see Figure 4 by Glogowsky (2018) and
Figure 1 by Goncalves and Mello (2018). Nevertheless, the IRS data do have friction errors
as the excess mass due to bunching is visibly dispersed in a small interval near the kink (for
example, Figure 5 by Weber (2016)). Therefore, to apply our procedures to the IRS data, we
first need to filter reported income out of friction error. A proper deconvolution theory must
be developed to tackle this problem, but it is beyond the scope of this paper. For now, we
simply need a practical way of removing friction error before applying the different bunching
estimators, so that they may be properly compared.
Following the intuition of Chetty et al. (2011), we fit a seventh-order polynomial to the
empirical CDF of reported income with friction errors ỹ. As does Saez (2010), we exclude
observations that lie within $1,500 of the kink and allow an intercept change at the kink.
The extrapolation of the fitted polynomial to the excluded region results in a CDF with a
jump discontinuity at the kink. This is an estimate for the CDF of income without friction
error, that is, Fy (y). The size of the discontinuity equals the bunching mass. We then rely
on the fact that y = Fy Fỹ−1 (ỹ) and use the estimated CDFs to transform ỹ into y.
Our filtering procedure is different from the polynomial strategy discussed in Section 2.3.
We simply aim at removing the friction error from the sample, while the the polynomial
strategy of Example 2 aims to remove friction error and recover the counterfactual
distribution of income, which requires much stronger restrictions according to Lemma 1. Our
filtering procedure works well in cases in which 1) the researcher has a good prior on the
support of the friction error distribution ($1,500 in this case), 2) the friction error affects
bunching individuals more than non-bunching individuals, or 3) the variance of the friction
32
error is small. A more general filtering method is deferred to future work.9
5.2
Estimates Across Methods
Table 1 reports estimates of the elasticity of taxable income using a classic bunching
method, non-parametric bounds, and Tobit models with covariates. Each of these estimates
relies on a different set of assumptions to identify the elasticity of taxable income, and
together they provide insights into which assumptions are most defensible in the context of
the EITC.
Column 1 reports our estimates of the elasticity of taxable income using a trapezoidal
approximation (Example 1).10 This method assumes the unobserved PDF is linear in the
bunching region, which prior literature believed to approximate non-linear distributions well.
In practice, the appropriateness of this approximation depends on the true distribution and
length of the bunching region, which are both unobserved. Linearity may be inappropriate if
the distribution is sufficiently non-linear or the bunching region is wide. Column 1
demonstrates substantial heterogeneity in estimates across different subsamples. In
particular, the elasticity estimate is 0.426 for the all filers sample, 0.854 for self-employed
individuals, 1.102 for self-employed married individuals, and 0.784 for self-employed not
married individuals.
The guidelines for implementation of our non-parametric bounds in Section 4.1 utilizes a
range of values for M that includes the maximum slope magnitude of fy . We reiterate that
M is unidentified and that the slope of fy provides a starting point. The bunching Stata
package consistently estimates the maximum slope of fy by taking the maximum slope in
the histogram of y across all consecutive bins. We find that the slope is never bigger than
0.5 across our subsamples. For a more conservative view, we report our non-parametric
9
Supplemental Appendix B.6 recomputes our estimates using the filtering procedure employed by Saez
(2010).
10
We estimate the PDF of the variables in logs rather than in levels, which simplifies the elasticity
formula based on the trapezoidal approximation in Example 1.
33
bounds using M = 0.5 and M = 1 in Table 1, Columns 2 and 3, and plot bounds for M up
to 2 in Figure 3. The vertical lines in these figures designate the minimum and maximum
slope, such that both the upper and lower bounds are finite numbers. The first line is the
smallest slope that allows a continuous PDF to be consistent with both the bunching mass
and observed income distribution. At the minimum slope, both lower and upper bounds are
equal to the estimate based on the trapezoidal approximation, reported in column 1.
As M increases, the set of possible PDF shapes in the bunching region becomes richer.
The second line is the maximum slope before the set of possible distributions allows for a
PDF that touches zero in the bunching interval. In that case, the bunching mass remains
constant for arbitrarily large ε, and the upper bound is infinity (Theorem 2).
A large range between lower and upper bounds in Figure 3 suggests the estimates change
substantially with the shape of the unobserved distribution. For example, the bounds are
uninformative for the self-employed married sample, even for small values of M . This
indicates that the data will not provide precise information on the elasticity unless the
researcher imposes further functional form restrictions on the distribution of n∗ . In contrast,
we learn the most in the case of all filers and self-employed not married, where the bounds
are narrower than in other subsamples for M = 0.5. The lower bound is always defined for
larger choices of M , which gives partial information on the elasticity without the need of
being precise with the choice of M . For the exceedingly high value of M = 2, the lower
bound is about 0.25 for all filers and 0.5 for the other three subsamples.
Columns 4–7 report our estimates of the Tobit model using the full sample and truncated
samples at 75%, 50%, and 25% of the data. Figures 4–7 complement these estimates by
graphing the actual distribution and the implied distribution from the Tobit estimates at
different levels of truncation in panels a through e. These estimates incorporate covariates,
including indicator variables for married, tax preparer used, real estate interest deduction,
employment status, contributed to charity, tax form used, and tax filing status. The fit of
the Tobit model generally improves as we truncate the sample closer to the kink, which
34
implies that the semi-parametric assumption of mixed normals is more reasonable locally
than globally. The minimum truncation necessary for a reasonable fit varies by subsample.
For example, for self-employed not married, the fit seems reasonable using 80% or less of the
data, but for all filers, the fit only becomes reasonable at around 20%. It is interesting to
observe that the Tobit with covariates fit the distribution better in narrower cuts of the data
than for all filers.
Panel f in Figures 4–7 graph the elasticity estimate as a function of the percentage of
data used. The estimates tend to plateau as the distribution fits improve. For example, in
the self-employed not married sample depicted in Figure 7, the estimates are all around 0.75,
using less than 80% of the data. It is worth pointing out that truncated samples with less
than 20% of the data lead to numerical issues, such as perfect collinearity of covariates and
lack of convergence in the likelihood maximization. This leads to imprecise estimates, as
indicated by an upward bend in left extremity of the curves depicted in panel f of Figures
4–7.
5.3
Comparisons Across Methods
Comparisons across methods provides insights into the reasonableness of different
assumptions used to estimate the elasticity. The trapezoidal approximation is always within
the bounds, because its estimate is based on a linear interpolation of the PDF in the
bunching region. The slope of such line equals the minimum slope for which the bounds are
defined. In contrast, the Tobit model using 100% of the data is often below the lower bounds,
but the Tobit distribution fails to fit the observed distribution of income globally. Truncated
Tobit estimates generally enter the bounds as the truncation window decreases and, as a
result, the fit of the Tobit distribution improves. For the all filers sample, an M larger than
1 is needed for the bounds to cover the Tobit estimate truncated at 25%. This reiterates our
previous discussion that the Tobit fit for all filers is poor until we use 20% or less of the data.
Consider self-employed married and self-employed not married filers. Figures 6 and 7
35
demonstrate that the bunching mass is larger for self-employed not married individuals than
for self-employed married individuals. This difference in bunching mass might lead a
researcher to conjecture that the elasticity is larger for self-employed not married individuals.
Whether this conjecture is true depends, however, on differences in the underlying
distribution of heterogeneity. Estimates based on the trapezoidal approximation contradict
that conjecture. The estimates in column 1 of Table 1 are larger for self-employed married
individuals than self-employed not married. The global distributional assumption of a
mixture of normals averaged over covariates produces a larger elasticity for self-employed
not married (column 4 in Table 1). However, the credibility of these Tobit estimates is
questioned by the poor fit shown in panel a of Figures 6 and 7. Truncating the sample
obtains a better fit, and we find that the elasticities are approximately the same for married
and not married. The disagreement across methods for these subsamples indicates that
assumptions on the distribution of heterogeneity are critical to obtain informative elasticity
estimates.
6
Conclusion
We show how to use bunching from piecewise-linear budget constraints to identify
elasticities, under conditions weaker than those used in the literature on kinks and notches.
The key theoretical point is that bunching is determined by the elasticity parameter and the
shape of an unobserved distribution. Additional assumptions or data are needed to identify
the elasticity.
We propose a suite of estimation techniques that allow researchers to tailor their
estimation to different assumptions and data variation. These include non-parametric
bounds and semi-parametric censored models with covariates. The non-parametric bounds
are the least restrictive method and also nest estimators from the previous literature.
These techniques have wide applicability, because piecewise-linear budget constraints are
common across fields, from public finance and labor, to industrial organization and
36
accounting. Our estimation strategies also provide a foundation for future advances in
techniques that will account for different empirical hurdles. Of particular interest are
extensions that consider optimization and friction errors, extensive margin responses, and
panel data methods.
7
Acknowledgements
The views expressed in this paper are those of the authors and do not necessarily reflect the
views of the Federal Reserve Board or the Federal Reserve System. We would like to thank
Matias Cattaneo, Bill Evans, Roger Gordon, Jim Hines, Dan Hungerman, Michael Jansson,
Henrik Kleven, Brian Knight, Erzo Luttmer, Byron Lutz, Dayanand Manoli, Magne
Mogstad, Whitney Newey, Andreas Peichl, Emmanual Saez, Dan Silverman, and Joel
Slemrod for valuable comments and discussions. The paper also benefited from feedback
received from seminar participants at the UCSD Workshop on Bunching Estimators,
Econometric Society, International Association for Applied Econometrics, International
Institute of Public Finance, National Tax Association, Dartmouth College, Federal Reserve
Board, and University of Michigan. Jessica C. Liu, Michael A. Navarrete, and Alexis M.
Payne provided excellent research assistance. All remaining errors are our own. Bertanha
acknowledges financial support received while visiting the Kenneth C. Griffin Department of
Economics, University of Chicago.
37
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Figure 1: Identification of the Elasticity in the Case of a Kink
(a) Distribution of Observed Income
(b) Counterfactual Distribution of Income in the
Absence of Kink
(c) Distribution of Ability Consistent with
Observed Income and Higher Elasticity
(d) Distribution of Ability Consistent with
Observed Income and Lower Elasticity
Notes: Panel 1a plots an example of PDF of y. The continuous portions are equal to the PDF of ability n∗
shifted by εs0 for y < k, and by εs1 for y > k, respectively. The shaded area represents a discrete mass point
with probability B = P (y = k), that is, the probability of bunching. Panel 1b shows the counterfactual PDF
of y0 , that is, the distribution of income if tax rates did not change at the kink. The PDF of y0 is continuous,
and equals the PDF of n∗ shifted by εs0 . It is also equal to the PDF of y before the kink, and to the shifted
PDF of y after the kink. However, the distribution of y does not reveal the shape of the PDF of y0 in the
bunching region (i.e. φ). The shaded area under φ integrates to the probability of bunching B. The last two
panels (Panels 1c-1d) display two different distributions of n∗ that generate the same distribution of income
y (Panel 1a) with two different elasticities, ε < ε̄, according to Equation 4. The PDF of n∗ outside of the
bunching region is equal to the PDF of y shifted by εs0 , if n∗ < k − εs0 ; or shifted by εs1 , if n∗ > k − εs1 .
Aside from B, the distribution of income does not contain any information about the shape of φ in the PDF
of n∗ . If we assume fn∗ is Lipschitz continuous with known constant, it is possible to derive upper and lower
bounds for φ, which correspond, respectively, to lower and upper bounds on the elasticity (Theorem 2).
43
Figure 2: Robustness of Tobit Estimates to Lack of Normality
(a) 100% of the data used for estimation
Data
Tobit model
0.20
0.00
0
2
4
6
Earnings (Log thousands of 2008 $)
0.20
0.00
0
2
4
6
Earnings (Log thousands of 2008 $)
8
0.20
1.25
1
0
2
4
6
Earnings (Log thousands of 2008 $)
.75
8
0
10
20
30
40
50
60
70
80
90
100
90
100
Percent of data used for estimation
(f) Elasticity by percent used
2
0.80
Earnings density (100 bins)
Earnings density (100 bins)
44
0.40
0.40
1.5
(e) 40% of the data used for estimation
Data
Tobit model
0.60
0.60
0.00
8
(d) 100% of the data used for estimation
0.80
1.75
Elasticity estimate
0.40
2
1.75
Elasticity estimate
0.60
(c) Elasticity by percent used
0.80
Earnings density (100 bins)
0.80
Earnings density (100 bins)
(b) 40% of the data used for estimation
0.60
0.40
0.20
0.00
1.5
1.25
1
0
2
4
6
Earnings (Log thousands of 2008 $)
8
.75
0
10
20
30
40
50
60
70
80
Percent of data used for estimation
Notes: The simulation experiment illustrates the robustness of Tobit estimates to deviations from the normality assumption (Assumption 14). The
experiment generates 500,000 observations of y, and two covariates (X1 , X2 ), assuming ε = 1, and n∗ |X1 , X2 ∼ Normal(β0 + β1 X1 + β2 X2 , σ 2 ) (see
details in Section 4.2.1). As in the EITC case, the kink point is k = 2.1494, with t0 = −0.34 and t1 = 0. The first exercise estimates a mid-censored
Tobit that is correctly specified with both covariates X1 and X2 . Panels (a) and (b) show the histogram of simulated data for y, and the best-fit Tobit
distributions for two truncation sizes, 100% and 40% of the sample used. Panel (c) displays the elasticity estimate as a function of the percentage
of data used in each truncated estimation, along with 95% confidence bands. The second exercise drops X2 and estimates a misspecified model.
Panels (d)-(f) are analogous to Panels (a)-(c), except that they use the estimates from the misspecified Tobit model, where n∗ |X1 is not normal. The
estimation truncated at 40% fits the distribution of y, and the elasticity converges to the true value (Lemma 2).
Table 1: Estimates Using U.S. Tax Returns 1995--2004
Statistical Model
All
Elasticity (ε)
Self-employed
Elasticity (ε)
Self-employed,
married
Elasticity (ε)
45
Self-employed,
not married
Elasticity (ε)
(1)
Trapezoidal
Approximation
(2)
Theorem 2
Bounds
M = 0.5
(3)
Theorem 2
Bounds
M=1
(4)
Tobit
Full Sample
(5)
Tobit
Trunc. 75%
(6)
Tobit
Trunc. 50%
(7)
Tobit
Trunc. 25%
(8)
0.426
(0.0289)
[0.376, 0.521]
[0.342, ∞]
0.195
(0.0001)
0.280
(0.0002)
0.291
(0.0002)
0.326
(0.0002)
Sample
details
Obs. 189.1m
Avg. $54.1k
Std. $131.1k
0.854
(0.0885)
[0.721, 1.294]
[0.639, ∞]
0.603
(0.0006)
0.790
(0.0008)
0.787
(0.0008)
0.796
(0.0009)
Obs.
Avg.
Std.
33.5m
$61.8k
$168.2k
1.102
(0.3081)
[0.718, ∞]
[0.587, ∞]
0.373
(0.0006)
0.586
(0.0010)
0.692
(0.0012)
0.722
(0.0013)
Obs.
Avg.
Std.
24.0m
$75.0k
$185.6k
0.784
(0.1024)
[0.741, 0.843]
[0.692, 0.974]
0.894
(0.0010)
0.749
(0.0009)
0.713
(0.0009)
0.753
(0.0014)
Obs.
Avg.
Std.
9.6m
$28.7k
$106.3k
Notes: The table shows estimates of the elasticity for four different subsamples of the IRS data, and using three different approaches discussed in the
paper. The first approach (column 1) uses the trapezoidal approximation to point-identify the elasticity (Example 1). We obtained non-parametric
estimates of the side limits of fy at the kink using the method of Cattaneo, Jansson, and Ma (2019). The estimate for the bunching mass equals the
sample proportion of y observations that equals the kink point (see discussion on friction errors in Section 5.1). We obtained standard errors using 100
bootstrap iterations. The second approach (columns 2 and 3) uses the same estimates of the bunching mass and side limits to compute partially
identified sets for the elasticity (Theorem 2). Upper and lower bounds are calculated for two choices of M, that is, the maximum slope of the PDF of
the unobserved heterogeneity n∗ . Column 4 has Tobit MLE estimates of the elasticity that utilizes the full sample of data, along with robust standard
errors. Columns 5 through 7 report truncated Tobit MLE estimates. As we move from column 5 to column 7, we restrict the estimation sample to
shrinking symmetric windows around the kink that utilizes 75% to 25% of the data. The set of covariates that enters the Tobit estimation is kept
constant across different truncation windows. It includes dummy variables such as marital and employment status, year effects, types of deductions or
social security benefits received, and whether the filer used a tax prep software.
Figure 3: Partial Identification Bounds for the Elasticity
(b) Self-Employed Filers
(a) All Filers
.07
.82
.09
2.25
Upper
Lower
Trapezoidal
1.75
1.5
1.25
1
.75
.5
.25
Upper
Lower
Trapezoidal
2
Elasticity estimate
Elasticity estimate
2
1.75
1.5
1.25
1
.75
.5
0
.5
1
1.5
Maximum slope of the unobserved density
.25
2
(c) Self-employed and Married Filers
0
1
.24
2
1.5
1.25
1
.75
1.54
Upper
Lower
Trapezoidal
2
Elasticity estimate
1.75
1.5
2.25
Upper
Lower
Trapezoidal
2
Elasticity estimate
.5
Maximum slope of the unobserved density
(d) Self-employed and Not Married Filers
.03 .15
2.25
.5
.25
.55
2.25
1.75
1.5
1.25
1
.75
.5
0
.5
1
1.5
Maximum slope of the unobserved density
.25
2
0
.5
1
1.5
Maximum slope of the unobserved density
2
Notes: Panels a through d display partially identified sets for the elasticity for all filers with one child, and
three other subsamples defined by employment and marital status. The y-axis has elasticity values between
lower and upper bounds given various choices of M on the x-axis, that is, the maximum slope magnitude of
the PDF of the unobserved heterogeneity n∗ (Theorem 2). Each panel has two vertical lines. The line on the
left corresponds to the smallest choice of M for which the bounds are defined. At the smallest M , upper and
lower bounds are equal to the elasticity estimate based on the trapezoidal approximation (Example 1). The
vertical line on the right corresponds to the largest choice of M for which the upper bound is finite. Higher
slopes allow for the possibility of PDFs that are zero in the bunching window. As a result, we may have a
finite bunching mass for any arbitrarily large elasticity.
46
Figure 4: Truncated Tobit - All Filers
Data
Tobit model
0.50
0.40
0.30
0.20
0.10
0.00
0
2
4
6
Earnings (Log thousands of 2008 $)
(c) 60% of the data used for estimation
0.70
0.70
0.60
0.60
0.50
0.40
0.30
0.20
0.10
0.00
8
0
2
4
6
Earnings (Log thousands of 2008 $)
(d) 40% of the data used for estimation
(e) 20% of the data used for estimation
0.70
0.70
0.60
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0
2
4
6
Earnings (Log thousands of 2008 $)
8
0.40
0.30
0.20
0.10
0
2
4
6
8
Earnings (Log thousands of 2008 $)
(f) Elasticity by percent used
.95
.85
0.50
0.40
0.30
0.20
0.10
0.00
0.50
0.00
8
Elasticity estimate
47
Earnings density (100 bins)
Earnings density (100 bins)
0.60
Earnings density (100 bins)
Earnings density (100 bins)
0.70
(b) 80% of the data used for estimation
Earnings density (100 bins)
(a) 100% of the data used for estimation
.75
.65
.55
.45
.35
.25
0
2
4
6
Earnings (Log thousands of 2008 $)
8
.15
0
10
20
30
40
50
60
70
80
Percent of data used for estimation
90
100
Notes: the figure displays best-fit Tobit distributions and elasticity estimates for various choices of a symmetric truncation window around the kink
point. Estimation uses the following dummy variables as covariates: marital and employment status, year effects, types of deductions or social security
benefits received, and whether the filer used a tax prep software. The set of included covariates is kept constant across different truncation windows.
Panels a through e show the histogram of income for all filers (bars), along with the best-fit Tobit PDF for each truncation window (line). The best-fit
PDF is constructed using the truncated Tobit likelihood averaged over covariate values in the sample. Panel f displays the Tobit elasticity estimate as
a function of the percentage of data used in estimation.
Figure 5: Truncated Tobit - Self-employed Filers
Data
Tobit model
0.50
0.40
0.30
0.20
0.10
0.00
0
2
4
6
Earnings (Log thousands of 2008 $)
(c) 60% of the data used for estimation
0.70
0.70
0.60
0.60
0.50
0.40
0.30
0.20
0.10
0.00
8
0
2
4
6
Earnings (Log thousands of 2008 $)
(d) 40% of the data used for estimation
(e) 20% of the data used for estimation
0.70
0.70
0.60
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0
2
4
6
Earnings (Log thousands of 2008 $)
8
0.40
0.30
0.20
0.10
0
2
4
6
8
Earnings (Log thousands of 2008 $)
(f) Elasticity by percent used
.95
.85
0.50
0.40
0.30
0.20
0.10
0.00
0.50
0.00
8
Elasticity estimate
48
Earnings density (100 bins)
Earnings density (100 bins)
0.60
Earnings density (100 bins)
Earnings density (100 bins)
0.70
(b) 80% of the data used for estimation
Earnings density (100 bins)
(a) 100% of the data used for estimation
.75
.65
.55
.45
.35
.25
0
2
4
6
Earnings (Log thousands of 2008 $)
8
.15
0
10
20
30
40
50
60
70
80
Percent of data used for estimation
90
100
Notes: the figure displays best-fit Tobit distributions and elasticity estimates for various choices of a symmetric truncation window around the kink
point. Estimation uses the following dummy variables as covariates: marital status, year effects, types of deductions or social security benefits received,
and whether the filer used a tax prep software. The set of included covariates is kept constant across different truncation windows. Panels a through e
show the histogram of income for self-employed filers (bars), along with the best-fit Tobit PDF for each truncation window (line). The best-fit PDF is
constructed using the truncated Tobit likelihood averaged over covariate values in the sample. Panel f displays the Tobit elasticity estimate as a
function of the percentage of data used in estimation.
Figure 6: Truncated Tobit - Self-employed and Married Filers
Data
Tobit model
0.50
0.40
0.30
0.20
0.10
0.00
0
2
4
6
Earnings (Log thousands of 2008 $)
(c) 60% of the data used for estimation
0.70
0.70
0.60
0.60
0.50
0.40
0.30
0.20
0.10
0.00
8
0
2
4
6
Earnings (Log thousands of 2008 $)
(d) 40% of the data used for estimation
(e) 20% of the data used for estimation
0.70
0.70
0.60
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0
2
4
6
Earnings (Log thousands of 2008 $)
8
0.40
0.30
0.20
0.10
0
2
4
6
8
Earnings (Log thousands of 2008 $)
(f) Elasticity by percent used
.95
.85
0.50
0.40
0.30
0.20
0.10
0.00
0.50
0.00
8
Elasticity estimate
49
Earnings density (100 bins)
Earnings density (100 bins)
0.60
Earnings density (100 bins)
Earnings density (100 bins)
0.70
(b) 80% of the data used for estimation
Earnings density (100 bins)
(a) 100% of the data used for estimation
.75
.65
.55
.45
.35
.25
0
2
4
6
Earnings (Log thousands of 2008 $)
8
.15
0
10
20
30
40
50
60
70
80
Percent of data used for estimation
90
100
Notes: the figure displays best-fit Tobit distributions and elasticity estimates for various choices of a symmetric truncation window around the kink
point. Estimation uses the following dummy variables as covariates: year effects, types of deductions or social security benefits received, and whether
the filer used a tax prep software. The set of included covariates is kept constant across different truncation windows. Panels a through e show the
histogram of income for self-employed and married filers (bars), along with the best-fit Tobit PDF for each truncation window (line). The best-fit PDF
is constructed using the truncated Tobit likelihood averaged over covariate values in the sample. Panel f displays the Tobit elasticity estimate as a
function of the percentage of data used in estimation.
Figure 7: Truncated Tobit - Self-employed and Not Married Filers
Earnings density (100 bins)
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0
2
4
6
Earnings (Log thousands of 2008 $)
(c) 60% of the data used for estimation
1.40
1.40
1.20
1.20
1.00
0.80
0.60
0.40
0.20
0.00
8
0
2
4
6
Earnings (Log thousands of 2008 $)
(d) 40% of the data used for estimation
(e) 19% of the data used for estimation
1.40
1.40
1.20
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0
2
4
6
Earnings (Log thousands of 2008 $)
8
0.80
0.60
0.40
0.20
0
2
4
6
8
Earnings (Log thousands of 2008 $)
(f) Elasticity by percent used
.95
.85
1.00
0.80
0.60
0.40
0.20
0.00
1.00
0.00
8
Elasticity estimate
50
Earnings density (100 bins)
Data
Tobit model
Earnings density (100 bins)
Earnings density (100 bins)
1.40
(b) 80% of the data used for estimation
Earnings density (100 bins)
(a) 100% of the data used for estimation
.75
.65
.55
.45
.35
.25
0
2
4
6
Earnings (Log thousands of 2008 $)
8
.15
0
10
20
30
40
50
60
70
80
Percent of data used for estimation
90
100
Notes: the figure displays best-fit Tobit distributions and elasticity estimates for various choices of a symmetric truncation window around the kink
point. Estimation uses the following dummy variables as covariates: year effects, types of deductions or social security benefits received, and whether
the filer used a tax prep software. The set of included covariates is kept constant across different truncation windows. Panels a through e show the
histogram of income for self-employed and not married filers (bars), along with the best-fit Tobit PDF for each truncation window (line). The best-fit
PDF is constructed using the truncated Tobit likelihood averaged over covariate values in the sample. Panel f displays the Tobit elasticity estimate as
a function of the percentage of data used in estimation.
A
A.1
Appendix
Identification with a Notch - Proof of Theorem 1
We present the proof of Theorem 1 in the more general case of multiple tax changes with
at least one notch (Sections B.1 and B.2 in the supplemental appendix). Let p ∈ {1, . . . , L}
be the index of the smallest notch Kp . As explained in the text, the presence of a notch may
remove the next tax change Kp+1 from the solution to the utility maximization problem
with multiple kinks and notches (Lemma B.1 in the supplemental appendix). Let
q ∈ {p + 1, . . . , L} be the index of the next tax change that appears in the solution.
Following the proof of Lemma B.1, the distribution of Y does not have any mass in the
interval (Kp ; YpI ] where YpI = NpI (1 − tq−1 )ε , and NpI is defined as part of the solution in
Equation B.4 in the supplemental appendix. The econometrician observes the value of YpI ,
which is between Kq−1 and Kq . The goal is to solve for ε using this information.
The proof of Lemma B.1 says NpI satisfies the equation below.
NpI (1 − tq−1 )1+ε + ε NpI
−1/ε
(Kp )
1+ε
ε
= (1 + ε) [Cp − Iq−1 + Kq−1 (1 − tq−1 )]
Use the fact that YpI = NpI (1 − tq−1 )ε and YpI
in the equation above to get
YpI (1 − tq−1 ) + ε YpI
− 1ε
− 1ε
(1 − tq−1 ) = NpI
− 1ε
and substitute these
1+ε
(1 − tq−1 )(Kp ) ε = (1 + ε) [Cp − Iq−1 + Kq−1 (1 − tq−1 )]
1ε
Kp
Cp − Iq−1 + Kq−1 (1 − tq−1 )
I
Yp + εKp
(A.1)
= (1 + ε)
YpI
1 − tq−1
The elasticity ε is identified if there exists an unique solution for ε in Equation A.1 as
function of YpI , Kp , Cp , Iq−1 , tq−1 . We know a solution exists, and we show it must be
unique. Consider the left-hand and right-hand sides of (A.1) as functions of ε. The solution
occurs at the value of ε where both of these functions intersect. Uniqueness is equivalent to
single-crossing of these functions.
The function on the right-hand side (RHS) of (A.1) has positive intercept equal to
[Cp − Iq−1 + Kq−1 (1 − tq−1 )]/(1 − tq−1 ). The function on the left-hand side (LHS) has
1ε
p
converges to zero as ε ↓ 0. The intercept of the
intercept equal to YpI because εKp K
YpI
LHS is strictly bigger than the intercept of the RHS:
Cp − Iq−1 + Kq−1 (1 − tq−1 )
1 − tq−1
I
Iq−1 + (Yp − Kq−1 )(1 − tq−1 ) ≷ Cp
YpI ≷
CpI ≷ Cp
where CpI
is strictly
where YpI
CpI > Cp .
is the consumption value on the budget frontier when income is equal to YpI which
greater than Cp . In fact, the consumer is indifferent between (CpI , YpI ) and (Cp , Yp )
> Yp . Since utility is strictly decreasing in Y and increasing in C, we must have
Therefore, YpI > [Cp − Iq−1 + Kq−1 (1 − tq−1 )] /(1 − tq−1 ).
51
The function on the RHS of (A.1) has positive slope equal to
[Cp − Iq−1 + Kq−1 (1 − tq−1 )]/(1 − tq−1 ). The function on the LHS has strictly positive
derivative for any positive ε,
Kp
YpI
1ε
which is strictly positive because Kp > 0,
derivative is strictly increasing with ε,
1ε
∂2
LHS = Kp
∂ε2
∂
LHS = Kp
∂ε
Kp
YpI
Kp
YpI
1ε
1
1 − ln
ε
Kp
YpI
∈ (0, 1), and
− 1ε
ln
Kp
YpI
> 0. The
2
1
Kp
ln
3
ε
YpI
∂
LHS as ε → ∞ is equal to Kp . Therefore, the
which is also strictly positive. The limit of ∂ε
slope of the LHS is positive, strictly increasing but always less than Kp . Next, we show that
Kp is strictly less than the constant slope of the RHS.
Cp − Iq−1 + Kq−1 (1 − tq−1 )
1 − tq−1
Iq−1 + (Kp − Kq−1 )(1 − tq−1 ) ≷ Cp .
Kp ≷
The value Cp∗ = Iq−1 + (Kp − Kq−1 )(1 − tq−1 ) is what consumption would be if income were
equal to Kp and the budget segment between Kq−1 and Kq were extrapolated back to Kp .
We know that the indifference curve touches this budget segment at one point (CpI , YpI ), and
every other point on the extrapolated budget segment has strictly lower utility. We also
know that (Cp , Kp ) is on such indifference curve, so that (Cp , Kp ) is strictly preferred to
(Cp∗ , Kp ). Therefore, Cp∗ < Cp , and Kp < [Cp − Iq−1 + Kq−1 (1 − tq−1 )]/(1 − tq−1 ), and the
slope of the function on the LHS of (A.1) is always less than the slope of the function of the
RHS.
In summary, the intercept of the function on the LHS of (A.1) is greater than the
intercept of the function on the RHS. Both functions are strictly increasing: the one on the
RHS has constant slope, and the one on the LHS has increasing slope that is smaller than
the slope of the RHS function. Therefore, the intersection of these two functions is unique.
A.2
Impossibility of Non-parametric Identification of the Elasticity - Proof of
Lemma 1
Consider the case of one kink k, with one tax change, s0 to s1 . It suffices to show that
for every ε > 0, there exists Fn∗ ,ε ∈ Fn∗ such that Fy = T (k, s0 , s1 , Fn∗ ,ε , ε) for fixed Fy , k,
s0 , and s1 . To show the existence of such an Fn∗ ,ε , fix arbitrary ε > 0 and then construct
Fn∗ ,ε as follows:
1. First, define a continuous function φ : [k − εs0 , k − εs1 ] → R++ such that: (a)
φ(k − εs0 ) = limu↑k fy (u); (b) φ(k − εs1 ) = limu↓k fy (u); and (c)
52
R
φ(u) du = Fy (k) − limu↑k Fy (u).
2. Second, compute the CDF Fn∗ ,ε by integrating the following PDF:
fy (εs0 + v) , v ∈ (−∞, k − εs0 )
fn∗ ,ε (v) = φ (v) ,
v ∈ [k − εs0 , k − εs1 ]
fy (εs1 + v) v ∈ (k − εs1 , +∞) .
A.3
Partial Identification with Non-parametric Restrictions - Proof of Theorem
2
First, let’s fix ε > 0. We look at all possible PDFs in Fn∗ and compute the maximum
and minimum integrals over the interval [n, n]. The length of this interval is ε(s0 − s1 ).
Thus, without loss of generality, we restrict our attention to fn∗ over the interval
[0, ε(s0 − s1 )] such that:
(i) fn∗ is continuous, and it connects the point (0, fy (k − )) to (ε(s0 − s1 ), fy (k + )) in the
(x,y) plane;
(ii) the absolute value of the slope of fn∗ is bounded by M.
+
−
y (k )|
. Suppose
First, start with fn∗ being a line. The magnitude of the slope is |fy (kε(s)−f
0 −s1 )
this magnitude is bigger than M . Then, any fn∗ satisfying (i) will have a slope magnitude
+ )−f (k − )|
y
higher than M at some point. Therefore, we need to look at ε ≥ ε1 where ε1 = |fy (k
.
M (s0 −s1 )
For fixed ε ≥ ε1 , the slope of the line will be less or equal to M . The maximum possible
area is attained when the function has the shape of a hat with two line segments that attain
the maximum slope. The first line segment starts at (0, fy (k − )) and has slope +M ; the
second line segment ends at (ε(s0 − s1 ), fy (k + )) and has slope −M . Call this function f n∗ .
These lines intersect at x∗ where
x∗ =
fy (k + ) − fy (k − ) + M ε(s0 − s1 )
.
2M
Note that x∗ is always such 0 ≤ x∗ ≤ ε(s0 − s1 ) because ε ≥ ε1 . Note that it is impossible to
find another fn∗ that satisfies (i), it is greater than f n∗ , and that has slope magnitude less or
equal than M . The maximum area is
A(ε) =
Z
ε(s0 −s1 )
f n∗ (v) dv
0
= (1/4M ) M 2 ε2 s20 − 2M 2 ε2 s0 s1 + M 2 ε2 s21 + 2M εfy (k − )
s0 − 2M εfy (k − )s1 + 2M εfy (k + )s0 − 2M εfy (k + )s1 − fy (k − )2 + 2fy (k − )fy (k + ) − fy (k + )2 )
The function A(ε) is strictly increasing with respect to ε over ε ≥ ε1 . In fact, the
derivative is ((s0 − s1 )(fy (k − ) + fy (k + ) + M ε(s0 − s1 ))/2 which is strictly positive.
The minimum possible area is attained when the function has the shape of an inverted
hat whose lines attain the maximum slope. That is, a combination of two line segments.
53
One that starts (0, fy (k − )) and has slope −M , and another that ends at (ε(s0 − s1 ), fy (k + ))
and has slope +M . Differently the hat function, the intersection (x∗∗ , y ∗∗ ) of this inverted
hat function may or may not be above the x-axis. That is, y ∗∗ may be negative, but fn∗ is
always positive. In that case, we simply set the function to zero in the region where it would
be negative. Call this function f n∗ .
The intersection occurs at
x∗∗ =
fy (k − ) − fy (k + ) + M ε(s0 − s1 )
.
2M
Note that x∗∗ is always such x∗∗ ≥ 0 because ε ≥ ε1 . The y-value of the intersection is
y
∗∗
fy (k − ) + fy (k + ) − M ε(s0 − s1 )
=
.
2M
and this is positive as long as ε ≤ ε2 where ε2 =
For ε1 ≤ ε ≤ ε2 , the minimum area is
A(ε) =
Z
|fy (k+ )+fy (k− )|
.
M (s0 −s1 )
Note also that ε1 < ε2 .
ε(s0 −s1 )
f n∗ (v) dv
0
= (−1/4M ) M 2 ε2 s20 − 2M 2 ε2 s0 s1 + M 2 ε2 s21 − 2M εfy (k − )s0
+2M εfy (k − )s1 − 2M εfy (k + )s0 + 2M εfy (k + )s1 − fy (k − )2 + 2fy (k − )fy (k + ) − fy (k + )2
The function A(ε) is strictly increasing with respect to ε over ε1 ≤ ε < ε2 . In fact, the
derivative is ((s0 − s1 ) ∗ (fy (k − ) + fy (k + ) − M ε(s0 − s1 )))/2 which is strictly positive once
we take into account ε < ε2 . The function A(ε) is constant with respect to ε over ε ≥ ε2 .
Therefore, we have characterized the maximum and minimum areas A(ε) and A(ε) for
any given ε. These areas are undefined if ε < ε1 , they are equal if ε = ε1 , they are strictly
increasing wrt ε and A(ε) ≤ A(ε) for ε ∈ (ε1 , ε2 ). For ε ≥ ε2 , A(ε) continues to grow wrt ε
but A(ε) stays constant at A(ε2 ). The expression for A(ε2 ) is (fy (k − )2 + fy (k + )2 )/2M .
Finally, we define the partially identified set.
Case I: If B < A(ε1 ) = A(ε1 ), there does not exist any function fn∗ consistent with any
elasticity ε, so the set is empty. The expression for A(ε1 ) = A(ε1 ) is
(|fy (k − ) − fy (k + )|(fy (k − ) + fy (k + )))/(2M ).
Case II: Suppose B ≥ A(ε1 ) and B < A(ε2 ). There is an interval range for ε such that
for any ε in this interval there exists a function fn∗ whose integral equals B. The minimum
possible elasticity solves A(ε) = B. That gives
1/2
2 [fy (k + )2 /2 + fy (k − )2 /2 + M B]
ε=
M (s0 − s1 )
− (fy (k + ) + fy (k − ))
.
The maximum possible elasticity solves A(ε) = B. That gives
−2 [fy (k + )2 /2 + fy (k − )2 /2 − M B]
ε=
M (s0 − s1 )
54
1/2
+ (fy (k + ) + fy (k − ))
Case III: Suppose B ≥ A(ε2 ). It is still possible to find a minimum elasticity that solves
A(ε) = B. However, for any elasticity ε ≥ ε we have A(ε) ≤ B, so ε is infinity.
A.4
Tobit Regression - Proof of Identification Lemma 2
By Assumption 17, there exists true values (ε, β, σ) such that
B = Fy (k + ) − Fy (k − ) =Gn∗ (k − εs1 ; β, σ, FX ) − Gn∗ (k − εs0 ; β, σ, FX )
Fy (u) =Gn∗ (u − εs0 ; β, σ, FX ) for ∀u < k
Fy (u) =Gn∗ (u − εs1 ; β, σ, FX ) for ∀u > k,
where FX is the true CDF of X. The MLE estimator is consistent for (ε̄, β̄, σ̄), and we have
that Gy (y; ε̄, β̄, σ̄, FX ) = Fy (y) ∀y. Thus,
Gn∗ (k − ε̄s1 ; β̄, σ̄, FX ) − Gn∗ (k − ε̄s0 ; β̄, σ̄, FX )
= Gn∗ (k − εs1 ; β, σ, FX ) − Gn∗ (k − εs0 ; β, σ, FX )
Gn∗ (u − ε̄s0 ; β̄, σ̄, FX ) = Gn∗ (u − εs0 ; β, σ, FX ) for ∀u < k
Gn∗ (u − ε̄s1 ; β̄, σ̄, FX ) = Gn∗ (u − εs1 ; β, σ, FX ) for ∀u > k
The parametric family created by Gn∗ (n; β, σ, FX ), with FX fixed at the truth, satisfies
(11)-(13) by assumption. Therefore, the equations above solve uniquely with
(ε̄, β̄, σ̄) = (ε, β, σ).
A.5
Censored Quantile Regression - Proof of Identification Lemma 3
Call D = I {Qτ (y | X) 6= k}. Let β(τ ) = [β0 (τ ), β1 (τ ), . . . , βd (τ )]′ . Define
β̃(τ ) = [β0 (τ ) + εs0 , β1 (τ ), . . . , βd (τ ), ε(s1 − s0 )]′ . Multiplying Equation 19 by D yields
DQτ (y | X) = DX̃ β̃(τ ).
Pre-multiplying it by X̃ ′ and taking expectations leads to
DX̃ ′ Qτ (y | X) = DX̃ ′ X̃ β̃(τ )
i
h
i
E DX̃ ′ Qτ (y | X) = E DX̃ ′ X̃ β̃(τ )
h
i−1 h
i
E DX̃ ′ Qτ (y | X) .
β̃(τ ) = E DX̃ ′ X̃
h
(A.2)
An infinite amount of data identifies the joint distribution of (y, X). This identifies the
function Qτ (y | X = x) for every x in the support of X, and the joint distribution of
(y, Qτ (y | X) , X, X̃, D). Therefore, β̃(τ ) is identified by Equation A.2. Finally,
ε = β̃d+1 (τ )/(s1 − s0 ).
55
‘‘BETTER BUNCHING, NICER NOTCHING’’
Marinho Bertanha, Andrew McCallum, Nathan Seegert
B
B.1
Supplemental Appendix for Online Publication
General Utility Maximization Problem with Multiple Kinks and Notches
To generalize the objective function in Equation 1, we update the budget set to have J
different tax regimes that change at cutoff points 0 < K1 < . . . < KJ on pre-tax labor
income Y . Each tax regime has income tax tj such that 0 ≤ t0 ≤ t1 ≤ . . . ≤ tJ < 1. There
are two possible tax changes. A change in tax rate is a kink. A lump-sum tax change is
called a notch. Agent type N ∗ maximizes utility U (C, Y ; N ∗ ) as follows
1+ 1ε
N∗
Y
max
C−
C,Y
1 + 1/ε N ∗
J
X
s.t. C =
I{Kj < Y ≤ Kj+1 } [Ij + (1 − tj ) (Y − Kj )] ,
(B.1)
(B.2)
j=0
where K0 = 0, KJ+1 = ∞, I{·} is the indicator function, the solution is always on the
budget frontier (Equation B.2), and we assume the agent resolves indifference by choosing
the smallest value of Y . The elasticity of income Y with respect to (1 − tj ) is equal to ε
when the solution is interior.
The budget frontier is continuous except when there is a notch. The limit of the budget
frontier when Y ↓ Kj is equal to Ij , but equal to Ij−1 + (1 − tj−1 ) (Kj − Kj−1 ) when Y ↑ Kj .
The size of the jump discontinuity at a notch location Kj is equal to
Ij − Ij−1 − (1 − tj−1 ) (Kj − Kj−1 ). The intercepts Ij and Ij−1 are assumed to be such that
jump discontinuities at notches are negative.
B.2
General Solution with Multiple Kinks and Notches
Lemma B.1 below provides a general solution to Problem B.1 with any combination of
kinks and notches.
Lemma B.1. Define N = ∪Jj=0 (Kj (1 − tj )−ε ; Kj+1 (1 − tj )−ε ] as the set of N ∗ values for
which the indifference
curves are tangent to the budget frontier. The function Y ∗ : N → R,
PJ
∗
∗
Y (N ) = j=0 I{Kj (1 − tj )−ε < N ∗ ≤ Kj+1 (1 − tj )−ε }N ∗ (1 − tj )ε , maps N ∗ values to the
Y values corresponding to such tangency points. Similarly, C ∗ (N ∗ ) is consumption on the
budget frontier (Equation B.2) when Y = Y ∗ (N ∗ ). Let Cj be the value of C ∗ (N ∗ ) whenever
Y ∗ (N ∗ ) = Kj , j = 1, . . . , J. For a notch-point Kj , define the value of NjI to be that of the
first indifference curve tangent to the budget frontier on the right of Y = Kj , such that the
utility level is equal to the utility of the notch-point Kj ,
I
∗
∗
∗
∗
∗
Nj = min N ∈ N : U (Cj , Kj ) = U (C (N ), Y (N )) .
(B.3)
1
In the case of a kink, the bunching interval is defined as [N j , N j ], where
N j = Kj (1 − tj−1 )−ε , and N j = Kj (1 − tj )−ε . In the case of a notch, the expression for N j
equals that of the kink case, but N j changes to NjI .
Note that the bunching intervals of two consecutive kinks do not overlap, that is,
Kj (1 − tj )−ε < Kj+1 (1 − tj )−ε . The same is not true for a kink or a notch Kj+1 that comes
right after a notch Kj , because NjI may be greater than Kj+1 (1 − tj )−ε depending on ε. In
this case, Y = Kj+1 does not appear in the solution. To account for that, construct a
subsequence {jl }Ll=1 of {1, . . . , J} such that: (i) j1 = 1; and (ii) for l ≥ 2, set jl to be the
smallest j such that N j > N jl−1 . Then, the solution to the maximization problem in (B.1) is
given by
∗
N (1 − tj1 −1 )ε , if 0 < N ∗ < N j1
Kj 1
, if N j1 ≤ N ∗ ≤ N j1
∗
∗
ε
N (1 − tj2 −1 ) , if N j1 < N < N j2
..
(B.4)
Y =
.
∗
∗
ε
N (1 − tjL −1 ) , if N jL−1 < N < N jL
Kj L
, if N jL ≤ N ∗ ≤ N jL
N ∗ (1 − tJ )ε , if N jL < N ∗ < ∞.
Proof. For every N ∗ > 0, there exists an unique solution on the budget frontier. If the
consumer is indifferent between two solutions, we assume the consumer takes the solution
¯
with less Y . The proof is by induction over J¯ = 0, 1, . . . , J. Denote the budget frontier BF J
by
J¯
X
I{K̄j < Y ≤ K̄j+1 } Ij + (1 − tj ) (Y − K̄j ) .
C=
j=0
= ∞.
where K̄j = Kj for j = 0, 1, . . . , J¯ and K̄J+1
¯
¯
¯
As we change the budget frontier from BF J to BF J+1 , KJ+1
takes a finite value strictly
¯
greater than KJ¯, and KJ+2
is set to ∞. If the solution to Problem B.1 with budget frontier
¯
¯
< ∞, then this is also the solution to Problem B.1 with budget
BF J is such that Y < KJ+1
¯
¯
¯
¯
J+1
. In fact, points on BF J dominate points on BF J+1 , and they coincide for
frontier BF
Y < KJ+1
¯ .
Part I: J¯ = 0, solve Problem B.1 with budget BF 0 .
This is a standard consumer maximization problem where the optimal choice for Y
occurs at the point the indifference curve is tangent to BF 0 . Therefore, for N ∗ > 0,
Y = N ∗ (1 − t0 )ε .
Part II: J¯ = 1, solve Problem B.1 with budget BF 1 .
The budget frontier BF 1 has two segments BF01 for 0 < Y ≤ K1 , and BF11 for K1 < Y .
If N ∗ < K1 (1 − t0 )−ε , then the solution of Part I, Y = N ∗ (1 − t0 )ε < K1 , is also the solution
in Part II. It remains to find the solution for N ∗ ≥ K1 (1 − t0 )−ε . These solutions must lie on
BF 1 for Y ≥ K1 because they strictly dominate those that lie to the left of K1 .
Case I : Suppose K1 is a kink.
Assume N ∗ is such that K1 (1 − t0 )−ε ≤ N ∗ ≤ K1 (1 − t1 )−ε . If the solution is interior to
BF11 , then it must be at a tangent point in which case Y = N ∗ (1 − t1 )ε . However,
2
Y = N ∗ (1 − t1 )ε ≤ K1 , a contradiction because this Y falls outside of the interior of BF11 .
Therefore, if N ∗ is such that N 1 = K1 (1 − t0 )−ε ≤ N ∗ ≤ K1 (1 − t1 )−ε = N 1 , then the
solution is Y = K1 . Suppose N ∗ > N 1 . Then, the solution is in the interior of BF11 , and it is
equal to Y = N ∗ (1 − t1 )ε .
Case II : Suppose K1 is a notch.
There is a jump-down discontinuity in BF 1 at K1 , and BF 1 is continuous from the left.
Consider the point (C, Y ) = (C1 , K1 ) on BF01 . Define Y D to be the value of Y such that the
corresponding C value on BF11 is equal to C1 . The jump-down discontinuity creates a
strictly dominated region on BF11 because the utility of (C1 , K1 ) is strictly greater than the
utility of any solution with Y ∈ (K1 , Y D ). Indifference between K1 and Y D is resolved
towards K1 by assumption. Therefore, we cannot have solutions to Problem B.1 with budget
BF 1 such that Y ∈ (K1 , Y D ].
e1I as being the solution of Problem B.1 with budget BF 1 (instead of
Define the point N
BF ). This is the smallest N ∗ for which Problem B.1 with budget BF11 has solution with
utility equal to U (C1 , K1 ).
e1I exists. To see that, note that for small N ∗ , the tangent point
First, a solution N
Y = N ∗ (1 − t1 )ε along BF11 falls in the dominated region Y ∈ (K1 , Y D ], and the utility is less
than U (C1 , K1 ); on the other hand, the utility at this tangent point increases with N ∗ , and it
e1I ≥ Y D (1 − t1 )−ε > K1 (1 − t1 )−ε .
eventually equals U (C1 , K1 ). The solution is such that N
e I is unique. To see that, solve for N ∗ in the equation below.
Second, the solution N
1
U (C1 , K1 ) = U I1 + N ∗ (1 − t1 )ε+1 − K1 (1 − t1 ) , N ∗ (1 − t1 )ε
where C = I1 + N ∗ (1 − t1 )ε+1 − K1 (1 − t1 ) is consumption on BF11 when Y = N ∗ (1 − t1 )ε .
Evaluating and rearranging the equality gives
N ∗ (1 − t1 )1+ε + ε(N ∗ )−1/ε (K1 )
1+ε
ε
= (1 + ε) [C1 − I1 + K1 (1 − t1 )]
The solution is unique because the derivative of the right-hand side is strictly positive given
e1I is the unique solution to Problem B.1 when the budget is
N ∗ > K1 (1 − t1 )−ε . Note that N
BF 1 .
e I (1 − t1 )ε . Suppose there is a solution to Problem B.1 with budget BF 1
Call Ye1I = N
1
such that Y D < Y ≤ Ye1I . This solution is interior to budget BF11 , so we must have
Y = N ∗ (1 − t1 )ε for some N ∗ . But such a solution cannot be a solution to Problem B.1 with
budget BF 1 because Y ≤ Ye1I and so dominated by (C1 , K1 ). Therefore, we cannot have
solutions to Problem B.1 with budget BF 1 such that Y ∈ (K1 , Ye I ].
It remains to characterize the solution when N ∗ is such that K1 (1 − t0 )−ε ≤ N ∗ . If N ∗ is
such that N 1 = K1 (1 − t0 )−ε ≤ N ∗ ≤ Ye I (1 − t1 )−ε = N 1 , the solution cannot be in the
interior of BF01 since Y = N ∗ (1 − t0 )ε ≥ K1 ; it cannot be in (K1 , Ye I ] either. Assume it is in
the interior of BF11 with Y > YeI . Since it is interior, it satisfies Y = N ∗ (1 − t1 )ε , but
N ∗ ≤ Ye I (1 − t1 )−ε which makes Y ≤ Ye I , a contradiction. Therefore, the solution to
Problem B.1 with budget BF 1 when N ∗ ∈ [N 1 ; N 1 ] is Y = K1 . Finally, suppose N ∗ > N 1 .
Then, the solution is in the interior of BF11 , and it is equal to Y = N ∗ (1 − t1 )ε .
Part III:
¯
Assume the solution of Problem B.1 with budget BF J and 1 ≤ J¯ < J is as
3
¯
¯ Show that (B.4) with J¯ + 1 solves Problem B.1 with budget BF J+1
in Equation B.4 with J.
.
J¯
∗
Consider Problem B.1 with budget BF and solution B.4 with L being L̄. If N is such
¯
< ∞, then Y also solves Problem B.1 with budget BF J+1 . Therefore, the
that Y < KJ+1
¯
¯
¯
solution to Problem B.1 with budget BF J+1 or budget BF J coincide for those values of N ∗ .
Note also that, if Kj is a notch and j < jL̄ , then the value of N j (defined in (B.3)) does not
¯
¯
change when the budget changes from BF J to BF J+1 . If KjL̄ is a notch, then the value N jL̄
may change (case IV below). In what follows, consider the last two budget segments of
¯
¯
¯
J+1
.
and BFJ+1
BF J+1 : BFJJ+1
¯
¯
is a kink
Case I : KjL̄ is a kink, KJ+1
¯
−ε
¯
> N jL̄ , so that J¯ + 1 is the
= KJ+1
In this case, jL̄+1 = J + 1 because N J+1
¯ (1 − tJ¯)
¯
¯ To see that, note that
smallest j such that N j > N jL̄ . It is also true that jL̄ = J.
consecutive intervals [N j , N j ] never overlap for kinks because
N j = Kj (1 − tj )−ε < Kj+1 (1 − tj )−ε = N j+1 . The upper limit of a kink interval j is strictly
smaller than the lower limit of a notch interval j + 1. However, the upper limit of a notch
interval j may be bigger than the lower limit of the next interval j + 1. Suppose jL̄ = J¯
¯ Then, any j such that jL̄ < j ≤ J¯ is not in the subsequence
were not true, that is, jL̄ < J.
{jl } because KjL̄ is a notch, and its interval overlaps with the j interval. But this is a
contradiction with KjL̄ being a kink point.
−ε
J¯
If N ∗ < KJ+1
¯ (1 − tJ¯) , then the solution B.4 with budget BF is Y < KJ+1
¯ , and Y
¯
J+1
∗
for that same value of N . It remains to
also solves Problem B.1 with budget BF
−ε
∗
characterize the solution when N ≥ KJ+1
¯ (1 − tJ¯)
−ε
∗
−ε
Assume N is such that N J+1
≤ N ∗ ≤ KJ+1
= KJ+1
= N J+1
¯ )
¯ (1 − tJ+1
¯ . As
¯ (1 − tJ¯)
¯
¯
J+1
. The solution must be at
seen in Part II, Case I, the solution cannot be interior to BFJ+1
¯
¯
J+1
∗
, and it equals to
KJ+1
¯ . Assume N > N J+1
¯ . Then, the solution is interior to BFJ+1
¯
ε
)
.
Y = N ∗ (1 − tJ+1
¯
is a notch
Case II : KjL̄ is a kink, KJ+1
¯
¯ We also have jL̄+1 = J¯ + 1 because the j interval
As seen in Part III, Case I, jL̄ = J.
[N j , N j ] of a kink does not overlap with the j + 1 interval of a notch.
−ε
J¯
If N ∗ < KJ+1
¯ (1 − tJ¯) , then the solution B.4 with budget BF is Y < KJ+1
¯ , and Y
¯
J+1
∗
for that same value of N . It remains to
also solves Problem B.1 with budget BF
−ε
∗
characterize the solution when N ≥ KJ+1
¯ (1 − tJ¯)
−ε
≤ N ∗ ≤ N J+1
is the
= KJ+1
Assume N ∗ is such that N J+1
¯ (1 − tJ¯)
¯ , where N J+1
¯
¯
¯
J+1
solution of Problem B.3 when the budget is BF
. As seen in Part II, Case II, the solution
¯
J+1
ε
Y cannot be in (KJ+1
. Therefore, the solution
¯ , N J+1
¯ (1 − tJ+1
¯ ) ] or in the interior of BFJ+1
¯
¯
J+1
∗
is Y = KJ+1
, and it equals to
¯ . Assume N > N J+1
¯ . Then, the solution is interior to BFJ+1
¯
ε
∗
Y = N (1 − tJ+1
¯ ) .
Case III : KjL̄ is a notch, N jL̄ < N J+1
¯
¯
For the notch KjL̄ , the solution N jL̄ to Problem B.3 when the budget is BF J does not
¯
change when the budget becomes BF J+1 precisely because N jL̄ < N J+1
¯ . In this case,
∗
−ε
∗
¯
jL̄+1 = J + 1. For N such that N jL̄ < N < KJ+1
¯ (1 − tJ¯) , the solution B.4 with budget
4
¯
¯
J+1
for that same value
BF J is Y < KJ+1
¯ , and Y also solves Problem B.1 with budget BF
−ε
∗
∗
of N . It remains to characterize the solution when N ≥ KJ+1
¯ (1 − tJ¯) .
∗
is a kink, and that N is such that
Assume KJ+1
¯
−ε
= N J+1
N J+1
(1
−
tJ¯)−ε ≤ N ∗ ≤ KJ+1
=
K
¯ . As seen in Part II, Case I, the
¯ )
¯ (1 − tJ+1
¯
¯
J+1
¯
J+1
∗
. The solution must be at KJ+1
solution cannot be interior to BFJ+1
¯ . Assume N > N J+1
¯ .
¯
¯
J+1
ε
, and it equals to Y = N ∗ (1 − tJ+1
Then, the solution is interior to BFJ+1
¯ ) .
¯
−ε
Assume KJ+1
is a notch, and that N ∗ is such that N J+1
= KJ+1
≤ N ∗ ≤ N J+1
¯
¯ (1 − tJ¯)
¯ ,
¯
¯
J+1
is the solution of Problem B.3 when the budget is BF
where N J+1
. As seen in Part II,
¯
¯
J+1
ε
Case II, the solution Y cannot be in (KJ+1
.
¯ ) ] or in the interior of BFJ+1
¯ (1 − tJ+1
¯ , N J+1
¯
∗
Therefore, the solution is Y = KJ+1
¯ . Assume N > N J+1
¯ . Then, the solution is interior to
¯
J+1
ε
∗
, and it equals to Y = N (1 − tJ+1
BFJ+1
¯ ) .
¯
Case IV : KjL̄ is a notch, N jL̄ ≥ N J+1
¯
ε
The indifference value for Y at N jL̄ is YjIL̄ = N jL̄ (1 − tJ¯)ε ≥ N J+1
¯ . If
¯ (1 − tJ¯) = KJ+1
J¯
N jL̄ = N J+1
¯ , the solution to Problem B.3 when the budget is BF remains unchanged when
¯
I
the budget becomes BF J+1 . If N jL̄ > N J+1
¯ , and the solution to Problem
¯ , then Yj > KJ+1
L̄
¯
¯
B.3 when the budget is BF J changes when the budget becomes BF J+1 . The value of N jL̄
ε
increases such that the new indifference point satisfies YjIL̄ = N jL̄ (1 − tJ+1
¯ ) .
is the last tax-change point
There does not exist a j such that N j > N jL̄ because KJ+1
¯
available and N J+1
≤ N jL̄ . Therefore, when constructing the solution of Problem B.1 with
¯
¯
J+1
budget BF
, the last term in the subsequence {jl } remains jL̄ .
The point KjL̄ is a notch, so Part II, Case II says that for N ∗ such that
ε
N jL̄ = KjL̄ (1 − tjL̄ −1 )−ε ≤ N ∗ ≤ N jL̄ , the solution Y cannot be in (KjL̄ , N jL̄ (1 − tJ+1
¯ ) ] or
¯
J+1
in the interior of BFJ+1
. Therefore, the solution is Y = KjL̄ . Assume N ∗ > N jL̄ . Then, the
¯
¯
J+1
ε
, and it equals to Y = N ∗ (1 − tJ+1
solution is interior to BFJ+1
¯ ) .
¯
B.3
Friction Errors and Failure of the ‘‘Polynomial Strategy’’
This section presents a counterexample that illustrates the failure of a common
identification strategy used in applied work to estimate the elasticity using kinks. For a
review, see Kleven (2016).
First, we set the parameters of the model. The true values are: ε = 1.5 (elasticity);
t0 = .2 and t1 = 0.3 (before and after tax rates); kink-point k = 0. The bunching interval is
[n, n] = [0.335, .535]. The distribution of the ability variable is assumed uniform,
n∗ ∼ U [−.565; 1.435]; that is, the support is centered at 0.435 and has length equal to 2.
The probability of bunching, or bunching mass B, is equal to 10% in this example. The
friction error e is also assumed uniformly distributed e ∼ U [−0.5; 0.5]. The value of labor
income observed by the researcher is ye = y + e, where y is a function of n∗ , ε, t0 , and t1 , as
described in Equation 4.
In the counterfactual scenario of no tax change, we have n = n, and the counterfactual
income with friction error is denoted ye0 . The counterfactual income without friction error is
y0 . Figure B.1a depicts the PDF of ye and ye0 .
5
A common identification strategy used in applied work is to fit a polynomial to the PDF
of ye excluding observations in the neighborhood of the kink k = 0, that corresponds to the
support of the measurement error (i.e. [−0.5; 0.5]). The estimated bunching mass is the area
between the PDF of ye and the polynomial fit extrapolated to the excluded neighborhood
around the kink. Figure B.1b illustrates the procedure. The figure shows that such strategy
fails to identify the true bunching mass, even when the polynomial fit of 7th order is perfect,
and we assume the researcher knows the support of e.
The last part of the estimation strategy uses the extrapolated polynomial to predict the
counterfactual PDF of y0 . Following Equation 6, identification of ε requires the
counterfactual PDF of y0 , without measurement error. Figure B.1c shows that the
polynomial strategy fails to retrieve the PDF of y0 . The PDF predicted by the polynomial
regression does not integrate to one, and thus it is not a PDF. If we divide the
polynomial-based PDF in Figures B.1b and B.1c by its integral, the PDF shifts up in the
graphs. The re-normalized PDF still misses the true fy0 , and the underestimation of B is
larger than before.
The polynomial strategy fails for two reasons:
1. The PDF of ye is not simply the PDF of y plus the PDF of e (Figure B.1a), but the
convolution between the two PDFs. While y0 and e have uniform distributions, with a
flat PDF, their convolution does not have a flat PDF. As a result, extrapolating the
polynomial to find the bunching mass and to predict the PDF of y0 is misleading;
2. The counterfactual distribution required for identification of the elasticity is the PDF
of y0 , and not the PDF of ye0 (Equation 6). Moreover, even if friction errors were not a
problem, it is not possible to use the distribution of y to back out the distribution of y0
for values of y0 inside [k, k + (s0 − s1 )ε]. The shape of the distribution of y0 is
unidentified when n∗ falls in the bunching interval (Figure 1).
B.4
Parametric Gaussian Family Identifies the Elasticity
We demonstrate how to verify conditions (11) - (13) in the parametric Gaussian case.
Suppose the distribution of n∗ follows a normal distribution withunknown mean µ and
where Φ denotes the
unknown variance σ 2 , such that Fn∗ (n) = Gn∗ (n; µ, σ 2 ) = Φ n−µ
σ
standard normal CDF.
Take (k, s0 , s1 , ε, µ, σ 2 ) arbitrary. The goal is to show that ε̄ = ε, µ̄ = µ, and σ̄ 2 = σ 2 are
the only solutions to the equalities below:
k − εs0 − µ
k − ε̄s1 − µ̄
k − ε̄s0 − µ̄
k − εs1 − µ
−Φ
=Φ
−Φ
(B.5)
Φ
σ
σ
σ̄
σ̄
u − ε̄s0 − µ̄
u − εs0 − µ
=Φ
for ∀u < k
(B.6)
Φ
σ
σ̄
u − εs1 − µ
u − ε̄s1 − µ̄
Φ
=Φ
for ∀u > k
(B.7)
σ
σ̄
6
Take (B.6), and apply Φ−1 (·) to both sides.
u − εs0 − µ
u − ε̄s0 − µ̄
=
,
σ
σ̄
∀u < k.
These are two lines that must have the same slope, 1/σ = 1/σ̄, and the same intercept
(εs0 + µ)/σ = (ε̄s0 + µ̄)/σ̄. These imply that σ̄ = σ, and ε̄s0 + µ̄ = εs0 + µ.
Similarly, (B.7) implies that ε̄s1 + µ̄ = εs1 + µ. Subtracting this last equation from the
previous one gives ε̄(s1 − s0 ) = ε(s0 − s1 ), which yields ε̄ = ε. Finally, εs1 + µ̄ = εs1 + µ
gives µ̄ = µ.
B.5
Implementation of Censored Quantile Regressions
The optimization problem in Equation 21 is computationally difficult. For the left (or
right) censored case, Chernozhukov and Hong (2002) proposed a fast and practical estimator
that consists of three steps. First, you fit a flexible Probit model that explains the
probability of no censoring; then, you select observations whose values of X lead to a
predicted probability of no censoring that is greater than 1 − τ . Second, you fit a quantile
regression of y on X using the selected observations in the first step; then, you select
observations whose values of X lead to a predicted quantile that is greater than k. Third,
repeat the second step using the observations selected at the end of the second step.
Chernozhukov and Hong (2002) demonstrate consistency and asymptotic normality of their
three-step estimator. Moreover, they show that the standard errors computed by the
quantile regression in the third step are valid.
Our case of middle censoring requires a straightforward modification of the method
proposed by Chernozhukov and Hong (2002). Inspired by their algorithm, we propose the
following implementation steps.
1. Create dummies δi− = I{yi < k} (not censored, left of k) and δi+ = I{yi > k} (not
censored, right of k). Fit two Probit models to estimate P[δi+ |Xi ] = Φ(Xi g + ) and
P[δi− |Xi ] = Φ(Xi g − ), where Φ denotes the cdf of a standard normal distribution, and
g ± are vectors of parameters. You may use powers and interactions of Xi to make this
stage as flexible as possible. Select two subsamples as follows. Compute the 10th
quantile of the empirical distribution of Φ(Xi ĝ + ) − (1 − τ ) conditional on
Φ(Xi ĝ + ) > 1 − τ . Let κ+
0 (τ ) be the 10th quantile of that distribution. The first
+
subsample is J0 (τ ) = {i : Φ(Xi ĝ + ) > 1 − τ + κ+
0 (τ )}. The second subsample is
−
(τ
)},
where
κ
(τ
) is the 10th quantile of the empirical
J0− (τ ) = {i : Φ(Xi ĝ − ) > τ + κ−
0
0
−
distribution of Φ(Xi ĝ ) − τ conditional on Φ(Xi ĝ − ) > τ . Create a dummy
Wi0 = I{i ∈ J0+ (τ )}.
2. Fit the quantile regression model Qτ (yi |Xi , Wi0 ) = Xi b(τ ) + Wi0 δ(τ ) using observations
in J0− (τ ) ∪ J0+ (τ ). Use the estimates of this quantile regression, that is b̂0 (τ ) and δ̂ 0 (τ ),
to create two subsamples as follows. The first subsample is
+
J1+ (τ ) = {i : Xi b̂0 (τ ) + δ̂ 0 (τ ) > k + κ+
1 (τ )}, where κ1 (τ ) is the 3rd quantile of the
empirical distribution of Xi b̂0 (τ ) + δ̂ 0 (τ ) − k conditional on Xi b̂0 (τ ) + δ̂ 0 (τ ) > k. The
−
second subsample is J1− (τ ) = {i : Xi b̂0 (τ ) < k + κ−
1 (τ )}, where κ1 (τ ) is the 97th
7
quantile of the empirical distribution of Xi b̂0 (τ ) − k conditional on Xi b̂0 (τ ) < k.
Create a dummy Wi1 = I{i ∈ J1+ (τ )}.
3. Fit the quantile regression model Qτ (yi |Xi , Wi1 ) = Xi b(τ ) + Wi1 δ(τ ) using observations
in J1− (τ ) ∪ J1+ (τ ) to obtain estimates b̂1 (τ ) and δ̂ 1 (τ ). The elasticity estimator is
ε̂ = δ̂ 1 (τ )/(s1 − s0 ).
B.6
Estimates with the Filtering Method of Saez (2010)
In this section, we recompute the estimates of Table 1 using a different filtering method.
Specifically, we employ the procedure used by Saez (2010) to obtain the bunching mass and
the side limits of the distribution of income without friction error Y . The procedure
implicitly defines a way to estimate the unobserved distribution of Y given the observed
distribution of income with friction error Ỹ . We refer the reader to Figure 2 by Saez (2010).
The first step is to construct a histogram-based estimate of the PDF fỸ , and then
average fỸ for Ỹ ∈ [K − 2δ, K − δ] ∪ [K + δ, K + 2δ], where K = 8, 580 is the kink point, and
δ = 1, 500 defines the excluded region. Call that average f¯. The bunching mass is estimated
by the area between two curves, fỸ and f¯. The continuous portion of fY equals fỸ , except
for the excluded region [K − δ, K + δ], where fY equalsf¯. We obtain
the CDFs FY and FỸ
from their PDF estimates. Finally, we rely on Y = FY FỸ−1 (Ỹ ) to transform Ỹ into Y .
8
Table B.1: Estimates Using U.S. Tax Returns 1995--2004
Statistical Model
All
Elasticity (ε)
Self-employed
Elasticity (ε)
Self-employed,
married
Elasticity (ε)
9
Self-employed,
not married
Elasticity (ε)
(1)
Saez (2010)
(2)
Theorem 2
Bounds
M = 0.5
(3)
Theorem 2
Bounds
M=1
(4)
Tobit
Full Sample
(5)
Tobit
Trunc. 75%
(6)
Tobit
Trunc. 50%
(7)
Tobit
Trunc. 25%
(8)
0.235
(0.0311)
[0.225, 0.249]
[0.210, 0.282]
0.124
(0.0002)
0.177
(0.0001)
0.182
(0.0001)
0.199
(0.0002)
Sample
details
Obs. 189.1m
Avg. $54.1k
Std. $131.1k
0.933
(0.0759)
[0.765, 1.298]
[0.663, ∞]
0.617
(0.0006)
0.809
(0.0008)
0.805
(0.0008)
0.822
(0.0009)
Obs.
Avg.
Std.
33.5m
$61.8k
$168.2k
0.391
(0.0823)
[0.328, 0.382]
[0.285, ∞]
0.187
(0.0004)
0.286
(0.0007)
0.330
(0.0008)
0.331
(0.0008)
Obs.
Avg.
Std.
24.0m
$75.0k
$185.6k
1.260
(0.1193)
[1.130, 1.508]
[1.008, ∞]
1.145
(0.0012)
0.991
(0.0011)
1.003
(0.0012)
1.270†
(0.0018)
Obs.
Avg.
Std.
9.6m
$28.7k
$106.3k
Notes: The table shows estimates of the elasticity for four different subsamples of the IRS data, and using three different approaches discussed in the
paper. The first approach (column 1) uses the trapezoidal approximation to point-identify the elasticity (Example 1). Estimates and standard errors
were computed using the publicly available code by Saez (2010) at the website of the American Economic Journal, Economic Policy. The second
approach (columns 2 and 3) computes partially identified sets for the elasticity (Theorem 2), using non-parametric estimates of the side limits of fy at
the kink, and the bunching mass. Side limits were estimated using the method of Cattaneo et al. (2019). The estimate for the bunching mass equals
the sample proportion of y observations that equals the kink point (see discussion in Section B.6 on friction errors). Upper and lower bounds are
calculated for two choices of M, that is, the maximum slope of the PDF of the unobserved heterogeneity n∗ . Column 4 has Tobit MLE estimates of the
elasticity that utilizes the full sample of data, along with robust standard errors. Columns 5 through 7 report truncated Tobit MLE estimates. As we
move from column 5 to column 7, we restrict the estimation sample to shrinking symmetric windows around the kink that utilizes 75% to 25% of the
data. The set of covariates that enters the Tobit estimation is kept constant across different truncation windows. It includes dummy variables such as
marital and employment status, year effects, types of deductions or social security benefits received, and whether the filer used a tax prep software.
†There are too few observations for the maximum likelihood estimator to converge when using 25% of the sample. This estimate uses 27% instead.
Figure B.1: Counterexample where ‘‘Polynomial Strategy’’ Fails
(b) Estimation of Bunching Mass
0.8
f
0.7
f y +e
0.8
y+e
fy+e
fitted polynomial
0.7
0
0.6
0.6
0.5
0.5
PDF
PDF
(a) Distribution of Income with Friction Error
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
-0.1
True Bunching Mass: 0.10
Est. Bunching Mass: 0.07
-0.1
-1
-0.5
0
0.5
1
1.5
-1
-0.5
y+e
0
0.5
1
1.5
y+e
(c) Counterfactual Distribution of Income
without Friction Error
0.8
fy
0.7
estimated
0
0.6
PDF
0.5
0.4
0.3
0.2
0.1
0
-0.1
-1
-0.5
0
0.5
1
1.5
y0
Notes: The population model of this example has ε = 1.5, t0 = .2, and t1 = 0.3 at kink k = 0. The
distribution of ability is assumed uniform, n∗ ∼ U [−.565; 1.435]. The probability of bunching is equal to 10%,
and the distribution of the friction error is e ∼ U [−0.5; 0.5]. The researcher observes ye = y + e, where y is a
function of n∗ , ε, t0 , and t1 , as described in Equation 4. Figure B.1a displays the PDF of ye and ye0 . Figure
B.1b displays the fitted 7th-order polynomial to the PDF of ye using observations in (−∞, −0.5) ∪ (0.5, ∞).
The bunching mass is estimated by the integral of the difference between fye and the fitted polynomial, inside
the excluded region. The polynomial strategy understimates the true bunching mass, and does not retrieve
the PDF of y0 (Figure B.1c).
10