What Have We Learned About Earnings Management?
Correcting Disinformation about Discontinuities*
David Burgstahler
Julius A. Roller Professor of Accounting
University of Washington/Seattle
Elizabeth Chuk
University of Southern California
October 4, 2012
Abstract
Earnings distributions commonly exhibit statistically significant discontinuities at
prominent performance benchmarks. Discontinuities at zero earnings are widely interpreted as
evidence of earnings management to avoid a loss, and discontinuities at zero earnings change or
zero earnings surprise as evidence of management to avoid an earnings decrease or negative
earnings surprise, respectively. In contrast, two recent papers by Durtschi and Easton (2005,
2009, hereafter DE) assert that discontinuities are instead explained by some combination of
prior researchers' choice(s) of sample selection and scaling as well as a systematic relation
between the sign of earnings and market prices. Resolution of the conflicting interpretations of
discontinuities is important because 1) it affects how investors, regulators, and scholars view
earnings management and 2) it demonstrates the importance of a close linkage between theory
and research design choices. We evaluate the three alternative explanations proposed by DE.
We point out that DE provide no evidence that their explanations create discontinuities, but only
evidence showing that their modified research designs eliminate discontinuities. We also
demonstrate why the research designs used by DE eliminate evidence of discontinuities when
alternative designs using the same data identify highly significant discontinuities. Finally, we
outline the key characteristics of the extensive body of evidence documenting discontinuities in
earnings distributions that are consistent with the theory that earnings are managed but are
generally inconsistent with artifactual theories of discontinuities.
JEL classification:
Key Words:
G14, M40
Earnings management, discontinuities
* We appreciate comments and suggestions from Bob Bowen, Ilia Dichev, Weili Ge, Dave
Guenther, Frank Hodge, Bjorn Jorgensen, Bill Kinney, Allison Koester, Dave Maber, Rick
Mergenthaler, Greg Miller, Shiva Rajgopal, Tatiana Sandino, Terry Shevlin, D. Shores, Bob
Trezevant, and workshop participants at the 2010 UBC, Oregon, Washington Accounting
Conference, University of Colorado, University of Illinois, University of Michigan, University of
Southern California, University of Texas, National University of Singapore, and Singapore
Management University.
1. Introduction
There is pervasive evidence of discontinuities in distributions of reported earnings at
prominent earnings benchmarks, where distributions comprise fewer observations immediately
left of the benchmark and more observations immediately right of the benchmark than would be
expected if the distribution was smooth.1 For example, Burgstahler and Dichev (1997a, hereafter
BD) show that distributions of earnings levels and distributions of earnings changes exhibit
discontinuities at zero. In addition, BD document that the strength of discontinuities varies with
the costs and benefits of meeting benchmarks. This evidence is widely interpreted as consistent
with the theory that managers take actions to ensure that earnings meet benchmarks, e.g., earnings
are managed to avoid losses and earnings decreases.2 This interpretation is further supported by
survey evidence in Graham, Harvey, and Rajgopal (2005) indicating that managers are willing to
incur real costs in order to meet benchmarks.
Two recent papers by Durtschi and Easton (2005, hereafter DE1, and 2009, hereafter DE2)
assert that inferences attributing discontinuities to earnings management are "erroneous." These
papers observe that discontinuities are eliminated when the research design is changed and assert
that discontinuities in earnings distributions are driven by “deflation, sample selection, and a
difference between the characteristics of profit and loss observations.” These assertions have led
many to question whether discontinuities represent compelling empirical evidence of earnings
management, or instead are due to the artifactual explanations advocated in DE1 and DE2
(hereafter referred to collectively as DE).
1
The smoothness assumption is discussed in more detail in Section 2 and in Burgstahler (2012).
While this paper focuses primarily on evidence of discontinuities in distributions of earnings and earnings changes,
there is also widespread evidence of similar discontinuities in distributions of earnings surprises, e.g., Degeorge, Patel,
and Zeckhauser (1999), Brown (2001), Matsumoto (2002), Brown and Caylor (2005), Burgstahler and Eames (2006).
2
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It is conceivable that discontinuities are eliminated when DE alter the research design used
in prior studies because the research design factors that they focus on somehow induce
discontinuities. However, an important alternative possibility is that the DE research design
choices mechanically obscure discontinuities. The purpose of this paper is to distinguish between
these competing interpretations of the effects reported in DE.
Research design choices are critical determinants of the power of tests and the
interpretation of results. For example, if a significant result is replicated using a design based on a
much smaller sample size, the result of the replication may not be statistically significant.
However, it is well-understood that an insignificant result from a replication that lacks power due
to small sample size does not contradict a significant result from a larger sample, does not confirm
the null hypothesis, and does not disprove the alternative hypothesis. The same principle applies
in interpreting results of DE research designs that reduce the power of discontinuity tests.
First, we show that the absence of discontinuities in unscaled earnings or earnings per
share reported in DE is a direct result of the DE research design choices and show how research
designs that correct these flaws show significant discontinuities in both unscaled earnings and
earnings per share. Thus, the DE results show how lack of scaling combined with inadequate
research design can eliminate discontinuities but do not show that scaling creates discontinuities –
lack of scaling combined with appropriate design yields highly significant discontinuities.
Second, we show why the DE speculation that discontinuities in distributions of pricescaled earnings are explained by small negative earnings being “pushed away from zero” by low
price scalers and small positive earnings being “drawn toward zero” by high price scalers is
incorrect. We first note that this argument only applies to price-scaled earnings at the zero
benchmark and cannot be applied to much of the discontinuity evidence reported in the literature,
including discontinuities for benchmarks other than zero, for other earnings variables such as
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earnings changes and earnings surprises, and discontinuities in unscaled earnings and in earnings
per share. We then show that discontinuities occur in segments of the population where prices for
small negative earnings are not lower than for small positive earnings and that these segments
account for almost all of the discontinuity in price-scaled earnings at the zero benchmark. Thus,
even for the narrow portion of discontinuity evidence where this explanation could conceivably
apply, the relation between price and earnings does not explain discontinuities.
Third, we document that firms with missing market prices differ substantially from firms
with market prices in a variety of ways that naturally translate into differences between the scaled
earnings distributions for firms with and without market prices. The missing price distributions
have a high proportion of small loss observations and show little or no evidence of discontinuities,
in contrast to distributions for firms with price, which have highly significant discontinuities.
These clear differences explain the DE results showing that aggregating the missing price
observations together with observations that have price weakens the discontinuity in the
aggregated distribution. However, the fact that distributions for firms with missing price are
different, and specifically that the missing price distributions lack discontinuities, is not an
explanation for discontinuities in the distributions for firms that have prices.
In summary, the DE results do not contradict previous evidence of discontinuities nor the
common interpretation that discontinuities are due to earnings management to meet benchmarks.
Instead, the DE research design choices reduce power and mechanically eliminate the significance
of discontinuities because they 1) fail to account for the extremely important effect of size as a
covariate, 2) do not fit with the theory that earnings are managed more frequently when the
amount of earnings management required to meet the benchmark is smaller, and 3) place
inordinate weight on segments of the population where there is little reason to expect evidence of
discontinuity given the research design. The DE results show only that these critical research
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design flaws can obscure discontinuities—there is no evidence showing that any of these research
design choices can create discontinuities.
The paper is organized as follows. Section 2 provides background and discusses research
design for empirical tests of earnings management to meet benchmarks. Section 3 provides
empirical evidence showing why the DE research designs predictably lead to non-significant
discontinuity evidence and Section 4 explains why artifactual explanations for discontinuities,
including the explanations advocated in DE, are generally unable to explain the body of
discontinuity that has been reported in the literature. Section 5 concludes.
2. Background
Earnings management has been a prominent topic in the accounting literature for many
years. Healy and Wahlen (1999) note: "Despite the popular wisdom that earnings management
exists, it has been remarkably difficult for researchers to convincingly document it." Earnings
management typically cannot be directly observed but must instead be inferred, either by 1) testing
whether earnings, or a component of earnings, differs from what would have been expected in the
absence of earnings management (for example, tests based on abnormal accruals models), or
2) testing whether the distribution of reported earnings differs from what would have been
expected in the absence of earnings management.
The second approach adopts an implicit model of the distribution of pre-managed total
earnings. Specifically, the model of pre-managed earnings adopted by BD is simply that the
distribution of pre-managed earnings should be smooth in the sense that the expected frequency in
an interval is approximately equal to the average of the expected frequencies in the two
immediately adjacent intervals.3 As explained below, earnings management to meet benchmarks
3
Other statistical tests for discontinuities rely on more restrictive assumptions about the form of the distribution.
Therefore, in these tests a significant test statistic could be due to departures from these assumptions rather than to a
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creates discontinuities in the distribution of reported earnings, providing evidence that some firms
have taken actions consistent with earnings without specifically identifying which firms have
taken these actions. Further, because this approach focuses on reported total earnings, the set of
actions included in the definition of earnings management is broad, including all accounting
(accrual and disclosure) actions and all "real" operating, investing, and financing actions that
affect reported earnings.4
2.1 Theory of earnings management to meet benchmarks
Economic theory suggests that managers take actions when they believe the benefits
exceed the costs. Two assumptions transform this general theory into a specific theory of earnings
management to meet benchmarks: 1) The benefit of managing earnings to meet a benchmark is
improved terms of transactions with stakeholders.5 2) The cost of managing earnings is increasing
in the amount of earnings management. Let EMj* denote the maximum amount by which
earnings for observation j can be managed to meet the benchmark at a cost less than or equal to the
discontinuity. For example, the test in Hayn (1995) requires the additional assumption that the distribution of premanaged earnings is normal and a significant test statistic could be due to departures from normality. Other examples
include the assumption in Bollen and Pool (2009) that the pre-managed distribution of hedge fund returns is described
by a fitted distribution derived from the histogram of reported hedge fund returns and the assumption in Chen et al
(2010) of a specific distributional form.
4
McNichols (2000) and Dechow, Richardson, and Tuna (2003) discuss the fact that discontinuity evidence does not
reveal which actions have been employed to manage earnings, a limitation that is also recognized in most reviews of
earnings management. For example, the review by Healy and Whalen (1999, p. 368) invokes a broad definition of
earnings management: "Earnings management occurs when managers use judgment in financial reporting and in
structuring transactions to alter financial reports to either mislead some stakeholders about the underlying economic
performance of the company or to influence contractual outcomes that depend on reported accounting numbers."
Dechow and Skinner (2000) rely on a similarly broad definition that encompasses actions ranging from pure
disclosure management to real operating, investing, and financing actions. Schipper (1989, p. 92) focuses on a
slightly more narrow definition of "disclosure management" defined as "purposeful intervention in the external
financial reporting process, with the intent of obtaining some private gain…" but notes that it would be only a minor
extension to broaden this definition to encompass "real" earnings management. In contrast, studies that construct
models of a specific component of earnings are able to focus on a more narrow definition of earnings management.
For example, Dechow, Richardson, and Tuna (2003) focus specifically on the discretionary accruals form of earnings
management.
5
As discussed in BD section 4, examples of improvements in terms of transactions when benchmarks are met include
lower wage and benefit demands from current and potential employees, better terms from suppliers and creditors, and
higher valuation by shareholders.
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benefits. When pre-managed earnings are below the benchmark by an amount less than EMj*,
earnings are managed upward so that reported (post-managed) earnings are above the benchmark.
The values of the EMj* may vary across observations and the values of EMj* are
determined by multiple factors, but firm size is an important factor. Because the benefits of
meeting the benchmark increase with the number and size of transactions with stakeholders, and
because larger firms have more and larger transactions with shareholders, the benefits of earnings
management increase with firm size. Consequently, the maximum amount of earnings that can be
managed at a cost less than the benefits, EMj*, also increases with firm size. Therefore, in any
sample that includes a broad range of sizes, it is essential to account for the effect of size on EMj*.
There may be additional factors that affect the benefits or costs of earnings management
and therefore also affect the EMj*. As benefits decrease or costs increase due to these other
factors, the value of EMj* will move toward zero. Fewer observations will be within EMj* of the
benchmark, earnings management will occur for fewer observations, and discontinuities will
disappear. Thus, we expect to find less evidence of discontinuities in any segment of the
population where firms realize lower benefits or face higher costs of earnings management.6 For
example, firms in financial difficulty are likely to have much lower rates of earnings management.
For these firms, the benefits of meeting an earnings target are likely lower (because concerns
about financial difficulty reduce any potential improvement in terms of transactions due to
meeting the earnings target) and the costs are likely higher (because financial difficulty is
associated with decreased flexibility to take actions that could increase reported earnings).
6
For example, BD provide evidence that discontinuities in earnings levels and changes are weaker among firms with
a weaker record of recent profitability, consistent with the hypothesis that these firms realize lower benefits or face
higher costs of managing to meet benchmarks. Similarly, Beatty et al. (2002) show that the rate of earnings
management is lower among private banks than among public banks, consistent with the hypothesis that private banks
have lower incentives to manage to meet benchmarks.
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2.2 Previous evidence of earnings management to meet benchmarks
Numerous anecdotes and conversations with practitioners suggest that earnings are
frequently managed upward to meet benchmarks and this theory is also supported by survey
evidence. Graham, Harvey and Rajgopal (2005) provide the most well-known and widely-cited
survey evidence that shows managers are willing to take actions, and incur significant costs, to
meet benchmarks. Thus, there are strong a priori reasons to believe that earnings are managed to
meet benchmarks.
This theory is further supported by a vast empirical literature documenting discontinuities
at a number of alternative benchmarks in a variety of countries and institutional settings. Using
U.S. data, studies document discontinuities around various earnings benchmarks, including the
profit/loss benchmark, creating a discontinuity in earnings levels at zero (Hayn, 1995; BD, 1997;
Degeorge et al., 1999); prior-year earnings, creating a discontinuity in earnings changes at zero
(BD, 1997; Degeorge et al., 1999; Beatty et al., 2002; Donelson et al., 2012); and analyst
forecasts, creating a discontinuity in earnings surprise at zero (Degeorge et al., 1999; Abarbanell
and Lehavy 2003; Burgstahler and Eames, 2006; Donelson et al., 2012). There is also strong
evidence using international data (Leuz, Nanda, and Wysocki, 2003; Suda and Shuto, 2005; Haw
et al., 2005). Evidence of discontinuities also exists for performance measures other than
earnings, such as debt covenant slack ratios (Dichev and Skinner, 2002), current ratios (Dyreng,
Mayew, and Schipper, 2012), and professional baseball batting averages and college entrance
exam scores (Pope and Simonsohn, 2010).
In sum, evidence of discontinuities at prominent benchmarks in distributions of earnings
and other performance metrics is found throughout the accounting literature as well as for nonearnings metrics in non-accounting literature. Collectively, the evidence shows that the
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discontinuities are not limited to any particular country or setting or benchmark and are robust to a
wide variety of research design choices.
3. DE "alternative explanations" for discontinuities
DE1 and DE2 assert that "deflation, sample selection, and a difference between the
characteristics of profit and loss observations" are "much more likely" explanations for
discontinuities than the widely-accepted theory that discontinuities are the result of earnings
management to meet benchmarks. Scholars must remain open to alternative theories that might
explain discontinuities, such as the DE "very evident" explanations. However, alternative theories
should be subjected to a careful evaluation of the theory's consistency with the body of empirical
evidence, including empirical evidence from the previous literature and additional empirical
evidence presented below.
3.1 Scaling as an alternative explanation
In this section, we evaluate the first alternative explanation proposed by DE, scaling,
which they summarize as follows:
“The arguments that net income must be deflated ‘in an attempt to homogenize firms’
(DPZ, p. 16) or ‘because firms are drawn from a broad range of firm sizes’ (BD, p. 102)
seem reasonable at first glance. But these studies do not provide a priori reasons why
heterogeneity or differences in firm size might affect the empirical analyses. More
importantly, the implicit assumption is that deflation will not distort the underlying
distribution of net income. Our results suggest that it does. Furthermore, since the
hypotheses in the extant literature focus on management of earnings to avoid small losses,
and since the arguments underlying these hypotheses do not suggest that earnings are
managed relative to beginning-of-period price, we question whether it is ever appropriate
to examine deflated earnings in an earnings management context.” (DE1 p. 559)
We begin by considering the research design rationale for scaling. Contrary to the DE1
assertions above, scaling does not rely on "the implicit assumption … that deflation will not distort
the underlying distribution of net income." Rather, the explicit purpose of scaling is to "distort the
underlying distribution" to achieve two goals: 1) to account for size as a covariate that explains
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variation in earnings (irrespective of earnings management) and 2) to account for the relation
between sizej and EMj*, the amount of earnings that can be managed at a cost lower than the
benefits.
First, effective research design must account for the strong relation between firm size and
earnings. Theory and empirical evidence clearly indicate that size explains a great deal of
variation in earnings. For example, the mean earnings reported by a $1 billion firm is far larger
than the mean earnings reported by a $10 million firm. Similarly, the standard deviation is
strongly related to size – the probability of an earnings observation $10 million above or below the
mean is far larger for a $1 billion firm than for a $10 million firm. We show that by failing to
account for the substantial effect of size, the DE design places inordinate weight on observations
for the smallest firms and, for reasons explained in the next few paragraphs, also results in low
power tests for the smallest firms.
Second, effective research design must also account for size-related differences in EMj*.
The exact forms of the functions relating benefits and costs to size are unknown, but we assume,
as a first approximation, that the ratio of benefits to costs remains constant as firm size increases.
For example, we assume that if increasing the size of the firm by a factor of 5 increases the value
of improved terms of transactions with stakeholders by a factor of 5, the amount of earnings that
can be managed at a cost less than the benefits also increases by a factor of 5. The constant ratio
assumption implies that EMj* increases in approximate proportion to size.7 In terms of dollars, the
constant ratio assumption implies the benefits of meeting the benchmark will be about 100 times
larger for a $1 billion firm than for a $10 million firm and therefore the maximum amount of
7
Determining the optimal histogram interval width in scaled earnings units is a research design issue, discussed in
detail in Section 3.
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earnings that can be managed to meet the benchmark at cost less than the benefits will be about
100 times larger for the $1 billion firm.
Earnings management to meet a benchmark will transform pre-managed earnings within
EMj* of the benchmark into reported earnings above the benchmark, creating a discontinuity in
the distribution of reported earnings. The prominence of the discontinuity increases with
1) increases in the product of the number of pre-managed earnings observations in the interval
below the benchmark and the rate of earnings management among those observations, creating a
trough in the histogram of reported earnings, and 2) increases in the number of observations
managed to the interval above the benchmark, creating a peak in the distribution of reported
earnings. Conversely, prominence of the trough and the peak dissipate as the numbers of
observations managed from the interval immediately below the benchmark and managed to the
interval immediately above the benchmark decrease.8 Thus, the research design must use an
interval width that corresponds as closely as possible to the EMj*. In particular, an interval width
that does not reflect the effect of size on the EMj* will obscure or eliminate any discontinuity due
to earnings management.
The following sections show that significant discontinuities exist in distributions of
unscaled earnings and earnings per share, contrary to the DE assertion that discontinuities are due
to scaling. For both unscaled earnings and earnings per share, the analysis is arranged in three
steps: First, an overview figure that shows the strong relation between earnings and size by
partitioning the population into quartiles based on size. Second, we examine discontinuities using
the same interval width for firms of all sizes for the individual size quartiles and in the aggregate,
showing how and why the DE design obscures evidence of discontinuities. Third, we examine
8
See Burgstahler 2012 for analysis of determinants of power of tests for discontinuities.
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discontinuities using interval widths that reflects size-related differences in EMj*, showing how
this research design leads to highly significant discontinuities.
3.1.1 Unscaled earnings
Figure 1 provides a descriptive overview of the effect of size on earnings distributions.
We expect to see a positive relation between size and both the mean and dispersion of earnings
distributions because larger firms tend to have larger expected future earnings and there is a
positive relation between future earnings and current earnings. We use MVE as the measure of
size in the analysis because 1) valuation models provide a direct theoretical link between MVE
and expected future earnings, and 2) we expect MVE to be more closely related to differences in
the EMj* than alternative measures of firm size, such as total revenues, total assets, and book
value of equity. Any of the size measures might be most closely related to benefits of earnings
management to meet benchmarks because they all reflect the number and size of transactions with
stakeholders (though in slightly different ways). However, costs of earnings management are
likely to be most closely related to market value of equity, under the additional assumption that the
flexibility to manage current earnings is proportional to the present value of expected future
earnings. Therefore, we rely primarily on market value of equity as the measure of size.9
Figure 1 shows distributions of unscaled earnings for a range that includes earnings of
most of the firms on Compustat, –$50 to $200 million, and a histogram interval width of $2.5
million.10 The four panels correspond to the four quartiles of firm size, where size is measured as
9
Most of the results reported below also hold for alternative size measures. Further, results previously reported in the
literature show that discontinuities are observed in distributions of earnings scaled by any of these alternative
measures of size. See for example, BD p. 102.
10
BD found that a large proportion of the small number of exact zero values of unscaled earnings (Compustat item
NI) were errors. To avoid incorrectly treating data errors as values that meet the benchmark of zero, all observations
with exact zero values of unscaled earnings are deleted throughout the analysis (though values exactly equal to zero
are relatively rare so this choice has no effect on our conclusions). Note that this issue arises for unscaled earnings but
not for EPS, where exact zero values are relatively common and are not subject to a high rate of errors.
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the market value of equity (MVE).11 Firms from the smallest size quartile have median market
value of $7.5 million while firms from the upper three quartiles have median market values of
$50.1 million, $235.2 million, and $1,983.1 million, respectively. The range of size across the
four quartiles is substantial. In round numbers, median market value increases across each quartile
by a multiple of about 5.12
Figure 1 confirms the expected strong relation between MVE and the location of unscaled
earnings and the strong relation between the dispersion and MVE. For the lowest MVE quartile,
the distribution of unscaled earnings has a negative median and mean and is highly concentrated in
the vicinity of the zero benchmark. For progressively larger MVE quartiles, the medians and
means become increasingly positive and the dispersion of the earnings distributions increases.
The dotted vertical lines in each panel indicate the range of unscaled earnings examined in the DE
research design, unscaled earnings observations between –$5 million and $7 million. Figure 1
shows that the preponderance of observations included in the DE design are from the first quartile
while only relative few of the observations from the larger MVE quartiles are included, a point we
explore in more detail in Figure 2.
Figure 2 Panels A-D show distributions of unscaled earnings using an interval width of
$100,000, the width used in the DE research design. For reasons explained above, using a constant
interval width of $100,000 for firms regardless of size is appropriate only if the maximum amount
of earnings that can be managed at a cost lower than the benefits is $100,000 regardless of firm
size. When EMj* is much smaller than $100,000, many of the observations in the interval
immediately below the benchmark will be too far below the benchmark to be managed and the rate
of earnings management will be small, obscuring the discontinuity. When EMj* is much larger
11
Because calculation of MVE requires market price, the results in Figures 2 and 3 are subject to a selection effect
that restricts the analysis to firms that have market prices. Selection effects are addressed thoroughly in Section 3.3.
12
The precise multiple is somewhat greater than 5 moving from Q1 to Q2 (6.7) or Q3 to Q4 (8.4) and slightly smaller
than 5 moving from Q2 to Q3 (4.7).
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than $100,000, many of the observations in intervals further below the benchmark will be
managed, again reducing or eliminating the discontinuity.
The results in Panels A-D are consistent with the predicted effects of using a single interval
width for firms of all size. Results in Panel B show highly significant evidence of a discontinuity,
consistent with what is expected if $100,000 is approximately the value of EMj* for firms in Q2.
In contrast, results in Panel A show little evidence of a discontinuity, consistent with expectations
if $100,000 is much larger than EMj* for the much smaller firms in Q1.13 Similarly, results in
Panels C and D do not show evidence of significant discontinuities, consistent with expectations if
$100,000 is much smaller than EMj* for the much larger firms in Q3 and Q4.
Figure 2 Panel E shows explicitly how the combination of the four quartile distributions in
Panels A–D results in an aggregate unscaled earnings distribution that does not show evidence of a
discontinuity.14 The bottom layer of each histogram bar corresponds to the very small number of
observations from Q4 (from Panel D), the next layer corresponds to the slightly larger number of
observations from Q3 (Panel C), the next layer to observations from Q2 (Panel B), and the top
layer to observations from Q1 (Panel A). The aggregate distribution is dominated by the large
number of observations from Q1, where the interval width is much larger than the corresponding
EMj*, resulting in insignificant evidence of a discontinuity due to earnings management to meet
the benchmark. Evidence of the highly significant discontinuity in Quartile 2, is obscured in the
aggregate distribution.
13
The basic problem when the peak falls in the immediate vicinity of the benchmark is that it is often not clear what
the pre-managed distribution should have looked like under the null hypothesis of no earnings management. In these
cases the distributional evidence is inconclusive — the distribution neither confirms nor contradicts the theory that
earnings are managed to meet the benchmark. It is incorrect to claim that a reported earnings distribution like the one
illustrated in Panel D provides evidence of the existence of earnings management to meet the benchmark, and it is
equally incorrect to claim that it provides evidence of the absence of earnings management to meet the benchmark.
Jacob and Jorgensen (2007) also discuss the inconclusiveness of tests under this situation.
14
The aggregate distribution here corresponds to distributions in DE1 Figure 3 Panel B and DE2 Figures 1, 6, and 8.
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We next consider distributions of unscaled earnings using a research design where interval
widths that reflect size-related differences in EMj*. We expect the maximum amount of earnings
that can be managed at cost lower than the benefits to be proportional to firm size, so we define
EM* relative to size. For example, we expect that earnings would almost never be managed by an
amount equal to 100% of market value but we would expect that earnings would almost always be
managed by an amount equal to .001% of market value. The exact proportion of market value to
use is unclear, but one directly relevant factor is materiality, an amount of financial statement error
that is permitted by auditing and reporting standards. Materiality is often discussed as a
proportion of earnings, such as 5% of earnings (Newton 1977; Leslie 1985; Nelson 1993; Bernard
and Pincus 1996; DeZoort, Hermanson, and Houston 2003; Nelson, Smith, and Palmrose 2005).
We can translate this materiality guideline into an approximate proportion of our size measure,
MVE. If MVE is typically a multiple of between 10 and 20 times long-run expected earnings,
then 5% of expected earnings is between .25% and .5% of MVE.15 Thus, this range represents an
amount of earnings management that could typically be accomplished at reasonably low cost
given materiality guidelines.16
While we want to choose interval widths consistent with the materiality guideline
calculation above and consistent with previous research, we also want to facilitate comparisons
with results from the DE design. Therefore, we use the DE interval width for the quartile where
the DE interval width is closest to .25% of the median quartile MVE. For each of the other
15
These amounts also correspond to the interval widths typically used in the previous literature. For example, BD
used interval widths of .5% of market value for distributions of earnings levels and .25% of market value for
distributions of earnings changes and subsequent studies have commonly used similar interval widths.
16
To see why the DE interval width of $100,000 is implausibly large for the smallest quartile and implausibly small
for the larger quartiles, note that $100,000 represents 1.33% of median market value of $7.5 million (and thus 13.3%
to 26.6% of expected future earnings). Thus, for many firms in the first quartile, the cost of managing earnings by
$100,000 is likely to be far higher than the benefits from meeting the benchmark. On the other hand, for firms in the
third and fourth quartiles where median market values are $235 million and $1.9 billion, respectively, $100,000
represents just .04% and .005% of market value, $100,000 represents an amount of earnings management whose cost
is likely to be far below the benefits.
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quartiles, we then look at the approximate ratio of MVE for the quartile to the MVE for the
quartile where we are using the DE interval width, and adjust the interval widths by that
approximate ratio. For example, in Figure 2, the $100,000 DE interval width is about .2% of
median MVE for Q2, so Q2 in Panel BB uses a $100,000 interval width. For the other quartiles,
we adjust the widths by multiples of 5 because the median market values in successive quartiles
increase by factors of approximately 5. For Q1 in Panel AA, the interval width is $20,000 (1/5 of
the Q2 width). For Q3 in Panel CC, the interval width is $500,000 (5 times the Q2 width). For
Q4 in Panel DD the interval width is $2,500,000 (25 times the Q2 width).17
Using interval widths that reflect size-related differences in the amount of earnings that can
be managed at low cost, the results in Panels AA-DD show highly significant discontinuities at the
zero earnings benchmark in all four quartiles. For the upper three quartiles in Panels BB–DD, the
benchmark is clearly to one side of the peak of the distribution and the interpretation is clearcut.
All three upper quartiles show highly significant discontinuities at the zero benchmark. For the
first quartile in Panel AA where the benchmark falls near the peak of the distribution, results must
be interpreted more cautiously. Nonetheless, the number of observations in the first interval to the
right of zero is abnormally large relative to the numbers in the surrounding intervals, and it is
highly unlikely that the large number in the first interval above the benchmark could be
attributable to random variation.
3.1.2 Earnings scaled by number of shares (earnings per share)
Next, we perform a parallel analysis for earnings per share (EPS) because DE1 "argue that
firms do not manage earnings deflated by financial variables; and, although it is possible that firms
manage earnings per share, the distribution of earnings per share does not show evidence of
17
These interval widths represent .27% of median market value for quartile 1, .21% for quartile 3, and .13% for
quartile 4.
Page 15
earnings management."18 Because there is no discontinuity in the distribution of EPS in DE1
Figure 1 Panel B but a distinct discontinuity in the distribution of price-scaled earnings in their
Figure 1 Panel A, DE1 erroneously infer that scaling causes the discontinuity. We show below
that the lack of discontinuity in their EPS distribution is, in fact, due to substantially the same
design flaws that exist in the DE analysis of unscaled earnings, though there are some subtle
differences in the interpretation of the results. We again start with an overview of the distributions
by quartile, where the quartiles are now defined in terms of the scaler advocated by DE, price per
share (PPS).
Differences across PPS quartiles do not simply reflect the effects of size, because PPS is a
size measure (MVE) scaled by an arbitrary number of shares. The theoretical effect of scaling by
arbitrary numbers of shares is indeterminate—it is possible for large MVE firms to have small
PPS due to a large number of shares and for small MVE firms to have large PPS due to a small
number of shares. The empirical effect of scaling by number of shares depends on the relation
between number of shares and size, as illustrated by two simplified examples. First, if all firms
have the same number of shares outstanding, then PPS is a fraction (1/Number of shares) of MVE,
so the distribution of earnings per share would correspond exactly to the distributions of unscaled
earnings in Figures 1 and 2 (though scaled by the number of shares). Second, if firms continually
adjusted their number of shares to maintain a constant PPS, then the number of shares would
reflect differences in market value, and earnings per share would correspond to the distributions in
Figures 1 and 2 (but scaled proportional to market value). Empirically, the relation between
number of shares and size is more complicated than either of these two examples, so we expect
distributions of EPS to differ from distributions of unscaled earnings, but in ways that are
impossible to specify ex ante.
18
See the summary of the DE1 arguments in DE2 footnote 19, page 1265.
Page 16
Figure 3 shows distributions of EPS by PPS quartile, and the dotted lines show the range of
EPS included in the distributions reported in DE, –$1.00 to $1.00. Similar to the results in Figure
1, as PPS increases, the EPS distributions shift to the right and become more dispersed. These
effects reflect the effects of size as well as the effect of scaling by number of shares. The
empirical result of the effect of size combined with scaling by number of shares is that firms with
larger PPS (MVE per share) tend to have more positive EPS (earnings per share), and firms with
larger PPS tend to have greater variability of EPS.
Figure 4 Panels A-D show the distributions of EPS for each of the four PPS quartiles,
using the DE interval width of $.01 and range of –$1.00 to $1.00. The $.01 interval width used in
the DE design does not allow for differences in the amount of EPS that can be managed at a cost
lower than the benefits and provides only limited evidence of discontinuities.19 Although both the
second and third quartiles provide visual evidence of discontinuities, the standardized differences
are insignificant or only moderately significant.
Figure 4 Panel E shows explicitly why the aggregate distribution of EPS reported in DE
does not show evidence of a discontinuity. The aggregate distribution is the sum of the Panel D
distribution (the bottom layer of each histogram bar), the Panel C distribution (the second layer),
and the Panels B and A distributions (the top two layers).20 As with unscaled earnings, the
preponderance of observations included in the DE EPS distribution are from the smallest size
quartile. In the intervals immediately adjacent to the zero benchmark, there are about 10 times as
many observations from Q1 as from Q2, and roughly 40 times as many as from Q3, and 80 times
as many as from Q4. Because it is dominated by observations from Q1, which does not contain a
19
The DE interval width of $.01 represents 1.33% of median PPS for firms in the first quartile, .18% for firms in the
second quartile, .066% for firms in the third quartile, and .03% for firms in the fourth quartile. Therefore, the interval
width of $.01 used in DE is too wide for the smallest PPS quartile firms and too narrow for the largest PPS quartile
firms.
20
The dashed line at the center of the distribution indicates the EPS=$.00 benchmark. The other dashed lines,
indicating EPS benchmarks at multiples of $.10, are discussed in the next paragraph.
Page 17
discontinuity because the interval width of $.01 is too wide for firms with median PPS of $.75, the
aggregate distribution does not contain a clear-cut discontinuity.
Before turning to results for the design that adjusts for differences in EMj*, we note
another prominent feature of the EPS distribution that is consistent with the theory that earnings
are managed to meet benchmarks but inconsistent with the DE alternative explanations for
discontinuities. Carslaw (1988), Thomas (1989), Das and Zhang (2003), and Grundfest and
Malenko (2009) posit and report evidence of earnings management to achieve rounding in EPS,
such as earnings management to increase earnings so that EPS rounds up from positive amounts
ending in .x9 to the next highest multiple of 10 cents per share. The dashed vertical lines in Figure
5 Panel E indicate the intervals corresponding to EPS at multiples of 10 cents per share (i.e., $.10,
$.20, …, $1.00 per share) and discontinuities at the various 10 cent per share benchmarks are
significant.21 Of particular note are the highly significant left and right standardized differences at
the $1.00 EPS benchmark, consistent with particularly strong incentives to manage earnings such
that earnings per share rounds up to a whole dollar.22 The significant discontinuities at non-zero
benchmarks cannot be explained by the scaling explanation offered by DE and, as discussed
further in Section 3.2 below, are also inconsistent with the other DE explanations. 23
Returning to the issue of discontinuities at the zero benchmark, we reexamine the EPS
distributions using interval widths that reflect size-related differences in EMj*. Applying the
criteria used in defining the interval widths in Figure 2 Panels AA-DD, the interval width for
21
Seven of ten left standardized differences are significantly negative at the .05 level while nine of ten right
standardized differences are significantly positive. A more powerful test for the "rounding up" form of earnings
management can be constructed based on the average of the standardized differences across the k individual
benchmarks and this test yields still more significant results. The standardized average computed from the ten left
standardized differences in Figure 5 is -7.53 and the standardized average for the ten right standardized differences is
8.85.
22
The discontinuity in the aggregate distribution at EPS= $1.00 is also visually apparent in each of the individual
quartile distributions shown in Panels B-D.
23
DE1 footnote 20 acknowledges that discontinuities exist at multiples of 10 cents per share but fails to recognize
that these discontinuities cannot be explained by any of the "alternative, much more likely, explanations" proposed in
DE.
Page 18
Quartile 2 is set to the $.01 interval width used in DE. For the other three quartiles, the widths and
ranges are, respectively, 1/5, 3, and 7 times the width and range used in Quartile 2, resulting in
interval widths of $.002, $.01, $.03, and $.07, roughly equal to .25% of median PPS for the four
quartiles of $.75, $5.34, $14.75, and $33.75, respectively.24 Figure 4 Panels AA-DD show the
resulting distributions of EPS. The results for Quartiles 2, 3, and 4 are significant and are
discussed first. Then we turn to the results for Quartile 1.
For Quartile 2 in Panel BB (which is by construction identical to Panel B except for
different vertical scales), there is clear visual evidence of a trough below the $.00 benchmark, and
the left standardized difference is significant at the .05 level. However, the results are consistent
with the conjecture that there are two benchmarks. The number of observations with EPS equal to
$.00 is distinctly larger than the number of observations with EPS equal to –$.01 and the number
of observations equal to $.01 is also distinctly larger than the number of observations with EPS
equal to $.00. This is the pattern expected with dual benchmarks where EPS=$.00 yields benefits
from achieving non-negative EPS while EPS=$.01 yields additional benefits from achieving
positive EPS.25 At the benchmark of .00, the left standardized difference of –2.168 is significant.
The right standardized difference of –.758 (by definition equal to the left standardized difference
24
The Compustat variable EPSFX, expressed in dollars and cents, is used as EPS for quartiles 2, 3, and 4. However,
the bin width of $0.002 for quartile 1 requires EPS measured in fractions of a cent. Therefore, for Panel AA, we
compute EPS as Compustat variable IBADJ divided by Compustat variable CSHFD.
25
To illustrate how dual benchmarks affect the standardized difference statistics, consider an example where
1) earnings management to avoid negative EPS leads to a sharply larger number of observations in the $.00 interval
than in the –$.01 interval, and 2) earnings management to achieve positive EPS also leads to a larger number of
observations in the $.01 interval than in the $.00 interval by the same amount. In this example, the left standardized
difference at $.00 is negative, because the number of observations with EPS=–$.01 is less than the average of the
numbers with EPS=–$.02 and EPS=$.00. However, because of the assumption that the discontinuities at $.00 and
$.01 are the same size, the number of observations with EPS=$.00 is equal to the average of the number of
observations with EPS= –$.01 and EPS=+$.01. Therefore, the right standardized difference at $.00 is zero (rather
than positive). Similarly, the left standardized difference at $.01 (and is by definition equal to the right standardized
difference at $.00) is zero. However, the right standardized difference at $.01 is positive because the number of
observations with EPS=$.01 is greater than the average of the number of observations with EPS= $.00 and EPS=$.02.
Page 19
at .01) is insignificant. At the benchmark of .01, the right standardized difference of 2.090 (not
reported in the figure) is significant.
For Quartiles 3 and 4, Panels CC and DD show highly significant discontinuities.26 For
Quartile 3, the left and right standardized differences are each significant at the .01 level and for
Quartile 4, the standardized differences are each significant at the .05 level.
For Quartile 1 in Panel AA, the evidence is inconclusive. The EPS distribution includes
many observations of $.00 EPS and, not unexpectedly for a set of firms with median PPS of $.75,
a larger proportion of small negative EPS than small positive EPS.27 Overall, the distribution for
firms in Quartile 1 neither confirms nor contradicts the theory that earnings are managed to meet
the benchmark.28
It is also important to recognize that very low PPS also proxies for financial difficulty for
institutional reasons. Firms with PPS<$1.00 are generally not allowed to trade on major
exchanges, and Table 2 shows that the inability to maintain PPS above this amount is also
associated with low revenue, low assets, and negative equity. Thus, firms in the first PPS quartile
are more likely to be experiencing financial difficulty, and also likely to have even smaller values
of EMj*. Table 2 shows that financial difficulty is especially strongly related to values of
PPS<$1.00. In fact, results in Appendix Figure A-3 using interval widths of $.005 show that the
26
Note that the interval widths used in Panels CC and DD group EPS=.00 and EPS=.01 into a single interval, so that
the first interval above the benchmark combines all observations meeting either benchmark.
27
Note that in the first quartile, the zero benchmark is somewhat ambiguous. For example, if the zero benchmark is
defined by EPS rounded to the nearest whole cent, the “zero” benchmark is –$.005. Thus, observations in the first
two intervals below the benchmark, which comprise small negative EPS between .000 and –$.002 and between –$.002
and –$.004, represent $.00 EPS when rounded to the nearest whole cent. This ambiguity will tend to obscure empirical
evidence of discontinuities.
28
Beaver et al. (2007) document that the number of shares outstanding is decreasing in earnings around zero. It is
possible that the EPS measure used by DE is deflated by a higher number of shares outstanding for loss firms than the
lower number of shares outstanding for profit firms. However, this explanation is unlikely to induce discontinuities in
EPS because we observe discontinuities at multiples of $0.10 in Figure 5 Panel E, even though there is no evidence in
DE or elsewhere in the literature that the number of shares is systematically greater at multiples of $0.09 per share
than at the corresponding multiples of $0.10 per share.
Page 20
lack of discontinuity in PPS Q1 is due primarily to firms with PPS<1.00, and there is a significant
discontinuity for firms in Q1 with PPS > $1.00.
3.1.3 Summary of the effects of scaling
In this section, we have shown why the distributions of unscaled earnings and EPS
reported by DE do not show discontinuities—the DE research designs fail to account for the
substantial effect of size as a covariate and fail to account for differences in the amount of
earnings that can be managed at a cost lower than the benefits. In contrast, a research design that
accounts for these two fundamental effects but without scaling results in significant discontinuities
in distributions of unscaled earnings and EPS.
Although the results above show clearly that discontinuities are not the result of scaling, it
is important to recognize that a research design based on scaled earnings has important conceptual
advantages. Scaling adjusts each observation by its own individual size, whereas partitioning into
size quartiles only partially accounts for the effect of size by holding size roughly constant within
each quartile. Further, scaling allows all observations to be aggregated into a single distribution,
resulting in a more powerful, unified test rather than four separate tests, each with one-fourth the
sample size.29 Thus, scaling maximizes the trough in the first interval below the benchmark.
Scaling also provides a larger effective sample size, allowing the discontinuity to be tested in a
single test, rather than separate distributions and separate tests that result from the second design
based on partitioning into size quartiles.
29
Increased power that results from combining results into a single distribution is illustrated by the still more
significant results discussed below for the aggregate distribution of market-value-scaled earnings in Figure 6.
Page 21
3.2 Relation between earnings and market prices as an alternative explanation
DE1 proposes a mechanism by which higher average prices for firms with small positive
earnings than for firms with small negative earnings could create a discontinuity in the distribution
of price-deflated earnings:
“These results suggest that the firms reporting a small profit tend to have a higher
beginning-of-year price, and, when net income is deflated by this higher price, relatively
more of the earnings observations are drawn toward zero, whereas the lower beginning-ofyear price for firms reporting a loss tends to push relatively more of the earnings
observations away from zero.” (p. 573)
In this section, we evaluate whether this mechanism represents a plausible alternative explanation
for discontinuities. Before turning to specifics, note that this is a variation of DE’s first
explanation in that it relies on scaling earnings by price. Because our results in Section 3.1 show
that scaling in general does not explain discontinuities, it is unlikely that scaling by price in
particular explains discontinuities.
Nonetheless, in this section, we evaluate the assertion in DE2 that this mechanism
represents “a simple explanation” for discontinuities in the distribution of net income deflated by
price. (DE2 p. 1251) We first point out the limited scope of this explanation – the relation
between earnings and price can explain discontinuities in only one narrow setting, discontinuities
at zero in distributions of price-scaled earnings. Second, we show that even in this narrow setting,
the relation between earnings and price does not explain discontinuities.
3.2.1 Settings where the relation between earnings and price cannot explain discontinuities
The DE mechanism requires 1) a benchmark of zero, 2) lower prices for observations
immediately below the benchmark than immediately above the benchmark, and 3) scaling by
price. All three of these elements are required; when any of the three elements is missing, the DE
mechanism cannot explain discontinuities.
Page 22
The significant discontinuities at the various non-zero (10 cent per share) benchmarks in
Figure 5 Panel E cannot be explained by the DE mechanism because the EPS variable is not scaled
by price. For zero benchmarks, the DE mechanism applies only when prices are substantially
lower immediately below the benchmark than above the benchmark. However, there is no
evidence in DE (or elsewhere in the literature) showing that beginning-of-the-year prices are lower
for small negative earnings changes versus small positive earnings changes or for small negative
earnings surprise versus small positive earnings surprise.30 Even for zero benchmarks where
prices are substantially lower below the benchmark than above, the DE mechanism requires
scaling by price, so the DE mechanism cannot explain the significant discontinuities in
distributions of unscaled earnings shown in Figure 2 Panels AA-DD nor in distributions of EPS
shown in Figure 4 Panels AA-DD. In sum, a large portion of the evidence of discontinuities
previously reported in the literature cannot be explained by the relation between earnings and
price, including discontinuities in unscaled EPS at non-zero benchmarks, discontinuities in
earnings changes and earnings surprises, and discontinuities in unscaled earnings and EPS at the
zero benchmark.
3.2.2 Discontinuities where the relation between price and earnings could be relevant
We nonetheless consider whether the relation between price and earnings can explain
discontinuities for the specific case of the zero benchmark in levels of price-scaled earnings.
Based on evidence that small negative unscaled earnings observations have lower prices than
small positive unscaled earnings observations, DE conclude that scaled negative observations are
pushed away from zero and scaled positive observations are drawn toward zero. However, this
observation shows only that the PPS associated with small negative EPS observations is typically
30
DE speculate that discontinuities in changes in earnings might be explained by the positive correlation between
earnings levels and earnings changes. However, for reasons explained in Section A1 of the appendix, the positive
correlation with earnings levels cannot explain discontinuities in changes in earnings.
Page 23
lower than the PPS associated with small positive EPS observations.31 This indirect evidence does
not show that scaling by price creates the discontinuity in distributions of scaled earnings.
To provide direct evidence on the effect of scaling on distributions of scaled earnings, we
examine distributions of scaled earnings. We begin by examining distributions of scaled earnings
within segments of the population where PPS is held roughly constant. Within quartiles of PPS,
there is less variation in PPS, so discontinuities within a quartile are less likely to be explained by
differences in PPS for small negative EPS than for small positive PPS. The values of PPS
separating the four quartiles are $2.58, $9.50, and $22.11. Therefore, the relative ranges of the
PPS scaler are smallest in Q2 and Q3.32 The relative range is large in Q4, where there are a few
extremely large values of PPS, but the vast majority of observations have PPS between $22 and
$100, so the relative range for the majority of the observations is on the order of 5. The largest
relative range of scalers is in Q1, where there are a relatively large number of observations with
PPS less than $.10, including values as low as $.01, and observations with PPS up to $2.50.
If the discontinuity in price-scaled earnings is due to the relation between price and
earnings as asserted by DE, there should be little evidence of discontinuities in the distributions
for Q2 and Q3, where the relative range of price scalers is narrow. Evidence in Q4 should be
stronger, as there is a broader range of prices, though for the majority of observations the range is
still fairly limited. The strongest evidence of discontinuities should be concentrated in Q1, where
there is a much wider relative range of prices used to scale earnings.
31
The explanation for the empirical relation between PPS and EPS is clear from our Figure 3, which shows that firms
from the smallest PPS quartile generate far more of the small negative EPS observations than do firms from the larger
PPS quartiles. Thus, small negative EPS observations are associated with smaller PPS.
32
Note that the DE "pushing" and "pulling" effect depends on the relative, rather than absolute, range of PPS within a
quartile. For example, the DE effect could be substantial in a quartile where some observations are scaled by PPS
=$.01 and others by PPS=$.50, but there would be almost no effect in a quartile where some observations are scaled
by PPS =$20.01 and others by PPS=$20.50.
Page 24
Figure 5 Panels A–D show histograms of price-scaled earnings for each of the size
quartiles Q1–Q4. Visual inspection of Figure 5 and the standardized differences show that the
results are inconsistent with what is expected if discontinuities in scaled earnings are explained by
the relation between earnings and price. The discontinuities are strongest in the second and third
quartiles, somewhat weaker in the fourth quartile, and weakest in the first quartile.
We also calculate another measure of the discontinuities to describe the proportion of the
discontinuity in the aggregate scaled earnings distribution due to each of the PPS quartiles,
dividing the difference between the number of observations in the interval above the benchmark
versus below the benchmark for each of the individual quartiles by the total difference in the
aggregate distribution. Using this measure, the three larger PPS quartiles account for 90% of the
discontinuity in the aggregate distribution.33
Figure 5 Panel E provides a graphical representation of the conclusions above, showing the
aggregate price-scaled earnings distribution as the sum of distributions from the four PPS
quartiles. The discontinuity in the aggregate distribution of scaled earnings is not due to
observations from Q1, where the effect proposed by DE should have the greatest effect on price
scaled earnings, but rather primarily to observations from Q2 and Q3 where the relation between
price and earnings should have the least effect.
Thus, the results show that the discontinuity in scaled earnings is concentrated primarily
within Q2 and Q3 where the relative range of the price scaler is most restricted and where we
would expect the relation between earnings and price to have the least effect. Nonetheless, it is
still possible that within these quartiles, the average price associated with small negative earnings
remains significantly smaller than the average price scaler associated with small positive earnings.
33
The aggregate distribution has 762 more observations in the interval right of the benchmark than in the interval left
of the benchmark, due to differences of 80, 286, 241, and 155, respectively, in the individual PPS quartiles. Thus, the
proportions of the total difference due to each of the four PPS quartiles are 10%, 38%, 32%, and 20%, respectively.
Page 25
Thus, it is conceivable that the DE mechanism operates within the PPS quartiles, so we next
examine the relation between earnings and price within each PPS quartile.
Our Figure 6 Panel A replicates DE1 Figure 6 and confirms the empirical result in DE
showing that small negative earnings observations are associated with lower PPS, aggregating
across firms of all sizes. However, Figure 6 Panel B shows that this aggregate relation does not
carry over to all the individual PPS quartiles. The horizontal dashed lines indicate the three PPS
values that separate the four PPS quartiles. Within each PPS quartile, three lines are plotted
showing the 25th, 50th, and 75th percentile of PPS for each portfolio of EPS.34 The results show
that the PPS associated with small negative EPS is not systematically lower than PPS associated
with small positive EPS within Q2, Q3, or Q4. Thus, the aggregate relation in Panel A does not
hold in any of the three upper quartiles and therefore cannot explain the discontinuity in these
three quartiles.35
For Q1, where there is a broader range of price scalers, there is some evidence that small
negative earnings observations tend to have smaller PPS than do small positive earnings
observations. Thus, it is conceivable that the relation between earnings and price contributes
something to the discontinuity in the distribution of scaled earnings in Q1. However, as we saw in
Figure 5, observations from Q1 play a very minor role in the discontinuity in the aggregate
distribution of scaled earnings.36
34
Note that because the number of observations in the plotted range of EPS decreases substantially across quartiles
(as can be seen in Figure 3), random variation in the plotted percentiles increases substantially across the four PPS
quartiles. For example, there are only a few small unscaled EPS observations in quartile 4, so the 25th, 50th, and 75th
percentile statistics show a large amount of random variation. However, the small number of unscaled earnings
observations from the larger price quartiles reflects the mistaken assumption in DE that small scaled earnings
observations are always the result of scaling small unscaled earnings observations. In fact, Figure 6 shows that there
are roughly comparable numbers of small scaled earnings observations across all four quartiles.
35
Results for the relation between MVE and unscaled earnings and scaled earnings observation by MVE quartiles are
not reported here, but they are qualitatively similar to results for PPS and EPS.
36
In fact, results in Appendix Figure A-3 shows that evidence of a discontinuity in scaled earnings from Q1 is due
primarily to firms with PPS > $1.00, and Figure A-4 shows that for Q1 firms with PPS>$1.00, there is essentially no
difference between the PPS associated with small negative EPS versus small positive EPS. Thus, even within Q1 the
Page 26
3.2.3 Summary
Contrary to the conclusions in DE, the evidence shows that the relation between earnings
and price does not explain the discontinuity in the aggregate distribution of net income deflated by
market capitalization. At most, the relation between earnings and price might contribute
marginally to the discontinuity for an extremely narrow segment of extant discontinuity evidence
– discontinuities in distributions of price-scaled earnings at zero for firms with PPS below $1.00.
To borrow a phrase from DE2 (p. 1251), the evidence shows that the relation between earnings
and price has "nothing to do with" the discontinuity in the aggregate distribution of price-deflated
earnings.
3.3 Sample selection as an alternative explanation
DE speculate that discontinuities in earnings distributions might be attributable to selection
effects. They note that scaling by market value requires beginning market price, and report two
related results: 1) A greater proportion of loss observations than profit observations are eliminated
when missing price observations are excluded from the sample, and 2) discontinuities become
weaker when missing price observations are included in the sample. Based on these results, DE
infer that sample selection effects explain discontinuities. For example, they conclude, based on
the distribution of earnings per share in DE1 Figure 2 Panel B, that the discontinuities observed in
distributions of scaled earnings are explained by selection criteria:
“It follows from figure 2, panel B that, since a larger proportion of loss observations are
deleted from the sample because of an inability to calculate beginning-of-year market
capitalization, the discontinuity of the (deflated) earnings distribution at zero may be
partially due to this sample selection criterion” (DE1, p. 566).
In this section, we evaluate sample selection as an alternative explanation.
relation between earnings and price cannot explain the discontinuity— almost all of the Q1 discontinuity in scaled
earnings comes from the portion of Q1 where the relation between earnings and price cannot explain the discontinuity.
Page 27
3.3.1 Inference from DE evidence of the effects of selection criteria
The effects of selection criteria can be characterized in terms of 1) the core population that
is always included, 2) an additional segment that may be excluded by selection criteria, and 3) the
extended population, comprising both the core and the segment. Using this terminology, the
extended population is the result of inclusion of the segment with the core or, equivalently, the
core is the result of exclusion of the segment from the extended population.
The effects of inclusion or exclusion of the segment depend on the relative strength of the
discontinuities in the core versus the segment. Define the segment as having a weaker
discontinuity than the core if it has a lower ratio of number of observations in the interval
immediately above the benchmark to the number in the interval immediately below the
benchmark. For a segment with a weaker discontinuity, there are direct and predictable effects.
Inclusion of the segment results in an extended distribution with a lower ratio and weaker
discontinuity than the core. Similarly, exclusion of the segment from the extended distribution
leaves the core, which has a higher ratio and stronger discontinuity than the extended distribution.
Finally, because the segment has a weaker discontinuity, exclusion of the segment also by
definition eliminates a larger proportion of observations below the benchmark than the proportion
in the core distribution.37
Figure 7 shows how these points apply to inferences from the evidence in DE on the effect
of inclusion or exclusion of missing price observations. For both unscaled earnings (Panel A) and
37
To illustrate these effects with a numerical example, consider a core with a strong discontinuity (with 75
observations in the interval immediately below the benchmark and 175 observations in the interval immediately above
the benchmark, and a ratio of 175 / 75 ≅ 2.33) and a segment with the weakest possible discontinuity, namely no
discontinuity (with 125 observations below and 125 observations above the benchmark, and a ratio of observations
above versus below the discontinuity of 125 / 125 = 1). Inclusion of the segment results in an extended distribution
with a weaker discontinuity and lower ratio than in the core. In the extended distribution, the ratio is
(175 +125) / (75 + 125) = 1.5; in the core, the ratio is 175 / 75 = 2.33. Thus, inclusion of the segment weakens the
discontinuity in the extended distribution. Similarly, exclusion of the segment results in elimination of a larger
proportion of observations below the benchmark than the proportion in the core distribution. Specifically, exclusion
of the segment eliminates125/200 = 62.5% of observations below the benchmark, a proportion that is larger than the
125/300 = 41.7% of observations from the extended distribution eliminated in the interval above the benchmark.
Page 28
EPS (Panel B), the distribution for the segment of firms with missing price has no discontinuity at
the benchmark and, in fact, has a higher proportion of small loss observations than small profit
observations. In contrast, the core distributions for firms that have price (i.e., distributions of
unscaled earnings in Figure 2 and EPS in Figure 4 above), have significant discontinuities, at least
for firms in the upper three size quartiles. Therefore, the direct effect of inclusion of missing price
observations is a weaker discontinuity in the extended distribution and the direct effect of
exclusion of missing price observations is elimination of a larger proportion of small loss
observations than the proportion in the core distribution.38
However, these two empirical facts are simply direct and predictable effects of a weaker
discontinuity for the segment of firms that are missing price. Contrary to assertions in DE, these
direct and predictable effects only show that distributions for the segment of missing price
observations have weaker discontinuities—they do not do not show that discontinuities in the core
of non-missing price observations are explained by sample selection. The explanation for weaker
discontinuities in the segment of missing-price observations is an open question, though it might
well be that firms with missing prices have lower incentives to manage earnings to meet the
benchmark. Whatever the explanation for the lack of discontinuity among firms with missing
price, the effects of sample selection cannot explain the discontinuity in the core distribution of
firms with price.
4. Evaluation alternative explanations for discontinuities
It is important for researchers to continue to explore the possibility that discontinuities in
earnings distributions might be explained by something other than earnings management.
38
These effects are further exaggerated by the DE research designs, because the number of missing price
observations in the small magnitude unscaled earnings intervals immediately below and above the benchmark far
exceeds the number of non-missing price observations in the DE intervals from the upper three quartiles in Figures 3
and 5.
Page 29
However, alternative explanations for discontinuities, such as the artifactual explanations proposed
by DE, must be evaluated based on their ability to explain the body of discontinuity evidence,
rather than just selected narrow elements of the evidence.
The introduction to DE2 unintentionally points to the key issues in evaluating alternative
explanations:
“Rigorous academic research cannot be based on just an appeal to the popularity of the
notion that the shapes of the earnings distributions are evidence of earnings management
because: (1) supporting evidence that this is so is sparse, perhaps even nonexistent and
(2) alternative explanations for the shapes of the distributions are often very evident. In this
paper, we elaborate on several alternative explanations.” (DE2 p. 1250)
Section 3 above exposes the fallacies in the "very evident" explanations advocated by DE. In this
section, we further examine the DE claim that previous evidence in the literature consistent with
the theory that earnings are managed to meet benchmarks is sparse.
Previous results summarized in Section 2 together with the results reported above establish
a number of key characteristics of the body of empirical evidence on discontinuities:
(1) Discontinuities appear in distributions of earnings, changes in earnings, and earnings
surprise (i.e., earnings less forecasts) at the zero benchmark, as well as at non-zero
benchmarks such as multiples of 10 cents per share.
(2) Discontinuities do not depend on the choice of scaler (discontinuities appear in earnings
scaled by market value, book value, assets, or revenue) nor do they depend on whether
earnings are scaled (discontinuities appear in scaled earnings, unscaled earnings, and EPS).
(3) Discontinuities frequently have a trough below the benchmark and a peak above the
benchmark, consistent with the idea that earnings management transforms pre-managed
earnings below the benchmark into reported earnings above the benchmark.
(4) Discontinuities are stronger among firms that are likely to realize lower benefits or face
higher costs of earnings management.
(5) Discontinuities appear in distributions of various forms of earnings that are widelyreported and commonly used in stakeholder decisions, such as earnings before
extraordinary items, or net income, or the numerator of earnings per share.
(6) Discontinuities do not appear in transformations of earnings that are not widely-reported
and commonly used in stakeholder decisions, such as earnings before special items or
earnings before depreciation or earnings before interest expense.
Page 30
Some explanations for discontinuities are rendered implausible by the first and third
characteristics. For example, the explanation in Beaver, McNichols, and Nelson (2007) based on
significantly higher tax rates for positive earnings than for negative earnings cannot explain
discontinuities in earnings changes, earnings surprises, or at 10 cent multiples of EPS, since there
is no evidence that tax rates are different above versus below these benchmarks.39 Further, as
Beaver, McNichols, and Nelson note in their introduction, the tax explanation cannot explain the
peak and trough that are commonly observed.
The last characteristic in the list above is often overlooked and its importance
underappreciated. The discontinuities reported by BD were discovered in scatterplots prepared for
an unrelated research issue addressed in Burgstahler and Dichev (1997b). Those scatterplots,
which have not been widely discussed, revealed an important fact: Discontinuities exist in
distributions of reported net income (as used in the plots in the working paper version of
Burgstahler and Dichev 1997b) but not in distributions of earnings before special items (as used in
the published version). In fact, further investigation at that time revealed that there are many such
transformations of earnings that exhibit no discontinuity at zero, such as earnings before
depreciation or earnings before research and development expense.40 The key characteristics of
transformations that eliminate the discontinuity are 1) the transformed earnings measure must not
be one that is widely-used in stakeholder decisions and 2) the omitted earnings component (or
39
Note also that the difference in tax rates above and below zero earnings levels that BMN assume in their simulation
is far greater than the actual difference shown in their Figure 2 Panel B, so the simulation exaggerates the possible
effect of the difference in tax rates.
40
The difference between the distributions before and after an omitted component of earnings may at first seem to
suggest that the omitted component causes, or at least contributes to, the discontinuity in reported earnings. However,
this is true only in the same limited sense that every component of earnings (and randomly generated noise) with nontrivial variability contributes to the discontinuity in reported earnings.
Following a similar line of reasoning, Beaver, McNichols, and Nelson (2007) suggest that because removal of the
special items component of earnings eliminates the discontinuity in the distribution of earnings, special items
"contribute to" the discontinuity.
Page 31
components) must have non-trivial variability.41 Further, Dechow, Richardson, and Tuna (2003)
explain and illustrate in their Figures 6(a) and 6(b) how subtracting a random noise variable with
non-trivial variability from reported earnings results in a transformed earnings ("earnings before
random noise") that does not include a discontinuity at zero. Artifactual explanations, such as
those advocated by DE, apply also to the transformed earnings measures, and thus cannot explain
why there are no discontinuities in distributions of the transformed measures. In contrast, this
result is exactly what is expected if reported earnings measures are managed, while the
transformed earnings measures are not.42
Jacob and Jorgensen (2007) report other evidence showing that discontinuities occur in
distributions of reported earnings measures but not in closely-related measures that are not widelyreported, consistent with earnings management but not with the DE explanations. They examine
distributions of "pseudo annual earnings" measures constructed by adding earnings from four
adjacent quarters ending at interim quarter-ends. Since each "pseudo earnings" number covers an
annual period (but not an annual fiscal period), the DE explanations should also apply to the
pseudo observations. However, Jacob and Jorgensen show that distributions of annual earnings
contain highly significant discontinuities while distributions of "pseudo annual earnings" do not.43
41
A component of earnings with "non-trivial" variability is a component that is not simply constant, or nearly
constant, across observations. For example, for firms that do not have special items, special items is zero and by
definition has trivial variability (and for these firms the distribution of earnings before special items is identical to the
distribution of earnings including special items). However, for firms with non-trivial special items, earnings before
special items has little or no discontinuity.
42
Jorgensen, Lee, and Rock (2012) report yet another example of evidence inconsistent with discontinuities being
attributable to artifacts. When the focal EPS measure was changed by SFAS 128, firms were required to report
retroactively restated Diluted EPS measures. The discontinuity at the zero benchmark appears in distributions of
changes in 'as reported' Primary EPS but not in changes in 'as restated' Diluted EPS, consistent with expectations if
reported earnings measures are managed to avoid small earnings decreases.
43
There is extensive discussion in DE2 explaining why annual earnings are different from pseudo annual earnings.
However, the issue is not whether annual earnings are different from pseudo earnings—Jacob and Jorgensen
constructed pseudo earnings with the specific purpose of constructing a variable different from annual earnings. The
critical issue is why the DE explanations do not create discontinuities in distributions of pseudo earnings. DE2 never
explain why scaling, selection, and the correlation between earnings and price do not create discontinuities in
distributions of pseudo earnings.
Page 32
Finally, alternative explanations for discontinuities in earnings distributions should also be
able to explain when discontinuities will occur. However, one of the notable aspects of the DE
papers is that they provide no evidence showing how any of their explanations create
discontinuities, but only evidence showing how discontinuities are obscured or eliminated by
changes in the research design. In evaluating proposed explanations for discontinuities, it is also
important to consider whether it is possible to show that the alternative explanations cause
discontinuities in distributions of variables other than earnings.
Therefore, we consider one more example of a variable where the effects that DE claim
cause discontinuities should create a discontinuity, yet the distribution of this variable does not
contain a discontinuity. This variable is the annual change in market value, a market measure
directly analogous to the accounting measure of the annual change in value, accounting earnings.
The annual change in market value and annual earnings have the same statistical correlation with
beginning price: Both unscaled earnings and unscaled changes in market value are positively
correlated with beginning market value for positive changes in market value and negatively
correlated for negative changes in value. Further, exactly the same scaling and selection effects
occur when the two variables are scaled by beginning market value to compute, respectively,
price-scaled annual earnings and annual returns. Therefore, if the DE effects of scaling, selection,
and correlation with price induce a discontinuity in distributions of price-scaled earnings, these
effects should also induce a discontinuity in the distribution of returns. Of course, no
discontinuity has been observed in the distribution of annual market returns.
DE2 also propose that the results in Jacob and Jorgensen (2007) are erroneous because under the integral method of
accounting, APB 28 “forbids firms from smoothing expenses across fiscal years” (DE2, P. 1267). However, Kerstein
and Rai (2007) directly test the assertion in DE2 that discontinuities in annual earnings are not driven by earnings
management but instead are driven by settling up in the fourth quarter of estimates from the first three quarters (DE2,
P. 1268). Kerstein and Rai (2007) find that an unusually high proportion of firms with small cumulative profits or
losses as of the end of the third quarter report small annual profits (as opposed to small annual losses) and conclude
that firms use the fourth quarter to manage earnings upward.
Page 33
In order to supplant the theory that discontinuities are explained by earnings management,
artifactual explanations must provide a better explanation for the body of empirical evidence. It is
difficult to account for the six characteristics outlined above using any artifactual explanation.
This is especially true for the fifth and sixth characteristics, since the artifactual explanation must
simultaneously apply to reported earnings and not apply to closely-related transformations of
earnings . Thus, it seems unlikely that any artifactual explanation, including the explanations
advocated in DE, will replace the theory that earnings are managed to meet benchmarks as the
most plausible explanation for discontinuities.
5. Conclusion
A vast body of literature documents discontinuities in earnings distributions at prominent
earnings benchmarks. Discontinuities occur for a variety of earnings variables (earnings levels,
earnings changes, and earnings surprises) using a variety of scalers (market value, book value,
revenue, or total assets) and additional evidence in this paper shows significant discontinuities for
unscaled earnings and unscaled earnings per share. Discontinuities occur for reported earnings
measures that commonly play a prominent role in decisions by investors and other stakeholders
and in the financial press (such as earnings before extraordinary items or net income or earnings
per share) while discontinuities do not occur for highly-correlated but less prominent earnings
measures (such as earnings before special items). Evidence in previous papers shows that the
discontinuities differ across segments of the population (for example, segments with differing
recent earnings history) and additional evidence in this paper shows how discontinuities differ
substantially with market capitalization and with price per share.
Discontinuity evidence has been widely interpreted as consistent with the theory that
managers take actions to ensure that earnings meet benchmarks. For example, discontinuities at
the zero benchmark in distributions of earnings (or earnings changes) are consistent with the
Page 34
theory that earnings are managed to avoid small losses (or small earnings decreases) while
discontinuities at zero earnings surprise are consistent with the theory that earnings and
expectations are managed to avoid small negative earnings surprise. This interpretation is
supported by survey evidence such as Graham, Harvey, and Rajgopal (2005) indicating that
managers are willing to incur real costs to meet benchmarks.
Two recent papers by Durtschi and Easton show that the discontinuity in distributions of
earnings can be eliminated by 1) using unscaled earnings or earnings scaled by number of shares,
and 2) adopting modified research designs that fail to account for size-related differences in the
amount of earnings that can be managed at low cost and shift the focus almost exclusively to firms
with low or missing price. Based on these results, DE erroneously infer that discontinuities in
earnings distributions are "driven by sample selection bias and scaling" and that "deflation, sample
selection, and a difference between the characteristics of profit and loss observations" are "much
more likely" explanations for discontinuities than the widely-accepted theory that discontinuities
are the result of earnings management to meet benchmarks.
The results and analysis presented in this paper demonstrate why these DE inferences are
erroneous. Based on theory and principles of research design, we show how the DE research
design choices obscure evidence of discontinuities by 1) failing to account for the extremely
important effect of size as a covariate, 2) failing to recognize that the amount of earnings that can
be managed at cost lower than the benefits increases with firm size, and 3) failing to recognize
substantial differences in discontinuities across segments of the population. As a result, the DE
designs move almost all the evidential weight to segments of the population where the
distributional evidence is predictably weak or inconclusive and away from segments of the
population where the effect is predictably strong. In contrast, research designs that correct these
flaws consistently show evidence of significant discontinuities. Thus, the extensive body of
Page 35
empirical evidence of discontinuities (including the empirical evidence presented in DE) is
consistent with the theory that earnings are commonly managed to meet benchmarks and
inconsistent with the alternative explanations offered in DE.
Previous research has commonly used discontinuities as a proxy for earnings management
(e.g., Marquardt and Wiedman 2004, Roychowdhury, 2006; Frank and Rego, 2006; Bergstresser
et al., 2006; Gleason and Mills, 2008; Dhaliwal, Gleason, and Mills, 2004). The assertions in DE
caused many researchers and some journal editors and reviewers to question the appropriateness
of this proxy. However, the results here reaffirm the validity of using discontinuity evidence to
identify firms that are more likely to have managed earnings. Earnings management remains the
only plausible explanation for the extensive body of evidence documenting discontinuities in
earnings distributions.
Page 36
Appendix
This appendix provides additional results related to two points. First, Section A.1 shows
why the DE assertion that a discontinuity in the distribution of earnings changes could be
explained by the positive correlation between earnings levels and earnings changes is incorrect.
Second, to extend the results in Figures 5 and 6 showing the relation between price and earnings
cannot explain 90% of the discontinuity in the aggregate price-scaled earnings distribution,
Section A.2 shows that the relation between price and earnings cannot explain most of the
remaining 10% of the discontinuity.
A.1 Discontinuities in earnings change distributions
DE assert that discontinuities in earnings changes can be explained by 1) the existence of
discontinuities in earnings levels, and 2) the positive correlation between earnings levels and
earnings changes:
"BMN argue that earnings changes are a noisy version of earnings levels; that is, the sign
of earnings changes (increase or decrease) is a correlated, but noisy, signal of the sign of
earnings levels (profit or loss). As a consequence, they predict that the discontinuity in
(price-deflated) earnings changes is largely driven by the same factors that determine the
discontinuity in the distribution of (price-deflated) earnings levels. Consistent with BMN’s
contention, we find that the Spearman correlation between earnings levels and earnings
changes is 0.39. Therefore, we suggest pricing differences and sample selection bias as
alternative explanations for the discontinuity in the frequency distribution of deflated
change in earnings."(DE1 p. 560)44
There are two major flaws in the assertion that discontinuities in earnings changes are due to
discontinuities in earnings levels combined with the positive correlation between changes and
levels.
First, there are numerous examples of variables that are even more highly correlated with
earnings levels than are changes in earnings, but nonetheless show no evidence of the significant
44
Note that this material from the BMN working paper does not appear in the published version, Beaver, McNichols,
and Nelson (2007).
Page 37
discontinuity that occurs in the distribution of earnings.45 Thus, empirical evidence clearly shows
that correlation with earnings levels is not sufficient to cause a discontinuity in another variable
that is correlated with earnings levels.
Second, and perhaps more importantly, even if earnings levels and earnings changes were
perfectly correlated, the discontinuity at zero in levels would translate into a discontinuity at zero
in changes only if a highly-specific relation between the means and dispersions of the distributions
also holds.46 Because there is no reason to believe this specific relation between means and
dispersions holds for distributions of earnings levels and earnings changes, there is no reason to
believe the discontinuity in the distribution of earnings at the zero benchmark explains the
discontinuity in the distribution of earnings changes at the zero benchmark.
It is nonetheless important to understand why the distribution of changes in EPS in DE1
Figure 4 Panel B does not include a clear discontinuity. Figure A-1 Panels A–D show
distributions of the change in EPS by PPS quartile. For larger PPS, the locations of the EPS
change distributions shift upward, and the dispersion also increases. These differences lead to
different evidence of discontinuities across quartiles. For Quartile 1, the peak of the distribution
of changes in EPS is immediately to the right of the zero benchmark so, even though the
standardized differences are highly significant, a cautious interpretation is that the evidence from
Quartile 1 is inconclusive. For Quartiles 2, 3, and 4, the peak is to the right of the zero
benchmark, so the results provide clearly interpretable evidence on whether earnings have been
managed. Results for Quartiles 2 and 3 have left standardized differences significant at the .01
45
Specific examples include earnings before special items (see Beaver, McNichols, and Nelson, 2007) or earnings
before randomly generated noise (see Dechow, Richardson, and Tuna 2003).
46
If earnings levels and changes are perfectly correlated, a discontinuity x standard deviations from the mean of
earnings levels translates into a discontinuity x standard deviations from the mean of earnings changes. Therefore, the
discontinuities occur at zero for both levels and changes only if there is a specific relation among the means and
standard deviations for the levels and change distributions. For example, if earnings levels and changes are perfectly
correlated and the ratio of the means is 7, the discontinuity at zero in the distribution of earnings translates into a
discontinuity at zero in the distribution of changes only if the ratio of the dispersions is also exactly 7.
Page 38
level for both quartiles and right standardized difference significant at the .01 and .05 levels,
respectively. The discontinuity in Quartile 4 yields a left standardized difference significant at the
.05 level though the right standardized difference is not significant.47
In addition to showing significant discontinuities in the three upper quartiles, the results in
Figure A-1 also demonstrate why the distribution of EPS changes in DE1 Figure 4 Panel B shows
no evidence of discontinuity. The DE1 distribution combines results from all four quartiles. In
the vicinity of zero, this aggregate distribution is dominated by the much larger number of
observations in Quartile 1 where there is inconclusive evidence of earnings management, resulting
in inconclusive evidence in the DE1 aggregate distribution, despite significant evidence of
discontinuities in the upper three quartiles.
Figure A-2 shows distributions of price-scaled changes in earnings for the four PPS
quartiles. The scaled earnings change distributions have similar means (and medians) across
quartiles, while the standard deviation decreases in higher market value quartiles. Evidence of
discontinuities varies across quartiles due to differences in the distributions. For Quartile 1, which
has the lowest number of observations in the vicinity of the zero benchmark, the location of the
peak of the distribution is unclear, so a cautious interpretation is that the evidence is inconclusive.
For Quartiles 2, 3, and 4, which account for the large majority of observations in the vicinity of
zero, the peak is more clearly several intervals to the right of the zero benchmark, and the results
provide more clearly interpretable evidence. Left and right standardized differences for Quartile 3
are significant at the .001 level, while the right standardized difference for Quartile 2, and the left
standardized difference for Quartile 4 are significant at the .05 level. Thus, the pattern of evidence
47
For reasons outlined in the earlier discussion of Figure 5, interval widths and ranges adjusted to reflect size
differences across the four quartiles should provide a more powerful assessment of whether earnings are managed to
avoid small decreases in earnings per share. Using adjusted interval widths does, in fact, provide more significant
evidence of discontinuities than the results reported in Figure A-3. However, to save space, we do not report these
alternative results.
Page 39
in the scaled earnings change distributions corresponds to the pattern in the unscaled EPS change
distributions. While the distribution for the lowest quartile is inconclusive, discontinuities occur
in the distribution of earnings changes for all three of the upper quartiles of PPS. Moreover,
almost all of the discontinuity in the aggregate distribution of scaled changes in earnings is due to
the three upper quartiles — about 92% of the difference between the number of observations in the
interval above the benchmark versus below the benchmark is due to the differences in the three
upper quartiles.
A.2 Discontinuities in scaled earnings for the smallest PPS quartile
Figure A-3 shows distributions of scaled EPS for firms from the first quartile of PPS, the
one segment where it could be argued that the relation between earnings and price might play
some role in discontinuities. Panel A shows the distribution for firms with PPS ≥ $1.00 and
Panel B shows the distribution for PPS<$1.00. The distribution for the entire first quartile has 80
more observations in the first interval above the benchmark than the first interval below the
benchmark. F or firms with PPS of $1.00 or more, the difference between the numbers of
observations in the interval immediately above versus below the zero benchmark is 67 and for
firms with PPS less than $1.00, the difference is 13. Thus, most of the discontinuity in the first
quartile is due to observations where PPS ≥ $1.00.
Figure A-4 shows the relation between PPS and EPS for first quartile firms with
PPS ≥ $1.00. For this segment of the first quartile, PPS for small negative EPS are only slightly
smaller than PPS for small positive EPS. Thus, the relation between price and earnings does not
explain the discontinuity in scaled earnings for firms in the first quartile with PPS ≥ $1.00. Thus,
the results in Figures A-3 and A-4 show that the relation between price and earnings can, at most,
explain no more than a trivial proportion of the discontinuity in the distribution of scaled earnings.
Page 40
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Table 1
Descriptive Statistics
N
1st Pctl
25th Pctl
50th Pctl
75th Pctl
99th Pctl
Mean
Std Dev
Skewness
Figures 1 and 2: Unscaled Earnings, By MVE Quartile
Panel A or AA
37,687
-40.226
-2.393
Panel B or BB
37,686
-94.648
-5.503
Panel C or CC
37,687 -262.915
-6.947
Panel D or DD
37,686 -1,533.650
32.180
Panel E
150,746 -429.257
-2.849
-0.398
0.965
9.100
95.352
2.027
0.579
4.548
20.865
297.764
24.234
14.300
27.669
98.249
6,428.000
2,460.190
-1.813
-2.867
0.703
359.243
74.085
6.402
16.017
46.728
996.787
330.489
-3.196
-3.045
-2.880
3.873
5.366
Figures 3 and 4: EPS, By PPS Quartile
Panel A or AA
37,909
-6.180
Panel B or BB
37,344
-6.740
Panel C or CC
37,588
-6.890
Panel D or DD
37,673
-7.340
Panel E
150,514
-6.750
-0.290
-0.600
0.080
0.940
-0.200
-0.060
0.010
0.730
1.720
0.230
0.010
0.400
1.260
2.680
1.200
1.130
2.200
3.680
11.910
6.570
-0.305
-0.271
0.465
1.861
0.426
0.915
1.284
1.527
2.407
1.756
-4.140
-2.308
-2.022
0.371
-0.295
Figure 5: Earnings Scaled By Price, By PPS Quartile
Panel A
37,667
-16.400
-0.497
Panel B
37,661
-1.514
-0.132
Panel C
37,668
-0.566
0.006
Panel D
37,661
-0.248
0.033
Panel E
150,657
-3.799
-0.092
-0.139
0.002
0.054
0.058
0.035
0.027
0.079
0.089
0.082
0.080
5.317
0.527
0.302
0.270
0.908
-0.564
-0.065
0.031
0.054
-0.102
2.222
0.283
0.121
0.066
0.548
-4.659
-2.374
-2.135
-1.093
-4.495
Figure 7: Unscaled Earnings and EPS For Missing Price Observations
Panel A
45,089 -265.140
-2.112
0.712
12.053
Panel B
35,962
-24.750
-0.340
0.030
0.650
1,143.000
24.930
36.611
0.000
162.810
4.368
4.780
-0.165
Table 2
Percentage of Sample With Missing, Zero, or Negative Values for Assets, Sales, and Book Value
% Firms With Missing Assets
% Firms With Zero Assets
% Firms With Negative Assets
% Firms With Missing Sales
% Firms With Zero Sales
% Firms With Negative Sales
% Firms With Missing BVE
% Firms With Zero BVE
% Firms With Negative BVE
MVE QTL 1 MVE QTL 2 MVE QTL 3 MVE QTL 4 Missing PPS
0.05%
0.07%
0.08%
0.04%
0.16%
0.91%
0.05%
0.01%
0.00%
0.51%
0.00%
0.00%
0.00%
0.00%
0.00%
0.05%
0.23%
0.15%
0.02%
0.07%
10.90%
4.38%
2.22%
0.35%
7.35%
0.18%
0.08%
0.06%
0.03%
0.09%
0.10%
0.13%
0.23%
0.38%
0.39%
0.14%
0.02%
0.01%
0.01%
0.17%
22.38%
6.49%
3.59%
2.30%
19.36%
PPS QTL 1
0.05%
0.99%
0.00%
0.03%
14.56%
0.15%
0.08%
0.18%
25.70%
PPS QTL 2
0.04%
0.05%
0.00%
0.02%
3.04%
0.11%
0.15%
0.01%
5.90%
PPS QTL 3
0.09%
0.00%
0.00%
0.07%
0.79%
0.07%
0.24%
0.01%
2.10%
PPS QTL 4
0.04%
0.00%
0.00%
0.03%
0.20%
0.03%
0.23%
0.00%
1.34%
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Histograms of unscaled earnings (Compustat item NI), partitioned into quartiles based on beginning market value of equity (the product of Compustat items PRCC_F and CSHO). The 25th, 50th, and
75th percentiles of MVE that define the four quartiles are $22.2m, $106.3m, and $586.8m, respectively. The center dashed line indicates the zero benchmark. All panels use the range -$50m to $200m of
unscaled earnings with interval widths of $2.5m. The outer dotted lines indicate the range -$5m to $7m used in DE.
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Histograms of unscaled earnings (Compustat item NI) partitioned into quartiles based on beginning market value of equity (the product of Compustat items PRCC_F and CSHO). All panels use interval
widths of $100,000 and the range of -$5m to $7m as in DE1 Figure 3 and DE2 Figures 1, 6, and 8. The vertical dashed line indicates the zero benchmark. Standardized differences are:
Panel A
Panel B
Panel C
Panel D
Left Std Diff
-0.107
-6.015
-0.665
-0.392
Right Std Diff
4.507
5.690
0.117
0.603
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Histogram of unscaled earnings (Compustat item NI), combining all four MVE quartiles from Panels A to D of Figure 2.
The lowest layer of the histogram is MVE Quartile 4 (Panel D), the second lowest layer is MVE Quartile 3 (Panel C),
the third lowest layer is MVE Quartile 2 (Panel B), and the top layer is MVE Quartile 1 (Panel A).
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Histograms of unscaled earnings (Compustat item NI), partitioned into quartiles based on beginning market value of equity (the product of Compustat items PRCC_F and CSHO). The range and interval widths are
adjusted across panels, such that the interval width is approximately 0.002 times the median market value of equity for the observations in the panel, resulting in interval widths of $20,000, $100,000, $500,000,
and $2,500,000 in Panels AA, BB, CC, and DD, respectively. The vertical dashed line indicates the zero benchmark. Standardized differences are:
Panel AA
Panel BB
Panel CC
Panel DD
Left Std Diff
-2.399
-6.015
-4.323
-2.772
Right Std Diff
4.998
5.690
3.536
2.503
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Histograms of EPS (Compustat item EPSFX), partitioned into quartiles based on beginning price per share (Compustat item PRCC_F). The 25th, 50th, and 75th percentiles of PPS that define the four
quartiles are $2.50, $9.31, and $22.00, respectively. All panels use the range -$7.00 to $7.00 of EPS with interval widths of $0.07 per share. The center dashed line indicates the zero benchmark. The outer
dotted lines indicate the range -$1.00 to $1.00 used in DE.
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Histograms of EPS (Compustat item EPSFX), partitioned into quartiles based on beginning price per share (Compustat item PRCC_F). All panels use the interval width of $.01 as in DE1 Figure 1
Panel B, Figure 2, and Figures 6 through 10. The vertical dashed line indicates the zero benchmark. The first (second) interval right of the dashed line corresponds to EPS of $0.00 ($0.01). Standardized
differences are:
Left Std Diff
Right Std Diff
Panel A
2.705
10.589
Panel B
-2.168
-0.758
Panel C
-1.646
-1.646
Panel D
-1.559
0.781
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Histogram of EPS (Compustat item EPSFX), combining all four PPS quartiles from Panels A to D of Figure 4. The lowest layer of the histogram is PPS Quartile 4 (Panel D), the second lowest layer is PPS Quartile 3 (Panel C),
the third lowest layer is PPS Quartile 3 (Panel B), and the top layer is PPS Quartile 1 (Panel A). For the zero benchmark, the left standardized difference is 1.426 and the right standardized difference is 9.472.
Left Std Right
Diff Std Diff
$0.10
-1.569
1.919
$0.20
-2.873
4.132
$0.30
-1.759
2.384
$0.40
-3.348
4.500
$0.50
-0.952
1.708
$0.60
-1.743
2.367
$0.70
-2.764
2.183
$0.80
-3.461
3.782
$0.90
-1.115
2.059
$1.00
-4.184
2.913
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Histograms of EPS (Compustat item EPSFX), partitioned into quartiles based on beginning price per share (Compustat item PRCC_F). The range and interval widths are adjusted across panels, such that the interval width
is approximately 0.002 times the median PPS for the panel. The vertical dashed line indicates the zero benchmark. In Panel AA, the first interval right of the dashed line represents EPS between [$0.00, $0.002). In Panel BB,
the first interval right of the dashed line represents EPS of $0.00. In Panel CC, the first interval right of the dashed line represents EPS between [$0.00, $0.02]. In Panel DD, the first interval right of the dashed line represents
EPS between [$0.00, $0.06]. Standardized differences are:
Left Std Diff
Right Std Diff
Panel AA
3.729
2.303
Panel BB
-2.168
-0.758
Panel CC
-3.327
3.010
Panel DD
-1.895
1.991
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Histogram of earnings scaled by market value of equity (Compustat item NI scaled by the product of PRCC_F and CSHO). The standardized differences are as follows:
Quartile 1
Quartile 2
Quartile 3
Quartile 4
Left Std Diff
-1.021
-7.607
-5.433
-2.793
Right Std Diff
2.083
5.705
3.907
0.704
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Histogram of earnings scaled by market value of equity (Compustat item NI scaled by the product of PRCC_F and CSHO),
combining all four PPS quartiles from Panels A to D of Figure 5. The aggregate standardized differences are:
Left Std Diff
-8.870
Right Std Diff
6.496
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Each figure plots the 25th (gray dotted line), 50th (black solid line), and 75th (gray solid line) percentiles of price per share for each one cent of EPS.
Panel A represents the aggregate distribution. In Panel B, the dashed horizontal lines separate the four PPS quartiles.
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Histograms of unscaled earnings (Compustat item NI) in Panel A and EPS (Compustat item EPSFX) in Panel B for firms with missing price per share. Standardized differences are:
Panel A
Panel B
Left Std Diff
6.55682
-4.9424
Right Std Diff
4.43835
14.7629
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Histograms of change in EPS (Compustat item EPSFX), partitioned into quartiles based on beginning price per share (Compustat item PRCC_F). The 25th, 50th, and 75th percentiles of PPS that define the four quartiles
are $2.60, $9.50, and $22.31, respectively. All panels use the same interval width of $.01 as in DE1 Figure 4. The vertical dashed line indicates the zero benchmark. Standardized differences are:
Panel A
Panel B
Panel C
Panel D
Left Std Diff
-7.543
-3.112
-2.549
-2.306
Right Std Diff
16.407
2.598
1.819
0.660
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Histograms of change in earnings scaled by beginning market value of equity (Compustat items NI and the product of PRCC_F and CSHO), partitioned into quartiles based on beginning price per share (Compustat
item PRCC_F). The 25th, 50th, and 75th percentiles of PPS that define the four quartiles are $2.60, $9.50, and $22.31, respectively. The vertical dashed line indicates the zero benchmark. Standardized differences are:
Panel A
Panel B
Panel C
Panel D
Aggregate
Left Std Diff
-2.478
-1.140
-3.713
-2.164
-4.785
Right Std Diff
4.065
2.008
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0.928
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This figure presents a closer look at PPS quartile 1 for the histogram of earnings scaled by market value of equity (Compustat items NI
and the product of PRCC_F and CSHO). Panel A contains firms in PPS quartile 1 with PPS of at least $1. Panel B contains firms in
PPS quartile 1 with PPS less than $1.
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This figure presents a closer look at PPS quartile 1 for plots of price per share versus earnings per share, further partitioning PPS quartile 1 from
Figure 6 Panel B into firms in PPS quartile 1 with PPS of at least $1 (Panel A) and firms in PPS quartile 1 with PPS less than $1 (Panel B).
The gray dotted line, black solid line, and gray solid line plot the 25th, 50th, and 75th percentiles of MVE, respectively.