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Modeling Conditional Earnings Discontinuities

Dmitri Byzalov

Sudipta Basu

Temple University

Abstract

We develop new tests of distribution discontinuity conditional on multiple explanatory variables

for detecting earnings management and analyzing its determinants. These tests combine

Burgstahler and Dichev’s (1997) intuition on benchmark-driven earnings management with a

flexible statistical model that addresses important limitations of the existing distribution

discontinuity tests. Our conditional discontinuity method offers large improvements in test

performance relative to both histogram-based tests of the existence of earnings discontinuity and

logit-based tests of the determinants of earnings discontinuity, and it changes some of the major

findings in the earnings discontinuity literature. Our method is flexible, robust, and easy to

implement in standard statistical software. Future research on distribution discontinuities could

benefit from adopting our conditional discontinuity tests.

Keywords: standardized difference test; zero benchmark; smooth distribution; nonlinear

interpolation; conditional distribution

JEL codes: M41, C20, C25

We thank Eric Allen (discussant), Ram Mudambi, Oleg Rytchkov, and seminar participants at Temple

University and the Hawaii Accounting Research Conference for helpful comments and suggestions.

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1. Introduction

We propose new tests of conditional distribution discontinuity for analyzing earnings

management and its determinants. Our tests embed Burgstahler and Dichev’s (1997) earnings

discontinuity logic in a flexible statistical model that can incorporate multiple determinants of the

earnings distribution and can be implemented using the Stata code we provide.1 Our conditional

discontinuity method addresses important limitations of the standard discontinuity tests, offers

large improvements in Type-I error and statistical power, and changes major findings in the

earnings discontinuity literature.

If some managers manipulate earnings to avoid reporting a loss, then the earnings distribution

has a discontinuity at the zero benchmark, with unusually few small losses and unusually many

small profits (Burgstahler and Dichev, 1997). If some managers avoid earnings decreases, then

there is a similar discontinuity for earnings changes. Burgstahler and Dichev document both

discontinuities in U.S. Compustat data.2 Their test relies on less restrictive assumptions than tests

of abnormal accruals (Jones, 1991) and real activities (Roychowdhury, 2006), thus yielding more

credible inferences. The distribution discontinuity approach is widely used to detect earnings

management (e.g., Matsumoto, 2002; Leuz et al., 2003; Barth et al., 2008) and to analyze

properties of the “suspect” observations just above the zero benchmark (e.g., Roychowdhury,

2006; Zang, 2012).

Burgstahler and Dichev’s (1997) test is based on the empirical histogram of earnings and cannot

easily incorporate multiple explanatory variables. A researcher can compare histograms between

two large partitions for a given variable (e.g., R&D intensive versus non R&D intensive firms),

1 The code is publicly available at http://astro.temple.edu/~dbyzalov/ 2 Degeorge et al. (1999) document another discontinuity associated with meeting or just beating analyst forecasts.

Durtschi and Easton (2005, 2009) and Beaver et al. (2007) argue that the earnings discontinuities might not reflect

earnings management, but Burgstahler and Chuk (2015) refute their arguments.

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but this test is prone to correlated omitted variable bias because it uses only one explanatory

variable at a time.3 Frankel et al. (2002), Matsumoto (2002), Cheng and Warfield (2005), and

others study earnings management determinants in a logit model for meeting or beating an earnings

benchmark. However, as we show in Section 3, the logit model yields faulty inferences because it

mistakes variation in the pre-managed earnings distribution for variation in the earnings

discontinuity.

To overcome these limitations of histogram- and logit-based tests, we incorporate Burgstahler

and Dichev’s (1997) intuition in a flexible statistical model of the conditional earnings distribution.

We estimate a smooth distribution of pre-managed earnings and an incremental effect of earnings

management just below and above the zero benchmark, where each model component is

conditioned on multiple explanatory variables. We use a flexible polynomial approximation to

avoid restrictive functional form assumptions. Earnings discontinuity is assessed using standard

tests for model coefficients. We develop a simple two-stage estimation approach that only requires

ordinary least squares (OLS) in each stage, which future researchers can use easily.4

Our conditional discontinuity test successfully detects earnings discontinuity in U.S. Compustat

data and is robust. When we incorporate multiple major determinants of earnings discontinuity in

our model, their measured impact differs qualitatively from estimates based on subsamples for

individual partitioning variables. This suggests that the standard subsample comparisons in the

earnings discontinuity literature (e.g., Burgstahler and Chuk, 2017) are vulnerable to correlated

omitted variable bias because they study only one partitioning variable at a time. While the

3 To implement the test with multiple explanatory variables, one would need subsamples that capture the joint variation

in these variables. This approach quickly becomes impractical. For example, if a sample is partitioned at the median

of five explanatory variables, then the total number of subsamples is 25 = 32. The tests will have low power due to

small subsample size, and it will be difficult to draw any conclusions from 32 histograms. 4 As an additional benchmark, we estimate the model in a single stage using maximum likelihood. However, this

estimation approach is more complex computationally, and it yields only a slight improvement in test power relative

to the two-stage approach. Therefore, we recommend that researchers use the simpler two-stage estimation.

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existence of this bias is not surprising, our conditional discontinuity method offers a practical

solution for future research.

We first compare the statistical performance of our conditional discontinuity test to the

Burgstahler and Dichev (1997) test in a simulation without explanatory variables. While both tests

have acceptable Type-I errors, our method improves statistical power by up to 29.6 percentage

points. This improvement arises because we combine information just below and above zero into

a single estimate of earnings discontinuity, whereas the standard test yields two separate (and

sometimes contradictory) estimates below and above zero. Thus, even when a researcher only

needs a basic test without explanatory variables, she could benefit from our method.

We next compare our conditional discontinuity test to the logit model in a simulation with a

single explanatory variable X that can affect both the pre-managed earnings distribution and the

probability of earnings management (e.g., R&D intensive firms might have both more volatile pre-

managed earnings and more frequent earnings management). The logit model has excessive Type-

I errors of 17.6–75.7% because it cannot separate the effect of X on the pre-managed earnings

distribution from the effect of X on the earnings management probability. In contrast, our method

reliably separates the two effects, yielding both valid Type-I errors and greater statistical power

than logit. Thus, in tests of earnings management determinants, our conditional discontinuity

method offers a critical improvement over the standard logit-based estimates.

Our method changes major prior findings. Using a generalized logit model for meeting or

beating analyst forecasts, Barton and Simko (2002) find that firms with higher beginning-of-period

net operating assets (NOA) are less likely to manage earnings, reflecting the role of the balance

sheet as an earnings management constraint. However, this finding is an artifact of the logit model.

When we control for the association between NOA and the pre-managed earnings distribution

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using our conditional discontinuity method, the effect of NOA on earnings discontinuity weakens

considerably and becomes insignificant. Thus, the prior findings reflect variation in the pre-

managed distribution rather than an actual earnings management effect.

We develop the conditional discontinuity method in Section 2, present the main results in

Section 3, re-visit prior findings in Section 4, and conclude in Section 5.

2. Statistical model of conditional earnings discontinuity at the zero benchmark

We present the model in the context of earnings management to avoid reporting small losses,

but the analysis generalizes to other distribution discontinuities.5 Let 𝐸𝐴𝑅𝑁∗ be “pre-managed”

earnings, which are known to managers but unobservable to researchers, scaled as needed (e.g.,

on a per-share basis or divided by the lagged market value of equity). Let 𝐸𝐴𝑅𝑁 be reported

earnings, which is 𝐸𝐴𝑅𝑁∗ plus potential earnings management to convert small losses into small

profits.6 Following Burgstahler and Dichev (1997), we assume that the distribution of pre-managed

earnings 𝐸𝐴𝑅𝑁∗ is smooth and interpret the discontinuity at zero in the distribution of reported

earnings 𝐸𝐴𝑅𝑁 as evidence of earnings management. We formalize pre-managed earnings as

𝐸𝐴𝑅𝑁∗~𝑓∗(𝐸𝐴𝑅𝑁∗|𝑋) (1)

where 𝑓∗(… ) is a smooth probability density function conditional on explanatory variables 𝑋 =

𝑋1 … 𝑋𝑀 (e.g., firm characteristics that affect the mean or variance of pre-managed earnings). We

omit the firm and year indexes for brevity.

5 For example, if earnings levels are replaced with earnings changes or consensus analyst forecast errors, then the

same model describes earnings management to avoid earnings decreases (Burgstahler and Dichev, 1997) or to meet

or beat analyst forecasts (Degeorge et al., 1999), respectively. Discontinuities in debt covenant slack ratios (Dichev

and Skinner, 2002), working capital ratios (Dyreng et al., 2017), reported hedge fund monthly returns (Bollen and

Pool, 2009), and marathon times (Allen et al., 2017) are amenable to similar analyses. 6 Other forms of earnings management such as earnings smoothing (e.g., Ronen and Sadan, 1981; Subramanyam,

1996; Leuz et al., 2003; Sivakumar and Waymire, 2003) or “big bath” (Healy, 1985) do not cause a discontinuity at

zero earnings. We treat them as a component of pre-managed earnings and focus on earnings management at the zero

benchmark.

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When pre-managed earnings 𝐸𝐴𝑅𝑁∗ are in a narrow interval [−𝐾, 0) just below zero, managers

decide whether to report managed earnings 𝐸𝐴𝑅𝑁 instead of 𝐸𝐴𝑅𝑁∗. Let 𝑚 ∈ {0,1} be the

earnings management choice, which a researcher cannot observe. When 𝑚 = 0, managers report

the pre-managed earnings, i.e., 𝐸𝐴𝑅𝑁 = 𝐸𝐴𝑅𝑁∗. When 𝑚 = 1, they report a small profit that is

drawn from a probability density function ℎ(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑋) in a narrow interval [0, 𝐾) just

above zero. This reporting process defines the conditional distribution of reported earnings

𝑓(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑋, 𝑚) = {𝑎 𝑚𝑎𝑠𝑠 𝑝𝑜𝑖𝑛𝑡 𝑎𝑡 𝐸𝐴𝑅𝑁 = 𝐸𝐴𝑅𝑁∗ 𝑖𝑓 𝑚 = 0

ℎ(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑋) 𝑓𝑜𝑟 𝐸𝐴𝑅𝑁 ∈ [0, 𝐾) 𝑖𝑓 𝑚 = 1(2)

Managers choose whether to manage earnings based on the expected costs and benefits of

reporting a small profit instead of a small loss. From a researcher’s perspective, this decision is

probabilistic, and the conditional probability of earnings management is

Pr(𝑚 = 1|𝐸𝐴𝑅𝑁∗, 𝑋) = {𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) 𝑖𝑓 𝐸𝐴𝑅𝑁∗ ∈ [−𝐾, 0)

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒(3)

where 𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) can vary with both 𝐸𝐴𝑅𝑁∗ (e.g., it is easier to conceal a loss of $1,000 than

$1,000,000) and 𝑋 (e.g., a firm with larger current assets can better manage earnings). The

conditional earnings management probability (3) is the main construct of interest for many

questions in earnings management research. This research primarily focuses on how earnings

management is affected by various economic and institutional factors, such as investor protection

(Leuz et al., 2003), accounting standards (Barth et al., 2008), reporting incentives (Burgstahler et

al., 2006), and auditor incentives (Frankel et al., 2002; Ashbaugh et al., 2003). A researcher can

include these and other factors in the 𝑋 vector in our model.

Degeorge et al. (1999) and Burgstahler and Eames (2003) present stylized theoretical models

of earnings management that resemble our model (1)–(3), but they use these models only as a

motivation for Burgstahler and Dichev (1997) histogram-based standardized difference tests.

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While these tests can detect an unconditional earnings discontinuity in a given sample, they limit

a researcher’s ability to study the determinants of earnings management, i.e., the variables that

affect the conditional earnings discontinuity. For example, if there is a large, statistically

significant discontinuity in partition A but not in partition B, then one can draw informal inferences

about the effect of the partitioning variable. However, if the discontinuity is statistically significant

in both partitions, then it is difficult to tell whether the partitioning variable has an effect (e.g.,

Burgstahler and Dichev, 1997, p. 111). More important, the standard histogram-based tests cannot

accommodate multiple explanatory variables, i.e., one cannot draw inferences about the impact of

a variable 𝑋1 after controlling for variables 𝑋2 … 𝑋𝑀. Therefore, these tests suffer from correlated

omitted variable bias with respect to 𝑋2 … 𝑋𝑀, even when the researcher has these data for the

sample. Frankel et al. (2002), Matsumoto (2002), Cheng and Warfield (2005), and others analyze

earnings discontinuity determinants by incorporating them as explanatory variables in a logit

model for meeting or beating an earnings benchmark. However, we show in Section 3.6 that this

logit model cannot separate variation in the earnings discontinuity from variation in the smooth

pre-managed earnings distribution. This severely distorts inferences about the determinants of

earnings discontinuity. Thus, the logit approach should not be used for this analysis.

To overcome these limitations of current methods, we directly estimate model (1)–(3) as

described next. The estimates of the conditional probability Pr(𝑚 = 1|𝐸𝐴𝑅𝑁∗, 𝑋) in (3) measure

the impact of multiple explanatory variables 𝑋 on earnings discontinuity and let us test hypotheses.

2.1. Predicted distribution of reported earnings in the model

To estimate model (1)–(3), we derive the predicted conditional earnings distribution

𝑓(𝐸𝐴𝑅𝑁|𝑋) in the model and then fit it to the data. We focus on the main intuitions and relegate

formal derivations to Appendix A. The computation of 𝑓(𝐸𝐴𝑅𝑁|𝑋) depends on whether reported

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earnings 𝐸𝐴𝑅𝑁 are in the small-loss interval (i.e., a potential trigger of earning management),

small-profit interval (i.e., a possible consequence of earnings management), or outside these

intervals (i.e., neither a cause nor a consequence of earnings management), as shown in Figure 1.

In each interval, we consider all values of the unobserved driving variables 𝐸𝐴𝑅𝑁∗ and 𝑚 that are

consistent with the observed 𝐸𝐴𝑅𝑁, and derive the probability distribution of 𝐸𝐴𝑅𝑁 based on the

underlying distributions of the driving variables 𝐸𝐴𝑅𝑁∗ and 𝑚 in the model.

First, suppose that a firm reports negative earnings below the small-loss interval [−𝐾, 0), i.e.,

𝐸𝐴𝑅𝑁 < −𝐾. This outcome can arise only if (a) the firm did not manage earnings to report a small

profit, and (b) unobserved pre-managed 𝐸𝐴𝑅𝑁∗ equals the observed 𝐸𝐴𝑅𝑁.7 Since 𝐸𝐴𝑅𝑁∗ < −𝐾

from the second condition, the probability of earnings management Pr(𝑚 = 1|𝐸𝐴𝑅𝑁∗, 𝑋) is zero

per (3), and thus the first condition holds with certainty. Therefore, the conditional distribution of

reported earnings 𝐸𝐴𝑅𝑁 reflects the density of pre-managed earnings 𝑓∗(𝐸𝐴𝑅𝑁∗|𝑋) at 𝐸𝐴𝑅𝑁∗ =

𝐸𝐴𝑅𝑁 without any adjustment for earnings management

𝑓(𝐸𝐴𝑅𝑁|𝑋) = 𝑓∗(𝐸𝐴𝑅𝑁|𝑋) 𝑓𝑜𝑟 𝐸𝐴𝑅𝑁 < −𝐾 (4𝑎)

Second, suppose that a firm reports a small loss, i.e., 𝐸𝐴𝑅𝑁 ∈ [−𝐾, 0). This outcome can arise

only if (a) the firm did not manage earnings, and (b) unobserved pre-managed 𝐸𝐴𝑅𝑁∗ equals the

observed 𝐸𝐴𝑅𝑁. Since 𝐸𝐴𝑅𝑁∗ ∈ [−𝐾, 0) from the second condition, there is a positive

probability of earnings management 𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) per (3), and the first condition holds with

probability 1 − 𝑃(𝐸𝐴𝑅𝑁∗, 𝑋). In other words, given its 𝐸𝐴𝑅𝑁∗, the firm could manage earnings

with some probability but did not. Therefore, the conditional distribution of reported earnings

reflects both the lack of earnings management and the density of pre-managed earnings

7 When pre-managed earnings are far below zero, managers might take a “big bath” (Healy, 1985) to build up slack

for future earnings management. Big baths do not cause a discontinuity at zero and are treated as a part of 𝐸𝐴𝑅𝑁∗.

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𝑓∗(𝐸𝐴𝑅𝑁∗|𝑋) at 𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁

𝑓(𝐸𝐴𝑅𝑁|𝑋) = [1 − 𝑃(𝐸𝐴𝑅𝑁, 𝑋)]𝑓∗(𝐸𝐴𝑅𝑁|𝑋) 𝑓𝑜𝑟 𝐸𝐴𝑅𝑁 ∈ [−𝐾, 0) (4𝑏)

Third, suppose that a firm reports a small profit, i.e., 𝐸𝐴𝑅𝑁 ∈ [0, 𝐾). This outcome can arise

in two ways. First, the firm could have a small pre-managed profit 𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁 without any

earnings management. The associated probability density function is 𝑓∗(𝐸𝐴𝑅𝑁|𝑋). Second, the

firm could have a small pre-managed loss 𝐸𝐴𝑅𝑁∗ ∈ [−𝐾, 0) and convert it into a profit through

earnings management. The associated probability density function conditional on 𝐸𝐴𝑅𝑁∗ is

ℎ(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑋)𝑃(𝐸𝐴𝑅𝑁∗, 𝑋), i.e., the density of managed earnings from (2) times the

probability of earnings management from (3). Because 𝐸𝐴𝑅𝑁∗ is unobservable in this scenario,

we integrate it out, i.e., we take the expectation of ℎ(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑋)𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) with respect

to the unobserved random variable 𝐸𝐴𝑅𝑁∗~𝑓∗(𝐸𝐴𝑅𝑁∗|𝑋) in the relevant interval [−𝐾, 0). The

conditional distribution of reported earnings reflects the total of these two paths to a small profit

𝑓(𝐸𝐴𝑅𝑁|𝑋) = 𝑓∗(𝐸𝐴𝑅𝑁|𝑋)

+ ∫ ℎ(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑋)𝑃(𝐸𝐴𝑅𝑁∗, 𝑋)𝑓∗(𝐸𝐴𝑅𝑁∗|𝑋)𝑑𝐸𝐴𝑅𝑁∗

𝐸𝐴𝑅𝑁∗∈[−𝐾,0)

= 𝑓∗(𝐸𝐴𝑅𝑁|𝑋) + 𝐺(𝐸𝐴𝑅𝑁, 𝑋) 𝑓𝑜𝑟 𝐸𝐴𝑅𝑁 ∈ [0, 𝐾) (4𝑐)

where 𝐺(𝐸𝐴𝑅𝑁, 𝑋) stands for the integral from the second line of equation (4c). Instead of

computing this integral inside the estimation loop, we directly estimate the function 𝐺(𝐸𝐴𝑅𝑁, 𝑋).8

Finally, suppose that a firm reports positive earnings above the small-profit interval [0, 𝐾), i.e.,

𝐸𝐴𝑅𝑁 > 𝐾. This outcome can arise only if (a) the firm did not manage earnings, and (b)

unobserved pre-managed 𝐸𝐴𝑅𝑁∗ equals the observed 𝐸𝐴𝑅𝑁. Since 𝐸𝐴𝑅𝑁∗ is positive from the

8 The integral in (4c) embeds the constraint that the probability mass that is removed below zero due to earnings

management in (4b) must equal the probability mass that is added above zero in (4c). We impose this constraint on

𝐺(𝐸𝐴𝑅𝑁, 𝑋) during estimation as described in Section 2.2.

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second condition, the probability of earnings management Pr(𝑚 = 1|𝐸𝐴𝑅𝑁∗, 𝑋) is zero per (3),

and thus the first condition holds with certainty. The conditional distribution of reported earnings

is thus

𝑓(𝐸𝐴𝑅𝑁|𝑋) = 𝑓∗(𝐸𝐴𝑅𝑁|𝑋) 𝑓𝑜𝑟 𝐸𝐴𝑅𝑁 > 𝐾 (4𝑑)

2.2. Empirical implementation and estimation

The main construct of interest in estimation is the conditional earnings management probability

(3) and its impact on reported earnings through equations (4b) and (4c). These estimates are

identified by the shape of the conditional earnings distribution in a narrow interval [−𝐾, 𝐾] near

zero (the two middle segments in Figure 1), combined with the smoothness assumption that ties

the distribution of pre-managed earnings in the interval [−𝐾, 𝐾] to the distribution in the adjacent

intervals. Therefore, we must accurately capture the earnings distribution inside [−𝐾, 𝐾] and in

the adjacent intervals. In contrast, earnings that are far from zero do not help identify the earnings

management parameters. Further, these earnings values could potentially confound the estimates

if the model’s earnings distribution is misspecified far from zero. To avoid this unnecessary

complication, we estimate the model only for earnings near zero. For example, when the small-

loss and small-profit interval width 𝐾 is set to 0.01 (i.e., 1% of the lagged market value of equity),

we restrict the estimation interval of earnings to [–0.04, 0.04) and discard observations outside this

interval (the results are robust to adjusting the width of the estimation interval). This approach

enables us to carefully model the conditional earnings distribution in the important range near zero

without risking faulty inferences due to potential misspecification far from zero.9

9 This estimation approach involves selection on the dependent variable. It should never be used to estimate the

conditional mean of earnings (e.g., in a regression for earnings) because the mean is determined by the entire earnings

distribution. However, when the objective is to characterize the shape of a conditional distribution in a particular

interval, this approach yields valid estimates, as we prove in Appendix B.

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We model the conditional probability density function of pre-managed earnings (1) using a

flexible polynomial approximation in the estimation interval 10

𝑓∗(𝐸𝐴𝑅𝑁∗ = 𝑧|𝑋) = 𝛼0(𝑋) + 𝛼1(𝑋) × 𝑧 + 𝛼2(𝑋) × 𝑧2 + ⋯ + 𝛼𝑃(𝑋) × 𝑧𝑃 (5𝑎)

where 𝑃 is the degree of the polynomial, and the polynomial coefficients 𝛼𝑝(𝑋) for 𝑝 = 0 … 𝑃 can

vary with 𝑋

𝛼𝑝(𝑋) = 𝛼𝑝,0 + 𝛼𝑝,1𝑋1 + ⋯ + 𝛼𝑝,𝑀𝑋𝑀 (5𝑏)

To complete the empirical model, we specify the conditional earnings management probability

𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) in (4b) and the incremental probability density function of managed earnings

𝐺(𝐸𝐴𝑅𝑁, 𝑋) in (4c). We consider two specifications. In Model I, 𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) and 𝐺(𝐸𝐴𝑅𝑁, 𝑋)

can vary with 𝑋, but they are flat (for a given 𝑋) throughout the entire small-loss and small-profit

intervals. We illustrate this specification in Panel A of Figure 2. The earnings distribution is shifted

downward equally for all small losses and is shifted upward equally for all small profits, where

the size of the shifts is conditional on 𝑋.

Model I

𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) = {𝜋0 + 𝜋1𝑋1 + ⋯ + 𝜋𝑀𝑋𝑀 𝑖𝑓 𝐸𝐴𝑅𝑁∗ ∈ [−𝐾, 0)

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒(6𝑎)

𝐺(𝐸𝐴𝑅𝑁, 𝑋) = {𝑔(𝑋) 𝑖𝑓 𝐸𝐴𝑅𝑁 ∈ [0, 𝐾)

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒(6𝑏)

where 𝑔(𝑋) is defined later in this subsection.

In Model II, 𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) and 𝐺(𝐸𝐴𝑅𝑁, 𝑋) vary with 𝑋, similar to Model I, and they also vary

with the size of the small loss or profit, as illustrated in Panel B of Figure 2. We implement this

specification by incorporating an adjustment factor 𝑞(… ) that varies with earnings.

10 The polynomial specification is more flexible than standard parametric distributions such as Normal or Student’s t.

It provides a simple, stable local approximation in our narrow estimation interval, but should not be used over much

broader intervals because the polynomial terms explode for earnings far from zero.

12

Model II

𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) = {𝑞(𝐸𝐴𝑅𝑁∗) × (𝜋0 + 𝜋1𝑋1 + ⋯ + 𝜋𝑀𝑋𝑀) 𝑖𝑓 𝐸𝐴𝑅𝑁∗ ∈ [−𝐾, 0)

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒(7𝑎)

𝐺(𝐸𝐴𝑅𝑁, 𝑋) = {𝑞(𝐸𝐴𝑅𝑁) × 𝑔(𝑋) 𝑖𝑓 𝐸𝐴𝑅𝑁 ∈ [0, 𝐾)

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒(7𝑏)

𝑞(𝑧) = 2 (1 −|𝑧|

𝐾) 𝑓𝑜𝑟 𝑧 = 𝐸𝐴𝑅𝑁∗ 𝑜𝑟 𝐸𝐴𝑅𝑁 (7𝑐)

where the adjustment factor 𝑞(𝑧) in (7c) equals 1 on average in both intervals, equals 2 at 𝑦 = 0,

and decreases linearly to 0 at the outer bound 𝐾 or −𝐾 of the respective interval.11 The earnings

management probability 𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) in (7a) is largest when pre-managed loss 𝐸𝐴𝑅𝑁∗ is just

below zero, and it decreases with the size of the loss. This decrease captures the difficulty of

concealing a larger loss through earnings management (e.g., Barton and Simko, 2002). Similarly,

the density of managed earnings 𝐺(𝐸𝐴𝑅𝑁, 𝑋) in (7b) is largest when the managed profit 𝐸𝐴𝑅𝑁 is

just above zero, and it decreases with the size of the profit. This decrease reflects the difficulty of

reporting a larger profit through earnings management and the smaller incremental benefit of

beating the target rather than just meeting it.

In both models, we compute 𝑔(𝑋) in (6b) and (7b) through the restriction that all small losses

that disappear due to earnings management must re-appear as small profits.12 Thus, 𝑔(𝑋) is

determined by the logic of the model and does not require any additional parameters. This logical

11 One could use a more flexible specification such as 𝑞(𝑧) = 2(1 − 𝜆1|𝑧|/𝐾) with a slope coefficient 𝜆1, but this

would complicate the estimation by introducing three-way interactions between the coefficients. We are primarily

interested in the effect of the explanatory variables 𝑋 on earnings management through the coefficients 𝜋, as opposed

to the exact functional form of 𝑞(𝑧). Therefore, we prefer to use a simple specification of 𝑞(𝑧) that can be combined

with a richer specification of 𝜋. 12 For each 𝑋, the function 𝐺(𝐸𝐴𝑅𝑁, 𝑋) in (6b) and (7b) must satisfy the constraint

∫ 𝑃(𝐸𝐴𝑅𝑁∗, 𝑋)𝑓∗(𝐸𝐴𝑅𝑁∗|𝑋)0

−𝐾𝑑𝐸𝐴𝑅𝑁∗ = ∫ 𝐺(𝐸𝐴𝑅𝑁, 𝑋)

𝐾

0𝑑𝐸𝐴𝑅𝑁, where the left-hand side represents the

missing small losses and the right-hand side represents the added small profits. The integral on the left-hand side has

an analytical expression as a higher-order polynomial and the integral on the right-hand side simplifies to 𝑔(𝑋) × 𝐾.

Given the coefficients 𝛼 and 𝜋, we compute 𝑔(𝑋) for each 𝑋 through these expressions.

13

restriction on 𝑔(𝑋) combines the information on small losses and small profits, yielding more

efficient estimates of the earnings management process. In contrast, standard histogram-based tests

report a left standardized difference for small losses and a right standardized difference for small

profits and do not combine them into a single, more efficient, test statistic. And as we show in

Section 3.6, the logit model for meeting or beating an earnings benchmark should not be used at

all because it yields biased—rather than just inefficient—tests.

We examine two estimation approaches. First, we implement maximum likelihood (ML)

estimation with a custom-written likelihood function that incorporates the restrictions from

equations (4a)–(7c).13 Appendix B provides the implementation details. ML estimation directly

fits the conditional density function of earnings from the model to the data. It is asymptotically

efficient (Wooldridge, 2002, Ch. 14), providing an upper bound of potential test performance.

However, because ML involves numerical search in a multi-dimensional parameter space, it can

require careful fine-tuning of the optimization options (e.g., starting values, optimization

algorithm, Hessian updating method) to achieve convergence, and it is susceptible to local

maxima. Therefore, while we use ML as a benchmark for potential test performance, we

recommend a simpler estimation approach for future research.

Our second estimation approach is slightly less efficient than ML, but it is easier to implement

and is more computationally robust because it does not rely on iterative numerical search. We

break down the estimation into two ordinary least squares (OLS) stages, where all of the model

restrictions are embedded in the construction of the explanatory variables in stage 2.14

13 Chen et al. (2010) estimate a mixed-Normal model of earnings discontinuity using maximum likelihood. However,

they only measure the average unconditional discontinuity in the data and do not incorporate explanatory variables.

Thus, they do not realize the primary benefit of this modeling approach. 14 OLS directly computes the estimates using matrix algebra, thus avoiding the numerical complications that arise in

iterative search-based methods such as ML or non-linear least squares. The separation into two stages lets us use OLS

because each stage is linear with respect to the relevant estimation parameters.

14

Because OLS cannot estimate a continuous density function, we discretize earnings into 𝐵 bins

and estimate a discrete distribution using firm-year-bin data. In what follows, we fully spell out

the variable indexes to clarify the data structure. For each firm-year observation indexed by 𝑖, 𝑡,

we define 𝐵 dummy dependent variables 𝑌𝑖,𝑡,𝑏 that equal 1 if earnings 𝐸𝐴𝑅𝑁𝑖,𝑡 are in bin 𝑏 and 0

otherwise. We also define bin-specific explanatory variables 𝑧𝑏 that represent the midpoint of each

bin, where the bin grid is the same across all firm-years and does not require the 𝑖, 𝑡 index. This

discretization translates the continuous earnings distribution into the bin-specific conditional

means 𝐸(𝑌𝑖,𝑡,𝑏|𝑋𝑖,𝑡, 𝑧𝑏) that can be estimated using OLS. To pool the relevant bins in estimation,

we stack the data. In other words, we replace each firm-year observation (indexed by 𝑖, 𝑡) in the

original sample with 𝐵 firm-year-bin observations (indexed by 𝑖, 𝑡, 𝑏) that share the firm-year

variables 𝑋𝑖,𝑡 and differ in the bin-specific variables 𝑌𝑖,𝑡,𝑏 and 𝑧𝑏. For each firm-year, we then use

all bin-level observations that are relevant for the respective estimation stage.

In the first stage, we estimate the pre-managed earnings distribution parameters 𝛼 in (5a). The

sample for stage 1 comprises all firm-years in the main estimation sample (e.g., data with earnings

in the ±0.04 range per our main definitions); for each firm-year, we use firm-year-bin observations

for all bins outside the small-loss and small-profit intervals.15 The regression model is

𝑌𝑖,𝑡,𝑏 = 𝛼0(𝑋𝑖,𝑡) + 𝛼1(𝑋𝑖,𝑡) × 𝑧𝑏 + 𝛼2(𝑋𝑖,𝑡) × 𝑧𝑏2 + ⋯ + 𝛼𝑃(𝑋𝑖,𝑡) × 𝑧𝑏

𝑃 + 𝜀𝑖,𝑡,𝑏 (8𝑎)

where the polynomial coefficients 𝛼𝑝(𝑋𝑖,𝑡) = 𝛼𝑝,0 + 𝛼𝑝,1𝑋𝑖,𝑡,1 + ⋯ + 𝛼𝑝,𝑀𝑋𝑖,𝑡,𝑀 for 𝑝 = 0 … 𝑃

follow (5b), 𝑋𝑖,𝑡 is the vector of explanatory variables for firm-year 𝑖, 𝑡, and 𝑧𝑏 is the midpoint of

15 For example, when the estimation interval width is 0.04, the small-loss and small-profit interval width is 0.01, and

the bin width is 0.0025, we generate 32 [=2×0.04/0.0025] bin dummies for each firm-year. Four of the bin dummies

correspond to the small-loss interval [−0.01, 0) and four correspond to the small-profit interval [0, 0.01). We omit

these bins in stage 1 and use the remaining 24 firm-year-bin dummies per firm-year in estimation. This bin selection

is unrelated to reported earnings for the firm-year. For example, when a firm reports a small profit or a small loss, the

bin-level data for the firm-year are included in the sample, but all 24 of the included bin dummies equal zero.

15

earnings bin 𝑏. OLS approximates the conditional mean 𝐸(𝑌𝑖,𝑡,𝑏|𝑋𝑖,𝑡, 𝑧𝑏), which for a binary

dependent variable is equivalent to 𝑃𝑟(𝑌𝑖,𝑡,𝑏 = 1|𝑋𝑖,𝑡, 𝑧𝑏), i.e., the probability that earnings are in

bin 𝑏 conditional on explanatory variables 𝑋𝑖,𝑡.16 This estimation logic follows the linear

probability model (e.g., Wooldridge, 2002, Ch. 15), and the stacking of bin-level data imposes the

polynomial structure (8a) across the bins. In a model without explanatory variables (i.e., 𝑋𝑖,𝑡 =

{}), stage 1 can be visualized as a polynomial that best fits the empirical histogram of earnings

outside the small-loss and small profit intervals, as illustrated in Panel A of Figure 3.17 When there

are one or more explanatory variables, stage 1 is analogous to fitting multiple earnings histograms

conditional on 𝑋𝑖,𝑡, and regression model (8a) pools the information for different values of 𝑋𝑖,𝑡.

In the second stage, we estimate the earnings management probability parameters 𝜋 in (6a) and

(7a) for Models I and II, respectively. The sample for stage 2 comprises all firm-years in the main

estimation sample; for each firm-year, we use firm-year-bin observations for bins inside the small-

loss and small-profit intervals, as illustrated in Panel B of Figure 3. The regression model is

𝑌𝑖,𝑡,𝑏 − �̂�𝑖,𝑡,𝑏 = (𝜋0 + 𝜋1𝑋𝑖,𝑡,1 + ⋯ + 𝜋𝑀𝑋𝑖,𝑡,𝑀) × 𝑊𝑖,𝑡,𝑏 + 𝑢𝑖,𝑡,𝑏 (8𝑏)

where �̂�𝑖,𝑡,𝑏 is the predicted probability of bin 𝑏 based on the pre-managed earnings distribution

from stage 1, which is subtracted from the dependent variable 𝑌𝑖,𝑡,𝑏 to isolate the net effect of

earnings management, and 𝑊𝑖,𝑡,𝑏 is a synthetic explanatory variable that specifies the earnings

16 The discrete bin probability (8a) approximates the integral of the continuous density function (5a) for the bin using

the function value at the bin midpoint. Therefore, the empirical coefficients 𝛼 in (8a) are not entirely equivalent to the

theoretical 𝛼 in (5a). Because the 𝛼-s are not individually interpretable (as parts of a polynomial), and their only job

is to approximate the underlying smooth distribution, we slightly abuse the notation and reuse 𝛼 in (8a) for brevity. 17 Labor economics researchers (e.g., Chetty et al., 2011; Kleven and Waseem, 2013; Kleven, 2016) developed a

related approach to examine discontinuities caused by kinks in the marginal tax rates for labor income. These papers

fit a flexible polynomial to the empirical histogram of labor income, using data away from the kink, and then measure

the difference between actual and predicted bin counts near the kink. However, similar to the Burgstahler and Dichev

(1997) test, the empirical approach in these economics papers cannot incorporate multiple explanatory variables. Our

estimation approach is also related to regression discontinuity methods (e.g., Wooldridge, 2002, Ch. 18), which use a

flexible smooth function of the covariates in a narrow interval around the discontinuity combined with a step function

at the discontinuity threshold.

16

management process. In Model I, 𝑊𝑖,𝑡,𝑏 is

𝑊𝑖,𝑡,𝑏 = {−�̂�𝑖,𝑡,𝑏 𝑖𝑓 𝑏𝑖𝑛 𝑏 𝑖𝑠 𝑎 𝑠𝑚𝑎𝑙𝑙 𝑙𝑜𝑠𝑠

−𝑚𝑒𝑎𝑛(𝑊𝑖,𝑡,𝑏′)𝑎𝑐𝑟𝑜𝑠𝑠 𝑎𝑙𝑙 𝑠𝑚𝑎𝑙𝑙 𝑙𝑜𝑠𝑠 𝑏𝑖𝑛𝑠 𝑏′ 𝑖𝑓 𝑏𝑖𝑛 𝑏 𝑖𝑠 𝑎 𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡

(8𝑐)

In Model II, 𝑊𝑖,𝑡,𝑏 is

𝑊𝑖,𝑡,𝑏 = {−𝑞(𝑧𝑏) × �̂�𝑖,𝑡,𝑏 𝑖𝑓 𝑏𝑖𝑛 𝑏 𝑖𝑠 𝑎 𝑠𝑚𝑎𝑙𝑙 𝑙𝑜𝑠𝑠

−𝑞(𝑧𝑏) × 𝑚𝑒𝑎𝑛(𝑊𝑖,𝑡,𝑏′)𝑎𝑐𝑟𝑜𝑠𝑠 𝑎𝑙𝑙 𝑠𝑚𝑎𝑙𝑙 𝑙𝑜𝑠𝑠 𝑏𝑖𝑛𝑠 𝑏′ 𝑖𝑓 𝑏𝑖𝑛 𝑏 𝑖𝑠 𝑎 𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡

(8𝑑)

where 𝑞(𝑧𝑏) is the adjustment factor from (7c).

These definitions of 𝑊𝑖,𝑡,𝑏 embed the logical restrictions of Models I and II. First, the total

incidence of earnings management in each small-loss bin must be proportional to the pre-managed

earnings distribution per (4b). We incorporate this proportionality restriction through the pre-

managed bin probability �̂�𝑖,𝑡,𝑏 in the first line of (8c) and (8d). This restriction is important because

it lets us separate the effect of 𝑋𝑖,𝑡 on the earnings management probability from the effect of 𝑋𝑖,𝑡

on the pre-managed distribution. Second, because managed small losses are observed as small

profits, the effect of earnings management on the small-profit bin probabilities must be fully offset

by its effect on the small-loss bin probabilities. This restriction is implemented though

−𝑚𝑒𝑎𝑛(𝑊𝑖,𝑡,𝑏′) in the second line of (8c) and (8d), which ensures that the small-profit and small-

loss effects sum to zero for each firm-year. Third, the earnings management probability for small

losses must be flat in Model I and must vary with the size of the loss in Model II per (6a) and (7a),

respectively, with analogous definitions (6b) and (7b) for small profits. To implement these

restrictions, (8b) incorporates a uniform (across bins) earnings management probability 𝜋0 +

𝜋1𝑋𝑖,𝑡,1 + ⋯ + 𝜋𝑀𝑋𝑖,𝑡,𝑀 per Model I, and (8d) multiplies it by the bin-specific adjustment factor

𝑞(𝑧𝑏) for Model II.

Because the model restrictions are embedded in the second-stage explanatory variables, we are

17

able to use simple, computationally stable OLS instead of iterative non-linear methods that rely on

numerical optimization. Because the variables in stage 2 are constructed using the coefficient

estimates from stage 1, the standard errors in stage 2 must be adjusted for the estimation noise

from stage 1. We derive this adjustment in Appendix C and implement it in our example Stata

codes.18 After adjusting the covariance matrix of the estimates, we use standard Stata post-

estimation commands for model coefficients to test for earnings discontinuity and its determinants.

3. Empirical results

3.1. Sample selection

We use U.S. Compustat data for 1988–2015. Following Burgstahler and Dichev (1997), we

discard financial firms and utilities (SIC codes 6000–6500 and 4400–4999, respectively) and

require data on current net income (Compustat item NI) and lagged market value of equity

(PRCC_F×CSHO). To maintain a consistent sample in all of the main tests, we also require data

on lagged total assets (item AT), lagged non-cash current assets and current liabilities (ACT–CHE

and LCT–DLC, respectively), and lagged cost of goods sold (COGS) for the computation of the

control variables. We impose additional data requirements in some of the tests. We measure

earnings as net income scaled by the lagged market value of equity.

Table 1 summarizes the sample construction. After imposing our main data requirements, the

full sample comprises 152,586 firm-years. Most of our tests only use the subsample with scaled

earnings in the [−0.04, 0.04) interval, which comprises 34,483 firm-years.

3.2. Earnings discontinuity estimates without explanatory variables

18 In untabulated simulations, standard errors without this adjustment are biased downward slightly, as expected, which

leads to moderate over-rejection in hypothesis tests. Our adjustment resolves this bias. Because the bin dummies are

mutually exclusive, the bin-level observations are correlated within each firm-year. Standard clustering schemes (e.g.,

one-way by firm or two-way by firm and year) fully address this within-firm-year correlation.

18

To provide a simple illustration of our method, first we estimate Models I and II without any

explanatory variables. These unconditional estimates parallel the standard Burgstahler and Dichev

(1997) histogram-based tests. We set the small-loss and small-profit interval width 𝐾 to 0.005,

0.01, and 0.015. We use a third-order polynomial (𝑃 = 3) for the probability density function (5a)

because higher-order polynomial terms are consistently insignificant in untabulated tests; the

results are robust to increasing 𝑃. The estimation parameters are the polynomial coefficients

𝛼0 … 𝛼3 and the earnings management probability coefficient 𝜋0.

Panel A of Table 2 presents the maximum likelihood (ML) estimates. ML is more numerically

complex than our main two-stage estimation approach, but it helps us choose the empirical

specification. First, after ML estimation, the Vuong (1989) test lets us assess the significance of

performance differences between non-nested Models I and II.19 Second, the ML estimates do not

depend on researcher choice of bin width because ML directly fits the continuous earnings

distribution. In all cases, Model II has better log-likelihood than Model I, and the difference is

significant in the Vuong test. The small-loss and small-profit interval width 𝐾 = 0.01 for Model

II yields higher log-likelihood than 𝐾 = 0.005 and 0.015, and it also outperforms 𝐾 = 0.0075

and 0.0125 in untabulated tests. Thus, the data favor Model II with 𝐾 = 0.01 (column 4). Per this

specification, a smaller “small loss” is more likely to trigger earnings management than a bigger

“small loss”, and a smaller “small profit” is more likely to be the result of earnings management

than a bigger “small profit”.20 The predicted earnings distribution in this model accurately fits the

empirical earnings histogram (Panel A of Figure 4).

19 The Vuong (1989) test is defined only for ML. Dechow (1994) derives the Vuong test for ordinary least squares

(OLS). However, her derivation hinges on the maximum likelihood interpretation of OLS, which is valid only for i.i.d.

Normal regression residuals and does not generalize to our two-stage estimation approach. 20 Consistent with our results, Figures 3 and 4 in Burgstahler and Dichev (1997) show that earnings discontinuity

spans several bins and is largest in the bins just below and above zero. This pattern better fits Model II than Model I.

19

We choose the default bin width for two-stage estimation as 2 × 𝐼𝑄𝑅 × 𝑁−1/3, following

Degeorge et al. (1999), where 𝑁 = 34,483 and 𝐼𝑄𝑅 = 0.0386 (untabulated). This formula yields

a recommended bin width of 0.0024, which we round to 0.0025 for convenience. Panel B of Table

2 presents our main two-stage estimates. Consistent with the ML results in Panel A, the adjusted

R2 for the two-stage estimates is highest in Model II with 𝐾 = 0.01 (column 4), and the predicted

earnings distribution for these estimates accurately fits the empirical earnings distribution (Panel

B of Figure 4).21 The earnings management probability coefficient for this specification is 𝜋0 =

0.120, which indicates that on average, 12% of small losses in the [−0.01, 0) interval are managed

upward and reported as small profits. The t-statistic on 𝜋0 (𝑡 = 10.49) indicates a statistically

significant discontinuity in the earnings distribution. For comparison, in a Burgstahler and Dichev

(1997) histogram-based test for the same sample, the standardized left difference is −9.06 and the

standardized right difference is 7.23. The model restriction that all of the managed small losses

from (4b) must become small profits in (4c) lets us combine the information below and above zero

into a single estimate 𝜋0, resulting in a more powerful test with a higher test statistic.22

Panel C of Table 2 examines alternative empirical settings for our two-stage estimation

approach. First, we set the bin width to 0.005 following Burgstahler and Dichev (1997). The

earnings management probability estimates remain similar (e.g., 𝜋0 = 0.118, 𝑡 = 10.10 for the

main specification in column 4). We next use a finer earnings discretization with bin width set to

0.001, as illustrated in Panel C of Figure 4. Because our method pools information from all relevant

21 The adjusted R2 appears low (less than 1%) because the estimation uses dummy dependent variables. For example,

when the conditional bin probability is 0.4, the bin-level dependent variable is 0 with probability 60% and 1 with

probability 40%. Even when the model perfectly captures the true bin probability, actual values for each firm-year-

bin observation will deviate from the predicted value (0.4) by either −0.4 or 0.6, reducing the R2. The polynomial

coefficients 𝛼0 … 𝛼3 are not comparable between Panels A and B because they are scaled differently in ML estimation

as explained in Appendix B; both sets of estimates predict a similar pre-managed distribution in Figure 4. 22 Under the maintained assumption that the model is correctly specified, the ML estimates are asymptotically more

efficient than the two-stage estimates. As expected, ML estimation in column 4 of Panel A yields a similar point

estimate (𝜋0 = 0.124) with an even higher test statistic (𝑡 = 12.27).

20

bins, we can use these narrow bins without sacrificing test power. The earnings management

probability estimate increases slightly to 𝜋0 = 0.122 (𝑡 = 10.74). Notably, reducing the bin width

to 0.001 improves the t-statistics in all columns, likely because the finer bin grid better captures

the underlying continuous distribution. The results are also robust to varying the order of the

polynomial approximation and the width of the estimation interval.23

3.3. Examples of unreasonable empirical design choices for our tests

The choice of the estimation interval for earnings in our method requires researcher judgment.

The estimation interval should (a) contain sufficient data to reliably estimate the smooth pre-

managed earnings distribution near the zero benchmark, and (b) be sufficiently compact to allow

accurate polynomial approximation of the local earnings density function. Figure 5 illustrates two

questionable empirical design choices that violate these guidelines. In Panel A, the estimation

interval [−0.015, 0.015) is only slightly wider than the small-loss and small-profit intervals. The

pre-managed earnings distribution is identified by data only in the two narrow intervals

[−0.015, −0.01) and [0.01, 0.015), which might lead to unreliable estimates. In Panel B, the

estimation interval [−0.2, 0.2) is too wide. The estimation procedure tries to fit the earnings

distribution over the entire interval [−0.2, 0.2) using a simple polynomial function (5a) and cannot

accurately fit the distribution in the important earnings range near zero. Therefore, the estimates

are likely unreliable. These unreasonable specifications should not be used.

23 In both models, the probability estimate 𝜋0 decreases with the width of the small-loss interval, because the total

incidence of earnings management is determined by (1) the probability that pre-managed earnings are in the small-

loss interval, and (2) the probability of earnings management conditional on the small loss. The first number increases

with the width of the small-loss interval; therefore, for the same total incidence of earnings management, the second

number must decrease. Burgstahler and Dichev (1997, p. 108) note a similar tradeoff in their estimates of the

prevalence of earnings management.

21

3.4. Type-I error and power in simulated discontinuity tests without explanatory variables

We conduct simulations to assess the Type-I error and statistical power of our method. In each

simulation, we generate an artificial sample of pre-managed earnings 𝐸𝐴𝑅𝑁∗ drawn from the

probability density function (5a) on the estimation interval [−0.04, 0.04), using the earnings

distribution parameters from the main empirical specification (column 4 in Panel A of Table 2). In

Type-I error simulations, the null hypothesis of no earnings management is true, and therefore we

do not manage simulated earnings. In test power simulations, we convert some of the small losses

into small profits per Model II with the true earnings management probability 𝜋0𝑡𝑟𝑢𝑒 = 0.025 and

0.05 (i.e., 2.5% or 5% of small losses are managed on average). We estimate the model on the

simulated sample, using both ML and the two-stage approach, and test the significance of the

earnings management probability estimate 𝜋0. Because the alternative hypothesis of earnings

management is only consistent with positive 𝜋0, we use one-tailed tests (untabulated two-tailed

tests yield similar results). We also conduct Burgstahler and Dichev (1997) standardized difference

tests. We repeat each simulation 1,000 times for sample size 𝑁 = 5,000 (less than 1/6 of our main

sample) and 𝑁 = 30,000 (slightly less than the main sample with scaled earnings in the

[−0.04, 0.04) interval).

Table 3 presents the simulated rejection rates at a nominal significance level of 5%. The rates

in columns 1 and 4 reflect Type-I error. The left standardized difference test in column 4 has a

slightly elevated Type-I error of 6.6%, while all other Type-I errors for all tests are in line with the

nominal level.24

Consistent with Burgstahler and Chuk’s (2014) simulations, the standardized difference test

24 When the rejection rate is 5%, the total number of rejections in 1,000 simulations is a binomial random variable

with 𝑛 = 1,000 and 𝑝 = 0.05. The corresponding 95% confidence interval for the rejection rate is [3.6%, 6.4%].

22

successfully detects earnings management. For example, when 𝜋0𝑡𝑟𝑢𝑒 = 0.025, earnings

management affects just 13 observations on average out of 𝑁 = 5,000 in column 2 and 80

observations on average out of 𝑁 = 30,000 in column 5.25 The rejection rates based on the left

(right) standardized difference are 22.0% and 59.3% (15.9% and 42.4%) for 𝑁 = 5,000 and

30,000, respectively. When 𝜋0𝑡𝑟𝑢𝑒 = 0.05, the rejection rates are 44.3% and 97.7% (36.5% and

92.3%), respectively.26

Our conditional discontinuity method further improves these rejection rates. Because ML

estimation is asymptotically efficient (Wooldridge, 2002), the ML rejection rates in Table 3

approximate the upper bound of test performance. ML estimation improves test power by 1.9–29.6

percentage points relative to the standardized difference tests, as represented by the black bars in

Figure 6. Because non-ML estimation in general is less asymptotically efficient than ML, the

improvement in test power for the two-stage approach is slightly smaller (1.3–29.3 percentage

points, as illustrated by the grey bars in Figure 6), as expected, but it is generally comparable to

that for ML. For example, for 𝜋0𝑡𝑟𝑢𝑒 = 0.05 and 𝑁 = 5,000 in column 3, the rejection rates

improve from 36.5–44.3% in the standardized difference tests to 58.9% in ML estimation and

52.2–57.5% in two-stage estimation. Notably, the performance gap between two-stage estimation

and ML vanishes for a finer bin grid (the darker grey bars in Figure 6). When the bin width is

0.005, the test power improvement for the two-stage approach is on average 78% as large as it is

25 Small pre-managed losses are approximately 10.7% of the 𝑁 earnings observations in the interval [−0.04, 0.04).

Therefore, when 𝜋0𝑡𝑟𝑢𝑒 = 0.025, the expected number of managed small losses per sample is approximately

0.107 × 0.025 × 5,000 = 13.38 for 𝑁 = 5,000 and 0.107 × 0.025 × 30,000 = 80.25 for 𝑁 = 30,000. 26 The discrepancy between the left and right standardized differences is likely driven by the curvature of the earnings

distribution. The computation of the standardized differences is based on linear interpolation, while the simulated

distribution of pre-managed earnings is convex, following the empirical estimates shown in Figure 4. When linear

interpolation is applied to a convex distribution, it overstates the missing earnings density below zero and understates

the excess earnings density above zero. These interpolation biases cause over-rejection in the left standardized

difference test and under-rejection in the right standardized difference test in this simulation.

23

for ML; when the bin width is reduced to 0.0025 and 0.001, the ratio increases to 92% and 98%,

respectively.27 Thus, two-stage estimation performs almost as well as ML without suffering from

the numerical complications of ML.

In summary, in both ML and two-stage estimation, our conditional discontinuity method offers

a sizable improvement in test power relative to the standardized difference tests. Thus, even a

researcher who only wants to conduct standard Burgstahler and Dichev (1997) discontinuity

analysis without any explanatory variables could benefit from our method.

3.5. Measuring the impact of observable determinants of earnings discontinuity

To illustrate how our conditional discontinuity method can accommodate multiple explanatory

variables, we measure the impact of several major determinants of earnings discontinuity from

prior research. Following Burgstahler and Dichev (1997), we use current asset intensity and current

liability intensity as proxies for a firm’s ability to manage earnings through working capital

manipulation. Following Burgstahler and Chuk (2017), we use COGS intensity and R&D intensity

as proxies for implicit claims by stakeholders such as customers, employees, and suppliers. These

implicit claims could create contracting incentives for earnings management. Because earnings

management can affect concurrent assets, liabilities, COGS, and R&D expense, we lag the

explanatory variables. A priori, we do not expect to overturn the fundamental intuition on these

variables; instead, our objective is to show how our conditional discontinuity method can produce

richer, more precise inferences.

27 Bin width = 0.001 increases the memory requirements and estimation time considerably, because each firm-year

observation in the original data is transformed into 80 [2×0.04/0.001] firm-year-bin observations in the two-stage

estimation, increasing the data size by a factor of 80 (vs. a factor of 32 for bin width = 0.0025 and 16 for bin width =

0.005). When the data size increases, the estimation time can increase more than proportionately because (1) the

estimation requires sorting the data, with computational burden proportional to 𝑁 × 𝑙𝑜𝑔𝑁 (Knuth, 1998), and (2) if

the memory requirements exceed the available random-access memory (RAM), the execution slows down

dramatically due to memory paging on the disk. To reduce the memory requirements and speed up estimation, a

researcher can discard unused variables from the sample prior to calling our two-stage estimation procedure.

24

Burgstahler and Dichev (1997) display data on current asset and current liability intensities near

the zero threshold but do not statistically test their impact. Burgstahler and Chuk (2017) partition

the sample based on the terciles of a single explanatory variable and conduct the standardized

difference test in each partition. For both R&D intensity and COGS intensity, they find a much

larger, more statistically significant discontinuity in the top tercile than in the bottom tercile,

consistent with their predictions. However, this comparison is interpretable only when the two

partitions have sharply different results; further, even in this best-case scenario, it is difficult to

assess whether the partitioning variable’s effect is significant.28 More important, this approach

uses only one partitioning variable at a time and cannot assess which variable has a larger net

effect. In contrast, our method directly incorporates multiple explanatory variables.

The estimates are presented in Table 4. To parallel Burgstahler and Chuk’s (2017) tercile

partitions, in Panel A we convert the explanatory variables into tercile dummies. The intercept 𝜋0

captures the baseline probability of earnings management when all of the included explanatory

variables are in the bottom tercile (the omitted base category), and the coefficients on the tercile

dummies capture the incremental probabilities. The t-statistics on these coefficients provide a

direct test of the differences between the bottom tercile and the second or third tercile.

Consistent with Burgstahler and Dichev’s (1997) intuition, the estimates in models with a single

explanatory variable in columns 1 and 2 show that earnings management probability increases

with both current asset intensity and current liability intensity, as indicated by the significant

positive coefficients on the tercile dummies CA2, CA3 and CL2, CL3, respectively. For example,

28 In general, when a coefficient is significant in partition A and is insignificant in partition B, the difference between

the two partitions is not necessarily significant (Gelman and Stern 2006). For example, the divergent results in

partitions A and B could arise because of differences in test power, even when the partitioning variable does not have

a systematic effect. A χ2 test can detect distribution differences between partitions but cannot tell whether they are

related to earnings discontinuity.

25

for the bottom tercile of current asset intensity in column 1, the predicted earnings management

probability is 2.8% [0.028]; for the second and third terciles, the probability increases to 17.7%

[0.028+0.149] and 21.4% [0.028+0.186], respectively, and both increases are significant at the 1%

level. Column 3 incorporates both current assets and current liabilities. The coefficients on CA2

and CA3 for current assets decrease slightly but remain significant. In contrast, for current

liabilities, the coefficient on CL2 decreases by more than half, and the coefficient on CL3 decreases

by more than two-thirds and becomes insignificant. Thus, after controlling for current assets, there

is a qualitative change in the inferences for current liabilities. This change indicates that the single-

variable estimates for current liabilities in column 2 have correlated omitted variable bias.

Consistent with Burgstahler and Chuk (2017), the estimates in models for individual

explanatory variables in columns 4 and 5 show a positive effect of COGS intensity and R&D

intensity on earnings management probability, as reflected in the significant positive coefficients

on COGS2, COGS3 and RD2, RD3, respectively. These estimates are robust to combining COGS

and R&D intensity in column 6.

In column 7, we include all of the variables concurrently. Compared to the two-variable

estimates in 6, the coefficients on COGS2, COGS3, and RD2 decrease by more than one-third, and

two of them (COGS2 and RD2) become insignificant. Thus, the inclusion of current assets and

current liabilities leads to a qualitative change in the inferences for COGS and R&D in column 7.

This evidence further confirms that the simpler tests in columns 1–6 have a correlated omitted

variable problem.29

Among the variables in column 7, current asset intensity has the largest net effect on the

29 Of course, if we include additional explanatory variables, then we will likely find correlated omitted variable bias

even in the combined model in column 7. We do not claim that this model contains “the right list of controls”; instead,

our objective is to introduce a more powerful method for incorporating multiple explanatory variables in discontinuity-

based tests and to show that the standard inferences change even after combining basic controls.

26

earnings management probability (10.5 percentage points for the top tercile relative to the bottom

tercile, after controlling for all other variables), and COGS has the second-largest effect (9.5

percentage points for the top tercile). The net effect of R&D intensity is considerably smaller (6.0

percentage points), while current liability intensity is largely irrelevant (3.9 percentage points and

insignificant at the 10% level). While this comparison of relative magnitudes is straightforward in

our method, it is difficult to implement (in the case of multiple explanatory variables) in standard

histogram-based tests.

Panel B of Table 4 examines the sensitivity of these estimates to empirical design choices. The

results are generally robust to alternative definitions of the small-loss and small profit intervals in

columns 2–3, different estimation intervals in columns 4–5, different bin widths in columns 6–7,

the use of continuous explanatory variables instead of tercile dummies in column 8, and ML

estimation in column 9.

In Panel C, we examine standard logit estimates for the small-profit dummy and compare them

to the estimates in our model. The logit estimates for the full sample in column 1 differ qualitatively

from the logit estimates for the subsample with scaled earnings near zero in column 2. For example,

the effect of current liabilities is significantly negative for the full sample, which contradicts the

theory (Burgstahler and Dichev, 1997), but it is significantly positive for the restricted subsample.

These estimates suggest that the standard logit estimates for the full sample (e.g., Matsumoto,

2002) are confounded by irrelevant observations with earnings far from zero.30 Even after

restricting the sample, the logit estimates in column 2 differ considerably from our Model II

30 The conditional probability of a small profit Pr(𝐸𝐴𝑅𝑁 ∈ [0, 𝐾)|𝑋) in the full sample can be decomposed as

Pr(𝐸𝐴𝑅𝑁 ∈ [0, 𝐾)|𝐸𝐴𝑅𝑁 ∈ [−0.04, 0.04), 𝑋) × Pr(𝐸𝐴𝑅𝑁 ∈ [−0.04, 0.04)|𝑋), where the first term is the

probability of a small profit for observations in the restricted subsample and the second term is the probability of

inclusion in the restricted subsample. In untabulated analysis, Pr(𝐸𝐴𝑅𝑁 ∈ [−0.04, 0.04)|𝑋) varies significantly with

the control variables. This indicates that the estimates for the full sample are influenced by the irrelevant parts of the

earnings distribution outside the [−0.04, 0.04) interval.

27

estimates in column 3 (e.g., the effect of COGS intensity is insignificant in the logit model but is

positive and significant in our Model II, consistent with the theory). It might be unclear a priori

which of these two estimates is better. Therefore, next we conduct a simulation horse race between

logit and our conditional discontinuity method.

3.6. Logit model for a small-profit dummy versus our conditional discontinuity method in

simulated tests

Barton and Simko (2002), Frankel et al. (2002), Matsumoto (2002), Ashbaugh et al. (2003),

Cheng and Warfield (2005), Jiang et al. (2010), and others study potential determinants of earnings

management by incorporating them as explanatory variables in a logit model with a dummy

dependent variable for meeting or beating an earnings benchmark.31 While we explicitly model

the earnings distribution in the relevant interval around zero, the logit model transforms this

distribution into a single dummy, thus discarding information. This information loss can reduce

test power. More important, it can confound the interpretation of the inferences drawn from the

logit model. The explanatory variables likely influence both the earnings management behavior at

zero and the distribution of pre-managed earnings near zero (e.g., R&D intensive firms might have

both stronger incentives to manage earnings and greater volatility of pre-managed earning). Both

effects influence the probability of observing a small profit. Conditional on explanatory variables

𝑋, this probability is

Pr(𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡|𝑋) = Pr(𝑚𝑎𝑛𝑎𝑔𝑒𝑑 𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡|𝑋)

+ Pr(𝑢𝑛𝑚𝑎𝑛𝑎𝑔𝑒𝑑 𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡|𝑋)(9)

31 Some authors (e.g., Barton and Simko, 2002; Matsumoto, 2002; Jiang et al., 2010) define the dependent variable as

“meeting or beating” the benchmark by any amount, while others (e.g., Frankel et al., 2002; Cheng and Warfield,

2005) restrict attention to “meeting or just beating” the benchmark. The “meet or beat” definition is less appropriate

because it can be confounded by economically irrelevant distribution changes far from the benchmark.

28

where both probabilities on the right-hand side can vary with 𝑋. The logit model estimates the

probability profile Pr(𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡|𝑋) using data on the small-profit dummy and 𝑋. It cannot

disentangle Pr(𝑚𝑎𝑛𝑎𝑔𝑒𝑑 𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡|𝑋) from Pr(𝑢𝑛𝑚𝑎𝑛𝑎𝑔𝑒𝑑 𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡|𝑋) because the

small-profit dummy does not contain any information on the nature of the observed small profit.

In other words, by construction the logit model cannot isolate the determinants of earnings

discontinuity. In contrast, the dependent variables in our conditional discontinuity method retain

information on the shape of the earnings distribution, which lets us disentangle the earnings

discontinuity at zero (the analog of Pr(𝑚𝑎𝑛𝑎𝑔𝑒𝑑 𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡|𝑋) in (9)) from the smooth

distribution of pre-managed earnings near zero (the analog of Pr(𝑢𝑛𝑚𝑎𝑛𝑎𝑔𝑒𝑑 𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡|𝑋)

in (9)). Therefore, our method can produce credible inferences on the determinants of earnings

discontinuity.

We illustrate these issues using simulation. For each observation in the simulated sample, we

generate an explanatory variable 𝑋 that equals 0 or 1 with probability 0.5 each (the results

generalize to a continuous 𝑋 and multiple 𝑋s in untabulated tests). Conditional on 𝑋, we compute

the polynomial coefficients 𝛼𝑝𝑡𝑟𝑢𝑒(𝑋) = 𝛼𝑝,0

𝑡𝑟𝑢𝑒 + 𝛼𝑝,1𝑡𝑟𝑢𝑒𝑋 for 𝑝 = 0 … 3 in the probability density

function (5a), where the parameters 𝛼𝑝,0𝑡𝑟𝑢𝑒 are calibrated based on our main empirical estimates in

Panel A of Table 2, and the remaining parameters depend on the simulation scenario. We draw

pre-managed earnings from the density function (5a) and manage some of the small losses with

average probability 𝜋0𝑡𝑟𝑢𝑒 + 𝜋1

𝑡𝑟𝑢𝑒𝑋 following equations (7a) and (7b). We estimate the models on

the simulated data, using both ML and two-stage estimation for our Model II, and test whether 𝑋

29

affects the small-profit probability in the logit model or earnings discontinuity in our model.32

Each simulation is repeated 1,000 times for sample sizes 𝑁 = 5,000 and 𝑁 = 30,000.

In Panel A of Table 5, we assume that 𝑋 does not affect the pre-managed earnings distribution

(i.e., 𝛼𝑝,1𝑡𝑟𝑢𝑒 = 0 for 𝑝 = 0 … 3). This is a special case in which the entire effect of 𝑋 flows through

Pr(𝑚𝑎𝑛𝑎𝑔𝑒𝑑 𝑠𝑚𝑎𝑙𝑙 𝑝𝑟𝑜𝑓𝑖𝑡|𝑋), and thus the logit estimates are not confounded by the

fundamental identification problem discussed above. In columns 1 and 4, the earnings

management probability does not vary with 𝑋 (i.e., 𝜋1𝑡𝑟𝑢𝑒 = 0), and the rejection rates represent

Type-I error. The Type-I errors for both logit and our Model II are consistent with the nominal

level of 5%. In columns 2–3 and 5–6, 𝑋 = 1 increases the earnings management probability by 5

and 10 percentage points (𝜋1𝑡𝑟𝑢𝑒 = 0.05 and 0.1, respectively). As expected, our Model II yields

more powerful tests than the logit model. For example, for 𝜋1𝑡𝑟𝑢𝑒 = 0.1 and 𝑁 = 5,000, Model II

detects a significant effect of 𝑋 on the earnings discontinuity in 43.0–48.2% of the simulations,

versus just 20.8% for the logit model. For 𝜋1𝑡𝑟𝑢𝑒 = 0.05 and 𝑁 = 30,000, the rates are 57.1–61.6%

versus 30.5%, respectively.33 Thus, even in this best-case scenario for the logit model, our

conditional discontinuity method offers a large improvement in test power because it uses earnings

information more efficiently.

32 In general, the alternative hypothesis for 𝑋 can include both positive and negative coefficient values. Therefore, we

use two-tailed tests. The intercept in the logit model cannot be interpreted as a measure of earnings discontinuity.

Therefore, we only focus on the effect of 𝑋. The small-profit dummy in the logit model equals 1 when simulated

earnings are between 0 and 1.0% of the lagged market value of equity, for consistency with the true earnings

management process in the simulated data. Our simulation protocol restricts simulated earnings to the interval

[−0.04, 0.04). Prior research often estimates logit models for earnings discontinuity using samples that span the entire

earnings distribution (e.g., Frankel et al., 2002; Matsumoto, 2002). However, this empirical design is both

unnecessary, because observations far from zero do not provide any information on earnings discontinuity at zero, and

dangerous, because the logit estimates can be confounded by model misspecification for earnings far from zero. By

restricting the estimation sample for the logit model to our main estimation interval [−0.04, 0.04), we give logit the

best chance to be competitive. In untabulated tests for an unrestricted range of simulated earnings, the logit model

suffers from additional biases when 𝑋 affects the proportion of earnings that fall within the [−0.04, 0.04) interval. 33 As expected, ML estimation of Model II has slightly higher test power than two-stage estimation. Because this

performance difference is slight, and two-stage estimation is much easier to implement and use than ML, we

recommend the two-stage approach for future research.

30

In Panel B, we set 𝛼2,1𝑡𝑟𝑢𝑒 = 2 and 𝛼3,1

𝑡𝑟𝑢𝑒 = 0.5, such that 𝑋 changes the pre-managed earnings

distribution as illustrated in Figure 7. The proportion of small pre-managed profits is 12.1% for

𝑋 = 0 in the left plot and 11.1% for 𝑋 = 1 in the right plot, as indicated by the shaded area. This

difference confounds the interpretation of the logit estimates. Even when 𝑋 does not play any role

in earnings management (i.e., 𝜋1𝑡𝑟𝑢𝑒 = 0) in columns 1 and 4, logit detects a significant effect of

𝑋 in 17.6% of the simulations for 𝑁 = 5,000 and 75.7% of the simulations for 𝑁 = 30,000. Thus,

a significant effect of 𝑋 in the logit model does not indicate that 𝑋 affects earnings management,

because the model cannot separate the earnings management effect from variation in the smooth

pre-managed distribution. In contrast, our Model II successfully separates the two effects, yielding

type-I errors that are consistent with the nominal significance level of 5%. In columns 2–3 and 5–

6, 𝑋 increases the earnings management probability (i.e., 𝜋1𝑡𝑟𝑢𝑒 > 0). The logit model fails to

detect this effect. For example, for 𝜋1𝑡𝑟𝑢𝑒 = 0.1 and 𝑁 = 5,000 in column 3, logit yields a rejection

rate of just 4.9%, versus 41.5–46.2% in our Model II. Even for 𝑁 = 30,000, the logit rejection

rate is only 5.2%, versus 98.9–99.6% in Model II.34 Thus, an insignificant effect of 𝑋 in the logit

model does not indicate that 𝑋 is irrelevant for earnings management.

In summary, researchers should never use the logit model to analyze earnings management at

the zero benchmark. Even when the estimates are interpretable in the special case in Panel A, the

logit model lacks statistical power. In the more general case in Panel B, the estimates are not

interpretable as evidence on earnings management because the logit model cannot separate the

effect on earnings discontinuity from the effect on the pre-managed earnings distribution.

34 The rejection rate in the logit model decreases with 𝜋1

𝑡𝑟𝑢𝑒, i.e., the model is less likely to detect a stronger effect of

𝑋. The reason is that the positive effect of 𝑋 on the small-profit probability through 𝜋1𝑡𝑟𝑢𝑒 partly offsets the negative

confounding effect of 𝑋 from Figure 7. Therefore, when 𝜋1𝑡𝑟𝑢𝑒 is larger, the logit model is less likely to produce a

significant negative estimate dominated by the confounding effect. Contrarily, when 𝜋1𝑡𝑟𝑢𝑒 is negative, it acts in the

same direction as the confounding effect, inflating the rejection rates in the logit model (untabulated).

31

4. Application: The balance sheet as an earnings management constraint

Using a generalized logit model, Barton and Simko (2002) find a negative association between

beginning-of-period net operating assets (NOA) and the probability of meeting or beating the

consensus analyst EPS forecast.35 They attribute this result to prior net asset overstatement acting

as a constraint on current earnings management. However, as we show in Section 3.6, the logit

model cannot distinguish variation in the pre-managed earnings distribution from variation in the

earnings discontinuity. This confounds inferences in the logit model. Therefore, we examine the

relation between NOA and earnings management using our conditional discontinuity method.

Our sample and variable definitions replicate Barton and Simko (2002). We combine quarterly

U.S. Compustat data with IBES analyst forecast data during 1993–1999 and discard financial firms

and utilities (SIC codes 6000–6999 and 4400–4999). The final sample comprises 48,054 firm-

quarter observations. The variable definitions are summarized in the notes to Table 6.

The dependent variables in all models are based on the EPS surprise, computed as actual EPS

for the quarter minus the most recent consensus mean EPS forecast for the quarter and rounded to

the nearest penny.36 Barton and Simko define the dummy dependent variable in the logit model as

“meet or beat” the benchmark (i.e., EPS surprise ≥ 0); however, this definition could be

confounded by large positive EPS surprises, which are unlikely to arise from benchmark-driven

earnings management. Therefore, we also use a “meet or just beat” dummy for EPS surprises of 0

and 1 cents following Cheng and Warfield (2005).

35 Barton and Simko examine multiple EPS surprise thresholds from –5 cents to +5 cents. They use a logit specification

for the probability of meeting or beating each threshold per their equation (2), and they estimate the model jointly for

multiple thresholds using generalized ordered logit estimation. This joint estimation is necessary only if a researcher

wants to conduct hypothesis tests across different thresholds; further, it is less robust than standard logit estimation

because the inferences for the economically important zero threshold could be confounded by model misspecification

for other thresholds. Therefore, we use a standard logit model for the zero threshold in our replication. 36 Analysts typically do not include fractions of a cent in their EPS forecasts (Dechow and You, 2012). Therefore,

Models I and II for unrounded data would need to control for analysts’ tendency to report whole numbers, which

would complicate the estimation and increase the risk of faulty inferences due to misspecification.

32

To adapt our Models I and II to analyst forecast data, we replace earnings in equations (5)–(7c)

with EPS surprises and use 1-cent bins based on the resolution of the EPS surprise data. We

interpret reported EPS surprises of 0 and 1 cents as potential outcomes of earnings management,

following Cheng and Warfield (2005), and interpret pre-managed EPS surprises of -1 and -2 cents

(i.e., the same number of bins for symmetry) as potential triggers of earnings management. We

restrict the estimation sample for Models I and II to EPS surprises within the [−5 𝑐𝑒𝑛𝑡𝑠, 5 𝑐𝑒𝑛𝑡𝑠)

interval, such that the pre-managed EPS surprise distribution is estimated based on 3 bins below

zero and 3 bins above zero (counting only the bins unaffected by earnings management).37

Columns 1–4 of Table 6 present the logit estimates. Column 1 replicates Barton and Simko’s

(2002) estimates (i.e., the probability of meeting or beating the consensus EPS forecast in the full

sample). Consistent with Barton and Simko, higher beginning-of-period NOA significantly

reduces the probability of meeting or beating the forecast (the coefficient on NOA is −0.027, 𝑡 =

−7.84; for comparison, Barton and Simko report −0.031, 𝑡 = −4.98 in their Table 5). Column 2

redefines the dependent variable as “meet or just beat” the EPS forecast. The coefficient on NOA

shrinks by one-third to −0.017 but remains significant (𝑡 = −4.58). We next restrict the sample

to EPS surprises in the [−5 𝑐𝑒𝑛𝑡𝑠, 5 𝑐𝑒𝑛𝑡𝑠) interval. Because benchmark-driven earnings

management converts small negative surprises into small non-negative surprises but does not

affect large surprises, the logit model is estimated more reliably on this restricted sample. For both

“meet or beat” in column 3 and “meet or just beat” in column 4, the logit coefficient on NOA

remains negative and significant (−0.034, 𝑡 = −7.75 and −0.013, 𝑡 = −3.00, respectively).

37 To maintain consistent notation with Sections 2 and 3, the estimation interval [−5 𝑐𝑒𝑛𝑡𝑠, 5 𝑐𝑒𝑛𝑡𝑠) is defined as a

half-open interval, where the lower bound (-5 cents) is included and the upper bound (+5 cents) is excluded. Under

this definition, the estimation sample includes five “miss” bins from -5 to -1 cents and five “meet or beat” bins from

0 to 4 cents, thus preserving symmetry in bins. The results are robust to including the 5-cent bin in the sample.

33

Columns 5–9 present the estimates for our conditional distribution discontinuity Models I and

II. In untabulated exploratory analysis across various specifications for the EPS surprise data,

Model I has consistently higher adjusted R2 than Model II, and the adjusted R2 for Model I is

maximized when we use a fourth-order polynomial. Therefore, we use Model I with 𝑃 = 4 as our

main specification for EPS surprises.

In column 5, the explanatory variables affect the earnings discontinuity but do not affect the

pre-managed distribution. Thus, similar to the logit model, this restricted version of Model I

attributes all changes in the EPS surprise distribution to earnings management. Beginning-of-

period NOA has a significant negative effect on earnings management (−0.013, 𝑡 = −3.15),

consistent with Barton and Simko (2002). In the full model in column 6, the explanatory variables

affect both the earnings discontinuity and the pre-managed distribution, thus letting us separate the

earnings management effect from the confounding variation in the pre-managed distribution. The

effect of NOA on earnings discontinuity weakens from −0.013 in column 5 to −0.007 in column

6 and is insignificant at the 10% level (𝑡 = −1.45). This insignificant result is robust to alternative

empirical specifications in columns 7–9. Notably, NOA has a significant effect on the pre-managed

distribution through the coefficients 𝛼0,𝑁𝑂𝐴 … 𝛼4,𝑁𝑂𝐴 (𝐹 = 9.92 … 11.84 in columns 6–9). In other

words, NOA is associated with the economic factors that affect the distribution of pre-managed

EPS surprises. In the simpler models in columns 1–5, this effect on the pre-managed distribution

is mistakenly attributed to earnings management. By disentangling these two effects, our full

conditional discontinuity model in columns 6–9 leads to qualitatively different conclusions about

the effect of beginning-of-period NOA on earnings management.

5. Conclusion

We propose conditional distribution discontinuity tests for studying earnings management that

34

overcome important limitations of the existing tests. A standard Burgstahler and Dichev (1997)

unconditional discontinuity test cannot incorporate multiple explanatory variables, which limits a

researcher’s ability to study the determinants of earnings management. Standard logit estimates

for meeting or beating an earnings benchmark cannot distinguish variation in earning discontinuity

from variation in the pre-managed earnings distribution, which leads to faulty inferences about the

determinants of earnings management. We develop a flexible statistical model that can incorporate

multiple explanatory variables and that can separate these variables’ effect on earnings

discontinuity from their effect on the pre-managed earnings distribution, and propose a simple

two-stage estimation procedure that is almost as efficient as maximum likelihood and is much

easier to implement and to use.

Simulation analysis shows that our conditional discontinuity method considerably outperforms

the standard tests in terms of both Type-I error and statistical power. Notably, we find that the

widely-used logit estimates for meeting or just beating an earnings benchmark suffer from large

Type-I errors and should never be used to study the determinants of earnings discontinuity. In

contrast, our method yields reliable inferences on these determinants.

Our method changes some of the major findings in prior accounting research. For example, we

show that Barton and Simko’s (2002) findings on the relation between beginning-of-period net

operating assets and earnings management reflect variation in the pre-managed earnings

distribution rather than an actual earnings management effect.

Our method is robust, easy to implement in standard statistical software, and can benefit

researchers who use distribution-discontinuity tests, whether in accounting or other fields.

35

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37

Large losses are reported when

pre-managed 𝐸𝐴𝑅𝑁∗ < −𝐾 (earnings management to avoid

losses is irrelevant in this interval)

Small losses are reported when

• pre-managed 𝐸𝐴𝑅𝑁∗ is in the

small-loss interval [−𝐾, 0)

AND

• managers did not manipulate

earnings to report a small

profit

Small profits are reported when

• pre-managed 𝐸𝐴𝑅𝑁∗ is in the

small-profit interval [0, 𝐾)

OR

• pre-managed 𝐸𝐴𝑅𝑁∗ is in the

small-loss interval [−𝐾, 0)

AND

managers manipulated earnings to report a small

profit

Large profits are reported when

pre-managed 𝐸𝐴𝑅𝑁∗ > 𝐾

(earnings management to avoid

small losses is irrelevant in this interval)

−𝐾 0 𝐾 reported earnings 𝐸𝐴𝑅𝑁

Fig 1. The four scenarios for observed earnings in the theoretical model (1)–(3)

38

Panel A: Model I

Panel B: Model II

Fig 2. Earnings distribution in Models I and II conditional on 𝑋

39

Panel A: Stage 1 — estimation of the pre-managed earnings distribution

Model I

Model II

Panel B: Stage 2 — estimation of the earnings management process

Fig 3. Our two-stage estimation approach, illustrated in a model without explanatory variables

The bars represent the empirical histogram of earnings. In a specification without explanatory variables, our estimation

for the firm-year-bin data can be visualized as fitting the model to the histogram bins. The first stage in Panel A

estimates the pre-managed earnings distribution by fitting a polynomial to the bins outside the small-loss and small-

profit intervals. The second stage in Panel B estimates the incremental effect of earnings management, relative to the

pre-managed distribution from stage 1 (the dashed line), using bins in the small-loss and small-profit intervals.

-.04 -.02 0 .02 .04

-.04 -.02 0 .02 .04 -.04 -.02 0 .02 .04

40

Panel A: Maximum likelihood estimates from column 4 in Panel A of Table 2

Panel B: Two-stage estimates from column 4 in Panel B of Table 2 (bin width = 0.0025)

Panel C: Two-stage estimates with a finer earnings discretization (bin width = 0.001)

Fig 4. Predicted earnings distribution based on the estimates in Table 2 and the empirical

earnings histogram for comparison

In Panel A, the histogram bin width does not affect the ML estimates. In Panels B and C, the two-stage estimates are

based on the same bin grid as the corresponding histogram.

-.04 -.02 0 .02 .04

-.04 -.02 0 .02 .04

-.04 -.02 0 .02 .04

41

Panel A: The estimation interval is too narrow. The estimates of the pre-managed earnings

distribution are based on a limited range of data and might be unreliable.

Panel B: The estimation interval is too wide. The cubic polynomial cannot accurately capture

the pre-managed earnings distribution.

Fig 5. Examples of questionable empirical design choices that should be avoided

-.04 -.02 0 .02 .04

-.2 -.18 -.16 -.14 -.12 -.1 -.08 -.06 -.04 -.02 0 .02 .04 .06 .08 .1 .12 .14 .16 .18 .2

42

Panel A: Improvement relative to the left standardized difference test

Panel B: Improvement relative to the right standardized difference test

Fig 6. The percentage point improvement in test power for our ML and two-stage estimates,

relative to the standardized difference tests, based on the simulation results in Table 3

0

2

4

6

8

10

12

14

16

N=5,000, π = 0.025 N=5,000, π = 0.05 N=30,000, π = 0.025 N=30,000, π = 0.05

Per

cen

tage

po

int

imp

rove

men

t in

re

ject

ion

ra

tes

ML 2-stage, bin width = 0.005 2-stage, bin width = 0.0025 2-stage, bin width = 0.001

0

5

10

15

20

25

30

35

N=5,000, π = 0.025 N=5,000, π = 0.05 N=30,000, π = 0.025 N=30,000, π = 0.05

Per

cen

tage

po

int

imp

rove

men

t in

re

ject

ion

ra

tes

ML 2-stage, bin width = 0.005 2-stage, bin width = 0.0025 2-stage, bin width = 0.001

43

𝑿 = 𝟎 𝑿 = 𝟏

Pr(small profit)=12.1%

Pr(small profit)=11.1%

Fig 7. The distribution of pre-managed earnings in the simulation in Panel B of Table 5

44

Table 1. Sample construction

Number of firm-year

observations in the sample

Full annual U.S. Compustat sample during 1988–2015 323,829

Discard financial firms and utilities 276,723

Discard observations with missing data on net income or market value of equity 197,399

Discard observations with insufficient data to construct the main variables 152,586

Restrict the sample to scaled earnings in the [−0.04, 0.04) interval 34,483

45

Table 2. Estimates without observable determinants of earnings management

Panel A: Maximum likelihood estimates as a benchmark for asymptotically efficient estimates The estimation interval is [−0.04, 0.04), cubic polynomial.

Small-loss and small-profit interval width is

𝐾 = 0.005 𝐾 = 0.01 𝐾 = 0.015

Model I Model II Model I Model II Model I Model II

(1) (2) (3) (4) (5) (6)

Polynomial coefficients in the probability density function (5a) of pre-managed earnings:

𝛼0a 1.000

(.)

1.000

(.)

1.000

(.)

1.000

(.)

1.000

(.)

1.000

(.)

𝛼1 14.446***

(23.15)

14.650***

(23.55)

13.371***

(20.74)

13.920***

(22.10)

12.224***

(17.57)

13.086***

(20.21)

𝛼2/10 b 21.689***

(15.06)

21.702***

(15.07)

21.639***

(15.03)

21.652***

(15.04)

21.643***

(15.03)

21.623***

(15.02)

𝛼3/100 b 6.053

(0.97)

4.270

(0.69)

15.352**

(2.41)

10.609*

(1.69)

25.060***

(3.70)

17.771***

(2.78)

Earnings management probability for small-loss observations:

𝝅𝟎 0.173***

(10.54)

0.163***

(11.96)

0.128***

(10.43)

0.124***

(12.27)

0.105***

(9.59)

0.102***

(11.84)

lnL 88,900.56 88,914.15 88,899.35 88,919.00 88,890.59 88,914.06

Vuong Z c 2.38** 3.31*** 4.15***

Panel B: Main two-stage estimates The estimation interval is [−0.04, 0.04), cubic polynomial, bin width is 0.0025.

Small-loss and small-profit interval width is

𝐾 = 0.005 𝐾 = 0.01 𝐾 = 0.015

Model I Model II Model I Model II Model I Model II

(1) (2) (3) (4) (5) (6)

Polynomial coefficients in the probability density function (5a) of pre-managed earnings:

𝛼0 0.028***

(101.69)

0.028***

(101.69)

0.028***

(80.86)

0.028***

(80.86)

0.029***

(59.93)

0.029***

(59.93)

𝛼1 0.417***

(19.86)

0.417***

(19.86)

0.399***

(18.46)

0.399***

(18.46)

0.392***

(16.48)

0.392***

(16.48)

𝛼2/10 b 0.596***

(12.95)

0.596***

(12.95)

0.598***

(11.87)

0.598***

(11.87)

0.534***

(8.96)

0.534***

(8.96)

𝛼3/100 b 0.052

(0.27)

0.052

(0.27)

0.209

(1.06)

0.209

(1.06)

0.264

(1.22)

0.264

(1.22)

Earnings management probability for small-loss observations:

𝝅𝟎 0.170***

(9.49)

0.161***

(10.14)

0.124***

(9.19)

0.120***

(10.49)

0.088***

(7.35)

0.093***

(9.69)

adj. R2 (%) 0.352 0.353 0.352 0.354 0.349 0.353

46

Panel C: Robustness checks — earnings management probability coefficient 𝜋0 for alternative

empirical definitions in two-stage estimation Unless stated otherwise, the estimation interval is [−0.04, 0.04), cubic polynomial, bin width is 0.0025.

Small-loss and small-profit interval width is

𝐾 = 0.005 𝐾 = 0.01 𝐾 = 0.015

Model I Model II Model I Model II Model I Model II

(1) (2) (3) (4) (5) (6)

Bin width

0.005 0.170***

(9.47)

0.170***

(9.47)

0.124***

(9.19)

0.118***

(10.01)

0.089***

(7.36)

0.093***

(9.49)

0.0025 0.170***

(9.49)

0.161***

(10.14)

0.124***

(9.19)

0.120***

(10.49)

0.088***

(7.35)

0.093***

(9.69)

0.001 0.170***

(9.51)

0.162***

(10.42)

0.124***

(9.23)

0.122***

(10.74)

0.089***

(7.41)

0.095***

(9.93)

Polynomial degree for the pre-managed distribution

2 0.169***

(9.49)

0.161***

(10.13)

0.120***

(9.09)

0.118***

(10.42)

0.082***

(7.39)

0.090***

(9.70)

3 0.170***

(9.49)

0.161***

(10.14)

0.124***

(9.19)

0.120***

(10.49)

0.088***

(7.35)

0.093***

(9.69)

4 0.172***

(9.46)

0.163***

(10.10)

0.126***

(9.15)

0.123***

(10.37)

0.089***

(7.31)

0.094***

(9.39)

Estimation interval width

0.03 0.168***

(9.31)

0.160***

(10.01)

0.123***

(8.77)

0.120***

(10.23)

0.088***

(6.33)

0.092***

(8.84)

0.04 0.170***

(9.49)

0.161***

(10.14)

0.124***

(9.19)

0.120***

(10.49)

0.088***

(7.35)

0.093***

(9.69)

0.05 0.166***

(9.44)

0.158***

(10.11)

0.118***

(9.00)

0.115***

(10.38)

0.081***

(7.16)

0.088***

(9.58)

0.06 0.159***

(9.38)

0.152***

(10.07)

0.110***

(8.78)

0.109***

(10.25)

0.073***

(6.83)

0.082***

(9.39)

The table presents the estimates of Models I and II for alternative empirical specifications. The dependent variable

𝐸𝐴𝑅𝑁 is net income (NI) scaled by the lagged market value of equity (PRCC_F×CSHO). *, **, and *** indicate

significance at the 1%, 5%, and 10% level, respectively, in two-tailed tests. Panel A presents the maximum likelihood

(ML) estimates. The t-statistics in parentheses in Panel A use the standard ML computation, which does not

incorporate clustering. Panels B and C present the estimates for our two-stage estimation approach. The t-statistics in

Panels B and C are clustered by firm and are adjusted for the first-stage estimation noise as described in Appendix C.

Unless stated otherwise, the estimation sample comprises 𝑁 = 34,483 firm-year observations with scaled earnings in

the [−0.04, 0.04) interval. The number of firm-year-bin observations in two-stage estimation is larger (e.g., for the

empirical definitions in Panel B, there are 2 × 0.04 0.0025⁄ = 32 bin-level observations per firm-year). Because the

two-stage estimates are clustered by firm, the larger sample size at the firm-year-bin level does not artificially inflate

the t-statistics. a The intercept 𝛼0 in ML estimation is set to 1 and the density function is rescaled as explained in Appendix B. This

rescaling is necessary only for the ML estimates in Panel A. We do not (and should not) use it in the two-stage

estimation in Panels B and C. Because of this scaling difference, the polynomial coefficients are not comparable

between ML and two-stage estimates. b The polynomial coefficients 𝛼2 on 𝐸𝐴𝑅𝑁2 and 𝛼3 on 𝐸𝐴𝑅𝑁3 are orders of magnitude larger than the other

coefficients because 𝐸𝐴𝑅𝑁2 and 𝐸𝐴𝑅𝑁3 are very small numbers (e.g., for the main estimation interval width of 0.04,

𝐸𝐴𝑅𝑁2 is less than 0.042 = 0.0016, and 𝐸𝐴𝑅𝑁3 is less than 0.043 = 0.000064 in absolute value). We rescale them

to ensure the stability of numerical optimization in ML estimation and we present the rescaled estimates 𝛼2/10 and

𝛼3/100 in the table. c A positive Vuong Z-statistic indicates that Model II performs better than Model I. The Vuong test is defined only

for ML estimates. Because our two-stage estimates do not have a maximum likelihood interpretation, we cannot use

Vuong tests in Panels B and C.

47

Table 3. Rejection rates in simulated earnings discontinuity tests

Simulated sample comprises 5,000

observations in the interval [-0.04, 0.04)

Simulated sample comprises 30,000

observations in the interval [-0.04, 0.04)

true earnings management probability is true earnings management probability is

𝜋0𝑡𝑟𝑢𝑒 = 0 𝜋0

𝑡𝑟𝑢𝑒 = 0.025 𝜋0𝑡𝑟𝑢𝑒 = 0.05 𝜋0

𝑡𝑟𝑢𝑒 = 0 𝜋0𝑡𝑟𝑢𝑒 = 0.025 𝜋0

𝑡𝑟𝑢𝑒 = 0.05

(1) (2) (3) (4) (5) (6)

Burgstahler and Dichev (1997) standardized difference test

left difference 6.2 22.0 44.3 6.6 59.3 97.7

right difference 4.6 15.9 36.5 3.7 42.4 92.3

Significance test for the earnings management probability 𝜋0 in our main specification (Model II with 𝐾 = 0.01)

ML estimation 5.1 23.9 58.9 5.6 72.0 99.8

Two-stage estimation with

bin width = 0.005 5.2 23.3 52.2 5.3 67.7 99.6

bin width = 0.0025 5.0 23.6 56.5 5.2 71.0 99.7

bin width = 0.001 5.6 23.9 57.5 5.4 71.7 99.8

The table presents the rejection rates in one-tailed tests with a 5% nominal significance level in 1,000 simulated

samples. The simulated distribution of pre-managed earnings follows the estimates from column 4 of Panel A in Table

2, and the simulated earnings management process follows Model II with the true earnings management probability

𝜋0𝑡𝑟𝑢𝑒 set to 0, 0.025, or 0.05. In estimation for the simulated data, the estimation interval is [-0.04, 0.04), the small-

loss and small-profit interval width is 0.01, and the bin width for earnings discretization in the two-stage method varies

from 0.005 to 0.001. For the Type-I errors in columns 1 and 4, the 95% confidence interval is 3.6% to 6.4%.

48

Table 4. Estimates of major determinants of conditional earning discontinuity

Panel A: Estimates for X converted into tercile dummies Model II, estimation interval is [−0.04, 0.04), cubic polynomial, small-loss and small-profit interval

width is 0.01, and bin width is 0.0025. Pred.

sign

CA only CL only CA+CL COGS only R&D only COGS+R&D Combined

(1) (2) (3) (4) (5) (6) (7)

Earnings management probability coefficients 𝜋0 … 𝜋𝑀

intercept 0.028

(1.61)

0.047***

(2.74)

0.020

(1.08)

0.045***

(2.60)

0.096***

(5.74)

0.013

(0.58)

-0.024

(-0.98)

CA2 + 0.149***

(5.68)

0.121***

(4.05)

0.092***

(3.00)

CA3 + 0.186***

(6.56)

0.150***

(3.56)

0.105**

(2.35)

CL2 + 0.128***

(5.04)

0.054*

(1.79)

0.054*

(1.78)

CL3 + 0.156***

(5.29)

0.042

(0.96)

0.039

(0.89)

COGS2 + 0.110***

(4.29)

0.103***

(3.98)

0.042

(1.51)

COGS3 + 0.166***

(5.52)

0.177***

(5.74)

0.095***

(2.67)

RD2 + 0.076**

(2.10)

0.066*

(1.80)

0.042

(1.14)

RD3 + 0.039*

(1.65)

0.063**

(2.56)

0.060**

(2.44)

𝛼0,0 … 𝛼𝑃,𝑀 included as stand-alone cubic polynomial terms and interactions with X

adj. R2 (%) 0.419 0.386 0.425 0.418 0.370 0.427 0.474

49

Panel B: Robustness checks — estimates for alternative empirical definitions

Pred.

sign

Main

specification

Small loss/profit interval Estimation interval Bin width

0.005

Bin width

0.001

Continuous

X

ML

estimates 𝐾 = 0.005 𝐾 = 0.015 [−0.03, 0.03] [−0.06, 0.06] (1) (2) (3) (4) (5) (6) (7) (8) (9)

Earnings management probability coefficients 𝜋0 … 𝜋𝑀

intercept -0.024

(-0.98)

-0.005

(-0.14)

-0.029

(-1.42)

-0.016

(-0.67)

-0.041*

(-1.81)

-0.027

(-1.09)

-0.018

(-0.73)

0.046**

(2.51)

-0.013

(-0.65)

CA2 + 0.092***

(3.00)

0.092**

(2.13)

0.092***

(3.56)

0.080**

(2.53)

0.083***

(2.92)

0.104***

(3.27)

0.088***

(2.86)

0.094***

(3.17)

CA3 + 0.105**

(2.35)

0.072

(1.14)

0.111***

(2.99)

0.108**

(2.29)

0.117***

(2.70)

0.108**

(2.32)

0.107**

(2.41)

0.143***

(3.48)

CL2 + 0.054*

(1.78)

0.112***

(2.68)

0.026

(1.02)

0.051*

(1.66)

0.082***

(2.91)

0.044

(1.40)

0.054*

(1.80)

0.055*

(1.91)

CL3 + 0.039

(0.89)

0.075

(1.21)

0.013

(0.36)

0.043

(0.93)

0.069*

(1.65)

0.037

(0.82)

0.035

(0.80)

0.015

(0.37)

COGS2 + 0.042

(1.51)

0.072*

(1.84)

0.029

(1.23)

0.042

(1.46)

0.036

(1.40)

0.034

(1.18)

0.038

(1.35)

0.040

(1.49)

COGS3 + 0.095***

(2.67)

0.109**

(2.20)

0.088***

(2.89)

0.080**

(2.20)

0.067**

(1.98)

0.094**

(2.57)

0.095***

(2.67)

0.060**

(2.03)

RD2 + 0.042

(1.14)

0.032

(0.63)

0.017

(0.57)

0.044

(1.15)

0.041

(1.18)

0.046

(1.21)

0.046

(1.25)

0.039

(1.21)

RD3 + 0.060**

(2.44)

0.038

(1.11)

0.059***

(2.85)

0.061**

(2.39)

0.072***

(3.10)

0.067***

(2.65)

0.059**

(2.41)

0.062***

(2.80)

CA + 0.059*

(1.96)

CL + 0.018

(0.31)

COGS + 0.078***

(3.19)

RD + 0.059

(0.69)

𝛼0,0 … 𝛼𝑃,𝑀 included as stand-alone cubic polynomial terms and interactions with X

adj. R2 (%) 0.474 0.472 0.471 0.386 a 0.597 a 0.977 a 0.186 a 0.426

lnL 89,511.54

N 34,483 34,483 34,483 24,685 55,761 34,483 34,483 34,483 34,483

50

Panel C: Model II estimates versus logit model estimates

Pred.

sign

Logit model

for the full sample

Logit model for subsample with

𝐸𝐴𝑅𝑁 ∈ [−0.04, 0.04)

Main specification of Model II

from Panel A

(1) (2) (3)

Logit coefficients, earnings management probability coefficients 𝜋0 … 𝜋𝑀 in Model II

intercept -3.254***

(-102.56)

-2.015***

(-60.78)

-0.024

(-0.98)

CA2 + 0.080*

(1.89)

0.061

(1.32)

0.092***

(3.00)

CA3 + 0.156***

(2.67)

0.215***

(3.32)

0.105**

(2.35)

CL2 + -0.301***

(-7.29)

0.072

(1.62)

0.054*

(1.78)

CL3 + -0.685***

(-12.17)

0.146**

(2.34)

0.039

(0.89)

COGS2 + -0.051

(-1.30)

-0.047

(-1.14)

0.042

(1.51)

COGS3 + -0.111***

(-2.57)

-0.032

(-0.67)

0.095***

(2.67)

RD2 + 0.079*

(1.64)

-0.006

(-0.11)

0.042

(1.14)

RD3 + 0.133***

(4.00)

0.115***

(3.25)

0.060**

(2.44)

𝛼0,0 … 𝛼𝑃,𝑀 – – included

lnL -20,505.23 -13,526.29

adj. R2 (%) 0.474

N 152,586 34,483 34,483

The table presents the estimates of Model II with explanatory variables 𝑋 for alternative empirical definitions. Unless

stated otherwise, the estimation sample comprises 𝑁 = 34,483 firm-year observations with scaled earnings in the

[−0.04, 0.04) interval. For brevity, we only tabulate the coefficients 𝜋 that determine the earnings management

probability as a function of 𝑋; each variable in 𝑋 also affects the untabulated polynomial coefficients 𝛼0(𝑋) … 𝛼𝑃(𝑋)

in the probability density function (5a) of pre-managed earnings.

EARN is net income (NI) scaled by the lagged market value of equity (PRCC_F×CSHO). Following Burgstahler and

Dichev (1997), CA is the ratio of non-cash current assets (ACT–CHE) to the market value of equity (PRCC_F×CSHO),

and CL is the ratio of current liabilities (LCT–DLC) to the market value of equity. Following Burgstahler and Chuk

(2017), COGS is the ratio of cost of goods sold (COGS) to total assets (AT), and RD is the ratio of R&D expense

(XRD) to total assets. We replace missing R&D with zero following Hirshleifer et al. (2012). CA2, CA3, CL2, CL3,

COGS2, COGS3, RD2, and RD3 are dummy variables for the second and third terciles of CA, CL, COGS, and RD,

respectively. 53% of observations have zero or missing R&D; to avoid an arbitrary division of these observations

between the bottom and middle terciles, we assign all of them to the bottom “tercile” of R&D and shrink the middle

tercile. All explanatory variables are lagged. a The adjusted R2 in columns 4–7 of Panel B is not comparable to the other columns because of differences in sample

size and/or bin dummy definitions.

51

Table 5. Simulations to assess the performance of our conditional discontinuity tests

relative to the logit model for a small-profit dummy

Panel A: 𝑋 does not affect the pre-managed earnings distribution (𝛼𝑝,1𝑡𝑟𝑢𝑒 = 0 for 𝑝 = 0 … 3)

Simulated sample comprises 5,000

observations in the interval [-0.04, 0.04)

Simulated sample comprises 30,000

observations in the interval [-0.04, 0.04)

true effect of X is true effect of X is

𝜋1𝑡𝑟𝑢𝑒 = 0 𝜋1

𝑡𝑟𝑢𝑒 = 0.05 𝜋1𝑡𝑟𝑢𝑒 = 0.1 𝜋1

𝑡𝑟𝑢𝑒 = 0 𝜋1𝑡𝑟𝑢𝑒 = 0.05 𝜋1

𝑡𝑟𝑢𝑒 = 0.1

(1) (2) (3) (4) (5) (6)

Logit 5.8 8.6 20.8 5.2 30.5 81.3 Significance test for the coefficient 𝜋1 on X in our main specification (Model II with 𝐾 = 0.01)

ML estimate 4.6 13.9 48.2 5.9 61.6 99.6

Two-stage estimate with

bin width = 0.005 4.2 12.9 43.0 4.7 57.1 99.0

bin width = 0.0025 4.5 13.7 45.9 5.2 59.5 99.6

bin width = 0.001 4.4 13.4 46.4 5.3 60.2 99.5

Panel B: 𝑋 changes the pre-managed earnings distribution as illustrated in Figure 7 (𝛼0,1𝑡𝑟𝑢𝑒 =

𝛼1,1𝑡𝑟𝑢𝑒 = 0, 𝛼2,1

𝑡𝑟𝑢𝑒 = 2, 𝛼3,1𝑡𝑟𝑢𝑒 = 0.5)

Simulated sample comprises 5,000

observations in the interval [-0.04, 0.04)

Simulated sample comprises 30,000

observations in the interval [-0.04, 0.04)

true effect of X is true effect of X is

𝜋1𝑡𝑟𝑢𝑒 = 0 𝜋1

𝑡𝑟𝑢𝑒 = 0.05 𝜋1𝑡𝑟𝑢𝑒 = 0.1 𝜋1

𝑡𝑟𝑢𝑒 = 0 𝜋1𝑡𝑟𝑢𝑒 = 0.05 𝜋1

𝑡𝑟𝑢𝑒 = 0.1

(1) (2) (3) (4) (5) (6)

Logit 17.6 7.3 4.9 75.7 26.5 5.2 Significance test for the coefficient 𝜋1 on X in our main specification (Model II with 𝐾 = 0.01)

ML estimate 4.6 15.1 46.2 5.4 58.8 99.6

Two-stage estimate with

bin width = 0.005 5.1 14.1 41.5 5.0 53.8 98.9

bin width = 0.0025 4.4 14.7 44.0 4.4 57.6 99.6

bin width = 0.001 4.4 14.4 44.3 5.3 57.9 99.5

The table presents the rejection rates in two-tailed tests of the slope coefficient on X in the logit model and the

coefficient 𝜋1 on X in Model II with a 5% nominal significance level in 1,000 simulated samples. The parameters

𝛼𝑝,0𝑡𝑟𝑢𝑒 are based on the estimates in column 4 of Panel A in Table 2, and 𝜋0

𝑡𝑟𝑢𝑒 = 0 for simplicity. The true earnings

management process follows Model II. For the Type-I errors in columns 1 and 4, the 95% confidence interval is 3.6%

to 6.4%.

52

Table 6. The relation between beginning-of-period net operating assets (NOA) and meeting or beating analyst forecasts

Logit for

the full sample

Logit for subsample with

𝐸𝑃𝑆 𝑠𝑢𝑟𝑝𝑟𝑖𝑠𝑒 ∈ [−5𝑐, 5𝑐) Restricted

Model I

Main estimates

for Model I

Robustness checks

meet or beat meet or just

beat by 1c meet or beat

meet or just

beat by 1c Model II

meet vs

miss by 1c

𝐸𝑃𝑆 𝑠𝑢𝑟𝑝𝑟𝑖𝑠𝑒 ∈ [−4𝑐, 4𝑐)

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Logit coefficients, earnings management probability coefficients 𝜋0 … 𝜋𝑀 in Models I and II

intercept -1.550***

(-24.39)

-1.483***

(-23.14)

-0.863***

(-10.62)

-0.824***

(-11.29)

-0.260***

(-3.78)

0.025

(0.21)

0.117

(1.21)

0.203**

(2.32)

0.206**

(2.29)

NOA -0.027***

(-7.84)

-0.017***

(-4.58)

-0.034***

(-7.75)

-0.013***

(-3.00)

-0.013***

(-3.15)

-0.007

(-1.45)

-0.004

(-0.97)

-0.002

(-0.52)

-0.001

(-0.30)

SHARES -1.155***

(-7.21)

0.608***

(4.17)

-0.917***

(-4.89)

0.322**

(2.01)

-0.046

(-0.25)

0.422

(1.07)

0.438

(1.42)

0.434**

(2.10)

0.389*

(1.92)

BIG5 0.058**

(1.99)

-0.012

(-0.41)

0.069*

(1.92)

-0.021

(-0.63)

0.011

(0.36)

0.024

(0.52)

0.021

(0.55)

0.012

(0.30)

0.007

(0.17)

PB 0.034***

(8.63)

0.030***

(8.88)

0.027***

(5.61)

0.011***

(2.97)

0.011***

(2.98)

-0.002

(-0.38)

-0.004

(-0.84)

-0.002

(-0.40)

-0.001

(-0.13)

LTGN_RISK 0.082***

(3.63)

0.002

(0.08)

0.144***

(5.19)

-0.025

(-1.00)

0.016

(0.66)

-0.036

(-0.94)

-0.045

(-1.48)

-0.032

(-1.01)

-0.022

(-0.71)

ANALYSTS 0.019***

(5.35)

0.016***

(4.56)

0.020***

(4.48)

0.015***

(3.99)

0.013***

(3.11)

0.010

(1.19)

0.006

(0.97)

0.004

(0.70)

0.002

(0.48)

PREV_MB 0.966***

(45.66)

0.527***

(22.52)

0.775***

(29.40)

0.393***

(15.37)

0.410***

(14.79)

0.253***

(7.12)

0.170***

(5.83)

0.173***

(5.99)

0.151***

(5.32)

CV_FORECAST -0.235***

(-7.38)

-0.082**

(-2.36)

-0.367***

(-7.69)

-0.184***

(-4.01)

-0.199***

(-4.99)

-0.120

(-1.34)

-0.105

(-1.37)

-0.176***

(-3.55)

-0.182***

(-3.51)

DOWN_REV -0.346***

(-15.37)

0.138***

(6.07)

-0.359***

(-12.84)

0.025

(1.00)

-0.106***

(-4.75)

0.014

(0.38)

0.037

(1.24)

0.007

(0.26)

-0.002

(-0.07)

SALES_GRW 0.528***

(20.50)

0.115***

(4.99)

0.461***

(13.63)

0.124***

(4.59)

0.162***

(6.31)

0.104*

(1.69)

0.055

(1.26)

0.042

(1.23)

0.034

(0.95)

ROE 2.200***

(17.89)

1.566***

(12.31)

1.235***

(6.99)

0.748***

(4.58)

0.558***

(3.71)

0.104

(0.47)

-0.005

(-0.03)

0.052

(0.28)

0.106

(0.49)

∆ROE 0.924***

(8.79)

-0.075

(-0.70)

0.514***

(3.35)

-0.098

(-0.70)

0.087

(0.72)

0.306

(1.58)

0.204

(1.32)

0.063

(0.40)

-0.019

(-0.10)

MKT_CAP 0.157***

(14.77)

-0.034***

(-3.21)

0.117***

(8.88)

-0.008

(-0.64)

0.021*

(1.82)

-0.001

(-0.03)

-0.009

(-0.57)

-0.015

(-0.99)

-0.017

(-1.12)

53

Logit for

the full sample

Logit for subsample with

𝐸𝑃𝑆 𝑠𝑢𝑟𝑝𝑟𝑖𝑠𝑒 ∈ [−5𝑐, 5𝑐) Restricted

Model I

Main estimates

for Model I

Robustness checks

meet or beat meet or just

beat by 1c meet or beat

meet or just

beat by 1c Model II

meet vs

miss by 1c

𝐸𝑃𝑆 𝑠𝑢𝑟𝑝𝑟𝑖𝑠𝑒 ∈ [−4𝑐, 4𝑐)

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Coefficients 𝛼 in the pre-managed distribution

𝛼0,𝑁𝑂𝐴 0.001

(0.59)

0.001

(0.59)

0.001

(0.76)

0.001

(0.58)

𝛼1,𝑁𝑂𝐴 -0.112***

(-5.35)

-0.112***

(-5.35)

-0.130***

(-6.53)

-0.150***

(-4.11)

𝛼2,𝑁𝑂𝐴 -0.282

(-1.01)

-0.282

(-1.01)

-0.209

(-1.30)

-0.170

(-0.41)

𝛼3,𝑁𝑂𝐴 0.497***

(4.10)

0.497***

(4.10)

0.596***

(5.18)

0.745**

(2.20)

𝛼4,𝑁𝑂𝐴 1.346

(1.34)

1.346

(1.34)

1.093*

(1.68)

0.672

(0.25)

other 𝛼𝑝,𝑗 polynomial

terms 𝛼𝑝,0 only

polynomial terms 𝛼𝑝,0

and interaction terms 𝛼𝑝,1 … 𝛼𝑝,𝑀 for all X

F-statistic for

𝛼0,𝑁𝑂𝐴 … 𝛼𝑃,𝑁𝑂𝐴

9.92*** 9.92*** 11.84*** 11.28***

lnL -28,377.16 -27,934.61 -18,807.09 -21,943.10

adj. R2 (%) 4.18 4.40 4.28 4.74 5.04

N 48,054 48,054 32,707 32,707 32,707 32,707 32,707 32,707 29,748

The table presents the estimates for the logit model and our Models I and II. In untabulated exploratory analysis across various specifications for EPS surprise data,

Model I has consistently higher adjusted R2 than Model II, which leads us to use Model I as the main specification for EPS surprises. The sample and variable

definitions follow Barton and Simko (2002). The sample is the intersection of quarterly Compustat and IBES data during 1993–1999. The dependent variables are

based on the EPS surprise, computed as actual EPS minus the latest consensus mean EPS forecast for the quarter, both rounded to the nearest penny. Both actual

EPS and consensus EPS forecast are from split-unadjusted data. The dependent variable “meet or beat” in columns 1 and 3 equals one for non-negative EPS

surprises of any size and equals zero otherwise. The dependent variable “meet or just beat” in columns 2 and 4 equals one for EPS surprises of 0 and 1 cents and

equals zero otherwise. The dependent variable for Models I and II in columns 5–9 is the EPS surprise in cents. In columns 5–7 and 9, we interpret EPS surprises

of 0 and 1 cents (-1 and -2 cents) as possible outcomes (triggers) of earnings management in Models I and II. In column 8, only EPS surprises of 0 cents (-1 cent)

are interpreted as possible outcomes (triggers) of earnings management. Unless stated otherwise, the estimation sample for Models I and II is restricted to EPS

surprises in the [−5 𝑐𝑒𝑛𝑡𝑠, 5 𝑐𝑒𝑛𝑡𝑠) interval. NOA is the ratio of net operating assets (CEQQ−CHEQ+DLCQ+DLTTQ) to sales (SALEQ) at the beginning of the

quarter. SHARES is total shares outstanding (CSHOQ). BIG5 is a dummy variable for a Big-5 auditor (AU=1,4,5,6,7). PB is the market-to-book ratio

(CEQQ/[PRCCQ*CSHOQ]). LTGN_RISK is a dummy variable for high litigation risk industries (SIC codes 2833–2836, 3570–3577, 3600–3674, 5200–5961,

7370–7374, 8731–8734). ANALYSTS is the number of analysts (NUMEST). PREV_MB is a dummy for meeting or beating the consensus forecast in the prior

quarter. CV_FORECAST is the coefficient of variation in the analysts’ most recent forecasts for the quarter. DOWN_REV is the number of downward forecast

revisions (total NUMDOWN during the quarter). SALES_GRW is sales (SALEQ) growth relative to quarter t–4. ROE is return on equity (NIQ/CEQQ). ∆ROE is

change in ROE relative to quarter t–4. MKT_CAP is the natural logarithm of the market value of equity (PRCCQ*CSHOQ).

54

Appendix A. Derivation of the distribution of observed earnings in model (1)–(3)

After integrating out the unobservables 𝐸𝐴𝑅𝑁∗ and 𝑚, the probability density function of

reported earnings can be rewritten as

𝑓(𝐸𝐴𝑅𝑁|𝑋) = ∫ ∑ 𝑓(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑚, 𝑋) Pr(𝑚|𝐸𝐴𝑅𝑁∗, 𝑋)

𝑚=0,1

𝑓∗(𝐸𝐴𝑅𝑁∗|𝑋)𝑑𝐸𝐴𝑅𝑁∗

𝐸𝐴𝑅𝑁∗

(𝐴. 1)

We simplify this expression for four distinct intervals of 𝐸𝐴𝑅𝑁.

Case 1: 𝐸𝐴𝑅𝑁 < −𝐾, i.e., a large loss. From (2), 𝑓(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑚, 𝑋) is non-zero only for

𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁 and 𝑚 = 0.38 Equation (A1.1) simplifies to

𝑓(𝐸𝐴𝑅𝑁|𝑋) = Pr(𝑚 = 0|𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁, 𝑋) 𝑓∗(𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁|𝑋) (𝐴. 2)

From (3), Pr(𝑚 = 0|𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁, 𝑋) = 1 because 𝐸𝐴𝑅𝑁 ∉ [−𝐾, 0). Therefore, (A.2)

simplifies to equation (4a) in Section 2

𝑓(𝐸𝐴𝑅𝑁|𝑋) = 𝑓∗(𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁|𝑋) (4𝑎)

Case 2: 𝐸𝐴𝑅𝑁 ∈ [−𝐾, 0), i.e., a small loss. From (2), 𝑓(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑚, 𝑋) is non-zero only

for 𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁 and 𝑚 = 0.39 Therefore, (A.1) simplifies to (A.2), similar to Case 1. From

(3), for 𝐸𝐴𝑅𝑁 ∈ [−𝐾, 0), Pr(𝑚 = 0|𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁, 𝑋) = 1 − 𝑃(𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁, 𝑋).

Therefore, (A.2) simplifies to equation (4b) in Section 2

𝑓(𝐸𝐴𝑅𝑁|𝑋) = [1 − 𝑃(𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁, 𝑋)]𝑓∗(𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁|𝑋) (4𝑏)

Case 3: 𝐸𝐴𝑅𝑁 ∈ [0, 𝐾), i.e., a small profit. From (2), 𝑓(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑚, 𝑋) is non-zero in

two situations: (a) 𝑚 = 0 and 𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁, and (b) 𝑚 = 1 and 𝐸𝐴𝑅𝑁∗ ∈ [−𝐾, 0). Therefore,

(A.1) becomes

38 When 𝑚 = 1 (i.e., managers manipulate earnings to report a small profit), the probability of reporting a large loss

is zero. When 𝑚 = 0 (i.e., no earnings management), reported 𝐸𝐴𝑅𝑁 must equal the pre-managed 𝐸𝐴𝑅𝑁∗. Because

𝑓(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑚, 𝑋) collapses to a mass point with weight 1, it drops out of the computation. 39 When 𝑚 = 1, the probability of reporting a small loss is zero. When 𝑚 = 0, reported 𝐸𝐴𝑅𝑁 must equal the pre-

managed 𝐸𝐴𝑅𝑁∗.

55

𝑓(𝐸𝐴𝑅𝑁|𝑋) = Pr(𝑚 = 0|𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁, 𝑋) 𝑓∗(𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁|𝑋) +

∫ 𝑓(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑚 = 1, 𝑋) Pr(𝑚 = 1|𝐸𝐴𝑅𝑁∗, 𝑋) 𝑓∗(𝐸𝐴𝑅𝑁∗|𝑋)𝑑𝐸𝐴𝑅𝑁∗

𝐸𝐴𝑅𝑁∗∈[−𝐾,0)

(𝐴. 3)

In the first line of (A.3), Pr(𝑚 = 0|𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁, 𝑋) = 1 per (3) because 𝐸𝐴𝑅𝑁 ∉

[−𝐾, 0). In the second line of (A.3), 𝑓(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑚 = 1, 𝑋) = ℎ(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑋) per (2)

and Pr(𝑚 = 1|𝐸𝐴𝑅𝑁∗, 𝑋) = 𝑃(𝐸𝐴𝑅𝑁∗, 𝑋) per (3). Therefore, (A.3) simplifies to equation (4c) in

Section 2

𝑓(𝐸𝐴𝑅𝑁|𝑋) = 𝑓∗(𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁|𝑋) +

∫ ℎ(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑋)𝑃(𝐸𝐴𝑅𝑁∗, 𝑋)𝑓∗(𝐸𝐴𝑅𝑁∗|𝑋)𝑑𝐸𝐴𝑅𝑁∗

𝐸𝐴𝑅𝑁∗∈[−𝐾,0)

(4𝑐)

Case 4: 𝐸𝐴𝑅𝑁 > 𝐾, i.e., a large profit. From (2), 𝑓(𝐸𝐴𝑅𝑁|𝐸𝐴𝑅𝑁∗, 𝑚, 𝑋) is non-zero only for

𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁 and 𝑚 = 0, and (A.1) simplifies to (A.2), similar to Cases 1 and 2. From (3),

Pr(𝑚 = 0|𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁, 𝑋) = 1 because 𝐸𝐴𝑅𝑁 ∉ [−𝐾, 0). Therefore, (A.2) simplifies to

equation (4d) in Section 2

𝑓(𝐸𝐴𝑅𝑁|𝑋) = 𝑓∗(𝐸𝐴𝑅𝑁∗ = 𝐸𝐴𝑅𝑁|𝑋) (4𝑑)

56

Appendix B. Implementation of maximum likelihood (ML) estimation

For each firm-year observation 𝑖, 𝑡, the log-likelihood is ln 𝑓(𝐸𝐴𝑅𝑁𝑖,𝑡|𝑋𝑖,𝑡) as defined in (4a)–

(7c), where 𝐸𝐴𝑅𝑁𝑖,𝑡 and 𝑋𝑖,𝑡 are from the data, and the coefficient vector comprises 𝛼0,0 … 𝛼𝑃,𝑀

and 𝜋0 … 𝜋𝑀. Maximum likelihood estimation finds the coefficients 𝛼0,0 … 𝛼𝑃,𝑀 and 𝜋0 … 𝜋𝑀 that

maximize the total log-likelihood for the sample.40 We code the log-likelihood as a user-defined

Stata function. The ml command in Stata takes this function as an input and handles the numerical

optimization and the computation of the standard errors.

We estimate the model for the subsample with earnings in a relatively narrow interval [−𝑅, 𝑅)

around zero (e.g., 𝐸𝐴𝑅𝑁 ∈ [−0.04, 0.04) for our main definitions). Although this approach

involves selection on the dependent variable, we show that it yields the same parameter estimates

as conventional maximum likelihood estimation on the full sample. The conventional ML

approach solves

max𝜃

∑ ln 𝑓(𝐸𝐴𝑅𝑁𝑖,𝑡|𝑋𝑖,𝑡)

𝑖=1…𝑁,𝑡=1…𝑇

=

= max𝜃

( ∑ ln 𝑓(𝐸𝐴𝑅𝑁𝑖,𝑡|𝑋𝑖,𝑡)

𝑖,𝑡: 𝐸𝐴𝑅𝑁𝑖,𝑡∈[−𝑅,𝑅)

+ ∑ ln 𝑓(𝐸𝐴𝑅𝑁𝑖,𝑡|𝑋𝑖,𝑡)

𝑖,𝑡: 𝐸𝐴𝑅𝑁𝑖,𝑡∉[−𝑅,𝑅)

) (𝐵. 1)

where 𝜃 is the full parameter vector, which comprises 𝛼0,0 … 𝛼𝑃,𝑀, 𝜋0 … 𝜋𝑀, and additional

parameters that determine the earnings distribution outside [−𝑅, 𝑅); 𝑓(𝐸𝐴𝑅𝑁𝑖,𝑡|𝑋𝑖,𝑡) is the

probability density function of reported earnings from equations (4a)–(4d), evaluated at the

parameter values 𝜃; and [−𝑅, 𝑅) is the earnings interval used in our subsample-specific estimation.

40 For each observation 𝑖, 𝑡, we normalize the probability density function (5a) by imposing the standard restriction

that the total probability mass ∫ 𝑓(𝐸𝐴𝑅𝑁|𝑋𝑖,𝑡)𝑑𝐸𝐴𝑅𝑁 of the conditional earnings distribution must equal 1. Without

this normalization, the ML estimation procedure would artificially drive the log-likelihood to infinity by increasing

the coefficients 𝛼 to infinity. This issue is specific to ML estimation. In our two-stage regression-based method, the

estimates minimize the distance between predicted and actual bin frequencies, and thus the actual bin frequencies

directly determine the scale of the density parameters, removing the need for an explicit normalization.

57

Suppose that a researcher uses a separate subset of parameters 𝜃𝑜𝑢𝑡𝑠𝑖𝑑𝑒 for the pre-managed

earnings distribution outside the estimation interval [−𝑅, 𝑅). From equations (4a)–(4d) in Section

2, all other components of 𝜃 (i.e., the pre-managed distribution parameters 𝛼0,0 … 𝛼𝑃,𝑀 for the

interval [−𝑅, 𝑅) and the earnings management parameters 𝜋0 … 𝜋𝑀) affect the likelihood only for

observations with reported earnings inside [−𝑅, 𝑅). Therefore, (B.1) can be rewritten as

max𝜋0…𝜋𝑀, 𝛼0,0…𝛼𝑃,𝑀

∑ ln 𝑓(𝐸𝐴𝑅𝑁𝑖,𝑡|𝑋𝑖,𝑡)

𝑖,𝑡: 𝐸𝐴𝑅𝑁𝑖,𝑡∈[−𝑅,𝑅)

+ max𝜃𝑜𝑢𝑡𝑠𝑖𝑑𝑒

∑ ln 𝑓(𝐸𝐴𝑅𝑁𝑖,𝑡|𝑋𝑖,𝑡)

𝑖,𝑡: 𝐸𝐴𝑅𝑁𝑖,𝑡∉[−𝑅,𝑅)

(𝐵. 2)

The first maximization in this expression is equivalent to our maximum likelihood estimation

procedure for the subsample with earnings in the interval [−𝑅, 𝑅). It fully determines all of the

parameters that we are interested in, i.e., 𝛼0,0 … 𝛼𝑃,𝑀 and 𝜋0 … 𝜋𝑀. Therefore, our estimation of

𝛼0,0 … 𝛼𝑃,𝑀 and 𝜋0 … 𝜋𝑀 on the restricted sample is equivalent to conventional maximum

likelihood estimation on the full sample with an additional parameter vector 𝜃𝑜𝑢𝑡𝑠𝑖𝑑𝑒 .

58

Appendix C. Derivation of the standard errors in our two-stage estimation method

Because the explanatory variables in the second-stage regression (8b) are constructed based on

the first-stage estimates �̂� = (�̂�0,0 … �̂�𝑃,𝑀)′ from (8a), the standard errors of �̂� = (�̂�0 … �̂�𝑀)′ in

the second stage should be adjusted for the first-stage estimation noise. The usual OLS standard

errors (with appropriate clustering) do not incorporate this adjustment and should not be used.

Using the method of moments representation of OLS (e.g., Wooldridge, 2002, Ch. 14), the

regression estimates �̂� and �̂� in the two stages (8a) and (8b) are defined by the moment conditions

ℎ̅(�̂�) =1

𝑁∑ (𝑌𝑖,𝑡,𝑏 − �̂�′𝑃𝑖,𝑡,𝑏)𝑃𝑖,𝑡,𝑏 = 0𝑖,𝑡,𝑏:

𝑏∉𝑠𝑚𝑎𝑙𝑙 𝑙𝑜𝑠𝑠/𝑝𝑟𝑜𝑓𝑖𝑡 𝑏𝑖𝑛𝑠

(𝐶. 1)

�̅�(�̂�, �̂�) =1

𝑁∑ (𝑌𝑖,𝑡,𝑏 − �̂�𝑖,𝑡,𝑏(�̂�) − �̂�′𝑄𝑖,𝑡,𝑏(�̂�))𝑄𝑖,𝑡,𝑏(�̂�) = 0𝑖,𝑡,𝑏:

𝑏∈𝑠𝑚𝑎𝑙𝑙 𝑙𝑜𝑠𝑠/𝑝𝑟𝑜𝑓𝑖𝑡 𝑏𝑖𝑛𝑠

(𝐶. 2)

where 𝑃𝑖,𝑡,𝑏 is the full vector of explanatory variables in stage 1 (i.e., 1, 𝑧𝑏 , 𝑧𝑏2 … 𝑧𝑏

𝑃 and its

interactions with 𝑋𝑖,𝑡,1 … 𝑋𝑖,𝑡,𝑀), �̂�𝑖,𝑡,𝑏(�̂�) is the predicted value from stage 1, and 𝑄𝑖,𝑡,𝑏(�̂�) is the

full vector of explanatory variables in stage 2 (i.e., 𝑊𝑖,𝑡,𝑏 and its interactions with 𝑋𝑖,𝑡,1 … 𝑋𝑖,𝑡,𝑀).

The estimation noise in �̂� affects the second-stage standard errors through both �̂�𝑖,𝑡,𝑏(�̂�) and

𝑄𝑖,𝑡,𝑏(�̂�) in (C.2).

The Taylor expansion of (C.1) and (C.2) around the true values 𝛼∗ and 𝜋∗ is

[ℎ̅(�̂�)

�̅�(�̂�, �̂�)] ≈ [

ℎ̅(𝛼∗)

�̅�(𝛼∗, 𝜋∗)] + [

∇𝛼ℎ̅ 0∇𝛼�̅� ∇𝜋�̅�

] [�̂� − 𝛼∗

�̂� − 𝜋∗] (𝐶. 3)

After combining (C.3) with (C.1) and (C.2), we have

[�̂� − 𝛼∗

�̂� − 𝜋∗] ≈ − [∇𝛼ℎ̅ 0∇𝛼�̅� ∇𝜋�̅�

]−1

[ℎ̅(𝛼∗)

�̅�(𝛼∗, 𝜋∗)] (𝐶. 4)

From (C.4), the asymptotic covariance matrix of the estimates �̂� and �̂� is

59

𝐶𝑜𝑣 (√𝑁 [�̂��̂�

]) = Γ′ΩΓ (𝐶. 5𝑎)

where

Γ = (−𝑝𝑙𝑖𝑚 [∇𝛼ℎ̅ 0∇𝛼�̅� ∇𝜋�̅�

])−1

(𝐶. 5𝑏)

Ω = 𝐶𝑜𝑣 (√𝑁 [ℎ̅(𝛼∗)

�̅�(𝛼∗, 𝜋∗)]) (𝐶. 5𝑐)

Because �̂� and �̂� converge in probability to the (unknown) true values, the matrices Γ and Ω can

be evaluated at �̂� and �̂� instead of 𝛼∗ and 𝜋∗. The covariance matrix Ω of the moment conditions

is clustered as needed. We incorporate the computation of (C.5a)–(C.5c) in our example Stata

codes.