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Rock Center for Corporate Governance at Stanford UniversityWorking Paper No. 65

Joseph A. Grundfest and Nadya Malenko

Quadrophobia: Strategic Rounding of EPS Data

Stanford Law SchoolJohn M. Olin Program in Law and Economics

Working Paper No. 388


2/65Electronic copy available at: http://ssrn.com/abstract=1474668Electronic copy available at: http://ssrn.com/abstract=1474668

Quadrophobia: Strategic Rounding of EPS Data

Joseph A. Grundfest yStanford Law School and The Rock Center for Corporate Governance

Nadya Malenko z

Graduate School of Business, Stanford University

Rock Center for Corporate Governance at Stanford University Working Paper No. 65

First version: April 2008

This version: October 2009

Abstract

We hypothesize that earnings management causes quadrophobia, the under-representation of the number four in the rst post-decimal digit of EPS data. We demonstrate that quadrophobia ispervasive, persistent, and follows economically rational patterns. Consistent with analyst coveragebeing a determinant of earnings management, quadrophobia increases (declines) when companiesgain (lose) analyst coverage, and is more frequent when earnings are close to analyst forecasts.

Persistent quadrophobes are more likely to restate nancials and to be sued in SEC proceedingsalleging accounting violations. Quadrophobia, even if itself legal, therefore appears to signal apropensity to engage in problematic accounting practices.

We are grateful to Anat Admati, Robert Daines, Ian Gow, Elaine Harwood, Daniel Ho, Alan Jagolinzer, David

Larcker, Andrei Malenko, Maureen McNichols, Roman Weil, Anastasia Zakolyukina, and participants of the StanfordLaw Review Symposium on corporate governance for helpful comments and suggestions; to Alan Jagolinzer, DavidLarcker, and Anastasia Zakolyukina for providing us with the data; and to the Rock Center for Corporate Governancefor nancial support. Earlier versions of this paper were circulated under Nadya Malenkos previous surname of Zhukova.

y Stanford University Law School, 559 Nathan Abbott Way, Stanford, CA 94305-8610, United States, email: [email protected].

z Graduate School of Business, Stanford University, 518 Memorial Way, Stanford, CA 94305-5015, United States,email: [email protected].


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1. Introduction

The exercise of managerial discretion is unavoidable when preparing nancial statements.

Inventory valuations, nancial asset writedowns, and the setting of accruals and reserves, are

among the dozens of decisions that compel the exercise of discretion. A common concern among

investors, regulators, and academics is that discretion can be exercised to obscure a rms actual

nancial performance, particularly through practices known as "earnings management." These

practices encompass a wide array of accounting techniques that can help earnings per share

(EPS) meet analyst expectations, maintain a history of smooth quarterly increases, or achieve

targets related to executive compensation bonus targets.An extensive literature examines the incidence and magnitude of earnings management, the

factors that induce rms to engage in the practice, and the eectiveness of various regulatory

measures designed to constrain the practice. This paper extends the literature by applying a

novel, simple, statistically robust measure of earnings management that analyzes the distri-

bution of the rst post-decimal digit in EPS data, reported in cents per share, for evidence

that management has "rounded up" its reported EPS results. It further extends the literature

by demonstrating that this form of earnings management, even if entirely legal, anticipates

problematic accounting practices that can lead to restatements or enforcement actions by the

Securities and Exchange Commission.

Because reported earnings per share in the United States are rounded to the nearest cent,

earnings of 13.4 cents are rounded down to 13 cents while earnings of 13.5 cents are rounded

up to 14 cents. The amount of accounting discretion required to increase rounded EPS by

one cent, all other factors equal, is at a local minimum when the rst digit to the right of the

decimal in EPS calculations is a four. Accordingly, if managers of publicly traded rms want

to increase their reported earnings by one cent, for whatever reason, then the number four

should be signicantly underrepresented in the rst post-decimal digit of EPS data. We call

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this pattern "quadrophobia." 1

Quadrophobia thus constitutes a specic form of earnings management that, in the abstract,

reects the exercise of accounting judgment over a quantitatively small number. Quadrophobia

can be practiced either in isolation or in conjunction with other techniques that can increase

reported EPS by one cent or more. The accounting judgments that lead to quadrophobia can,

depending on context and intent, either be entirely legal, or they can constitute a violation

of the federal securities laws. Further, even if the techniques used to generate quadrophobia

are, in and of themselves, entirely legal, they can be correlated with other forms of accounting

conduct that the Securities and Exchange Commission (SEC) nds problematic.

We study the incidence of quadrophobia in quarterly and annual earnings reports for publicly

traded rms over the seventeen year period spanning 1980 through 2006. We demonstrate that

quadrophobia is pervasive and statistically signicant among rms with analyst coverage. It is

much less prevalent, though still signicant, among rms without coverage. To test the causality

of this relation we examine the subsample of rms with analyst coverage and demonstrate that

the probability of quadrophobia in any given rm increases (decreases) when analyst coverage is

initiated (dropped). This nding suggests, but because of the possibility of unobserved variables

does not conclusively establish, that analyst coverage causes quadrophobia and is not merely

correlated with the phenomenon. This nding also supports the intuition that quadrophobia is

animated by managements desire to meet or exceed analyst earnings estimates. Consistent with

this hypothesis, we demonstrate that quadrophobia is particularly pronounced when reported

earnings are close to analyst forecasts.

Quadrophobia also follows an economically rational pattern. It is more apparent when1Equivalent reasoning suggests that the number ve should be overrepresented in the rst post-decimal

digit of positive EPS reported in cents; a pattern we call "quintophilia." When dealing with negative EPS data,however, rms have an incentive to avoid the number ve in the rst post-decimal digit, because rounding wouldthen increase negative reported earnings. By the same logic, rms with negative earnings have an incentiveto over-represent the number four in the rst post-decimal digit. Firms with negative EPS should thereforedisplay "quintophobia" and "quadrophilia." These additional patterns are explored in greater detail in Table 3and Section 4.1 below.

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market-to-book ratios are high, suggesting that rounding will cause a greater boost in stock

price values. Firm size has a non-monotonic eect, with mid-size rms more likely to engage in

rounding behavior than smaller or larger rms. This size eect suggests that the smallest rms

tend not to manage rounding behavior, and that the largest rms either nd it more dicult

to engage in managed rounding because it requires the exercise of accounting discretion over a

larger absolute dollar amount, or have tighter controls over this form of conduct. Quadrophobia

is also more apparent when EPS has a small absolute value, and a penny a share is a larger

percentage of reported EPS.

While size, market-to-book ratio, and analyst coverage are important determinants of whether

a company will engage in rounding, an interesting question is whether quadrophobia is ran-

domly distributed among issuers with common characteristics or whether it is concentrated in

a subset of companies. Our data support the latter hypothesis. We nd that quadrophobia is

persistent: companies with a history of quadrophobia are likely to remain quadrophobes. It

therefore appears that individual managements are more likely to engage in strategic rounding

behavior, all other factors equal.

Although quadrophobia is pervasive and persistent, it can reect the exercise of legitimate

accounting judgment and is not necessarily related to violation of accounting standards. To

determine the extent to which this form of earnings management is benign, we examine whether

rounding behavior is associated with a higher probability of potentially problematic accounting

conduct. We nd that quadrophobes are more likely to restate nancials and to be named

as defendants in SEC Accounting and Auditing Enforcement Releases (AAER). Thus, even if

quadrophobia itself represents legitimate accounting discretion, it appears to be practiced bymanagements that are more likely to engage in problematic practices that raise the possibility

of violations of accounting norms or federal securities laws. Indeed, there is no reported enforce-

ment action by the SEC that targets rounding behavior per se . The nding that quadrophobia

anticipates restatements and AAER proceedings therefore suggests that quadrophobic manage-

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ments are more likely to engage in other problematic forms of accounting conduct, and does

not support the conclusion that quadrophobia, in and of itself, violates the federal securities

laws.

Given this nding, we examine whether audit oversight and regulation constrain quadropho-

bia to a statistically measurable degree. A growing literature examines whether the Sarbanes

Oxley Act of 2002 (SOX) has had a mitigating eect on earnings management (see Section 2

for a brief review of this literature). Our data are inconclusive regarding the eects of SOX or

of the increasingly vigorous enforcement regime following the Enron and WorldCom frauds on

the incidence of quadrophobia. While the incidence of quadrophobia declines after Sarbanes

Oxleys enactment, the decline is a continuation of a pre-existing trend consistent with a longer-

term shift toward the targeting of pro-forma EPS rather than GAAP EPS by managements

and analysts alike. Because these trends are concurrent, available data do not allow us to

disentangle the eects of increased reliance on pro-forma EPS from the confounding eects of

changes in the regulatory and enforcement regime. We therefore cannot draw rm conclusions

as to the eects, if any, of Sarbanes Oxley or of increased enforcement in the wake of Enron

and WorldCom.

We also compare the incidence of quadrophobia in unaudited quarterly data with its inci-

dence in audited annual data, in order to test whether the audit process has a deterrent eect

on quadrophobia. We nd that the presence of auditors appears to have a historically contin-

gent eect. Prior to 2001, quadrophobia was as prevalent in the audited annual EPS data as in

the unaudited quarterly data, suggesting that auditors did not deter the practice. Since 2001,

however, the data are inconclusive. Although quadrophobia is less prevalent in the auditedannual data than in the quarterly data after 2001, the presence of concurrent pro forma eects

impairs our ability to draw rm conclusions regarding the eect of audit oversight.

Quadrophobias policy implications have not been explored in the literature. On the one

hand, the dollar amounts involved in quadrophobic earnings management can be quite small.

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In 2006, the most recent year in our sample, the mean (median) aggregate amount of earnings

over which management would have to exercise discretion in order to move quarterly EPS by

a tenth of a cent was $149,000 ($31,000), or 0.15% (0.41%) of the companys total quarterly

revenue. 2 If the focus is on the quantitative materiality of the dollars at issue, it is easy to

conclude that quadrophobia may be an intriguing pattern in the data, but does not constitute

a meaningful problem in the market.

Securities and Exchange Commission Sta Accounting Bulletin 99 (SAB 99), however, sup-

ports an alternative interpretation. SAB 99 articulates a qualitative interpretation of materi-

ality and suggests that even minimal dollar amounts can be material if they assist in meeting

EPS expectations, are likely to aect stock price, or are related to executive compensation,

among other criteria. Our regression ndings that quadrophobia anticipates restatements and

AAER proceedings are consistent with the broader concern expressed in SAB 99 because, even

if quadrophobia always reects a legitimate exercise of accounting discretion over a quantita-

tively small number, it appears to be practiced by managements that are more likely to engage

in other problematic accounting practices. This evidence is not inconsistent with the stronger

assertion that the exercise of discretion that leads to quadrophobia is in itself improper, but

available data do not allow us to test that stronger hypothesis. Securities enforcement agen-

cies, auditors, and others with an interest in the integrity of nancial statements could therefore

reasonably conclude that quadrophobes warrant enhanced scrutiny.

The remainder of the paper is organized as follows. Section 2 discusses prior research. Section

3 describes the data and presents descriptive statistics for our sample. Section 4 describes our

methodology. Section 5 presents the results of multivariate analysis and demonstrates theimportance of analyst coverage for rounding behavior as well as the presence of economically

2An increase of $0.001 in earnings per share requires increasing aggregated earnings by N*$0.001, whereN is the number of shares outstanding. The average (median) quarterly earnings for companies with positiveearnings in 2006 were $98 million (7.5 million) and the average (median) number of shares outstanding was 149million (31 million).

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rational patterns in the incidence of quadrophobia. Section 6 reports tests of persistence.

Section 7 examines the eect of auditors and the passage of SOX on rounding behavior. Section

8 studies the relation between quadrophobia and the incidence of restatements and AAER

proceedings and class action securities litigation. Section 9 discusses public policy implications.

Section 10 oers concluding remarks.

2. Prior literature

This paper is related to three distinct bodies of research on earnings management: studies of

distributional patterns in reported earnings, research on the relation between analyst coverageand earnings management, and the literature on nancial reporting practices in the post-Enron

and WorldCom period.

Several studies explore whether the distribution of earnings around certain thresholds is

smooth, as would be expected in the absence of human intervention, or exhibits discontinuities

consistent with earnings management (see, e.g., Burgstahler and Dichev, 1997; Degeorge, Patel,

and Zeckhauser, 1999). Carslaw (1988) examines patterns in the second from left-most digitin reported aggregate earnings data. He nds zeros are overrepresented and nines are under-

represented, and interprets the data as evidence that companies round up earnings to achieve

investors cognitive reference points of N 10k . Similar to Carslaw (1988), our paper provides

evidence that rms engage in rounding, however, we focus on rounding in earnings per share

and not on aggregate earnings. Thomas (1989) documents unusually high frequencies of EPS

gures divisible by ve and ten cents. While we also examine distributional patterns in EPS

data, our focus is on the rst post-decimal digit rather than the last cent as in Thomas (1989).

Our paper is most closely related to Das and Zhang (2003) who demonstrate that numbers

below ve are underrepresented (overrepresented) in the rst post-decimal digit in positive (neg-

ative) EPS. Our paper diers from Das and Zhang (2003) in several major respects. First, we do

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not aggregate digits below and above ve and instead focus on digit four, under-representation

of which is most pronounced. Second, we distinguish between basic and diluted EPS and

demonstrate that diluted EPS should be used to identify rounding behavior after the passage

of FAS 128. We also extend the analysis of rounding behavior in several important directions,

demonstrating persistence and the eect of analyst coverage on rounding, establishing a rela-

tion between quadrophobia and alleged violations of accounting norms and of securities laws,

and examining the eects of SOX and related enforcement activity on rounding.

Our paper also contributes to the literature that explores the use of earnings management to

meet or exceed analyst expectations. Payne and Robb (2000) demonstrate that discretionary

accruals are signicantly positive when pre-managed earnings are below the consensus analysts

forecast. Similarly, Matsumoto (2002) nds that discretionary accruals are more likely to be

positive in rm-quarters where reported earnings meet or exceed earnings forecasts than in

quarters where reported earnings fall short of expectations. Burgstahler and Eames (2006),

Matsumoto (2002), and Bartov, Givoly, and Hayn (2002) present evidence that, in addition

to accrual-based earnings management, rms also use expectations earnings management by

lowering market expectations to produce a positive earnings surprise or to avoid a negative

earnings surprise. Our ndings are consistent with this literature in that we demonstrate that

quadrophobia is particularly pronounced when reported earnings are close to analyst forecasts.

In addition, we show that rms with analyst coverage are signicantly more likely to engage in

rounding than rms without coverage, and provide evidence that is consistent with this relation

being causal.

Finally, our paper contributes to a growing literature that examines whether the incidenceof earnings management declined subsequent to passage of SOX and following heightened en-

forcement eorts triggered by the Enron and WorldCom frauds of 2001 and 2002. Lobo and

Zhou (2006), Cohen, Dey, and Lys (2008), Koh, Matsumoto, and Rajgopal (2008), and Bartov

and Cohen (2008) show a signicant decline in accrual earnings management in the post-SOX

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period. While these papers rely on estimates of discretionary accruals to identify changes in

earnings management, our paper focuses on the analysis of distributional patterns in reported

earnings. In this respect, our paper is more closely related to Aono and Guan (2007), who

examine the distribution of the second digit in reported annual earnings, applying the tech-

niques developed in Carslaw (1988), and nd a signicant post-SOX decrease in that measure

of earnings management. In contrast, as explained in greater detail below, our analysis is incon-

clusive as to whether SOX has had a statistically signicant impact on earnings management

as manifested in rounding behavior.

3. Data

We obtain all rm-year and rm-quarter observations from COMPUSTAT industrial annual

and quarterly les for the period spanning 1980 to 2006. We eliminate all observations with

missing net income data, data describing the number of shares used to calculate EPS, or

observations that show negative total assets. The resulting sample of 788,567 rm-quarter

observations and 213,390 rm-year observations covers 22,460 companies. Sample sizes aresmaller for several analyses because of limited availability of other variables of interest.

Analyst data covering the same period are obtained from the I/B/E/S Summary data-

base. For each observation we capture the most recent consensus forecast prior to the earnings

announcement date. Consensus analyst forecasts are available for approximately 32% of rm-

quarter observations and 40% of rm-year observations.

Table 1 presents descriptive statistics for the entire quarterly data sample, the sample of

rm-quarter observations for which a corresponding consensus analyst forecast is available in

I/B/E/S, and the sample of observations for which an I/B/E/S forecast is unavailable. The

sample rms have median total assets of $96 million and median market capitalization of $75

million. As is apparent from Table 1, the subsamples have strikingly dierent characteris-

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tics: rms with analyst coverage are signicantly larger, (whether measured by assets, market

capitalization, or sales), have higher return on assets, and lower leverage. These ndings are

consistent with prior literature describing the characteristics of rms with analyst coverage (see,

e.g., Bhushan (1989)).

[Table 1 here]

Because we are interested in the rst post-decimal digit of EPS expressed in cents, we

cannot use EPS data provided by Compustat; those data are already rounded to the nearest

cent. To obtain the unrounded EPS expressed in cents, we multiply income after extraordinary

items (Compustat data item 10 + Compustat data item 26) by 100 and divide by the number

of common shares used to calculate EPS (data items 15 and 124 for basic and diluted EPS,

respectively).

4. Distributional patterns in EPS data

Our basic hypothesis is that earnings management manifests itself in the numerical distrib-ution of the rst post-decimal digit of EPS expressed in cents per share. More precisely, EPS

data are rounded up to the next highest cent if the rst post-decimal digit is ve through

nine, and rounded down to the next lowest cent if that digit is one through four. Because the

amount of earnings management required to obtain an extra rounded cent of reported EPS is

minimized, all other factors equal, when the rst post-decimal digit is a four, earnings man-

agement through rounding should cause a statistically signicant under-representation of the

number four in the rst post-decimal digit of EPS expressed in cents per share. Consistent with

this hypothesis, the numbers one through three should also be under-represented in the rst

post-decimal digit, and the numbers ve through nine should be over-represented. The pattern

should reverse for negative earnings because companies then have an incentive to decrease the

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absolute value of negative EPS data.

4.1. Methodology and the uniform distribution expectation

As an initial matter, to test whether the number four is under-represented, we must specify

a null hypothesis that describes the baseline distribution of numbers in the rst post-decimal

digit that prevails in the absence of earnings management. Common intuition suggests that

any number is equally likely to appear as the rst digit following the decimal and that the

numbers zero through nine should therefore be uniformly distributed in the rst post-decimal

digit of EPS reported in cents.

This intuition cannot, however, be accepted at face value. Benfords Law (Benford (1938))

suggests that the incidence of the numbers zero through nine in specic digits of nancial

and other data sets can be far from uniform, with the number one being signicantly over-

represented as the leading signicant digit. Nigrini (1999) demonstrates that Benfords law

could be applied to discover certain forms of accounting and expense related fraud. Carslaw

(1988) and Thomas (1989) present evidence that Benfords Law applies to the analysis of

aggregate earnings data.

Thomas (1989) notes, however, that Benfords Law does not apply to earnings per share

data. Indeed, even if the distributions of both aggregate earnings and the number of shares are

consistent with Benfords law, there is no reason to expect the ratio of these numbers to follow

Benfords law as well. Further, because average quarterly EPS in our sample is 38 cents per

share, and because Benfords Law indicates that the distribution of the n-th digit approaches

the uniform distribution exponentially fast as n approaches innity (see Hill (1995)), even if

Benfords Law were to apply, it would, on average, suggest a largely uniform distribution of

the number four in the rst post-decimal digit of EPS expressed in cents.

To test whether the uniform distribution is an appropriate null hypothesis, we initially

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examine the incidence of the numbers zero through nine in the rst post-decimal digit of

per-share accounting data that do not regularly attract market attention and for which there

is no incentive to manage through rounding or otherwise. We then compare those data to

patterns observed in reported EPS. We focus on sales per share, and operating income per

share calculated both before and after depreciation. 3 We consider only positive gures because

we expect the pattern to reverse for negative gures.

Before proceeding with this analysis, we note that the distribution of the rst post-decimal

digit is likely to diverge from the uniform for reasons unrelated to Benfords Law if there is

a large proportion of relatively small EPS numbers in our sample. Indeed, for gures below

0.1 cents the rst post-decimal digit is always equal to zero. The presence of rms with such

low levels of earnings per share, or sales or operating income per share, would therefore bias

upwards the frequency of zeros and bias downwards the frequency of all other digits even if

the underlying distribution is in fact uniform. In each of the tests reported below we therefore

restrict attention to the sample where the respective per share gure is greater than 0.1 cents.

This constraint eliminates less than 1% of all observations. Similarly, the rst post-decimal

digit is always zero if the respective per share gure is an integer. However, the number of

observations with integer gures per share is very small (less than 700 observations), and our

results are not sensitive to the inclusion or exclusion of these observations.

To test the null hypothesis that the frequency of any number in the rst digit following the

decimal equals the expected frequency p0, we apply the two-sided z-statistic

z =j p p0j

q p0 (1 p0 )

n

, where n is the sample size and p is the frequency of the number in the sample. Under the null

hypothesis, z has a standard Normal distribution. Similarly, to compare the frequency of any3We also consider a number of other series of per-share data, including, for example, cash holdings per share

and assets per share, and nd similar results. The analysis is available from the authors on request.

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number in two dierent samples ( p1 and p2), we use the two-sided z-statistic

z 2 =j p1 p2j

q p(1 p)( 1n 1 + 1n 2 ), where n 1; n 2 are the sample sizes and p is the frequency of the number in question in the

combined sample of size n 1 + n 2. Under the null hypothesis that the frequency in the two

samples is the same, z 2 is also a standard Normal variable (Fleiss et al. (2003)).

Because our main hypothesis is that earnings management causes signicant under-representation

of fours, we initially analyze the frequency of the number four in the rst post-decimal digit

in the selected data sets and then expand the analysis to consider the distribution of other

numbers in the same post-decimal digit. Fig. 1 illustrates the time-series distribution of four in

the rst post-decimal digit for sales per share, per-share operating income before depreciation,

per-share operating income after depreciation, and earnings per share, each expressed in cents.

The solid lines represent the actual frequency with which the number four appears in the rst

post-decimal digit, while the dotted lines correspond to 95% condence intervals around the

expected frequency of 0.1.

[Figure 1 here]

The frequency of the number four in the rst post-decimal digit of sales per share, operating

income before depreciation per share, and operating income after depreciation per share is,

in each case, statistically indistinguishable from 0.1 for each year in the sample. The EPS

data, however, dier dramatically. As hypothesized, the number four is there signicantly and

consistently under-represented. The lowest observed frequency was 0.0754 in 1998, indicatingthat almost one quarter of the fours expected in the absence of earnings management were

missing.

We next extend our analysis to the distribution of other digits. To conserve space and for

ease of presentation we present the data only for 1994, the mid-point of our sample. Results

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for that year are representative of the sample as a whole, and equivalent analyses for all other

calendar years in the sample are available from the authors on request.

Table 2 presents the average frequency of numbers zero through nine in the rst post-decimal

digit of per share sales, and per share operating income before and after depreciation for the 1994

data. Consistent with the hypothesis of a uniform distribution, the frequency of each number

is statistically indistinguishable from 0.1 at the 5% level for each of the three accounting gures

per share.

[Table 2 here]

Table 3 reports corresponding results for earnings per share in 1994 for the sample as a

whole, as well as for the analyst coverage and non-coverage subsamples. Because earnings

management through rounding causes distributional patterns to dier for observations with

positive and negative earnings, we present the frequency of each digit separately for positive

and negative EPS in panel A and panel B respectively. Again, results for 1994 are representative

of the sample as a whole, and equivalent analyses for all other calendar years in the sample are

available from the authors on request.

The rst row of each panel of Table 3 presents the average frequency of each number in the

rst post-decimal digit of EPS reported in 1994. T-statistics for the null hypothesis that the

frequency is equal to 0.1 are reported in parentheses. Considering the sample as a whole, the

distribution for positive earnings displayed in Panel A diers dramatically from the uniform,

and indicates a signicant under-representation of numbers two, three and especially four. In

sharp contrast, the numbers ve through nine, and zero, are signicantly over-represented. Thispattern is entirely consistent with our earnings management hypothesis.

Dividing the sample into rms with and without analyst coverage demonstrates that quadropho-

bia is particularly pronounced in rms with analyst coverage. Analyst forecasts of earnings per

share are an important benchmark, and the desire to meet or exceed this benchmark can pro-

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vide powerful incentives for managements to engage in rounding behavior. 4 Firms with analyst

coverage report a four in the rst post-decimal digit of EPS in cents in only 7.36% of observa-

tions, whereas rms without analyst coverage also under-represent the number four, but do so

less dramatically, reporting a four in 8.45% of observations. The last row of Panel A in Table

3 reports the t-statistic for the dierence in frequencies between the analyst and non-analyst

samples, and indicates that the dierence in the incidence of the number four between rms

with and without coverage is signicant at the 1% level. Similar patterns are also observed in

connection with the numbers two and three.

The negative EPS data reported in Panel B are less dramatic but entirely consistent with

the earnings management hypothesis. The degree of under and over-representation is neither

as statistically signicant nor as large as observed among observations with positive EPS. The

number ve is now under-represented, as predicted. The dierence in under-representation of

that number between the analyst and non-analyst subsamples is statistically signicant at the

1% level, again suggesting that analyst coverage is related to earnings management through

rounding. The number four is, as predicted, over-represented, although the dierence between

the analyst and non-analyst subsamples is insignicant.

To be sure, the data presented in Table 3 suggest a variety of patterns unrelated to quadropho-

bia. The over-representation of the number zero in rms with positive earnings is signicant

and powerful, but is statistically indistinguishable in the analyst and non-analyst subsamples.

There is also a particularly powerful tropism to the number nine, the most over-represented

number in the sample, and rms with analyst coverage are more likely to report a nine than

rms without coverage. The incidence of the number ve among rms with positive earnings,which one might hypothesize should be most frequently over-represented because it is the eas-

iest number to reach for purposes of rounding, does not appear to be particularly signicant,4The literature indicates that the stock market heavily punishes companies for missing earnings expectations

and rewards rms for exceeding expectations. See, e.g., Kasznik and McNichols (1999), Bartov, Givoly, andHayn (2002), and Bhojraj et al. (2009).

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especially in comparison with the over-representation of other numbers.

Each of these and related observations can support additional hypotheses. The following

analysis, however, focuses exclusively on the incidence of the number four in the rst post-

decimal digit of companies with positive earnings. We reserve for further research an exploration

of the factors that inuence the distribution of other numbers in the rst post-decimal digit of

positive EPS or the distribution of numbers in negative EPS data.

[Table 3 here]

4.2. Rounding in basic and diluted EPS

Further evidence that managements employ rounding as a form of earnings management

arises in connection with an analysis of EPS data surrounding adoption of FAS 128 in 1997.

Prior to adoption of FAS 128, companies were required to report primary EPS, which included

the dilutive eect of certain stock-based awards only if such inclusion diluted EPS by at least 3%

(the materiality threshold). In addition, companies that exceeded the materiality threshold

were also required to report fully diluted EPS, which included all potentially dilutive securities.Because primary EPS included a certain amount of dilution, reporting primary and fully diluted

EPS disclosed only a partial range of dilution to readers of nancial statements. Moreover, there

was evidence that the complex calculation of primary EPS was not fully understood and not

always consistently applied by reporting companies (see Statement of Financial Accounting

Standards No. 128). For these reasons, FAS 128 replaced primary EPS with basic EPS (the

number that excludes any potential dilution from the calculation of EPS) and required dual

representation of basic and diluted EPS on the income statement for all companies regardless

of capital structure.

Prior to adoption of FAS 128, primary EPS was likely to be the main measure used by

analysts and other readers of nancial statements because a large percentage of reporting

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companies did not have to report fully diluted EPS. On the other hand, after the adoption of

FAS 128, analysts are likely to focus on diluted EPS because it is more informative to investors

than basic EPS. 5 It is exceedingly unlikely that managements are able simultaneously to round

basic and diluted EPS unless they coincide. Managements interested in earnings management

through rounding should therefore display quadrophobia in primary EPS prior to the 1997

adoption of FAS 128 and thereafter shift to quadrophobia in diluted EPS. Fig. 2(a) and 2(b)

are consistent with this hypothesis. As before, the solid lines correspond to the frequency of

number 4 in the rst post-decimal digit of positive EPS expressed in cents, and the dotted lines

correspond to 95% condence intervals around 0.1.

Fig. 2(a) demonstrates that the frequency of the number four in diluted EPS after 1997 was

as low as in primary EPS before 1997, while the frequency of four in basic EPS is substantially

higher than in diluted EPS. This is consistent with the hypothesis that after the adoption of

FAS 128 most rms shifted their focus to rounding of diluted EPS. Fig. 2(b) provides additional

supporting evidence. It considers the sample of observations where the basic and diluted EPS

gures dier from each other by at least 0.3 cents. For these observations we nd no evidence

of post-1997 rounding in basic EPS and signicant evidence of rounding in diluted EPS. These

data suggest that rounding in basic EPS observed in Fig. 2(a) is caused by the subsample of

observations where the two gures coincide.

The data thus indicate that FAS 128 caused a shift in rounding behavior from primary EPS

to diluted EPS. Our analysis, therefore, measures rounding using primary EPS for all years

prior to 1997 and diluted EPS thereafter, including the data already presented in Section 3.

[Figure 2 here]5Some respondents to the EPS Prospectus noted that that they did not nd basic EPS to be a useful statistic

and thought that users would focus only on diluted EPS (see Statement of Financial Accounting Standards No.128). See also Jennings, LeClere and Thompson (1997), who provide some evidence that diluted EPS is a moreuseful EPS measure than basic EPS.

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5. Quadrophobia and analyst coverage

The univariate analyses conducted to this point suggest that analyst coverage is correlated

with quadrophobia. A more rigorous multivariate test of this hypothesis requires identication

of additional explanatory variables that can also be correlated with quadrophobia. We nd

that even after correcting for several additional explanatory variables that are signicantly

correlated with quadrophobia in an economically rational manner, analyst coverage remains the

most signicant explanatory factor. We then extend the analysis by examining the incidence

of quadrophobia at companies that gain and lose analyst coverage. We nd that the addition

of analyst coverage increases quadrophobia and the elimination of analyst coverage decreasesquadrophobia, even after adjusting for other explanatory variables. This treatment eect is

consistent with but cannot conclusively establish that analyst coverage causes and is not merely

correlated with quadrophobia. We further rene the analysis to respond to the observation that

reporting companies and analysts often focus on pro-forma EPS, rather than on GAAP EPS.

Because I/B/E/S data do not track denitions of pro-forma EPS relied upon by companies and

analysts, we focus on the subset of rms for which we can demonstrate that analyst forecasts

track GAAP EPS. The data indicate that analyst coverage has a particularly powerful impact

on this subset of observations, and that the relation between analyst coverage and quadrophobia

could well be stronger than indicated by our analysis because of noise introduced by this pro-

forma eect. Put another way, in addition to rounding in GAAP EPS, there could be additional

rounding in pro-forma EPS, but available data do not allow us to measure this form of rounding.

5.1. Company characteristics and rounding behavior

Several factors other than analyst coverage can explain quadrophobia. A companys size

could, for example, inuence rounding behavior. If smaller issuers do not expect much of a

following in the market, then they would expect little benet from the markets perception of an

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increase of one cent in earnings per share and would have little incentive to engage in strategic

rounding. At the other extreme, if larger companies are more intensely scrutinized by auditors

and regulators, they may nd it harder to apply accounting discretion. Larger companies also

have more shares outstanding and would therefore have to identify a larger aggregate amount

of earnings over which to exercise discretion in order to increase EPS by a tenth of a cent. We

therefore specify a model that tests for a non-monotonic relation between rounding behavior

and company size.

Firms with better growth opportunities have a greater incentive to engage in rounding

behavior because they are more likely to have potential interest in raising new capital from

public markets. Incentives to round EPS gures are stronger when the absolute value of EPS

is small, because a one cent increase in earnings is more important for a company if its EPS is

otherwise two cents than if it is two dollars. Further, based on preliminary ndings in Table 3,

we expect analyst coverage to be an important determinant of quadrophobia, and hypothesize

that the closer earnings are to the consensus analyst forecast, the greater the incentive to engage

in strategic rounding. Put another way, the incentive to engage in quadrophobia is greater if

earnings are a penny or two away from the consensus forecast than if they are so far below or

above the forecast that an additional penny has little marginal eect on market expectations.

We apply probit analysis to test these hypotheses. For robustness checks, we replicated the

analysis with logit analysis and obtained essentially identical results. The dependent binary

variable is set to one if the rst post-decimal digit in EPS reported in cents is four and zero

otherwise. A negative coecient on an explanatory variable thus implies that fours are less

common (i.e., quadrophobia), and quadrophobia is more pronounced as the explanatory variableincreases.

Our proxy for rm size (SIZE) is the logarithm of total assets. In unreported results we also

consider the logarithms of market capitalization and of sales as alternative proxies and obtain

similar results. Given our discussion above, we include both linear and quadratic terms of rm

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size in our set of explanatory variables. We use market-to-book ratio (M/B) as our proxy for

the rms growth opportunities. Market-to-book ratio is calculated as the sum of total assets

and market value of equity minus the book value of equity divided by total assets, measured

as of the end of the quarter for which earnings are announced. For robustness checks, we also

considered the price-to-earnings ratio as a proxy for growth opportunities and obtained similar

results. The variable EPS is the companys earnings per share. Each continuous variable is

winsorized at 1% and 99% to mitigate the inuence of outliers. The binary variable ANALYST

is set to one if the consensus analyst forecast is available for the corresponding rm-quarter

observation and zero otherwise. The binary variable MEET is set equal to one if the dierence

between the consensus forecast and actual EPS reported by I/B/E/S is less than two cents and

is set to zero otherwise. Finally, we include year xed eects to control for time trends, if any,

in earnings management behavior.

Results of the estimation for the sample as a whole are presented in the rst panel of Table

4. Coecients are reported in the rst column, t-statistics are reported in parentheses, and the

corresponding marginal eects are reported in the second column to the right of the coecients.

Most variables have expected signs and are highly signicant. The coecients on the two

size variables indicate that there is a non-linear U-shaped relation between rm size and the

frequency of the digit four, with larger and smaller companies less likely to engage in rounding

behavior. Higher market-to-book ratios are also positively correlated with the incidence of

quadrophobia, as previously suggested. EPS is positively correlated with the frequency of

the number four, which implies that companies with low earnings per share have stronger

incentives to engage in rounding than companies with high earnings per share. ANALYST isboth statistically and economically signicant with a marginal eect equal to -0.017. In other

words, the frequency of the number four for companies with analyst coverage is on average

lower by almost 0.02 than for companies without analyst coverage. This result is consistent

with the univariate analysis of Table 3 (showing a frequency for the number four in the rst

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post-decimal digit of 0.074 for the sample with analyst coverage and of 0.085 for the sample

without analyst coverage). Finally, the coecients on the year xed eects, which we do not

report for the sake of brevity, are consistent with the pattern in Fig. 1. In particular, the

incidence of quadrophobia is especially pronounced in the mid-1990s but declines signicantly

starting in 2001. Section 7 examines this time trend in greater depth and provides additional

explanations for the observed variation of rounding behavior over time.

Given that analyst coverage is an important covariate of quadrophobia, we repeat the analy-

sis on the two subsamples that we considered in the previous section: the rst subsample con-

tains rm-quarter observations for which a corresponding consensus analyst forecast is available

in I/B/E/S, and the second sample consists of remaining observations.

Results for the sample without analyst coverage, reported in the last panel of Table 4, are

close to those of the sample as a whole: size is non-monotonically related to the frequency of

rounding, and companies with higher market-to-book ratios and smaller EPS are more likely

to engage in rounding.

For the sample with analyst coverage we include the MEET variable. Results of the estima-

tion are presented in the second panel of Table 4. The scale of EPS remains positively correlated

with the frequency of four and market-to-book ratio remains negatively correlated but becomes

insignicant. SIZE, however, has a dierent eect in this subsample. The frequency of rounding

is no longer a U-shaped function of size. If the regression specication includes both linear and

quadratic terms, both are insignicant (specication (2) of the panel), but if the specication

includes the linear term alone the correlation is positive and signicant. This nding suggests

that larger companies with analyst following are less likely to engage in strategic rounding. Itis consistent with our earlier observation in connection with the analysis of Table 1 that com-

panies covered by analysts are, on average, signicantly larger than companies without analyst

following. Small companies, which gave rise to a negative relation between size and frequency

of rounding in the entire sample, are less likely to receive analyst coverage and hence, only the

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positive eect remains. Finally, the variable MEET is highly signicant, both statistically and

economically. The marginal probability of MEET is -0.02, suggesting that the frequency of four

in rm-quarter observations where reported EPS are close to the consensus analyst forecast is

0.02 lower than on the sample average.

[Table 4 here]

We examine the relation between quadrophobia and analyst expectations in greater depth in

Fig. 3. The horizontal axis measures the dierence between reported EPS and the consensus an-

alyst forecast, each expressed in cents per share. The histogram presents the average frequency

of the number four in the rst post-decimal EPS digit for all observations with a given dierencebetween reported and consensus EPS. The dotted line denotes the lower bound of the 95% con-

dence interval around 0.1. Fig. 3 conrms the results of our multivariate tests: quadrophobia

is especially pronounced when the gure reported by the company is close to analyst expecta-

tions. The frequency of four is as low as 0.065 for situations in which analyst forecasts equal

reported EPS. As the dierence between the consensus forecast and reported EPS increases,

regardless of whether the company misses or exceeds expectations, the frequency of rounding

declines. Evidently, the incentive to engage in quadrophobia seems particularly powerful when

the result of the rounding causes the rms to meet or come close to analyst expectations.

[Figure 3 here]

5.2. Initiation and cessation of analyst coverage

The multivariate analysis conducted to this point establishes a statistically signicant corre-lation between quadrophobia and analyst coverage, but does not establish that analyst cover-

age causes quadrophobia. The data do, however, permit a more sophisticated treatment eect

analysis that provides at least partial support for the hypothesis that the observed relation

represents causation and not mere correlation.

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To implement this analysis we observe that most companies in our sample were not followed

by analysts over the entire trading period. Some rst received analyst coverage after several

years of trading, while others lost coverage at a certain point in time. The introduction of

analyst coverage is analogous to treating a patient with an active compound. Cessation

of analyst coverage is analogous to withdrawing treatment. The decision to initiate analyst

coverage is, however, endogenous and could be triggered by the same unobservable factors that

provide incentives to engage in rounding (e.g. investor interest in the company). Nevertheless,

if for any given company, after correcting for the explanatory variables already described, the

probability of quadrophobia is higher during the period of analyst coverage than it is for the

same company before gaining or after losing coverage, then the data support a nding of

causality, although that nding cannot be conclusive because of the potential omitted variable

problem.

We therefore consider the patterns of quadrophobia related to the institution and cessation

of coverage. Approximately 80% of companies with analyst coverage have a reporting history

that precedes the initiation of analyst coverage. The average pre-coverage history is ten years

long. In contrast, it is rarer for analysts to drop rms from coverage in our sample, and only

20% of companies that have coverage wind up losing it. The average post-coverage history for

these companies is two years long.

We extend the analysis by introducing two additional binary variables. BeforeCov equals

one if the rm-year observation belongs to a period before analyst coverage was initiated and

AfterCov equals one if the rm-year observation belongs to a period after analysts dropped

coverage. A rm-year observation with both binary variables equal to zero corresponds to theperiod when the rm was followed by analysts. The hypothesis that analyst coverage causes

quadrophobia implies a positive relation between the frequency of the number four and both

binary variables.

Panel 1 of Table 5 presents results of a probit regression of a binary variable set to one if four

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is the rst post-decimal EPS digit on a constant and the two analyst coverage binary variables.

T-statistics are reported in parentheses and marginal probabilities are reported to the right of

coecients. Both BeforeCov and AfterCov are positive and highly signicant with marginal

eects of 0.01 and 0.02 respectively. The average frequency of digit four in EPS reported by

the company thus decreases by 0.01 after the initiation of analyst coverage and increases by

0.02 after coverage is dropped. With an average frequency of digit four in a random sample

of 0.1, the dierence of 0.01 is economically large, consistent with the hypothesis that analyst

following causes quadrophobia.

It is possible, however, that changes in company characteristics could co-determine analyst

coverage and quadrophobia. To separate the eect of analyst coverage from other potential

determinants of rounding behavior, and to address the potential bias resulting from these

omitted variables, we include two sets of additional control variables in our regressions. In

particular, following the literature on the determinants of analyst coverage (see, e.g., Bhushan,

1989; McNichols and OBrien, 2001), we include rm size, performance (measured by the

market-to-book ratio), price, and leverage as control variables.

Results of these estimates are reported in panels 2 and 3 of Table 5. Panel 2 introduces

rm size and market-to-book ratio, and panel 3 adds price and leverage to the set of control

variables. Each continuous variable is winsorized at 1% and 99% to mitigate the inuence

of outliers. The eect of analyst coverage remains as powerful in the presence of either set of

control variables: coecients on both BeforeCov and AfterCov are positive and signicant with

marginal eects of 0.01 and 0.02, as previously. The data thus further support the conclusion

that analyst coverage causes quadrophobia and is not merely correlated with it.

[Table 5 here]

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5.3. The pro-forma eect

The analysis has to this point proceeded on the assumption that companies and analysts

target EPS data prepared in accordance with GAAP. It is, however, common knowledge thatanalysts issue forecasts based only on recurring income, excluding one-time gains and losses, and

occasionally on other non-GAAP measures of performance. The resulting EPS gure is called

pro-forma EPS or street EPS. Companies often encourage analysts to consider non-GAAP

EPS, and would then have an incentive to target pro-forma rather than GAAP EPS gures to

meet analyst expectations (see Doyle and Soliman (2002) for evidence and discussion). SEC

regulations clearly permit reporting of non-GAAP EPS and only since 2003 have required a

reconciliation of these data to GAAP (Conditions for Using non-GAAP Financial Measures,

January 22, 2003, SEC 2003).

To the extent that companies target non-GAAP EPS measures, quadrophobia should be

apparent in the non-GAAP metrics and not in the GAAP measures that are the focus of our

analysis. It follows that our ndings to this stage are likely conservative and understate the

prevalence of quadrophobia among all publicly reporting companies.

The data necessary to examine the extent of rounding in pro forma EPS are not readily

available. I/B/E/S provides EPS data reported by the company adjusting it to the method

used by the majority of analysts (data item ACTUAL). However, these pro-forma EPS data

are already rounded to the nearest cent and therefore do not support a calculation of the rst

post-decimal digit before rounding. Nevertheless, we can test the hypothesis that companies

target pro forma rather than GAAP EPS using our data on GAAP EPS. We begin with the

observation that it is generally dicult simultaneously to round up GAAP and pro forma EPS

when they are suciently dierent from each other. Therefore, if we can identify a subset of

observations for which we expect pro forma estimates to be suciently close to GAAP EPS, we

would then expect to nd a higher degree of quadrophobia in those data than for observations

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where there is cause to believe that pro forma EPS dier materially from GAAP data.

We accordingly divide our sample of observations with analyst coverage into two subsamples.

The rst consists of all rm-quarter observations where actual EPS reported by I/B/E/S (pro

forma EPS) coincides with EPS calculated from Compustat (GAAP EPS) when rounded to the

nearest cent. This subsample captures observations for which we hypothesize analyst forecasts

were tied to GAAP estimates. The second includes rm-quarter observations for which actual

EPS reported by I/B/E/S diers from the rounded GAAP EPS. This subsample captures

observations for which we hypothesize analyst forecasts were tied to non-GAAP pro forma

estimates. Our results indicate that rounding in the GAAP consistent subsample is much

stronger than in the second subsample where companies and analysts are likely targeting pro

forma estimates that dier from GAAP. The average frequencies of number four in the two

subsamples over the entire sample period are 0.058 and 0.086 respectively. The dierence is

economically and statistically signicant.

6. Persistence

The evidence established to this point indicates that rounding behavior is pervasive and

follows an economically rational pattern. It does not, however, distinguish between the pos-

sibility that quadrophobia is driven by a subset of rms that repeatedly round EPS, and the

possibility that rounding behavior occurs randomly among rms with certain characteristics.

In other words, is rounding persistent?

Persistence suggests that quadrophobia in any given period should be most pronounced

among rms with a history of quadrophobia in prior periods, thereby implying positive auto-

correlation in quadrophobia. To test this hypothesis we analyze each rm-quarter observation

and examine the rst post-decimal EPS digit reported by each rm in the quarters prior to the

current quarter under examination. Based on that analysis we construct a quadrophobia score,

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Q it , which measures the extent of rounding by rm i prior to quarter t. More specically, Q it

is small if there were few fours in the rst post-decimal digit in prior quarters, suggesting that

a history of rounding behavior is more likely. If quadrophobia is persistent then Q it should

be positively correlated with the frequency of digit four in EPS reported by rm i in quarters

subsequent to quarter t.

We construct several quadrophobia scores, each distinguished by the number of previous

quarters that enter the calculation. More precisely, Q (N )it is a binary variable set equal to one

if there was at least one four in quarters (t; t 1 : : : t N + 1) , and zero otherwise. Higher

values of N indicate that more past quarters are included in the score. We therefore expect

that scores with higher values of N will have greater predictive power. We only consider only

those rm-quarter observations for which EPS in all N consecutive quarters are available. Our

sample size therefore declines with N.

Table 6 presents results of univariate tests for Q (N )it ; N = 1 ; 2; 5; 10; 20 and 40. For each N

we divide the sample into two subsamples corresponding to values of Q (N )it being zero and one.

For each subsample we calculate the average frequency of number four in the rst post-decimal

digit of EPS reported in quarters t + 1 ; t + 2 ; t + 3 , calculated separately for each quarter.

Results for the subsample with Q (N )it = 0 are reported in the rst row and results for Q(N )it = 1

are reported in the second row. The third row presents t-statistics for the null hypothesis that

the frequencies in the two subsamples are equal.

If quadrophobia is persistent then rms that have historically not reported fours in the

rst post-decimal digit should continue not to report fours. Table 6 strongly conrms this

hypothesis: the frequency in the second row is consistently higher than in the rst row andthat dierence is always highly signicant. Predictive power declines slightly as we predict

further into the future but remains strong. In unreported results we performed the analysis for

ten future quarters and obtained similar results.

As expected, predictive power of Q (N )it increases with N: the dierence in frequencies between

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the two subsamples increases from approximately 0.01 for N = 1 ; 2 and 5 (the past ve quarters)

to about 0.02 as we consider the horizon of ve or ten years ( N = 20 and N = 40 respectively).

The absence of fours over the horizon of ten years is relatively strong evidence of earnings

management and the conditional frequency of four for the subsample of companies with Q (40)it =

0 is as low as 0.056. Put more starkly, if a company has for ten years failed to report a four

in the rst post-decimal digit of its GAAP EPS data, there is only slightly better than a 5%

chance that it will report a four in that digit in any of the next three quarters.

We also repeat the analysis on two subsamples corresponding to companies that never re-

ceived analyst coverage and those with coverage at some point in our sample. The frequen-

cies in the analyst subsample are lower than in the non-analyst subsample for both values of

quadrophobia scores, consistent with our previous results. Moreover, past rounding behavior

appears to be a stronger predictor of future rounding behavior in the analyst subsample than

in the non-analyst subsample.

These analyses do not, however, consider the possibility that persistence as measured in

these data merely reects stability of EPS data rather than earnings management through

rounding. In particular, if a rms earnings are stable over time, then persistence will simply

reect that stability even if quadrophobia is absent. More precisely, if the dierence in EPS

between two subsequent quarters is smaller than 0.1 cents, then the rst post-decimal digit in

these EPS gures is likely to be the same, leading to a positive autocorrelation in the frequency

of number four.

To test for this possibility, we search for all pairs of consecutive quarters with the same

post-decimal digit of reported EPS. These observations are rare and constitute less than 5% of the sample. We exclude these pairs from our sample and repeat the analysis on the remaining

observations. We also consider alternative denitions of persistence in EPS and exclude ob-

servations with a dierence in consecutive EPS of less than 0.05 or 0.1 cents per share. The

results remain unchanged, conrming that the positive correlation is driven by persistence of

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rounding behavior and not persistence in the levels of EPS.

We also perform robustness checks using several other denitions of quadrophobia scores.

First, we extend the denition to combine digits three and four: a Q-score equals zero if and

only if there were no threes or fours in the rst post-decimal digits of EPS in the past N quarters.

Our results are robust to this specication but the predictive power of this quadrophobia score

is slightly weaker. Second, instead of considering a binary variable, we introduce a variable

that takes N + 1 values: ~Q (N )it equals k if there are exactly k fours in the past N quarters,

k 2 f 0; 1; :::N g. However, because it is rare that there are fours in more than two consecutive

quarters in our sample, the sample size for values of ~Q (N )it greater than two is very small and the

predictive power of this test is low. The results of these robustness checks are available from the

authors upon request, and all are consistent with the nding of persistence in quadrophobia.

[Table 6 here]

Quadrophobia is also persistent even after controlling for other determinants of rounding

behavior. Table 7 presents the results of multivariate probit regressions that repeat the analysis

presented in Table 4 with an additional binary explanatory variable Q (5)t1

that equals one if

there was at least one four in the rst post-decimal digit of EPS reported by rm i in quarters

(t 1; t 2; : : : t 5), and equals zero otherwise. The results are similar if we consider other

quadrophobia scores.

All explanatory variables retain the same signs and approximately the same marginal eects

as before. The quadrophobia score is, in addition, highly statistically and economically signif-

icant with a marginal probability of 0.012, consistent with our univariate analysis. Repeating

the analysis on the sample of companies with and without analyst coverage conrms the hy-

pothesis that persistence is stronger for the analyst subsample: the marginal eects of Q (5)t 1 are

0.015 and 0.008 respectively.

[Table 7 here]

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7. Eect of audit oversight and regulation on rounding

As discussed in Section 2, the academic literature is mixed as to whether earnings manage-

ment has decreased in the wake of heightened regulatory and auditor scrutiny resulting from

the Enron and WorldCom frauds and the passage of the Sarbanes Oxley Act of 2002.

This section addresses that question from two distinct perspectives. First, we observe that

quarterly nancial statements are typically unaudited, but that the annual statement is typi-

cally subject to a full audit. 6 If the audit process itself reduces the ability to exercise managerial

discretion through rounding, then the incidence of quadrophobia in annual data should be lower

than in quarterly data. Second, we test for time trends in the data to determine whether theincidence of quadrophobia has declined in a statistically measurable manner since 2002.

The analysis to this point has examined the frequency of rounding in quarterly EPS data.

Unlike quarterly nancial statements, annual statements are audited and hence, comparing the

extent of quadrophobia in quarterly and annual gures allows us to test whether and to what

extent, if any, the audit process itself deters managers from engaging in rounding behavior. We

also analyze separately each of the individual quarters: if the presence of auditors constrains

rounding, then we expect quadrophobia to be less pronounced in nancial statements for the

fourth quarter, which are prepared at the time of the annual audit.

Fig. 4(a) illustrates the incidence of rounding in each of the four quarters. Fig. 4(b)

repeats the analysis for annual EPS gures. We performed identical analyses on the analyst

coverage subsample and obtained similar results. For brevity, we present only the results for

the entire sample. As before, the solid lines represent the frequency of number four in the rst

post-decimal digit and the dotted lines correspond to the upper and lower bounds of the 95%

condence intervals around 0.1.6Quarterly statements are subject to a review pursuant to 17 C.F.R. part 210. 10-01(d) that does not involve

a level of scrutiny approaching that employed in an audit. Auditors are not held liable for alleged misstatementsin quarterly nancials that have been reviewed but not audited. (Lattanzio v. Deloitte & Touche LLP 476 F.3d147 (2d Cir. 2007))

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Fig. 4(a) demonstrates that the extent of rounding is similar across the rst three scal

quarters, but is much weaker in the fourth quarter. Specically, the average frequency of

number four over the entire sample period is 0.081 for each of the rst three quarters, while for

the fourth quarter it equals 0.088. We perform a formal statistical test in panel A of Table 8,

which presents the dierence in the average frequency of number four for each pair of quarters.

T-statistics for the test of the null hypothesis that the frequency is the same for each pair of

quarters are reported in parentheses. In addition to considering the entire sample period, we

also divide our sample into two subsamples, corresponding to the pre-SOX (1980-2001) and

post-SOX (2002-2006) periods.

[Figure 4 here]

Table 8 conrms our initial observation that quadrophobia is equally prevalent among the

rst three quarters but is signicantly less prevalent in fourth quarter nancial statements. The

dierence in the incidence of quadrophobia between the fourth quarter and any of the rst three

quarters over the entire sample period is about 0.006, which is economically and statistically

signicant. In contrast, the dierence between any of the rst three quarters is less than 0.001,

which is not signicant. This nding holds both in the pre-SOX and the post-SOX periods.

While this pattern is consistent with the fact that fourth quarter nancial statements are

prepared at the time of the audit unlike other quarterly statements, two other explanations

of this fourth quarter eect are also possible. First, as discussed in Section 5, our analysis of

GAAP measures is likely to underestimate the true prevalence of quadrophobia if companies are

targeting non-GAAP pro-forma estimates used by analysts. This conservative bias is expectedto be the strongest for the fourth quarter data, where the incidence of non-recurring items is

the highest (see Burgstahler, Jiambalvo, and Shevlin (1999) for related evidence). Consistent

with this hypothesis, Bradshaw and Sloan (2002) nd that the dierence between GAAP and

pro-forma EPS is greater for the fourth quarter than for each of the rst three quarters.

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Another plausible explanation for the lower prevalence of quadrophobia in the fourth quarter

is a substitution eect. As demonstrated below, there is considerable evidence of rounding in

annual EPS. Because earnings in the four quarters sum up to annual earnings, it will generally

be dicult for the company simultaneously to round up both the fourth quarter and the annual

EPS. If investors place more emphasis on annual rather than fourth quarter gures, companies

may choose to forgo the opportunity to round up to report a higher fourth quarter EPS gure

and instead choose to round up to report a higher annual EPS gure.

Fig. 4(b) indicates that quadrophobia was common in annual data in the pre-SOX period:

the average frequency of the number four over the 1980-2001 period is 0.08. Moreover, as panel

B of Table 8 demonstrates, the extent of rounding in annual data in the pre-SOX period is

comparable to the extent of rounding in the rst three quarters. Although the frequency of

four in annual EPS is slightly higher than in each of the rst three quarters, this dierence

is signicant only for the rst quarter. These ndings suggest that although annual nancial

statements are audited, the presence of auditors does not have a strong mitigating eect on

rounding behavior prior to the passage of SOX.

The pattern, however, is rather dierent in the post-SOX period. Fig. 4(b) suggests that

the frequency of the number four in annual EPS data is statistically indistinguishable from

10% starting in 2001. In contrast, SOX does not appear to have a similarly strong eect on

rounding of quarterly gures: although the frequency of number four has increased, especially

in the second quarter, it remains signicantly smaller than 10% for the rst three quarters.

[Table 8 here]

To examine the eects of audit oversight in greater depth, Table 9 repeats our multivariate

analysis on the sample that combines quarterly and annual data. As previously, the dependent

binary variable is set to one if the rst post-decimal digit in EPS reported in cents is four

and set to zero otherwise. To account for the dierence between annual and quarterly data, we

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introduce several binary variables. The variable QTR i is set to one if and only if the observation

corresponds to quarterly data in scal quarter i, and the variable Annual is set to one if and only

if the observation corresponds to annual data. Finally, we introduce the variable Post-SOX,

which is set to one if the observation belongs to the period 2002-2006 and set to zero otherwise.

Model (1) of Table 9 compares the prevalence of quadrophobia in quarterly and annual

data and in dierent quarters. We exclude the variable QTR 1, making it the base group, and

include the dummy variables for the last three scal quarters and annual data. The results

support our univariate analysis: the frequency of number four in the second and third quarters

is statistically indistinguishable from that in the rst quarter. The likelihood ratio test that

both coecients are jointly equal to zero is not rejected: the corresponding p-value is 0.38. In

contrast, the extent of rounding in the fourth quarter and annual data is signicantly smaller

than in the rst quarter. As is clear from Fig. 4 and Table 8, however, the dierence between

annual data and the rst three quarters is substantial only in the last years of our sample.

This is further conrmed in model (3) of Table 9, which shows that the variable Annual is no

longer signicant when we include an interaction term between Annual and Post-SOX as an

additional explanatory variable.

The eect of the audit process on the incidence of rounding behavior thus appears to be

historically contingent. Prior to 2001, quadrophobia was as prevalent in the audited annual EPS

data as in the unaudited quarterly data. Since 2001, however, quadrophobia is less prevalent in

the audited annual data than in the unaudited quarterly data, and remains signicant only in

quarterly data. The data therefore are consistent with the hypothesis that the audit process has

recently become more eective in inhibiting earnings management that leads to quadrophobia.As we discuss below, however, this evidence is also consistent with the substitution of pro-

forma earnings management for GAAP earnings management in annual data, and it is hard to

disentangle these eects.

We next examine the eect of the passage of SOX on the incidence of rounding by including

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the variable Post-SOX in model (2) of Table 9. Consistent with the univariate results, Post-

SOX is statistically signicant with a marginal eect of about 0.001. To account for the

potential dierence in the eect of SOX on rounding in quarterly and annual data, we introduce

interaction terms between Post-SOX and quarterly and annual dummy variables in model (3).

Post-SOX remains positive and signicant, implying that the passage of SOX had a mitigating

eect on quadrophobia in both annual and individual quarter nancial statements. Interaction

terms between Post-SOX and individual quarter indicators are not signicant and the likelihood

ratio test that that they are jointly equal to zero is not rejected (the p-value is 0.94). In

other words, the decline in rounding following the passage of SOX was very similar in all four

scal quarters. The interaction term between Post-SOX and Annual is positive and highly

signicant, indicating that the decline in rounding was much stronger in audited annual data,

consistent with the univariate results. Moreover, as noted above, the variable Annual is no

longer signicant, suggesting that the dierence between quarterly and annual data is only

apparent after 2001.

To test whether the passage of SOX changed the relation between the incidence of rounding

and company characteristics such as size, market to book ratio, and the presence of analyst

coverage, model (4) of Table 9 compares the regressions in the pre and post-SOX period by

including interaction terms between Post-SOX and each independent variable of model (1). Our

results indicate that the relation between the incidence of rounding and company size, market

to book ratio, and analyst coverage did not change in the post-SOX period: the interaction

terms between size, M/B, Analyst and PostSOX are not signicantly dierent from zero, both

individually and jointly. The interaction term between EPS and post-SOX is negative andsignicant, suggesting that the decline in rounding in the post-SOX period has been more

pronounced in companies with small values of EPS.

[Table 9 here]

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Viewed in isolation, these results appear consistent with the hypothesis that passage of

SOX in 2002 and heightened enforcement following the Enron and WorldCom frauds had a

moderating inuence on rounding behavior. The eect appears to be particularly pronounced

in annual data, consistent with heightened auditor scrutiny in the post-SOX period. However,

quadrophobia remains pervasive in unaudited quarterly statements, implying that this form of

earnings management has not completely disappeared. These ndings could, however, over-

simplify the situation. As is apparent from a simple visual inspection of Fig. 4(b), there has

been a gradual decrease in the incidence of quadrophobia in the annual data since the 1990s,

well before adoption of the Sarbanes Oxley Act, or the Enron and WorldCom frauds. There

also appears to be no sudden or sustained change in the trend, post Sarbanes Oxley. A similar,

though less pronounced, trend is observed in quarterly data (see Fig. 1).

The substitution eect between GAAP and pro-forma EPS previously discussed provides a

plausible explanation for this trend in a manner unrelated to Sarbanes Oxley. If managers have

been gradually moving away from targeting GAAP EPS to targeting pro- forma gures, we

could expect a gradual increase in the frequency of number four in GAAP EPS over time. This

hypothesis is consistent with the evidence in Bradshaw and Sloan (2002), who examine earnings

announcements disclosures and show an increasing emphasis of managers on pro forma measures

over GAAP measures over the last twenty years. The fact that the trend is more pronounced in

annual rather than quarterly data is further consistent with the observation that non-recurring

items giving rise to a divergence between GAAP and pro-forma nancials are more likely to

appear in annual than quarterly data.

Publicly available data sets do not track pro-forma EPS, and the structure of the I/B/E/Sdatabase makes it impossible to discern with precision when analyst forecasts are of GAAP or

pro-forma EPS. Rigorous formal tests of the pro-forma substitution hypothesis are therefore

not feasible. We can, however, examine the subset of observations where there is a cause to

believe that GAAP and pro-forma measures are suciently close that no substitution eect

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is at work. We divide the annual analyst coverage sample into two subsamples, depending on

whether pro-forma EPS reported by I/B/E/S and GAAP EPS reported by Compustat coincide

when rounded to the nearest cent (see Section 5 for additional details). We nd no decline in

the frequency of rounding in annual data in the GAAP consistent subsample. The average

frequency of the number four during the 2002-2006 period is 7.88% for this subsample and the

frequency in statistically dierent from 0.1 in each year. This nding is inconsistent with the

hypothesis that the passage of SOX has had a modulating eect on rounding behavior in annual

data, and that the presence of auditors has recently become more eective in constraining this

form of earnings management. However, this split of the data is likely to introduce noise into

our estimates and does not allow us to draw any rm conclusions.

We therefore suggest that the evidence regarding the eect of auditors and SOX on rounding

behavior is mixed and inconclusive. While heightened enforcement in the post-SOX period could

have reduced the incidence of quadrophobia, particularly in audited annual data, we are unable

formally to disentangle this eect from the pre-existing trend in the data that is consistent with

the substitution of pro-forma earnings management for GAAP earnings management.

8. Does quadrophobia anticipate alleged violations of ac-

counting standards and of federal securities laws?

Quadrophobia can reect the exercise of legitimate accounting discretion, or it can result

from a violation of accounting standards. It can also be the consequence of legitimate, but

aggressive, discretion correlated with other conduct that violates accounting standards. Simply

demonstrating that quadrophobia is pervasive and persistent is therefore insucient to establish

its relation to questionable accounting practices.

We therefore examine the serial correlation between quadrophobia and three measures of

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accounting misconduct: restatements, SEC enforcement actions alleging accounting violations,

and class action securities fraud litigation. We show that persistent quadrophobia presages

future restatements and accounting-driven enforcement actions, but that the relation between

quadrophobia and class action securities fraud litigation is more complex. These ndings are

consistent with the view that quadrophobia, even if it reects the legitimate exercise of ac-

counting judgment, tends to be practiced by management teams that are more likely to engage

in potentially problematic forms of accounting conduct.

The restatement data are provided by Glass Lewis and Co. and cover 4010 restatements

led between 2003 and 2007.7 The Accounting and Auditing Enforcement Release data are

from the SEC website and cover 134 enforcement actions instituted between 2004 and 2007.

The class action securities fraud litigation data are from Woodru Sawyer and Co. and cover

2941 lawsuits spanning the period 1981-2007. The restatement data set denes the period that

was restated, the AAER data set denes the fraud period, and the securities class action data

set denes the period over which the alleged fraud was uncorrected in the market. We use these

data to dene the alleged violation period for all three types of data.

To examine whether quadrophobia anticipates potentially problematic accounting practices

we conduct three separate sets of probit regressions in which the dependent variable measures

the incidence of restatements, SEC enforcement actions, or class action securities fraud liti-

gation. For each type of event, the dependent variable for a rm-quarter pair (i,t) is set to

zero if the rm never experiences this event after quarter t, or if the alleged violation period

for this event starts later than N years after quarter t, and set to one if the alleged violation

period starts within N years from quarter t. We consider several values of N in the denitionof the dependent variable. By increasing N we improve our ability to dierentiate between

rms that engage in potentially problematic accounting practices and those that do not. For7The Glass Lewis and Co. data set includes restatements led to correct accounting errors as dened by

Accounting Principles Board (APB) opinion 20 and does not include restatements for changes in accountingprinciples and restatements led to make minor wording changes or typographical errors.

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example, when N equals one, rms that engage in misreporting two years from the current

period and rms that never engage in misreporting are classied identically. However, when

N equals ve, a rm has to avoid an allegation of misreporting for ve years before it can be

classied identically with rms that are never subject to such allegations. The predictive power

of all explanatory variables should therefore increase with N.

For each N we restrict our sample to rm-quarter

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