An Improved Test for Earnings Management Using Kernel Density Estimation

^†An earlier version of this paper was circulated and presented under the title ‘Identifying Discontinuities in Distributions of Earnings by Kernel Density Estimation’. Paper accepted by Steven Young. Additional materials are available in an online Supplement at the journal's Taylor and Francis website.

Abstract

This paper describes improvements on methods developed by Burgstahler and Dichev (1997, Earnings management to avoid earnings decreases and losses, Journal of Accounting and Economics, 24(1), pp. 99–126) and Bollen and Pool (2009, Do hedge fund managers misreport returns? Evidence from the pooled distribution, Journal of Finance, 64(5), pp. 2257–2288) to test for earnings management by identifying discontinuities in distributions of scaled earnings or earnings forecast errors. While existing methods use preselected bandwidths for kernel density estimation and histogram construction, the proposed test procedure addresses the key problem of bandwidth selection by using a bootstrap test to endogenise the selection step. The main advantage offered by the bootstrap procedure over prior methods is that it provides a reference distribution that cannot be globally distinguished from the empirical distribution rather than assuming a correct reference distribution. This procedure limits the researcher's degrees of freedom and offers a simple procedure to find and test a local discontinuity. I apply the bootstrap density estimation to earnings, earnings changes, and earnings forecast errors in US firms over the period 1976–2010. Significance levels found in earlier studies are greatly reduced, often to insignificant values. Discontinuities cannot be detected in analysts’ forecast errors, while such findings of discontinuities in earlier research can be explained by a simple rounding mechanism. Earnings data show a large drop in loss aversion after 2003 that cannot be detected in changes of earnings.

Acknowledgements

The author would like to thank Bernhard Gegenfurtner for sparking the idea and Lucile Faurel and participants at the 2010 American Accounting Association's meeting, the 2012 British Accounting and Finance Association's meeting and the 2012 European Accounting Association's meeting for valuable comments. Helpful suggestions by the editor, Steven Young, and two anonymous referees, which considerably improved this paper, are gratefully acknowledged.

Notes

^† An earlier version of this paper was circulated and presented under the title ‘Identifying Discontinuities in Distributions of Earnings by Kernel Density Estimation’. Paper accepted by Steven Young. Additional materials are available in an online Supplement at the journal's Taylor and Francis website.

¹ More precisely, a discontinuity in the sense of this paper is a point at which a density function is discontinuous, jumping from a region of lower density to one of higher density or vice versa.

² For a cross-section of studies discussing empirical evidence for earnings management and alternative explanations for discontinuities in standardised earnings, earnings per share, earnings increases, or analysts’ forecast errors, see Burgstahler and Chuk (2012), Beatty et al. (2002), Burgstahler and Eames (2003), Beaver et al. (2003, 2007), Cohen and Lys (2003), Dechow et al. (1995, 2003), Glaum et al. (2004), Brown and Caylor (2005); Coulton et al. (2005), Durtschi and Easton (2005); Burgstahler and Eames (2006), Gore et al. (2007), Pinnuck and Lillis (2007), Talha et al. (2008), Tung et al. (2008), and Charoenwonga and Jiraporn (2009).

³ I refer to bootstrapping as a procedure that draws repeated samples with replacement (resampling) from an empirical distribution to establish the uncertainty of a statistic of this distribution. In this paper, bootstrapping is used to estimate the uncertainty about the empirical cumulative distribution at the point of a suspected discontinuity.

⁴ ‘Earnings and changes of earnings’ both refer to scaled, or standardised, earnings. To enhance readability, I omit ‘scaled’ when referring to earnings scaled by market capitalisation.

⁵ The empirical distribution and kernel density estimate shown in the right panel correspond to Figure 2.

⁶ Bollen and Pool (2009, p. 2270) use a smoothed bootstrap to test the robustness of their findings. Although it shares the name, their bootstrapping approach should not be confused with the one proposed here. While they employ bootstrapping as part of a discontinuity test after assuming a reference distribution (by preselecting a bandwidth), I use it to construct a plausible reference distribution.

⁷ This can be shown analytically for some cases, such as a uniform distribution and uniform kernel. If a distribution contains several and possibly smaller discontinuities, a graphical representation of the difference between empirical distribution and integrated KDE may reveal more potential locations for a discontinuity test.

⁸ Deflating by market capitalisation is standard practice in the literature (Burgstahler and Dichev, 1997; Dechow et al., 2003; Beaver et al., 2007). It is necessary to account for firm size, as size is related to the distribution of net income and to the extent of potential earnings management. Results are robust to scaling by book value of equity or total assets. For an analysis using undeflated net income, see Section 4.3.

⁹ Excluding zero earnings follows the literature, see Burgstahler and Dichev (1997), Bollen and Pool (2009), and Brown et al. (2010).

¹⁰ For a counterargument of why scaling is necessary and a detailed discussion of Durschi and Easton's (2005, 2009) findings, see Burgstahler and Chuk (2012).

¹¹ A prior version of this paper included results for discontinuities in standardised earnings of German firms. In these data, significance peaked in 2000 and has been declining since, but is still detectable in 2008. If a Bonferroni correction for multiple hypothesis tests is applied, discontinuities during the periods 1996–1998 and 2006–2008 are insignificant. This pattern resembles stock market movements over the period 1991–2008, which is supported by evidence provided by Günther et al. (2009): They suggest that neither voluntary nor mandatory adoption of International Financial Reporting Standards (IFRS) around 2005 unambiguously leads to higher earnings quality and that capital market phases are an important determinant of earnings quality.

¹² If the bootstrap KDE approach detects more discontinuities than Rule-of-thumb KDE, then this could be due to bandwidths being larger, but that does not necessarily lead to higher significance in yearly subsamples.

¹³ While the difference peaks at zero in 2001, no such anomaly can be found in 2009. Although time-specific events are hard to disentangle from other confounding time effects, it might be worthwhile to investigate whether the apparent reduction in earnings management after 2003 could be due to the Sarbanes–Oxley Act of 2002.

¹⁴ Results using Burgstahler and Dichev's (1997) and Bollen and Pool's (2009) tests are similar to those reported in the respective studies. Exact test statistics are available from the author.