European Accounting Review, Volume 23, Issue 4, 2014, pp. 559-591.

**Abstract: **The methods proposed by Burgstahler and Dichev (1997) and Bollen and Pool (2009) to test for earnings management have been used extensively in the literature. This paper proposes a more general test procedure based on kernel density estimation using a kernel bandwidth selected by a bootstrap test. Its main advantage over prior methods is the construction of a kernel density estimate that cannot be globally distinguished from the empirical distribution, which greatly reduces an upward bias in test statistics found in earlier results. It 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 U.S. firms over the period 1976-2010. Results confirm earlier findings of discontinuities in the whole sample of earnings and earnings changes, but not in all subsamples. There is a large drop in loss aversion after 2002, which cannot be detected in earnings changes. Discontinuities in analysts’ forecast errors found by earlier research are more likely to be caused by rounding errors than by earnings management.

Published in the European Accounting Review. If you cannot access the article, please let me know.

A working paper version is available at SSRN (latest version: August 2013).

## Short presentation of method and key findings

The main motivation why we need to move from Burgstahler and Dichev’s method of testing discontinuities to a more refined approach is presented in this document: Earnings_management_KDE.pdf.

## Estimation algorithms

The current version of the test procedure is implemented in R and Stata. The Stata code is available from the journal as a supplemental file, the R version can be downloaded here. The R version is about two or three times faster than the Stata one. If you are investigating earnings management or plan to implement the algorithm, please get in touch. Please also let me know if you find any errors or if you develop the algorithm further.

## Interpretation of failures to converge (AKA: commonly found problems)

If the data generating process for the data to be tested for a discontinuity does not produce a distribution that has a step discontinuity but some other kind of discontinuity, the algorithm may fail to converge. The algorithm typically fails to converge for “distributions” like Durtschi & Easton’s (2005) unscaled EPS, in which individual observations are not drawn from the same underlying distribution. In the case of earnings per share, this produces a spike at zero (and an asymptotic discontinuity in theory), not a step discontinuity as in typical earnings management. This behaviour of the algorithm is also discussed in the paper. Non-convergence may thus be interpreted as a signal that the distribution tested is degenerate and does not have a step discontinuity.

The same non-convergence behaviour can be observed, if for some reason a large point mass exists somewhere in the region to be tested (typically anywhere on the real line). For example, numerical values as indicators for missing values or a large number of zeros caused by rounding or other reasons may lead to non-convergence. This is again caused by the underlying distribution not having a step discontinuity.

Last updated: May 2, 2015.