Incorrect Inferences When Using Residuals as Dependent Variables

• Date: 01 June 2018
• Author: WEI CHEN,PAUL HRIBAR,SAMUEL MELESSA
• DOI: http://doi.org/10.1111/1475-679X.12195
• Published date: 01 June 2018

DOI: 10.1111/1475-679X.12195

Journal of Accounting Research

Vol. 56 No. 3 June 2018

Printed in U.S.A.

Incorrect Inferences When Using

Residuals as Dependent Variables

WEI CHEN,

∗PAUL HRIBAR,

∗AND SAMUEL MELESSA

∗

Received 20 April 2015; accepted 17 July 2017

ABSTRACT

We analyze a procedure common in empirical accounting and finance re-

search where researchers use ordinary least squares to decompose a depen-

dent variable into its predicted and residual components and use the residu-

als as the dependent variable in a second regression. This two-step procedure

is used to examine determinants of constructs such as discretionary accru-

als, real activities management, discretionary book-tax differences, and abnor-

mal investment. We show that the typical implementation of this procedure

generates biased coefficients and standard errors that can lead to incorrect

inferences, with both Type I and Type II errors. We further show that the

magnitude of the bias in coefficients and standard errors is a function of the

correlations between model regressors. We illustrate the potential magnitude

of the bias in accounting research in four commonly used settings. Our results

indicate significant bias in many of these settings. We offer three solutions to

avoid the bias.

JEL codes: C18;G10;G30;M40;M41

Keywords: two-stage; residuals; coefficient bias; discretionary accruals; real

earnings management

∗Tippie College of Business, University of Iowa.

Accepted by Christian Leuz. We appreciate helpful comments and suggestions from

Ted Christensen, Michael Drake, Kyle Peterson, Kris Ramesh, Jacob Thomas, two anony-

mous reviewers, and workshop participants from the University of Iowa and the BYU

Accounting Research Symposium. We thank the Tippie College of Business, University

of Iowa for financial support. An online appendix to this paper can be downloaded at

http://research.chicagobooth.edu/arc/journal-of-accounting-research/online-supplements.

751

Copyright C, University of Chicago on behalf of the Accounting Research Center,2018

752 W.CHEN,P.HRIBAR,AND S.MELESSA

1. Introduction

Many studies in accounting and finance examine the determinants of the

abnormal, discretionary, or unexplained components of various variables.

Examples include discretionary accruals, unexplained audit fees, excess

compensation, discretionary book-tax differences, and abnormal stock re-

turns. The typical procedure used in these studies is to first estimate the

discretionary or unexpected component (e.g., discretionary accruals) as

the residual from an ordinary least squares (OLS) regression, where the

predicted value represents the normal, nondiscretionary, or expected com-

ponent. The unexpected component (i.e., the first-stage residual) is then

used as the dependent variable in a second-step OLS regression designed to

test hypotheses about its determinants. For example, numerous studies es-

timate discretionary accruals as the residual from the Jones [1991] model.

A second model is then estimated by regressing discretionary accruals on a

variable of interest and a set of control variables to test a hypothesis about

the cause of earnings management. Our survey of five recent years of ac-

counting research (2011 to 2015) published in Contemporary Accounting Re-

search, Journal of Accounting and Economics, Journal of Accounting Research, Re-

view of Accounting Studies, and The Accounting Review found 61 studies using

this procedure in settings such as corporate investment, tax aggressiveness,

management forecasts, and earnings management (see table 1).

Studies using this procedure typically do not include the independent

variables from the first-step regression as additional independent variables

in the second-step regression. Of the 61 studies identified in our survey,

only one study (Hsu and Pourjalali [2015]) includes the first-step indepen-

dent variables in the second-step regression. One likely reason for the omis-

sion is the belief that because the dependent variable is the residual from

a first-step regression, and therefore orthogonal to the first-step regressors,

the first-step regressors do not need to be included in the second-step re-

gression (e.g., Dehaan, Hodge, and Shevlin [2013]). In this paper, we eval-

uate this two-step regression procedure and show that when the first-step

regressors are not included in the second-step regression, the two-step pro-

cedure generates biased estimates of the second-step regressors and can

result in Type I and Type II errors. We point out that there is no economet-

ric justification for this two-step procedure and emphasize that the most

straightforward way to avoid the bias generated by the procedure is to sim-

ply estimate the model in a single regression.1

We use the Frisch–Waugh–Lovell Theorem and simulations to demon-

strate that this two-step procedure produces biased coefficients and

t-statistics. We initially consider a simple model in which the residuals from

a first-step regression are regressed on only a single variable of interest, that

is, the second-step regression contains no control variables. We show that

1In section 5 we offer two equivalent procedures that also avoid the bias.

INCORRECT INFERENCES USING RESIDUALS AS DEPENDENT VARIABLES 753

TABLE 1

Studies Using Two-Step Regression Procedureand Published Between 2011 and 2015 in Contemporary

Accounting Research, Journal of Accounting and Economics, Journal of Accounting Research, Review of

Accounting Studies, and The Accounting Review

Variable Decomposed

(Influential Studies) List of Studies

Accruals (Jones [1991]; DeFond

and Jiambalvo [1994]; Dechow,

Sloan, and Sweeney [1995])

Ali and Zhang [2015]; Chan et al. [2015]; Dhaliwal

et al. [2015]; Huang et al. [2015]; Hou et al.

[2015]; Ke, Lennox, and Xin [2015]; Baber,

Krishnan, and Zhang [2014]; Badolato,

Donelson, and Ege [2014]; Francis, Pinnuck, and

Watanabe [2014]; Franz, Hassabelnaby,and Lobo

[2014]; Kim, Mauldin, and Patro [2014]; Lennox

and Li [2014]; Ecker et al. [2013]; Francis,

Michas, and Seavey [2013]; Francis and Michas

[2013]; Wongsunwai [2013]; Dyreng, Hanlon,

and Maydew [2012]; Kim, Park, and Wier [2012];

Lennox and Li [2012]; Ayers, Ramalingegowda,

and Yeung [2011]; Cahan, Zhang, and Veenman

[2011]; Das, Kim, and Patro [2011]; McInnis and

Collins [2011]; Michas [2011]

Real activities (Roychowdhury

[2006])

Ali and Zhang [2015]; Brochet, Loumioti, and

Serafeim [2015]; Chan et al. [2015]; Chen et al.

[2015]; Franz, Hassabelnaby, and Lobo [2014];

Kim, Park, and Wier [2012]; Lee [2012]

Permanent, book-tax differences

(Frank, Lynch, and Rego

[2009])

McGuire, Wang, and Wilson [2014]; Badertscher,

Katz, and Rego [2013]; McGuire, Omer, and

Wang [2012]; Rego and Wilson [2012]

Audit fees (Hribar, Kravet, and

Wilson [2014])

Beck and Mauldin [2014]; Hribar, Kravet, and

Wilson [2014]; Dehaan, Hodge, and Shevlin

[2013]; Messier et al. [2011]

Earnings Allee and Deangelis [2015]; Causholli, Chambers,

and Payne [2014]; Ahmed, Neel, and Wang

[2013]; Jayaraman [2012]

Compensation (Core, Guay, and

Larcker [2008]a)

Kim, Mauldin, and Patro [2014]; Armstrong, Gow,

and Larcker [2013]; Chhaochharia, Kumar, and

Niessen-Ruenzi [2012]

Investment (Biddle, Hilary, and

Verdi [2009]a)

Chen, Young, and Zhuang [2013]; Chen et al.

[2011]

Tax aggressive Armstrong et al. [2015]

Change in cash Chen and Shane [2014]

Response time for investment

choices

Farrell, Goh, and White [2014]

Performance-based pay Grabner [2014]

Affiliated trading Hsu and Pourjalali [2015]

New hires Jung, Lee, and Weber [2014]

Restructuring costs Lee [2014]

Change in compensated absences Kido, Petacchi, and Weber [2012]

Management forecast error Gong, Li, and Wang [2011]

Predicted bias in managers’

forecast

Merkley, Bamber,and Christensen [2013]

Bond ratings Mansi, Maxwell, and Miller [2011]

(Continued)