Two-Stage Regression Analysis and Biased Estimates in Accounting Research: An Application of the Frisch-Waugh-Lovell Theorem

We analyze a procedure common in empirical accounting and finance research where researchers use ordinary least squares to decompose a dependent variable into its predicted and residual components and use the residuals as the dependent variable in a second regression. This two-step procedure is used to examine determinants of constructs such as discretionary accruals, real activities management, discretionary book-tax differences, and abnormal 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 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.