244 British Journal of Mathematical and Statistical Psychology (2011), 64, 244–258 C  2010 The British Psychological Society The British Psychological Society www.wileyonlinelibrary.com A comparison of two-stage procedures for testing least-squares coefficients under heteroscedasticity Marie Ng 1∗ and Rand R. Wilcox 2 1 Institute for Health Metrics and Evaluation, University of Washington, Seattle, USA 2 Department of Psychology, University of Southern California, Los Angeles, USA This study explores the performance of several two-stage procedures for testing ordinary least-squares (OLS) coefficients under heteroscedasticity. A test of the usual homoscedasticity assumption is carried out in the first stage of the procedure. Subsequently, a test of the regression coefficients is chosen and performed in the second stage. Three recently developed methods for detecting heteroscedasticity are examined. In addition, three heteroscedastic robust tests of OLS coefficients are considered. A major finding is that performing a test of heteroscedasticity prior to applying a heteroscedastic robust test can lead to poor control over Type I errors. 1. Introduction Consider a typical linear regression model: where X is the n × p + 1 design matrix,  = [ 0 ,  1 ,...,  p ] ′ , and  is the error term. An assumption typically made in (1) is that  is homoscedastic; in other words, Var( ) =  2 . When this assumption holds, the standard error of the ordinary least-squares (OLS) estimate of ˆ  is given by a simple formula, Var( ˆ ) = (X ′ X) −1  2 . The classic t test and F test can be applied to make statistical inferences regarding the ˆ s. However, in applied settings, the assumption of homoscedasticity is often violated. Even though the OLS estimates remain unbiased in such situations, the usual formula for estimating the standard error of ˆ  is no longer correct. Consequently, the classic test statistics can have poor control over Type I errors or low statistical power (e.g., White, 1980; Wilcox, 1996). Two general strategies have emerged to tackle the issue of heteroscedasticity. One focuses on the detection of violations of the homoscedasticity assumption. The other ∗ Correspondence should be addressed to Dr Marie Ng, Institute for Health Metrics and Evaluation, University of Washington, 2301 5th Avenue Suite 600, Seattle, WA 98121, USA (e-mail: marieng@u.washington.edu). DOI:10.1348/000711010X508683