Evaluating the Effect of Model Misspecification on Likelihood Ratio Test Reliability and Type I Error Rates

Posted: Mar 28, 2009

• Nathan Howard
• Nora Hayes
• Paisley Wood

Abstract

The likelihood ratio test stands as one of the most fundamental and widely employed statistical procedures in scientific research, providing a principled framework for hypothesis testing across diverse domains including genetics, econometrics, psychometrics, and machine learning. Its theoretical foundation rests upon the elegant asymptotic properties derived under the assumption of correctly specified statistical models, where under the null hypothesis, the test statistic follows a chi-square distribution with degrees of freedom determined by the difference in parameter dimensionality between nested models. This mathematical elegance, however, belies a critical vulnerability: the sensitivity of LRT performance to violations of the model specification assumption. In practical applications, researchers frequently confront situations where the true data-generating process remains unknown or only partially understood, leading to inevitable model misspecification through omitted variables, incorrect distributional assumptions, inappropriate functional forms, or neglected dependencies. Despite extensive theoretical work establishing the asymptotic robustness of certain statistical procedures, the finite-sample behavior of likelihood ratio tests under misspecification remains inadequately characterized. The prevailing literature often treats misspecification as a binary phenomenon—either present or absent—while in reality, misspecification exists along a continuum of severity and manifests in diverse forms. This research addresses this gap by systemat