Prediction/Estimation With Simple Linear Models: Is It Really That Simple?

Consider the simple normal linear regression model for estimation/prediction at a new design point. When the slope parameter is not obviously nonzero, hypothesis testing and information criteria can be used for identifying the right model. We compare the performances of such methods both theoretically and empirically from different perspectives for more insight. The testing approach at the conventional size of 0.05, in spite of being the “standard approach,” performs poorly in estimation. We also found that the frequently told story “the Bayesian information criterion (BIC) is good when the true model is finite-dimensional, and the Akaike information criterion (AIC) is good when the true model is infinite-dimensional” is far from being accurate. In addition, despite some successes in the effort to go beyond the debate between AIC and BIC by adaptive model selection, it turns out that it is not possible to share the pointwise adaptation property of BIC and the minimax-rate adaptation property of AIC by any model selection method. When model selection methods have difficulty in selection, model combining is a better alternative in terms of estimation accuracy.This work was completed when the author was on leave from Iowa State University and was a New Direction Visiting Professor at the Institute for Mathematics and its Applications (IMA) at the University of Minnesota. The fundings from both IMA and ISU are greatly appreciated. The work was also partly supported by NSF CAREER grant DMS0094323. The author thanks Xiaotong Shen and Hannes Leeb for very helpful discussions. The paper also benefited from the questions and comments from the participants at the statistics seminars the author gave at the University of Minnesota and Duke University. The author is very grateful to the anonymous reviewers and the co-editor Benedikt Pötscher for carefully reading earlier versions of the paper, bringing my attention to several closely related previous and current results, and making many very valuable suggestions, which significantly improved the paper in both content and presentation.