Hodges’s superefficient estimator is a shrinkage estimator that, at first sight, appears to be better than the usual estimator, the sample mean, when the sample size is large. However, this turns out to be an illusion, since the performance of this estimator is actually much worse than the usual estimator when the sample size is large. Kabaila (Econom Theory 11:537–549, 1995) described how to extend the concept of superefficiency to confidence intervals and prediction intervals. Post-model-selection, the lasso and other similar estimators, such as SCAD, are also shrinkage estimators. Kabaila (Econom Theory 11:537–549, 1995), Leeb and Pötscher (Econom Theory 21:21–59, 2005), Leeb and Pötscher (J Econom 142:201–211, 2008), and Pötscher (Sankhya A 71:1–18, 2009) have shown that a similar superefficiency phenomenon can arise for both these estimators and the confidence intervals and prediction intervals based on them. The frequentist model averaged estimators proposed by Buckland et al. (Biometrics, 53:603–618, 1997) are also shrinkage estimators. We consider the case of two nested linear regression models. Using the finite-sample results of Kabaila et al. (J Stat Plann Inference 207:10–26, 2020), we prove that both these estimators and the confidence intervals based on them display a superefficiency phenomenon and that, despite initial appearances, they are actually much poorer than the usual estimators and confidence intervals.