In a [recent thread][1], use of adjusted $R^2$ ($R^2_{adj.}$) is mentioned in the context of model selection, e.g. > The adjustment was invented as a solution to problems caused by variable selection **Questions:** 1. Is there any justification for using $R^2_{adj.}$ for model selection? 2. Does it have any optimality properties in the context of model selection? For example, AIC is an efficient criterion and BIC is a consistent one, but $R^2$ does not coincide with any of them and so makes me wonder if it can be optimal in any other sense. [1]: https://stats.stackexchange.com/questions/408249/adjusted-r2-for-model-with-only-one-independent-variable