The adjusted R2 was introduced in chapter 5. The usual R2 is ‘adjusted’ to impose a small penalty when a variable is added to the model. Adding a variable with any correlation to y always reduces SSE and increases the size of the usual R2. With the adjusted version, the improvement in fit may be outweighed by the penalty and it could become smaller as variables are added. The formula is:
This sometimes referred to as “R-bar squared,” (i. e., R2 ) although in gretl it is called “adjusted R-squared.” The biggest drawback of using R2 as a model selection rule is that the penalty it imposes for adding regressors is too small on average. It tends to lead to models that contain irrelevant variables. There are other model selection rules that impose larger penalties for adding regressors and two of these are considered below.