I am going through the lab portion on variable subset selection methods in An Introduction to Statistical Learning. On page 249 it says:
Finally, we perform best subset selection on the full data set, and select the best ten-variable model. It is important that we make use of the full data set in order to obtain more accurate coefficient estimates. Note that we perform best subset selection on the full data set and select the best ten- variable model, rather than simply using the variables that were obtained from the training set, because the best ten-variable model on the full data set may differ from the corresponding model on the training set.
I am a bit confused on this point because it would seem that we should use the validation approach to select the particular variables that give us the lowest test error, then estimate the final model using those variables on the full data set. Here, it seems that we are only using the validation approach to select the number of variables that should be in the final model, and then using the full data set to select the particular variables.
What is the advantage of selecting the particular variables using the full data set? I understand why we would want to estimate the coefficients for the final model using as many data points as possible. But isn't the particular variables to include (and not just the number of variables) the kind of hyper-parameters that we would want to tune using a validation approach?