The way Schweser describes returns-based analysis, Page 141, Book 3, it seems like both the selection return (the error term in the regression of portfolio against the index) and 1 minus the style fit (i.e., 1 minus R squared) are the same thing. It seems they both measure return due to active management. What am I missing?
A statistician would explain this better, but I’ll give it a bash: if you regress the portfolio returns on the index, the error term of the regression contains the variation in the portfolio returns that can not be explained by variation in the index returns. The implication is that it therefore contains variation which is not due to style (the index reflects style – e.g. it could be a large-cap growth index). So, it explains the variation that is due to active management. R-squared measures the goodness-of-fit of the regression. i.e. how well the index variation explains the portfolio returns variation. So, (1-R-squared) reflects the bits NOT explained by the regression/the index. In other words, the bits due to active management (not the bits due to style).
I was confused on this as well. Check out question 8 CFAI exam 2007.