as you increase the number of variables why does R2 increase and the F-stat and R2 adjusted either increase or decrease?
Because you can keep adding variables even though the relationship is spurious. It all comes down to the relationship of the performance of the S&P to the growth of a tree theory. If you have a fundamental factor model that tries to predict the return of the S&P and you add a coefficient related to the growth of a tree, R^2 will increase.
If I ask you whether your salary is related to your level of education, you will probably say yes, somewhat (a weak R2). Now add years of experience. You would say yes, that help explain salary. Add gender. Yeah, that adds to it too. Add person’s height, weight, looks, etc, even seemingly unrelated ones, like color of your car or your knowledge of Greek history…they will probably increase R2, but are they all justified? I’m not sure about the other statement about F-stat and adjusted R2 decreasing.
adj-R2 = R2*(n-1)/(n-k-1) so it adjusts for the number of independent variables (k). addition of the variable could either cause R^2 to increase or decrease. So given that the additional variable definitely decreases the denominator, but the numerator portion could either increase or decrease - the direction of movement of adj-R^2 is not clear. It could either go up or down.
cpk, adjusted R2 cannot go up. Look at the correct formula: Adj R2 = 1-[(n-1)/(n-k-1)] (1-R2)
Dreary Thanks for correcting the formula. I only have Schweser Notes of 2009. I see on Pg 187 in the middle of the page… they make the following statement: 1. when there is more than 1 indep. var - adj_R^2 <= R^2 2. Adding an indep. variable may increase R^2. If R^2 increased - Ra^2 (given the (1-r^2) in the formula) would decrease. But if R^2 decreased - Ra^2 might increase. If R^2 gets low enough -> Ra^2 may be negative or even 0. So Ra^2 might increase or decrease.
"If R^2 gets low enough -> Ra^2 may be negative or even 0. " If it is negative it is treated as zero , in the footnotes in bottom of page in CFAI text