11.61.* Adjusted R2 is defined as R2 adj

= 1 − s2 s2y

, where s2 is the estimated conditional variance and s2y is the sample variance of y, both of which are unbiased. This relates to ordinary R2 by R2 adj

= R2 −

p n − (p + 1)

(1 − R2).

(a) Suppose R2 = 0.339 for a model with p = 2 explanatory variables, as in Table 11.5.

Find R2 adj when n = 10, 40 (as in the text example), and 1000. Show that R2 adj approaches R2 in value as n increases.

(b) Show that R2 adj < 0 when R2 < p/(n − 1). This is undesirable, and R2 adj is equated to 0 in such cases. (Also, unlike R2, R2 adj can decrease when we add an explanatory variable to a model.)