Let ¯b∗

i denote the estimated standardized regression coefficient when xi is treated as the response variable and y as an explanatory variable, controlling for the same set of other variables. Then, ¯b∗

i need not equal b∗

i. The squared partial correlation between y and xi, which is symmetric, equals b∗

i

¯b

∗

i .

(a) Explain why the partial correlation must fall between b∗

i and ¯b∗

i . (Note: When a =

√

bc, a is said to be the geometric average of b and c.)

(b) Even though b∗

i does not necessarily fall between −1 and +1, explain why b∗

i

¯b

∗

i cannot exceed 1.