Sarah performs a regression of the return on a mutual fund (y) on four predictors plus an intercept. She uses monthly returns over 105 months. Her software calculates the F statistic for the regression as F = 20.0, but then it quits working before it calculates the value of [math]R^2[/math] . While she waits on hold with the help desk, she tries to calculate [math]R^2[/math] from the F-statistic.

Determine which of the following statements about the attempted calculation is true.

• There is insufficient information, but it could be calculated if she had the value of the residual sum of squares (RSS).
• There is insufficient information, but it could be calculated if she had the value of the total sum of squares (TSS) and RSS.
• [math]R^2 = 0.44 [/math]
• [math]R^2 = 0.56 [/math]
• [math]R^2 = 0.80 [/math]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: C

Even though the formula for R² involves RSS and TSS, she just needs their ratio, which can be obtained from F.

[[math]] \begin{aligned} F &= \frac{(TSS − RSS ) / 4}{RSS / (105 − 4 − 1)} = 20 \\ \frac{TSS-RSS}{RSS} &= 20(4)/100 = 0.80 \\ \frac{TSS}{RSS} &= 1.80 \\ \frac{RSS}{TSS} &= \frac{1}{1.80} \\ R^2 &= 1- \frac{RSS}{TSS} \\ &= 1- \frac{1}{1.80}\\ &= \frac{0.80}{1.80} \\ &= 0.44 \end{aligned} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.