$1.$ a$)$ Why do you disagree with Jack$$s comments about the uselessness of the regression due to the low R$-$squared?

$($b$)$ Can you think of a situation in which a useless regression has a high R$-$squared?

$($c$)$ There are techniques to determine the validity of a regression model$$in particular, whether the relationship is linear and the error terms display equal variance $($homoskedasticity$).$ Does the regression in Table $1$ violate either of these two assumptions? Justify your answer

Jack is soon convinced that a low R$-$squared does not render his regression useless and begins bombarding you with questions. Use Table $1$ in the readings to answer Questions $2$ through $6.$

$2.($a$)$ Estimate the excess return $($RET$)$ of the funds that Bob and Putney currently manage. Assume that Princeton$$s average composite SAT score is $1355,$ while Ohio State$$s is $1042.$ Between Bob and Putney, who is expected to obtain higher returns at their current funds, and by how much?

$($b$)$ Between Bob and Putney, who is expected to obtain higher returns if hired by AMBTPM, and by how much?

$3.($a$)$ Can you prove at the $5$ percent significance level that if Bob had attended Princeton instead of Ohio State, then the return of his current fund would be greater?

$($b$)$ Can you prove at the $10$ percent level of significance that if Bob were managing a growth fund instead of a growth and income fund, then he would achieve at least $1$ percent higher average returns?

$4.($a$)$ Does the regression in Table $1$ provide strong evidence for the claim that fund managers with MBAs perform worse than managers without MBAs? What is being held constant in this comparison? Discuss.

$($b$)$ It has been suggested that fund managers without MBAs get higher expected returns because they invest in riskier stocks. If this were true, what effect would including an independent variable, Beta $($with higher values corresponding to higher levels of systematic risk in the fund$$s portfolio$),$ have on the coefficient of MBA in the regression of Table $1?$

$5.($a$)$ What is the lowest level of significance at which you can prove that the manager$$s age has a negative impact on his or her fund$$s performance holding the type of the fund, the manager$$s education, and years of experience at the fund constant?

$($b$)$ A survivorship bias is thought to be present in analyzing fund manager performance in which a younger manager$$s survival in the industry is more closely linked to his$/$her performance than an older manager$$s survival. In other words, if a new manager does not perform successfully, he or she is not tolerated in the industry for long, but a more experienced manager may be forgiven a year or two of poor performance. Would the presence of this survivorship bias dampen or exacerbate the effect seen in Part $($a$)?$

$6.($a$)$Streamline$$ the regression given in Table $1,$ that is$,$ eliminate all variables that are not significant at the $15$ percent level. Write down the new regression equation and check whether the specification satisfies the assumptions of linearity and homoskedasticity.

$($b$)$ Compare the coefficient of AGE in the new and the old regressions. What can explain the sign $($direction$)$ of the change in this estimator? Discuss

$7.($a$)$ You receive the prospectus of a growth fund started in the current year by a Princeton alum. What is the estimated RET $($excess return relative to the return of the benchmark market portfolio$)$ for this fund?

$($b$)$ Are you confident that this fund will $$beat the market$,$ that is$,$ provide a return in excess of that of the benchmark market portfolio? Which standard error do we have to use in order to answer this question?

$($c$)$ Suppose that you manage to identify a large number of growth funds started recently by Princeton graduates. By investing equally in all of these funds, how likely is it that your return will exceed that of the benchmark market portfolio by more than $1\mathrm{.}5$ percent? Which standard error did you use in your answer?

$8.$ Suppose that you gain access to a much larger sample of random observations of the same variables that you have in the current dataset. Do you expect that any of your answers to Parts $($a$)($c$)$ of Question $8$ will change, and if so$,$ how? Discuss.

$9.($a$)$ Based on the dataset, can you prove at the $5$ percent level of significance that among fund managers with the same educational background and same experience with the same fund, those managing growth and income funds are, on average, older?

$($b$)$ Using the regression developed in part a$,$ provide an $80$ percent confidence interval for the average age difference between managers who graduated from the same college in the United States and have managed a growth fund for the same number of years but differ in whether or not they have an MBA. Are the $($otherwise comparable$)$ managers with MBAs younger or older, on average? Discuss $($conjecture$)$ why this is the case.

$10.$ Based on your analysis of the case, which candidate do you support for AMBTPM$$s job opening: Bob or Putney? Discuss.