A model that allows major league baseball player salary to differ by position is
$$
\log (\text {salary})
$$ $$
=\beta_{0}+\beta_{1} \text { years }+\beta_{2} \text { gamesyr }+\beta_{3} \text { bavg }+\beta_{4}
$$
$$
+\beta_{5} \text { rbisyr }+\beta_{6} \text { runsyr }+\beta_{\text { fldperc }}+\beta_{\text { s allstar }}
$$
$$
\begin{array}{l}{+\beta_{9} \text { frstbase }+\beta_{10} s c n d b a s e+\beta_{11} \text { thrdbase }+\beta_{12} \text { shrtstop }} \\ {+\beta_{13} \text { catcher }+u}\end{array}
$$
where outfield is the base group.
$$
\begin{array}{l}{\text { (i) State the null hypothesis that, controlling for other factors, catchers and outfielders earn, on }} \\ {\text { average, the same amount. Test this hypothesis using the data in MLB } 1 \text { and comment on the }} \\ {\text { size of the estimated salary differential. }}\end{array}
$$
$$
\begin{array}{l}{\text { (ii) State and test the null hypothesis that there is no difference in average salary across positions, }} \\ {\text { once other factors have been controlled for. }} \\ {\text { (iii) Are the results from parts (i) and (ii) consistent? If not, explain what is happening. }}\end{array}
$$

Answer

i. $H_0$:$ \beta_{13}$=0. Salary differential is large. ii. $H_0: \beta_9$ to $\beta_13 =0$. Can't reject at 10 % level. iii. Roughly consistent.

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Video Transcript

another hypotheses that controlling for other factors catchers and outfielders earn. On average, the same amount can be written as follows. Beta 13 is the coefficient of catcher dummy. So when you regret the whole equation, you can look at the T statistic produced by their statistical software, right? The regression results give beta 13 hat roughly 0.254 with a standard errors. Earth 0.31 The T start is about 1.94 and the P value against a two sided alternative. These now hypotheses, um, against by the 13th. Uh huh, Not equal zero. This is a two solid alternative. The P value is just over 0.5 Therefore, we would reject the null hypotheses at about the 5% level controlling for the performance and experience variables. The estimated salary differential between catchers and outfielders is large on the order of 100 times E to the power of point 251 minus one, which is roughly 29% Yeah, In part two, we have a joint non hypothesis with a nine equal beta 10 up to beta 13 equals zero. The F statistic with five and 339 degrees of freedom is 1.78 and it's P value is the bow 0.117 We are unable to reject the null at the 10% level. Our three part one and two are roughly consistent. The evidence against the joint now in Part two, is weaker because we're testing along with the marginally significant catcher, several others in significant variables, especially third base and shortstop, which has an absolute T statistic well below one.

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