Labor economists have extensively researched the determinants of earnings. Investment in human capital, measured in years of education, and on the job training are some of the most important explanatory variables in this research. You decide to apply earnings functions to the field of sports economics by finding the determinants for baseball pitcher salaries. You collect data on 455 pitchers for the 1998 baseball season and estimate the following equation using OLS and heteroskedasticity-robust standard errors:

\begin{aligned}\widehat { \text { Ln(Eami } ) = } & 12.45 + 0.052 \times \text { Years } + 0.00089 \times \text { Innings } + 0.0032 \times \text { Saves } \\& ( 0.08 ) ( 0.026 ) \quad \quad\quad\quad\quad\quad( 0.00020 )\quad\quad\quad\quad\quad(0.0018) \\& - 0.0085 \times \text { ERA, } R 2 = 0.45 , \text { SER } = 0.874 \\& ( 0.0168 )\end{aligned}

where Earn is annual salary in dollars, Years is number of years in the major leagues, Innings is number of innings pitched during the career before the 1998 season, Saves is number of saves during the career before the 1998 season, and ERA is the earned run average before the 1998 season.
(a)What happens to earnings when the pitcher stays in the league for one additional year? Compare the salaries of two relievers, one with 10 more saves than the other. What effect does pitching 100 more innings have on the salary of the pitcher? What effect does reducing his ERA by 1.5? Do the signs correspond to your expectations? Explain.
(b)Are the individual coefficients statistically significant? Indicate the level of significance you used and the type of alternative hypothesis you considered.
(c)Although you are quite impressed with the fit of the regression, someone suggests that you should include the square of years and innings as additional explanatory variables. Your results change as follows:

\begin{array} { l } \widehat { \operatorname { Ln } \left( \text { Eam } _ { i } \right) } = 12.15 + 0.160 \times \text { Years } + 0.00268 \times \text { Innings } + 0.0063 \times \text { Saves } \\\begin{array} { l l l } \quad\quad\quad\quad\quad\quad( 0.05 ) & ( 0.039 )\quad\quad\quad \quad ( 0.00030 ) \quad\quad\quad\quad\quad ( 0.0010 )\end{array} \\- 0.0584 \times E R A - 0.0165 \times \text { Years } 2 - 0.00000045 \times \text { Innings } ^2 \\\begin{array} { l l l } \quad\quad\quad( 0.0165 ) \quad\quad\quad\quad ( 0.0026 )\quad \quad ( 0.00000012 )\end{array} \\\\R ^ { 2 } = 0.69 , S E R = 0.666 \\\end{array}

What is her reasoning? Are the coefficients of the quadratic terms statistically significant? Are they meaningful?
(d)Calculate the effect of moving from two to three years, as opposed to from 12 to 13 years.
(e)You also decide to test the specification for stability across leagues (National League and American League)by including a dummy variable for the National League and allowing the intercept and all slopes to differ. The resulting F-statistic for restricting all coefficients that involve the National League dummy variable to zero, is 0.40. Compare this to the relevant critical value from the table and decide whether or not these additional variables should be included.

For the Polynomial Regression Model

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