Suppose You Are Interested in Testing Whether Athletes Receive Better

Suppose you are interested in testing whether athletes receive better grades in high school but you are considered that the decision to compete in athletics is endogenous because athletes tend to have higher levels of motivation which results in the performing better at school work but cannot be measured and included as an independent variable in the regression.After collecting data for a sample of 32,416 high school seniors,you estimate the following instrumental variable model

Second-Stage:

$\begin{array} { l } \widehat { \text { GPA} } _ { i } =& 1.253 + &0.29 \text { Athletıc} ~~ \widehat { part}icıpation _{i} +& 0.02 \text { Income } _ { i } - &0.076 \text { Male } _ { i } \\&( 0.034 ) &( 0.08 )& (0.002)& (1.382) \\\end{array}$

First-Stage:

$\begin{array} { l } \text { Athletıc} ~~ \widehat { part}icıpation _{i}=&0.03+&.213 \text { Height }_{i}+&0.006 \text { Income }_{i}+&0.15 \text { Male }_{i} \\&(0.002)&(0.014)&(0.001)&(0.05) \\\end{array}$

a)Which of the two models is the regression model you are interested in estimating? Why is it not appropriate to estimate this model using OLS?
b)What two attributes does height need to have to make it a valid instrument? Do you think these hold in this circumstance?
c)Explain the intuition behind the first stage model.
d)Comment on the results above.
e)Why is it necessary to have such a large sample size when estimating an instrumental variable model? Explain.