Question

# 4. Now assume that the true relationship is: \text { Income }_{i}=\beta_{0}+\beta_{1} \text { Education }_{i}+\beta_{2} \text { Ability }_{i}+\mu_{i}(2) Plug equation (1) into equation (2) to get \text { Income }_{i}=\beta_{0}+\beta_{1} \text { Education }_{i}+\beta_{2}\left[\delta_{0}+\delta_{1} \text { Education }_{i}+e_{i}\right]+\mu_{i}(3) The final step is to rearrange (3) so that the form looks like: \text { Income }_{i}=a+b * \text { Education }_{i}+c \text { a. }[5 \text { Points }] \text { Report to me the value of } b \text { in terms of } \beta_{1}, \beta_{2}, \text { and } \delta_{1} \text { . } ^^20Income^^20_i=\vec{\beta}+\bar{\beta}^^20Education^^20_i+\tilde{\mu}_i \text { b. [8 Points] Assume that } b=\vec{\beta}_{1} \text { in the relationship } How does the correlation between ability and education, and the correlation between ability ^^20Note:^^20\hat{\delta_1}=\frac{\text{ Cov( } \text{ Bducation Abulity })}{\text{ Var } \text{ (Rducation) }} \text { and income affected the biasness of } \bar{\beta}_{1} \text { ? When is } \bar{\beta}_{1}=\beta_{1} \text { true? }  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8  Fig: 9  Fig: 10  Fig: 11  Fig: 12