$($a$)$ The following model examines how the return to education $($profits from schooling$)$ and the gender pay gap have changed from $1978$ to $1985$ using the US data. The results are presented as follows $($standard errors in the parentheses$)$:

$)=(()2)=()=($

where $ln($wage$)$ is the logarithmic term of hourly nominal wage, educ is the years of education, exper refers to the years of experience and exper ${?}^2$ is its squared term. The variable $y85$ is a dummy variable equal to $1$ if the observation comes from $1985$ and $0$ if it comes from $1978,$ union is a dummy variable equal to $1$ if the person belongs to a union, and $0$ otherwise, female is the gender dummy variable equal to $1$ if the person is female otherwise equal to $0.$ Both $y85$ educ and $y85$ female are interaction variables.

$($i$)$ Interpret the return to education in $1985$ and $1978,$ and comment on your findings.

$[3]$

$($ii$)$ Interpret the gender gap, report both the approximate and exact changes. Does the gender gap increase over years?

$[3]$

$($iii$)$ Suppose you suspect that there exists a structural change across the time periods of $1978$ and $1985.$ What type of test do you plan to implement for the structural change? Explain clearly how you will carry out the test.

$[5]$

$($iv$)$ Interpret the coefficient on $y85?($Be careful here: you must account for the interaction terms $y85$educ and $y85$ female.$)$

$[2]$

$($v$)$ Let us assume that we refit the above equation and let all wages be measured in $1978$ dollars. In particular, define the real wage as rwage which equals to the wage for $1978$ and rwage equals to wage$/1\mathrm{.}65$ for $1985.$ Then use $ln($rwage$)$ in place of $ln($wage$)$ in fitting the model. Which coefficient$($s$)$ will differ from those in the above equation? Will the $R^2$ from the new regression be the same as $0\mathrm{.}426?$ Explain your answer.

$[4]$

$($vi$)$ There was a notable fall in union membership from $1978$ to $1985.$ Describe and discuss how you will test whether union participation changed from $1978$ to $1985.$

$[2]$