Is the relationship between experience and wages constant over one's lifetime? We will investigate this question using a quadratic model. The data file cps5_small contains 1200 observations on hourly wage rates, experience, and other variables from the March 2013 Current Population Survey (CPS). 

a. Create the variable EXPER30 \(=\) EXPER -30 . Describe this variable. When is it positive, negative or zero?

b. Estimate by least squares the quadratic model \(W A G E=\gamma_{1}+\gamma_{2}(E X P E R 30)^{2}+e\). Test the null hypothesis that \(\gamma_{2}=0\) against the alternative \(\gamma_{2} eq 0\) at the \(1 \%\) level of significance. Is there a statistically significant quadratic relationship between expected WAGE and EXPER30?

c. Create a plot of the fitted value \(\widehat{W A G E}=\hat{\gamma}_{1}+\hat{\gamma}_{2}(E X P E R 30)^{2}\), on the \(y\)-axis, versus EXPER30, on the \(x\)-axis. Up to the value \(E X P E R 30=0\) is the slope of the plot constant, or is it increasing, or decreasing? Up to the value EXPER30 \(=0\) is the function increasing at an increasing rate or increasing at a decreasing rate?

d. If \(y=a+b x^{2}\) then \(d y / d x=2 b x\). Using this result, calculate the estimated slope of the fitted function \(\widehat{W A G E}=\hat{\gamma}_{1}+\hat{\gamma}_{2}(E X P E R 30)^{2}\), when \(E X P E R=0\), when \(E X P E R=10\), and when \(E X P E R=20\).

e. Calculate the \(t\)-statistic for the null hypothesis that the slope of the function is zero, \(H_{0}: 2 \gamma_{2}\) \(E X P E R 30=0\), when \(E X P E R=0\), when \(E X P E R=10\), and when \(E X P E R=20\).