I'm confused about the last question. I will quickly go through the beginning parts, so you can skip to question 3 at the very bottom as it may not be necessary.
- Estimate linear regression for log wages on education ,experience, and experience squared. $$ \begin{array}{cccccc} & \text { intercept } & \text { educ } & \text { exper } & \text { expersquared } & \hat{\sigma} \\ \text { estimate } & 4.0163 & 0.0923 & 0.0791 & -0.0020 & .410 \\ \text { s.e. } & (0.2223) & (0.0076) & (0.0249) & (0.0009) & \end{array} $$
- Predict the effect on average log wage of increasing everybody's education level by one year if regression model is $$ \log (\text { wage })_{\mathrm{i}}=\beta_{0}+\beta_{1} \times \operatorname{educ}_{\mathrm{i}}+\beta_{2} \times \operatorname{exper}_{\mathrm{i}}+\beta_{3} \times \operatorname{exper}_{\mathrm{i}}^{2}+\varepsilon_{\mathrm{i}} $$ Then the effect of increasing education level of individual i by one year is $$ \theta_{i}=\beta_{1}-\beta_{2}-\beta_{3} \cdot\left(2 \cdot \operatorname{exper}_{\mathrm{i}}-1\right) $$ because one year additional education implies one year less work experience.
The effect we are interested in is $$ \theta=\beta_{1}-\beta_{2}-\beta_{3} \cdot(2 \cdot \overline{\operatorname{exper}}-1) $$ And we can calculate this effect by running the regression with a redefined set of covariates, as follows: $$ \beta_{1}=\theta+\beta_{2}+\beta_{3} \cdot(2 \cdot \overline{\operatorname{exper}}-1) $$ we can write the regression function as $$ \begin{array}{l}\log (\text { earnings })_{i}=\beta_{0}+\left(\theta+\beta_{2}+\beta_{3} \cdot(2 \cdot \overline{\operatorname{exper}}-1)\right) \times \operatorname{educ}_{i} \\ \quad+\beta_{2} \times \operatorname{exper}_{i}+\beta_{3} \times \operatorname{exper}_{i}^{2}+\varepsilon_{i}\end{array} $$ or $$ \begin{array}{c}\log (\text { earnings })_{i}=\beta_{0}+\theta \times \operatorname{educ}_{i}+\beta_{2} \times\left(\operatorname{exper}_{i}+\operatorname{educ}_{i}\right) \\ \quad+\beta_{3} \times\left(\operatorname{exper}_{i}^{2}+(2 \cdot \overline{\operatorname{exper}}-1) \cdot \operatorname{educ}_{i}\right)+\varepsilon_{i}\end{array} $$ I am omitting the Table of estimates from running this regression
Question 3. Predict the effect on the average level of earnings of the following policy: increase the level of education for those who currently have earnings below 12 years of education to 12, and leave the level of education for others unchanged. Hint: Use the formula for the mean of the lognormal distribution.
The parameter of interest is the expected difference in average earnings where we change the education level of those with less than twelve years of education to exactly twelve years of education. The expected level of earnings (use the formula for the mean of a log-normal distribution) for someone with education level educ and experience exper is $$ \exp \left(\beta_{0}+\beta_{1} \cdot \text { educ }+\beta_{2} \cdot \operatorname{exper}+\beta_{3} \cdot \operatorname{exper}^{2}+\sigma^{2} / 2\right) $$
Hence the parameter of interest is $$ \begin{array}{c}\theta=\frac{1}{N} \sum_{i \mid \operatorname{educ}_{i}<12}\left(\exp \left(\beta_{0}+12 \beta_{1}+\beta_{2}\left(\operatorname{exper}_{i}-12+\text { educ }_{i}\right)+\beta_{3}\left(\operatorname{exper}_{i}-12+\mathrm{educ}_{i}\right)^{2}+\frac{\sigma^{2}}{2}\right)\right. \\ \left.\quad-\exp \left(\beta_{0}+\beta_{1} \mathrm{educ}_{i}+\beta_{2} \operatorname{exper}_{i}+\beta_{3} \operatorname{exper}_{i}^{2}+\frac{\sigma^{2}}{2}\right)\right)\end{array} $$
My question is that I don't understand the first top part of this last expression. Where does the (exper $_{i}-12+$ educ $_{i}$) term come from multiplying each beta?
The mean of a lognormal distribution $\log X \sim \text{Normal}(\mu, \sigma^2)$ is $\exp(\mu+\frac 1 2 \sigma^2)$ so for $$\log(wage_i)\sim \text{Normal}(\beta_0+\beta_1 educ+\beta_2 exper+\beta_3 exper^2, \sigma^2)$$
the mean of $wage$ is $\exp(\beta_0+\beta_1 educ+\beta_2 exper+\beta_3 exper^2 + \frac 1 2 \sigma^2)$. Notice this is actually the mean of $wage$ and not $\log (wage)$. Now you ask about $exper-12+educ$ or $exper-(12-educ)$. $12-educ$ is the number of years of education left for twelve years of education for a person who has less than $12$ years of education. So to increase the education years to $12$, the work experience must be subtracted by this much. So the net effect is on the mean of $wage$ is
$$\frac 1 N\sum_{i|educ_i<12}^N\left[\exp\left\{\beta_0+\beta_1(12)+\beta_2(exper-(12-educ))+\beta_3(exper-12+educ)^2+\frac 12 \sigma^2\right\}-\exp\left\{\beta_0+\beta_1educ+\beta_2 exper+\beta_3 exper^2+\frac 1 2\sigma^2\right\}\right]$$