schooling earnings 0 12.5572 1 12.37059 2 12.28345 3 12.53942 4 13.60204 5 13.69685 6 14.46392 7 15.48792 8 16.58223 9 17.75331 10 18.31583 11 18.93581 12 20.69255 13 22.40492 14 23.55711 15 25.12731 16 30.9133 17 30.25502 18 31.35148 19 31.24208 20 34.62525

The file "SchoolEarnings.csv" contains data on earnings and schooling. The data is from a 1991 study by Angrist and Krueger entitled "Does Compulsory School Attendance Affect Schooling and Earnings?"published in the Quarterly Journal of Economics. Angrist and Krueger obtained their data from the Integrated Public Use Microdata Series from the 1980 U.S. Census. The study specifically focused on men born between 1930 and 1939, so they were 41-50 years old at the time of data collection. Each row corresponds to a large group of men with the same level of schooling. The variable "schooling" indicates the years of schooling for each group. 12 corresponds with high school completion, and 16 corresponds with college completion. The variable "earnings" indicates the average annual earnings (in 1980 U.S. dollars) for the men in each group.

Question: Now run a second-order polynomial regression (i.e. include schooling and schooling squared as predictors). Report the coefficients. Repeat the exercise with the trend line from the previous exercise, but this time instead of choosing the "linear" trendline, choose the "exponential" trendline. This will plot the predicted values from the second-order polynomial regression to the scatterplot. Include the plot with your answer.

Do these predictions meaningfully differ from the predictions you got using the linear regression?