Alright, so I'm taking an applied econometrics course this summer and I'm getting a little stumped on an assignment where we have to compare four regression models for the same set of data. They are, in dependent-independent variable form:

a) linear-linear
b) log-linear
c) log-log
d) log-quadratic - ie. ln(Y)=beta_0+beta_1(X)+beta_2(X^2)+...u

I'm using EViews to do the regression calculations, and it's all OLS regressions. I don't have any problems computing and interpreting the regression outputs, but comparing the regression models is getting the better of me.

I know that model d, the log-quadratic, is not a worthwhile model since the coefficients beta_1 and beta_2 are both not statistically significant.

However, between the other three models a, b, and c, there's nearly no difference in them (that I can tell at least). All the variable coefficients are significant at any level (p-val=0), and the adjusted R-squared values extremely similar (0.199500, 0.200342, 0.200453).

Doing a scatterplot graph of the independent and dependent variable of question doesn't help much since the independent variable is age, and only goes from 25-35, while the dependent variable is average hourly earnings. The scatterplot ends up looking like a bunch of lines based on age (since Age is a discrete variable, not totally continuous), and there's no discernible trend that I can see, except for maybe a very small uptick of high hourly wages and ages over 32 or 33.

So, my question to the brightest minds of BG is if I've got linear-linear, log-linear, and log-log models, all telling me basically the same thing, how the hell do I compare them?

A second, less important question is about polynomial regressions. Say for a quadratic regression, why do you include the variable multiple times? ie Y=beta_0+beta_1(X)+beta_2(X^2)+beta_3(X^3)...+u where X is the same dependent variable. Why not just include the variable of question once at the exponent you think it is?

Thank you for any responses, I know there's some math folks out there but I'm sure someone's done stats :D