Do you think I should apply a transformation to my independent variables? I have done a simple linear regression on my two standardized independent variables and standardized dpendent variable. In the residual plot there is a distinct quadratic pattern left after the two explanatory variables have done their work. I feel like applying a quadratic transformation to my independents, in order to add the R squared of the residual quadratic pattern to the R squared of the model. What should I do exactly to achieve this?  


Thanks!
 A: It looks like there are two things going on:


*

*The response is not linear in the predictors (or at least not in some important predictors). 

*The spread increases as the mean increases, perhaps in proportion to the mean.
There are several things that might be done. 
a) you might consider something like a GLM; the most obvious first model to try would be a gamma GLM with log-link, but you might try an identity link and some quadratic terms.
b) you might try some monotonic transformation of $y$. The obvious one to try is the log transformation. 
c) another possible transformation that's sometimes suitable with a single predictor, or where the heteroskedasticity is mainly related to a single predictor is to divide $y$ by $x$ and regress it on $x$. This makes some sense when the conditional response is close to symmetric.
d) you could add quadratic terms in your predictors (you wouldn't just square them, you'd keep the linear terms as well; though you might want to consider orthogonal polynomials). However, this would completely fail to deal with the heteroskedasticity. If you're mostly interested in testing, ne possibility would be to use some form of heteroskedasticity consistent standard errors (/sandwich estimator), but the heteroskedasticity that's present is very simple in form, and it seems like overkill to use something with so many implicit variance parameters.
It's important to have a reasonable model for the conditional distribution (especially the mean and variance, though as noted in (d), something might be done in the case of variance); if you don't, the inferences you seek to make will not have the properties you want.
