Non-linear relationship among independent variables in linear model Assume that we have a linear model with 4 independent variables, and 2 of them have a strong non-linear relationship (between them). How this fact could affect my model ($R^2$, or implications on the robustness of the model)? 
It is a question that popped up to my mind while working with linear regressions. Any tips?
 A: When you refer to a "linear model" I take that to mean an ordinary least squares model with linear adjustment for four regressors. If the relationship with the predictors is in fact non-linear then there are a number of ways forward: 
First solution: just leave them as-is. The resulting model gives biased predictions and has heteroscedastic errors, but so what? Maybe the model performance is good enough, and has the added benefit of being simple and easy to communicate. The R^2 is not interpreted as a fraction of variance "explained" but as the mean-squared error of predictions. The risk of doing this is that, when validating the model in independent data, there may be a shift in distribution of the regressors, and domains in which the heteroscedastic model performed poorly may be overrepresented.
Second solution: include higher order terms or breakpoints (splines). Splines are highly adept polynomial approximations that can capture most forms of non-linearity. The benefit is that they are easy to fit, and they are possible to interpret. They need not dramatically complicate the model. The problem is that they can be somewhat difficult to communicate. For instance, with a quadratic trend, there are many features of interest such as the location of the inflection point, and the magnitude of trend at various levels of the regressor(s). It is a subtle point to calculate these things and present their confidence intervals.
