Testing if variables have a non-linear relationships with the dependent variable

Question

How can I find evidence that a independent variable has a non-linear relationship with the dependent variable? Can I possibly achieve this by squaring all the independent variables and estimate a model containing both the squared variables and the original variables and then if the coefficients on the squared variables are a value close to 0, then we can assume that the variable has no non-linear relationship with the dependent variable?

Also, how can I find what the effect of using the logs of both the dependent and independent variables in the above case will do? Wouldn't the estimate the model be not valid as it contains multicollinearity? So is would it be more appropriate if I remove the linear terms and just estimate the model using the logs of the squared variables?

Answer 1

This is not the evidence, but the default way to test the nonlinear relationships is by fitting polynomials. The idea's rooted in Taylor expansion, that any function can be represented as a polynomial of high degree.

Omitting the main effect is generally not a good idea unless you can support it by external knowledge. For instance, you know that based on physics of the phenomenon, that the relationship is symmetrical. In this case you would not have a linear term (main effect), because that would not be symmetrical, so you may go for polynomials of even orders.