I'm running a model where both a 2nd or 3rd order polynomial would seem to fit the data. I'm trying to decide which one to use. In the quadratic model (panel fixed effects) the first and second order of the variable are significant but less so compared to a model with a cubic included (y=x+x^2+x^3+e). On the other hand, in my case the quadratic specification would be more intuitive. And more generally, how should the significance of the polynomials be interpreted? If I've understood correctly, the significance of the highest polynomial determines whether the model specification is reasonable but all the lower order polynomials should be included irrespective of their significance.