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I have been analyzing panel data for a while now using different methods (Generalized Linear Models, fractional polynomials and GAMs).

If we just ignore GMMs for now, I have come to find that Fractional polynomials are more intuitive than GAMs as they fit higher order polynomials which are easier and more intuitive for most readers to interpret/understand than splines. In addition, it is easier to compute the first and second derivative function to identify periods of significant change and Maximum or Minimum values of the polynomial function. Finally, I think that it is more likely you won't end up overfitting the data.

Would you agree with this? What are just briefly the advantages of one over the other one in your opinion?

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In my mind they are equally intuitive. What matters, whether using nonparametric smoothers, fractional polynomials, or splines, is that the quality of fit per degree of freedom spent is good on the average. Note that fractional polynomials have a few disadvantages: $X$ must be $>0$ and if you shift $X$ by adding a constant the form of the fractional polynomial needed for a good fit can change substantially.

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