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I carried out regression with 7th degree and 8th degree polynomials. As expected, the residual sum of squares for 8th degree polynomial regression is less than that of 7th degree polynomial regression. On the other hand, the mean absolute percentage error (MAPE) of the 8th degree polynomial regression is higher than that of 7th degree polynomial regression. Note that the degrees of freedom is also considered for taking the average for MAPE; however the result is same if the average is taken w.r.t the number of observations.

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  • $\begingroup$ First, what were you expecting? I don't see any contradiction here, because there doesn't seem to be an obvious relation between MAPE and MSE. $\endgroup$ – Aksakal Aug 25 '15 at 15:55
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    $\begingroup$ Without knowing anything about MAPE, since it is a measure of predictive accuracy, what happened makes perfect sense. Your results are saying that the 8th degree (is more of) an overfit, and hence it's predictive accuracy should be less than the degree 7 solution $\endgroup$ – aginensky Aug 25 '15 at 17:07
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I am unfamiliar with any definition of the MAPE that includes the degrees of freedom (which would be hard, since the MAPE is often used to assess time series model forecasts, where the notion of "degree of freedom" is not entirely clear).

Are you asking about in-sample or out-of sample accuracy, both for and ? High-degree polynomials are almost certain to overfit, yielding spuriously good in-sample fitting accuracy but catastrophic out-of-sample accuracy. Here are a few examples with toy data and zero-th to third degree polynomials - I shudder to think of what an eighth degree polynomial would do here:

overfitting

If you truly want to model nonlinearities, consider splines. Harrell's Regression Modeling Strategies has a very helpful overview.

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