# Which if these two models works better?

I have this time series I want to perform polynomial regression on, to estimate the trend.

To start, I tried using only a second order polynomial, these are the results (AIC=30.37105)

We can see how the curve doesn't fit really well, mostly in the center. Checking the residuals after the seasonal component has been estimated (using four dummy variables, one for each season, added to the previous linear model), it is obvious that there is still some auto correlation.

The auto-correlation function shows that the residuals don't seem like only white noise.

Then I tried using a fourth order polynomial. (AIC=27.60569)

We can see that adjusted R^2 has increased and that all the coefficient are still significant (not as significant as before). These are the residuals, again, after the seasonal component has been estimated using four dummies added to the model above

And the auto-correlation function on them

This seems much better to me. (AIC for the second model with the dummies is lower too)

However one of my teachers at University told me that it makes no sense to use a polynomial of order four in a situation like this because the second model is surely overfitting. Please note he still told me that the model of order 2 isn't good either (by looking at the residuals), and we will see better ways to handle time series, but I still thought the model of order 4 was an improvement.

Things I don't understand:

1. How can I detect that the second model is overfitting? (Is it? I would say that it is fitting better, mostly looking at the residuals and AIC)
2. Why is it always suggested not to use polynomials of order higher than 3-5? (Thinking that polynomial regression isn't that useful for out of sample extrapolation anyway, I would say that at least I should try to create a model that fits the data in sample well)

The time series: y=c(9.328161, 9.262356, 10.23992, 9.937655, 10.90888, 11.055, 11.09276, 10.51899, 11.59662, 11.98787, 11.66901, 11.30161, 11.60235, 11.7764, 11.8741, 11.24226, 11.70456, 11.98945, 11.99912, 11.15315, 11.82424, 11.88189, 12.11866, 11.44164, 11.92071, 11.888770, 11.6900, 11.10875, 11.08220, 11.37809, 10.84132, 10.100, 9.998741, 10.05856, 9.600)

I think when you're instructor introduces more-time series modeling methods you will get a better intuition for the problems with your models in general.