Evaluating ARIMA models with the AIC

Question

Having come across ARIMA/seasonal ARIMA recently, I am wondering why the AIC is chosen as an estimator for the applicability of a model. According to Wikipedia, it evaluates the goodness of the fit while punishing non-parsimonious models in order to prevent overfitting. Many grid search functions such as auto_arima in Python or R use it as an evaluation metric and suggest the model with the lowest AIC as the best fit.

However, in my case, choosing a simple model (with the lowest AIC -> small amount of parameters) just results in a model, that strongly follows previous in-sample observations and performs very badly on the test sample data. I don't see how overfitting is prevented just by choosing a small number of parameters...

ARIMA(1,0,1)(0,0,0,53); AIC=-16.7

Am I misunderstanding something? What could be a workaround to prevent this?

Answer 1

In the case of an ARIMA model whatever the parameters of the model are it will follow past observations, in the sense that you predict next values given previous values from your data. Now, auto.arima just tries some models and gives you the one with the lowest AIC by default or some other information criterion e.g BIC. This does not mean anything more than what the definition of those criteria are: so the model with the lowest AIC is the one that gives minimizes the AIC function. In case of time series analysis after you make sure that time series is stationary, I would recommend that you examine the ACF and PACF plots of your time series and read this

P.S I don't get this straight orange line in your plot after the dashed vertical line.

Answer 2

We usually use some form of cross-validation to protect against overfitting. It is well known that leave-one-out cross-validation is asymptotically equivalent to AIC under some assumptions about normality etc. Indeed, back when we had less computing power, AIC and other information criteria were handy exactly because they accomplish something very similar to cross-validation analytically.

Also, note that by their nature ARMA(1,1) models -- and other stationary ARMA models for that matter -- tend to converge to a constant rather quickly. The easiest way to see this is to write down the expressions of y_t+1, y_t+2 as a function of y_t. You will see that the expression has exponentials of numbers less than 1 (your AR and MA parameters), which quickly converge to zero as t grows. Also see this discussion.

The reason why your 'observed' data (to the left of the dashed line) does not exhibit this behaviour is that for each period you get a new realisation of random error term epsilon_t. On the right hand side, you do not get these realisations of random shocks, but instead they are replaced with their expressed value 0.