Overview of selection method for p-order of AR($p$) model

Question

I would like to get an overview over the most common methods for estimating the lag order of an AR($p$) process.

The common literature states:

1. Akaike information criterion (AIC)
2. Bayesian information criterion (BIC)
3. Hannan–Quinn information criterion (HQC)
4. Akaike's final prediction error (FPE) criterion
5. Criterion autoregressive transfer function (CAT) %of Parzen.
6. Minimum description length (MDL)
7. Resiudal variance (RV) method

I kindly ask to refer to other methods and maybe highlight their advantages compared to AIC or BIC.

Answer 1

You can use auto-correlation plots/partial auto-correlation plots. See the Box-Jenkins wikipedia page. The idea is that if you process is an AR(q) process, after q lags the partial-autocorrelation plot, should go to zero. Partial-autocorrelation plot basically is a plot that compares the correlation of $x(t)$ and $x(t+k)$, except you only take the component of $x(t)$ and $x(t+k)$ that is orthogonal to all the intervening lags $(x(t)...x(t+k-1))$ (ie you take the residual of a regression). You test when that plot dies down to zero--using confidence intervals for example--and that gives you the number of lags.

The advantage of this method is that it is theoretically justified and you can test whether your process is not only AR(q) but ARIMA(p,d,q) with slight modifications. The downsides is this only works if your time series is ARIMA. If it is not it could give a good approximation but all theoretical underpinnings are thrown out the door. Information Criterion works for any type of time series process which give it more flexibility than partial auto-correlation (like lets say you wanted to build a structural time series model or a neural net). Unlike Information Criterions, autocorrelation plots don't return a score so it is difficult to tell how much worse an AR(q+1) fitted graph is with an AR(q) graph, but it does tell you which one is correct so maybe that is all you need.

Answer 2

You seem to be concerned with evaluating a specific model whereas in my opinion you should be concerned with identifying a useful model . See this discussion and it's threads to learn more Interpretation of p-values in ARIMA model . At the the end of the model identification process , I would suggest the AIC criterion .

This is NOT to say I would use the AIC model on an inadequate model that perhaps ignored the Gaussian Assumptions.