I am wrapping up my dissertation and I am fitting ARFIMA models on a log(RV) process. I'd love to hear your thoughts about my deductions and arguments and if they are complete rubbish or not. I would welcome and appreciate any other opinions, comments and/or criticism.
RV here stands for the realized volatility which is a persistent process and the ACF and PACF are shown below. I use RV15 meaning that the RV was calculated from 15-min bars. I should stress that I use log(RV) rather than RV because log(RV) is almost normal (I can provide histogram and QQ plots if you want to see them).
I proceeded to calculate the AIC and BIC for a 5x5 matrix of (p,q) coefficients. I also then calculated a 10x10 matrix. I won't paste the matrix here, I'll just summarise the results:
- For the 5x5 AIC matrix, the lowest AIC is for p=4,q=0.
- For the 5x5 BIC matrix, the lowest BIC is for p=0,q=0.
- For a 10x10 AIC matrix, the lowest AIC (up to 4sf) occur for p=10,q=6 and for p=5,q=10.
- For a 10x10 BIC matrix, the lowest BIC is for p=0,q=0.
I explain this with arguments similar to this answer - that is, AIC will prefer a higher dimensional model to model the even higher dimensional reality while BIC would prefer a simpler model out of the given set to avoid overfitting.
However, I make the point that the ACF and PACF plots suggest log(RV) has no MA(q) process (slowly decaying ACF and PACF which is zero after lag 4 or so). Therefore, I suggest that the true model is ARFIMA(p,d,0). I then fitted ARFIMA(p,d,0) for p up to 30 and found that the model with the lowest AIC is ARFIMA(4,d,0) and the lowest BIC is ARFIMA(0,d,0) (i.e. just the long-memory property).
However, my last point (about no MA(q) component) does not exactly fit with the AIC for the 10x10 matrix which found that the best fitting models had a substantial MA(q) component. I am a bit sceptical about models with such a large number of parameters and so I would tend to argue that a model such as ARFIMA(10,d,6) or ARFIMA(5,d,10) would be overfitted. I don't have enough experience to tell if these arguments are valid.
