# Examining the dependence of the fractional difference parameter in ARFIMA(0,d,0) vs bar size for Realized Volatility

Realized volatility is a long-memory process and so I fitted an ARFIMA(0,d,0) to log(RV15) where RV15 is realized volatility calculated from 15-min bars. I proceeded to examine how changing the bar size from which RV is constructed impacts the $$d$$ parameter in ARFIMA(0,d,0). Below are the plots. It seems that the parameter $$d$$ decreases with increasing the bar size (see left figure). Lower $$d$$ parameter means a "shorter" memory time series (see right figure).

I cannot properly explain why this happens. That is, I cannot sit down and verbalise it to explain it to someone else. And even with my explanation, I'm not sure it is correct.

My explanation is the following. Using higher resolution when calculating RV means more information about intraday trading patterns is summarised in the RV. This increase in information contained in the RV allows the value from 20 days ago to better explain the value today. This also captures patterns that repeat themselves and they increase the correlation whereas the lower resolution does not capture these patterns as well.

• Is it really shorter memory in actual hours or days? Isn't that d simply "periods". Meaning for daily data you have d=1 is 1 day, for 15 min, d=96 represents 1day (assuming the underlying trades 24h a day). Intuitively, if you had milliseconds, the price a millisecond before will almost surely be very close.id you had monthly data, there is no logical reason why prices should be close to each other. Aug 14 at 13:58
• Yes, it is shorter memory in actual days because RV is a daily series. The value for each day is calculated from the intraday bars. But for example, RV5 and RV60 would have exactly the same number of data points (one for each day) even though the 5-min return series is 12 times longer than the 60-min return series. Does this answer your question?
– s5s
Aug 14 at 16:31
• Hi @s5s. Out of curiosity, are you doing a paper and do you have an advisor you can ask for guidance? You might be able to validate your explanation via a simulation study. That is, if there's more information inherent in RV by using high-frequency data and that is the reason for a shorter memory process (lower $d$), then you should observe the same pattern for RVs on a simulated process with different magnitudes of noise (eg. you can additively add a normal random variate on the return process, with different sd's acting as different magnitudes of noise). [1/2]
– Pleb
Aug 15 at 15:31
• First, estimate RV on the simulated process (eg. GBM) and observe whether RV converge towards $\sigma_{GBM}$ without any added noise process. Secondly, estimate RV on the same process for different magnitudes of noise. Thirdly, use the RVs to estimate the ARFIMA(0,d,0). If you get lower $d$-values for the estimated RVs with little to no noise, as opposed to the other RVs with large magnitude of noise, then it strengthens your above statement. You can try this, if you cannot find any relevant papers explaining the pattern. If you have already found your answer, please post it below 😊. [2/2]
– Pleb
Aug 15 at 15:33