# Process Transforms (Fractional Difference)

Let's say I have a process $$X_t$$ with unknown variance process $$V_t$$.
Then, I write $$\mathrm{EMA}[X_t]$$ to be the 5 sec exponential moving average of $$X_t$$.

Consider the transformation $$\sum (X_t-\mathrm{EMA}[X_t]).$$

What can we say about the transformation in terms of the original $$X_t$$ and $$V_t$$? (Doesn't have to be proof, but I do like proofs)

As far as I know, a process minus it's exponential moving average is equivalent to fractional differencing. Summing the differential produces a process that is highly correlated to X, and has a more normalized (smoothed) variance process.

If you could point me in the right direction to study something like this I would be very appreciative.

• Hi: I can't help but where did you get the info that a process minus it's EWMA is equivalent to fractional differencing. I'm not saying that's wrong but rather I'm just wondering where you got that from ? – mark leeds Feb 22 '20 at 12:01
• Hi Mark. I am just trying to understand what theories apply to a process transformation like this. We could assume that the mean and variance are constant for X. A colleague informed me that fractional differencing is relevant for a problem like this, though I am unfamiliar with that set of techniques. – NEO ULTRA Feb 22 '20 at 20:38
• Hi: are you trying to do someting with financial returns ? if so, email me off list and I can send you something tha could be relevant ? If not, then I don't think I can help because, taking a process and subtracting the ewma from it, doesn't ring a bell to me except for this one specific paper I'm thinking of. Actually, I don't have time this second, but tonight I'll try to find it and send the link so that anyone can read it if they're interested. – mark leeds Feb 23 '20 at 21:49
• Hi: Like I said, if you're analyzing returns ( keep in mind that an EWMA is always equivalent to a moving average of window length $n$ for $\lambda = \frac{2}{n+1}$ ), then the paper at this link could be useful. papers.ssrn.com/sol3/papers.cfm?abstract_id=2604942. Other than this possible relevance, I've never heard of the ewma being subtracted from a time series process which is why I asked where you got that from. It could be true for sure. I just don't understand it and have never come across it. Thanks. – mark leeds Feb 23 '20 at 23:54