# Understanding fractional-differencing formula

I have a time series $y_t$ and I would like to model it as an ARFIMA (a.k.a. FARIMA) process. If $y_t$ is integrated of (fractional) order $d$, I would like to fractionally-difference it to make it stationary.

Question: is the following formula defining fractional-differencing correct?

$\Delta^d y_t := y_t - d y_{t-1} + \frac{d(d-1)}{2!} y_{t-2} - \frac{d(d-1)(d-2)}{3!} y_{t-3} + ... +(-1)^{k+1} \frac{d(d-1) \cdot ... \cdot (d-k)}{k!} y_{t-k} + ...$

(Here $\Delta^d$ denotes fractional-differencing of order $d$.)

I base the formula on this Wikipedia article on ARFIMA, Chapter ARFIMA($0,d,0$), but I am not sure if I got it correctly.

Yes it seems to be correct. The fractional filter is defined by the binomial expansion:

$\Delta^{d}=\left(1-L\right)^{d}=1-dL+\frac{d\left(d-1\right)}{2!}L^{2}-\frac{d\left(d-1\right)\left(d-2\right)}{3!}L^{3}+\cdots$

Note that $L$ is the lag operator and that this filter cannot be simplified when $0<d<1$. Now consider the process:

$\Delta^{d}X_{t}=\left(1-L\right)^{d}X_{t}=\varepsilon_{t}$

Expanding, we get:

$\Delta^{d}X_{t}=\left(1-L\right)^{d}X_{t}=X_{t}-dLX_{t}+\frac{d\left(d-1\right)}{2!}L^{2}X_{t}-\frac{d\left(d-1\right)\left(d-2\right)}{3!}L^{3}X_{t}+\cdots=\varepsilon_{t}$

which can be written as:

$X_{t}=dX_{t-1}-\frac{d\left(d-1\right)}{2!}X_{t-2}+\frac{d\left(d-1\right)\left(d-2\right)}{3!}X_{t-3}-\cdots+\varepsilon_{t}$

See Asset Price Dynamics, Volatility and Prediction by Stephen J. Taylor (p. 243 in the 2007 ed.) or Time Series: Theory and Methods by Brockwell and Davis for further references.

• My trouble was getting from the general definition of the filter (as you have) to applying the filter on a particular $y_t$. I know it must be obvious, but could you perhaps include a step showing how to get from your formula to mine? Commented Apr 10, 2015 at 7:34
• See my edited answer. Commented Apr 10, 2015 at 9:21
• is it just me or is the addition/subtraction alternating? Commented Nov 8, 2020 at 16:22