Fractional Differencing Weights, Lag Operators, and Memory Retention
Summary
The document explains how fractional differencing applies the lag operator to a price series and produces a weighted sum of current and past observations. Expanding the operator $(1-B)^d$ with the generalized binomial theorem gives the weights: each power of the lag operator shifts the series by that many time steps. The weights are therefore the coefficients in the expansion, rather than a separate object with an independent relationship to the lag operator.
The accompanying graph plots those coefficients across lag counts for different values of the differencing order. First differencing, at d=1, removes dependence on lags beyond the immediately previous observation, while fractional orders below one retain contributions from a longer history. This illustrates the proposed trade-off between making a series stationary and retaining information about its past. The answer also cautions that choosing d is asset-sensitive: an Augmented Dickey-Fuller example gives different results across assets and asset classes, making a single setting difficult to select for a broad portfolio. The discussion explains the concept but does not establish that fractional differencing will improve a trading strategy.
Key ideas
- Applying the lag operator's fractional power to a series yields binomial coefficients as lag weights.
- Each lag-operator power shifts the observation backward by the corresponding number of periods.
- First differencing has zero weights beyond the immediately preceding observation, while fractional orders retain longer history.
- The coefficient graph illustrates how memory retention changes with the differencing order.
- The suitable order may vary across assets, and the document does not show that the method improves trading results.
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# Backshifting Price Timeseries with Memory Preservation
# Backshifting Price Timeseries with Memory Preservation
In Advances in Financial Machine Learning the author makes a case for fractionally differentiated price returns in chapter 5. The reason is to both maintain memory and to generate a stationary time series. The author argues that the first order derivative of prices, price returns, achieve stationarity but experience memory loss. He defines the backshift operator $B$ as
$B^{k}X_{n} = X_{n-k}$
and then applies the binomial expansion on this
$(1-B)^{d} = \sum_{n=1}^{\infty} {d \choose k} (-B)^{k}$
This then related to how memory can be preserved when d is a positive real
$\hat X_{t} = \sum_{k=0}^{\infty} w_{k} X_{t-k}$
Where the weights, $w$ are expanded using the same binomial method as $B$
Other than the binomial expansion, what is the relationship between $B$ and $w$ and the relevance of the graph?
This is the caption of the graph
FIGURE 5.1 𝜔k (y-axis) as k increases (x-axis). Each line is associated with a particular value of d ∈ [0,1], in 0.1 increments.
## Answer by Pleb (score 4, accepted)
https://quant.stackexchange.com/a/63359
#### Other than the binomial expansion, what is the relationship between $B$ and $w$ ?
As I understand correctly, there is no immediate relationship between the lag operator $B$ and the weights $w$. However, going from the fractional differencing operator, $(1-B)^d$ to $\hat{X}_t$ can be done as follows: \begin{align} (1-B)^d X_t &= \sum_{k=0}^{\infty} \begin{pmatrix} d \\ k\\ \end{pmatrix} (-B)^k \cdot X_t\\ &=X_t \sum_{k=0}^{\infty} (-B)^k \prod_{i=0}^{k-1}\frac{d-i}{k-i}\\ &= X_t \left(1-d\cdot B + \frac{d(d-1)}{2!}B^2 - \frac{d(d-1)(d-2)}{3!}B^3 + \cdots\right)\\ &= X_t - d\cdot X_{t-1} + \frac{d(d-1)}{2!} X_{t-2} - \frac{d(d-1)(d-2)}{3!} X_{t-3} + \cdots \\ &= \sum_{k=0}^{\infty} \omega_kX_{t-k}\\ &= \hat{X}_t \end{align} where $\omega=\left\{1,-d,\frac{d(d-1)}{2!},-\frac{d(d-1)(d-2)}{3!},\ldots\right\}$ is the binomial expansion, as noted. Furthermore, we have used (per definition) the lag-operator applied to $X_t$, namely $B^k X_t = X_{t-k}$, in the second last equality. I believe it was the authors intention to simplify the notation and reformulate it into something more recognizing for people within finance.
#### What is the relevance of the graph?
The graph shows you the memory-retainment (in terms of $k$ included lags of log-prices, $X_{t-1},X_{t-2},\ldots$ used to differencing the price-process) of fractional differenced processes, for different values of $d\in [0,1]$. As the graph depicts, there exist a memory cut-off for the first-differenced price process ($d=1$), in the sense that the history of the previous prices is disregarded when transforming the process into a stationary time-series. This is also one of the motivations for using fractional differencing (FD), since log-prices $X_t$, depends on many previous lags and not only the first lag, $X_{t-1}$. In order to preserve this dependence structure (memory), we resort to FD for $d<1$. If we depict the fractionally differentiated returns for two values of $d=\{0.5,1\}$ you get: \begin{align} &d=1: \qquad \; \: \: \hat{X}_t=X_t - X_{t-1}\\ &d=0.5: \qquad \hat{X}_t=X_t - 0.5X_{t-1} - 0.125 X_{t-2} - 0.0625 X_{t-3} -\cdots, \end{align} and conclusively for $d=1$, we have the standard first-differencing of the log-prices, and clearly see a cut-off at $k=2$, since $\omega_k=0$ for all $k\geq 2$. This does not happen for $d=0.5$, since $\omega = \{1,-0.5,-0.125,-0.0625,\ldots\}$.
Extra information: In one of his slidesets he argues that there exists a trade-off between stationarity and memory by choosing $d$ accordingly. However, as seen in his own example (example 2: He uses an Augmented Dickey-Fuller test to test the time-series under different values of $d$), the choice of $d$ is clearly sensitive to the choice of asset and asset-class. This makes it hard to hone in on one optimal choice of $d$, if you are a portfolio manager working with a large set of assets.
I hope this helps!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.