Fractional Differentiation

Ordinary differencing can remove both persistence and useful long-run information. Fractional differentiation introduces a continuous order and lets the data retain a controlled amount of memory.

The central relation

The fractional difference operator is

(1-B)^dX_t=\sum_{k=0}^{\infty}\omega_kX_{t-k},\qquad \omega_0=1,\qquad \omega_k=-\omega_{k-1}\frac{d-k+1}{k}.

$X_t$ is the value at time $t$ , $B$ moves a value back one period, $d$ is the difference order, and $\omega_k$ is the weight on lag $k$ . The recursive weights determine how quickly older observations lose influence.

Balancing memory retention with statistical stability

The first figure displays the lag weights for several orders. The second checks whether the selected order remains useful outside the sample used to choose it.

Fractional-difference weights across lags for several difference orders. — Small difference orders retain longer memory because the lag weights decay more slowly. Larger orders place less weight on distant observations.

Out-of-sample stationarity and memory diagnostics across fractional difference orders. — The stability plot checks the selected order on later observations instead of relying only on the period used for selection.

The order is a modeling choice. Passing one stationarity check does not guarantee stable behavior under another sample or another notion of memory.

Notebooks


1. Constructing fractional differences from binomial weights
2. Preserving memory with a fixed-width operator
3. Selecting the difference order for out-of-sample stability