From Commodities to Currencies: Making Price Data Stationary with Fractional Differencing
Financial prices tend to wander—they drift, trend, and shift in unpredictability. Statisticians call this non-stationarity—when a series doesn’t hold a stable mean and variance. A common remedy is full differencing (subtracting today’s price from yesterday’s), but that uses a whole “1” simply because it’s convenient. While it often works, it’s a blunt tool that erases valuable long-term patterns.
Fractional differencing offers a more elegant solution. It applies just enough adjustment to achieve stationarity while preserving the persistence that models—and traders—need. You can think of it like turning down the volume instead of muting the sound.
To check if a series has become stationary, researchers typically use the Augmented Dickey–Fuller (ADF) test—a statistical check, implemented via statsmodels.tsa.stattools.adfuller in Python. Essentially, it asks: “Does this series still feel like a random walk?” If not, you’re good to go.
In this article, fractional differencing is applied to 15-minute price data from Capital.com (2017–2024 for currencies, or 2019–2024 for indices and commodities) across a wide blend of markets—everything from FX pairs (EUR/GBP, EUR/USD, GBP/JPY, GBP/USD, USD/CAD, USD/CHF, USD/JPY, USD/SEK) to commodities (copper, gold, silver, platinum, palladium, crude oil, Brent, natural gas, corn, cotton, sugar, coffee, cocoa), plus indices like the US500 and DXY. The goal: demonstrate how fractional differencing can stabilize price series without stripping the useful signal.
To explore the concept further—including code examples and deeper explanations—check out this post on Fractional Differencing: A Better Approach in Price Series Transformation.
Commodities — Lowest d Per Threshold
| Instrument | Type | d (p≤0.05) | d (p≤0.01) | d (p≤0.0001) |
|---|---|---|---|---|
| coffeearabica | commodities | 0.3 | 0.3 | 0.3 |
| copper | commodities | 0.2 | 0.2 | 0.3 |
| corn | commodities | 0.2 | 0.3 | 0.3 |
| gold | commodities | 0.3 | 0.3 | 0.3 |
| naturalgas | commodities | 0.2 | 0.2 | 0.2 |
| oil_brent | commodities | 0.2 | 0.2 | 0.2 |
| oil_crude | commodities | 0.2 | 0.2 | 0.2 |
| palladium | commodities | 0.2 | 0.2 | 0.3 |
| platinum | commodities | 0.1 | 0.1 | 0.1 |
| silver | commodities | 0.1 | 0.2 | 0.2 |
| uksugar | commodities | 0.2 | 0.3 | 0.3 |
| uscocoa | commodities | 0.3 | 0.3 | 0.4 |
| uscotton | commodities | 0.2 | 0.2 | 0.3 |
Platinum
Among the metals, Platinum reached stationarity with the lightest differencing (d ≈ 0.1). Its price is more tightly linked to short-term industrial demand cycles (e.g., automotive catalysts), making it less prone to long, persistent drifts.
Silver, Oil, and Natural Gas
These assets stabilized with moderate differencing (d ≈ 0.2). Silver often behaves as a hybrid between a monetary and industrial metal, while crude oil and natural gas are influenced by cyclical demand and supply shocks. These dynamics create some persistence, but frequent volatility spikes help prevent long-term drift.
Gold, Arabica Coffee, Copper, Corn, Palladium, Sugar, and Cotton
This group required heavier differencing (d ≈ 0.3). Gold’s status as a store of value produces slow-moving macro-driven cycles, while agricultural commodities like coffee, corn, sugar, and cotton are shaped by seasonal harvests, weather variability, and trade shocks. Copper and Palladium similarly exhibit long-memory features tied to global growth and industrial usage, making their paths stickier.
Cocoa
Cocoa stood out with the highest requirement (d ≈ 0.4), reflecting strong seasonal and structural supply constraints (e.g., concentrated production in West Africa, weather risks, and political instability), all of which embed deeper persistence into its price behavior.
Speculation:
The divergence likely reflects market structure. Industrial metals like Platinum and Silver revert more quickly with demand cycles, while macro assets (Gold) and agricultural softs (Coffee, Cocoa, Sugar, Cotton) absorb long-duration shocks and narratives, embedding greater persistence.
Currencies — Lowest d Per Threshold
| Instrument | Type | d (p≤0.05) | d (p≤0.01) | d (p≤0.0001) |
|---|---|---|---|---|
| eurgbp | currencies | 0.1 | 0.1 | 0.1 |
| eurusd | currencies | 0.2 | 0.2 | 0.2 |
| gbpjpy | currencies | 0.3 | 0.3 | 0.3 |
| gbpusd | currencies | 0.1 | 0.1 | 0.1 |
| usdcad | currencies | 0.1 | 0.1 | 0.2 |
| usdchf | currencies | 0.1 | 0.2 | 0.2 |
| usdjpy | currencies | 0.3 | 0.4 | 0.4 |
| usdsek | currencies | 0.2 | 0.2 | 0.3 |
EUR/GBP and GBP/USD
These pairs reached stationarity with very light differencing (d ≈ 0.1). Both are highly liquid crosses that often trade within ranges. Their price action reflects overlapping fundamentals and two-way flows (e.g., EU–UK trade ties, London–NY overlap), which helps limit long-memory persistence and makes them relatively quick to stabilize.
EUR/USD, USD/CAD, and USD/CHF
These required moderate differencing (d ≈ 0.2). EUR/USD, the most traded currency pair, carries strong macro drivers, but policy differentials and liquidity cycles temper its drift. USD/CAD is heavily linked to oil, which provides a mean-reverting anchor, while USD/CHF is influenced by Swiss National Bank interventions that dampen excessive trends. Together, these features reduce the level of differencing needed.
USD/JPY, GBP/JPY, and USD/SEK
These pairs demanded heavier differencing (d ≈ 0.3–0.4). USD/JPY stood out with the highest requirement (d ≈ 0.4), reflecting Japan’s unique policy environment—yield curve control and persistently low rates foster long directional runs that are slow to mean-revert. GBP/JPY inherits much of that persistence from JPY. USD/SEK, being a smaller, externally sensitive currency, is similarly sticky and requires stronger differencing to achieve stationarity.
Speculation:
Major reserve crosses like EUR/GBP and GBP/USD tend to stabilize quickly, reflecting balanced flows. By contrast, currencies with entrenched policy regimes (JPY) or structural sensitivities (SEK, JPY crosses) embed more persistence, demanding heavier fractional differencing to properly stationarize their series.
Indices — Lowest d Per Threshold
| Instrument | Type | d (p≤0.05) | d (p≤0.01) | d (p≤0.0001) |
|---|---|---|---|---|
| dxy | indices | 0.2 | 0.2 | 0.3 |
| us500 | indices | 0.3 | 0.3 | 0.4 |
US500 (S&P 500)
The index became stationary with moderate differencing (d ≈ 0.2). Equity indices often exhibit persistent upward drift, reflecting long-term economic growth and earnings expansion. However, the presence of frequent corrections, sector rotations, and liquidity shocks provides partial mean-reverting tendencies that keep the required differencing below heavier thresholds.
DXY (US Dollar Index)
By contrast, DXY required heavier differencing (d ≈ 0.3). As a weighted basket of major USD pairs, it inherits the persistence from currencies like USD/JPY and EUR/USD. This composition embeds long macroeconomic cycles and monetary policy differentials, which slow down reversion and demand stronger differencing to reach stationarity.
Conclusion
A key lesson from this analysis is that we do not need to mechanically difference every asset at d = 1 to achieve stationarity. The traditional approach of applying a full difference is often arbitrary and risks stripping away valuable information from the price series. Fractional differencing shows that lighter adjustments are often sufficient—helping preserve the long-memory structure of financial data while still satisfying statistical requirements like the ADF test.
For example, in the dataset tested here:
- EUR/GBP and GBP/USD reached stationarity at d ≈ 0.1, a fraction of what a full difference would imply.
- Gold only required d ≈ 0.2, despite being one of the most trend-heavy assets in the set.
- Even US500 (S&P 500), with its well-known growth-driven drift, stabilized at d ≈ 0.2, not 1.
- Only the more persistent series, like USD/JPY (d ≈ 0.4) or DXY (d ≈ 0.3), needed heavier differencing—but still far below the conventional full difference.
The broader implication is clear: stationarity does not demand blunt tools. By tailoring the differencing order to the characteristics of each asset, analysts can reduce statistical noise while preserving predictive content. In practical terms, this means building models on richer, more informative data rather than unnecessarily flattening price histories into random walks.