Oil's lead over Bitcoin depends on wavelet pre-filtering

Oil's reported lead over Bitcoin in medium- to long-horizon equity risk, roughly 1.5 to 6 months, depends on filtering returns before fitting the VAR. Drissi, Ben Mbarek and Sliti find that lead in their band-filtered decomposition. In their own appendix, the standard method produces a ranking that is flat and statistically indistinguishable from zero. The paper shows how heavily spillover attribution can depend on an implementation choice. Its evidence does not support an oil-led risk budget.

How the triangle is built

The authors give the horizon split an economic explanation. Oil affects listed firms through costs; in exporting economies, national income and energy-sector earnings provide another route. Contracts and budgets slow the transmission, so equity prices should respond over weeks and months. Bitcoin has no claim on corporate cash flows. Its route into equities runs through shared investors, with margin calls after crypto crashes and changes in risk appetite transmitting the shock. Those effects should arrive quickly and fade quickly.

Brent and Bitcoin sit upstream in the design as shock transmitters. Eight equity indices receive those shocks. Four represent net oil importers (EURO STOXX 50, Nikkei 225, BSE30, Shanghai Composite), and four represent exporters (S&P/TSX, MOEX, Tadawul, FTSE ADX). Daily observations run from 8 January 2020 to 13 June 2024. Intersecting ten trading calendars reduces the sample to 671 common days. The daily global geopolitical risk (GPR) index supplies the high-risk regime, defined as its top tercile, above roughly 137.5.

The analysis combines three tools. Wavelet coherence measures co-movement by horizon, while wavelet local projections trace responses to identified shocks. Frequency-domain connectedness supplies the headline measure, the Substitution Index. For each market and horizon band, it subtracts oil's share of equity forecast-error variance from Bitcoin's share, then divides by their sum. The index ranges from -1 (oil-led) to +1 (Bitcoin-led). Its bands cover 8-16, 16-32, 32-64 and 64-128 trading days. The VARs use rolling 250-day windows and a 20-day forecast horizon.

The index measures relative variance attribution, and the authors also decline to turn their findings into a portfolio rule.

Does the pre-filter create the lead?

The baseline first band-filters each return series using a maximal overlap discrete wavelet transform (MODWT), which separates returns into horizon bands. It then fits a VAR for each band. Baruník and Křehlík's standard procedure fits one VAR to unfiltered returns, then integrates the spectral decomposition over each band.

The authors ran that standard version. It produces nearly identical index values across bands. In the 32-64 day band, the 90% bootstrap bands include zero in 91 to 100% of rolling windows, depending on the market. They include zero in at least 90% of windows in the other three bands.

The abstract acknowledges the dependence on implementation. Both patterns change with the way the decomposition is computed, it says, so the authors present them as descriptive. It continues: "In the baseline, whether oil or Bitcoin leads, and whether the two act as substitutes or as complements, depends on the investment horizon and on the level of geopolitical risk."

The phrase "in the baseline" carries the defence.

That baseline is the implementation that fails the check. The local projections have the same weakness. Oil-shock responses roughly double in absolute value when GPR is high, according to the authors, but they say those regressions were not re-estimated without the pre-filter.

The exporter gap changes direction

Exporters appear more oil-led in the band-filtered results. With an unfiltered VAR, the 32-64 day band gives exporters an average of -0.06 and importers -0.30. Importers therefore appear more oil-led. The difference is not significant (p = 0.31). Lagging transmitter returns for markets closing before Brent or Bitcoin yields p = 0.79; using two-day returns yields p = 0.34. The sign changes again with 375- or 500-day windows.

The upper rows of the robustness table retain the pre-filter and report that exporter negativity survives. The last four rows change either the decomposition or the wavelet inference, and neither pattern survives. Four countries per group are too few to build country-level risk budgets around this contrast.

Five cycles, little long-period support

At the top of the 64-128 day band, the authors note, 671 days contain only about five complete cycles. For return-based coherence in that band, Brent-S&P/TSX and Brent-FTSE ADX each flag 0.42 of pixels pointwise. Both fall to 0.00 under the areawise test. That test controls false patches across the entire map; every long-period Brent-equity region fails it. Bitcoin-S&P/TSX survives with 0.11 (8-16 days), 0.12 (16-32) and 0.07 (32-64).

Using return coherence and the areawise test reverses the ordering for two of four pairs. The authors themselves describe the long-horizon oil coherence as descriptive.

Before anyone builds a monitor, two inputs warrant another look. The 34 Bitcoin shock days and the GPR tercile both use cutoffs drawn from full-sample distributions. Those cutoffs work for description but are unavailable in real time. Bitcoin's daily returns also have an ARCH LM p of 0.999 and a Ljung-Box p of 0.690. These are odd figures for the asset and may reflect returns calculated across irregular calendar gaps.

We could not reproduce any of this. Our data lacks the EIA Brent spot series and the daily GPR index. Substituting WTI or US equities would change the question being asked.

Two hypotheses remain worth monitoring. One concerns short-horizon Bitcoin-equity coupling in Canada. The other asks whether oil-shock responses really double above a GPR of 137.5 when the regressions use unfiltered, lagged-aligned returns.