The Geometric Dickey–Fuller Test: Unit Root Testing via Clifford Algebra with Applications to Finance, Macroeconomics, and Predictive Maintenance

We propose the Geometric Dickey–Fuller (GDF) test, a nonparametric unit root

test grounded in the Clifford algebra Cl(3,0). The test embeds a univariate time

series in a three-dimensional state space (level, velocity, acceleration), computes the

geometric product of consecutive state vectors, and extracts bivector components

that measure rotation in the level–velocity and level–acceleration planes. We prove

that the expected bivector in the level–velocity plane equals zero if and only if the

process has a unit root (Proposition 1), establishing a formal bridge between Clifford

algebra and the Dickey–Fuller framework. The GDF test combines four geometric

channels—mean bivector, mean acceleration bivector, directional asymmetry, and

bivector persistence—via Non-Parametric Combination (NPC) of permutation-based

p-values.

Beyond its power as a unit root test, GDF provides channel-level diagnostics with

no analogue in existing tests: it distinguishes linear from nonlinear mean-reversion,

symmetric from asymmetric dynamics, and persistent from independent reversion

episodes. We introduce GDF-GLS, which applies Elliott–Rothenberg–Stock GLS

detrending before geometric analysis, achieving competitive power with DF-GLS for

trending series. We validate the test through extensive Monte Carlo simulations (size,

power against linear and nonlinear alternatives, robustness to GARCH errors, heavy

tails, and structural breaks) and document a critical sensitivity to moving-average

error structure that requires pre-whitening.

Empirical applications span three domains. In macroeconomics, we benchmark

GDF against ADF on the Nelson–Plosser dataset (14 series, 1860–1970) and the

FRED-QD database (245 series, 1959–2025), finding 82.0% agreement and iden-

tifying 19 series where GDF’s geometric channels detect stationarity that ADF

misses, of which 8 survive pre-whitening for moving-average structure—including

nonfarm payroll, industrial production, and government consumption. In pairs

trading, channel fingerprints distinguish tradeable from untradeable pairs using real

equity data, and rolling-window analysis reveals how the cointegration structure of

GLD/GDX mutates over time. In predictive maintenance, rolling GDF monitoring

on NASA C-MAPSS turbofan data achieves 94-cycle average lead time before engine

failure, with channel diagnostics that characterize the type of degradation.

The paper situates GDF within the historical arc from the Cowles Commission’s

structural program through Box–Jenkins, Sims, and Granger–Engle–Johansen,

arguing that geometric algebra recovers interpretive content at the unit root testing

stage—a stage where the discipline had given it up.

Keywords: unit root testing, Clifford algebra, geometric algebra, bivector, non-

parametric test, permutation test, time series, pairs trading, predictive maintenance