Spectral Foundations of Geometric Unit Root Testing: Bivectors, Stationarity, and the BeveridgeNelson Decomposition

The geometric DickeyFuller (GDF) test of Vázquez Broquá and Sudjianto (2026a) uses bivector means in the Cliord algebra Cl(3,0) to test for a unit root, but its theoretical justication is limited to the AR(1) case. This paper provides the spectral foundation that the GDF test requires for general processes. Our main result establishes that the mean bivector in the e1e2 plane is strictly nega- tive for any stationary process with a continuous spectral density, via the identity γ(1)−γ(0) =π −π(cos λ−1)f(λ) dλ<0, and converges to zero at rate Op(T−1/2) under a unit root. This extends the GDF from AR(1) to AR(p), ARMA, and long-memory processes with d < 1/2. We then connect the geometric product de- composition to the BeveridgeNelson (1981) permanenttransitory partition: the scalar part of the product captures the permanent component (spectral mass at fre- quency zero) while the bivector captures the transitory component (spectral mass away from zero). As a secondary result, we show that the scalar mean converges to a nite limit that is monotonically increasing in the trend signal-to-noise ratio |β|/σ, providing a continuous measure of trend strength. Monte Carlo experiments with 1,000 replications across 12 data-generating processes conrm all theoretical predictions.