Partial Cointegration

Classic cointegration is an all-or-nothing test: either the spread between two assets is fully mean-reverting (stationary) or it isn't. Real pairs are rarely that clean — a spread between two related but not identical companies often behaves like a mix of both, part of it oscillating around a level in a way you could trade, and part of it drifting off permanently for reasons that have nothing to do with mean reversion, like one company slowly growing faster than the other.

Partial cointegration models the spread as the sum of two pieces: a stationary, mean-reverting component and a random-walk component that never comes back. Formally the spread is written $S_t = R_t + M_t$, where $M_t$ is a random walk and $R_t$ is a mean-reverting process, and the model estimates what fraction of the total variance belongs to each piece. A spread that's 90% mean-reverting is a much better trading candidate than one that's 90% random walk, even though a plain cointegration test might reject both, or accept both, without telling you which is the better bet.

This matters for pair selection because it separates two different failure modes that a simple pass/fail cointegration test conflates: a pair can fail a strict cointegration test simply because a small permanent drift is present alongside real mean reversion, and discarding it outright throws away a genuinely tradeable relationship along with the noise.