The Kalman Filter: Dynamic Hedge Ratios in State-Space Form

The Engle-Granger and Johansen estimators treat the cointegrating vector $\gamma$ as a constant estimated once over a formation window. For many pairs this is adequate, but when the economic relationship between two firms drifts — leverage ratios change, product mix shifts, a stock reclassifies into a different sector — the static $\gamma$ becomes stale. Trades entered on a mis-estimated hedge ratio carry unintended directional exposure, and the spread no longer mean-reverts to the expected equilibrium.

The Kalman filter solves this by treating $\gamma_t$ as a time-varying state that is tracked and updated daily using every new price observation. The filter predicts the spread given today's hedge ratio, observes the actual price, and updates the estimate — balancing prior belief against new evidence in an optimal (minimum-variance) way. It is the continuous-time analog of rolling window OLS, but without the sharp cut-off and with an explicit model for how $\gamma_t$ evolves.

Learning objectives. By the end of this note you should be able to: