How is the Kalman filter used in finance through state-space models?

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AcadiFi · Accepted Answer

A state-space model has two equations. The measurement equation y_t = Z_t alpha_t + epsilon_t links observables to the unobserved state alpha_t...

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The Kalman filter recursively computes the conditional mean and covariance of the state given observations up to time t. Two steps each period: predict (project state forward using T_t) and update (blend prediction with new observation weighted by the Kalman gain K_t).

At Starling, dynamic CAPM beta can be modeled as alpha_t = beta_t, with y_t = r_{stock,t} - r_f = beta_t (r_m,t - r_f) + epsilon_t and beta_t = beta_{t-1} + eta_t. The filter outputs a smoothed time series of beta that reacts to new data without the window-choice arbitrariness of rolling OLS.

Applications extend to yield-curve factor estimation (Nelson-Siegel with time-varying factors), pairs-trading spreads with drifting hedge ratios, and macro nowcasting with mixed-frequency data. For AcadiFi candidates preparing Level III: remember Kalman assumes linearity and Gaussian noise. Nonlinear extensions include the Extended Kalman Filter (linearize locally) and the Unscented Kalman Filter (propagate sigma points). For non-Gaussian or highly nonlinear problems, particle filters dominate — see the next question.

How is the Kalman filter used in finance through state-space models?

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