State-Space Filter

Filtering time-series state-space

Bottom Line

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The Kalman filter estimates latent fair value and momentum from noisy observed prices using a two-state linear model with RTS backward smoothing.

Avg Momentum

Trend Participation

Innovation Var

Band Width

Tickers Tracked

Action Framework

Momentum + & Low Innovation

Trend is reliable. Position with the filter's direction. High conviction.

Momentum + & High Innovation

Trend exists but uncertainty is elevated. Reduce position size, widen stops.

Momentum - & Low Innovation

Controlled decline. Defensive positioning appropriate. Systematic, not panicked.

Momentum - & High Innovation

Regime uncertainty. Reduce exposure, wait for filter to stabilize before re-entry.

Overview

The Kalman filter treats the observed daily closing price as a noisy measurement of two latent states: log fair value and momentum (the rate of change of fair value). By framing the problem as linear-Gaussian state estimation, the filter continuously updates its beliefs about the true price level while discounting transient noise, microstructure effects, and short-horizon volatility.

What It Estimates

Latent fair value — the de-noised log-price level the market is centered on
Momentum state — the current drift or trend in fair value
Uncertainty bands — the filter's confidence in its estimates via the covariance matrix

Strengths

Optimal linear filter (minimum variance)
Recursive: updates in O(1) per timestep
Transparent uncertainty quantification
Natural framework for missing data

Limitations

Assumes linear dynamics and Gaussian noise
Parameter estimation (Q, R) requires calibration
May lag during structural breaks
Single regime model (no switching)

How It Works

The filter operates in a predict-update cycle, processing one observation at a time.

Observe noisy price. Read the day's closing price $y_t$.
Predict via state transition. $\hat{x}_{t|t-1} = A \hat{x}_{t-1|t-1} + C u_t$, $P_{t|t-1} = A P_{t-1|t-1} A' + Q$.
Update with Kalman gain. $\hat{x}_{t|t} = \hat{x}_{t|t-1} + K_t e_t$.
Smooth with RTS backward pass. Refine all historical estimates using future data.

Key Equations

State Vector

$$\mathbf{x}_t = \begin{bmatrix} \log p_t \\ \mu_t \end{bmatrix}$$

State Transition

$$\mathbf{x}_{t+1} = \mathbf{A}\,\mathbf{x}_t + \mathbf{C}\,u_t + \mathbf{w}_t, \quad \mathbf{w}_t \sim \mathcal{N}(0, \mathbf{Q})$$

Kalman Gain

$$\mathbf{K}_t = \mathbf{P}_{t|t-1}\,\mathbf{H}^\top \left(\mathbf{H}\,\mathbf{P}_{t|t-1}\,\mathbf{H}^\top + R_t\right)^{-1}$$

RTS Smoother

$$\hat{\mathbf{x}}_{t|T} = \hat{\mathbf{x}}_{t|t} + \mathbf{L}_t\left(\hat{\mathbf{x}}_{t+1|T} - \hat{\mathbf{x}}_{t+1|t}\right)$$

Diagnostics

Filter health is assessed through the innovation sequence $e_t = y_t - H \hat{x}_{t|t-1}$.

Autocorrelation test — Ljung-Box on innovations
Normality test — Jarque-Bera; heavy tails indicate non-Gaussian dynamics
Q/R ratio — governs responsiveness vs smoothness tradeoff

Key References

Kalman, R. E. (1960). A New Approach to Linear Filtering and Prediction Problems.
Rauch, H. E., Tung, F., and Striebel, C. T. (1965). Maximum Likelihood Estimates of Linear Dynamic Systems.
Harvey, A. C. (1990). Forecasting, Structural Time Series Models and the Kalman Filter.
Durbin, J. and Koopman, S. J. (2012). Time Series Analysis by State Space Methods.