State-space models and the Kalman filter

Tracking a hidden state

Many quantities that matter cannot be seen directly: the fair value of a stock behind noisy quotes, the hedge ratio between two assets that drifts through time, the current level of volatility. A state-space model describes them with two equations. The state evolves by itself, $x_t=Fx_{t-1}+w_t$ with noise variance $Q$. What you observe is a noisy view of it, $y_t=Hx_t+v_t$ with noise variance $R$. The Kalman filter combines the two in a loop that gives the best linear estimate of the state from the data seen so far, together with its uncertainty $P$. It is the exact solution when the noises are Gaussian and the model is linear.

The formulas above are the scalar local-level model, where the hidden level is a random walk ($F=H=1$). Each step has two parts. Predict: the best guess for the new state is the old estimate, and the uncertainty grows by $Q$ to $P^{-}=P+Q$. Update: the gain $K=P^{-}/(P^{-}+R)$ decides how far to move toward the new observation. A large $R$ (a noisy measurement) gives a small $K$, so the filter trusts its own prediction; a small $R$ gives $K$ near 1, so it follows the data. After the update the uncertainty falls to $(1-K)P^{-}$. The prediction error $y-\hat x^{-}$, called the innovation, should be white noise if the model is right, and $Q$ and $R$ are usually estimated by maximising the likelihood built from the innovations.

• State equation $x_t=Fx_{t-1}+w_t$ and observation equation $y_t=Hx_t+v_t$ with noise variances $Q$ and $R$.
• Kalman gain $K=P^{-}/(P^{-}+R)$: near 0 means trust the model, near 1 means trust the data.
• Uses: a time-varying hedge ratio for pairs, a fair value behind noisy prices, volatility states; innovations should be white noise.

Lab · what's graded

• The predict step10 pts
• The Kalman gain15 pts
• The updated estimate15 pts
• The updated uncertainty15 pts
• A noisier measurement10 pts
• A very noisy measurement10 pts
• Match the vocabulary15 pts
• A drifting hedge ratio10 pts