State Space Models and Kalman Filtering for Adaptive Estimates
Summary
The document introduces linear state space models, where an underlying state evolves over time and observations provide noisy, indirect information about it. It defines the state and observation equations, their transition and measurement noise, and the initial state distribution. A changing hedge ratio between related assets is given as a finance example.
It explains the Kalman filter as a Bayesian sequence of prediction and updating steps: use the prior state estimate to forecast an observation, compare that forecast with the new observation, then adjust the state estimate using the forecast error. It also distinguishes prediction, filtering of current states, and smoothing of past states with later information. The article describes how estimated means and covariances support multi-step forecasts. Its presentation is theoretical and notation-heavy; it does not provide a trading test or demonstrate predictive performance, and it notes that simpler time-series models may be preferable when they suffice.
Key ideas
- State space models represent hidden states that change over time and observations contaminated by measurement noise.
- The Kalman filter updates state estimates as new observations arrive using a Bayesian framework.
- Prediction, filtering, and smoothing answer different questions about future, current, and past states.
- A time-varying hedge ratio in a pairs trade is one potential financial application.
- Forecast distributions include uncertainty as well as expected values.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.