The Kalman filter is a state space model for estimating an unknown (‘hidden’) variable using observations of related variables and models of those relationships. The Kalman filter is underpinned by Bayesian probability theory and enables an estimate of the hidden variable in the presence of noise.
There are plenty of tutorials online that describe the mathematics of the Kalman filter, so I won’t repeat those here (this article is a wonderful read). Instead, I’ll show you how to implement the Kalman filter framework to provide a dynamic estimate of the hedge ratio in a pairs trading strategy. I’ll provide just enough math as is necessary to follow the implementation.
To implement our Kalman filter, we need four variables:
- A vector of our observed variable
- A vector of our hidden variable
- A state transition model (which describes how the hidden variable evolves from one state to the next)
- An observation model (a matrix of coefficients for the other variable – we use a hedge coefficient and an intercept)
For our hedge ratio/pairs trading application, the observed variable is one of our price series p1 and the hidden variable is our hedge ratio, β. The observed and hidden variables are related by the familiar spread equation:p1=β∗p2+ϵ, where ϵ is noise (in our pairs trading framework, we are essentially making bets on the mean reversion of ϵ). In the Kalman framework, the other price series, p2 provides our observation model.
We also need to define a state transition model that describes the evolution of β from one time period to the next. If we assume that β follows a random walk, then our state transition model is simply βt=βt−1+ω
Visit Robot Wealth website to read the rest of the article, and to download the sample R code.
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