advanced-algorithmic-trading

Chapter 13
State Space Models and the Kalman
Filter
Thus far in our analysis of time series we have considered linear time series models including
ARMA, ARIMA as well as the GARCH model for conditional heteroskedasticity. In this chapter
we are going to consider a more general class of models known as state space models. The
primary benefit of these models is that unlike the ARIMA family their parameters can adapt
over time.
State space models are very general and it is possible to put the models we have considered
to date into a state space formulation. However in order to keep the analysis straightforward it
is often better to use the simpler representation previously described.
The general premise of a state space model is that we have a set of states that evolve in time
(such as the hedge ratio between two cointegrated pairs of equities) but our observations of these
states contain statistical noise (such as market microstructure noise), and hence we are unable
to ever directly observe the "true" states.
The goal of the state space model is to infer information about the states, given the observations,
as new information arrives. A famous algorithm for carrying out this procedure is the
Kalman Filter, which we will discuss at length in this chapter.
The Kalman Filter is ubiquitous in engineering control problems such as guidance & navigation,
spacecraft trajectory analysis and manufacturing. However it is also widely used in
quantitative finance.
In engineering, for instance, a Kalman Filter will be used to estimate values of the state,
which are then used to control the system under study. This introduces a feedback loop–often
in real-time.
Perhaps the most common usage of a Kalman Filter in quantitative trading is to update
hedging ratios between assets in a statistical arbitrage pairs trade. We will consider such an
example in this, and subsequent, chapters.
Generally there are three types of inference that are of interest when considering state space
models:
• Prediction - Forecasting subsequent values of the state
• Filtering - Estimating the current values of the state from past and current observations
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