Odd Result from Computing Correlation Matrix from Kalman Filter Posteriori Covariance Estimate

Question

I am using a Kalman Filter to estimate the return dynamics of a forwards curve on a particular commodity. My state space is the initial forwards values, and an initial guess of the drift functions for each forward (initially just the first observed price change). These are for OTC products, my estimates for uncertainty are based on market consensus standard deviations. My Estimate of the Q matrix (process covariance noise) is based on the average variance, and my estimate for the R matrix (covariance observation noise) is based on each days market consensus standard deviation.

From the resulting Posteriori Covariance matrix, I estimate the correlation matrix. The result is constant correlation. Basically the off diagonal is roughly 72% almost everywhere. I am not sure why this should be the linearly optimal result, or perhaps there is a flaw in my understanding of the Kalman filter?

I can not provide the data I used for this experiment, but I am more than happy to answer clarifying questions if it would help the community to better answer my question: Particularly, what could be the cause of observational constant correlation?

Answer 1

Let your linear system be defined by the latent multivariate state variable $x_t$, progressing in an AR(1) fashion, and let your observation at time step $t$ be linear in the latent state:

$$ \begin{align} x_{t+1}&=Ax_t+u_t\\ y_{t}&=Hx_t+v_t\\ \end{align} $$ with $u_t\sim N(0,Q)$ and $v_t\sim N(0,R)$ time-and-state-independent process noise.

Let $K_t$ denote the Kalman gain and $P_{t,t}$ denote the updated estimation error covariance.

As this system is time-invariant ($A,H,Q,R$ are constants), it will result in

• a steady state Kalman Gain $K_t\to K_{\infty}$
• a steady state updated estimation error covariance $P_{t,t}\to P_{\infty}$

for $t$ sufficiently large* and whenever the spectral radius of $A$ is smaller than 1.

Below, you find a couple of references (... I googled them, maybe some more thorough research is needed on your end):

A quite recent introduction to the KF

Search for 'steady' in here

Section 2.2 of these leture nodes

* Sufficiently large means usually some small number; in my experiments the filter usually converges after 10 steps or so.

HTH?