Skewed and fat-tailed error process in Kalman filter

Question

I'm trying to filter a time series, of which I occasionally can observe the state variable variable but not always. I also have a noisy measure of this state variable all the time. By picking the time points when I have both, I can compute the error between the two, which gives me a sense of what the noisy measure is like.

The distribution of error seems to be both skewed and fat-tailed. Based on my rudimentary understanding of Kalman filter, this type of error process would violate the the assumption of normally distributed errors, and using just mean and variance would be biased because the first two moments of the probability distribution are not suffice to characterize the whole thing. I have two questions.

(1) if I don't have other good approaches for now and insist on using Kalman filter, what risk am I taking? Or in other words, how does this kind of distribution affect the accuracy of Kalman filter.

(2) Since the parameters of the measurement equation has to be estimated as well (both coefficients and error variance), I was thinking about using Box-Cox transformation to fix the skew and fit it to a t-distribution. Can I then estimate the parameters online using MLE of t-distribution.

Answer 1

The definition of a state space model is a hidden/unobserved markov chain $x_t$, and observed data $y_t$. I don't know if you can go breaking the rules so early on like that. First hunch: try a model like this for each security. It might accomplish what you want to accomplish, while not breaking the rule that states have to be hidden.

$$ y_t = c + x_t + v_t $$ $$ x_t = x_{t-1} + w_t $$ where $y_t \in \mathbf{R}^3$, $x_t \in \mathbf{R}^1$, $w_t \overset{iid}{\sim} \text{Normal}(0, Q)$, $v_t \overset{iid}{\sim} \text{Normal}(0, R)$.

At regularly sampled intervals $$ \left[ \begin{array}{c} y_{1,t}\\ y_{2,t} \\ y_{3,t} \end{array}\right] = \left[ \begin{array}{c} ask_t\\ execution_t \\ bid_t \end{array}\right] $$ and $c = (\epsilon/2, 0, -\epsilon/2)$ so $\epsilon$ is the bid-ask spread. Finally $$ R = \left[ \begin{array}{ccc} r_{1,1} & 0 & 0\\ 0 & r_{2,2} & 0 \\ 0 & 0 & r_{3,3} \end{array}\right] $$ so you would expect $r_{2,2}$ to be equal to or atleast very close to $0$, and the other two you might constrain to be the same. Anyway the parameters you would have to estimate would be $(c, R, Q)$, which is do-able.

Also, nowhere am I saying that the state is observable. But I am kind of working around that by saying $y_{2,t} = x_t + \{\text{very little noise}\}$, and we're thinking about $x_t$ as the 'true' price (you kind of have to defend your interpretation of 'true' here).

Then do diagnostics on the residuals of this model. You will probably have better looking ones. Just a guess, though. I don't work in this domain. If it's still a problem you can make $v_t$ have fatter tails.

Possible extensions:

1. Let $c$ change overtime (call it $c_t$)
2. Sample at irregularly sampled intervals
3. better state dynamics.
4. model multiple securities all at once
5. Let the variances be stochastic or maybe just changing deterministically in time
6. Executions happen on the NBBO, which means sometimes $y_{1,t} = y_{2,t}$ which doesn't jive with your choice of $c$. So $c_t$ might even need to be switching or something.