From Exponential to Polynomial: An Exact Filter for High-Dimensional MSM Models

In this paper we propose a new formulation of the Bayesian Filter as used in the discrete-time Markov-Switching-Multifractal (MSM) model of volatility based on existing permutation symmetry within the likelihood structure. We show both analytically and empirically that such a formulation leads to a reduction in time complexity from $O(D^k)$ to $O(k^D)$ thereby significantly reducing the computational bottleneck associated with dimensionality. We compare the agreement between the naive and sector filters and find that while there are significant disagreements, the ground-truth recovery of the latter seems to improve on the former.