ISBN 978-3-8439-2976-9

978-3-8439-2976-9, Reihe Mathematik

Elisabeth Leoff
Stochastic Filtering in Regime-Switching Models: Econometric Properties, Discretization and Convergence

174 Seiten, Dissertation Technische Universität Kaiserslautern (2016), Softcover, A5

Zusammenfassung / Abstract

Regime-switching models for asset returns are widely used in finance and statistics. In the classical Hidden Markov Model (HMM) the values of the drift process are controlled by a Markov chain with finite state space. This Markov chain is assumed to be unobservable and describes the underlying state of the economy. In the Markov Switching Model (MSM) the volatility jumps with the Markov chain, as well. Due to the unobservable chain, for applications one has to use stochastic filtering theory to estimate its current state.

In this work we consider HMM and MSM in discrete and continuous time. We analyze their econometric properties by investigating Stylized Facts and see the limitations of the HMM due to its constant volatility.

Representations of the autocorrelation functions in both models will be derived, which depend on the transition probabilities of the chain. We demonstrate the existence of volatility clustering in the MSM, which is not present in the HMM. For the leverage effect in the MSM we prove a similar representation as for the autocorrelations and derive a heuristic choice of parameters that leads to leverage effect in the 2-state model.

We then consider the connections between the discrete- and continuous-time models by investigating the issues of consistency and convergence in regime-switching models. We demonstrate that the discrete-time HMM-filter corresponds to the robust discretization of the continuous-time filter and prove a stability result for a sequence of SDE's. With this we show that the continuous-time HMM-filter converges to the vector of the invariant distribution if what we call the signal-to-noise matrix converges to 0. The behavior of the discrete-time filters for increasing number of grid points will also be considered.

Following this, we prove that there is no filtering problem in the continuous-time MSM. To obtain a continuous-time model with better econometric properties than the HMM, that still allows for a filtering problem, the Filterbased-volatility Hidden Markov Model (FB-HMM) is introduced. We prove that in a sense this is the optimal approximation of the MSM and show the filtering equations in this model. To analyze its consistency we prove that already the Euler discretization converges. We conclude by considering the stylized facts of the model and show that they are similar to those in the MSM, albeit a bit less pronounced. In this sense the FB-HMM lies between the HMM and the MSM.