Moment Based Regression Algorithms for Drift and Volatility Estimation in Continuous-Time Markov Switching Models

We consider a continuous time Markov switching model (MSM) which is widely used in mathematical finance. The aim is to estimate the parameters given observations in discrete time. Since there is no finite dimensional filter for estimating the underlying state of the MSM, it is not possible to compute numerically the maximum likelihood parameter estimate via the well known expectation maximization (EM) algorithm. Therefore in this paper, we propose a method of moments based parameter estimator. The moments of the observed process are computed explicitly as a function of the time discretization interval of the discrete time observation process. We then propose two algorithms for parameter estimation of the MSM. The first algorithm is based on a least-squares fit to the exact moments over different time lags, while the second algorithm is based on estimating the coefficients of the expansion (with respect to time) of the moments. Extensive numerical results comparing the algorithm with the EM algorithm for the discretized model are presented. Copyright © 2008 The Authors. Journal compilation © Royal Economic Society 2008

[1]  John B. Moore,et al.  Hidden Markov Models: Estimation and Control , 1994 .

[2]  S. Frühwirth-Schnatter,et al.  Markov Chain Monte Carlo Methods for Parameter Estimation in Multidimensional Continuous Time Markov Switching Models , 2010 .

[3]  Bent E. Sørensen,et al.  GMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study , 1996 .

[4]  Ulrich G. Haussmann,et al.  Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain , 2004, Finance Stochastics.

[5]  E. Nummelin.  General irreducible Markov chains and non-negative operators: Preface , 1984 .

[6]  James D. Hamilton.  A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle , 1989 .

[7]  E. Nummelin.  General irreducible Markov chains and non-negative operators: List of symbols and notation , 1984 .

[8]  Robert J. Elliott,et al.  American options with regime switching , 2002 .

[9]  Vikram Krishnamurthy,et al.  Time discretization of continuous-time filters and smoothers for HMM parameter estimation , 1996, IEEE Trans. Inf. Theory.

[10]  R. Varga,et al.  Proof of Theorem 4 , 1983 .

[11]  L. Hansen.  Large Sample Properties of Generalized Method of Moments Estimators , 1982 .

[12]  Robert J. Elliott,et al.  Estimating the implicit interest rate of a risky asset , 1994 .

[13]  N. Shephard,et al.  Econometric analysis of realized volatility and its use in estimating stochastic volatility models , 2002 .

[14]  Xin Guo.  An explicit solution to an optimal stopping problem with regime switching , 2001, Journal of Applied Probability.

[15]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[16]  Robert J. Elliott,et al.  Drift and volatility estimation in discrete time , 1998 .

[17]  Kyriakos Chourdakis,et al.  Continuous Time Regime Switching Models and Applications in Estimating Processes with Stochastic Volatility and Jumps , 2002 .

[18]  Loukia Meligkotsidou,et al.  Filtering Methods for Mixture Models , 2007 .

[19]  P. Fearnhead,et al.  Particle filters for partially observed diffusions , 2007, 0710.4245.

[20]  Bent E. Sørensen,et al.  A Continuous-Time Arbitrage-Pricing Model With Stochastic Volatility and Jumps , 1996 .

[21]  Giuseppe Cavaliere.  Stochastic Volatility: Selected Readings , 2006 .

[22]  Robert J. Elliott,et al.  Exact adaptive filters for Markov chains observed in Gaussian noise , 1994, Autom..

[23]  P. Fearnhead,et al.  Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion) , 2006 .

[24]  Allan Timmermann,et al.  Moments of Markov switching models , 2000 .

[25]  Robust continuous-time smoothers without two-sided stochastic integrals , 2002, IEEE Trans. Autom. Control..

[26]  Robert J. Elliott,et al.  New finite-dimensional filters and smoothers for noisily observed Markov chains , 1993, IEEE Trans. Inf. Theory.

[27]  Ming-Deh A. Huang,et al.  Proof of proposition 2 , 1992 .