HMM based scenario generation for an investment optimisation problem

This is the post-print version of the article. The official published version can be accessed from the link below - Copyright @ 2012 Springer-Verlag.The Geometric Brownian motion (GBM) is a standard method for modelling financial time series. An important criticism of this method is that the parameters of the GBM are assumed to be constants; due to this fact, important features of the time series, like extreme behaviour or volatility clustering cannot be captured. We propose an approach by which the parameters of the GBM are able to switch between regimes, more precisely they are governed by a hidden Markov chain. Thus, we model the financial time series via a hidden Markov model (HMM) with a GBM in each state. Using this approach, we generate scenarios for a financial portfolio optimisation problem in which the portfolio CVaR is minimised. Numerical results are presented.This study was funded by NET ACE at OptiRisk Systems

Efficient Gibbs Sampling for Markov Switching GARCH Models

We develop efficient simulation techniques for Bayesian inference on switching GARCH models. Our contribution to existing literature is manifold. First, we discuss different multi-move sampling techniques for Markov Switching (MS) state space models with particular attention to MS-GARCH models. Our multi-move sampling strategy is based on the Forward Filtering Backward Sampling (FFBS) applied to an approximation of MS-GARCH. Another important contribution is the use of multi-point samplers, such as the Multiple-Try Metropolis (MTM) and the Multiple trial Metropolize Independent Sampler, in combination with FFBS for the MS-GARCH process. In this sense we ex- tend to the MS state space models the work of So [2006] on efficient MTM sampler for continuous state space models. Finally, we suggest to further improve the sampler efficiency by introducing the antithetic sampling of Craiu and Meng [2005] and Craiu and Lemieux [2007] within the FFBS. Our simulation experiments on MS-GARCH model show that our multi-point and multi-move strategies allow the sampler to gain efficiency when compared with single-move Gibbs sampling.Comment: 38 pages, 7 figure

Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime

An autoregressive process with Markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable Markov chain. In this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite. Consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden Markov chain conditional on the observations.Comment: Published at http://dx.doi.org/10.1214/009053604000000021 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Solution of Markov-switching Rational Expectations Models

This paper describes a method for solving a class of forward-looking Markov-switching Rational Expectations models under noisy measurement, by specifying the unobservable expectations component as a general-measurable function of the observable states of the system, to be determined optimally via stochastic control and filtering theory. Solution existence is proved by setting this function to the regime-dependent feedback control minimizing the mean-square deviation of the equilibrium path from the corresponding perfect-foresight autoregressive Markov jump state motion. As the exact expression of the conditional (rational) expectations term is derived both in finite and infinite horizon model formulations, no (asymptotic) stationarity assumptions are needed to solve forward the system, for only initial values knowledge is required. A simple sufficient condition for the mean-square stability of the obtained rational expectations equilibrium is also provided.Rational Expectations, Markov-switching dynamic systems, Dynamic programming, Time-varying Kalman filter

A Generalized ARFIMA Process with Markov-Switching Fractional Differencing Parameter

We propose a general class of Markov-switching-ARFIMA processes in order to combine strands of long memory and Markov-switching literature. Although the coverage of this class of models is broad, we show that these models can be easily estimated with the DLV algorithm proposed. This algorithm combines the Durbin-Levinson and Viterbi procedures. A Monte Carlo experiment reveals that the finite sample performance of the proposed algorithm for a simple mixture model of Markov-switching mean and ARFIMA(1, d, 1) process is satisfactory. We apply the Markov-switching-ARFIMA models to the U.S. real interest rates, the Nile river level, and the U.S. unemployment rates, respectively. The results are all highly consistent with the conjectures made or empirical results found in the literature. Particularly, we confirm the conjecture in Beran and Terrin (1996) that the observations 1 to about 100 of the Nile river data seem to be more independent than the subsequent observations, and the value of differencing parameter is lower for the first 100 observations than for the subsequent data.Markov chain; ARFIMA process; Viterbi algorithm; Long memory.

Classical Estimation of Multivariate Markov-Switching Models using MSVARlib

This paper introduces an upgraded version of MSVARlib, a Gauss and Ox- Gauss compliant library, focusing on Multivariate Markov Switching Regressions in their most general specification. This new set of procedures allows to estimate, through classical optimization methods, models belonging to the MSI(M)(AH)-VARX ``intercept regime dependent'' family. This research enhances the first package MSVARlib 1.1, which has been deeply inspired by the works of Hamilton and Krolzig. Not to mention the extension to a generalized multivariate regression framework, it notably augments the range of models with a possibly unlimited finite number of Markov states, offers automatic or manual intialization procedures and adds new statistical tests. The first part of this article provides the basic theoretical grounds of the related Markov-switching models. Following sections give some illustrations of the programs through univariate and multivariate examples. One is based on a non-linear reading of the american unemployment rate. A second study is focused on coincident stochastic models of US recessions and slowdowns. The paper concludes on possible extensions and new applications. Detailed guidelines in appendices and tutorial programs are provided to help the reader handling the Gauss package and the joined replication files.Multivariate Markov-Switching Regressions, Hidden markov Models, Non linear regressions, Open source Gauss library, Business cycle, EM algorithm, Kittagawa-Hamilton Filtering, Recession Detection Models, MSVAR, MS-VAR, Hamilton's Model, Krolzig MSVAR library,Filtered probabilities, Smoothed probabilities.