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This paper introduces a comprehensive class of models known as Markov-Switching Threshold Stochastic Volatility (MS-TSV) models, specifically designed to address asymmetry and the leverage effect observed in the volatility of financial time series. Extending the classical threshold stochastic volatility model, our approach expresses the parameters governing log-volatility as a function of a homogeneous Markov chain with a finite state space. The primary goal of our proposed model is to capture the dynamic behavior of volatility driven by a Markov chain, enabling the accommodation of both gradual shifts due to economic forces and sudden changes caused by abnormal events. Following the model's definition, we derive several probabilistic properties of the MS-TSV models, including strict (or second-order) stationarity, causality, ergodicity, and the computation of higher-order moments. Additionally, we provide the expression for the covariance function of the squared (or powered) process. Furthermore, we establish the limit theory for the Quasi-Maximum Likelihood Estimator (QMLE) and demonstrate the strong consistency of this estimator. Finally, a simulation study is presented to assess the performance of the proposed estimation method. [ABSTRACT FROM AUTHOR]

In this work, we consider doubly Markov switching AR models, where analytic tractability and flexibility are quite simply a competitive advantage, which becomes an attractive tool for modeling economic and financial time series. In these models, the parameters are allowed to depend on an unobservable time-homogeneous Markov chain with finite state space. So, we discuss some basic probabilistic properties of D - M S A R p , q model such as conditions ensuring the existence of strict, second-order stationarity solution, causal and ergodic solution and its moments properties. The quasi-maximum likelihood estimator of the parameters in the model is shown to be strongly consistent. [ABSTRACT FROM AUTHOR]

Markov-switching (MS) models are becoming increasingly popular as efficient tools of modeling various phenomena in different disciplines, in particular for non Gaussian time series. In this articlept", we propose a broad class of Markov-switchingBILINEAR–GARCHprocesses (MS−BLGARCHhereafter) obtained by adding to aMS−GARCHmodel one or more interaction components between the observed series and its volatility process. This parameterization offers remarkably rich dynamics and complex behavior for modeling and forecasting financial time-series data which exhibit structural changes. In these models, the parameters of conditional variance are allowed to vary according to some latent time-homogeneous Markov chain with finite state space or “regimes.” The main aim of this new model is to capture asymmetric and hence purported to be able to capture leverage effect characterized by the negativity of the correlation between returns shocks and subsequent shocks in volatility patterns in different regimes. So, first, some basic structural properties of this new model including sufficient conditions ensuring the existence of stationary, causal, ergodic solutions, and moments properties are given. Second, since the second-order structure provides a useful information to identify an appropriate time-series model, we derive the expression of the covariance function of forMS−BLGARCHand for its powers. As a consequence, we find that the second (resp. higher)-order structure is similar to some linear processes, and henceMS−BLGARCH(resp. its powers) admit anARMArepresentation. This finding allows us for parameter estimation viaGMMprocedure proved by a Monte Carlo study and applied to foreign exchange rate of the Algerian Dinar against the single European currency. [ABSTRACT FROM AUTHOR]

In this article we introduce Markov-switching bilinear (MS-BL) process in order to model time series that exhibit structural breaks but behave (locally) strong nonlinear features. One advantage of this new class of models is its capability to take into account the important characterization of non-Gaussianity behaviour, and its ability to capture certain phenomena commonly observed in practice, such as limit cycles, self-excitation, asymmetric distribution, leptokurtosis and Taylor effect. In this class of models, which includes many popular growth curve models, the parameters are allowed to change over time according to an unobservable time-homogeneous Markov chain with finite state space. Our attention is focussed firstly on some basic issues concerning this class including sufficient conditions ensuring the existence of strictly stationarity and ergodic solutions and the existence of finite order moments. As a consequence, we find that the-structures of the process and its powers are similar to that of an ARMA with some uncorrelated innovation. Secondary, we invest the fundamental problems linked with MS-BL models, i.e. parameters estimation by a minimum distance estimator (MDE). More precisely, we provide the detail on the asymptotic properties of MDE, in particular, we discuss conditions for its consistency and asymptotic normality. Numerical experiments on simulated datasets are presented to highlight the theoretical results. [ABSTRACT FROM AUTHOR]

This article investigates some probabilistic properties and statistical applications of general Markov-switching bilinear processesthat offer remarkably rich dynamics and complex behaviour to model non-Gaussian data with structural changes. In these models, the parameters are allowed to depend on unobservable time-homogeneous and stationary Markov chain with finite state space. So, some basic issues concerning this class of models including necessary and sufficient conditions ensuring the existence of ergodic stationary (in some sense) solutions, existence of finite moments of any order and-mixing are studied. As a consequence, we observe that the local stationarity of the underlying process is neither sufficient nor necessary to obtain the global stationarity. Also, the covariance functions of the process and its power are evaluated and it is shown that the second (respectively, higher)-order structure is similar to some linear processes, and hence admitrepresentation. We establish also sufficient conditions for themodel to bemixing and geometrically ergodic. We then use these results to give sufficient conditions formixing of a family ofprocesses. A number of illustrative examples are given to clarify the theory and the variety of applications. [ABSTRACT FROM PUBLISHER]