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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 [Formula omitted] 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., Author(s): Ahmed Ghezal [sup.1] Author Affiliations: (1) Department of Mathematics and Computer Sciences, University Center of Mila, , Mila, Algeria Introduction An autoregressive model with Markov switching has been introduced [...]

This paper presented a comprehensive study of a three-dimensional nonlinear system of difference equations, which can be reduced to a two-dimensional bilinear system. The system monitored the evolution of three sequences (P m) , (Q m) , (R m) , governed by recursive relations. We investigated the solvability of this system and provided general closed-form solutions for various parameter conditions. Furthermore, the simulations provided valuable insights into the dynamic behavior of animals, modeled using recursive difference equations. The model encapsulated essential behavioral metrics, represented by the variables P , Q , and R , which corresponded to individual actions, social interactions, and environmental stressors, respectively. These variables adapted dynamically in response to internal and external influences, illustrating the system's sensitivity to various behavioral and environmental conditions. [ABSTRACT FROM AUTHOR]

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In this study, we explored an asymmetric multivariate stochastic difference volatility model that extends various probabilistic and statistical properties previously discussed in the literature. We rigorously established that the model exhibits periodic stationarity and periodic ergodicity. Additionally, we delved into the robust consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator (QMLE), providing a comprehensive analysis of its theoretical underpinnings. Finally, we demonstrated the practical applicability of our major findings through a series of pertinent applications. This work not only contributes to the existing body of knowledge on stochastic volatility modeling, but also opens new avenues for further research in this domain. [ABSTRACT FROM AUTHOR]

In this paper, we focus on obtaining the closed-form solution for the following bidimensional system of higher-order rational difference equations: u(1) n+1 = 1 3 - 2u(2) n-m, u(2) n+1 = 1 3 - 2u(1) n-m, n,m N0, where the initial values u(1) -j and u(2) -j, j {0, 1, ...,m} are real numbers not equal to 3/2. We show that the solutions of this system are associated with Mersenne numbers and/or Mersenne-Lucas numbers. It is shown that the global stability of positive solutions of this system holds. Finally, we provide numerical examples to illustrate our results. [ABSTRACT FROM AUTHOR]

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 the current paper, we explore some necessary probabilistic properties for the asymptotic inference of a broad class of periodic bilinear– GARCH processes (P – BLGARCH) obtained by adding to the standard periodic GARCH models one or more interaction components between the observed series and its volatility process. In these models, the parameters of conditional variance are allowed to switch periodically between different regimes. This specification lead us to obtain a new model which is able to capture the asymmetry and hence leverage effect characterized by the negativity of the correlation between returns shocks and subsequent shocks in volatility patterns for seasonal financial time series. So, the goal here is to give in first part some basic structural properties of P – BLGARCH necessary for the remainder of the paper. In the second part, we study the consistency and the asymptotic normality of the quasi-maximum likelihood estimator (QMLE) illustrated by a Monte Carlo study and applied to model the exchange rate of the Algerian Dinar against the US-dollar. [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]