On stability of nonlinear AR processes with Markov switching

Yao, Jianfeng; Attali, Jean-Gabriel

doi:10.1017/s000186780000999x

Cited by 14 publications

(12 citation statements)

References 14 publications

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“…7] which, due to our slightly weaker hypothesis (4), are the following facts. Following the proof of Lemma 26, observe that, due to (1), (28) and (27), on…”

Section: A2 Asymptotic Behavior Of the Log-likelihoodmentioning

confidence: 92%

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Consistency of the maximum likelihood estimate for non-homogeneous Markov–switching models

Ailliot¹,

Pène

2015

ESAIM: PS

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Many nonlinear time series models have been proposed in the last decades. Among them, the models with regime switchings provide a class of versatile and interpretable models which have received a particular attention in the literature. In this paper, we consider a large family of such models which generalize the well known Markov-switching AutoRegressive (MS-AR) by allowing non-homogeneous switching and encompass Threshold AutoRegressive (TAR) models and prove the consistency of the maximum likelihood estimator under general conditions. We show that these conditions apply to specific but representative models with non-homogeneous Markov switchings. The famous MacKenzie River lynx dataset is used to illustrate one of these models.

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“…7] which, due to our slightly weaker hypothesis (4), are the following facts. Following the proof of Lemma 26, observe that, due to (1), (28) and (27), on…”

Section: A2 Asymptotic Behavior Of the Log-likelihoodmentioning

confidence: 92%

“…It is a key step to prove the consistence of the MLE (see Theorem 2). Various authors have studied the ergodicity of MS-AR ( [28], [27], [11]) and TAR ( [6], [3]) models. A classical approach to prove the ergodicity of a non-linear time series consists in establishing a drift condition.…”

Section: Properties Of the Markov Chainmentioning

confidence: 99%

Consistency of the maximum likelihood estimate for non-homogeneous Markov–switching models

Ailliot¹,

Pène

2015

ESAIM: PS

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“…The MS-NAR model is called sublinear if conditions E2 and E3 hold. In Proposition 2.1, we summarize some results, given by Yao and Attali (1999), for sublinear MS-NAR model. Proposition 2.1 (Yao and Attali).…”

Section: E1mentioning

confidence: 92%

“…It is somewhat complex to determine the stability of the MS-NAR model. In this section, we recall known results given by Yao and Attali (1999). Our aim is to summarize the sufficient conditions that ensure the existence and uniqueness of a strictly stationary ergodic solution for the model, as well as the existence for the respective stationary distribution of a moment of order s > 1.…”

Section: Stability and Existence Of Momentsmentioning

confidence: 99%

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A Robbins–Monro Algorithm for Non‐Parametric Estimation of NAR Process with Markov Switching: Consistency

Fermín

Rios

Rodríguez

2017

Journal Time Series Analysis

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We consider nonparametric estimation for autoregressive processes with Markov switching. In this context, the Nadaraya-Watson type estimator of regression funtions is interpreted as solution of a local weighted least-square problem, which does not closed-form solution in the case of hidden Markov switching. We introduce a nonparametric recursive algorithm to approximate the estimator. Our algorithm restores the missing data by means of a Monte-Carlo step and estimate the regression function via a Robbins-Monro step. Consistency of the estimator is proved using the strong α-mixing property of the model. Finally, we present some simulations illustrating the performances of our nonparametric estimation procedure.

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HiddenMarkov Models

Rydén

2004

Encyclopedia of Actuarial Science

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Hidden Markov Models (HMMs) are a popular and widespread tool for modeling a large range of time series data and for generating sequences of dependent observations. HMMs have been applied successfully to various complex problems, being especially effective with those requiring a huge amount of measured data, such as workload access patterns in computer storage systems and pattern recognition in speech, handwriting and music, to name but a few. Nowadays, the theory of HMMs is well developed in the case of finite, discrete processes. However in many applications, such as storage workload modelling [1], which we discuss in some detail, discrete models are not ideal and we prefer a continuoustime approach. The aim of this project is to develop such a new theory in continuous time, based on the discrete HMMs, in particular an adaptation of the so-called Baum-Welch algorithm to facilitate the efficient computation of the relevant parameters of a HMM. Therefore this thesis first introduces the main concepts related to HMMs in the discrete case, then transfers its focus to a particular application, namely workloads in FLASH memory systems, while trying to spell out the limitations of the discrete approach. Finally a continuous-time approach is developed that leads to the implementation of a new, but similar, version of the Baum-Welch algorithm in a very simple context. The substantially increased complexity inherent in a continuous-time analysis, compared with discrete time, obviates the numerical analysis of models with more than two hidden states in the timescale of this project, but the theoretical and algorithmic extension to any number of hidden states is explained as "future work".

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On stability of nonlinear AR processes with Markov switching

Cited by 14 publications

References 14 publications

Consistency of the maximum likelihood estimate for non-homogeneous Markov–switching models

Consistency of the maximum likelihood estimate for non-homogeneous Markov–switching models

A Robbins–Monro Algorithm for Non‐Parametric Estimation of NAR Process with Markov Switching: Consistency

HiddenMarkov Models

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