Probabilistic Properties of a Nonlinear ARMA Process with Markov Switching

Abstract

We consider a nonlinear autoregressive moving average (ARMA) process with Markov switching and find sufficient conditions for strict stationarity, geometric ergodicity, and the existence of moments of the process with respect to the stationary distribution. Functional central limit theorem is also obtained.

Consistency of quasi-maximum likelihood estimator for Markov-switching bilinear time series models

“…These regression functions may be linear as well as non-linear were investigated in order to capture the probabilistic and statistical properties of such models. For instance, MS-ARMA: Francq and Zakoïan (2001), MS-nonlinear ARMA: Lee (2005) and Yao and Attali (2000), MS-GARCH: Francq and Zakoïan (2005) among others. As important subclass of models bound with these regression functions is the so-called discrete-time bilinear process (X t , t ∈ Z) generated by the following stochastic difference equation…”

On the Markov-switching bilinear processes: stationarity, higher-order moments andβ-mixing

“…Þ almost surely (a:s:) where f is a measurable function from R 1 to R. Such solutions are called causal. The mentioned properties were studied recently for the MS-ARMA by FZ [10], Stelzer [24] and by Lee [17]. For the MS-GARCH by FZ [9], Liu [19] and Abramson and Cohen [1].…”

“…in each 'regime') linear or nonlinear models were investigated in order to capture the probabilistic and statistical properties of such models. For instance, MS-ARMA: Francq and Zakoïan (hereafter FZ) [10] and Stelzer [24], MS-nonlinear ARMA and bilinear processes: Lee [17], Yao and Attali [26] and Bibi and Aknouche [4], MS-GARCH: FZ [9], Hass et al [13], Liu [19] among others.…”

A Functional Central Limit Theorem for an ARMA(p, q) Process with Markov Switching

“…One way to prove the FCLT is to use the Markovian structure of the model to obtain mixing properties. If W t is ϕ-irreducible weak Feller and top Lyapunov exponent of A(u t ) is negative, then the process Y t is geometrically ergodic and β-mixing and the FCLT for x t is obtained (Lee, 2005). The stationarity and geometric ergodicity for vector valued MSARMA(p, q) process with a general state space parameter chain are examined by Stelzer (2009).…”

“…Functional central limit theorem(FCLT) is applied for statistical inference in time series to establish the asymptotics of various statistics concerning, for example a test for stability such as CUCUM or MOSUM and unit root testing. Probabilistic properties of MSARMA(p, q) models have been studied in, e.g., Francq and Zakoïan (2001), Yao and Attali (2000), Yang (2000), Lee (2005), and Stelzer (2009).…”