Markov-switching BILINEAR − GARCH models: Structure and estimation

ABSTRACT

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-switching BILINEAR–GARCH processes (MS − BLGARCH hereafter) obtained by adding to a MS − GARCH model 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 for MS − BLGARCH and for its powers. As a consequence, we find that the second (resp. higher)-order structure is similar to some linear processes, and hence MS − BLGARCH (resp. its powers) admit an ARMA representation. This finding allows us for parameter estimation via GMM procedure proved by a Monte Carlo study and applied to foreign exchange rate of the Algerian Dinar against the single European currency.

KEYWORDS:

• ARMA representation
• GMM
• Markov-switching BILINEAR-GARCH processes
• Stationarity.

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62M10
• Secondary 62M05

Acknowledgments

We would like to thank Prof. N. Balakrishnan editor of the journal, for his attention, encouragement, and valuable advice. Also, we are very grateful to two anonymous referees for reading the article very carefully and making many constructive remarks and suggestions.