Gaussian mixture vector autoregression

This paper proposes a new nonlinear vector autoregressive (VAR) model referred to as the Gaussian mixture vector autoregressive (GMVAR) model. The GMVAR model belongs to the family of mixture vector autoregressive models and is designed for analyzing time series that exhibit regime-switching behavior. The main difference between the GMVAR model and previous mixture VAR models lies in the definition of the mixing weights that govern the regime probabilities. In the GMVAR model the mixing weights depend on past values of the series in a specific way that has very advantageous properties from both theoretical and practical point of view. A practical advantage is that there is a wide diversity of ways in which a researcher can associate different regimes with specific economically meaningful characteristics of the phenomenon modeled. A theoretical advantage is that stationarity and ergodicity of the underlying stochastic process are straightforward to establish and, contrary to most other nonlinear autoregressive models, explicit expressions of low order stationary marginal distributions are known. These theoretical properties are used to develop an asymptotic theory of maximum likelihood estimation for the GMVAR model whose practical usefulness is illustrated in a bivariate setting by examining the relationship between the EUR–USD exchange rate and a related interest rate data.