Ebook Modeling and Estimation of The Synchronization in Multivariate Regime-Switching Models
Synchronization of cyclical behavior of economic and financial time series variables has generated considerable research interest in recent years. Typical examples include the synchronization of business cycles in different countries or regions (see Artis et al., 1997, 2004, among others) and the synchronization of bull and bear phases in financial markets (see Edwards et al., 2003; Bekaert et al., 2005, among others). Much attention in this research area has been focused on the extreme cases of independence and perfect synchronization of cycles. In the first case, the cycles in two variables are purely idiosyncratic, while in the second case the two variables are driven by a single common cycle such that regime shifts occur contemporaneously. A wide range of econometric techniques has been developed to examine these polar cases, ranging from nonparametric test statistics (see Harding and Pagan, 2006), to unobserved components models (see Koopman and Harvey, 1997), to Markov Switching Vector AutoRegressive (MS-VAR) models (see Krolzig, 1997).
Perhaps not surprisingly, it is often found that neither independence nor perfect synchronization are adequate representations of the cyclical dynamics in economic and financial variables. A variety of mechanisms may lead to the intermediate case of ’imperfect synchronization’. For example, it may be that cycles in individual variables are partly due to common components and partly due to idiosyncratic factors. This seems relevant when considering, for example, business cycles in different countries or regions, Kose et al. (see 2003); Del Negro and Otrok (see 2008). A second possibility is that the different variables in fact share a single common cyclical component, but are subject to different phase shifts. This approach is appropriate in the context of business cycles for jointly modeling the cyclical behavior of leading and coincident indicator variables. Similarly, in financial markets shifts from bull to bear phases and vice versa may occur earlier in some assets than in others due to differences in liquidity or gradual information diffusion among other reasons.
The possibility that imperfect synchronization is due to different phase shifts of a single common cycle has been explored by Koopman and Azevedo (2008) using unobserved components models, and by Hamilton and Perez-Quiros (1996) and Segers et al. (2009) using MS-VAR models.
In this paper, we adopt the framework of MS-VAR models for describing imperfect synchronization due to different phase shifts of a single common cycle. We develop a general framework where we can simultaneously estimate the degree of synchronization together with the regimes. We extend the models of Hamilton and Perez-Quiros (1996) (HP henceforth) and Segers et al. (2009) (SPV henceforth) in two important directions. First, we allow for multiple regimes and consider the possibility of different phase shifts for the different regimes. Second, we allow for a regime-dependent covariance matrix. Despite the formulation of the relationship between the cycles in SPV encompasses all types of imperfect synchronization extending the model in HP, this was possible by exploiting the binary nature of indicator variables.
Therefore, this formulation, consequently the model, is only applicable when the number of regimes is two. For a general model, one needs to incorporate a restriction-free formulation for modeling the relationship between multicategorical variables into the model. We provide an algorithm for that purpose that allows us to model various degrees of synchronization regardless to the number of regimes. Another complication arises when the regime dependent heteroscedasticity is present. In this case, as the timing of the regimes are not identical, estimation of the covariance matrix as a whole is not feasible. Both of the models in HP and SPV by-passed this complication by assuming regime-independent covariance matrix dynamics. However, considering the lack of consensus of model specification in business cycle analysis as well as ample evidence of volatility clustering in financial data, this assumption might be quite restrictive. To circumvent this, we treat each element of the covariance matrix separately using the decomposition of the covariance matrix into correlations and standard deviations.
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