Abstract
Abstract We derive sufficient conditions for the existence of second and fourth moments of Markov switching multivariate generalized autoregressive conditional heteroscedastic processes in the general vector specification. We provide matrix expressions in closed form for such moments, which are obtained by using a Markov switching vector autoregressive moving-average representation of the initial process. These expressions are shown to be readily programmable in addition of greatly reducing the computational cost. As theoretical applications of the results, we derive the spectral density matrix of the squares and cross products, propose a new definition of multivariate kurtosis measure to recognize heavy-tailed features in financial real data, and provide a matrix expression in closed form of the impulse-response function for the volatility. An empirical example illustrates the results.
Highlights
We derive sufficient conditions for the existence of second and fourth moments of Markov switching multivariate generalized autoregressive conditional heteroscedastic processes in the general vector specification
We propose a new definition of multivariate kurtosis measure for multivariate Markov switching (MS) generalized autoregressive conditional heteroscedastic (GARCH) models
2.2 Multivariate Kurtosis for MS GARCH Models Having matrix expressions for the unconditional second and fourth moments of the multivariate MS GARCH process x 1⁄4 ðxtÞ, we can introduce a new kurtosis measure for such time series, which changes in regime, and derive a matrix formula in closed form for it
Summary
Let us consider the general M-state MS m-dimensional GARCH(p, q) model [in short, MS(M) GARCH(p, q)]: xt 1⁄4 H1t =2gt;. The following result has been proved by Hafner (2003, theorem 1) for multivariate standard GARCH models. In the case of multivariate mixed GARCH models, the fourth moment structure has been derived by Bauwens, Hafner, and Rombouts (2007, formula 18). Their formula relates with that given in Theorem 3: both expressions for vec ðRyÞ have the matrix GK on the left side, followed by matrix products that look alike. As remarked in the introduction, the formula for the fourth moments in Theorem 3 is nonrecursive and without the use of an infinite summation as in Hafner (2003) for standard vector GARCH models. We illustrate some theoretical implications of our results
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