Bayesian Estimation of the Markov-Switching GJR(1, 1) Model with Student-t Innovations
In this chapter, we address the problem of estimating GARCH models subject to structural changes in the parameters; namely, the Markov-switching GARCH models (henceforth MS-GARCH). In this framework, a hidden Markov sequence {S t with state space {1,…,K}
- PDF / 1,534,305 Bytes
- 45 Pages / 595.276 x 841.89 pts (A4) Page_size
- 60 Downloads / 221 Views
his chapter, we address the problem of estimating GARCH models subject to structural changes in the parameters; namely, the Markov-switching GARCH models (henceforth MS-GARCH). In this framework, a hidden Markov sequence {st } with state space {1, . . . , K} allows discrete changes in the model parameters. Such processes have received a lot of attention in recent years as they provide an explanation of the high persistence in volatility (i.e., nearly unit root process for the conditional variance) observed with single-regime GARCH models [see, e.g., Lamoureux and Lastrapes 1990]. Furthermore, the MS-GARCH models allow for a quick change in the volatility level which leads to significant improvements in volatility forecasts, as shown by Dueker [1997], Klaassen [2002], Marcucci [2005]. Following the seminal work of Hamilton and Susmel [1994], different parametrizations have been proposed to account for discrete changes in the GARCH parameters [see, e.g., Dueker 1997, Gray 1996, Klaassen 2002]. However, these parametrizations for the conditional variance process lead to computational difficulties. Indeed, the evaluation of the likelihood function for a sample of length T requires the integration over all K T possible paths, rendering the estimation infeasible. As a remedy, approximation schemes have been proposed to shorten the dependence on the state variable’s history. While this difficulty is not present in ARCH type models, lower order GARCH specification of the conditional variance offers a more parsimonious representation than higher order ARCH models.
110
7 MS-GJR(1, 1) Model with Student-t Innovations
In order to avoid any difficulties related to the past infinite history of the state variable, we adopt a recent parametrization due to Haas et al. [2004]. In their model, the authors hypothesize K separate GARCH(1, 1) processes for the conditional variance of the MS-GARCH process {yt }. The conditional variances at time t can be written in vector form as follows: β1 α01 α11 h1t−1 h1t 2 2 2 2 2 β h ht . α0 α1 2 . = . + . yt−1 + . t−1 (7.1) . .. . . . . . . . . hK t
α0K
α1K
βK
hK t−1
where denotes the Hadamard product, i.e., element-by-element multiplication. The MS-GARCH process {yt } is then simply obtained by setting: yt = εt (hst t )1/2 where εt is an error term with zero mean and unit variance. The parameters α0k , α1k and β k are therefore the GARCH(1, 1) parameters related to the kth state of the nature. Under this specification, the conditional variance is solely a function of the past data and current state st , which avoids the problem of infinite history. In the context of the Bayesian estimation, this allows to simulate the state process in a multi-move manner which enhances the sampler’s efficiency. In addition to its appealing computational aspects, the MS-GARCH model of Haas et al. [2004] has conceptual advantages. In effect, one reason for specifying Markov-switching models that
Data Loading...
