A discussion on the innovation distribution of Markov regime-switching GARCH model

Markov Regime-Switching Generalized Autoregressive Conditional Heteroskedastic (MRS-GARCH) model is a widely used approach to model the financial volatility with potential structural breaks. The original innovation of the MRS-GARCH model is assumed to follow the Normal distribution, which cannot accommodate fat-tailed properties commonly existing in financial time series. Many existing studies point out that this problem can lead to inconsistent estimates. To overcome it, the Student's t-distribution and General Error Distribution (GED) are the two most popular alternatives. However, a recent study points out that the Student's t-distribution lacks stability. Instead, this research incorporates the a-stable distribution in the GARCH-type model. The issue of the a-stable distribution is that its second moment does not exist. To solve this problem, the tempered stable distribution, which retains most characteristics of the a-stable distribution and has defined moments, is a natural candidate. In this paper, we conduct a series of simulation studies to demonstrate that MRS-GARCH model with tempered stable distribution consistently outperform that with Student's t-distribution and GED. Our empirical study on the S&P 500 daily return volatility also generates robust results. Therefore, we argue that the tempered stable distribution could be a widely useful tool for modelling the financial volatility in general contexts with a MRS-GARCH-type specification.