Stock market indices are volatile by nature, and sudden shocks are known to affect volatility patterns. The autoregressive conditional heteroskedasticity (ARCH) and generalized ARCH (GARCH) models neglect structural breaks triggered by sudden shocks that may lead to an overestimation of persistence, causing an upward bias in the estimates. Different regime-switching models that have abrupt regime-switching governed by a Markov chain were developed to model volatility in financial time series data. Volatility modelling was also extended to spatially interconnected time series, resulting in spatial variants of ARCH models. This inspired us to propose a Markov switching framework of the spatio-temporal log-ARCH model. In this article, we discuss the Markov-switching extension of the model, the estimation procedure and the smooth inferences of the regimes. The Monte-Carlo simulation studies show that the maximum likelihood estimation method for our proposed model has good finite sample properties. The proposed model was applied to 28 stock indices data that were presumably affected by the 2015-2016 Chinese stock market crash. The results showed that our model is a better fit compared to that of the one-regime counterpart. Furthermore, the smoothed inference of the data indicated the approximate periods where structural breaks occurred. This model can capture structural breaks that simultaneously occur in nearby locations.

翻译：股票市场指数本质上具有波动性，已知突发冲击会影响波动模式。自回归条件异方差（ARCH）模型及其广义形式（GARCH）忽略了突发冲击造成的结构性断裂，这可能导致对持久性的过度估计，从而引起估计值的向上偏差。为对金融时间序列数据的波动性进行建模，研究者开发了多种由马尔可夫链控制的突变体制切换模型。波动性建模也被扩展至空间互联的时间序列，由此产生了ARCH模型的空间变体。这启发我们提出了时空对数ARCH模型的马尔可夫切换框架。本文讨论了该模型的马尔可夫切换扩展、估计方法以及体制的平滑推断。蒙特卡洛模拟研究表明，本文所提模型的最大似然估计方法具有优良的有限样本性质。我们将该模型应用于可能受2015-2016年中国股市崩盘影响的28个股指数据。结果表明，与单体制对应模型相比，我们的模型拟合效果更优。此外，数据的平滑推断揭示了结构性断裂发生的大致时间段。该模型能够捕捉邻近位置同时发生的结构性断裂。