Abstract We assume that equity returns follow multi-state threshold autoregressions and generalize existing results for threshold autoregressive models presented in Knight and Satchell 2011. “Some new results for threshold AR(1) models,” Journal of Time Series Econometrics 3(2011):1–42 and Knight, Satchell, and Srivastava (2014) for the existence of a stationary process and the conditions necessary for the existence of a mean and a variance; we also present formulae for these moments. Using a simulation study, we explore what these results entail with respect to the impact they can have on tests for detecting bubbles or market efficiency. We find that bubbles are easier to detect in processes where a stationary distribution does not exist. Furthermore, we explore how threshold autoregressive models with i.i.d trigger variables may enable us to identify how often asset markets are inefficient. We find, unsurprisingly, that the fraction of time spent in an efficient state depends upon the full specification of the model; the notion of how efficient a market is, in this context at least, a model-dependent concept. However, our methodology allows us to compare efficiency across different asset markets.

中文翻译：

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特定的市场效率低下的时间比例是多少？…当股票价格遵循阈值自回归时分析市场效率频率的方法

摘要我们假设股权收益遵循多状态阈值自回归，并概括了Knight和Satchell 2011中提出的阈值自回归模型的现有结果。“阈值AR（1）模型的一些新结果，”《时间序列计量经济学》 3（2011）： 1–42以及Knight，Satchell和Srivastava（2014）提出了平稳过程的存在以及存在均值和方差的必要条件；我们还介绍了这些时刻的公式。通过模拟研究，我们探讨了这些结果对检测气泡或市场效率的测试可能产生的影响。我们发现，在不存在平稳分布的过程中，气泡更易于检测。此外，我们探索了如何使用ii进行阈值自回归模型 d触发变量可以使我们确定资产市场效率低下的频率。毫不奇怪，我们发现在有效状态下花费的时间比例取决于模型的完整规格；至少在这种情况下，市场效率如何取决于模型的概念。但是，我们的方法允许我们比较不同资产市场的效率。