Economic and financial time series can feature locally explosive behavior when a bubble is formed. The economic or financial bubble, especially its dynamics, is an intriguing topic that has been attracting longstanding attention. To illustrate the dynamics of the local explosion itself, the paper presents a novel, simple, yet useful time series model, called the stochastic nonlinear autoregressive model, which is always strictly stationary and geometrically ergodic and can create long swings or persistence observed in many macroeconomic variables. When a nonlinear autoregressive coefficient is outside of a certain range, the model has periodically explosive behaviors and can then be used to portray the bubble dynamics. Further, the quasi-maximum likelihood estimation (QMLE) of our model is considered, and its strong consistency and asymptotic normality are established under minimal assumptions on innovation. A new model diagnostic checking statistic is developed for model fitting adequacy. In addition two methods for bubble tagging are proposed, one from the residual perspective and the other from the null-state perspective. Monte Carlo simulation studies are conducted to assess the performances of the QMLE and the two bubble tagging methods in finite samples. Finally, the usefulness of the model is illustrated by an empirical application to the monthly Hang Seng Index.

翻译：经济与金融时间序列在泡沫形成时可能呈现局部爆炸性行为。经济或金融泡沫，特别是其动态机制，是一个长期吸引学界关注的有趣课题。为刻画局部爆炸现象本身的动态特征，本文提出一种新颖、简洁且实用的时间序列模型——随机非线性自回归模型。该模型始终满足严格平稳性与几何遍历性，能够再现许多宏观经济变量中观测到的长期波动或持续性特征。当非线性自回归系数超出特定区间时，模型会呈现周期性爆炸行为，从而可用于描述泡沫动态。进一步地，本文研究了模型的拟极大似然估计（QMLE），并在关于新息的最小假设下建立了其强相合性与渐近正态性。针对模型拟合充分性，本文发展了一种新的模型诊断检验统计量。此外，从残差视角和零状态视角出发，分别提出了两种泡沫标记方法。通过蒙特卡洛模拟研究，评估了拟极大似然估计及两种泡沫标记方法在有限样本中的表现。最后，通过月度恒生指数的实证应用验证了该模型的有效性。