  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.
