Abstract

Economic and financial time series can feature locally explosive behav

ior 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

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 dynam-

ics. 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.

Information

Preprint No.SS-2025-0450
Manuscript IDSS-2025-0450
Complete AuthorsXuanling Yang, Dong Li, Ting Zhang
Corresponding AuthorsXuanling Yang
Emailsyangxl@buct.edu.cn

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Acknowledgments

The authors are very grateful to the anonymous referees, the associate editor, and the co-editor for their constructive suggestions and comments,

leading to a substantial improvement in the presentation and contents.

Yang’s research is supported by the Fundamental Research Funds for the

Central Universities (ZY2513). Li’s research is partially supported by the

NSFC (No.72471127). Zhang’s research is supported by the U.S. NSF DMS-

2412661.

Supplementary Materials

The Supplementary Material contains part of simulation results and all

technical proofs of theorems and propositions in the article.

Supplementary materials are available for download.