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Bubble Modeling and Tagging: A Stochastic Nonlinear Autoregression Approach

Xuanling Yang, Dong Li, Ting Zhang

TL;DR

The paper introduces the SNAR model to capture locally explosive financial bubbles within a causal, stationary framework, and develops a QMLE with strong consistency and asymptotic normality under mild conditions. It advances model diagnostics with a self-weighted portmanteau-type statistic and proposes two bubble-tagging approaches—residual-based and null-based—that accommodate transient and persistent bubble dynamics. Through simulations, the QMLE and tagging methods show robust finite-sample performance under heavy-tailed and non-Gaussian innovations. An empirical application to the Hang Seng Index demonstrates detectable local explosiveness and identifies major crisis periods, illustrating the practical utility of SNAR for bubble analysis and tagging.

Abstract

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.

Bubble Modeling and Tagging: A Stochastic Nonlinear Autoregression Approach

TL;DR

The paper introduces the SNAR model to capture locally explosive financial bubbles within a causal, stationary framework, and develops a QMLE with strong consistency and asymptotic normality under mild conditions. It advances model diagnostics with a self-weighted portmanteau-type statistic and proposes two bubble-tagging approaches—residual-based and null-based—that accommodate transient and persistent bubble dynamics. Through simulations, the QMLE and tagging methods show robust finite-sample performance under heavy-tailed and non-Gaussian innovations. An empirical application to the Hang Seng Index demonstrates detectable local explosiveness and identifies major crisis periods, illustrating the practical utility of SNAR for bubble analysis and tagging.

Abstract

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.
Paper Structure (17 sections, 6 theorems, 56 equations, 6 figures, 7 tables)

This paper contains 17 sections, 6 theorems, 56 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Probabilistic Properties of the SNAR Model
  3. Quasi-Maximum Likelihood Estimation
  4. Model Diagnostic Checking
  5. Bubble Tagging
  6. A Residual-Based Method for Bubble Tagging
  7. A Null-Based Method for Bubble Tagging
  8. Simulation Studies
  9. An empirical example
  10. Conclusions
  11. Technical Proofs
  12. Proof of Theorem \ref{['ssge']}
  13. Proof of Theorem \ref{['thm.consistent']}
  14. Proof of Theorem \ref{['thm.asym']}
  15. Proof of Theorem \ref{['thm.rho']}
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose that $\mathrm{(i)}$$\{\varepsilon_t\}$ is i.i.d. and independent of i.i.d. binary variables $\{s_t\}$ with $0\leq p_0<1$, and $\mathrm{(ii)}$$\varepsilon_1$ has a positive density on $\mathbb{R}$ with $\mathbb{E}(\log^+|\varepsilon_1|)<\infty$. Then there exists a strictly stationary, nonant

Figures (6)

  • Figure 1: Simulated paths of model \ref{['eq.model']} with $\varepsilon_t\sim\mathcal{N}(0,6^2)$, $p_0=0.977$, and (a) $\phi_0=1.025$ and (b) $\phi_0=1.05$.
  • Figure 2: The strict stationarity region $\{(p, \phi): \,\phi\in\mathbb{R}, \,p\phi^2<1, \,0\leq p<1\}$ of $y_t$ with finite second moment.
  • Figure 3: The histogram of $\sqrt{n}(\widehat{\phi}_n-\phi_0)$ with the sample size $n=400$. The left column panel corresponds to Case I, i.e., $y_t$ is weakly stationary; the middle to Case II, and the right to Case III, i.e., $y_t$ has an infinite variance, respectively. The upper row panel is when $\varepsilon_t\sim\mathcal{N}(0, 1)$, the middle when $\varepsilon_t\sim$ the Laplace distribution, and the lower when $\varepsilon_t\sim \mathrm{st}_5$, respectively.
  • Figure 4: (a) Real HSI prices with the fitted linear trend (the dotted line); (b) $y_t$ series.
  • Figure 5: Selected dates of $\widehat{s}_t=0$ by Rules 1--4.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 1
  • Proposition 2