Change-point estimation for Weibull time series with copula-based Markov models

Technical Summary: Change-point estimation for Weibull time series with copula-based Markov models

Problem Statement
The paper addresses the challenge of offline change-point estimation in time series data characterized by nonlinear serial dependence and nonnegative observations, such as event times and volatility measures. Traditional change-point methodologies often rely on assumptions of independent observations or linear dependence structures (e.g., autoregressive models), which are frequently inadequate for capturing the asymmetric and nonlinear dependence structures prevalent in practical time series. The authors aim to develop a framework capable of simultaneously detecting structural changes in both marginal distributions and the underlying dependence structure.

Methodology
The authors propose a copula-based Markov chain model with Weibull marginal distributions. The methodology is structured as follows:

• Model Structure: The time series $\{X_t\}$ is modeled as a two-stage process with a single change point $\tau$. Before $\tau$, observations follow a Weibull distribution with parameters $(\lambda_0, k_0)$; after $\tau$, they follow a Weibull distribution with parameters $(\lambda_1, k_1)$.
• Dependence Modeling: Serial dependence is modeled using copulas. The study specifically incorporates the Clayton copula to capture asymmetric lower-tail dependence and the Joe copula for upper-tail dependence. The joint distribution of consecutive observations is defined via the copula function $C_\alpha$ applied to the marginal cumulative distribution functions. A specific transition parameter $\alpha_{01}$ is assumed for the pair $(X_\tau, X_{\tau+1})$ at the change point, treated as a fixed, known quantity due to identifiability issues with a single observation pair.
• Estimation Procedure:
  • Maximum Likelihood Estimation (MLE): The log-likelihood function is derived based on the conditional densities of the Markov chain. Since the likelihood is not differentiable with respect to the discrete change point $\tau$, a two-stage profile likelihood approach is employed.
  • Newton–Raphson Algorithm: For a fixed $\tau$, the remaining parameters are estimated numerically using the Newton–Raphson algorithm. To ensure numerical stability and handle parameter constraints (e.g., $\lambda, k > 0$), a re-parameterization technique is adopted (e.g., transforming $\lambda$ to $\log(\lambda)$).
  • Interval Estimation: Confidence intervals for both model parameters and the change point are constructed using a parametric bootstrap Monte Carlo procedure.
• Empirical Application: The method is applied to two datasets: the daily VIX index during the COVID-19 pandemic (2019–2020) and the interarrival times of VIX values exceeding a threshold of 30 over a longer period (2018–2024).

Key Results
Extensive simulation studies and empirical analyses were conducted to evaluate the performance of the proposed estimators:

• Finite-Sample Performance: Simulation results indicate that the proposed estimators perform well in terms of Root Mean Square Error (RMSE) and Relative Error (RE). The estimation of the change point $\tau$ is particularly accurate, with RE values remaining small even under moderate model misspecification.
• Robustness to Misspecification: The study investigated scenarios where the assumed copula family differed from the true data-generating process. While the estimation of tail dependence parameters ($\alpha$) proved sensitive to copula misspecification (showing increased RE), the estimation of the change point $\tau$ remained relatively stable and robust.
• Sample Size and Dependence: Estimation accuracy generally improved with larger sample sizes. Stronger dependence structures slightly increased the RMSE for parameter estimates but did not significantly degrade the performance of the change-point estimator.
• Empirical Findings (VIX Data):
  • In the analysis of the VIX index during the COVID-19 pandemic, the model identified a structural change point around late February 2020 (specifically $\hat{\tau} = 121$ for the Clayton model and $\hat{\tau} = 120$ for the Joe model), aligning with the onset of the pandemic and subsequent market volatility.
  • Model selection via the Akaike Information Criterion (AIC) favored the Clayton copula-based Markov model, suggesting that the VIX series exhibited stronger lower-tail dependence (downside risk) during the crisis period.
  • The analysis of interarrival times for VIX values exceeding 30 also detected a structural change around early March 2020, reinforcing the finding that the pandemic induced a shift in both the marginal distribution and the serial dependence structure of market fear.

Significance and Claims
The paper claims that the proposed framework offers a flexible and reliable approach for change-point estimation in nonnegative time series with nonlinear dependence. By decoupling marginal distributions from dependence structures via copulas, the method allows for the modeling of asymmetric tail behaviors that linear models cannot capture.

The authors emphasize that while the estimation of dependence parameters requires careful model selection to avoid misspecification bias, the primary goal of detecting the structural change point remains robust even if the copula family is not perfectly specified. The empirical application to financial volatility data demonstrates the practical utility of the approach in identifying regime shifts associated with major global events, highlighting the importance of accounting for asymmetric tail dependence in financial time series analysis during periods of market stress.

Future Directions
The paper suggests several avenues for future research, including:

• Extending the framework to accommodate multiple change points.
• Investigating alternative copula families (e.g., Gaussian, Gumbel, survival-Gumbel) to capture different dependence forms.
• Incorporating more flexible marginal distributions to improve adaptability.
• Exploring Bayesian approaches or simulation-based methods to address computational demands in more complex copula structures.