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Estimating Time-Varying Tail Dependence with an SJC Copula

Article Quant Q&A · Author: Kondo
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Summary

The document presents a dynamic specification for the upper tail dependence of a bivariate time-varying symmetrized Joe–Clayton copula. The coefficient is updated from its previous value and a lagged measure of disagreement between two transformed series, with a link function keeping the estimate within the unit interval. The question asks how much data is needed to estimate the intercept and dynamic coefficients reliably, and how to choose the observation count.

No answer or empirical guidance is provided. The material therefore introduces a modeling idea and a sample-size concern, but does not establish a reliable minimum number of observations. In practice, adequacy would depend on the data, dependence strength, model specification, and estimation uncertainty; the excerpt supplies no tests or results to resolve those issues. Its main relevance is to dependence modeling for risk analysis or multivariate financial research.

Key ideas

  • The upper tail dependence coefficient is modeled as a time-varying quantity.
  • Its dynamics use the previous coefficient and recent differences between transformed series.
  • A transformation function constrains the coefficient to the interval from zero to one.
  • The excerpt asks about sample size but gives no estimate or reliability criterion.

Tags

StatisticsRisk managementMulti-asset
Full text
# Estimating time-varying tail dependence for Archimedean copulas


# Estimating time-varying tail dependence for Archimedean copulas












Patton (2006) defines the upper tail dependence coefficient for a time-varying bivariate SJC copula as $$\tau^u_t=\Lambda \left(\omega_u + \beta_u \tau^u_{t-1}+\alpha_u \frac{1}{10}\sum^{10}_{i=1}|u_{t-i}-v_{t-i}| \right)$$ where $\tau^u_t$ is the upper tail dependence coefficient at time $t$, $u_t$ and $v_t$ and the univariate transformations, and $\Lambda$ is a transformation function needed to keep $\tau^u_t$ in $[0,1]$.

Although I'm working on an extension for multivariate copulas, I'd like to use a similar equation. My (simple) question though (which I guess holds for way simpler cases like an ARMA(1,1) for any time series): how many observations do I need to estimate $\omega_u$, $\alpha_u$ and $\beta_u$; what is the value of $d$ in $(\tau^u_t)_{t\in\{1, ..., d\}}$ in order to have a reliable estimate?

Patton (2006): http://www.christoffersen.com/CHRISTOP/2007/Patton_IER_2006.pdf

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.