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Large Skew-t Copula Models and Asymmetric Dependence in Intraday Equity Returns

Lin Deng, Michael Stanley Smith, Worapree Maneesoonthorn

Abstract

Skew-t copula models are attractive for the modeling of financial data because they allow for asymmetric and extreme tail dependence. We show that the copula implicit in the skew-t distribution of Azzalini and Capitanio (2003) allows for a higher level of pairwise asymmetric dependence than two popular alternative skew-t copulas. Estimation of this copula in high dimensions is challenging, and we propose a fast and accurate Bayesian variational inference (VI) approach to do so. The method uses a generative representation of the skew-t distribution to define an augmented posterior that can be approximated accurately. A stochastic gradient ascent algorithm is used to solve the variational optimization. The methodology is used to estimate skew-t factor copula models with up to 15 factors for intraday returns from 2017 to 2021 on 93 U.S. equities. The copula captures substantial heterogeneity in asymmetric dependence over equity pairs, in addition to the variability in pairwise correlations. In a moving window study we show that the asymmetric dependencies also vary over time, and that intraday predictive densities from the skew-t copula are more accurate than those from benchmark copula models. Portfolio selection strategies based on the estimated pairwise asymmetric dependencies improve performance relative to the index.

Large Skew-t Copula Models and Asymmetric Dependence in Intraday Equity Returns

Abstract

Skew-t copula models are attractive for the modeling of financial data because they allow for asymmetric and extreme tail dependence. We show that the copula implicit in the skew-t distribution of Azzalini and Capitanio (2003) allows for a higher level of pairwise asymmetric dependence than two popular alternative skew-t copulas. Estimation of this copula in high dimensions is challenging, and we propose a fast and accurate Bayesian variational inference (VI) approach to do so. The method uses a generative representation of the skew-t distribution to define an augmented posterior that can be approximated accurately. A stochastic gradient ascent algorithm is used to solve the variational optimization. The methodology is used to estimate skew-t factor copula models with up to 15 factors for intraday returns from 2017 to 2021 on 93 U.S. equities. The copula captures substantial heterogeneity in asymmetric dependence over equity pairs, in addition to the variability in pairwise correlations. In a moving window study we show that the asymmetric dependencies also vary over time, and that intraday predictive densities from the skew-t copula are more accurate than those from benchmark copula models. Portfolio selection strategies based on the estimated pairwise asymmetric dependencies improve performance relative to the index.
Paper Structure (63 sections, 82 equations, 19 figures, 7 tables, 2 algorithms)

This paper contains 63 sections, 82 equations, 19 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Skew-t Copulas
  3. Copula models
  4. Three skew-t copulas
  5. GH skew-t copula
  6. SDB skew-t copula
  7. AC skew-t copula
  8. Asymmetric dependence
  9. Factor copula
  10. Variational Inference for the AC Skew-t Copula
  11. Likelihood and Extended Likelihood
  12. Priors and Posterior
  13. Hybrid variational inference
  14. Simulation
  15. Design
  16. ...and 48 more sections

Figures (19)

  • Figure 1: Maximum asymmetric dependence $\max_{\text{\boldmath$\delta$},\rho} \left\{\Delta(u) \right\}$ of the three skew-t copulas (a) AC, (b) SDB, and (c) GH. Results are given for degrees of freedom $\nu = 4, 6, 8, 10, 30$, and plotted as a function of $u$.
  • Figure 2: Comparison of the posterior mean estimates of the pairwise Spearman correlations $\rho^S_{ij}$ computed exactly using MCMC (horizontal axis) and approximately using VI (vertical axis). Panel (a) plots these for the 5-dimensional skew-t copula in Case 1, and panel (b) for the 30-dimensional skew-t copula in Case 2.
  • Figure 3: Comparison of the posterior mean estimates of the pairwise 5% quantile dependence metrics $\lambda_{LL}(0.05)$ and $\lambda_{UR}(0.05)$ for Case 1 computed exactly using MCMC (horizontal axis) and approximately using VI (vertical axis).
  • Figure 4: Comparison of the posterior mean estimates of pairwise metrics $\lambda_{LL}(0.05)$ and $\lambda_{UR}(0.05)$ for Case 2 computed exactly using MCMC (horizontal axis) and approximately using VI (vertical axis). Results are given for three VI implementations where 1, 10 and 25 random walk Metropolis-Hastings draws were used to obtain a draw at step (b) of Algorithm \ref{['alg:hvi']}, with almost identical values.
  • Figure 5: Low Volatility Period results from the AC Skew-t copula model for variables (VIX, BAC, JPM). Panels (a,e,g) give marginal predictive distributions for the series, and panels (b,c,f) give bivariate predictive distributions for the variable pairs, all on 23 Dec. 2017 at 09:45. Estimated quantile dependence plots for the pairs are given in panels (d) BAC--VIX, (g) JPM--VIX, and (h) JPM--BAC. Each quantile dependence plot visualizes asymmetry along the major diagonal by plotting $\lambda_{\tiny LL}(u)$ & $\lambda_{\hbox{\tiny UR}}(1-u)$ against $u$ (blue & red lines), and along the minor diagonal by plotting $\lambda_{\tiny LR}(u)$ & $\lambda_{\hbox{\tiny UL}}(1-u)$ versus $u$ (yellow & purple lines). Empirical quantile dependence plots are given for comparison (black lines).
  • ...and 14 more figures