Variational Bayes Estimation of Discrete-Margined Copula Models With Application to Time Series

ABSTRACT

We propose a new variational Bayes (VB) estimator for high-dimensional copulas with discrete, or a combination of discrete and continuous, margins. The method is based on a variational approximation to a tractable augmented posterior and is faster than previous likelihood-based approaches. We use it to estimate drawable vine copulas for univariate and multivariate Markov ordinal and mixed time series. These have dimension rT, where T is the number of observations and r is the number of series, and are difficult to estimate using previous methods. The vine pair-copulas are carefully selected to allow for heteroscedasticity, which is a feature of most ordinal time series data. When combined with flexible margins, the resulting time series models also allow for other common features of ordinal data, such as zero inflation, multiple modes, and under or overdispersion. Using six example series, we illustrate both the flexibility of the time series copula models and the efficacy of the VB estimator for copulas of up to 792 dimensions and 60 parameters. This far exceeds the size and complexity of copula models for discrete data that can be estimated using previous methods. An online appendix and MATLAB code implementing the method are available as supplementary materials.

KEYWORDS:

• Data augmentation
• Drawable vine
• Heteroscedasticity
• Multivariate ordinal and mixed time series
• Sparse variational approximation
• Stochastic gradient ascent

Acknowledgments

We thank two anonymous referees and an associate editor for their comments that helped improve the article greatly, including suggestions on how to improve the variational approximations.

Notes

1 These are the conditional distribution functions of $U_t|U_s=u_s,\dots ,U_{t-1}=u_{t-1}$ evaluated at u_t, and $U_s|U_{s+1}=u_{s+1},\dots ,U_t=u_t$ evaluated at u_s, respectively.

2 We are grateful to an anonymous referee who suggested that this may be an important consideration.