Variational Bayes estimation of discrete-margined copula models with application to time series

Rubén Loaiza-Maya, Michael Stanley Smith

Research output: Contribution to journalArticleResearchpeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)523-539
Number of pages17
JournalJournal of Computational and Graphical Statistics
Issue number3
Publication statusPublished - 3 Jul 2019
Externally publishedYes


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

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