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Neural Inference Functions for Margins for Time Series Copula Models

Daniel Fynn, David Gunawan, Andrew Zammit-Mangion

Abstract

Copula models are widely employed in multivariate time series analysis because they permit flexible modelling of marginal distributions independently of the dependence structure, which is fully characterised by the copula function. However, Bayesian inference with these models becomes computationally demanding as the number of variables in the time series increases. Motivated by the classical inference functions for margins (IFM) approach, we propose a new neural-network based inference framework for estimating parameters in copula models, termed the neural inference functions for margins (N-IFM). N-IFM enables rapid parameter estimation for new data, fast sequential prediction, and efficient model comparison via time-series validation. We assess the performance of N-IFM using both simulated and real datasets and compare it to Hamiltonian Monte Carlo, demonstrating substantial computational gains with comparable inferential accuracy.

Neural Inference Functions for Margins for Time Series Copula Models

Abstract

Copula models are widely employed in multivariate time series analysis because they permit flexible modelling of marginal distributions independently of the dependence structure, which is fully characterised by the copula function. However, Bayesian inference with these models becomes computationally demanding as the number of variables in the time series increases. Motivated by the classical inference functions for margins (IFM) approach, we propose a new neural-network based inference framework for estimating parameters in copula models, termed the neural inference functions for margins (N-IFM). N-IFM enables rapid parameter estimation for new data, fast sequential prediction, and efficient model comparison via time-series validation. We assess the performance of N-IFM using both simulated and real datasets and compare it to Hamiltonian Monte Carlo, demonstrating substantial computational gains with comparable inferential accuracy.
Paper Structure (37 sections, 51 equations, 29 figures, 16 tables, 5 algorithms)

This paper contains 37 sections, 51 equations, 29 figures, 16 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Copula models
  4. Implicit copula models
  5. Gaussian and $t$ copulas
  6. Factor correlation matrix
  7. Marginal time series model
  8. Neural Posterior Inference
  9. Classical inference functions for margins
  10. Neural inference functions for margins
  11. Neural approximation for marginal models
  12. Neural approximation for copula models
  13. Neural inference functions for margins (N-IFM) algorithm
  14. Measure of predictive performance
  15. Approximate predictive density
  16. ...and 22 more sections

Figures (29)

  • Figure 1: Inference framework for the marginal models in N-IFM.
  • Figure 2: Inference framework for the copula model in N-IFM.
  • Figure 3: Posterior distributions of selected GARCH(1,1) parameters and Gaussian copula parameters estimated using the N-IFM, HMC and HMC-IFM methods for simulated data generated from a one-factor Gaussian copula model with GARCH(1,1) marginals with Gaussian errors, comprising $D = 20$ series of length $T = 1000$. Results from N-IFM, HMC and HMC-IFM are compared with the true parameter values, indicated by the vertical dotted lines.
  • Figure 4: Selected one-step-ahead posterior predictive densities obtained from N-IFM, HMC and HMC-IFM for a one-factor Gaussian copula model with GARCH(1,1) marginals with Gaussian errors, comprising $D = 20$ series of length $T = 1000$. Diagonal panels show the marginal predictive distributions, while off-diagonal panels display the corresponding bivariate predictive distributions. In the diagonal panels, the vertical lines indicate the true parameter values, while in the off-diagonal panels, the dotted lines represent the true correlation between each pair of series.
  • Figure 5: Selected one-step-ahead posterior predictive densities obtained from N-IFM, HMC and HMC-IFM for a four-factor Gaussian copula model with GARCH(1,1) marginals with Student's-$t$ errors, comprising $D = 20$ series of length $T = 1000$ for the industry dataset. Diagonal panels show the marginal predictive distributions, while off-diagonal panels display the corresponding bivariate predictive distributions. In the diagonal panels, the vertical lines indicate the true parameter values.
  • ...and 24 more figures