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Copula-Based Time Series for Non-Gaussian and Non-Markovian Stationary Processes

Sven Pappert, Harry Joe

Abstract

In the copula-based approach to univariate time series modeling, the finite dimensional temporal dependence of a stationary time series is captured by a copula. Recent studies investigate how copula-based time series models can be generalized to have long-term autoregressive effects. We study a generalization that comes from a Markov sequence of order p and a q-dependent sequence. We derive the relation of the model to Gaussian-ARMA models and to the Gaussian-GARCH(1,1) model. We investigate distributional properties of the process and discuss the maximum likelihood estimation (MLE). Additionally we analyze the copula moving aggregate process of order one, or MAG(1), as it is a basic building block. Last we test the model in probabilistic forecasting studies on US inflation and German wind energy production.

Copula-Based Time Series for Non-Gaussian and Non-Markovian Stationary Processes

Abstract

In the copula-based approach to univariate time series modeling, the finite dimensional temporal dependence of a stationary time series is captured by a copula. Recent studies investigate how copula-based time series models can be generalized to have long-term autoregressive effects. We study a generalization that comes from a Markov sequence of order p and a q-dependent sequence. We derive the relation of the model to Gaussian-ARMA models and to the Gaussian-GARCH(1,1) model. We investigate distributional properties of the process and discuss the maximum likelihood estimation (MLE). Additionally we analyze the copula moving aggregate process of order one, or MAG(1), as it is a basic building block. Last we test the model in probabilistic forecasting studies on US inflation and German wind energy production.

Paper Structure

This paper contains 17 sections, 6 theorems, 41 equations, 11 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Copula-ARMA generalization
  3. Relation to Other Models
  4. Properties of the model
  5. Estimation
  6. Simulation and Numerical Experiments
  7. Forecasting Real Data
  8. Discussion
  9. Supporting Results
  10. Special cases of the CoARMA$(1,1)$ process
  11. Proofs
  12. Proof of Proposition \ref{['prop:GaussianARMA']}
  13. Proof of proposition \ref{['prop:ARCH(1)']}
  14. Proof of proposition \ref{['prop:GARCH(1,1)']}
  15. Proof of proposition \ref{['prop:Distributional_MAG(1)']}
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\{U_t\}_{t\in\mathbb{Z}}$ be defined through Eq. Eq:Model(p,q)_updating_Eq with latent process $W_t \sim U(0,1)$ and innovations $\varepsilon_t \stackrel{iid}{\sim} U(0,1)$. Let the copula corresponding to the AR-component be given as a $(p+1)$-dimensional stationary Gaussian D-vine copula, whi $\blacktriangleleft$$\blacktriangleleft$

Figures (11)

  • Figure 1: Contour plots of a stationary D-vine corresponding to an autoregressive copula-based time series model of order $p = 3$.
  • Figure 2: Contour plots of a MAG D-vine corresponding to a $q$-dependent copula-based time series model of order $q = 3$. Only dependencies involving variable $1$ are different from independence.
  • Figure 3: Numerical and simulation-based Spearman's rho, upper and lower 5%-tail dependence coefficients and order of the copula of consecutive observations from a Gaussian-MAG$(1)$, along with the dependence measures of the original copula $K_{21}$.
  • Figure 4: Numerical and simulation-based Spearman's rho, upper and lower 5%-tail dependence coefficients and order of the copula of consecutive observations from a Gumbel-MAG$(1)$, along with the dependence measures of the original copula $K_{21}$.
  • Figure 5: Numerical and simulation-based Spearman's rho, upper and lower 5%-tail dependence coefficients and order of the copula of consecutive observations from a Clayton-MAG$(1)$, along with the dependence measures of the original copula $K_{21}$.
  • ...and 6 more figures

Theorems & Definitions (21)

  • Proposition 1: Recovering Gaussian ARMA
  • proof
  • Example 1: Gaussian-MAG$(1)$
  • Example 2: Gaussian-MAG$(2)$
  • Example 3: Gaussian copulas and $p=1$, $q=1$ in Eq. \ref{['Eq:Model(p,q)_updating_Eq']}
  • Example 4: Gaussian copulas and $p=2$, $q=1$ in Eq. \ref{['Eq:Model(p,q)_updating_Eq']}
  • Proposition 2: Recovering ARCH$(1)$, cf. dias2024garch
  • proof
  • Proposition 3: Recovering GARCH$(1,1)$
  • proof
  • ...and 11 more