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Statistical inference for Levy-driven graph supOU processes: From short- to long-memory in high-dimensional time series

Shreya Mehta, Almut E. D. Veraart

TL;DR

We introduce Levy-driven graph supOU processes to model high-dimensional time series with graph-structured dependencies, enabling a smooth transition between short- and long-memory regimes and flexible marginals. The approach relies on a drift parametrisation $Q(\theta)=-\bigl(I_d+ c\bar{A}^T\bigr)\theta_2$ and two mixing specifications for $\pi(\theta_2)$: a sum of exponentials or a Gamma law, which control memory properties. An inference framework based on the generalized method of moments (GMM) is developed, featuring a two-step procedure to identify the mixing component and $c$ from the scaled autocovariance and then recover the Lévy-basis moments; asymptotic theory covers consistency and normality under weak dependence. The method is validated by simulations and applied to wind capacity factors in Europe, demonstrating practical usefulness and competitive finite-sample performance, with accompanying software release.

Abstract

This article introduces Levy-driven graph supOU processes, offering a parsimonious parametrisation for high-dimensional time-series, where dependencies between the individual components are governed via a graph structure. Specifically, we propose a model specification that allows for a smooth transition between short- and long-memory settings while accommodating a wide range of marginal distributions. We further develop an inference procedure based on the generalised method of moments, establish its asymptotic properties and demonstrate its strong finite sample performance through a simulation study. Finally, we illustrate the practical relevance of our new model and estimation method in an empirical study of wind capacity factors in an European electricity network context.

Statistical inference for Levy-driven graph supOU processes: From short- to long-memory in high-dimensional time series

TL;DR

We introduce Levy-driven graph supOU processes to model high-dimensional time series with graph-structured dependencies, enabling a smooth transition between short- and long-memory regimes and flexible marginals. The approach relies on a drift parametrisation and two mixing specifications for : a sum of exponentials or a Gamma law, which control memory properties. An inference framework based on the generalized method of moments (GMM) is developed, featuring a two-step procedure to identify the mixing component and from the scaled autocovariance and then recover the Lévy-basis moments; asymptotic theory covers consistency and normality under weak dependence. The method is validated by simulations and applied to wind capacity factors in Europe, demonstrating practical usefulness and competitive finite-sample performance, with accompanying software release.

Abstract

This article introduces Levy-driven graph supOU processes, offering a parsimonious parametrisation for high-dimensional time-series, where dependencies between the individual components are governed via a graph structure. Specifically, we propose a model specification that allows for a smooth transition between short- and long-memory settings while accommodating a wide range of marginal distributions. We further develop an inference procedure based on the generalised method of moments, establish its asymptotic properties and demonstrate its strong finite sample performance through a simulation study. Finally, we illustrate the practical relevance of our new model and estimation method in an empirical study of wind capacity factors in an European electricity network context.

Paper Structure

This paper contains 19 sections, 9 theorems, 79 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Graph supOU processes
  3. Notation
  4. Lévy bases and multivariate supOU processes
  5. Definition and properties of graph supOU processes
  6. Parametric models of graph supOU type -- Bridging between long and short memory
  7. Choosing a sum of exponentials for $\pi(\theta_2)$
  8. Choosing a Gamma law for $\pi(\theta_2)$
  9. Parameter estimation for the graph supOU process
  10. A two-step estimation procedure
  11. Generalised Method of Moments (GMM) Estimator
  12. Asymptotic theory of the GMM estimator for graph supOU processes
  13. Consistency
  14. Weak dependence properties of graph supOU processes
  15. Asymptotic normality
  16. ...and 4 more sections

Key Result

Theorem 2.2

(Theorem 3.1, Bar11) Let $\Lambda$ be an $\mathbb{R}^d$-valued Lévy basis on $M_d^-\times\mathbb{R}$ with generating quadruple $(\gamma,\Sigma,\nu,\pi)$ satisfying $\int_{\|x\|>1} \ln(\|x\|)\nu(dx)<\infty$ and assume there exist measurable functions $\rho:M_d^-\rightarrow \mathbb{R}^+\setminus\{0\}$

Figures (5)

  • Figure 1: Results from the simulation study: Figures \ref{['fig:sim_alpha']} and \ref{['fig:sim_c']} and depict the boxplots for the estimates of $\alpha$ and $c$, respectively, for different lags $h$ used in the scaled autocovariance. The outliers are not depicted to improve readability. The solid red line presents the true parameter value in both cases. Figure \ref{['fig:sim-error']} displays the corresponding error measures. Figure \ref{['fig:p_vio']} displays the violin plots for the estimates of $\alpha$ and $c$ for fixed lag (35). Finally, Figures \ref{['fig:estmean_median_sim']} and \ref{['fig:estvar_median_sim']} display the heatmap for the medians of the estimated means and variances over the 1000 Monte Carlo runs.
  • Figure 2: Graph structure: Figure \ref{['fig:map']} shows the 24-node network in Portugal; Figure \ref{['fig:normA']} contains the heatmap of the column normalised adjacency matrix associated with the graph.
  • Figure 3: Plots of the 24-dimensional time series of wind capacity factors in Portugal: Heatmap of the 24 hourly time series ( \ref{['fig:heatmap-original']}); boxplots of the ACFs (\ref{['fig:empacfs-original']}); time series of node 22 (Lisbon) (\ref{['fig:Lisbondata-original']}). The figures in the second row show the corresponding plots based on the deseasonalised and detrended data: Heatmap of the deseasonalised 24 hourly time series ( \ref{['fig:heatmap']}); boxplots of the ACFs of the deseasonalised time series (\ref{['fig:empacfs']}); deseasonalised time series of node 22 (Lisbon) (\ref{['fig:Lisbondata']}).
  • Figure 4: Estimation of $\alpha$ and $c$: Figure \ref{['fig:alphac-vec']} provides graphs of $\hat{\alpha}$ and $\hat{c}$ based on different choices of the lag $h$. Figures \ref{['fig:Fit']} (based on 40 lags) and \ref{['fig:Fit100']} (based on 100 lags) show the empirical eigenvalue function (red) together with the fitted counterparts based on a graph OU (green) and two graph supOU specifications, the gamma model (blue) and a sum of two weighted exponentials (orange).
  • Figure 5: Estimated mean and variance of the Lévy basis: Figure \ref{['fig:estmean']} Estimated mean $\widehat{\mu_L}$, Figure \ref{['fig:estvar']} estimated variance $\widehat{{\boldsymbol \sigma}_L^2}$, and Figure \ref{['fig:empvar']} empirical covariance $\widehat{\mathrm{Var}(X)}$.

Theorems & Definitions (27)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Example 2.8
  • Remark 2.9
  • Example 3.1
  • Example 3.2
  • ...and 17 more