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Nonparametric Vector Quantile Autoregression

Alberto González-Sanz, Marc Hallin, Yisha Yao

TL;DR

This work advances time-series forecasting by formulating a genuine multivariate nonparametric vector quantile autoregression (QVAR) based on center-outward quantiles from measure transportation. It defines predictive center-outward quantile maps and regions that yield exact conditional coverage, and develops a nonparametric estimator using a Nadaraya–Watson conditional measure and optimal transport on a spherical grid, with provable consistency and dimension-dependent rates. The methodology is validated through simulations that reveal nonlinear and heteroskedastic structures, and an EEG application demonstrates its utility for detecting multivariate brain connectivity patterns across clinical groups. Overall, the approach provides a flexible, distributional forecast framework for high-dimensional time series, enabling richer risk assessment and prediction beyond mean-based models.

Abstract

Prediction is a key issue in time series analysis. Just as classical mean regression models, classical autoregressive methods, yielding L$^2$ point-predictions, provide rather poor predictive summaries; a much more informative approach is based on quantile (auto)regression, where the whole distribution of future observations conditional on the past is consistently recovered. Since their introduction by Koenker and Xiao in 2006, autoregressive quantile autoregression methods have become a popular and successful alternative to the traditional L$^2$ ones. Due to the lack of a widely accepted concept of multivariate quantiles, however, quantile autoregression methods so far have been limited to univariate time series. Building upon recent measure-transportation-based concepts of multivariate quantiles, we develop here a nonparametric vector quantile autoregressive approach to the analysis and prediction of (nonlinear as well as linear) multivariate time series.

Nonparametric Vector Quantile Autoregression

TL;DR

This work advances time-series forecasting by formulating a genuine multivariate nonparametric vector quantile autoregression (QVAR) based on center-outward quantiles from measure transportation. It defines predictive center-outward quantile maps and regions that yield exact conditional coverage, and develops a nonparametric estimator using a Nadaraya–Watson conditional measure and optimal transport on a spherical grid, with provable consistency and dimension-dependent rates. The methodology is validated through simulations that reveal nonlinear and heteroskedastic structures, and an EEG application demonstrates its utility for detecting multivariate brain connectivity patterns across clinical groups. Overall, the approach provides a flexible, distributional forecast framework for high-dimensional time series, enabling richer risk assessment and prediction beyond mean-based models.

Abstract

Prediction is a key issue in time series analysis. Just as classical mean regression models, classical autoregressive methods, yielding L point-predictions, provide rather poor predictive summaries; a much more informative approach is based on quantile (auto)regression, where the whole distribution of future observations conditional on the past is consistently recovered. Since their introduction by Koenker and Xiao in 2006, autoregressive quantile autoregression methods have become a popular and successful alternative to the traditional L ones. Due to the lack of a widely accepted concept of multivariate quantiles, however, quantile autoregression methods so far have been limited to univariate time series. Building upon recent measure-transportation-based concepts of multivariate quantiles, we develop here a nonparametric vector quantile autoregressive approach to the analysis and prediction of (nonlinear as well as linear) multivariate time series.

Paper Structure

This paper contains 28 sections, 10 theorems, 86 equations, 20 figures, 1 table.

Table of Contents

  1. Introduction
  2. Outline of the paper.
  3. Center-Outward Quantiles
  4. Quantile functions
  5. Conditional quantile functions
  6. Prediction quantile functions and regions
  7. Estimation and Prediction
  8. Empirical prediction quantiles
  9. Consistency
  10. Consistency rates
  11. Numerical Applications
  12. Simulated examples
  13. A real data analysis
  14. Appendix: Proofs
  15. Measurability and the control of probability contents
  16. ...and 13 more sections

Key Result

Lemma 2.1

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and denote by $\mathcal{G}$ a sub-$\sigma$-field of $\mathcal{A}$. Then, (i) $\mathbb{Q}_{ X\vert \mathcal{G} }$ is $\mathcal{G}\otimes \mathcal{B}(\mathbb{B}^d)$-measurable and (ii) $\mathbb{F}_{ X\vert \mathcal{G} }$ is $\mathcal{G}\ot

Figures (20)

  • Figure 1: (Case 1) Simulated trajectories of the first (red) and second (blue) components of $X$ for $T=10,000$
  • Figure 2: (Case 1) The empirical conditional center-outward quantile contours of orders $\tau=~\!0.2$ (dark green), 0.4 (green), 0.8 (light olive), and conditional median (red) at randomly selected time points with different sample sizes $T=800,000$ (upper right panel), $T=80,000$ (lower left panel), and $T=40,000$ (lower right panel). The upper left panel provides the corresponding theoretical conditional contours and medians computed via equation \ref{['case1_theoretic_quantiles']}. Kernel bandwidths were chosen as $h=0.5 \times \text{average pairwise distance}$.
  • Figure 3: (Case 1) Estimated conditional center-outward quantile contours and medians for fixed sample size $T=800,000$, based on kernel bandwidths $h=\ell\times \text{average pairwise distance}$, with $\ell = 0.2$ (upper left panel), $\ell =0.5$ (upper right panel), $\ell =1.2$ (lower left panel), and $\ell =3.0$ (lower right panel).
  • Figure 4: (Case 1) The estimated one-step-ahead conditional quantile contours and medians at selected current values. The central panel shows the estimated center-outward quantiles of orders $\tau=0.2$ (dark green), 0.4 (green), 0.8 (light olive), the center-outward median (red), and the sample mean (light blue) of the (unconditional) stationary distribution, and the eight current values (orange) at which quantile prediction is implemented in the surrounding panels. The surrounding panels show the one-step predictive center-outward quantile contours of order $\tau=0.2$ (dark green), 0.4 (green), 0.8 (light olive), the conditional center-outward median (red), and the conventional VAR(1) one-step-ahead mean prediction (blue) at these eight particular current values.
  • Figure 5: (Case 2) Simulated trajectories of the first (red) and second (blue) components of $X$ for $T=10,000$.
  • ...and 15 more figures

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Definition 2.3
  • Remark 2.1
  • Lemma 2.2
  • Remark 3.1
  • Lemma 3.1
  • Definition 3.1: $\alpha$-mixing
  • Definition 3.2: Geometric ergodicity
  • ...and 12 more