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White noise testing for functional time series via functional quantile autocorrelation

Ángel López-Oriona, Ying Sun, Hanlin Shang

TL;DR

The paper develops a robust framework for testing serial dependence in functional time series by introducing functional quantile autocorrelation (FQA) and an omnibus grid-based statistic. It derives asymptotic distributions under strong white noise for both known and estimated quantile curves and proves consistency against stationary ergodic alternatives, all without requiring finite moments. Through extensive simulations, the FQA omnibus demonstrates accurate finite-sample size and superior power to existing methods, especially under nonlinear, tail, or outlier-degenerate dependence. An application to high-frequency intraday stock returns shows practical effectiveness in finance, where complex dependence and heavy tails are common. Overall, the approach broadens the functional time series toolkit with a robust, computation-friendly method for detecting general serial dependence.

Abstract

We introduce a novel class of nonlinear tests for serial dependence in functional time series, grounded in the functional quantile autocorrelation framework. Unlike traditional approaches based on the classical autocovariance kernel, the functional quantile autocorrelation framework leverages quantile-based excursion sets to robustly capture temporal dependence within infinite-dimensional functional data, accommodating potential outliers and complex nonlinear dependencies. We propose omnibus test statistics and study their asymptotic properties under both known and estimated quantile curves, establishing their asymptotic distribution and consistency under mild assumptions. In particular, no moment conditions are required for the validity of the tests. Extensive simulations and an application to high-frequency financial functional time series demonstrate the methodology's effectiveness, reliably detecting complex serial dependence with superior power relative to several existing tests. This work expands the toolkit for functional time series, providing a robust framework for inference in settings where traditional methods may fail.

White noise testing for functional time series via functional quantile autocorrelation

TL;DR

The paper develops a robust framework for testing serial dependence in functional time series by introducing functional quantile autocorrelation (FQA) and an omnibus grid-based statistic. It derives asymptotic distributions under strong white noise for both known and estimated quantile curves and proves consistency against stationary ergodic alternatives, all without requiring finite moments. Through extensive simulations, the FQA omnibus demonstrates accurate finite-sample size and superior power to existing methods, especially under nonlinear, tail, or outlier-degenerate dependence. An application to high-frequency intraday stock returns shows practical effectiveness in finance, where complex dependence and heavy tails are common. Overall, the approach broadens the functional time series toolkit with a robust, computation-friendly method for detecting general serial dependence.

Abstract

We introduce a novel class of nonlinear tests for serial dependence in functional time series, grounded in the functional quantile autocorrelation framework. Unlike traditional approaches based on the classical autocovariance kernel, the functional quantile autocorrelation framework leverages quantile-based excursion sets to robustly capture temporal dependence within infinite-dimensional functional data, accommodating potential outliers and complex nonlinear dependencies. We propose omnibus test statistics and study their asymptotic properties under both known and estimated quantile curves, establishing their asymptotic distribution and consistency under mild assumptions. In particular, no moment conditions are required for the validity of the tests. Extensive simulations and an application to high-frequency financial functional time series demonstrate the methodology's effectiveness, reliably detecting complex serial dependence with superior power relative to several existing tests. This work expands the toolkit for functional time series, providing a robust framework for inference in settings where traditional methods may fail.
Paper Structure (18 sections, 6 theorems, 34 equations, 4 figures, 3 tables)

This paper contains 18 sections, 6 theorems, 34 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Functional quantile autocorrelation
  4. Asymptotic properties of the sample FQA and the omnibus statistic
  5. Asymptotic behavior of the sample FQA and the omnibus statistic under independence when the quantile curve is known
  6. Asymptotic behavior of the sample FQA and the omnibus statistic under independence when the quantile curve is unknown
  7. Consistency of the sample FQA and the omnibus statistic
  8. Construction of serial dependence tests
  9. Tests at fixed levels and thresholds
  10. Omnibus tests over grids of parameters
  11. Simulation study
  12. Implementation details for the omnibus test
  13. Competing tests
  14. Finite-sample size under the null hypothesis
  15. Finite-sample power under serial dependence
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\boldsymbol{\mathcal{X}}=(\mathcal{X}_1, \mathcal{X}_2, \ldots, \mathcal{X}_T)$ be realizations of the stochastic process $\{\mathcal{X}_t, t \in \mathbb{Z}\}$ and fix $\tau \in (0,1)$. For $k=1,\ldots, T$, we have where $\overline{\mathcal{A}}_{\tau}^{k}=\{u \in \mathcal{I}: \mathcal{X}_k(u) \le \widehat{q}_\tau(u)\}$ and $\breve{\mathcal{A}}_{\tau}^{k}=\{u \in \mathcal{I}: \mathcal{X}_k(u)

Figures (4)

  • Figure 1: Empirical rejection rates of five tests (lag $l=1$) for four settings under the alternative hypothesis (FAR(1) scenario), considering different noise distributions. Rejection rates are plotted as a function of a parameter ($c$) controlling the strength of serial dependence. A significance level of $\alpha=0.05$ is considered.
  • Figure 2: Empirical rejection rates of five tests (lag $l=1$) for four settings under the alternative hypothesis (TFAR(1) scenario), considering different noise distributions. Rejection rates are plotted as a function of a quantity ($|c_1| + |c_2|$) controlling the strength of serial dependence. A significance level of $\alpha=0.05$ is considered.
  • Figure 3: Empirical rejection rates of the proposed omnibus test (lag $l=1$) for two settings under the alternative hypothesis (FAR(1) and TFAR(1) scenarios with Brownian motion noise), for different values of $T$. Rejection rates are plotted as a function of the parameters controlling the strength of serial dependence. A significance level of $\alpha=0.05$ is considered.
  • Figure 4: First 100 curves of the functional time series of log-returns for the six companies.

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5