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Asymptotically uniformly most powerful tests for diffusion processes with nonsynchronous observations

Teppei Ogihara, Futo Ueno

TL;DR

This work addresses hypothesis testing for diffusion processes observed under nonsynchronous sampling by developing a quasi-likelihood ratio framework with adaptive estimation of drift and diffusion coefficients. The resulting tests $T_n^1$ for the diffusion parameter block and $T_n^2$ for the drift parameter block are shown to be asymptotically $ ext{chi-square}$ under the null and consistent under alternatives, with an asymptotic uniformly most powerful property among asymptotically invariant tests. The paper also introduces fast computation techniques, including block decomposition and a log-determinant acceleration, to enable scalable application to high-frequency data. Numerical experiments demonstrate that the proposed tests outperform nonparametric counterparts like Hayashi–Yoshida in size control and power, particularly under local alternatives, highlighting practical advantages for financial econometrics and other high-frequency diffusion settings.

Abstract

This paper introduces a quasi-likelihood ratio testing procedure for diffusion processes observed under nonsynchronous sampling schemes. High-frequency data, particularly in financial econometrics, are often recorded at irregular time points, challenging conventional synchronous methods for parameter estimation and hypothesis testing. To address these challenges, we develop a quasi-likelihood framework that accommodates irregular sampling while integrating adaptive estimation techniques for both drift and diffusion coefficients, thereby enhancing optimization stability and reducing computational burden. We rigorously derive the asymptotic properties of the proposed test statistic, showing that it converges to a chi-squared distribution under the null hypothesis and exhibits consistency under alternatives. Moreover, we establish that the resulting tests are asymptotically uniformly most powerful. Extensive numerical experiments corroborate the theoretical findings and demonstrate that our method outperforms existing nonparametric approaches.

Asymptotically uniformly most powerful tests for diffusion processes with nonsynchronous observations

TL;DR

This work addresses hypothesis testing for diffusion processes observed under nonsynchronous sampling by developing a quasi-likelihood ratio framework with adaptive estimation of drift and diffusion coefficients. The resulting tests for the diffusion parameter block and for the drift parameter block are shown to be asymptotically under the null and consistent under alternatives, with an asymptotic uniformly most powerful property among asymptotically invariant tests. The paper also introduces fast computation techniques, including block decomposition and a log-determinant acceleration, to enable scalable application to high-frequency data. Numerical experiments demonstrate that the proposed tests outperform nonparametric counterparts like Hayashi–Yoshida in size control and power, particularly under local alternatives, highlighting practical advantages for financial econometrics and other high-frequency diffusion settings.

Abstract

This paper introduces a quasi-likelihood ratio testing procedure for diffusion processes observed under nonsynchronous sampling schemes. High-frequency data, particularly in financial econometrics, are often recorded at irregular time points, challenging conventional synchronous methods for parameter estimation and hypothesis testing. To address these challenges, we develop a quasi-likelihood framework that accommodates irregular sampling while integrating adaptive estimation techniques for both drift and diffusion coefficients, thereby enhancing optimization stability and reducing computational burden. We rigorously derive the asymptotic properties of the proposed test statistic, showing that it converges to a chi-squared distribution under the null hypothesis and exhibits consistency under alternatives. Moreover, we establish that the resulting tests are asymptotically uniformly most powerful. Extensive numerical experiments corroborate the theoretical findings and demonstrate that our method outperforms existing nonparametric approaches.

Paper Structure

This paper contains 18 sections, 22 theorems, 209 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Settings and the quasi-likelihood ratio test statistics
  4. Consistency of the quasi-likelihood ratio tests
  5. The AUMPI property of the test statistics
  6. Fast computation of $\mathbb{H}_n^l$
  7. Size reduction of $S_n$ by dividing observations into blocks
  8. Fast computation of $\log \det S_n$
  9. Numerical Experiments
  10. Test for $\sigma$ with time-invariant diffusion coefficients
  11. Test for $\sigma$ with time-varying diffusion coefficients
  12. Test for $\theta$
  13. Proofs of the Main Results
  14. Preparations
  15. Proof of Theorems \ref{['thm:sigma0']} and \ref{['thm:sigma1']}
  16. ...and 3 more sections

Key Result

Theorem 2.1

Suppose that Assumptions (A1)--(A4) and (A6) hold. Under $H_{0}^{(\sigma)}$,

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 29 more