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Rank tests for time-varying covariance matrices observed under noise

Markus Reiß, Lars Winkelmann

TL;DR

This work develops nonparametric rank tests for time-varying spot covariance matrices $Σ(t)$ observed with microstructure noise. It introduces localized spectral covariance estimators on sub-intervals $I_{t,h}$ with a mixing parameter $M$, derives a central limit theorem and non-asymptotic control, and constructs local and global test procedures for testing $H_0: \mathrm{rank}(Σ(t))\le r$ against sparse alternatives. The paper establishes sharp, minimax-optimal detection rates governed by Hölder smoothness $β$ and a spectral gap $\underline{λ}_r$, with faster rates when a gap is present; it also provides simulation-based critical values and a real-data application to German government bonds. Overall, the framework enables fast, robust detection of time-varying factor structure in high-frequency financial data under observation noise, with explicit finite-sample and asymptotic performance guarantees.

Abstract

We consider a $d$-dimensional continuous martingale $X(t)$ with quadratic variation matrix $\langle X\rangle_t=\int_0^t Σ(s)\,ds$ and develop tests for the rank of its spot covariance matrix $Σ(t)$, $t\in[0,1]$. The process $X$ is observed under observational noise, as is standard for microstructure noise models in high-frequency finance. We test the null hypothesis ${\mathcal H}_0:rank(Σ(t))\le r$ against local alternatives ${\mathcal H}_{1,n}:λ_{r+1}(Σ(t))\ge v_n$, where $λ_{r+1}$ denotes the $(r+1)$st eigenvalue and $v_n\downarrow 0$ as the sample size $n\to\infty$. We construct test statistics based on eigenvalues of carefully calibrated localized spectral covariance matrix estimates. Critical values are provided non-asymptotically as well as asymptotically via maximal eigenvalues of Gaussian orthogonal ensembles. The power analysis establishes asymptotic consistency for a separation rate $v_n\thicksim (\underlineλ_r^{-1/(β+1)}n^{-β/(β+1)})\wedge n^{-β/(β+2)}$, depending on the Hölder-regularity $β$ of $Σ$ and a possible spectral gap $\underlineλ_r\ge 0$ under ${\mathcal H}_0$. A lower bound shows the optimality of this rate. We discuss why the rate is much faster than conventional estimation rates. The theory is illustrated by simulations and a real data example with German government bonds of varying maturity.

Rank tests for time-varying covariance matrices observed under noise

TL;DR

This work develops nonparametric rank tests for time-varying spot covariance matrices observed with microstructure noise. It introduces localized spectral covariance estimators on sub-intervals with a mixing parameter , derives a central limit theorem and non-asymptotic control, and constructs local and global test procedures for testing against sparse alternatives. The paper establishes sharp, minimax-optimal detection rates governed by Hölder smoothness and a spectral gap , with faster rates when a gap is present; it also provides simulation-based critical values and a real-data application to German government bonds. Overall, the framework enables fast, robust detection of time-varying factor structure in high-frequency financial data under observation noise, with explicit finite-sample and asymptotic performance guarantees.

Abstract

We consider a -dimensional continuous martingale with quadratic variation matrix and develop tests for the rank of its spot covariance matrix , . The process is observed under observational noise, as is standard for microstructure noise models in high-frequency finance. We test the null hypothesis against local alternatives , where denotes the st eigenvalue and as the sample size . We construct test statistics based on eigenvalues of carefully calibrated localized spectral covariance matrix estimates. Critical values are provided non-asymptotically as well as asymptotically via maximal eigenvalues of Gaussian orthogonal ensembles. The power analysis establishes asymptotic consistency for a separation rate , depending on the Hölder-regularity of and a possible spectral gap under . A lower bound shows the optimality of this rate. We discuss why the rate is much faster than conventional estimation rates. The theory is illustrated by simulations and a real data example with German government bonds of varying maturity.
Paper Structure (13 sections, 20 theorems, 161 equations, 2 figures, 1 table)

This paper contains 13 sections, 20 theorems, 161 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic results
  3. Local tests under the null hypothesis
  4. Power against local alternatives and detection rate
  5. Global test
  6. Optimality of the detection rate
  7. Implementation and examples
  8. Further results and proofs
  9. Proof of the central limit theorem
  10. Eigenvalue concentration
  11. Empirical eigenvalue concentration under ${\mathcal{H}}_0$
  12. Lower eigenvalue deviation bounds
  13. Proof of the lower bound

Key Result

Theorem 2.1

Consider the asymptotics $M\to\infty$, while $t,h,n$ may vary arbitrarily with $M$. For each $v\in\mathop{\mathrm{\mathbb R}}\nolimits^d$ with $\lVert v \rVert=1$ assume Then the following central limit theorem holds: Moreover, we have ${\mathcal{C}}(\widehat{\Sigma}^{t,h})\lesssim M^{-1}((\lVert \Sigma_{\ell,\ell'} \rVert_{L^\infty(I_{t,h})})_{\ell,\ell'=1,\ldots,d}+\varepsilon_{n,h}^2M^2I_d)^{

Figures (2)

  • Figure 1: Left: Simulations under ${\mathcal{H}}_0$. Distribution of the largest eigenvalue of GOE in dimension 9 (red), simulation-based (SIM) distribution (blue), and the simulated second normalized eigenvalue (histogram), based on 20,000 Monte Carlo rounds. Coloured bars on the $x$-axis mark 90%, 95%, and 99% quantiles. Right: rejection frequencies of the global, simulation-based test under ${\mathcal{H}}_1$ (1,000 repetitions).
  • Figure 2: Data example. Left: Mid-quotes of 10 German zero coupon bonds with maturities between 3 and 30 years on 4 May 2023. Right: corresponding test statistic and critical values (horizontal dashed lines) of the global test.

Theorems & Definitions (52)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Example 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • ...and 42 more