dorsal/arxiv
View SchemaRank tests for time-varying covariance matrices observed under noise
| Authors | Markus Reiß, Lars Winkelmann |
|---|---|
| Categories | |
| ArXiv ID | 2601.08353vv1 |
| URL | https://arxiv.org/abs/2601.08353 |
| License | http://creativecommons.org/licenses/by-nc-sa/4.0/ |
Abstract
We consider a $d$-dimensional continuous martingale $X(t)$ with quadratic variation matrix $\langle X\rangle_t=\int_0^t \Sigma(s)\,ds$ and develop tests for the rank of its spot covariance matrix $\Sigma(t)$, $t\in[0,1]$. The process $X$ is observed under observational noise, as is standard for microstructure noise models in high-frequency finance. We test the null hypothesis ${\mathcal H}_0:rank(\Sigma(t))\le r$ against local alternatives ${\mathcal H}_{1,n}:\lambda_{r+1}(\Sigma(t))\ge v_n$, where $\lambda_{r+1}$ 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 $v_n\thicksim (\underline\lambda_r^{-1/(\beta+1)}n^{-\beta/(\beta+1)})\wedge n^{-\beta/(\beta+2)}$, depending on the H\"older-regularity $\beta$ of $\Sigma$ and a possible spectral gap $\underline\lambda_r\ge 0$ 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.
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"abstract": "We consider a $d$-dimensional continuous martingale $X(t)$ with quadratic variation matrix $\\langle X\\rangle_t=\\int_0^t \\Sigma(s)\\,ds$ and develop tests for the rank of its spot covariance matrix $\\Sigma(t)$, $t\\in[0,1]$. The process $X$ is observed under observational noise, as is standard for microstructure noise models in high-frequency finance. We test the null hypothesis ${\\mathcal H}_0:rank(\\Sigma(t))\\le r$ against local alternatives ${\\mathcal H}_{1,n}:\\lambda_{r+1}(\\Sigma(t))\\ge v_n$, where $\\lambda_{r+1}$ 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 $v_n\\thicksim (\\underline\\lambda_r^{-1/(\\beta+1)}n^{-\\beta/(\\beta+1)})\\wedge n^{-\\beta/(\\beta+2)}$, depending on the H\\\"older-regularity $\\beta$ of $\\Sigma$ and a possible spectral gap $\\underline\\lambda_r\\ge 0$ 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.",
"arxiv_id": "2601.08353",
"authors": [
"Markus Rei\u00df",
"Lars Winkelmann"
],
"categories": [
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"license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
"title": "Rank tests for time-varying covariance matrices observed under noise",
"url": "https://arxiv.org/abs/2601.08353",
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