Rank tests for time-varying covariance matrices observed under noise

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.