Rank tests for time-varying covariance matrices observed under noise

We consider a $d$-dimensional continuous martingale $X(t)$ with quadratic variation matrix $\langle X\rangle_t=\int_0^t Σ(s)\,ds$ 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.

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