Rank tests for time-varying covariance matrices observed under noise

Abstract

We consider a $d$-dimensional continuous martingale $X(t)$ with quadratic variation matrix $⟨X{⟩}_t={\int }_0^t\Sigma (s)ds$ 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 $v_n\sim ({\underline{\lambda }}_r^{-1/(\beta +1)}{n}^{-\beta /(\beta +1)})\wedge n^{-\beta /(\beta +2)}$, depending on the Hölder-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.