Inference on the maximal rank of time-varying covariance matrices using high-frequency data
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We study the rank of the instantaneous or spot covariance matrix Σ_X(t) of a multidimensional continuous semi-martingale X(t). Given high-frequency observations X(i/n), i=0,…,n, we test the null hypothesis rank(Σ_X(t))≤ r for all t against local alternatives where the average (r+1)st eigenvalue is larger than some signal detection rate v_n. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and a spectral gap of Σ_X(t). We establish explicit matrix perturbation and concentration results that provide non-asymptotic uniform critical values and optimal signal detection rates v_n. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.
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