Inference on the maximal rank of time-varying covariance matrices using high-frequency data

TL;DR

This paper develops a statistical method to test and estimate the maximal rank of time-varying covariance matrices in high-frequency financial data, accounting for bias and providing optimal detection rates.

Contribution

It introduces a new sequential testing approach for rank estimation of covariance matrices using high-frequency data, with explicit bias correction and data-driven critical values.

Findings

Established non-asymptotic critical values for rank testing.

Achieved optimal signal detection rates.

Validated methods through simulations and real data application.

Abstract

We study the rank of the instantaneous or spot covariance matrix $\Sigma_X(t)$ of a multidimensional continuous semi-martingale $X(t)$. Given high-frequency observations $X(i/n)$, $i=0,\ldots,n$, we test the null hypothesis $rank(\Sigma_X(t))\le r$ 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 $\Sigma_X(t)$. 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.