A Note on a Statistical Hypothesis Testing for Removing Noise by the Random Matrix Theory and Its Application to Co-Volatility Matrices

It is well known that the bias called market microstructure noise will arise, when estimating realized co-volatility matrix which is calculated as a sum of cross products of intraday high-frequency returns. An existing conventional technique for removing such a market microstructure noise is to perform eigenvalue decomposition of the sum of cross products matrix and to identify the elements corresponding to the decomposed values which are smaller than the maximum eigenvalue of the random matrix as noises. Although the maximum eigenvalue of a random matrix follows asymptotically Tracy-Widom distribution, the existing technique does not take this asymptotic nature into consideration, but only the convergence value is used for it. Therefore, it cannot evaluate quantitatively such a risk that regards accidentally essential volatility as a noise. In this paper, we propose a statistical hypothesis test for removing noise in co-volatility matrix based on the nature in which the maximum eigenvalue of a random matrix follows Tracy-Widom distribution asymptotically.