"Multivariate realised kernels: consistent positive semi-definite estimators of the covariation of equity prices with noise and
non-synchronous trading"

OLE E. BARNDORFF-NIELSEN
	Department of Mathematical Sciences, University of Aarhus

PETER REINHARD HANSEN
	Department of Economics, Stanford University

ASGER LUNDE
	Aarhus School of Business, University of Aarhus

NEIL SHEPHARD
	Oxford-Man Institute, University of Oxford
	
Abstract

We propose a multivariate realised kernel to estimate the ex-post covariation of log-prices. We show this new 
consistent estimator is guaranteed to be positive semi-definite and is robust to measurement noise of certain types
and can also handle non-synchronous trading. It is the first estimator which has these three properties which are 
all essential for empirical work in this area. We derive the large sample asymptotics of this estimator and assess 
its accuracy using a Monte Carlo study. We implement the estimator on some US equity data, comparing our 
results to previous work which has used returns measured over 5 or 10 minutes intervals. We show the new 
estimator is substantially more precise.

Key words: HAC estimator, Long run variance estimator; Market frictions; Quadratic variation; Realised
variance.

JEL Classifications: C01, C14, C32