The algebra of two scales estimation, and the S-TSRV: High frequency estimation that is robust to sampling times

In this paper, we derive a new algebraic property of two scales estimation in high frequency data, under which the effect of sampling times is canceled to high order. This is a particular robustness property of the two scales construction. In general, irregular, asynchronous, or endogenous times can cause problems in estimators based on equidistant observation of (trade or quote) times. The new algebraic property can be combined with pre-averaging, giving rise to the smoothed two-scales realized volatility (S-TSRV). We derive a finite sample solution to controlling edge effects and for handling irregular and endogenous observation times and asynchronously observed multivariate data. In connection with this development, we use the algebraic approach to define a version of the S-TSRV which has particularly small edge effect in microstructure noise. The main result of the paper is a representation of the statistical error of the estimator in terms of simple components. As an application of this representation, the paper develops a central limit theory for multivariate volatility estimators. The approach can also handle leads and lags in the signal process.