Assessment of Uncertainty in High Frequency Data: The Observed Asymptotic Variance

The availability of high frequency financial data has generated a series of estimators based on intra-day data, improving the quality of large areas of financial econometrics. However, estimating the standard error of these estimators is often challenging. The root of the problem is that traditionally, standard errors rely on estimating a theoretically derived asymptotic variance, and often this asymptotic variance involves substantially more complex quantities than the original parameter to be estimated.

Standard errors are important: they are used to assess the precision of estimators in the form of confidence intervals, to create “feasible statistics” for testing, to build forecasting models based on, say, daily estimates, and also to optimize the tuning parameters.

The contribution of the papers is to provide an alternative and general solution to this problem, which we call Observed Asymptotic Variance. It is a general nonparametric method for assessing asymptotic variance (AVAR). It provides consistent estimators of AVAR for a broad class of integrated spot parameters, such as the standard ones for variance (volatility) and covariance, and also for more complex estimators, such as, of leverage effects, high frequency betas, and semi-variance. The spot parameter process can be a general semi-martingale, with continuous and jump components. The observed AVAR can be implemented with either a two- or multiscale/regression method, depending on the size of the edge effects. Its construction works well in the presence of microstructure noise, and when the observation times are irregular or asynchronous in the multivariate case. The observed AVAR device is related to observed information in likelihood theory, but in this case it is non-parametric and uses the high-frequency data structure.