In theory, the sum of squares of log returns sampled at high frequency estimates their variance. When market microstructure noise is present but unaccounted for, however, we show that the optimal sampling frequency is finite and derives its closed-form expression. But even with optimal sampling, using say 5-min returns when transactions are recorded every second, a vast amount of data is discarded, in contradiction to basic statistical principles. We demonstrate that modeling the noise and using all the data is a better solution, even if one misspecifies the noise distribution. So the answer is: sample as often as possible.

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 this paper 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 parameters Θ = ∫ θt dt, where the spot parameter process θ can be a general semimartingale, with continuous and jump components. The observed AVAR is implemented with the help of a two-scales method. Its construction works well in the presence of microstructure noise, and when the observation times are irregular or asynchronous in the multivariate case. The methodology is valid for a wide variety of estimators, including the standard ones for variance and covariance, and also for more complex estimators, such as, of leverage effects, high frequency betas, and semivariance.

The econometric literature of high frequency data often relies on moment estimators which are derived from assuming local constancy of volatility and related quantities. We here study this local-constancy approximation as a general approach to estimation in such data. We show that the technique yields asymptotic properties (consistency, normality) that are correct subject to an ex post adjustment involving asymptotic likelihood ratios. These adjustments are derived and documented. Several examples of estimation are provided: powers of volatility, leverage effect, and integrated betas. The first order approximations based on local constancy can be over the period of one observation or over blocks of successive observations. It has the advantage of gaining in transparency in defining and analyzing estimators. The theory relies heavily on the interplay between stable convergence and measure change, and on asymptotic expansions for martingales.

In theory, the sum of squares of log returns sampled at high frequency estimates their variance. When market microstructure noise is present but unaccounted for, however, we show that the optimal sampling frequency is finite and derives its closed-form expression. But even with optimal sampling, using say 5-min returns when transactions are recorded every second, a vast amount of data is discarded, in contradiction to basic statistical principles. We demonstrate that modeling the noise and using all the data is a better solution, even if one misspecifies the noise distribution. So the answer is: sample as often as possible.