Abstract:
High frequency inference has generated a wave of research interest among econometricians and practitioners, as indicated from the increasing number of estimators based on intra-day data. However, we also witness a scarcity of methodology to assess the uncertainty -- standard error-- of the estimator. The root of the problem is that whether with or without the presence of microstructure, standard errors rely on estimating the asymptotic variance (AVAR), and often this asymptotic variance involves substantially more complex quantities than the original parameter to be estimated.
Standard errors are important: they are used both to assess the precision of estimators in the form of confidence intervals, to create ``feasible statistics" for testing, and also when building forecasting models based on, say, daily estimates.
The contribution of this paper is to provide an alternative and general solution to this problem, which we call {\it Observed Asymptotic Variance}. It is a general nonparametric method for assessing asymptotic variance (AVAR), and it provides consistent estimators of AVAR for a broad class of integrated parameters $\Theta = \int \theta_t dt$. The spot parameter process $\theta$ can be a general semimartingale, with continuous and jump components. The construction and the analysis of $\widehat{AVAR}{(\hat \Theta)}$ work well in the presence of microstructure noise, and when the observation times are irregular or asynchronous in the multivariate case. The edge effect -- phasing in and phasing out the information on the boundary of the data interval -- of any relevant estimator is also analyzed and treated rigorously.
As part of the theoretical development, the paper shows how to feasibly disentangle the effect from estimation error $\hat \Theta - \Theta$ and the variation in the parameter $\theta$ alone. For the latter, we obtain a consistent estimator of the quadratic variation (QV) of the parameter to be estimated, for example, the QV of the leverage effect.
The methodology is valid for a wide variety of estimators, including the standard ones for variance and covariance, and also for estimators, such as, of leverage effects, high frequency betas, and semi-variance.