Inference for Continuous Semimartingales Observed at High Frequency

Abstract

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 given. Several examples of estimation are provided: powers of volatility, leverage effect, integrated betas, bipower, and covariance under asynchronous observation. The first order approximations in this study can be over the period of one observation, or over blocks of successive observations. The advantage of blocking is a gain 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.

Nonparametric Bayesian volatility learning under microstructure noise

“…2.1.5 for an intuitive exposition. Formal results why this works in specific settings rely on Girsanov's theorem, see, e.g., Gugushvili et al (2020), Hoffmann et al (2012, Mykland and Zhang (2009). Further reasons why one would like to set b = 0 are that b is a nuisance parameter, in specific applications its appropriate parametric form might be unknown, and finally, a single observed time series is not sufficient to learn b consistently (see Ignatieva & Platen 2012).…”

“…There exists a large body of statistics and econometrics literature on nonparametric volatility estimation under microstructure noise. See, e.g., Hoffmann et al (2012), Jacod et al (2009), Mykland and Zhang (2009), Reiß (2011), Sabel et al (2015); a recent overview is Mykland and Zhang (2012). The literature predominantly deals with estimation of the integrated volatility t 0 s 2 (u)du, although inference on s has also been studied in several of these references.…”

On the Estimation of High-Dimensional Integrated Covariance Matrix Based on High-Frequency Data With Multiple Transactions

“…(2005), Mykland and Zhang (2009), Aït-Sahalia et al (2010), Aït-Sahalia et al (2011), Zhang et al (2005), and Zhang (2006); the realized kernel suggested by Barndorff-Nielsen et al (2008) and Barndorff-Nielsen et al (2011); the quasi-maximum likelihood method studied by Xiu (2010); and the pre-averaging approach proposed by Jacod et al (2009) and Li (2013).…”

Realized semicovariances: Empirical applications to volatility forecasting and portfolio optimization

“…Realized variance, the most commonly used realized measure, is constructed by summing up squared intra-daily returns. There is an extensive statistical theory for this subject derived in papers by Barndorff-Nielsen and Shephard (2002), Meddahi (2002), Andersen et al (2003), and Mykland and Zhang (2009), among others.…”