Under the two important modern financial market features of noise and non-synchronicity for multiple assets, for consistent estimators of the integrated covariations, we adopt the two-time scale average realized volatility matrix (ARVM) which is a matrix extension of the two-time scale realized volatilities of Zhang et al. (2005). An asymptotic normal theory is provided for the two-time scale ARVM and resulting realized covariations. The asymptotic normality is not directly applicable in practice to construct statistical methods owning to nuisance parameters. To bypass the nuisance parameter problem, two-stage stationary bootstrapping is proposed. We establish consistencies of the bootstrap distributions, and construct confidence intervals and hypothesis tests for the integrated covariance, regression coefficient and correlation coefficient. The validity of the stationary bootstrap for the high frequency heterogeneous returns is proved by showing that there exist parameters of the stationary bootstrap blocks so that the bootstrap consistencies hold. The proposed bootstrap methods extend the i.i.d. bootstrapping methods for realized covariations by Dovonon et al. (2013), that are confined to synchronous noise-free sampling. For high frequency noisy asynchronous samples, a Monte-Carlo experiment shows better finite sample performances of the proposed stationary bootstrap methods based on the two-time scale ARVM estimator than the wild blocks of blocks bootstrap methods of Hounyo (2017), based on pre-averaged truncated estimator.
Bibliographical noteFunding Information:
This paper was greatly indebt by constructive comments of two unknown referees and the editor. This work was supported by the National Research Foundation of Korea ( NRF–2015–1006133 ) and ( NRF-2016–4008780 ).
- High frequency data
- Market microstructure noise
- Non-synchronous trading
- Realized covariations
- Stationary bootstrap
- Two-time scale estimator