Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

In the classical mathematical finance framework it is standard to model asset price processes by continuous-time Itô processes. Paradoxically, increasing liquidity and trading activity with concomitant high-frequency data have stimulated a new angle on financial modelling of high-frequency intraday returns. An adequate and tractable model capturing the main stylized facts of typical data describes the dynamics of asset prices as a superposition of an underlying continuous-time Itô process and an independent discrete noise component induced by market microstructure effects.

 

The estimation of covariance matrices is a key issue in risk management and portfolio optimization. The entrywise (bivariate) estimation problem of non-diagonal entries of the covariance matrix in a model with synchronous sampling times comprises basically the same features and structure as the univariate volatility estimation and thus is broadly understood by the progress in the research in this field during the last decade.

 

A central problem when estimating covariance matrices is that asset prices are recorded at non-synchronous observation times, at which trading for one or a few particular assets take place. There has been progress in recent years in estimating the quadratic covariation of a bivariate non-synchronoulsy observed Itô process in the absence of noise. For the estimation of covariance matrices in the simultaneous presence of noise and non-synchronous trading, recent methods have been suggested. The estimator of Barndorff-Nielsen et al (2011) assures positive semi-definiteness, but does not attain the optimal convergence rate found in Bibinger(2011a) and shows a finite sample size bias due to non-synchronicity which is relevant for most applications.

 

A convenient approach suitable for the application in a high-dimensional setting including non-synchronous sampling designs with probable crucial differences in the observation frequencies of different assets, and minimal loss of efficiency in the entrywise estimation, is not available so far and hence one main goal of the project.

 

It is a completely original but crucial question to find a suitable modelling of dependent noise components in a multidimensional setting with non-synchronous observations. The theory for the estimation techniques is so far, carried out on simplifying independence assumptions which are unrealistic from an economic perspective. It will be analyzed if the methods perform well for more general endogenous sampling designs and noise processes or if they have to be modified.

 

Since we have to apply a smoothing and regularization technique to cope with noise and ensure positive semi-definiteness at the cost of a slower convergence rate, the impact of the non-synchronous sampling design is less striking than for the non-noisy case. Therefore, we believe that the complicated influence of interpolations, on which asymptotic variances of the Hayashi-Yoshida estimator hinge on, will not affect the asymptotics of the estimation in our setup any more. This leads to new aspects and open questions for the role and possible treatment of non-synchronous observation schemes in the high-dimensional case that should be illuminated within the IRTG.

 

 

 

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