Vast volatility matrix estimation for high-frequency financial data.(English)Zbl 1183.62184

Summary: High-frequency data observed on the prices of financial assets are commonly modeled by diffusion processes with micro-structure noise, and realized volatility-based methods are often used to estimate the integrated volatility. For problems involving a large number of assets, the estimation objects we face are volatility matrices of large size. The existing volatility estimators work well for a small number of assets but perform poorly when the number of assets is very large. In fact, they are inconsistent when both the number, $$p$$, of the assets and the average sample size, $$n$$, of the price data on the $$p$$ assets go to infinity.
This paper proposes a new type of estimators for the integrated volatility matrix and establishes asymptotic theory for the proposed estimators in a framework that allows both $$n$$ and $$p$$ to approach to infinity. The theory shows that the proposed estimators achieve high convergence rates under a sparsity assumption on the integrated volatility matrix. The numerical studies demonstrate that the proposed estimators perform well for large $$p$$ and complex price and volatility models. The proposed method is applied to real high-frequency financial data.

MSC:

 62P05 Applications of statistics to actuarial sciences and financial mathematics 91B25 Asset pricing models (MSC2010) 62G20 Asymptotic properties of nonparametric inference 65C60 Computational problems in statistics (MSC2010) 91G70 Statistical methods; risk measures 62H12 Estimation in multivariate analysis 62G05 Nonparametric estimation 62M05 Markov processes: estimation; hidden Markov models
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