"Sound and Fury": Nonlinear Functionals of Volatility Matrix in the Presence of Jump and Noise

This paper resolves a pivotal open problem on nonparametric inference for nonlinear functionals of volatility matrix. Multiple prominent statistical tasks can be formulated as functionals of volatility matrix, yet a unified statistical theory of general nonlinear functionals based on noisy data remains challenging and elusive. Nonetheless, this paper shows it can be achieved by combining the strengths of pre-averaging, jump truncation and nonlinearity bias correction. In light of general nonlinearity, bias correction beyond linear approximation becomes necessary. Resultant estimators are nonparametric and robust over a wide spectrum of stochastic models. Moreover, the estimators can be rate-optimal and stable central limit theorems are obtained. The proposed framework lends itself conveniently to uncertainty quantification and permits fully feasible inference. With strong theoretical guarantees, this paper provides an inferential foundation for a wealth of statistical methods for noisy high-frequency data, such as realized principal component analysis, continuous-time linear regression, realized Laplace transform, generalized method of integrated moments and specification tests, hence extends current application scopes to noisy data which is more prevalent in practice.