Functional stable limit theorems for quasi-efficient spectral covolatility estimators

Abstract

We consider noisy non-synchronous discrete observations of a continuous semimartingale with random volatility. Functional stable central limit theorems are established under high-frequency asymptotics in three setups: one-dimensional for the spectral estimator of integrated volatility, from twodimensional asynchronous observations for a bivariate spectral covolatility estimator and multivariate for a local method of moments. The results demonstrate that local adaptivity and smoothing noise dilution in the Fourier domain facilitate substantial efficiency gains compared to previous approaches. In particular, the derived asymptotic variances coincide with the benchmarks of semiparametric Cramér-Rao lower bounds and the considered estimators are thus asymptotically efficient in idealized subexperiments. Feasible central limit theorems allowing for confidence bounds are provided.

Change-point inference on volatility in noisy Itô semimartingales

“…Though the possible extensions appear to be relevant for applications, we work in the framework formulated in Assumption 2.3, mainly due to the lack of groundwork sufficient for the present work. Since we exploit some ingredients from previous works on spectral volatility estimation, particularly the form of the efficient asymptotic variance based on Altmeyer and Bibinger (2015), a generalization of our results requires non-trivial generalizations of these ingredients first. Furthermore, more general noise processes ask for extensive work on the estimation of the local long-run variance replacing (9).…”

“…The construction of the test is based on a combination of the techniques by Altmeyer and Bibinger (2015) and Bibinger et al (2017). In order to do so, we pick a sequence h n with h n ∝ n −1/2 log (n)…”

“…First of all, taking into account that the asymptotic order of the weights (48) do not depend on i, , we may consider them as a function w j = w j (σ 2 ) of the spot volatility. Calculating the first derivative, w j as pursued on page 40 in Altmeyer and Bibinger (2015), we get the upper bound…”

“…In this work we will extend the methods and results presented in Bibinger et al (2017) in order to construct a general test for volatility jumps based on the model (5). Our statistics are functionals of spectral spot volatility estimates building up on the local Fourier method of moments in Altmeyer and Bibinger (2015) extending the volatility estimation approach introduced by Reiß (2011). While several linear estimators for functionals of the volatility have by now been generalized to noise-robust approaches, the considered change-point test is based on maximum statistics and its extension to an efficient method under noise requires new techniques.…”

Estimation of the discontinuous leverage effect: Evidence from the NASDAQ order book

“…Reiß (2011) introduces spectral estimation of the quadratic variation from noisy observations. Bibinger et al (2014) and Altmeyer and Bibinger (2015) establish the asymptotic efficiency of spectral estimators of the integrated volatility matrix in the multivariate case with noisy and nonsynchronous observations. Bibinger et al (2017) propose a related spot volatility estimator.…”