Abstract
The Blumenthal–Getoor (BG) index characterizes the jump measure of an infinitely active Lévy process. It determines sample path properties and affects the behavior of various econometric procedures. If the process contains a diffusion term, existing estimators of the BG index based on high-frequency observations achieve rates of convergence which are suboptimal by a polynomial factor. In this paper, a novel estimator for the BG index and the successive BG indices is presented, attaining the optimal rate of convergence. If an additional proportionality factor needs to be inferred, the proposed estimator is rate-optimal up to logarithmic factors. Furthermore, our method yields a new efficient volatility estimator which accounts for jumps of infinite variation. All parameters are estimated jointly by the generalized method of moments. A simulation study compares the finite sample behavior of the proposed estimators with competing methods from the financial econometrics literature.
Highlights
Models for continuous time stochastic processes with jumps have gained increased interest in the statistical literature, most prominently in financial econometrics where they are used as a model for asset prices
The jump behavior of these processes Xt can be broadly characterized in terms of the jump activity index, given by
The method of moments is a standard technique for estimation in parametric models
Summary
Models for continuous time stochastic processes with jumps have gained increased interest in the statistical literature, most prominently in financial econometrics where they are used as a model for asset prices As the jump activity index is a central property of infinite activity jump models, it is natural to consider statistical estimation of its precise value Recent interest in this topic has been initiated by Aıt-Sahalia and Jacod (2009), who study the estimation of α based on discrete high-frequency observations Xi/n, i = 1, . In the considered high-frequency setting, the optimal rate of convergence for estimating α is conjectured to be nα/4, up to logarithmic factors This lower bound is justified by the results of Aıt-Sahalia and Jacod (2012), who study the diagonal entries of the Fisher matrix of a fully parametric submodel consisting of the sum of a Brownian motion and a symmetric α-stable Levy motion.
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