Estimating the Jump Activity Index Under Noisy Observations Using High-Frequency Data

It is widely accepted that the high-frequency data are contaminated by microstructure noise, whose effect on the statistical inference has been of increasing interest in the literature. Much of it, however, has focused on the integrated volatility. In this article, we investigate another important characteristic, namely, the jump activity index (JAI) of a discretely sampled semi-martingale corrupted by microstructure noise. We point out that ignoring the microstructure noise can have a disastrous effect on the estimation of the JAI. Consequently, we propose a two-stage procedure to estimate the JAI. It first reduces the effect of noise by local smoothing and then estimates the index from the smoothed data. The asymptotic properties such as consistency and asymptotic normality are given. Simulations are conducted to evaluate the performance of the procedure. Finally, we implement our estimators to some real datasets.