  Statistical inference for stochastic processes based on high-frequency
observations has been an active research area for more than two decades. One of
the most well-known and widely studied problems has been the estimation of the
quadratic variation of the continuous component of an It\^o semimartingale with
jumps. Several rate- and variance-efficient estimators have been proposed in
the literature when the jump component is of bounded variation. However, to
date, very few methods can deal with jumps of unbounded variation. By
developing new high-order expansions of the truncated moments of a locally
stable L\'evy process, we propose a new rate- and variance-efficient volatility
estimator for a class of It\^o semimartingales whose jumps behave locally like
those of a stable L\'evy process with Blumenthal-Getoor index $Y\in (1,8/5)$
(hence, of unbounded variation). The proposed method is based on a two-step
debiasing procedure for the truncated realized quadratic variation of the
process and can also cover the case $Y<1$. Our Monte Carlo experiments indicate
that the method outperforms other efficient alternatives in the literature in
the setting covered by our theoretical framework.
