and
|_ ( m— 1) rj
' = ∑ (X(j+1)/m - Xj/m)2 • (5)
j=1
Note that Sn2(Xi/n) is a nonparametric estimator of the volatility process evaluated at Xi/n; Florens-
Zmirou (1993) has established consistency and the asymptotic distribution of a scaled version of
(4) when the variance process follows (1).7 Recently, Sn2 (Xi/n) has been used by Bandi and Phillips
(2003), in the context of fully nonparametric estimation of diffusion processes; their asymptotic
theory is based on both the time span going to infinity and the discrete interval between successive
observations going to zero. This is because they are interested in the joint estimation of the drift
and variance diffusion terms.8
Conversely, our objective is to distinguish between the cases in which volatility is a measurable
function of the observable process, and the one in which it depends on some other state variable.
Therefore we remain silent about the drift term, and we only consider asymptotic theory in terms
of the discrete interval approaching zero. In fact, on a finite time span the contribution of the drift
term is asymptotically negligible.
Notice that Sn2 (Xi/n) is a consistent estimator of the instantaneous variance only under the null
hypothesis. Therefore, also its average over the sample realization of the process on a finite time
span, 1 /n ∑P'n∣-1)rj Sn2(Xi∕n), is a consistent estimator of integrated volatility only under the null
hypothesis.
RVm,r , which is known as realized volatility, has been proposed as a measure for volatility
concurrently by Andersen, Bollerslev, Diebold and Labys (2001), Andersen, Bollerslev, Diebold
and Ebens (2002) and Barndorff-Nielsen and Shephard (2002). The properties of realized volatility
have been extensively analyzed by Barndorff-Nielsen and Shephard (2002, 2004a,b), Andersen,
Bollerslev, Diebold and Labys (2003), Barndorff-Nielsen, Graversen and Shephard (2004) (see also
Andersen, Bollerslev, Meddahi, 2004a,b, and Meddahi, 2002, 2003). Realized volatility is a “model
free” estimator of the quadratic variation of the processes defined in (1) and (2), and is consistent for
the integrated (daily) volatility under both hypotheses. Barndorff-Nielsen and Shephard (2004a)
have shown that a scaled and centered version of RVm,r weakly converges to a mixed normal
distribution when the log price process follows a continuous semimartingale, a result which we will
use in the proof of our Theroem 1. The reason why we use two different sample frequencies in the
7The estimator Sn2 (Xi/n) has been also used by Corradi and White (1999) in order provide a test for the correct
specification of the variance process, regardless of the drift specification. Within the class of one-factor models, a
more general test, also allowing for time non-homogeneity, has been suggested by Dette, Podolskij and Vetter (2004).
8Bandi and Phillips (2003) consider a slightly modified version of Sn (Xi∕n), with a generic kernel K ( ∙ ) replacing
the indicator function. See also Jiang and Knight (1997).