# Econometric methods

Our database contains daily (close to close) financial returns

,

and a corresponding sequence of daily realised measures

.

Realised measures are theoretically sound high frequency, nonparametric based estimators of the variation of the price path of an asset during the times at which the asset trades frequently on an exchange. Realised measures ignore the variation of prices overnight and sometimes the variation in the first few minutes of the trading day when recorded prices may contain large errors. The background to realised measures can be found in the survey articles by Andersen, Bollerslev and Diebold (2008) and Barndorff-Nielsen and Shephard (2007).

We way the statistics reported in the library are generated is spelt out in Shephard and Sheppard (2009). Here we give a brief summary.

The simplest realised measure is realised variance

where

and are the times of trades or quotes (or a subset of
them) on the t-th day. The theoretical justification of this
measure is that if prices are observed without noise then as it consistently estimates the quadratic
variation of the price process on the *t*-th day. It was
formalised econometrically by Andersen, Bollerslev, Diebold and
Labys (2001) and Barndorff-Nielsen and Shephard (2002).

In practice market microstructure noise plays an important part and the above authors use 1-5 minute return data or a subset of trades or quotes (e.g. every 15th trade) to mitigate the effect of the noise. Hansen and Lunde (2006) systematically study the impact of noise on realised variance. If a subset of the data is used with the realised variance, then it is possible to average across many such estimators each using different subsets. This is called subsampling. When we report RV estimators we always subsample them to the maximum degree possible from the data as this averaging is always theoretically beneficial especially in the presence of modest amounts of noise.

Three classes of estimators which are somewhat robust to noise
have been suggested in the literature: preaveraging (Jacod, Li,
Mykland, Podolskij and Vetter(2007)), multiscale Zhang (2007) and
Zhang, Mykland and Ait-Sahalia (2005)) and realised kernel
(Barndorff-Nielsen, Hansen, Lunde and Shephard (2008))

Here we focus on the realised kernel in the case where we use a
Parzen weight function. It has the familiar form of a HAC type
estimator (except there is no adjustment for mean and the sums are
not scaled by their sample size)

,

where

and *k(x)* is the Parzen kernel function. It is necessary
for *H* to increase with the sample size in order to
consistently estimate the increments of quadratic variation in the
presence of noise. We follow precisely the bandwidth choice of
*H* spelt out in Barndorff-Nielsen, Hansen, Lunde and
Shephard (2009), to which we refer the reader for details. This
realised kernel is guaranteed to be non-negative, which is quite
important as some of our time series methods rely on this
property.