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\begin{document}
	

\title{\textbf{In-sample Asymptotics and Across-sample Efficiency Gains for High Frequency Data Statistics\\
	Online Appendix}}
\author{ \textbf{Eric Ghysels}\thanks{University of North Carolina, Department of Economics, Gardner Hall CB 3305, Chapel Hill, NC 27599-3305, \texttt{e-mail:} eghysels@unc.edu.} \and \textbf{Per Mykland}\thanks{University of Chicago, Department of Statistics, 5734 University Avenue, Chicago, Illinois 60637,
	 \texttt{e-mail:} mykland@galton.uchicago.edu.} \and
	\textbf{Eric Renault}\thanks{Department of Economics, University of Warwick, Coventry, CV4 7AL, United Kingdom, \texttt{e-mail:} 
		Eric.Renault@warwick.ac.uk.} }
\date{{\normalsize June 8, 2021}}
\maketitle
\begin{abstract}
	{\footnotesize We revisit in-sample asymptotic analysis extensively
		used in the realized volatility literature. We show that there are
		gains to be made in estimating current realized volatility from
		considering realizations in prior periods. The weighting schemes also relate to Kalman-Bucy filters, although our approach is
		non-Gaussian and model-free. We derive theoretical results for a
		broad class of processes pertaining to volatility, higher moments and leverage. The paper also contains a Monte Carlo simulation study showing the benefits of across-sample combinations.}
\end{abstract}

\thispagestyle{empty}

\newpage

\onehalfspacing \setcounter{page}{1}




In this Online Appendix we provide technical details pertaining to the results in \citeasnoun{GMR_2019}.  
Section \ref{qv-no} covers the proof of Proposition \ref{GMR-prop:rv}. 
Additional volatility examples are covered in Section \ref{sec:addvolex}, whereas examples beyond volatility appear in Section \ref{beyond}. Section \ref{sec:kalman} provides insights on the connection of our analysis with the Kalman filter. Finally, Section \ref{foregains} discusses a forecasting example.



\setcounter{equation}{0}

\section{Proof of Proposition \ref{GMR-prop:rv} \label{qv-no}}



We study the details for across-sample efficiency gains pertaining to the first Example \ref{GMR-ex:rv} for the standard realized volatility (RV), obtained by summing squared intra-daily returns, yielding the so called realized
variance, namely:
\begin{equation}
\hat{\Theta}_{n,t}=\sum_{t-1 < t_{i+1} \le t }(X_{t_{i+1}} - X_{t_{i}})^2.
\label{RVBlock}
\end{equation}
We study here the case of equidistant sampling, $t_{i}-t_{i-1}=\Delta $ $%
t_{i}=1/n.$
When the sampling frequency increases, i.e.\ $n$ $\rightarrow $ $\infty ,$
then the realized variance converges uniformly in probability to the
increment of the quadratic variation i.e.\
\begin{equation}
\lim_{n\rightarrow \infty } \hat{\Theta}_{n,t} \rightarrow ^{p}  \int_{t-1}^{t} \theta(s) ds.
\end{equation}
To streamline the notation we will drop the superscript $n.$ %
\citeasnoun{Barn-Shep-2002}, \citeasnoun{jacodprotter98} and
\citeasnoun{zhang2001} show that the error of realized variance is
asymptotically
\begin{equation}
\frac{n^{1/2}(\hat{\Theta}_{n,t}- \int_{t-1}^{t} \theta(s) ds)}{\sqrt{V_t}} \overset{d}{\rightarrow} N\left( 0,1\right)  \label{AsyErrox_RV}
\end{equation}
where $V_{t}$ = $2\int_{t-1}^{t}\sigma(s)^{4}ds$ is called (twice) the integrated quarticity. 
It should be noted that in the case of no leverage effect, the result in (\ref{AsyErrox_RV}) follows directly
from the simplified example in Section \ref{GMR-intro-example} of \citeasnoun{GMR_2019}. 


We want to estimate $\Theta_t$ = $\int_{t-1}^{t}\sigma^2(s)ds$ and take advantage of observations on the previous day, summarized by $\hat{\Theta}_{n,t-1},$ the
estimator of $\Theta_{t-1}.$  The key assumption is that the two estimators $\hat{\Theta}_{n,\tau},$ $\tau$ = $t-1$ and $t$ have an asymptotic accuracy of the same order of magnitude and are asymptotically independent, for a given volatility path, namely:
\begin{eqnarray}
\frac{n^{1/2}(\hat{\Theta}_{n,t-1}-\Theta_{t-1})}{\sqrt{V_{t-1}}} & \overset{d}{\rightarrow } & N(0,1)  \notag \\
\frac{n^{1/2}(\hat{\Theta}_{n,t}- \Theta_t)}{\sqrt{V_t}} &\overset{d}{\rightarrow } & N(0,1)  \label{basic-limit}
\end{eqnarray}
and the joint asymptotic distribution is the product of the marginals. We consider possible improvements of our
estimator of $ \int_{t-1}^{t} \theta(s) ds,$ assuming for the moment that we know the correlation
coefficient:
\begin{equation}  \label{lag1}
\varphi_n =\frac{Cov(\hat{\Theta}_{n,t},\hat{\Theta}_{n,t-1})}{Var(\hat{\Theta}_{n,t-1})}
\end{equation}
and the unconditional expectation $E(\hat{\Theta}_{n,t})$ = $E(\hat{\Theta}_{n,t-1})$ = $E(\int_{t-1}^{t}\sigma^2(s)ds).$ Note that
equation (\ref{lag1}) does \textit{not} imply that our analysis is confined to AR(1) models. Instead, equation (\ref{lag1}) only reflects the fact that we condition predictions on a single lag $\hat{\Theta}_{t-1}.$ Equation (\ref{lag1}) is essentially a general version of equation (\ref{GMR-fore}). There may be potential gains from considering more lags, as the underlying models would result
in higher order dynamics. Yet, for our analysis we currently focus
exclusively on prediction equations with a single lag. Higher order equations are a straightforward extension discussed later. In analogy with equation (\ref{GMR-phizeroH}) we also need:
\begin{equation}  \label{lag1nohat}
\varphi_n^0 =\frac{Cov(\Theta_{n,t},\Theta_{n,t-1})}{Var(\Theta_{n,t-1})}.
\end{equation}

The theory presented in the sequel will mirror the development in Section \ref{GMR-intro-example} of \citeasnoun{GMR_2019},
but be valid even when volatility is not piecewise constant.


Consider the best linear forecast of using (only) $\hat{\Theta}_{n,t-1}:$
\begin{equation}
\label{realizedpredwithoutn}
\breve{\Theta}_{n,t\mid t-1}=\varphi _{n}\hat{\Theta}_{n,t-1}+(1-\varphi
_{n})E\left( \Theta _{t}\right).
\end{equation}
Note that this \textit{realized forecast} is infeasible in practice and, to make it feasible, estimators of $\varphi $ and $E(\int_{t-1}^{t}\sigma^2(s)ds)$ are required. These estimators will be based on past time series of realized volatilities: $\hat{\Theta}_{n,\tau},$ $\tau$ = $t-1,...,t-T+1.$  The estimation error on these coefficients will be made negligible when $(T/n)$ goes to infinity.


Our goal is to combine the two measurements $\hat{\Theta}_{n,t}$ and
$\breve{\Theta}_{n,t\mid t-1}$ of $ \int_{t-1}^{t} \theta(s) ds$ to define a new estimator:
\begin{equation}
\hat{\Theta}_{n,t}(\omega_t )=(1-\omega_t )\hat{\Theta}_{n,t}+\omega_t
\breve{\Theta}_{n,t\mid t-1}.
\end{equation}
Intuitively, the more persistent the volatility process, the more $\hat{\Theta}_{n,t\mid
	t-1}$ is informative about $\hat{\Theta}_{n,t}$ and the larger the optimal weight $\omega_t$ should be. Note that the weight depends on $t,$ as indeed its
computation will be volatility path dependent. To characterize such an optimal choice, one may apply a conditional control variable principle, given the volatility path. The criterion to minimize will be the conditional mean squared error:
\begin{equation}  \label{contr0}
E_{\sigma }\left[ \hat{\Theta}_{n,t}(\omega _{t})-\Theta _{t}\right]
^{2}=E_{\sigma }\left\{ \hat{\Theta}_{n,t}-\Theta _{t}-\omega _{t}\left( 
\hat{\Theta}_{n,t}-\breve{\Theta}_{n,t\mid t-1}\right) \right\} ^{2}.
\end{equation}
Then, the problem to solve is obviously nearly identical to the one
considered in Section \ref{GMR-intro-example} of \citeasnoun{GMR_2019}, so that:
\begin{equation}
\hat{\Theta}_{n,t}(\omega _{n,t}^{\ast })=\hat{\Theta}_{n,t}-\omega
_{n,t}^{\ast }\left( \hat{\Theta}_{n,t}-\breve{\Theta}_{n,t\mid t-1}\right) 
\label{estimator}
\end{equation}
will be an optimal improvement of $\hat{\Theta}_{n,t}$ if $\omega_{n,t}^{\ast }$\ is defined according to the following control variable
formula:
\begin{equation}
\label{control2}
\omega _{n,t}^{\ast }=\frac{Cov_{\Theta }\left[ \hat{\Theta}_{n,t},\hat{%
		\Theta}_{n,t}-\breve{\Theta}_{n,t|t-1}\right] }{E_{\Theta }\left[ \hat{%
		\Theta}_{n,t}-\breve{\Theta}_{n,t|t-1}\right]^2 }=\frac{Var_{\Theta
	}[\hat{\Theta}_{n,t}]}{E_{\Theta }\left[ \hat{\Theta}_{n,t}-\breve{\Theta}_{n,t|t-1}\right]^2 }. 
\end{equation}

Note that $\omega _{n,t}^{\ast }$\ has been shrunk with respect to the
conditional regression coefficient $\hat{\Theta}_{n,t}$ on $(\hat{\Theta}_{n,t}$ - $\breve{\Theta}_{n,t\mid t-1}).$ This is due to the need to take into account the non-zero
mean of $(\hat{\Theta}_{n,t}-\breve{\Theta}_{n,t\mid t-1})$ given the volatility
path.

A closed form formula for the optimal weights is obtained by computing
moments, given the
volatility path, according to the asymptotic distribution appearing in (\ref{basic-limit}).
Then, given the volatility path, we have:
\begin{eqnarray*}
	E_{\sigma }\left( \hat{\Theta}_{n,t}-\breve{\Theta}_{n,t\mid t-1}\right) 
	&=&\Theta _{t}-\varphi _{n}\Theta _{t-1}-(1-\varphi _{n})E\left[ \Theta _{t}%
	\right]  \\
	Var_{\sigma }\left( \hat{\Theta}_{n,t}-\hat{\Theta}_{n,t| t-1}\right)  &=&%
	\frac{V_{t}}{n}+\varphi _{n}^{2}\frac{V_{t-1}}{n}+o\left( \frac{1}{n}\right) 
	\\
	Cov_{\sigma }\left[ \hat{\Theta}_{n,t},\hat{\Theta}_{n,t}-\hat{\Theta}%
	_{n,t\mid t-1}\right]  &=&Var_{\sigma }\left[ \hat{\Theta}_{n,t}\right] =%
	\frac{V_{t}}{n}+o\left( \frac{1}{n}\right).
\end{eqnarray*}
Therefore, $\omega _{n,t}^{\ast }$ defined by (\ref{estimator}) can be described by:%
\begin{eqnarray}
\label{eq:optimalweight}
\frac{1}{\omega _{n,t}^{\ast }} &=&1+\varphi _{n}^{2}\frac{V_{t-1}}{V_{t}}+n%
\frac{B_{F,n}^{2}(t)}{V_{t}}+o\left( \frac{1}{n}\right)   \label{optRV} \\
B_{F,n}(t) &=&\Theta _{t}-\varphi _{n}\Theta _{t-1}-(1-\varphi _{n})E\left[
\Theta _{t}\right].   \notag
\end{eqnarray}

The above yields the result in equation (\ref{GMR-optRV}) of Proposition \ref{GMR-prop:rv}. Note that, in order to understand this optimal weight, it is useful to rewrite it as follows (as a general version of (\ref{GMR-eq:optweightspecialcase}) in \citeasnoun{GMR_2019}):%@@@
\begin{eqnarray*}
	\frac{1}{\omega _{n,t}^{\ast }} &=&1+\varphi _{n}^{2}\frac{V_{t-1}}{V_{t}}+n%
	\frac{B_{F,n}^{2}(t)-B_{I}^{2}(t)}{V_{t}}+n\frac{B_{I}^{2}(t)}{V_{t}}%
	+o\left( \frac{1}{n}\right)  \\
	B_{I}(t) &=&\Theta _{t}-\varphi ^{0}\Theta _{t-1}-(1-\varphi ^{0})E\left[
	\Theta _{t}\right]  \\
	\varphi ^{0} &=&\frac{Cov\left[ \Theta _{t},\Theta _{t-1}\right] }{Var\left(
		\Theta _{t}\right) }=\varphi _{n}+O\left( \frac{1}{\sqrt{n}}\right).
\end{eqnarray*}




Moving to Example \ref{GMR-ex:bipower}, in
order to separate the jump and continuous sample path components in quadratic variation,
(and thus estimate $ \int_{t-1}^{t} \sigma^2(s) ds$), 
\citeasnoun{Barn-Shep-2003f} and \citeasnoun{Barn-Shep-2004}introduce the concept of bi-power
variation (BPV) defined as:
\begin{equation}
\hat{\Theta}_{n,t}(k)=\frac{\pi}{2}\sum_{j=k+1}^{n}\left\vert
X_{n,t_j}\right\vert \left\vert X_n,{t_{j-k}}\right\vert.  \label{BPV}
\end{equation}
Henceforth we will, without loss of generality, specialize our discussion to the case $k$ = 1, and therefore drop it
to simplify notation. \citeasnoun{Barn-Shep-2004} establish the
sampling behavior of ${\hat{\Theta}}_{n,t}$ as $n$ $\rightarrow
$ $\infty ,$ and show that under suitable regularity conditions:
$\plim_{n\rightarrow \infty }\hat{\Theta}_{n,t}(k)$ = $\int_{t-1}^t\sigma(s)^{2} ds.$
Therefore, in the presence of jumps, ${\hat{\Theta}}_{n,t}$
converges to the continuous path component of $ \int_{t-1}^{t} \theta(s) ds$ and is not affected by jumps. The sampling error of the bi-power variation is
$n^{\alpha}\left( \hat{\Theta}_{n,t}-\int_{t-1}^{t}\sigma\left(
s\right)^2 ds\right)/\sqrt{\nu _{bb}V_t}$ $\sim$ $N\left( 0, 1\right),$
where $\nu _{bb}$ % =  $(\pi/4)^2$ + $\pi$ - 5 
$\approx$ 2.6 and under the null where there are no jumps. Based
on these results, \citeasnoun{Barn-Shep-2003f} and
\citeasnoun{Barn-Shep-2004} introduce a framework to test for jumps
based on the fact that $QV$ consistently estimates the quadratic
variation, while $\hat{\Theta}$ consistently estimates the
integrated variance, even in the presence of jumps. Therefore, the
difference between the BPV and the scaled RV  is
sum of squared jumps (in the limit). Once we have identified the
jump component, we can subtract it from the realized variance and we
will have the continuous part of the process.

Using the arguments presented earlier we can improve estimates of
both RV and BPV. This should allow us to
improve estimates of integrated volatility as well as improve the
performance of tests for jumps. To do so we introduce:
\begin{equation}
\hat{\Theta}_{n,t}(\omega_t^{\ast
})=\hat{\Theta}_{n,t}-\omega_t^{\ast
}(\hat{\Theta}_{n,t}-\hat{\Theta}_{n,t\mid t-1}). \label{estimatorBPV}
\end{equation}
It will be an optimal improvement of $\hat{\Theta}_{n,t}$ when
$\omega_t ^{\ast }$ is again defined according to the 
control variable formula (\ref{control2}) where $QV$ is replaced by
$BPV.$ Note that we do not assume the same temporal dependence for
$QV$ and $BPV,$ as the projection of $QV$ on its past (one lag) and
that of $BPV$ on its own past (one lag) in general do not coincide.





\setcounter{equation}{0}

\section{Additional Volatility Examples \label{sec:addvolex}}

\begin{example}{\sc (Two-Scales Realized Volatility)}
	\label{ex:classical-tsrv} The observations are as in (\ref{GMR-eq:micro}) of the paper.
	The classical Two-Scales Realized Volatility (TSRV; \citeasnoun{twoscales05}, \citeasnoun{yacperlan2011}) has a convergence rate of  $\alpha = 1/6$. The cited papers show
	consistency and stable convergence. 
\end{example}



\begin{example}{\sc (Multi-Scale and Kernel Realized Volatility)}
	\label{ex:msrv}
	The observations are again as in (\ref{GMR-eq:micro}). The convergence rate is $\alpha = 1/4$. Conditions [i] and [iia] (and in particular [ii]) have been shown for
	Multi-Scale Realized Volatility (MSRV, \citeasnoun{lanmsrv06}) and 
	Realized Kernel estimators (RK, \citeasnoun{barndorffnielsenhansenlundeshephard08}). 
	The two methods are close in view of \citeasnoun{bibingermykland}.
	The rate is efficient since it also occurs in the parametric case (see \citeasnoun{gloterjacod}),
	and also see the non-parametric bound in \citeasnoun{reiss2011}.
\end{example}




In the interest of brevity we discuss the above Example \ref{ex:classical-tsrv} in more detail. The efficient price $X_t$ is now latent and observed with microstructure noise, as in equation
(\ref{GMR-eq:micro}) of the paper.
The analysis in the earlier
sections for the no-jump case goes through with these modifications.
In the case of the TSRV, 
\begin{equation*}
V_t ~ = ~ c \frac{4}{3} \int_{t-1}^t \sigma^4_u du + 8 c^{-2} \nu^4 ,
\end{equation*}
where $\nu^2$ is the variance of the noise, and $c$ is a smoothing
parameter. For the more complicated case of the MSRV and the kernel
estimators, we refer to the publications cited.\footnote{For estimation of the continuous part of $\Theta_t$ only, apart from bi- and multipower, the
	most fully developed theory involves removal by truncation, see
\citeasnoun{mancini01},  \citeasnoun{fanwang}, \citeasnoun{yacjacod07}, 
\citeasnoun{yacjacod08}, \citeasnoun{yacjacod09}, \citeasnoun{yacjacod12}, \citeasnoun{jacodtodorov10}, \citeasnoun{syleemykland08}, \citeasnoun{syleemykland-witherror12}, \citeasnoun{jingkongliumykland12}, as well as the work cited in the previous paragraph. The work by \citeasnoun{huangtauchen06} provides a
	complete theory, but under the assumption that the microstructure
	noise is Gaussian. Irregular observations can be handled using the concept of quadratic
	variation of time (see e.g.\ \citeasnoun{myklandzhang06}, \citeasnoun{MZdataclean16}, \citeasnoun{MZCalgerbra18}).}
In the case where there are both jumps and microstructure, there are
two different targets that can be considered, either the full
quadratic variation $\Theta_t$, or only its continuous part.
For estimation of the full quadratic variation, the estimators from the
continuous case remain consistent, and retain the same rate of convergence
as before. The asymptotic variance $V_t$ needs to be modified. The results
in this paper for the no-jump case therefore remain valid.\footnote{See, \citeasnoun{jacodprotter2012},
	\citeasnoun{yacjacod14}, \citeasnoun{MZdataclean16}, \citeasnoun{MZCalgerbra18}. } 





For other approaches to the estimation of volatility under microstructure noise, see, {\it inter alia},  \citeasnoun{yacpmrfs05}, \citeasnoun{bandirussell06} and
\citeasnoun{hansenlunde06}, \citeasnoun{jacodlimyklandpodolskijvetter09}, \citeasnoun{podolskijvetter09}, \citeasnoun{dachengxiu10}, and \citeasnoun{bibingerreiss14}. The case of bi-power variation can be also extended to measures
involving more general functions, as in \citeasnoun{B-NJS-2004}.
Provided such measure feature persistence we can apply the above
analysis in a more general context. One particular case of interest
is power variation, typically more persistent than quadratic
variation or related measures, as discussed in detail in
\citeasnoun{ForsGhys2003b}.

\setcounter{equation}{0}



\section{Beyond Realized Volatility\label{beyond}}

The set of examples discussed in this section go beyond measures of quadratic variation.


\begin{example}{\sc (Estimation of Covariance from Asynchronous Observations)} 
	\label{ex:asyn} 
	There are a number of different ways of handling covariances, including the 
	\citeasnoun{hayashiyoshida05} estimator, see also \citeasnoun{podolskijvetter09bipower}, \citeasnoun{christensenpodolskijvetter-asynchronous},  and \citeasnoun{bibingervetteronHY} for micro-structure, jumps,
	and asymptotic distributions. Alternatives include the Previous-Tick estimator (\citeasnoun{lancov2011}, \citeasnoun{bibingermykland}), and Quasi-Likelihood (\citeasnoun{shephardxiu}). The
	estimator in \citeasnoun[p.\ 172-175]{myklandzhang12} is a hybrid of Hayashi-Yoshida and Quasi-Likelihood. Consistency and stable convergence hold here again
	with varying rates $\alpha$.
\end{example} 



\begin{example}{\sc (Block Estimation of Higher Powers of Volatility)}
	\label{ex:highpow}
	The parameter of interest is $\theta(s) = g(\sigma(s)^2)$, with $g$ not being the identity function. To make estimators approximately or fully efficient, one can use block estimation.
	In the absence of microstructure noise, the convergence rate is $\alpha = 1/2$
	(\citeasnoun[p.\ 1421-1426]{myklandzhang_Mfinite}), \citeasnoun[p.\ 224-229]{doubleGauss2011},  \citeasnoun{jacodrosenbaum-sqrtn} and  \citeasnoun{jacodrosenbaum13}).
	If microstructure noise is present, the convergence rate is $\alpha = 1/4$ (\citeasnoun[p.\ 512-554]{jacodprotter2012}).\footnote{See also \citeasnoun{bgjps06}, \citeasnoun[p.\ 138]{myklandzhang12} and \citeasnoun{renault2015efficient} for related developments.}
\end{example}

Next, we are concerned with systems on the form $dV_t = \beta_t d X_t + dZ_t$, where $V_t$ and $X_t$ can be observed at high frequency, either with or without microstructure noise. Moreover, $X_t$ can be multidimensional. 
The coefficient process $\beta_t$ can either be the ``beta'' from portfolio optimization, with $Z_t$ in the role of idiosyncratic noise, or $\beta_t$
can be the hedging ``delta'' for an option, with $Z_t$ as tracking error. 
Nonparametric estimates can be used directly, or for forecasting, or for model checking.


\begin{example}{\sc (High Frequency Regression, and ANOVA)}
	\label{ex:anova}
	The regression problem seeks to estimate or make tests about $\int_{t-1}^{t} \beta(s) ds$ (\citeasnoun[p.\ 1424-1426]{myklandzhang_Mfinite},
	\citeasnoun{kalnina-betas},
	\citeasnoun[p.\ 268-273]{lanmodsel12}, \citeasnoun{reisstodorovtauchen}). 
	The ANOVA problem seeks to f $[Z,Z]_{t}$ 
	(\citeasnoun{zhang2001} and \citeasnoun{myklandzhang06}). Convergence rates are as for realized or other powers of volatility, with $\alpha = 1/2$ when there is no microstructure noise, and
	$\alpha = 1/4$ otherwise. 
\end{example}

\setcounter{equation}{0}

\section{Connection with Kalman Filter \label{sec:kalman}}

In-sample asymptotics applies to situations where the period $t$
variable of interest, say $Y_{t} $, is observed by a sequence of
noisy measurements $X_{t}$ which are, when the number $n$ of
intraday observations goes to infinity, asymptotically unbiased
and normally distributed with a possibly time dependent and random
variance. In other words, adopting a commonly used abuse of
notation, we have for large $n$ a measurement equation:
\begin{equation}
\label{measu}
X_{t} =Y_{t}+c_{t}Z_{t}  \qquad Z_{t}  \sim N(0,1). 
\end{equation}
Traditional Kalman filtering applies to a Gaussian measurement
equation like (\ref{measu}) augmented with a Gaussian transition
equation:
\begin{equation}
\label{transit} 
Y_{t} = a Y_{t-1}+g+\sqrt{H_{t}}u_{t} \qquad 
u_{t}  \sim N(0,1).  
\end{equation}
Note that the Kalman filter considered here is slightly more
general than the traditional one, since we allow for conditional
heteroskedasticity in equations (\ref{measu}) and (\ref{transit}).
Conditioning on the values of the variables $c_{t}$ and
$H_{t}$ allows us to consider heteroskedasticity as predetermined
and to use standard Bayes formula to obtain recursion formulas for
the filtering distribution, that is the conditional normal
distribution of $Y_{t}$ given $(X_{t},X_{t-1}, \ldots
,X_{1}).$ If $N(Y_{t}^{\ast },K_{t})$ stands for this
conditional distribution (the superscript $n$ is omitted for
convenience), the Kalman recursion formulas (see e.g.\ \citeasnoun{williams1991pm}, p.\ 168) can be written:
\begin{eqnarray}
\frac{1}{K_{t}} &=&\frac{1}{a
	^{2}K_{t-1}+H_{t}}+\frac{1}{c_{t}^{2}}
\label{Kal} \\
\frac{Y_{t}^{\ast }}{K_{t}} &=&\frac{a Y_{t-1}^{\ast
	}+g}{ a ^{2}K_{t-1}+H_{t}}+\frac{X_{t}}{c_{t}^{2}}.  \nonumber
\end{eqnarray}
Hence, the filtered values are defined as:
\begin{equation}
Y_{t}^{\ast }=(1-\omega _{t})X_{t}+\omega _{t}( a
Y_{t-1}^{\ast }+g) \label{kalfil}
\end{equation}
with:
\begin{equation}
\omega _{t}=\frac{c_{t}^{2}}{c_{t}^{2}+ a ^{2}K_{t-1}+H_{t}}.
\label{weightKal}
\end{equation}
As will be discussed in detail later, our way to take advantage of
past information to improve upon the measurement $X_{t}$ is quite
similar in spirit to the Kalman-Bucy filter with however a couple of
important differences. First, we do not deal with a Gaussian
transition equation. The processes we consider, like realized
variance or bi-power variation (defined later) are often constrained
to be positive (and therefore cannot be Gaussian). However, even more
importantly, we want to remain model free. Therefore, instead of the
model (\ref{transit}), we will simply refer to a linear projection
of the variable $Y_{t}$ on some of its lagged values. $Y_{t\mid
	t-1}$ will denote such an optimal linear predictor for a given
number of lags. If for instance we decide to use only one lag,
$\alpha $ and $g$ will be defined as (maintaining a model-free
setting):
\begin{eqnarray*}
	Y_{t\mid t-1} &=& a Y_{t-1}+g \\
	Y_{t} &=&Y_{t\mid t-1}+u_{t} \\
	E(u_{t}) &=&0, \qquad Cov(u_{t},Y_{t-1})=0.
\end{eqnarray*}
Note that the structure of equation (\ref{measu}) is maintained
but in a model-free context, up to some regularity conditions
ensuring its asymptotic validity.

Second, without a Gaussian transition equation we have to give up
the Bayes formula which enables us to compute conditional
expectations and variances like (\ref{Kal}). Moreover, maintaining
the model-free setting, we do not want to define a filtered value
function of {\it all} past information
$(X_{t},X_{t-1},...,X_{1}),$ but instead only
for a fixed number of lags every single day $t$. For instance, in
the case of one lag, our filtered value will be defined by:
\begin{equation}
\tilde{Y}_{t}=(1-\omega _{t}^{\ast })X_{t}+\omega _{t}^{\ast }(a
X_{t-1}+g). \label{weightus}
\end{equation}
Note that the actual filtered value will be slightly different due
to estimation error, an issue that also appears in traditional
Kalman filtering and that will be discussed later. It is important
to note that our filtered value $\tilde{Y}_{t}$ is different from the
Kalman filter $Y_{t}^{\ast }$ in two respects. First, it combines
the current measurement $X_{t}$ with the measured optimal forecast
$a X_{t-1}+g$ \ based on yesterday's information (measured
counterpart of $Y_{t\mid t-1}= a Y_{t-1}+g$) and not on the
filtered counterpart $a Y_{t-1}^{\ast }+g$. Second, we have
shown that an optimal weight for minimizing the conditional mean
squared error is given by:
\begin{equation}
\omega _{t}^{\ast }=\frac{c_{t}^{2}}{c_{t}^{2}+ a
	^{2}c_{t-1}^{2}+H_{t}}. \label{weightopt}
\end{equation}
Note that our optimal weight (\ref{weightopt}), albeit similar to
the Kalman one (\ref{weightKal}), is smaller since
$c_{t}^{2}>K_{t}$ (using equation (\ref{Kal})). Intuitively, our
weighting schemes give less weight to past information since we
summarize past information by the measured counterpart $a
X_{t-1}+g$ (with conditional variance $a ^{2}c_{t-1}^{2}$) of
the past forecast $Y_{t\mid t-1}=a Y_{t-1}+g,$ whereas the
Kalman filter weights use the more accurate filtered counterpart
$a Y_{t-1}^{\ast }+g$\ (i.e.\ with smaller conditional
variance $a ^{2}K_{t-1}<$ $a ^{2}c_{t-1}^{2}$).


\section{A Forecasting Example \label{foregains}}

We mainly focus on the \textit{measurement} of high frequency data
related processes such as quadratic variation, bi-power variation
and quarticity. Yet, forecasting future realizations of such
processes is often the ultimate goal. The purpose of this
subsection is to discuss the impact on forecasting performance of
our improved measurements.

We start from the observation that standard volatility
measurements feature a measurement error that can be considered,
at least asymptotically, as a martingale difference sequence.
Therefore, suppose we want to forecast $Y_{t+1}$, using past
observations $(X_s),s\leq t$ which are noisy measurements of
past $Y^{\prime }s.$ The maintained martingale difference
assumption implies that:
\[
Cov[Y_{t}-X_{t},X_s]=0,\forall s<t.
\]
Suppose now that we also consider past "improved" observations: $Z(s)$ =
$(1-\omega )X_s+\omega Y(s)^{\ast},$ where $Y(s)^{\ast
}$ is an unbiased linear predictor of $X_s.$\footnote{We
consider here weights that are not time varying - unlike in the
previous subsection. Hence, we are assessing here the impact on
forecasting performance of fixed weights $\omega$. Optimally
chosen time varying weights should ensure at least a comparable
forecasting performance. We will refer to the latter as
conditional schemes, in contrast to unconditional ones that are also discussed in subsection
\ref{GMR-feasible} of the paper.} Since $Y_{t+1}^{\ast }$ is an unbiased linear
predictor of $X_{t+1}:$
\[
X_{t+1}=Y_{t+1}^{\ast }+v_{t+1}^{\ast },E(v_{t+1}^{\ast
})=0,Cov[v_{t+1}^{\ast },Y_{t+1}^{\ast }]=0.
\]
Suppose we have a preferred forecasting rule
for $(X_{t})$ (based on say an ARFIMA model for QV such as in %
\citeasnoun{abdl-03}) and let us denote  it as $Y_{t+1}^{X}.$
This alternative unbiased linear predictor of $X_{t+1}$ would be such that:
\[
X_{t+1}=Y_{t+1}^{X}+v_{t+1}^{X},E(v_{t+1}^{X})=0,Cov[v_{t+1}^{X},Y_{t+1}^{X}]=0.
\]
It is natural to assume in addition that:
\[
Cov[v_{t+1}^{X},Y_{t+1}^{\ast }]=0
\]
hence, the predictor $Y_{t+1}^{\ast }$ does not allow us to
improve the preferred predictor $Y_{t+1}^{X}.$ Consider now a
modified forecast $Y_{t+1}^Z,$ based on past and current improved observations ${Z(s), s \le t}.$ By the definition of $Z$ we have:
\[
Y_{t+1}^{Z}=(1-\omega )Y_{t+1}^{X}+\omega Y_{t+1}^{\ast }.
\]
It is easy to show that the forecasting errors obtained from
respectively $Y_{t+1}^{X}$ and $Y_{t+1}^{Z}$ satisfy:
\begin{equation}  \label{forecasterrors}
Var(Y_{t+1}^{X}-Y_{t+1})-Var(Y_{t+1}^{Z}-Y_{t+1}) = \omega
^{2}(Var(v_{t+1}^{X}) - Var(v_{t+1}^{\ast })).
\end{equation}
This result has the following implication: using the proxy $(Z)$
instead of the proxy $(X)$ we will not deteriorate the forecasting
performance, except if we build on purpose the proxy $(Z)$ from a
predictor $(Y^{\ast })$ less accurate than the preferred predictor
$(Y^{X}).$ More generally, using a proxy $Z$ computed with time varying weights optimally chosen in a conditional
setting should indeed improve the forecasting performance.

The above result pertains to a simple linear forecasting setting. We expect that improvements of measurement are going to be even more important nonlinear settings. The most prominent
example where the objective of interest is a nonlinear function of future volatility is option pricing. The future path of volatility until
some time to maturity of the derivative contract determines the
current option price through a conditional expectation of a
nonlinear payoff function. A simplified example is the model of
\citeasnoun{hull2005poa} where the price of a European call option
of time to maturity $h$ and moneyness $k$ equals:
\[
C_t(h,k) = E_t[BS(\frac{1}{h}\int_t^{t+h}\sigma^2(u) du, k, h)]
\]
and $BS(\sigma^2,k, h)$ is the Black-Scholes option price formula.
Note that the above equation assumes no leverage and no price of
volatility risk.\footnote{Note that volatility is not priced in \citeasnoun{hull2005poa}, but nevertheless it is still a conceptually interesting example. \label{refIIIhull}}  This is a common example of derivative pricing
that will be studied later via simulation. It will be shown that
the improved volatility measurement has a significant impact on
option pricing.



The conclusions regarding the ranking
of weights appearing in Section \ref{GMR-simulation} of the paper change when we turn our attention to forecasting gains associated with the example discussed in this section.
We consider two
experiments. The first is a linear forecasting exercise:
\[
IV_{t+1} = a + bX_t + \varepsilon_{t+1}^X
\]
where the regressor $X_t$ is either one of the following:
$IV_t,$ $\hat{\Theta}_t$ and corrected $\hat{\Theta}_{n,t}$ using the optimal weighting
$\omega^{\ast}_{vt}$ and
$\omega^{\ast}_{ut}$
schemes with one-day lag of information. Obviously the infeasible benchmark
is the regression involving $IV$ as regressor. Hence, we compare how
close the feasible raw $\hat{\Theta}_{n,t}$ and corrected measure perform in
comparison. We do this for 5 and 10 minute sampling schemes.

To appraise the more realistic and interesting nonlinear forecasting
setting we consider the following prediction problem:
\[
\log(BS^{imp}_{t+k}(ATM,TTM)) = a + b \log(X_t) +
\varepsilon_{kt}^{X,TTM}
\]
where k = 1 day, 5 days, 20 days. The $BS^{imp}_{t+k}(ATM,TTM)$ is the Black-Scholes implied volatility generated for a sample of data obeying the stochastic volatility dynamics of the Heston model mentioned in the first subsection. We selected the Heston because we know how to price options, and hence compute Black-Scholes implied volatilities. We picked three times-to-maturity (TTM), namely 22, 44
and 66 days, corresponding to one-, two- and three-month options. Finally, we focused exclusively on at-the-money options (ATM) since
those are typically accurately priced and liquid.


The simulation evidence is reported in Table \ref{Heston} and focuses exclusively on the second
experiment described above - as the first one did not yield many differences between the various estimation schemes.
There is now a clear difference between the ranking in Table \ref{GMR-MSEGARCHDIFF} in the main paper and that
in Table \ref{Heston}. In the case of filtering, we find that
$\omega^{\ast}_{vt}$ is slightly better than $\omega^{\ast}_{ut}.$ Hence, the conditional weighting scheme with realized $U_t,$ i.e.\
$\omega^{\ast}_{ut}$ outperforms $\omega^{\ast}_{vt},$ and therefore as far as forecasting a nonlinear function goes, the
ranking is reversed.  Although the differences are not as significant, there
is a clear pattern of dominance which we did not observe in Table \ref{GMR-MSEGARCHDIFF} pertaining to volatility appearing in the paper.






\begin{table}
	\caption{Forecasting Log Black-Scholes Implied Volatilities  - Heston Model}
	\label{Heston}
	\tablexplain{We consider:
		\[
		\log(BS^{imp}_{t+k}(ATM,TTM)) = a + b \log(X_t) +
		\varepsilon_{kt}^{X,TTM}
		\]
		where k = 1 day, 5 days, 20 days. The $BS^{imp}_{t+k}(ATM,TTM)$ is
		the Black-Scholes implied volatility generated by a Heston model for
		at-the-money options (ATM) with times-to-maturity (TTM) 22, 44
		and 66 days.
	}
	{\footnotesize
		\begin{ctabular}{llcccc}
			\\
			& & IV &  QV  &  QV & QV  \\
			$X_t$ & &  & Corrected($\omega^{\ast}_{vt}$) & Corrected($\omega^{\ast}_{ut}$) & \\
			\\
			& &\multicolumn{4}{c}{ $R^2$ 1-day ahead forecast, ATM} \\
			TTM: 22days &   5 Min &   ${0.98}$ & ${0.93}$ &    ${0.94}$ &  ${0.93}$ \\
			& 10 Min &   ${0.98}$ &    ${0.88}$ &    ${0.91}$ &    ${0.90}$ \\
			\\
			TTM: 44days &  5 Min &  ${0.98}$ &    ${0.91}$ &    ${0.94}$ &    ${0.93}$ \\
			&  10 Min &   ${0.98}$ &    ${0.86}$ &    ${0.91}$ &    ${0.90}$ \\
			\\
			TTM: 66days &  5 Min &   ${0.98}$ &    ${0.93}$ &    ${0.94}$ &    ${0.93}$ \\   &   10 Min &   ${0.98}$ &    ${0.92}$ &    ${0.94}$ &    ${0.93}$ \\
			\\
			& & \multicolumn{4}{c}{$R^2$ 5-day ahead forecast, ATM}\\
			TTM: 22days &   5 Min  &  ${0.91}$ & ${0.86}$ & ${0.88}$ & ${0.87}$ \\
			& 10 Min &   ${0.91}$ &    ${0.82}$ &    ${0.85}$ &    ${0.84}$ \\
			\\
			TTM: 44days &  5 Min  & ${0.91}$ & ${0.86}$ &  ${0.88}$ & ${0.87}$  \\
			&  10 Min    &   ${0.91}$ &    ${0.82}$ &    ${0.85}$ &    ${0.84}$ \\
			\\
			TTM: 66days &  5 Min & ${0.91}$ &  ${0.85}$ &    ${0.87}$ &    ${0.87}$   \\
			&   10 Min &   ${0.91}$ &    ${0.82}$ &    ${0.85}$ &    ${0.84 }$ \\
			\\
			& &\multicolumn{4}{c}{$R^2$ 20-day ahead forecast, ATM}\\
			TTM: 22days &  5 Min &  ${0.72}$ &  ${0.68}$ &  ${0.70}$ &  ${0.69}$   \\
			& 10 Min &   ${0.72}$ &  ${0.65}$ &    ${0.67}$ &    ${0.66}$  \\
			\\
			TTM: 44days &  5 Min  &  ${0.72}$ &  ${0.68}$ &  ${0.69}$ & ${0.69 }$  \\
			&  10 Min   &   ${0.72}$ &    ${0.65}$ &    ${0.67}$ &    ${0.67}$  \\
			\\
			TTM: 66days &  5 Min & ${0.72}$ &  ${0.68}$ &  ${0.69}$ & ${0.69}$    \\
			&   10 Min  &  ${0.72}$ &    ${0.64}$ &    ${0.67}$ &    ${0.67}$     \\
			\\
		\end{ctabular}}
	\end{table}
	
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