On covariation estimation for multivariate continuous Itô semimartingales with noise in non-synchronous observation schemes

On covariation estimation for multivariate continuous It\^o semimartingales with noise in non-synchronous observation schemes

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Introduction

\setcounter{page}{1} In the past years there has been a considerable development of statistical methods for stochastic processes observed at high frequency. This was mainly motivated by financial applications, where the data, such as stock prices or currencies, are observed very frequently. It is well known that under the no-arbitrage assumption price processes must follow a semimartingale (see e.g. DS). However, at ultra high frequencies the financial data is contaminated by {\it microstructure noise} such as rounding errors, bid-ask bounds and misprints. This fact prevents us from using classical power variation based methods (see e.g. BGJPS or J2 among many others) to infer the characteristics of a semimartingale.

A standard model for a continuous It\^o semimartingale observed with errors is given by

where $(X_t)_{t\geq 0}$ is a $d$-dimensional process ({\it true price}) of the form

with $(a_s)_{s\geq 0}$ being an $\mathbb{R}^d$-valued c\`agl\`ad process, $(\sigma_s)_{s\geq 0}$ being an $\mathbb{R}^{d\times d^{\prime}}$-valued c\`agl\`ad volatility and $W$ representing a $d^{\prime}$-dimensional Brownian motion, and the $d$-dimensional error process $\varepsilon$ ({\it microstructure noise}) is iid with

independent of $X$. Throughout this work an asterisk denotes the transpose of a matrix.

The aim of this paper is to estimate the covariation matrix of $X$ over some interval, say $[0,1]$, i.e.

based on non-synchronous noisy observations ($Y=(Y^1, \ldots, Y^d)$)

where $0=t_0^k < \cdots<t_{n_k}^k=1$ are partitions of the interval $[0,1]$ with $\max_{1\leq i\leq n_k}|t_i^k - t_{i-1}^k|\rightarrow 0$ as $n_k\rightarrow \infty$ for all $1\leq k\leq d$. The univariate counterpart of this problem has been studied intensively in the literature. Let us mention the {\it two-scale approach} of ZMA (see Z for its more efficient multi-scale version), the {\it realised kernel method} proposed in BHLS and the {\it pre-averaging concept} originally introduced in PV2 (and further studied in JLMPV, JPV, PV1 in various settings) among others. These methods can be extended to the multivariate case in a rather straightforward manner if the observations are synchronous.

When the underlying data is non-synchronous, things are less obvious, as we are faced with two challenges at the same time: We have to de-noise the data as before, but we also need to apply a certain synchronization technique to create a new set of observations from which appropriate estimators for $[X]$ can be computed. For the multivariate realised kernel method, BHLS2 proposed to cope with non-synchronous data by applying the {\it refreshing time method} first, which synchronizes the observations via a previous tick method. In a second step, a noise robust estimator is constructed from this new data set. Similar in spirit is the extension of the multi-scale estimator due to B, where synchronous observations are obtained using the pseudo-aggregation algorithm of PA first. The resulting covariance estimator then becomes a multi-scale version of the Hayashi-Yoshida estimator from HY, which originally has been introduced to deal with non-synchronicity in semimartingale models without noise.

Both approaches have their drawbacks, however: (a) Using the previous tick approach (which generates pseudo data points) may lead to inconsistent estimators for certain observation schemes; this phenomenon has been noticed in HY in the setting of a pure diffusion; (b) After any of the synchronization techniques there remain at most $\min_{1\leq k\leq d} (n_k)$ data points, which amounts in throwing away a lot of data. In the no-noise case, this is usually no problem, as for the Hayashi-Yoshida estimator exactly those observations are dropped that bear no additional information on the covariance, but for noisy data they still can be used to wipe out the noise.

To avoid these afore-mentioned drawbacks, we propose to combine a synchronization technique and a concept for de-noising as well, but in reverse order: We apply the pre-averaging approach, which is designed to locally diminish the influence of the noise, first, and use the Hayashi-Yoshida method afterwards. Our estimator, denoted by $HY^n$, has the following important properties: \\ \\ (i) In general, we use all observations $Y_{t_i^k}^k$; \\ \\ (ii) The estimator has the optimal convergence rate $n^{-1/4}$; \\ \\ (iii) The estimation method is robust to certain dependence structures of the noise process. This property is important for practical applications as the economic theory typically does not provide any insight on modeling the noise. \\ \\ The main idea of the construction of $HY^n$ comes from CKP, where we indicated its consistency, but did not provide the complete asymptotic theory. In this paper we now prove a stable central limit theorem for $HY^n - [X]$. From a technical point of view, the conditions we use on the observation scheme $t_i^k$ are rather mild, but on the other hand there is no empirical evidence that such assumptions are reasonable in financial practice. However, a thorough analysis involving e.g.\ random observations times is beyond the scope of our paper. Furthermore, we explain how to estimate the (random) asymptotic covariance matrix that appears in the central limit theorem to obtain a {\it feasible} result (which may be used in practice to construct confidence regions). We would like to emphasize again that the construction of our estimator is not completely obvious (as there are several ways of combining the Hayashi-Yoshida method and the pre-averaging approach, which may result in different properties) and that the proof of the main result, which is based on a certain blocking technique, martingale inequalities and a stable central limit theorem for semimartingales, is more advanced than in the univariate setting.

This paper is organized as follows: in Section (ref) we introduce the set up and explain the construction of $HY^n$. The main results of the paper including the consistency of $HY^n$ and the associated stable central limit theorems are presented in Section (ref). Section (ref) deals with estimation techniques for the conditional variance, while in Section (ref) we show some numerical results to illustrate the finite sample properties of our estimator. Section (ref) is devoted to proofs, and some tedious parts are relegated to an Appendix in Section (ref).

The set up

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We start by introducing an appropriate filtered probability space on which our noisy process $Y$ is defined. Let \newline $(\Omega^{(0)} ,\mathcal{F}^{(0)},(\mathcal F_t^{(0)})_{t\in [0,1]},\mathbb{P}^{(0)})$ be an arbitrary space on which the true price process $X$ lives, such that all involved process $a$, $\sigma$ and $W$ are adapted. Now we consider a second filtered probability space $(\Omega^{(1)} ,\mathcal{F}^{(1)},(\mathcal F_t^{(1)})_{t\in [0,1]},\mathbb{P}^{(1)})$, where $\Omega^{(1)}$ is the set of functions from $[0,1]$ to $\mathbb{R}^d$ and $\mathcal F^{(1)}$ is the product $\sigma$-field of the Borel $\sigma$-algebras $\mathcal A_t$ on $\mathbb{R}^d$, indexed by $t \in [0,1]$. We define on it the noise process $\varepsilon =(\varepsilon_t)_{t\in [0,1]}$ as follows: let $Q$ be a probability law on $\mathbb{R}^d$ (the marginal law of $\varepsilon$) and set $\mathbb{P}^{(1)}$ as $\mathbb{P}^{(1)}=\otimes_{t\in [0,1]} P_t$ with $P_t=Q$ for all $t\in [0,1]$. Now, $(\varepsilon_t)_{t\in [0,1]}$ is defined as the canonical process on $(\Omega^{(1)} ,\mathcal{F}^{(1)},(\mathcal F_t^{(1)})_{t\in [0,1]},\mathbb{P}^{(1)})$ with $(\mathcal F^{(1)}_t)_{t\in [0,1]}$ being the canonical filtration. The process $Y$ in ((ref)) lives on the product space $(\Omega ,\mathcal{F},(\mathcal F_t)_{t\in [0,1]},\mathbb{P})$ given by:

We remark that the probability space on which the process $\varepsilon$ lives is rather minimal; a precise definition of it is required for the stable convergence results, however. The process $Y$ is defined in continuous time just for convenience, although the mapping $(\omega,t)\rightarrow Y_t(\omega )$ is not $\mathcal F \otimes \mathcal B([0,1])$-measurable.\newline \newline Now we introduce the assumptions on the sampling scheme. \\ \\ {\it Assumption (T):} The observation times $t_i^k$, $i=0,\ldots, n_k,$ $k=1, \ldots, d$ satisfy the following conditions:

Let us shortly comment the above assumptions. Condition (T1) makes the explicit computation of the asymptotic covariance matrix in the forthcoming central limit theorem possible. Condition (T3) implies that the observation numbers $n_k$ have the same order. Condition (T2) means that the points of the $l$th grid do not lie dense between any two successive points of the $k$th grid, i.e. the number of points $t_j^l$ that lie in the interval $[t_{i-1}^k, t_i^k]$ is uniformly bounded by a constant for all $1\leq k,l\leq d$ (cf. Lemma (ref) for a closely related result). When these last two conditions (similar number of observations and uniform boundedness of the number of points $t_j^l$ that belong to $[t_{i-1}^k, t_i^k]$) are fulfilled we say that the sampling schemes are {\it comparable}. Finally, condition (T4) means that the number of common points can be negligible compared to $n$ (if $m_{kl}=0$) or it can be of order $n$ (if $m_{kl}>0$).

We want to emphasize that the full force of Assumption (T) is only required for the proof of the central limit theorem! For the consistency result and the rate of convergence it suffices to assume that the grids $(t_i^k)$, $k=1, \ldots, d,$ are comparable. In particular, the representation ((ref)) and the condition (T4) are not required.

Now we explain the construction of our estimator $HY^n$. First, we choose a window size $k_n$ as

for some constant $\theta >0$. In the next step we choose a positive weight function $g:[0,1]\rightarrow \mathbb{R}$ with $g(0)=g(1)=0$, which is piecewise $C^1$ with piecewise Lipschitz derivative $g'$ and $\int_0^1 g^2(x)dx>0$. For any $d$-dimensional stochastic process $V=(V^1, \ldots, V^d)$ we define the quantity

which we call {\it pre-averaging in tick time}. The name refers to the fact that we use the same amount of data to construct $\overline V_{t_i^k}^k$ for all $1\leq k\leq d$; alternatively one could perform the {\it pre-averaging in calendar time} by using the same time interval for all coordinates $V^k$, but with different number of observations in each time window. The latter approach would result in different properties of the estimator.

As discussed in JLMPV, JPV or PV2 the local averages technique performed in ((ref)) diminishes the influence of the noise process $\varepsilon$ to some extent (but not completely) and helps us to get information about $\Sigma$. In the last step, as proposed in CKP, we define a Hayashi-Yoshida type estimator based on pre-averaged observations by

with $\psi=\int_0^1 g(x) dx$, and set $HY^n=(HY^n_{kl})_{1\leq k,l\leq d}$. In CKP we have already indicated the consistency of $HY^n$. The aim of this paper is to provide the complete asymptotic theory to be able to construct confidence regions for the quadratic covariation $[X]$.

The asymptotic theory

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We start with the consistency of the estimator $HY^n$ which has been shown in CKP.

As we remarked above the full force of Assumption (T) is not required for the proof of Theorem (ref); it is just the comparability of sampling times which matters (see CKP for more details). Two remarks are in order.

In order to describe the weak limit associated with $HY^n - [X]$ we need to introduce various notations. \\ \\ {\it Notation.} Let us first extend the weight function $g$ to the whole real line by setting $g(x)=0$ for $x \not \in [0,1]$. We set for $x\in [0,1]$

where $f_k$ resp. $m_k$ are given in ((ref)) resp. ((ref)). Now we define two sets of functions, namely

and

for $s\in \mathbb{R}$, $1\leq k,k',l,l'\leq d$ and $u\in [0,1]$. Notice that when for example the number of joint points between the $k$th and $k'$th grid is negligible compared to $n$ (which can only hold for $k \not=k'$) then $m_{kk'}=0$. In this case we have $\overline \gamma_{kl,k'l'} \equiv \widetilde \gamma_{kl,k'l'} \equiv0$.\\ \\ Before we present the stable central limit theorem let us recall the notion of stable convergence. A sequence of random variables $Z^n$ on $(\Omega, \mathcal F, \mathbb{P})$ converges stably in law towards $Z$, written $Z_n \stackrel{d_{st}}{\longrightarrow} Z$, with $Z$ being defined on an extension $(\Omega', \mathcal F', \mathbb{P}')$ of the original probability space $(\Omega, \mathcal F, \mathbb{P})$, iff for any bounded, continuous real-valued function $g$ and any bounded $\mathcal{F}$-measurable random variable $V$ it holds that $\mathbb{E}[ g(Z_n) V] \rightarrow \mathbb{E}'[ g(Z) V]$ as $n\rightarrow \infty$. We refer to AE, REN or JS for more details on stable convergence. The next theorem is the main result of our paper, and its proof is postponed to Section (ref).

The rate of convergence $n^{-1/4}$ is known to be optimal for the parametric analogue of our estimation problem (i.e. when the process $\Sigma$ is constant); see e.g. B or GJ. We remark that the covariance matrix $\Psi$ of the noise process $\varepsilon$ always appears in the representation of $V$ as $\overline \gamma_{kk,kk}(u), \widetilde \gamma_{kk,kk}(u)>0$ for all $1\leq k\leq d$.

Estimation of variance

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To transform the probabilistic result of Theorem (ref) into a feasible statistical one, we need to find a consistent estimator of the conditional covariance matrix $V$ defined by ((ref)). We will introduce three different approaches to solve this task -- a general one, which works in arbitrary dimensions and does not require information of the time transforming functions; a second estimator, which uses local estimates of the volatility $\Sigma $; a third one tuned for the one-dimensional case, where the variance becomes particularly simple as seen in Remark (ref). All proofs are given in Section (ref).

Let us begin with the first estimator, for which we benefit from related work in M, where an estimator for the variance of the usual Hayashi-Yoshida estimator in the no-noise case was constructed. We introduce a second auxiliary sequence $\beta_n = \varpi n^{\eta} + o(n^{\eta})$, $\varpi > 0, \eta \in (0,1)$, and compute for each $\alpha \in \{0, \ldots [n/\beta_n]-1 \}$ the statistic

which is essentially the same quantity as $HY^n_{kl}$, but we only sum over time points $t_i^k$ from the smaller interval $B_n(\alpha) = [\frac{\alpha \beta_n}n,\frac{(\alpha+1)\beta_n}n)$. We set

This estimator is based on a local estimation of the covariance of $HY^n_{kl}$ and $HY^n_{k'l'}$. In order to obtain reasonable estimates for this covariance on the interval $B_n(\alpha)$, we use $HY^n_{kl}(\alpha) HY^n_{k'l'}(\alpha)$ to mimic the covariance of interest plus the product of the expectations of both factors. The latter bias is corrected by quantities like $HY^n_{kl}(\alpha) HY^n_{k'l'}(\alpha-1)$, where we use the usual “conditional independence” of increments of $Y$ over disjoint intervals. $V^{n,1}_{kl,k'l'}$ is now constructed as a symmetrized version of these local estimates, and we sum up over all $a$ afterwards to obtain a global one.

A drawback of this construction is that we need an additional condition on the process $\sigma$. In order for $HY^n_{kl}(\alpha)$ and $HY^n_{kl}(\alpha-1)$ to estimate the same quantity up to an error small enough, one usually postulates that $\sigma$ is an It\^o semimartingale itself. Under a furher assumption on $\eta$ we have the following theorem.

As mentioned above, the second estimator uses local estimates of the volatility $\Sigma$ and the covariance matrix $\Psi$ of the noise, and we assume knowledge of the time-transforming functions $f_k$ and $f_{kl}$, which in practice have to be approximated via the observed time points.

We start with the construction of the estimator of $\Sigma_s$. We define $HY^n ([0,t])=(HY^n_{kl} ([0,t]))_{1\leq k,l\leq d}$ for $t\in [0,1]$ by

which is consistent for the integrated covariation matrix up to time $t$. As the volatility process $(\Sigma_s)_{s\in [0,1]}$ is left-continuous, it is a natural idea to estimate $\Sigma_s$ via

for some sequence $l_n$ with $l_n\rightarrow 0$, $\sqrt{n} l_n \rightarrow \infty$ and $s\in [l_n,1]$ (for $s\in [0,l_n]$ we set $\Sigma_{s,n}=\Sigma_{l_n,n}$). The condition $\sqrt{n} l_n \rightarrow \infty$ is required to guarantee a sufficient amount of asymptotically uncorrelated summands in the definition of $\Sigma_{s,n}$.

The estimation of the covariance matrix $\Psi$ is somewhat easier. Recall that $(t_p^{kl})_{1\leq p\leq n_{kl}}$ denotes the set of common points of the $k$th and the $l$th grid, and define $i(p,k,l)=i$ with $t_i^k= t_p^{kl}$ for arbitrary $k,l=1, \ldots, d$. The estimator of $\Psi^{kl}$ is now given as

The intuition behind this estimator is rather simple. First of all, since the increments of $X$ at highest frequency converge to $0$ almost surely, the process $Y$ can be replaced by $\varepsilon$ without any changes in the limit. For this reason the estimator $\Psi_{n}^{kl}$ converges to $\Psi^{kl}$ almost surely by the strong law of large numbers (applied to the iid process $\varepsilon $) if $n_{kl}\rightarrow \infty$. When the sequence $n_{kl}$ does not diverge to $\infty$ then the convergence does not hold, but we have $n_{kl}/n\rightarrow m_{kl}=0$. Thus the corresponding functions $\overline \gamma$ and $\widetilde \gamma$ vanish as well, and this will be sufficient for the estimation of $V$. \\ \\ After all we obtain the following result.

Let us finally focus on the one-dimensional case and recall the asymptotic variance in ((ref)). As noted before, we do not have to care about any of the $\kappa$'s from ((ref)), as they can directly be computed from our choice of $g$. Using the univariate version of the estimator in $(\ref{psiest})$ for $\Psi$ (which is consistent now) and the Hayashi-Yoshida type estimator $HY^n$ for $\int_0^1 \sigma_u^2 du$, all we need to find is a feasible estimator for the rescaled integrated quarticity $\int_0^1 \frac{\sigma_u^4}{f'(u)} du $. Among several possibilities (including yet another Hayashi-Yoshida type one) we have decided to go with a pre-averaged version of realized quarticity. Thus we set

and define

The result precisely reads as follows.

In order to present a feasible central limit theorem associated with Theorem (ref) we vectorize the quantities $HY^n$ and $[X]$, i.e.

where vec is the vectorization operator that stacks columns of a matrix below one another, and set

with $1\leq k,l\leq d^2$ and $b= 1,2,3$. Now, the properties of stable convergence imply the following result, which can be directly applied for the construction of confidence regions.

Numerical study

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Here, we supplement the above asymptotic results based on $n \to \infty$ with a finite sample analysis by using Monte Carlo experiments. We simulate a bivariate stochastic volatility model with noise, as was also conducted in previous work of BHLS2 and CKP.

More specifically, to simulate efficient log-prices we consider

where $B^{(i)} \perp\!\!\!\perp W$. Throughout, we work with $i = 1,2$. Note that $\rho^{(i)} \sigma_{t}^{(i)} \text{d}B_{t}^{(i)}$ represents an idiosyncratic shock, while $\sqrt{1 - [\rho^{(i)}]^2} \sigma_{t}^{(i)} \text{d}W_{t}$ is a common factor.

The model for the diffusive volatility is specified as: $\sigma_{t}^{(i)} = \exp (\beta_0^{(i)} + \beta_1^{(i)} \varrho_{t}^{(i)})$, where each of the $\varrho_{t}^{(i)}$ processes conform with Ornstein-Uhlenbeck dynamics: $\text{d} \varrho_{t}^{(i)} = \alpha^{(i)} \varrho_{t}^{(i)} \text{d}t + \text{d}B_{t}^{(i)}$. This assumption means that the innovations of $\rho^{(i)} \sigma_{t}^{(i)} \text{d}B_{t}^{(i)}$ and $\text{d} \sigma_{t}^{(i)}$ are perfectly correlated, while the covariation between $\text{d}X_{t}^{(i)}$ and $\text{d}\varrho_{t}^{(i)}$ is equal to $\rho^{(i)} \sigma_{t}^{(i)} \text{d}t$. Finally, note that the model allows the two underlying price processes $X_{t}^{(1)}$ and $X_{t}^{(2)}$ to be correlated in the magnitude of $\sqrt{1 - [\rho^{(1)}]^2} \sqrt{1 - [\rho^{(2)}]^2}$.

We carry out our numerical experiments by using the following parametrization, assumed to be identical across the two volatility factors: $(a^{(i)}, \beta_0^{(i)}, \beta_1^{(i)}, \alpha^{(i)}, \rho^{(i)}) = (0.03, -5/16, 1/8,-1/40,-0.3)$, so that $\beta_0^{(i)} = [\beta_1^{(i)}]^{2} / [2 \alpha^{(i)}]$. This choice of parameters implies that integrated volatility has been normalized, in the sense that $\mathbb{E} \Bigl( \int_{0}^{1} [\sigma_s^{(i)}]^2 \text{d}s \Bigr) = 1$.

We simulate 10,000 paths of this model over the interval $[0,1]$, which we partition into $N = 23,400$ subintervals of equal length $1 / N$. In constructing noisy prices $Y^{(i)}$, we first generate a complete high-frequency record of $N$ equidistant observations of the efficient price $X^{(i)}$ using a standard Euler scheme.\footnote{Note that the Ornstein-Uhlenbeck process permits an exact discretization (see, e.g., GM). We use that fact here to avoid committing errors in working out the discrete time distribution of $\text{d} \varrho^{(i)}$ over time steps of size $1 / N$.} The initial values for the $\varrho_{t}^{(i)}$ processes at each simulation run are drawn randomly from their stationary distribution, which is $\varrho_{t}^{(i)} \sim N(0,[-2\alpha^{(i)}]^{-1})$.

Next, we add simulated microstructure noise $Y^{(i)} = X^{(i)} + \varepsilon^{(i)}$ by taking

where $\gamma$ is the so-called noise ratio parameter. This choice means that the variance of the noise process increases with the level of volatility of $X^{(i)}$, as documented by BR. $\gamma$ takes the value 0.50, which is a typical level of noise (e.g., COP).

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Finally, in order to extract non-synchronous data from the complete synchronous high-frequency record, we proceed as follows (for reference, please see Figure (ref)). We consider three settings. In scenario 1), the sampling times of $Y^{(2)}$ form a subset of the observation grid of $Y^{(1)}$, but $Y^{(1)}$ is observed more frequently. Here, we use $n_{1} = 4,680$ and $n_{2} = 2,340$. In scenario 2), we take $n_{1} = n_{2} = 4,680$, but shift the observation times of $Y^{(2)}$ to lie midway between those of $Y^{(1)}$. Finally, in scenario 3), we generate random observation times using two independent Poisson processes with intensity $\lambda_{1}$ and $\lambda_{2}$. Here $\lambda_{i}$ denotes the average waiting time for new data from process $Y^{(i)}$, so that a typical simulation will have $N / \lambda_{i}$ observations of $Y^{(i)}, i = 1,2$. We set $\lambda_{1} = 5$ and $\lambda_{2} = 10$, which implies that the first asset is trading twice as fast as the second. Note that because we are simulating in discrete time, it is possible to see common points in the last setting, as depicted in the chart.

The choice of the remaining tuning parameters are the following: We use $\theta = 0.15$ and set $k_{n} = \lceil \theta \sqrt{n} \rceil$, where $\lceil x \rceil$ is the ceil function. Moreover, to estimate the variance appearing in the CLT of $HY_{kl}^{n}$, we use $V^{n,1}_{kl,kl}$ defined in (ref) with $\varpi =1$ and $\eta = 7 / 12$.

Our initial numerical experimentations show that the raw estimator from Eq. (ref) is slightly downward biased in finite samples. This is familiar from related estimators, such as CKP, where an additional factor is applied to correct for the loss of summands induced by pre-averaging. Here, the problem is slightly more delicate, but nonetheless a relatively simple device can be used to adjust the estimator. In particular, we generate a bivariate Brownian motion $(B^{(1)}, B^{(2)})$ with a known correlation $\rho$ (throughout, we use $\rho = 1$), where the coordinates of these two processes are identical to $(Y^{(1)}, Y^{(2)})$. We then estimate $R_{kl}^{n} = \mathbb{E}[HY_{kl}^{n}]$ across 10,000 repetitions using the data from $B^{(1)}$ and $B^{(2)}$ and divide the original statistic $HY_{kl}^{n}$ (based on data from $Y^{(1)}$ and $Y^{(2)}$) by $R_{kl}^{n} / \rho$. A similar procedure can be used to bias correct the estimator of variance.

Simulation results

In Table (ref), we present the relative bias and root mean squared error of our pre-averaged Hayashi-Yoshida estimator. As a comparison, we also computed the modulated realised covariance (MRC) of CKP based on refresh time sampling. As the table reveals, both estimators are unbiased (after bias correction) in all three scenarios. $HY_{22}^{n}$ does retain a slight bias in those scenarios, where $n_2$ is small, but the bias is less than half a percent. The rmse of $HY^{n}$ is larger than what we observe for the MRC, when the estimation target is a variance component; this observation is in line with the theoretical comparison of Remark (ref). This is particularly true for the slow-trading asset $Y^{(2)}$ in scenarios one and three. However, the rmse of $HY^{n}_{12}$ is smaller than the rmse of the MRC in all scenarios. This is explained by the fact that refresh time sampling essentially uses the slowest frequency and therefore highlights the advantages of $HY^n$.

Next, we turn to the accuracy of the asymptotic approximation, where we focus on estimation of integrated covariance, $\Sigma_{12}$. In Figure (ref), we plot the simulated finite sample distribution of the standardized $HY_{12}^{n}$ for the three setups considered here, where the variance of the estimator is accessed by $V^{n,1}_{12,12}$ as described above. Although the approximation is not perfect, the goodness of the fit is surprisingly good taking the relatively small sample into account. Also, the ordering is as expected with the second scenario offering the best approximation to the standard normal (where $n_{1} = n_{2} = 4,680$). Moreover, while the average number of observations is identical in scenario one and three, the randomness of the spacings in the latter setting slightly deteriorates the tracking of the standard normal.

Proofs

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Let $C>0$ denote a generic constant which may change from line to line; we also write $C_p>0$ if a constant depends on an external parameter $p$. For the sake of simplicity we will sometimes keep the dependence of some quantities on certain parameters unreflected if things are clear from the context. Also some notations might have a different meaning in different subsections, e.g. the quantity $R_n(p)$ stands for a generic asymptotically negligible random variable in Sections (ref)--(ref). \\ \\ We remark that all our theoretical results (Theorems (ref), (ref), (ref), (ref), (ref)) are {\it stable under localization}, i.e. if they are valid for bounded coefficients then they remain valid for locally bounded coefficients. This means, since the processes $a$ and $\sigma$ are c\`agl\`ad, thus locally bounded, we can assume without loss of generality:

See e.g. Section 3 in BGJPS for more details.

The second important step in all proofs is the approximation

which means that we may pretend that $a=0$ identically and that the volatility $\sigma$ is constant over the small intervals $[t_i^k, t_{i+k_n}^k]$. Indeed, we will show that such an approximation does not affect any of our theoretical results.

Before we start proving our main results let us state some simple lemmas which concern the observation times $t_i^k$ and the pre-averaging quantities $\overline Y_{t_i^k}^k$. In what follows we use the decomposition

We also decompose the statistic $HY^n$ as

with

{\it Proof:} To compute the cardinality of the above set we need to calculate $n_k(f_k(b) -f_k(a))$, which is an upper bound for the number of points falling into $[a,b]$, up to adding one. The mean value theorem and conditions (T2), (T3) imply that $$n_k(f_k(b) -f_k(a)) =n_k(f_k)' (\xi ) (b-a)\leq Cn(b-a),$$ where $\xi$ is some point between $a$ and $b$. $\hfill\Box$ \\ \\ The above lemma basically states that the amount of time points $t_i^k$ contained in $[a,b]$ is of the same order as in the equidistant case for all $k$.

{\it Proof:} These estimates are shown separately for $\overline N_{t_i^k}^k$, $\overline D_{t_i^k}^k$ and $\overline \varepsilon _{t_i^k}^k$. They are a simple consequence of the boundedness of the processes $a$ and $\sigma$, the Burkholder inequality and Lemma (ref). See e.g. Section 5.4 from JLMPV for a detailed computation in the equidistant case. $\hfill\Box$

Proof of Theorem (ref)

Because the summands in the definition of the estimator $HY^n$ are highly correlated, the main idea of the proof is to apply a similar method as for the proof of the central limit theorem for $m$-dependent data. Roughly speaking, we will collect all summands of $HY^n$ in big and small blocks. The function of the small blocks is to ensure the (conditional) asymptotic independence of the big blocks, and their contribution will become negligible in the limit.

Let us start with the formal definition of big and small blocks. For some $p>0$, we set

where $b$ is larger than $M\max_{1\leq k\leq d} (m_k^{-1})$ and $z=0, \ldots, [\frac{n}{(p+b)k_n}]-1$. The constant $b$ is chosen in this way to ensure that the quantities $\overline Y_{t_i^k}^k$, $\overline Y_{t_j^l}^l$ with $t_i^k\in B_z(p)$, $t_j^l\in B_{z'}(p)$ and $z \not = z'$ do not use the same data, at least for $n$ large enough (see the proof of Lemma (ref)). This fact leads to the asymptotic conditional independence of the big blocks. The notion of big blocks comes from the fact that the length of $B_z(p)$ is always $pk_n/n$, where we later let $p\rightarrow \infty$, which is large compared to the length $bk_n/n$ of small blocks $S_z(p)$.

We will perform the proof in several steps. In a certain sense we will prove the statement in a reverse order. The road map of the proof is as follows:

The central limit theorem for the big blocks

Whenever $t_i^k\in A_z(p)$, $t_j^l\in A_{z'}(p)$ for $A=B$ or $A=S$ (see ((ref))), we set

Here we follow the same approximation as in ((ref)), except the volatility process is now frozen in the beginning of the block $A_z(p)$ resp. $A_{z'}(p)$. We define $M_n^{kl} (p) =\sum_{z} \zeta_{zn}^{kl} (p)$ with

As $M_n^{kl} (p)$ is a quadratic form of $Y=X+\varepsilon$, we have a straightforward decomposition

where $M_n^{kl} (X,p)$ denotes the diffusion part of $M_n^{kl} (p)$, $M_n^{kl} (\varepsilon ,p)$ stands for the noise part of $M_n^{kl} (p)$ and $M_n^{kl} (X,\varepsilon ,p)$ is the mixed part of $M_n^{kl} (p)$, which will be used in the following sections. In these we will show that the quantities $M_n (p)$ and $L^n = n^{1/4}(HY^n - [X])$ are asymptotically equivalent, i.e.

for all $\delta >0$. Thus, it is sufficient to prove the following result which completes this section.

Proof: By Theorem IX.7.28 from JS it is sufficient to show that ($1\leq k,l,k',l'\leq d$)

to conclude the stable convergence $M_n (p) \stackrel{d_{st}}{\longrightarrow} M(p)$ as $n\rightarrow \infty.$ The statement (i) is proved in the Appendix. To show (ii) we remark that the increments of $W$ involved in $\zeta_{zn}^{kl}$ are independent of $\mathcal F_{\min B_z(p)}$. On the other hand, the quantity $\zeta_{zn}^{kl} (p) (W_{\max B_z(p)}^{k'} - W_{\min B_z(p)}^{k'})$ is an odd function of $W$ and $(W,\varepsilon) \stackrel{d}{=} (-W,\varepsilon)$ since $W,\varepsilon $ are independent, which implies that $$\mathbb{E}[\zeta_{zn}^{kl} (p) (W_{\max B_z(p)}^{k'} - W_{\min B_z(p)}^{k'})| \mathcal F_{\min B_z(p)}]=0.$$ Next, to show (iii) we observe that for fixed $p$ the number of summands involved in the definition of $\zeta_{zn}^{kl} (p)$ is $O(k_n^2)$. Due to Lemma (ref) and since $z=0, \ldots, [\frac{n}{(p+b)k_n}]-1$ we immediately deduce that $$\sum_{z} \mathbb{E}[|\zeta_{zn}^{kl} (p)|^4]\leq C_p \frac{n}{(p+b)k_n} n k_n^8 (k_n)^{-8} n^{-2} \leq \frac{C_p}{k_n}\rightarrow 0.$$ Part (iv) is shown in JLMPV for an analogous situation (see Lemma 5.7 therein). This completes the proof of the first statement of Theorem (ref). The second statement is again proved in the Appendix. $\hfill\Box$

Negligibility of the small blocks

In this section we still consider the approximative quantities $\alpha_{ij}^{kl} (p)$ from ((ref)) and show that the term $\widetilde{M}_n^{kl} (p) =\sum_{z} \widetilde \zeta_{zn}^{kl} (p)$ with $ \widetilde \zeta_{zn}^{kl} (p) = \sum_{i=1}^5 \widetilde \zeta_{zn}^{kl} (i,p)$ given as

is negligible in the sense of ((ref)). This representation holds for $p>b$ (see ((ref)) for the definition of the constant $b$), which we assume without loss of generality. As in ((ref)), we have the decomposition

into the $X$-part, the mixed part and the $\varepsilon$-part, which will be used in the following sections. Let us consider the term $\sum_{z} \widetilde \zeta_{zn}^{kl} (1,p)$. First of all, we remark that the summands $\widetilde \zeta_{zn}^{kl} (1,p)$ are uncorrelated (as $z$ runs) and the number of summands is of order $n/(pk_n)$. Furthermore, there are $O(k_n^2)$ summands in the definition of $\widetilde \zeta_{zn}^{kl} (1,p)$. Thus, we conclude from Lemma (ref) that

Hence, we obtain

for all $\delta >0$. The same assertion holds for $\widetilde{M}_n^{kl} (p)$, as counting the number of non-zero $\alpha_{ij}^{kl} (p)$ for $t_i^k$ and $t_j^l$ from disjoint blocks shows that the upper bound in ((ref)) is valid for $\widetilde \zeta_{zn}^{kl} (q,p)$ as well, $q = 2, \ldots, 5$. $\hfill\Box$

The approximation of the diffusion part I

We start with the decomposition of the diffusion part of the estimator $HY^n$. Set $ HY^n_{kl}[X] = HY^n_{kl}[D] + HY^n_{kl}[D,N] +HY^n_{kl}[N] $ with

where the processes $D$ and $N$ are given in ((ref)). In this section we will show that drift part $D$ of $X$ does not influence the central limit theorem, i.e.

We start with the term $HY^n_{kl}[D]$. Note that $HY^n_{kl}[D]$ contains $O(nk_n)$ non-zero summands (due to Lemma (ref)). Lemma (ref) and the Cauchy-Schwarz inequality imply that each summand satisfies $\mathbb{E}[|\overline D_{t_i^k}^k\overline D_{t_j^l}^l|]\leq C n^{-1}.$ Thus, $\mathbb{E}[|HY^n_{kl}[D]|]\leq C n^{-1/2},$ which implies $HY^n_{kl}[D]=o_{\mathbb P} (n^{-1/4})$.

The treatment of $HY^n_{kl}[D,N]$ is a bit more delicate. We set

and define

where $\mbox{id}$ denotes the identity function on $\mathbb{R}$. The latter approximates $\xi_{ij}^n$ by freezing the process $a$ in a small time interval. Let us set

We first show that $\widetilde{HY}^n_{kl}[D,N]=o_{\mathbb P} (n^{-1/4})$. Observe that

Due to Lemma (ref) the above sum contains $O(nk_n^3)$ non-zero summands, because the $ \widetilde \xi_{ij}^n$'s are martingale differences. Moreover, we have $\mathbb{E}[|\widetilde \xi_{ij}^n|^2]\leq C n^{-3/2}$ due to Lemma (ref). Thus, we conclude $\mathbb{E}[|\widetilde{HY}^n_{kl}[D,N]|^2] \leq C n^{-1},$ which implies that $\widetilde{HY}^n_{kl}[D,N]=o_{\mathbb P}(n^{-1/4})$. In a second step we show that $HY^n_{kl}[D,N] - \widetilde{HY}^n_{kl}[D,N] =o_{\mathbb P}(n^{-1/4}).$ For this purpose, for any c\`agl\`ad bounded multivariate process $f$, we denote by $N_\delta^f(t)$ the number of jumps of $f$ bigger than $\delta >0$ before time $t$. Furthermore, we define

Roughly speaking, $m_{\eta, \delta } (f)$ is a modulus of continuity of $f$ on intervals of at most length $\eta$, which do not contain jumps bigger than $\delta$. For $f$ as above, we obviously have $\lim_{\delta \rightarrow 0} \limsup_{\eta\rightarrow 0} m_{\eta, \delta } (f)=0, \mathbb P-a.s.$ Observe that

As we mentioned the above sum contains $O(nk_n)$ summands. We have

The right-hand side of the above inequality is bounded since the process $a$ is bounded by $C n^{-1/2}$. Consequently, distinguishing between the two situations, where $a$ has or does not have jumps bigger than $\delta$ in the interval $[t_{i+h-1}^k, t_{i+h}^k]$, we obtain the inequality

Using Lemma (ref) and Cauchy-Schwarz inequality we deduce that

Due to the dominated convergence theorem we conclude that

Thus $HY^n_{kl}[D,N] - \widetilde{HY}^n_{kl}[D,N] =o_{\mathbb P}(n^{-1/4}).$ Summarizing all results of this section we get

meaning that the martingale part $N$ is the dominating term in the decomposition of $HY^n_{kl}[X]$. $\hfill\Box$

A decomposition for the martingale part

Having proved in the previous section that $HY^n[X]$ can be replaced by $HY^n[N]$ without affecting the limit, we proceed with a further decomposition of $HY^n[N]$. In this section we will show that $HY^n[N]$ is essentially an unbiased estimator of $\int_0^1 \Sigma_s ds$. Recall that

By definition we have

with

Now, we will write the above quantity as a sum of martingale differences plus bias. For this purpose we need some additional notations. We decompose $ E_{ij}^{hh'}= \cup_{r=1}^4 E_{ij}^{hh'}(r)$ with

On $E_{ij}^{hh'}(1)$ we deduce by It\^o formula:

and similar decompositions are obtained on $E_{ij}^{hh'}(q)$, $q=2,3,4$, and we denote them by $\sum_{r=1}^5 \mu_{ij}^{hh'}(q,r)$. Notice that all terms $\mu_{ij}^{hh'}(q,r)$ are martingale differences for $1\leq q,r\leq 4$, while $\mu_{ij}^{hh'}(q,5)$ gives the bias for all $1\leq q\leq 4$. We define

for $1\leq q\leq 4,1\leq r\leq 5$. Now, a simple reordering shows that

where the error in the first identity is due to border effects, and the second identity uses $\psi=\int_0^1 g(x) dx$.

Thus, we conclude that

where

We remark again all terms $\eta_{ij}^{kl}$ are now sums of martingale differences. $\hfill\Box$

The approximation of the diffusion part II

In this section we will justify the approximation

where $M_n (X,p)$ and $\widetilde M_n(X,p)$ are defined by ((ref)) and ((ref)) respectively, for some $R_n^{kl}(p)$ with

for all $\delta >0$. This means that the diffusion part $n^{1/4}\Big(HY^n_{kl}[N] - \int_0^1 \Sigma_s^{kl} ds \Big) $ of our statistic is asymptotically equivalent to the sum of the diffusion parts of big and small blocks. Recalling the estimate ((ref)) from the previous section, it is easy to show

where $\widetilde{\eta}_{ij}^{kl}$ is defined in the same way as $\eta_{ij}^{kl}$ (see ((ref))) except the process $N^k$ (resp. $N^l$) is replaced by $(\sigma_{\min A_z(p)} W)^k$ (resp. $(\sigma_{\min A_{z'}(p)} W)^l$) when $t_i^k\in A_z(p)$ for some $z$ (resp. $t_j^k\in A_{z'}(p)$ for some $z'$) and $A=B$ or $A=S$. Note that the only difference compared to proving ((ref)) lies in the fact that $M_n^{kl} (X,p) + \widetilde M_n^{kl} (X,p)$ is unbiased by construction.

Recall that the quantity $\eta_{ij}^{kl}$ (resp. $\widetilde{\eta}_{ij}^{kl}$) consists of 17 summands. Hence, we have the decomposition $R_n^{kl}(p) = \sum_{r=1}^{17} R_n^{kl}(p,r).$ As an example we will only consider the treatment of the first summand, i.e.

where $\overline{\mu}_{ij}$ is defined by ((ref)). We conclude that

where $1\leq h,h',q,q'\leq k_n$ and either $h=q, h'=q'$ or $$(t_{i+h-1}^k, t_{i+h}^k]\cap (t_{j+q'-1}^l, t_{j+q'}^l] \not= \emptyset, \qquad (t_{i+q-1}^k, t_{i+q}^k]\cap (t_{j+h'-1}^l, t_{j+h'}^l] \not= \emptyset,$$ as otherwise the expectation vanishes. We remark that the above sum contains $O(k_n^2)$ terms. Now we follow the same strategy as in Section (ref). First, we note that

where the number of non-zero summands is $O(nk_n^3)$. Using the Cauchy-Schwarz inequality and the same approximations as at the end of Section (ref), we deduce that

for any $\delta >0$. Thus, for any fixed $p$, we have (by choosing $n$ large and then $\delta$ small) $ \lim_{n\rightarrow \infty} \mathbb{E}[|R_n^{kl}(p,1)|^2] =0.$ Hence, ((ref)) for any $\delta >0$, and we are done. $\hfill\Box$

The approximation of the mixed part

In this section we will prove that

where $M_n (X,\varepsilon ,p)$ and $\widetilde M_n(X,\varepsilon ,p)$ are defined by ((ref)) and ((ref)) respectively, $HY^n_{kl}[X,\varepsilon ]$ is given by ((ref)) and some $R_n^{kl}(p)$ with ((ref)) for all $\delta >0$. This proof is easier than the proofs in previous sections, because the processes $X$ and $\varepsilon$ are independent. We first show that

is a negligible sequence. Using Lemma (ref) and proceeding as in the treatment of the term $\widetilde{HY}^n_{kl}[D,N]$ from ((ref)) we deduce that $\mathbb{E}[|HY^n_{kl}[D,\varepsilon ]|^2]\leq Cn^{-1}.$ Hence, $n^{1/4} HY^n_{kl}[D,\varepsilon ] \stackrel{\mathbb{P}}{\longrightarrow} 0.$ Next, we obtain that

Using again Lemma (ref), the independence between $\varepsilon$ and the components of $X$, and similar methods as for $R_n^{kl}(p,1)$ in the previous section, we conclude that

for any $\delta >0$. Thus, for any fixed $p$, we have $\lim_{n\rightarrow \infty} \mathbb{E}[|R_n^{kl}(p,1)|^2] =0,$ and hence ((ref)) for any $\delta >0$, and we are done. $\hfill\Box$

The noise part and the final identity

Finally, we will show that

where $M_n (\varepsilon,p)$ and $\widetilde M_n(\varepsilon,p)$ are defined by ((ref)) and ((ref)) respectively, for some $R_n^{kl}(p)$ with ((ref)) for all $\delta >0$. This is a relatively easy exercise, because by definition we just need to prove that $n^{1/4} \mathbb{E}[HY^n_{kl}[\varepsilon ]] = o(1).$ By reordering the statistic $HY^n_{kl}$ we obtain that

for some constants $a_{ij}^{kl}(n)$ with $|a_{ij}^{kl}(n)|\leq C$. A simple calculation shows that $$a_{ij}^{kl}(n) = \left( \sum_{j=0}^{k_n-1} g \Big(\frac{j+1}{k_n} \Big) -g \Big(\frac{j}{k_n} \Big) \right)^2 = (g(1)-g(0))^2=0$$ except for those $t_i^k$ and $t_j^l$ that are among the first and last $O(n^{1/2})$ summands. Hence, $n^{1/4} \mathbb{E}[HY^n_{kl}[\varepsilon ]] = o(1)$ and we deduce that

for all $\delta >0$. \\ \\ Finally, let us put things together. In Sections (ref)--(ref) we have proved the identity

for some $R_n(p)$ and we have shown (see Section (ref)) that

for all $\delta >0$. On the other hand, we have proved in Section (ref) that

and, for $p\rightarrow \infty$:

This completes the proof of Theorem (ref). $\hfill\Box$

Consistency of the variance estimators

Proof of Theorem (ref)

It is obviously enough to prove the result for the unsymmetrized estimator

only, and we introduce two approximating versions of $HY^n_{kl}(\alpha)$ first, namely

where we have set

as in ((ref)), and the $W^{\nu}$ denote the independent components of the $d'$-dimensional Brownian motion $W$. Since $\sigma$ is assumed to be an It\^o semimartingale itself, the error due to replacing $\overline {Y}_{t_i^k}^k$ by $\overline {Z(\alpha)}_{t_i^k}^k$ is small: Let $t_i^k \in B_n(\alpha)$. Then

Lemma (ref) and Lemma (ref) give $E|HY^n_{kl}(\alpha)| \leq C \beta_n/n$, thus it is simple to deduce $E|HY^n_{kl}(\alpha) - \widetilde{HY}^n_{kl}(\alpha)| \leq C (\beta_n/n)^{3/2}$, and analogously for $\overline{HY}^n_{kl}(\alpha)$, so using $\eta < 2/3$ we obtain $\widetilde{V}^{n,1}_{kl,k'l'} - \overline{V}^{n,1}_{kl,k'l'} = o_{\mathbb P} (1)$ with

The remainder of the proof is simple now. Without loss of generality let $\beta_n > 4bk_n$ hold, so only $\overline{HY}^n_{k'l'}(\alpha)$ and $\overline{HY}^n_{k'l'}(\alpha+1)$ might share increments of $Y$. Then we obtain

by conditional independence, and we are left with

Write $V_{kl,k'l'} = \int_0^1 r_u du$, where the process $r$ is given by the right hand side of ((ref)). From the same arguments as in Lemma (ref) and Lemma (ref) in the Appendix plus using $\eta > 1/2$ we obtain

uniformly in $\alpha$, and the proof is complete. $\hfill\Box$

Proof of Theorem (ref)

From the proof of Theorem (ref) we have

uniformly in $s$. Therefore the discussion on $\Psi_{n}^{kl}$ shows that we are left to prove

which by left-continuity is obvious as well. $\hfill\Box$

Proof of Theorem (ref)

All we need to prove is

Since $\sigma$ is c\`agl\`ad, we know from the proof of Theorem 1 in PV1 that we may replace $|\overline Y_{t_i}|^4$ by $|\sigma_{t_i} \overline W_{t_i} + \overline \varepsilon_{t_i}|^4 $ without affecting the limit. We have

and similar identities hold for $6 |\overline W_{t_i}|^2 |\overline \varepsilon_{t_i}|^2$ and $|\overline \varepsilon_{t_i}|^4$ as well. The result follows easily now from a Riemann approximation. $\hfill\Box$