Spectral analysis of high-dimensional spot volatility matrix with applications

Spectral analysis of high-dimensional spot volatility matrix with applications

Introduction

During the past decades, it has been a hot topic for investigating the asymptotic spectral analysis of sample covariance matrices when the data dimension increases to infinity at the same rate as the sample size increases (Such a scenario is called large dimensional setting). It is shown that the theory differs from the classical limiting theory in the fixed dimension. To be specific, the limiting spectral distributions of the population covariance matrix and the sample covariance matrix will be governed by the so-called Mar$\check{\text{c}}$enko-Pastur equation (S1995), instead of the empirical central limit theorem at the fixed dimension. Interested readers can refer to B2010 for a comprehensive introduction of the large dimensional random matrix theory. However, most existing works assume that the samples are independent and identically distributed, which is not satisfied in many situations, especially for time series data. In this paper, the data type to our concern is the high-frequency data, which is neither independent nor identically distributed (Such a point will be elaborated in Section (ref)).

The computational technology has developed fast and has been widely applied in the financial market in recent years, making the high-frequency trading strategies possible and generating massive high-frequency data. As a result, developing statistical models and econometric methods for the high-frequency data has been experiencing exponential growth, both by practicers and researchers. The key objectives that attract most attention from statisticians and econometricians are the volatility of a single asset and the volatility matrix for the multivariate case (both of these two quantities have the integrated version and the spot version), since they play pivotal roles in many areas of financial economics, including asset and derivative pricing, portfolio allocation, risk management, hedging, and etc. Various methods of estimation and statistical inference regarding to these mentioned quantities have been proposed, and a bulk of related references can be found in AJ2014. We notice that most of the existing literature focuses on one-dimensional case or fixed dimension, while the consideration of high dimension allowing the dimensionality tend to infinity is rather rare in comparison. It is well known that the fact of high dimension is a common feature for vast datasets in this big data era, thus the extension of classical theory under fixed dimension to this scenario becomes an important research direction in statistics. For the high-frequency data, when both the number of assets and the sample size of the transactional price data of the assets go to infinity, the estimation of integrated and spot volatility matrix has been considered in WZ2010, TWC2013, TWZ2013, KKLW2018, KLW2018, K2018, DLX2019, BMN2019, BDLW2022, LLZ2024, and references therein. Instead of directly estimating the integrated and spot volatility matrix, their spectral analysis also has wide application in multivariate hypothesis testing, principal component analysis, and factor analysis, and is becoming a hot topic recently (See, e.g. AX2019, CMZ2020, AX2017, K2023, KLLL2023.). For example, as suggested in CMZ2020, investors should be better off investing in a statistically estimated principal component rather than an index fund. Under the large dimensional regime, namely the data dimension and the sample size tend to infinity at the same rate, ZL2011 studied the limiting spectral distribution of the realized volatility matrix estimator. They found that its limiting behavior is greatly affected by the time variability of the volatility matrix process, and the empirical result for independently and identically distributed samples established in large dimensional random matrix can not be directly used. Moreover, they proposed a time-variation adjusted realized volatility matrix estimator, whose limiting spectral distribution depends solely on that of the targeting integrated volatility matrix via a Mar$\check{\text{c}}$enko-Pastur equation, making the inference of the latter one to be possible. The same problem was also considered by HP2014, who obtained the explicit form of the moments of the limiting spectral distribution of the realized volatility matrix estimator by using the tools from graph theory. Based on ZL2011, XZ2018 considered the further presence of market microstructure noise in the high-frequency data and dealt with its effect by pre-averaging technique. Furthermore, for the specific volatility matrix process considered in ZL2011, YZC2021 established the central limit theorem for the linear spectral statistics of the time-variation adjusted realized covariance matrix estimator proposed in ZL2011. Based on this, they proposed several test statistics for the identity and sphericity hypotheses of the integrated volatility matrix.

In this paper, our focus is the spot volatility matrix, which can quantify the co-variation pattern of the price processes of multiple assets at any given time second. Unlike the integrated volatility matrix, which is defined over a fixed time interval, say a day, the spot volatility matrix is a more general and flexible object, leading itself to wider range of applications. {For example, in practice, most financial and macroeconomic time series exhibit time-varying volatility, and the expected volatility matrix in the near future can be very different from the average of the volatility matrix over a long time horizon. To reduce the bias of the forecasted volatility matrix, which plays a pivotal role in mean-variance efficient portfolio theory, we need to shorten the learning period to better capture the dynamics of the spot volatility matrix (See, e.g. FLY2012, JLT2024, and references therein).} The first question we are trying to figure out is the relationship between the limiting spectral distribution of the spot volatility matrix and the one of the realized spot volatility matrix estimator, under the infill setting of high-frequency data, namely, the length of time interval between two consecutive observations decreases to 0. Meanwhile, we allow the data dimension tends to infinity at the same rate as the sample size increases. For identically and independently distributed data, it is well known that the limiting spectral distribution of the population covariance matrix is linked with one of the sample covariance matrix by a Mar$\check{\text{c}}$enko–Pastur equation through Stieltjes transform when the date dimension and the sample size tend to infinity at the same rate (S1995). Our analysis shows that such a result remains valid for the spot volatility matrix and its estimator under some regular conditions of smoothness for the volatility matrix process. Furthermore, we also investigate the central limit theorem for the linear spectral statistics of the spot volatility matrix, which serves as a second-order limiting result and has wide applicability in multivariate statistical inference problems. As for possible applications of our established theoretical results, we consider two hypothesis testing problems on if the spot volatility matrix is equal or proportional to a given matrix, which are called identity test and sphericity test respectively. To this end, we propose several test statistics in the large dimensional regime and demonstrate their finite sample performances via simulation studies.

The rest of the article is organized as follows. In Section (ref), we introduce the framework of high-frequency data, the related assumptions and present the realized spot volatility matrix estimator. In Section (ref), we present the asymptotic results regarding to the empirical spectral distribution and the linear spectral statistics of the realized spot volatility matrix estimator. For the application of identity test and sphericity test for the spot volatility matrix, we propose several test statistics and present their asymptotic properties in Section (ref). Section (ref) concludes the paper. In a separate supplementary material, we conduct simulation studies to verify our established theory and demonstrate the finite sample performance of the proposed test statistics.

Notations: At the end of this introduction part, we introduce some notation and definitions used throughout the paper. For any matrix $\Sigma$, we use $\Sigma^{(i,j)}$, $\Sigma^{\mathsf{T}}$, $\text{tr} (\Sigma)$ to denote its $(i,j)$-th element, transpose, and trace, respectively. We use $\lambda_i^{\Sigma}$, $\lambda_{\min}^{\Sigma}$, and $\lambda_{\max}^{\Sigma}$ to denote the $i$-th (in descending order), the minimum, and the maximum eigenvalues of a $p-$dimensional squared matrix $\Sigma$, respectively. Furthermore, the notation $\lambda(\Sigma) = (\lambda_1^{\Sigma},...,\lambda_p^{\Sigma})^{\mathsf{T}}$ means the function extracting the eigenvalues of $\Sigma$ as a $p$-dimensional vector, in a non-increasing order. {Meanwhile, we let $\|\Sigma\|$ and $\|\Sigma\|_{F}$ be the spectral norm and Frobenius norm of $\Sigma$, namely $\|\Sigma\| := \sqrt{\lambda_{\max}^{\Sigma^{\mathsf{T}} \Sigma}}$ and $\|\Sigma\|_{F}:= \sqrt{\sum_{i=1}^{p} (\lambda_{i}^{\Sigma})^2}$. } For any vector $a$, we use $a^{(j)}$ to denote its $j-$th entry. Specifically, we define $\mathbf{0}_{p}, \mathbf{1}_{p}$ as an $p-$dimensional vector with each entry being 0 and 1, respectively, and $\mathbb{I}_p$ as a $p \times p$ identity matrix. The formula $A \preceq B$ means that $B-A$ is strictly positive definite. For any interval $I \subset [0, \infty)$ and any metric space $S$, $D(I; S)$ stands for the space of c$\grave{\text{a}}$dl$\grave{\text{a}}$g functions from $I$ to $S$. We use the notation $\longrightarrow^d, \longrightarrow^{p}$ as the convergence in distribution and convergence in probability respectively, and $N(\mu,\sigma^2)$ as the normal distribution with mean $\mu$ and variance $\sigma^2$. {The notation $\lfloor x\rfloor$ stands for $\max \{k \in \mathbb{Z}: k\leq x\}$.}

Setup and assumptions

Without the loss of generality, we define all the processes involved in the time interval $[0,1]$, where the time unit may be one day, one month, or one year. On the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{0\leq t\leq 1}, \mathbf{P})$, we denote the underlying efficient log-price process of the assets as $\{X_t\}_{0\leq t\leq 1}$. Moreover, we assume $X \in \mathbb{R}^{p}$, with $p$ being the total number of assets under consideration. According to the fundamental theorem of asset pricing (See, e.g. DS1994), in a frictionless financial market with no-arbitrage opportunity, the price process of an asset is necessarily to be a semimartingale. In accordance with this, we assume that $X$ follows a continuous Brownian semimartingale and can be written as

In the above model, $b \in \mathbb{R}^{p}$ is the drift term being progressively measurable and locally bounded, $B$ is a $q$-dimensional standard Brownian motion and its dimension $q$ usually means the number of risk factors involved in financial asset prices $X$ or the dimension of the state space of a continuous-time factor model (See, e.g. JM2013rank, AX2017 and many others), $\sigma$ is a $p\times q$ matrix-valued stochastic process generating adapted and c$\grave{\text{a}}$dl$\grave{\text{a}}$g path almost surely. In (ref), we allow $\sigma$ and $B$ to be mutually dependent with any general structure, thus depicting the leverage effect in finance. We define the spot volatility matrix at any given time point $t\in[0,1]$ as $c_t = \sigma_t \sigma_t^{\mathsf{T}}$ and its integral version, the integrated volatility matrix, as $\int_{0}^{1} c_tdt$\footnote{Without causing confusion, we will call both the processes $\sigma$ and $c$ as volatility process, from here and after.}. Both measurements quantify the joint variational strength among different assets and play pivotal roles in financial economics. In this paper, we are interested in exploring the local behavior of the volatility matrix process, thus we mainly focus on the spot volatility matrix. More importantly, our framework allows the dimensionality $p$ to diverge with the sample size, in contrast to most existing high-frequency studies where $p$ is treated as fixed. Besides, the factor number $q$ can be either fixed or divergent at a proper rate as $p\rightarrow \infty$. For notational simplicity, we omit the dependence of the dimension parameters $p$ and $q$ for $\sigma_t, c_t$. We assume that our model (ref) satisfies the following assumption:

In practice, the continuous sample path of $X_t$ over $[0,1]$ is unobservable, and it can only be observed at some discrete time points. Throughout this paper, we consider the equidistant time grids scattered over the interval $[0,1]$ and write them as $\{i/n: i=0,1,...,n\}$, where $n$ is the total number of observations. We see that as $n$ increases, the time length between two consecutive observations shrinks, resulting in the so-called high-frequency data. Our theoretical results will be established under the infill asymptotic setting of $n\rightarrow \infty$. With the definition $\Delta_i^n X = X_{i/n} - X_{(i-1)/n}$ for $i=1,..., n$, it is well known that the integrated volatility matrix $\int_{0}^{1} c_td_t$ can be estimated by the realized volatility matrix estimator $\widehat{IV}^n$:

And its local version of spot volatility matrix estimator at any given time $t \in [0,1]$ can be written as

where $k_n$ is a sequence of integers tending to infinity as $n \rightarrow \infty$, and meanwhile $k_n/n \rightarrow 0$. A complete introduction and theoretical analyses of these two estimators can be found in AJ2014, when the dimension $p$ is fixed as $n\rightarrow \infty$.

{ It is worthy noting that under (ref), $\{\Delta_i^n X: i=1,...,n\}$ are neither independent nor identically distributed. To see this, if the stochastic processes $b$ and $\sigma$ are independent of the Brownian motion $B$, since $\Delta_i^n X = \int_{(i-1)/n}^{i/n} b_sds + \int_{(i-1)/n}^{i/n} \sigma_sdB_s $, we have

where $MN$ stands for mixed normal distribution. Since $b$ and $c$ can be both autocorrelated and time-varying, $\Delta_i^n X$'s are not independent. In random matrix theory, the spectral properties of the sample covariance matrix have been established by using $i.i.d.$ data under the large dimensional regime (B2010). In this paper, we are interested in extending the theory to the study of spot volatility matrix based on similar technique called Stieltjes transform. Another possible alternative method to solve this problem is to treat the spot volatility matrix estimator (ref) as matrix-valued realization of $X$ driven by the stochastic differential equation (ref) and investigate its spectral properties as in MP2022, SYY2020, SYY2021, SYY2022 and references therein. The details remain to be explored. }

Asymptotic results

In this section, we present the results of empirical spectral distribution (ESD) and linear spectral statistics (LSS) for the realized spot volatility matrix estimator $\widehat{c_{t}}^{n}$. The definitions of ESD and LSS will be given later. In the sequel, we first review the empirical results of ESD and LSS for covariance matrix under the large dimensional regime, which are followed by the extension to spot volatility matrix. From here and after, for $i=1,..., n$, we define $\Delta_i^n S= S_{i/n} - S_{(i-1)/n}$ for a general process $S$. We use the notation $C$ for a general positive constant, which may take different values from line to line.

Empirical spectral distribution (ESD)

Let $M$ be a $p\times p$ random, symmetric and nonnegative definite matrix, whose eigenvalues are real and denoted as $\{\lambda_i^{M}: i=1,..., p\}$. Then, the empirical spectral distribution (ESD) of $M$ is defined as

where $\mathbf{I}_{\{\cdot\}}$ is the indicator function. In large dimensional random matrix theory, the relationship between the limiting spectral distribution of the population covariance matrix and the ESD of the sample covariance matrix is given by the Mar$\check{\text{c}}$enko–Pastur equation through the Stieltjes transform.

Specifically, for the special case of $\Sigma_p = \sigma^2 \mathbb{I}_p$ with some positive random variable $\sigma^2$, the limiting spectral distribution $F^{y, H}$ in the above proposition has an explicit expression called Mar$\check{\text{c}}$enko-Pastur law. Conditional on the ratio index $y$ and the scale index $\sigma^2$, the probability density function of $F^{y,H}$ is given by

where $a=\sigma^2(1-\sqrt{y})^2, b=\sigma^2(1+\sqrt{y})^2$, and a point mass $1-1/y$ at the origin if $y>1$. Let $\underline{m}(z)$ be the Stieltjes transform of $\underline{F}^{y,H} = y F^{y,H} + (1-y) \mathbf{1}_{[0, \infty)}$, which is the limiting ESD of $\underline{S}_p := 1/n \times \sum_{l=1}^{n} (\mathbf{Z}_l^{(p)})^{\mathsf{T}} \Sigma_p \mathbf{Z}_l^{(p)}$. Then, the equation (ref) can be equivalently written as

which is called Silverstein's equation, proposed in S1995.

As for the high-frequency setting considered in this paper, if there is no drift term and the volatility process is constant in (ref), namely

then, for $i=1,..., n$, $\sqrt{n} \Delta_i^n X$ are independent and identically distributed with the distribution $(\sigma_0 \sigma_0^{\mathsf{T}})^{1/2} \cdot N(\mathbf{0}_{p},\mathbb{I}_p)$. As a direct application of Proposition (ref), the limiting spectral distributions of the integrated volatility matrix $\int_{0}^{1} c_td_t$ and the realized volatility matrix estimator (ref) are determined by (ref) if $p/n \rightarrow y$ as $n \rightarrow \infty$. For the general stochastic volatility model as in (ref), since the $i.i.d.$ condition for the increments $\{\Delta_i^n X:i=1,...,n\}$ is usually violated, as a consequence, the theoretical results established in Proposition (ref) cannot be directly applied. In fact, in Proposition 3 of ZL2011, with the consideration of the special case

where $\gamma_t$ is a time-varying and nonrandom scalar, it is shown that the ESD of the realized volatility estimator (ref) does not converge to Mar$\check{\text{c}}$enko–Pastur law. Based on this special example, we see that, under the high-frequency setting, the relationship between the limiting spectral distributions of the integrated volatility matrix $\int_{0}^{1} c_td_t$ and the realized volatility matrix estimator (ref) is not governed by (ref) anymore. ZL2011 pointed out that this is caused by the time-variability of the volatility process and they did some detailed analyses on how the time-variability in the volatility process affects the limiting spectral distribution of the realized volatility matrix estimator. Moreover, for model (ref) satisfying the following class $\mathcal{C}$:

where, $\gamma_t$ is a one-dimensional random process belongs to $ D([0,1]; \mathbb{R})$, $\Lambda_p$ is a $p\times p$ nonrandom matrix satisfying $\text{tr}(\Lambda_p \Lambda_p^{\mathsf{T}}) = p$, they proposed an alternative estimator of the integrated volatility matrix that can be used to infer the limiting spectral distribution of the integrated volatility matrix. For general stochastic volatility model (ref) without any restriction, the relationship between the limiting spectral distributions of the integrated volatility matrix and its realized estimator remains unknown.

For the spot volatility matrix, similar to the above analyses, if the condition (ref) is satisfied, Proposition (ref) implies that the limiting spectral distributions of the spot volatility matrix $c_t$ and the realized spot volatility matrix estimator (ref) are determined by (ref), if the dimension $p$ and the sample size $k_n$ satisfy $p/k_n \rightarrow y$ as $k_n \rightarrow \infty$. Our first finding is that, with Assumption (ref) and under some suitable asymptotic conditions, such a result remains true for the general stochastic volatility model (ref). Specifically, we have

Proof. We define

and, for $j=1,...,k_n$,

and

From the above definitions, we see that $\widehat{c_{t}}^{n} = UU^{\mathsf{T}}$ and $\widetilde{c_{t}}^{n} = VV^{\mathsf{T}}$. According to Lemma 2.7 in B1999, we have

where $L(F,G)$ is the L$\acute{\text{e}}$vy distance between the two probability distribution functions $F$ and $G$. For $j=1,...,k_n$ and $i=1,...,p$, under Assumption (ref) and since $p/k_n = O(1)$, we have

and similarly

Based on the above results, we can further obtain

Plugging them into (ref) results in $\mathbb{E} \left[L(F^{\widehat{c_{t}}^{n}}, F^{\widetilde{c_{t}}^{n} })\right] \leq C(\frac{q}{n} + q^2\left(\frac{p}{n}\right)^{2\gamma} ) \rightarrow 0$, if condition $(\romannumeral1)$ is satisfied. Thus, Markov's inequality implies that $L(F^{\widehat{c_{t}}^{n}}, F^{\widetilde{c_{t}}^{n} })$ converges to 0 in probability, namely, the empirical spectral distributions of $\widehat{c_{t}}^{n}$ and $\widetilde{c_{t}}^{n}$ converge to the same distribution $F^{\bar{p}, H_t}$.

On the same probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{0\leq t\leq 1}, \mathbf{P})$ where $X$ is defined, we define, for $p=1,2,...$ and $1 \leq l\leq k_n$, $\mathbf{Z}_{l}^{(p)} = (Z_l^{(p,j)})_{1\leq j\leq p}$ with $Z_l^{(p,j)}$ being $i.i.d.$ standard normal random variables and independent of $B$. Let $c_t^{1/2}$ be the (nonnegative) square root matrix of $c_t$ and

Condition on $\mathcal{F}_t$, under which $\sigma_t, c_t$ are measurable, we have $\widetilde{c_{t}}^{n} \overset{d}= S_p$ since their components are $i.i.d.$ and satisfy $\sigma_{t} \sqrt{n}\Delta_i^n B \overset{d}= c_t^{1/2} \mathbf{Z}_l^{(p)} \sim N(\mathbf{0}_{p},c_t)$, thus the ESD $F^{S_p}$ converges to the same limiting spectral distribution of $\widetilde{c_{t}}^{n}$, that is $F^{\bar{p},H_t}$. According to Proposition (ref), the spectral distribution function $F^{\bar{p},H_t}$ is driven by (ref) and (ref), with $H$ and $F^{y,H}$ replaced by $H_t$ and $F^{\bar{p},H_t}$ respectively. The proof of the theorem is finished. $\hfill \square$

To see how fast can the dimensionality $p$ increases, we now take a deep analysis on condition $(\romannumeral1)$. Without loss of generality, we assume $p= O(n^{a}), q= O(n^{b})$ with $0< a\leq 1, 0\leq b <1$, then the condition turns to requiring $b+a\gamma - \gamma < 0$. We conclude that:

Linear spectral statistics (LSS)

For any random symmetric and nonnegative definite matrix $M$, its linear spectral statistics (LSS) is defined as

where we recall that $\{\lambda_i^{M}: i=1,..., p\} $ is the real spectrum of $M$, $F^{M}$ is the ESD of $M$, and $f(\cdot)$ is a function defined on $[0,\infty)$. LSS is important in multivariate statistical inference since many statistics for the population parameters can be expressed in such a form (A2003). With the definitions and notation given in Proposition (ref), a general LSS based on the sample covariance matrix $S_p$ can then be written as $\hat{\theta} = \int f(x)dF^{S_p}(x)$. Proposition (ref) describes the asymptotic distributional behavior of the eigenvalues of the sample covariance matrix, thus the point-wise limits of the eigenvalue statistics are the integrals of the corresponding functions with respect to the limiting spectral distribution $F^{y, H}$ in (ref). Namely, we have $\hat{\theta} \rightarrow \theta:= \int f(x)dF^{y, H}(x)$, almost surely, which serves as a first-order convergence result for LSS. Furthermore, for making statistical inferences or conducting hypothesis testing problems on the population parameter, we need the second-order convergence result of CLT for LSS. To this end, we let $F^{y_n, H_p}$ be the distribution defined by (ref) and (ref) with the parameters $y, H$ replaced by $y_n:=p/n, H_p:= F^{\Sigma_p}$, respectively, and define $G_p:= p( F^{S_p} - F^{y_n, H_p} )$. Our target is investigate the fluctuation of the following scaled and centralized LSS:

We remark that $G_p$ is defined as the difference between the sample-based ESD $F^{S_p}$ and $F^{y_n, H_p}$, instead of $F^{y, H}$. This is because, as explained in BS2004, on one hand, the convergence rate of $y_n\rightarrow y$ and $H_p \rightarrow H$ can be arbitrarily slow; on the other hand, from the perspective of statistical inference, $H_p$ can be viewed as a description of the current population and $y_n$ is the ratio of dimension to sample size for the current sample, such a consideration is more realistic. The empirical result in the large dimensional random matrix field shows that (ref) converges to a Gaussian distribution, whose expectation and covariance are obtained by using the Stieltjes transform.

As a direct application of Proposition (ref), under the special case of constant volatility model (ref), after directly replacing $\Sigma_p$ and $S_p$ with the integrated covariance matrix $\int_{0}^{1} c_td_t$ and its realized estimator $\widehat{IV}^n$ in (ref) respectively, we see that the random vector defined as in (ref) will converge to a Gaussian distribution with expectation and covariance given in (ref) and (ref), if $y_n:= p/n \rightarrow y>0$ as $n\rightarrow \infty$.

Before presenting our LSS result for the spot volatility matrix, we make some remarks as follows. We note that, in the above proposition, the population covariance matrix $\Sigma_p$ is restricted to be nonrandom, while such a limitation is not required for the first-order limiting result in Proposition (ref). We do not need such a restrictive nonrandom condition for the volatility process $\sigma$ in (ref), even for the investigation of LSS for the spot covariance matrix. Similar to the analysis in Remark (ref), we can approximate the population covariance matrix of $\Delta_i^n X$ with $i= \lfloor tn \rfloor +1,...,\lfloor tn \rfloor +k_n$ in (ref) by a common covariance matrix $c_{t}$, which can be seen as a nonrandom matrix at time $t$. Thus, the nonrandom assumption for $\sigma$ is not necessary for us.

For the spot volatility process in model (ref), we define, for fixed $t \in[0,1]$,

where $z_n:=p/k_n$ and $H_{p,t} := F^{c_t}$, whose definition is given in Theorem (ref). We have

Proof. Recall that $\widehat{c_{t}}$ and $\widetilde{c_{t}}$ are defined as in (ref) and (ref), we further define $\widetilde{G}_{p,t} = p( F^{\widetilde{c_{t}}^{n}} - F^{z_n, H_{p,t}} )$ and will show that

With $U$ and $V$ defined as in (ref), we can obtain

By using the results in (ref), we have

According to YBK1988, we have, almost surely,

and since $\widehat{c_{t}}^{n} = UU^{\mathsf{T}}$ and $\widetilde{c_{t}}^{n} = VV^{\mathsf{T}}$, we have

After plugging (ref)--(ref) into (ref) and using condition $(\romannumeral1)$, we can obtain (ref). Similarly, by defining $S_p$ as in (ref) and following the analyses therein, directly applying Proposition (ref) yields that the random vector $\left(\int f_1(x)\widetilde{G}_{p,t}(x),..., \int f_k(x)\widetilde{G}_{p,t}(x)\right)$ forms a tight sequence in $p$ and converges in distribution to a Gaussian vector $(X_{f_1},...,X_{f_k})$, whose means and covariance function are given by (ref) and (ref) respectively. And (ref) implies that the same result holds for the random vector (ref), which finishes the proof of the theorem. $\hfill \square$

To the best of our knowledge, allowing every entry of $\Sigma_{p}$ in Proposition (ref) to be random and extending related theory has not been considered, even in the large dimensional random matrix community. Partial of this problem have been solved if $\Sigma_{p}$ has some extra structures. For example, for the elliptical correlated model with $\Sigma_{p} = w \Lambda_p$, where $w>0$ is a scalar random variable and $\Lambda_p$ is a $p\times p$ nonrandom matrix of full rank (See, e.g. HLLZ2019, HLLZ2025 and references therein.). In YZC2021, they considered the elliptically distributed samples, which are similar to the high-frequency increments under the class $\mathcal{C}$ of (ref). They established the central limit theorem for the LSS of the sample covariance matrix by using the self-normalized observations, such an estimator was proposed in ZL2011 and its limiting spectral distribution was also investigated therein. In Theorem (ref), we do require the mentioned structure for the volatility process $c$ and all of its entries can be random. This also inspires us that, establishing the LSS result similar to Proposition (ref) for the sample covariance matrix when the entries of $\Sigma_{p}$ are random is also possible, at least it can be realized by restricting the variance of the entries.

Similar to the analyses for Theorem (ref), we assume $p= O(n^{a})$ with $0< a < 2/3$ and $q= O(n^{b})$ with $0\leq b <1$, then condition $(\romannumeral1)$ in Theorem (ref) turns to requiring $b+3a/2+2a\gamma - 2\gamma < 0$. We conclude that:

Application

In this section, we apply the theoretical results established in the last section to conduct hypothesis testing problems of identity test and sphericity test for the spot volatility matrix $c_t$ for $t\in[0,1]$.

Identity test

We first consider testing if the spot volatility matrix $c_{t}$ is equal to a given matrix $\Sigma$ or not. Assume $\Sigma$ is invertible, we note that such a testing problem is equivalent to the identity test after multiplying the observations by $\Sigma^{-1/2}$. Thus, without loss of generality, we consider the following hypothesis testing problem for some $t\in[0,1]$:

Before giving out our test statistic, we review the related results for the empirical covariance matrix under fixed dimension first. Suppose that $\bm{x}$ follows a $p$-dimensional multivariate Gaussian distribution $N(\mu_p, \Sigma_p)$ and we want to test if $\Sigma_p = \mathbb{I}_p$ or not. Let $(\bm{x}_1,...,\bm{x}_n)$ be a set of mutually independent samples from $\bm{x}$, then the likelihood ratio test (LRT) statistic is defined as

where

is the sample covariance matrix with $\bar{\bm{x}} = \frac{1}{n} \sum_{i=1}^{n} \bm{x}_i$. For fixed $p$, as $n \rightarrow \infty$, the classical theory from the multivariate statistical analysis states that (ref) converges to $\chi_{1/2p(p+1)}^2$ distribution if $\Sigma_p = \mathbb{I}_p$ (A2003). But under the large dimensional regime, such a test statistic is infeasible, as justified by the empirical studies in BJYZ2009. Moreover, based on the central limit theorem results for LSS in Proposition (ref), BJYZ2009 studied the statistic (ref) and demonstrated that (ref) converges to a normal distribution when $p/n\rightarrow y \in (0,1)$ as $n\rightarrow \infty$\footnote{{For the critical case of $y=1$, JJY2012 and JY2013 studied the statistic (ref) by using the Selberg integral and analyzing its moments respectively, their results demonstrate that (ref) still converges to a normal distribution, but the expectation and variance are different with the ones obtained for $y\in (0,1)$. If $y\in (1,\infty)$, the statistic (ref) is degenerate and not applicable since $|S_n| = 0$ when $p>n$, thus the statistics has to be modified. To this end, QLY2023 proposed a quasi-LRT test statistic and investigated its limiting distribution based on the LSS of a special estimator called re-normalized sample covariance matrix.}}. Similarly, applying our theoretical results in Theorem (ref), we can propose a test statistic for testing the hypotheses in (ref), as follows.

Proof. According to the proof of Theorem (ref), almost surely, $\widehat{c_{t}}^{n}$ has the same asymptotic distribution with $S_p$ defined in (ref). We define

Under the null hypothesis $H_0$ of $c_t = \mathbb{I}_p$, we have

and the first term converges to a Gaussian vector with mean $m(g)$ and variance $v(g)$, which are calculated in the proof of Theorem 3.1 in BJYZ2009. A direct application of Slutsky's theorem yields the conclusion (ref). $\hfill \square$

Conclusion in Theorem (ref) gives us the following test statistic:

Noticing that in the above theorem (Also for the results in BJYZ2009), the limiting value of the ratio between the dimension $p$ and the sample size $k_n$, $\bar{p}$, is constrained to be smaller than 1, which may not be satisfied for some applications in practice. When the dimension $p$ can be larger than the sample size $n$ in the large dimensional regime, under the $i.i.d.$ setting, LW2002 proposed the following test statistic:

and established its asymptotic properties. Similarly, for the spot volatility matrix estimator using the high-frequency data, we can define

and we can obtain

Proof. We define

with $S_p$ defined in (ref). According to (ref) in the proof of Theorem (ref), we have $p(\widetilde{W}_n - \widehat{W}_n) = o_p(1)$, thus $p(\widetilde{W}_n - \widehat{W}_n) \longrightarrow^p 0$. Moreover, it holds that $k_n\widehat{W}_n - p \longrightarrow^{d} N(1,4)$, as given by Proposition 7 in LW2002, where the required conditions are satisfied under our setting. This finishes the proof of Theorem (ref). $\hfill \square$

With Theorem (ref), we can obtain another test statistic

Sphericity test

In high-frequency financial econometrics, testing if the spot volatility matrix $c_t$ equals to a know covariance matrix may be too restrictive and has limited applications. But knowing the covariance matrix up to a constant can be good enough in some applications. For example, the minimum variance portfolio is given by $\Sigma^{-1}\mathbf{1}_{p}/(\mathbf{1}_{p}^{\mathsf{T}} \Sigma \mathbf{1}_{p})$, where $\Sigma$ is the population covariance matrix of all assets used for the investment. The optimal portfolio is therefore invariant to scaling in the covariance matrix. This inspires us to conduct the following sphericity hypothesis testing problem:

where $k^2$ is an unknown scalar parameter.

For the sphericity test, in traditional multivariate analysis based on the sample covariance matrix in $\eqref{S_n}$, when the dimension $p$ is fixed, the classical likelihood ratio test statistic can be found in Section 8.3.1 in M1982, and J1971 proposed a test statistic by using the self-normalized sample covariance matrix and its spectrum. In high-dimensional situations, similarly to the discussion in Section (ref), the former one is only valid when the dimension is smaller than the sample size, while the latter one does not have such a restriction. Thus, the statistic in J1972 is more favorable and has been extensively studied in recent years: See, for example, LW2002, WY2013, TLL2015 for the linear transform model, which corresponds to taking $\gamma_t$ in (ref) as a constant; ZPFW2014, PV2016 for the elliptical model if replacing $\gamma_t$ in (ref) by a random variable independent of time $t$. Similarly, we will modify the test statistic in J1972 under our high-frequency setting in large dimension. We do not consider other statistics and only note that many other ones can be found in HLLZ2019, YZC2021 and references therein.

We recall that the test statistic in J1972 based on the sample covariance matrix (ref) is given by

For the spot volatility matrix estimator using the high-frequency data, after replacing $S_n$ with $\widehat{c_{t}}^{n}$, we can obtain

For the above estimator, we have

Proof. The conclusion can be obtained by following the proof of Theorem (ref) and using Proposition 3 in LW2002. $\hfill \square$

With Theorem (ref), we can obtain the following statistic for the sphericity testing problem:

Simulation studies

In this supplementary material, we conduct some simulation studies to evaluate the finite sample performance of the realized spot volatility matrix estimator and verify the theoretical results established in the paper.

With the underlying efficient log-price process $\{X_t\}_{0 \leq t\leq 1}$ generated from (ref), we consider two different scenarios for the time-varying volatility process, deterministic or stochastic, with the following forms:

and

In the above models, $W$ is a one-dimensional standard Brownian motion independent of $B$, $B^{(1)}$ is the first entry of $B$ and $\rho$ controls the correlation strength between $X$ and $\sigma$, $\tilde{b} \in \mathbb{R}$ and $\tilde{\sigma} \in \mathbb{R}$ are the drift process and volatility process of the volatility process $\sigma$ respectively. For simplicity, we let $\tilde{\sigma} \equiv \text{r}_2$, which is constant, and $b, \tilde{b} \equiv 0$, since the drift processes play no roles asymptotically. The deterministic volatility and corresponding parameter setting are also considered in ZL2011 for experimental study. For the stochastic volatility model, It$\hat{\text{o}}$ semimartingale like $X$ is used and is popular in high-frequency literature. We consider different setting for $\text{r}_1 = 0, 0.0004, 0.0008$ in (ref) and $\text{r}_2= 0, 0.01, 0.02$ in (ref), these parameters control the variational intensity of the volatility process. Without loss of generality, we focus our analysis at the time point $t=0$ and fix $n=4680$, corresponding to 5-second observational data within a 6.5-hour trading day in the real financial market. We consider $k_n = \lfloor \sqrt{n} \rfloor= 68$ and $p/k_n \equiv \overline{p}$ for different values of $\bar{p} = 0.5 ,1, 1.5$ (Correspondingly, $p=34,68,102$).

We first present some simulation studies to illustrate the behavior of the empirical spectral distribution of the realized spot volatility matrix estimator, for both the deterministic volatility model (ref) and the stochastic volatility model (ref). In Figure (ref)--(ref), we use red solid lines to represent the cumulative distribution function of Mar$\check{\text{c}}$enko-Pastur law provided by Theorem (ref), which is also the limiting spectral distribution of the realized spot volatility matrix. Moreover, results for different dimensions $p=34,68,102$ are reported when multiple variational parameters $r_1$ and $r_2$ are used for the deterministic and stochastic volatility models, respectively. Furthermore, different levels of correlation strength $\rho=0.4, 0.8$ is also considered for the stochastic volatility model. From the figures, we see that all the empirical spectral distributions are close to corresponding theoretical Mar$\check{\text{c}}$enko-Pastur law, which verifies the conclusions in Theorem (ref).

Next, we consider the following hypothesis testing problem:

or equivalently

by using the test statistics proposed in Section (ref), and use the deterministic volatility (ref) for illustration. With fixed $r_1 = 0.0008$, we generate a total number of 1000 sample paths and calculate “BJYZ-test-spot" statistic for $\bar{p}=0.5$ and “LW-test-spot" statistic for $\bar{p} = 0.5, 1, 1.5$. The Q-Q plots of the resulting estimates are shown in Figure (ref)--(ref). We see that the distributions of the estimates are close to standard normal distribution, which verifies the central limit theorems established in Theorem (ref) and Theorem (ref). For the hypothesis testing problem (ref), the empirical sizes of “BJYZ-test-spot" and “LW-test-spot" statistics at various significance levels are presented in Table (ref). We observe that when the null hypothesis is true, the sizes of all test statistics are close to corresponding nominal levels. Moreover, for $\bar{p} = 0.5$, we see that “BJYZ-test-spot" generally outperforms “LW-test-spot", which suggests that the former one should be used if we have prior knowledge of $\bar{p}<1$. For both test statistics under different settings of significance level and $\bar{p}$, the size performance is relative worse for larger value of $r_1$. This is to our expectation since as the magnitude of $r_1$ increases, the sample data will gradually deviate from the identically distributed case. For the evaluation of power performance, we consider the alternative hypothesis of $c_0 = \text{diag}(0.0009\mathbb{I}_{\lfloor sp \rfloor}, 0.0004\mathbb{I}_{p-\lfloor sp \rfloor})$ for $s=0.45, 0.6, 0.75$, namely the diagonal matrix $c_0$ has respectively $s$ of its diagonal elements being 0.0009 whereas the rest $(1-s)$ part are 0.0004. Meanwhile, the same data generating scheme as model (ref) is used with $r_1=0, 0.0002, 0.0004$. Table (ref) presents the finite sample performances of the test statistics under different settings. The simulation results demonstrate satisfying power performance when $s$ is not larger than 0.6. Besides, we can also see that “BJYZ-test-spot" generally outperforms “LW-test-spot" when $\bar{p} = 0.5$, and for both statistics, the power result is relatively worse for larger value of $r_1$. These findings have already been obtained for size performance, and the reasons are same. We use the same model and settings for “J-test-spot" statistic proposed in Section (ref) for the sphericity test, and present the Q-Q plots of the estimates in Figure (ref), when $\bar{p} = 0.5, 1, 1.5$. We see that the distributions of the estimates are close to standard normal distribution, which verifies the central limit theorems established in Theorem (ref). As for the sphericity hypothesis testing problem of (ref), the same statistic was considered in YZC2021 and evaluated with simulation studies under class $\mathcal{C}$ (ref), thus we do not repeat the experiment.

Conclusion and future work

In this article, we conduct spectral analysis for the realized spot volatility matrix estimator by using the high-dimensional and high-frequency data, including its limiting spectral distribution and linear spectral statistics. Our theoretical results demonstrate that, under some general conditions, the limiting spectral distributions of the spot volatility matrix and the realized spot volatility matrix estimator are governed by a regular Mar$\check{\text{c}}$enko-Pastur equation. Furthermore, the central limit theorem for linear spectral statistics of the realized spot covariance matrix estimator is established. The result is similar to the one for independent and identically distributed sample in classical large-dimensional random matrix theory. As applications, we perform identity and sphericity tests for the spot volatility matrix and propose some test statistics by using our established theory. Then, the theoretical conclusions and the finite sample performance of the proposed statistics are justified by our simulation studies.

We note that the spectral theory developed for the spot volatility matrix in this paper could potentially be used to solve some other problems. For example, we may consider to test if $c_{t_1} = c_{t_2}$ for $0 \leq t_1< t_2\leq 1$. Another problem is that, in ZL2011 and YZC2021, the spot volatility matrix is restricted to satisfy a special structure of class $\mathcal{C}$ as in (ref), while such a structure remains to be testified by using the real data. From a high-frequency perspective, we may extend the analysis to the scenario with the presence of both jumps and market microstructure noise. {In this paper, we do not consider the jumps for simplicity. Nevertheless, as studied by Jacod and Protter (2012) (Theorem 13.3.3), the spot volatility estimator remains consistent when the jumps are not too frequent. For a general L$\acute{\text{e}}$vy jump part, a truncated version for the estimator is available, we refer to Jacod and Protter (2012) for more details.} Those mentioned problems are out the scope of this paper and will be considered in our future works.