A nonparametric test for diurnal variation in spot correlation processes

A nonparametric test for diurnal variation in spot correlation processes

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Introduction

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Correlation percolates through financial economics. It is a critical ingredient in the determination of optimal portfolio weights in a markowitz:52a mean-variance asset allocation problem, where the asset return correlations also determine a lower bound on diversification. Moreover, the correlation between the return of an asset and the return of the market portfolio is paramount in single- and multi-factor capital asset pricing models fama-french:15a, sharpe:64a, where it is used to calculate the so-called beta, which is an important driver of the premium over the risk free rate earned as a compensation by investing in the risky asset. In addition, correlation is also employed in risk management and hedging.

It has long been recognized that correlations are time-varying, and the vast majority of parametric models to describe interday correlation allow it to change dynamically engle:02a, noureldin-shephard-sheppard:12a. The properties of the correlation process have also been traversed in detail with nonparametric analysis from high-frequency data. This is typically done by studying a realized measure of the daily integrated covariance, which is mapped into a correlation estimate, e.g. ait-sahalia-fan-xiu:10a and boudt-cornelissen-croux:12a.

Surprisingly, relatively little is known about the behavior of correlation at the intraday horizon. This stands in sharp contrast to the volatility of individual equity returns that is known to evolve as a U- or reverse J-shaped curve with notably higher volatility near the opening and closing of the stock exchange than around noon harris:86a, wood-mcinish-ord:85a. Several estimators of the intraday volatility curve have emerged over the years, e.g. andersen-bollerslev:97b, andersen-bollerslev:98b propose a parametric model for periodicity in volatility, whereas boudt-croux-laurent:11b and christensen-hounyo-podolskij:18a develop nonparametric jump- and microstructure noise-robust estimators from high-frequency data that verify the existence of a pervasive structure in the intraday volatility.

The most common setup for describing the dynamic of spot volatility of an asset log-return at the interday and intraday horizon is a multiplicative time series model:

where $\sigma_{sv,t}$ is a stationary process meant to capture stochastic volatility, whereas $\sigma_{u,t}$ is a deterministic component intended to capture diurnal variation and assumed to be a constant time-of-day factor (i.e., $\sigma_{u,t}= \sigma_{u,t-1}$).\footnote{In recent work, andersen-thyrsgaard-todorov:19a suggest that the intraday volatility curve may be time-varying, see also andersen-su-todorov-zhang:24a.}

In a bivariate setting, any systematic evolution in the volatility is automatically transferred to the covariance process, $c_{t}^{XY} = \sigma_{t}^{X} \sigma_{t}^{Y} \rho_{t}$, where $\sigma_{t}^{X}$ and $\sigma_{t}^{Y}$ represent the spot volatility of asset $X$ and $Y$, whereas $\rho_{t}$ is their correlation. If the individual return variation of $X$ and $Y$ follows (ref), the covariance inherits an “imputed” diurnal pattern:

However, observing (ref) suggests that there may be an additional source of diurnal variation in the covariance, since the dynamic of the spot correlation further affects it. As in (ref), we can capture a recurrent behavior in the spot correlation as follows:

where $\rho_{sc,t}$ and $k_{u,t}$ are interpreted as above. In the modified setting of (ref), the breakdown of the covariance into its component parts is now given by

To the extent that correlations vary systematically within a day, we should expect the actual and imputed diurnal covariance curve to deviate bibinger-hautsch-malec-reiss:19a. To get a first impression of this, we begin with an inspection of Panel A in Figure (ref) in our empirical application in Section (ref), where we compare the average imputed and actual intraday covariance curve calculated pairwise for all constituents of the Dow Jones Industrial Average and a proxy for the market portfolio of aggregate movements in the U.S. equity market over the sample period 2010--2023. We observe a striking discrepancy between the two, most notably in the early morning and late afternoon. This provides strong evidence of this effect in the high-frequency data. Looking at it in terms of the correlation process in Panel B of the figure, we locate a very significant upward-sloping intraday correlation curve, which increases monotonically during the trading session in an almost piecewise linear fashion. This is consistent with allez-bouchaud:11a and concurrent work of hansen-luo:23a. There are large jumps in the correlation around the release of macroeconomic information, which corresponds to an influx of systematic risk to the market.\footnote{The presence of diurnal variation in the correlation also has implications for the parametric modeling of intraday spot covariance. In particular, one has to account for this effect to extract the stationary component of the covariance process. A “naive” approach with the imputed diurnal covariance based on the idiosyncratic intraday volatility curve---amounting to asset-wise deflation---is insufficient to get a covariance free of systematic intraday evolution.}

In this paper, we construct a testing procedure to detect diurnal variation in a correlation process. It distills local estimates of the spot correlation, after the high-frequency return series has been devolatized to remove the effect of idiosyncratic volatility (both deterministic and stochastic), thus isolating the correlation process, while also controlling for potential price jump variation. If there are systematic changes in the spot correlation estimates, the test statistic grows large and rejects the null hypothesis of no diurnal variation. This is related to, but different from, previous work by reiss-todorov-tauchen:15a for testing a constant beta. Overall, in our empirical high-frequency data, we implement the test statistic on a month-by-month basis and find that the proposed test statistic rejects the null hypothesis most of the times, thus confirming the circumstantial evidence from Figure (ref). Furthermore, we provide anecdotal evidence about how macroeconomic news and corporate earnings announcements affect the intraday correlation curve.

To highlight the exploitation of predictable dynamics in the correlation, we adopt the standpoint of a trader who hedges a long exposure in single stocks via the market portfolio. We report a nontrivial effect by incorporating diurnal correlation into the risk management process, relative to ignoring it, yielding a drop in combined portfolio variance of about twenty percent. It also delivers a much more stable hedge ratio during the course of the trading day, helping to reduce transaction costs derived from warehousing the risk.

The roadmap of the paper is as follows. In Section (ref), we present the model and list the assumptions required to extract an intraday correlation curve from a bivariate time series of high-frequency data. In Section (ref), we develop our point-in-time correlation estimator. In Section (ref), we propose a testing procedure, which can be employed to uncover the existence of diurnal variation in the correlation process. We derive the required asymptotic distribution theory, which is based on a functional central limit theorem. In Section (ref), we elaborate on the relaxation of a crucial assumption. We also show how our framework can be extended to a conditional version that incorporates relevant information that may help to determine the functional form of the diurnal correlation curve. In Section (ref), we inspect the small sample attributes of our framework via Monte Carlo simulation. In Section (ref), we apply it to a large panel of equity data. In Section (ref), we conclude. We relegate proofs and supplemental results to the Appendices.

Theoretical setup

We suppose a filtered probability space $( \Omega, \mathcal{F}, (\mathcal{F}_{t})_{t \geq 0}, P)$ describes a bivariate continuous-time log-price process $Z = (X,Y)^{ \top}$, where $( \mathcal{F}_{t})_{t \geq 0}$ is a filtration and $^{ \top}$ is the transpose operator.\footnote{Our analysis extends to $d$-dimensional processes in an obvious fashion.} $Z$ is observed on $[0,T]$, where $T$ is the number of days in the sample and the subinterval $[t-1,t]$ is the $t$th day, for $t = 1, \ldots, T$. We assume $Z$ is recorded discretely at the equidistant time points $t_{i} = t-1+i/n$, for $i=0,1, \ldots, n$, so a total of $nT$ increments are observed with a time gap of $\Delta = 1/n$. Throughout, the asymptotic theory is infill and long-span, i.e. we look at limits in which the time gap between consecutive observations goes to zero ($\Delta \rightarrow 0$ or $n \rightarrow \infty$) and the sample period increases ($T \rightarrow \infty$).

In absence of arbitrage (or rather a free lunch with vanishing risk) $Z$ is a semimartingale delbaen-schachermayer:94a. We suppose $Z$ is of the It\^{o}-type, which is a process with absolutely continuous components. Then, we can write the time $t$ value of $Z$ as follows:

where $Z_{0} = (X_{0},Y_{0})^{ \top}$ is $\mathcal{F}_{0}$-measurable,

where $(a_{t})_{t \geq 0}$ is a predictable and locally bounded drift, $( \sigma_{t})_{t \geq 0}$ is an adapted, c\`{a}dl\`{a}g volatility matrix, while $(W_{t})_{t \geq 0}$ is a bivariate standard Brownian motion with $\langle W^{X}, W^{Y} \rangle_{t} = 0$, where $\langle \cdot, \cdot \rangle$ denotes the predictable part of the quadratic covariation process.

$J_{t}$ is a pure-jump process, for which we impose the following restriction. \\[-0.25cm]

Assumption (J): $\quad$ $J_{t}^{ \wp}$ is such that

where $\mu^{ \wp} $ is an integer-valued random measure on $\mathbb{R}_{+} \times \mathbb{R}$ with compensator $\nu^{ \wp} ( \mathrm{d}t, \mathrm{d}x) = \chi_{t}^{ \wp} \mathrm{d}t \otimes F^{ \wp}( \mathrm{d}x)$, $\chi_{t}^{ \wp}$ is an adapted c\`{a}dl\`{a}g process, and $F^{ \wp}$ is a measure on $\mathbb{R}$. Here, and in the remainder of the article, the superscript $\wp$ notation is used to represent that the derived stochastic process is associated with $\wp$, where $\wp$ is either $X$ or $Y$. \\[-0.25cm]

We also assume that the stochastic volatility processes are It\^{o} semimartingales. \\[-0.25cm]

Assumption (V): $\quad$ $\sigma_{t}^{ \wp}$ is of the form:

where $( \tilde{a}_{t}^{ \wp})_{t \geq 0}$, $( \tilde{ \sigma}_{t}^{ \wp})_{t \geq 0}$, $( \tilde{ \nu}^{ \wp})_{t \geq 0}$, are adapted, c\`{a}dl\`{a}g stochastic processes, $\tilde{W}_{t} = ( \tilde{W}_{t}^{X}, \tilde{W}_{t}^{Y})^{ \top}$ is a bivariate standard Brownian motion, independent of $W$, but such that $\tilde{W}_{t}^{X}$ and $\tilde{W}_{t}^{Y}$ can be correlated. At last, $\tilde{ \mu}^{ \wp}( \mathrm{d}t, \mathrm{d}x)$ is the jump counting measure of $\sigma_{t}^{ \wp}$ with compensator $\tilde{ \chi}^{ \wp}_t \mathrm{d}t \otimes \tilde{F}^{ \wp}( \mathrm{d}x)$, where $\tilde{ \chi}_{t}^{ \wp}$ is an adapted c\`{a}dl\`{a}g process, and $\tilde{F}^{ \wp}$ is a measure on $\mathbb{R}$. \\[-0.25cm]

The above constitutes a more or less nonparametric framework for modeling arbitrage-free price processes, which accommodates most of the models employed in practice. Mainly, we exclude semimartingales that are not absolutely continuous, but this is not too restrictive.\footnote{An example of a continuous local martingale that has no stochastic integral representation is a Brownian motion time-changed with the Cantor function (or devil's staircase), see ait-sahalia-jacod:18a and barndorff-nielsen-shephard:04a.} Note that we integrate over the jump size distribution directly with respect to the Poisson random measure. Hence, we are assuming that the jump processes are of finite variation.\footnote{In general, the Poisson random measure needs to be compensated (i.e. converted to a martingale) for jump processes of infinite variation to ensure that the summation (over the small jumps) is convergent.} They may be infinitely active, but they should be absolutely summable. We add more regularity to the jump processes below. Furthermore, Assumption (V) excludes the possibility that volatility can be rough, e.g. that it is driven by a fractional Brownian motion with a Hurst exponent less than a half, which has been a recurrent theme in the recent literature bolko-christensen-pakkanen-veliyev:23a, fukasawa-takabatake-westphal:22a, gatheral-jaisson-rosenbaum:18a, shi-yu:23a, wang-xiao-yu:23a.

It is possible to expand our results to a more general setting. For instance, to cope with infinite variation jumps we can apply the Laplace transform-based estimator of liu-liu-liu:18a or the debiased truncation-based estimator in boniece-figueroa-lopez-zhou:25a. To handle roughness, we can rely on estimators of spot volatility that are robust to this assumption, such as the Fourier transform-based estimator from mancino-mariotti-toscano:24a or the truncation-based estimator of christensen-thyrsgaard-veliyev:19a. Then, we can directly plug-in such consistent estimators into our diurnal correlation framework. However, we do not pursue these extensions here.

In the maintained framework, the continuous part of the quadratic covariation process of $Z$ is absolutely continuous with respect to the Lebesgue measure, so it has a derivative:

and instantaneous correlation:

where $\wp^{c}$ is the continuous part of $\wp$.

We need to make some additional assumptions, starting with one for the correlation reminiscent to equation (ref) for the stochastic volatility process. \\[-0.25cm]

Assumption (C1): $\quad$ The spot correlation $\rho_{t}$ factors as:

where $\rho_{sc,t}$ is a stochastic process and $k_{u,t}$ is a deterministic component. \\[-0.25cm]

In Assumption (C1) only the left-hand side of (ref) is identified, so the scale of one of the terms on the right-hand side needs to be fixed. We add such an identification condition in Assumption (C2). Furthermore, note that as the diurnal component is not a correlation in itself, there is nothing to stop it from venturing outside $(-1,1)$, so long as the overall product of the diurnal and stochastic component does not.

In view of equation (ref) and (ref), the spot covariance is the product of a stochastic process and a deterministic component, where the latter captures diurnal variation:

Note that for $X=Y$, $k_{u,t} = \rho _{sc,t} = 1$. Hence, our paper generalizes christensen-hounyo-podolskij:18a to a multivariate context.

In view of Assumption (C1), the spot covariance matrix factors as follows:

where $\odot$ denotes the Hadamard product.

We further impose that: \\[-0.25cm]

Assumption (C2): $\quad$ $( \sigma_{u,t}^{ \wp})_{t \geq 0}$ and $( k_{u,t})_{t \geq 0}$ are bounded, Riemann integrable, one-periodic functions such that $\int_{t-1}^{t} \sigma_{u,s}^{X} \sigma_{u,s}^{Y} k_{u,s} \mathrm{d}s = 1$. \\[-0.25cm]

Assumption (C3): $\quad $ $\sigma _{sv,t}^{ \wp} > 0,$ $\sigma _{u,t}^{ \wp} > 0$, $\rho_{sc,t} \neq 0$ and $k_{u,t} \neq 0$, for all $t \geq 0$ except on a set with Lebesgue measure zero. \\[-0.25cm]

Assumption (C2) adds some regularity on $\sigma_{u,t}^{ \wp}$ and $k_{u,t}$. The requirement on the definite integral of the diurnal covariance function is a natural generalization from the univariate framework, where it reduces to the standard identification condition $\int_{t-1}^{t} ( \sigma_{u,s}^{ \wp})^{2} \mathrm{d}s = 1$. We also suppose that $\sigma_{u}^{ \wp}$ and $k_{u}$ are recurrent, i.e. $\sigma_{u,t}^{ \wp} = \sigma_{u,t-1}^{ \wp}$ and $k_{u,t} = k_{u,t-1}$ for all $t\geq 1$, so that these functions are consistently estimable from a long enough sample of high-frequency data. While the latter is not uncommon in the literature, it is a strong assumption that encounters problems in practice, since empirical evidence suggests that the intraday volatility curve may be time-varying andersen-thyrsgaard-todorov:19a. We relax this part of the assumption in Section (ref) to allow for much more general dynamics in these processes. Assumption (C3) presupposes that both correlation components are bounded away from zero, except on a set of Lebesgue measure zero, since we evidently cannot identify $\rho_{sc,t} \neq 0$ if $k_{u,t} = 0$, and vice versa. The condition allows the correlation process to cross zero in a continuous fashion, provided it does not get “stuck” at the origin. For example, this holds if the driving force of the stochastic correlation is a Brownian motion, for which the zero set is uncountably infinite but of Lebesgue measure zero.

As our asymptotic theory is based on both $n \rightarrow \infty$ and $T \rightarrow \infty$, we cannot activate the localization procedure for high-frequency data described in jacod-protter:12a to bound various processes, so instead we impose a related condition: \\[-0.25cm]

Assumption (C4): The drift term $a^{\wp}$ is Lipschitz continuous (in mean square), i.e. $E \big[|a_{t}^{ \wp} - a_{s}^{ \wp}|^{2}] \leq C|t-s|$, for any $s, t\in[0, \infty)$ and a positive constant $C$ (that does not depend on $s$ and $t$),

Moreover, $F^{ \wp}( \mathbb{R}) < \infty$, $\tilde{F}^{ \wp}( \mathbb{R}) < \infty$, $\int_{ \mathbb{R}} |x|^2 \tilde{F}^{ \wp}( \mathrm{d}x) < \infty$, and

Assumption (C4) follows Assumption I of andersen-su-todorov-zhang:24a for the univariate case; see also Assumption 1 of andersen-tan-todorov-zhang:25a. It restricts the jump processes to be of finite activity, but this can be relaxed, as shown in the Supplementary Appendix of their paper. Moreover, the moment conditions are also stricter than necessary.

The last set of assumptions concerns the stationarity and ergodicity of the stochastic volatility and correlation processes. \\[-0.25cm]

Assumption (C5): For any positive integer $s > 0$ and $\tau \in [0, 1)$, $\sigma_{sv,s-1+ \tau}^{ \wp}$ and $\rho_{sc,s-1+ \tau}$ are functions (depending on $\tau$) of $M_{s-1+ \tau}$, where $(M_{t})_{t \geq 0}$ is a multivariate Markov process, which is stationary, ergodic and $\alpha$-mixing with mixing coefficient

where $\mathcal{G}_{t} = \sigma(M_{u} \mid u \leq t)$ and $\mathcal{G}^{t} = \sigma(M_{u} \mid u \geq t)$ are the “backward”- and “forward”-looking $\sigma$-algebras, such that $\alpha_{s} = O(s^{-q- \ell})$ for some $q>0$ and an arbitrarily small constant $\ell > 0$. \\[-0.25cm]

Assumption (C5) follows Assumption II of andersen-su-todorov-zhang:24a and Assumption $H_{0}$ in the recent contribution of andersen-tan-todorov-zhang:25a. The astute indexation ensures that subsets of the volatility and correlation, separated by an integer-valued index set, can be time-dependent through a transformation of a multivariate Markov process. The remaining parts are standard regularity conditions for inference with weakly dependent processes. In particular, the decay rate $q$ of the sequence of mixing coefficients is restricted further to establish consistency and, more so, for a functional CLT.

Assumptions (C1) -- (C5) are sufficient to identify both volatility and correlation components $\sigma_{sv,t}^{ \wp }$, $\sigma_{u,t}^{ \wp}$, $\rho_{sc,t}$, and $k_{u,t}$.

To construct our hypothesis we partition the sample space $\Omega$ into

and $\Omega_{ \mathcal{H}_{a}} = \Omega _{ \mathcal{H}_{0}}^{ \complement}$. The null is then defined as $\mathcal{H}_{0}: \omega \in \Omega _{\mathcal{H}_{0}}$, i.e. it consists of paths with no diurnal correlation. The alternative is $\mathcal{H}_{a}: \omega \in \Omega_{ \mathcal{H}_{a}}$. As usual in time series analysis, the premise here is that we cannot repeat the experiment. We can access discrete high-frequency data from a single path. On this basis, the goal is to decide which subset our realization lies in. We note that an equivalent representation of null hypothesis is the following: $\Omega_{ \mathcal{H}_{0}} = \{ \omega : \int_{0}^{1}( k_{u,t}-1)^{2} \mathrm{d}t = 0\}$.

Spot correlation estimator

To implement our testing procedure, we first need an estimator of the spot correlation coefficient, which we construct from a standard localized estimator of the continuous part of the quadratic covariation process.

We represent the log-price increments of $Z$ as follows:

for $t = 1, \ldots ,T$ and $i=1, \ldots, n$.

The road forward is to split the sample into smaller blocks consisting of $k_{n}$ log-price increments. We suppose $k_{n}$ is a divisor of $n$ for notational convenience, which implies that there are $n/k_{n}$ blocks per day. Over the $j$th block on day $t$, we define $\tau_{j} = \frac{j-1}{n/k_n}$ and set

for $t = 1, \dots, T$ and $j = 1, \dots, n/k_{n}$, $\mathcal{A}_{t, \tau_{j}}^{ \wp, n}= \{| \Delta_{(t-1)n + \ell}^{n} \wp| \leq v_{n,t,j}^ {\wp} \}$, with

where $\alpha_{n,t,j}^{ \wp} = \alpha^{ \wp} BV_{n,t,j}^{ \wp}$ such that $\alpha^{ \wp } > 0$, $\varpi \in (0,1/2)$, and

Equation (ref) is the realized covariance of barndorff-nielsen-shephard:04a upgraded with the truncation device of mancini:09a. The latter removes returns that originate from the jump component of the log-price process. This ensures that $\hat{c}_{t, \tau_{j}}$ is consistent for the continuous part of the quadratic covariation, i.e. integrated covariance, over the block. The threshold is a function of a localized bipower variation estimator barndorff-nielsen-shephard:04b, so the truncation is time-varying and adapts to the level of intraday volatility. This is important, because failure to capture the dynamic of the volatility process can cause problems for inference boudt-croux-laurent:11b.

It is convenient to work with a statistic defined on the whole interval $[0,T]$, which we do by setting $\hat{c}_{t, \tau} \equiv \hat{c}_{t,\tau_j}$, for $\tau \in [\tau_j, \tau_{j+1})$.

To proceed, we estimate the intraday curve in the spot covariance and transform this into an estimate of the diurnal component in the correlation process. We propose to scale an estimator targeting the average spot covariance at a particular time-of-the-day with another estimator of the unconditional covariance over the whole day, where the latter serves as a normalization to adhere to Assumption (C2), i.e.

with

where $A \oslash B$ is the Hadamard division.

An estimator of the deterministic component of the intraday correlation is the following:

It is worthwhile to note that $\hat{k}_{u, \tau}$ can equivalently be written as

where

The next result derives the probability limit of the various estimators.

The proof relies on a double-asymptotic setting with $n \rightarrow \infty$ and $T \rightarrow \infty$. Intuitively, to retrieve the stationary expectation of the covariance process, the time horizon has to increase. In this regard, the requirement on the memory of the process is rather weak and merely states that the autocorrelation function has to be absolutely summable. As $n \rightarrow \infty$, on each block the realized covariance converges to the integrated covariance. The condition $k_{n} \rightarrow \infty$ with $n / k_{n} \rightarrow \infty$ says that we reduce the time span of such a block at a sufficiently slow rate so there is an accumulation of log-returns inside each estimation window. Taken together, this implies that realized covariance collapses to the latent point-in-time covariance and---after conversion---that our estimator of the diurnal component of the spot correlation process is consistent.

Testing procedure

In this section, we construct our testing procedure to discriminate between the null and alternative hypothesis. We develop a test statistic that accommodates the general setting for the spot covariance process (as outlined in Assumptions (C1) -- (C5)).

Test statistic

We begin with a preliminary functional central limit theorem (CLT) concerning the asymptotic distribution of the diurnal covariance estimator from (ref). We define the Hilbert space:

equipped with the usual inner product $\langle \cdot\,, \cdot \rangle$ and the induced norm $\left\lVert \cdot\right\rVert$. We use the notation $x_{n} \asymp y_{n}$ to represent that, as $n \rightarrow \infty$, $1/C \leq x_{n} / y_{n} \leq C$ for some positive constant $C$.

This theorem extends Theorem 1 of andersen-su-todorov-zhang:24a from the univariate to the multivariate setting. Compared to Theorem (ref), we impose a faster rate of decay on the sequence of mixing coefficients.

In Assumption (C5), we require the random component of the correlation process to follow the same stationarity condition imposed on the volatility process. Consequently, the random component of the covariance process satisfies this condition, allowing us to select the orders of $T$ and $k_{n}$ as in the univariate case. Condition (ref) further restricts the growth of $T$ and $k_{n}$ relative to $n$. Such constraints also appear in closely related work on long-span estimation with high-frequency data; see, e.g., equation (9) in andersen-su-todorov-zhang:24a and equation (5) in andersen-tan-todorov-zhang:25a. As they explain, when the truncation parameter $\varpi$ is set close to $1/2$, the resulting bounds on $c$ and $d$ are weakest. That is, the choice of $c$ can be any number in (0,2), making the length of the time period very flexible. Moreover, once $c$ is chosen, the optimal choice of $d$ has been discussed in andersen-su-todorov-zhang:24a. In particular, when $c>1/2$, the optimal choice of $d$ is $(2-c)/3$. On the other hand, if we take the optimal convergence rate for spot volatility; namely, if $d$ is close to $1/2$, then $c \leq 1$, indicating that $T$ cannot grow faster than $n$, implying that high-frequency sampling should increase at least as fast as the time span. We refer to Section 5 of andersen-su-todorov-zhang:24a for a detailed discussion of the bias-variance tradeoff.

By applying the functional delta rule to (ref) with $g(x,y,z) = y(xz)^{-1/2}$, it follows that

where, as shown in Appendix (ref),

Hence, it follows that under the null hypothesis (where $k_{u, \tau} \equiv 1$):

Now, we propose our test statistic:

The next theorem helps to explain the behavior of $\mathcal{N}^{ \mathrm{inf.}}$.

Theorem (ref) implies that $\mathcal{N}^{ \mathrm{inf.}} \rightarrow \infty$ under $\mathcal{H}_{a}$, so a test based on it is consistent. Note that part (a) of the theorem holds irrespective of whether $k_{u,t} = 1$ (i.e., there is no diurnal variation in the correlation) or not.

The asymptotic variance matrix is latent and has to be replaced with an estimator. Note that $\mathcal{H}$ is a mean zero Gaussian process with covariance kernel:

According to Theorem (ref), we can estimate $c_{u, \tau}^{X}$, $c_{u, \tau}^{XY}$ and $c_{u, \tau}^{Y}$ with $\hat{c}_{u, \tau}^{X}$, $\hat{c}_{u, \tau}^{XY}$ and $\hat{c}_{u, \tau}^{Y}$, respectively, and likewise for terms with index $\kappa$. We propose a standard HAC-based estimator of $\Gamma_{ \kappa, \tau}$:

where

$H_{T}$ is the lag length, $\omega$ is a kernel andrews:91a, and

It should be noted that the expectation of $A_{t, \kappa}^{Z}$ is zero for $h = 0, \dots, H_{T}$. The following result then gives the consistency of $\widehat{ \Gamma}_{ \kappa, \tau}$.

Hence, we arrive at the following estimator of $C( \kappa, \tau)$:

Now, define $\widehat{ \mathcal{H}}$ to be an $\mathcal{F}$-conditional $\mathcal{L}^{2}$-valued mean zero Gaussian process with covariance kernel $\widehat{C}$, as defined in (ref). We can then show that $\widehat{ \mathcal{H}}$ converges in law to $\mathcal{H}$ (in $\mathcal{L}^{2}$).

The CLT in Theorem (ref) again generalizes the associated Theorem 6 in andersen-su-todorov-zhang:24a to the multivariate case. Compared to Theorem (ref), it imposes the additional rate conditions $c + d > 1-16 \varpi / 7$ and $d > (3-8 \varpi)/3$. The requirement $c>(3-8 \varpi)/3$ is stronger than $d > 1-4\varpi$ in Theorem (ref), but for $\varpi \geq 3/8$ it is automatically satisfied. The same observation applies to the condition $c + d > 1-16 \varpi / 7$.

We can simulate the asymptotic distribution of the nonpivotal test statistic, $\left\lVert \mathcal{H}\right\rVert^{2}$. We partition the interval $[0,1]$ into $m$ subintervals of equal length, where $m = n/k_{n}$, and consider an $m$-dimensional normal random vector $( \widehat{ \mathcal{H}}_{ \tau_{1}}, \dots, \widehat{ \mathcal{H}}_{ \tau_{m}})^{ \top}$ with mean zero and conditional covariance matrix $\widehat{C} = ( \widehat{C}_{ \tau_{i}, \tau_{j}})_{1 \leq i,j \leq m}$, where $\tau_{j} = j/m$ for $j=1, \dots, m$. Next, observe that

where $( \lambda_{j})_{j=1}^{m}$ are the eigenvalues of $\widehat{C}$ and $( \chi_{j}^{2})_{j=1}^{m}$ are independent $\chi^{2}(1)$-distributed random variates, defined on an extension of the original probability space and independent from $\mathcal{F}$. Since $\widehat{C}$ is an estimate of a covariance matrix, it can possess negative eigenvalues in practice. We therefore follow andersen-su-todorov-zhang:24a and retain only those terms in (ref) that are associated with positive eigenvalues. The above process delivers one possible outcome and can be repeated as many times as necessary to get an acceptable approximation to the law of $\left\lVert \mathcal{H}\right\rVert^{2}$.

Extensions

Stochastic diurnal correlation

In Assumption (C1), we restricted the intraday curve in the correlation process to be deterministic. To allow for a more general structure that incorporates stochastic diurnal correlation, we follow andersen-su-todorov-zhang:24a and suppose instead that for $Z \in \{ X, Y, XY \}$,

In contrast to before, (ref) only restricts the calender effect in correlation to be present in expectation. This implies that, on average, the function is periodic as in Assumption (C2), such that it remains consistently estimable in the in-fill and long-span limit by ergodicity.

The diurnal correlation function is now given by:

We can construct a test of the hypothesis

In this setting, it readily follows that

Moreover, the following theorem establishes a functional CLT.

We propose the following infeasible test statistic:

where $\bar{k}_{u} = \frac{1}{n/k_{n}} \sum_{j=1}^{n/k_{n}} \hat{k}_{u, \tau_{j}}$. It has the following properties.

Again, we can design a standard HAC-based estimator of $\Gamma_{ \kappa, \tau}$:

where

with $H_{T}$ and $\omega$ defined as above. Accordingly, we can construct an estimator of the covariance kernel as follows:

As before, we can simulate the asymptotic distribution of the test statistic by partitioning the interval $[0,1]$ into $m$ subintervals of equal length, where $m = n/k_{n}$. We generate the $m$-dimensional normal random vector $( \widehat{ \mathcal{H}}_{ \tau_{1}}, \dots, \widehat{ \mathcal{H}}_{ \tau_{m}})$ with mean zero and conditional covariance matrix $\widehat{C} = ( \widehat{C}_{ \tau_{i}, \tau_{j}})_{1 \leq i,j \leq m}$, now based on (ref), where $\tau_{j} = j/m$ for $j=1, \dots, m$. Next, we set

In the context of random diurnal volatility and correlation, the decompositions in (ref) and (ref) are lost, and the identification condition in Assumption (C2) becomes meaningless. So the hypothesis $k_{u,t} \equiv k_{u}$ (a constant) for $t \in (0,1)$ does not imply that $k_{u,t} \equiv 1$. Therefore, we employ the equivalent condition of $k_{u,t} \equiv k_{u}$ for $t \in (0,1)$, namely $k_{u,t} - \int_{0}^{1} k_{u,t} \mathrm{d}t \equiv 0$, to create the modified test statistic in (ref), which is different from the previous one. That being said, although the new test statistic is of course also available for testing with a deterministic diurnal correlation function, it is not identical to (ref), because the old version of the test statistic incorporates the extra information provided by the identification condition in Assumption (C2).

Incorporating conditioning information

In this section, we follow andersen-thyrsgaard-todorov:19a by showing how our theoretical framework can be generalized to a conditional version that incorporates some additional information that may help to explain the form of the diurnal correlation function, such as the release of important news announcements; an idea that we explore further in the empirical application. To this end, we redefine the random variables $\tilde{c}_{u, \tau_{j}}$ and $ \begingroup \def\mathaccent#c##2{ \kern0.8\dimexpr\macc@kerna \overline{\kern-0.8\dimexpr\macc@kerna\macc@nucleus\kern0.2\dimexpr\macc@kerna} \kern-0.2\dimexpr\macc@kerna } \macc@depth\@ne \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax} \macc@set@skewchar\relax \let\mathaccentV\macc@nested@a \macc@nested@a\relax111{c} \endgroup _{u, \tau_{j}}$ in (ref) as follows:

where $\mathcal{B}_{t-1}$ is an $\mathcal{F}_{t-1}$-adapted random set. Provided appropriate stationarity, ergodicity, and mixing conditions hold, we can deduce a straightforward extension of Theorem (ref):

where $\mathcal{W}^{ \mathcal{B}} = ( \mathcal{W}^{ \mathcal{B}}_{1}, \mathcal{W}_{2}^{ \mathcal{B}}, \mathcal{W}^{ \mathcal{B}}_{3})^{ \top}$, and the $\mathcal{W}^{ \mathcal{B}}_{i}$'s are $\mathcal{L}^{2}$-valued mean zero Gaussian processes with covariance matrix function:

Here, with $Z_{1}, Z_{2} \in \{ X, Y, XY \}$,

for $\kappa, \tau \in [0,1]$, and

Thus, we can proceed as above to construct both point estimates of $k_{u,t}$ and the test statistic. We omit a formal proof of this result, as it follows directly from Theorem (ref).

Small sample comparisons

In the above, we developed a procedure to detect diurnal variation in a correlation process. We continue with a Monte Carlo exploration to gauge the finite sample properties of the proposed test statistic in a controlled environment.

We simulate a bivariate jump-diffusion process on the time interval $[0,T]$. It has a continuous part, which is given by

where $W_{t}^{ \wp}$ is a standard Brownian motion.\footnote{Throughout this section, the driving stochastic processes are assumed to be mutually independent, unless explicitly stated otherwise.} This implies a conditional spot covariance $\mathbb{E} \big( \mathrm{d}X_{t}^{ \mathrm{c}} \mathrm{d} Y_{t}^{ \mathrm{c}} \mid \mathcal{F}_{t} \big) = \sigma_{t}^{X} \sigma_{t}^{Y} \rho_{t} \mathrm{d}t$ with correlation $\rho_{t}$.

The idiosyncratic volatility $\sigma_{t}^{ \wp} = \sigma_{sv,t}^{ \wp} \sigma_{u,t}$ is modeled as:

where $c_{sv,t}^{ \wp} \equiv ( \sigma_{sv,t}^{ \wp})^{2}$.

$\sigma_{sv,t}$, has a heston:93a-type dynamic. As in christensen-thyrsgaard-veliyev:19a, we set $\lambda = 0.05$, $c_{0} = 1$, and $\xi = 0.2$. We allow for a leverage effect by taking $\text{corr}( \mathrm{d}W_{t}^{ \wp}, \mathrm{d}B_{t}^{ \wp}) = -\sqrt{0.5}$. Furthermore, in line with our empirical work the intraday volatility curve is V-shaped. We take $C = 0.5$ and $A = 2.0$, which renders volatility about twice as large at the start and end of the unit interval than in the middle.\footnote{We also inspected a superposition of exponential functions: $\sigma_{u,t} = C + A e^{-a_{1}t} + B e^{-a_{2} (1-t)}$, where $A = 0.75$, $B = 0.25$, $C = 0.88929198$, and $a_{1} = a_{2} = 10$ andersen-dobrev-schaumburg:12a, hasbrouck:99a. The odd value of $C$ is such that $\int_{0}^{1} \sigma_{u,t}^{2} \mathrm{d}t = 1$. This delivers an inverse J-shaped curve, which agrees better with Panel A of Figure (ref) in our empirical application. However, the results are basically unchanged compared to those we report here and are available at request.}

As required by Assumption (C1) we decompose $\rho_{t} = \rho_{sc,t} k_{u,t}$, where the diurnal correlation component $k_{u,t}$ is an affine deterministic function of $t$:

We assume that $b = 2(1-a)$.\footnote{Taken together, the functional form of $\sigma_{u,t}$ and $k_{u,t}$ imply that $\int_{0}^{1} \sigma_{u,t}^{2} \mathrm{d}t = \int_{0}^{1} k_{u,t} \mathrm{d}t = \int_{0}^{1} \sigma_{u,t}^{XY} \mathrm{d}t = 1$.} As such, the null hypothesis of no diurnal variation in $\rho_{t}$ is equivalent to the restriction $\mathcal{H}_{0}: a = 1$, whereas the alternative is $\mathcal{H}_{a}: a \neq 1$. We examine $a = (1.00, 0.95, \dots, 0.80)$. Apart from being convenient, the non-decreasing linear form is also a decent description of the diurnal pattern observed in the correlation processes investigated in Section (ref). Our parametric model further prefixes $k_{u,0.5} = 1$, which is consistent with prevailing evidence in Panel B of Figure (ref) in that section. The domain of $a$ is also shown in the figure. The lowest value $a = 0.8$---or $b = 0.4$---is small relative to the slope $\hat{b} = 0.8062$ estimated from that dataset, so our results should be conservative.

The stochastic correlation process follows:

with $\rho_{sc,0} \in (-1,1)$.

The above SDE can be constructed via a Fisher transformation of $\rho_{sc,t}$ teng-ehrhardt-gunther:16a: $\displaystyle P_{sc,t} = \operatorname{arctanh}( \rho_{sc,t}) = \frac{1}{2} \ln \left( \frac{1 + \rho_{sc,t} }{1 - \rho_{sc,t}} \right)$. Suppose $P_{sc,t}$ is a modified Gaussian Ornstein-Uhlenbeck process $\mathrm{d} P_{sc,t} = - \tilde{ \kappa} \left( \tanh \left(P_{sc,t} \right) - \tilde{ \rho} \right) \mathrm{d}t + \tilde{ \sigma} \mathrm{d} \tilde{B}_{t}$ with $\tilde{ \kappa}, \tilde{ \sigma} > 0$ and $\tilde{ \rho} \in (-1,1)$. An application of It\^{o}'s Lemma to the inverse $\displaystyle \rho_{sc,t} = \tanh(P_{sc,t}) = \frac{ \exp(2P_{sc,t})-1}{\exp(2P_{sc,t})+1}$ then delivers (ref) with $\sigma^{2} = \tilde{ \sigma}^{2}$, $\kappa = \tilde{ \kappa} + \tilde{ \sigma}^{2}$ and $\displaystyle \rho = \frac{ \tilde{ \kappa}}{ \tilde{ \kappa} + \tilde{ \sigma}^{2}} \tilde{ \rho}$. If the parameters satisfy the “Feller”-type condition $\displaystyle \kappa > \frac{ \sigma^{2}}{1 \pm \rho}$, $\rho_{sc,t}$ is stationary with state space $(-1,1)$, i.e. the probability mass at the boundary goes sufficiently fast to zero as $\rho_{sc,t} \rightarrow \pm 1$, such that the barriers are not attainable (nor attractive). This is suitable for a dynamic correlation model.

We set $\kappa = 1.5$, $\sigma = 0.3$, and $\rho = (0.2, 0.4, 0.6)$. This implies that the above condition is fulfilled in every scenario. Our choices of $\rho$ incur a weak to strong positive association between $X$ and $Y$ in line with the descriptive statistics of the unconditional sample correlation coefficient presented in Table (ref) in the empirical investigation. On the one hand, the intermediate and largest value of $\rho$ are in line with what we observe there, whereas the lowest value of $\rho$ is beneath the 1. quartile of the sample correlation between every asset pair. On the other hand, the former rarely lead to a negative spot correlation, whereas the unconditional distribution of the latter has a nontrivial amount of probability mass below zero (i.e., the chance of observing a negative correlation is around 0.1 for $\rho = 0.2$, whereas it is close to zero otherwise). This is intended to show the impact of weak correlation on our test statistic, since in this case $\rho_{sc,t}$ can linger about zero with a higher chance. Moreover, examining a smaller numeric value of $\rho$ is relevant for other asset classes. In any case, we draw the initial condition $\rho_{sc,0}$ at random from the stationary distribution of $\rho_{sc,t}$, which is illustrated in Panel A of Figure (ref) for $\rho = 0.6$.\footnote{The stationary density is given by $\displaystyle f_{ \rho}(x) = \frac{m}{2^{c}}(1+x)^{a+b}(1-x)^{a-b}$, for $x \in (-1,1)$, where $\displaystyle a = \frac{\kappa - 2 \sigma^{2}}{ \sigma^{2}}$, $\displaystyle b = \frac{ \kappa \rho}{ \sigma^{2}}$, and $\displaystyle c = \frac{ \kappa}{ \sigma^{2}}$. $m$ is a normalizing constant, such that $\int_{-1}^{1} f_{ \rho}(x) \mathrm{d}x = 1$, which can be expressed analytically via the hypergeometric and gamma function.} A realization of the full-blown continuous-time dynamics of $\rho_{sc,t}$ in this case is shown in Panel B.\footnote{We employ full truncation to enforce that $\rho_{t}$ remains in $(-1,1)$.}

We add a pure-jump component to the continuous sample path of log-price, which is simulated as a compound Poisson process:

where $q_{t}^{ \wp}$ is the jump size at time $t$ and $N_{t}^{ \wp}$ is a Poisson process with intensity $\lambda_{J}$. We draw $q_{t}^{ \wp} \sim N(0, \sigma_{J}^{2})$ with $\displaystyle \sigma_{J} = \sqrt{ \frac{1}{ \lambda_{J}} \frac{p_{J}}{1-p_{J}} c_{0}}$, so the quadratic jump variation is proportional to the average diffusive variance. $p_{J}$ controls how much of the second-order variation in the log-price process that is due to the jump component. We assume $\lambda_{J} = 0.2$ and $p_{J} = 0.1$, such that a jump is observed in every fifth replication, while accounting for 10% of the quadratic variation, on average. This conforms with empirical evidence on jump testing ait-sahalia-jacod-li:12a, ait-sahalia-xiu:16a, bajgrowicz-scaillet-treccani:16a.

We discretize the system with an Euler scheme and a baseline step of $\mathrm{d}t = 1/23{,}400$. This represents the “continuous-time” foundation from which we extract a coarser sample of size $n = $ 26, 39, 78, 390, 780, 1,560, and 4,680, equidistant log-price increments over each interval $[t-1,t]$, for $t = 1, \dots, T$ and $T = $ 5, 22, and 66. The former can be interpreted as discretely sampling a process every 900, 600, 300, 60, 30, 15, and 5 seconds, while the latter corresponds to observing such high-frequency data over a week, month, and quarter.\footnote{In practice, recording a price at 5- or 15-second intervals induces a nontrivial amount of microstructure noise in the estimation. Hence, $n = 1{,}560$ or $n = 4{,}680$ is a much larger sampling frequency than we feel comfortable with in the empirical application. It is mainly added to illustrate the convergence properties of our test.}

In practice, high-frequency estimation of the correlation between asset returns is known to diminish as the sampling frequency goes up, because the observed data are asynchronous, i.e. lack alignment in time epps:79a. To gauge the importance of this, we also consider a scenario, where $X$ and $Y$ are observed at irregularly spaced sampling times. We simulate the number of observations on day $t$ as $n_{t}^{ \wp} \sim \text{Poi}( \lambda_{n})$, where $\lambda_{n} = 4{,}680$, such that the average daily number of data points is equal to the largest value of $n$ from the equidistant setting.\footnote{To put this in perspective, the choice of $\lambda_{n}$ is merely a quarter of the average daily number of trades in the least liquid asset considered in our empirical application (17,920 for TRV, as shown in Table (ref)), so it exacerbates the degree of asynchronicity we encounter there.} Conditional on $n_{t}$, we select the observation grid as a random sample without replacement of size $n_{t}$ from $0, \mathrm{d}t, \dots, 1$ and proceed as above, but using previous-tick imputation to construct an equidistant (and synchronous) sample of size $n$.\footnote{The refresh time approach of barndorff-nielsen-hansen-lunde-shephard:11a was another option.}

A total of 10,000 replica are made. As described in Section (ref), in each simulation we divide the available high-frequency data $(\Delta_{(t-1)n+i}^{n} X)_{i = 1, \dots,n \text{ and } t=1, \dots, T}$ and $(\Delta_{(t-1)n+i}^{n} Y)_{i = 1, \dots,n \text{ and } t=1, \dots, T}$ into non-overlapping subsets of size $k_{n} = $ 13, 13, 26, 130, 195, 390, and 963, corresponding to $n/k_{n} = $ 2, 3, 3, 3, 4, 4, and 5, so the number of blocks is rising slowly with $n$, as required by the rate condition from Theorem (ref). Indeed, because the testing procedure explores the properties of the covariance process, a casual robustness check suggests that is preferable with a smaller number of blocks consisting of a larger number of increments, than vice versa, as it is important to get a good approximation of its intraday dynamic.

We calculate the jump-robust bipower variation and relieve log-returns from the jump component by blockwise truncation of increments that are numerically above $v_{n} = q \sqrt{ \text{BV}}n^{- \varpi}$ with $q = 5$ and $\varpi = 0.49$. Hence, our procedure labels a log-return as a jump if it exceeds about five diffusive standard deviations.

To compute the test statistic, we implement the HAC estimator of the asymptotic covariance matrix with a lag length $H_{T} = [T^{1/3}]$ and a Parzen kernel to ensure positive semi-definiteness.\footnote{We also experimented with a Bartlett kernel, but that did not lead to substantial changes.} The results are robust to the concrete choice of lag length, so long as it is not exceedingly large. To evaluate the test statistic, we draw $9{,}999$ realizations of $\widehat{ \mathcal{Z}}$ and extract an appropriate quantile from the induced empirical distribution function.

The outcome of the exercise is presented in Tables (ref) -- (ref), which show rejection rates of the testing procedure at the $\alpha = 0.01$ level of significance.\footnote{The corresponding analysis at the 5% and 10% significance levels are reported in Appendix (ref).} The various intraday sample sizes appear in rows and diurnal correlation slopes in columns, while the different values of $T$ are reported in Panels A -- C, respectively. In addition, the left-hand (right-hand) side of each table is for the equidistant (irregular) sampling scheme.

The column headings with $a = 1.00$ refer to the null hypothesis and we look at those to begin with. We observe that for $T = 5$ the test is somewhat oversized, as the rejection rates are higher than the nominal level. With such a small $T$, the time-averaged block-wise realized covariance is inevitably going to be a very crude measure of the associated time-of-day spot covariance, which introduces some distortion. At $T = 22$, the rejection rates have already settled around the anticipated value at the 1% nominal level, but we still see a slight overrejection. The latter can arise from discrepancies between the sampling distribution of the test statistic for a finite number of blocks and that predicted by the asymptotic theory. Of course, it can potentially also be attributed to our choice of tuning parameters in the implementation. By and large, however, the numbers line up with the asymptotic distribution theory under the null. We therefore leave the pursuit of more optimal tuning parameters to a future endeavor.

Moving to the right toward columns with $a \neq 1$, which defines our alternative, we observe a monotonic rise in the rejection rates as $a$ gets smaller, which steepens the slope of the intraday correlation curve, and as the sample size increases (either $n$ or $T$). This is as prescribed by the asymptotic theory from Section (ref). Note that for commonly employed intraday sample sizes (e.g. $n = 78$ or $n = 390$) and a month worth of high-frequency data (i.e. $T = 22$), the power is often rather good. This is compelling, since our naive configuration with a straight line understates the evolution of the nonlinear curve observed in practice.

To gauge the effect of changing $\rho$, i.e. the average degree of asset return correlation, we note that a lower value leads to a decrease in the rejection rates. That is, weak correlation is detrimental to both the size and power of the test statistic. This effect is rather substantial for $\rho = 0.20$ compared to $\rho = 0.60$, but as expected we do observe a sustained and significant improvement with increasing $n$ and $T$ or a reduction in $a$.

At last, we inspect the robustness of the test statistic to random sampling times. As consistent with the analysis for varying $\rho$ in the previous paragraph, we learn that irregularly spaced data reduces the rejection rates vis-\`{a}-vis the equidistant setting. This can be ascribed to the Epps effect, which induces an attenuation bias in the estimated level of the correlation process. Indeed, the discrepancy gets more pronounced as the sampling frequency $n$ is increased relative to the intensity of the counting process $\lambda_{n}$, which causes a gradual worsening of the synchronization problem. However, whereas the drop in power remains present even with larger $n$ so long as we look at a small value of $T$, the effect is much less pronounced for data stretching over even a modest time period. This suggests that this problem should not be a big concern in practice.

In summary, the test statistic has acceptable size control and decent power in most of the settings that are relevant to our empirical application, which we turn to next.

Empirical application

We conduct an assessment about the presence of diurnal variation in the empirical correlation process by studying a vast dataset covering an extended time frame and a broad selection of companies from the large-cap segment of the US stock market.

Data description

At our disposal are high-frequency data from the members of the Dow Jones Industrial Average index, as of the August 31, 2020 recomposition. In addition, we include the SPDR (formerly known as Standard & Poor's Depository Receipts) S&P 500 trust, listed under the ticker symbol SPY. The latter is an exchange-traded fund that aims to replicate the total return of the S&P 500 index (before expenses). Its price development is therefore representative of market-wide changes in the valuation of US equities.

We downloaded a time series of transaction and quotation data for each security from the NYSE Trade and Quote (TAQ) database for the sample period January 4, 2010 to April 28, 2023. Prior to our investigation, we preprocessed the raw high-frequency data with a standard filtering algorithm to remove outliers barndorff-nielsen-hansen-lunde-shephard:09a, christensen-oomen-podolskij:14a.

The US stock market is open for trading from 9:30am to 4:00pm on normal business days. However, on a regular basis most venues halt trading at an earlier time in observance of upcoming holidays. This is, for example, done before Independence Day, Thanksgiving, and Christmas Eve. In such instances, the trading session is shortened and the exchanges close at 1:00pm. As the diurnal correlation pattern on those days can be expected to deviate substantially from that on a regular business day with a usual trading schedule, we remove them from the sample. Furthermore, we purge the Flash Crash of May 6, 2010 due to its highly irregular volatility that exerts a disproportional effect on our estimation procedure. As a result, the empirical investigation is based on the $T = 3{,}325$ days remaining in our sample.

In Table (ref), we present a list of ticker symbols and descriptive statistics of the associated high-frequency data.

We construct a 60-second equidistant transaction price series from the cleaned high-frequency data using the previous-tick rule of wasserfallen-zimmermann:85a, so we collect $n = 390$ high-frequency returns per day for each asset. Although the asymptotic theory requires $n \rightarrow \infty$ and the amount of tick-by-tick data is an order of magnitude larger---as evident from column “$N$” in Table (ref)---a 60-second window is the smallest time gap at which the data can be perceived noise-free, as gauged by the ait-sahalia-xiu:19a Hausman test for microstructure noise. We compute their test statistic at the daily horizon and report the rejection rate in the “H” column in Table (ref).\footnote{Thanks to Dacheng Xiu for making Matlab code to implement the test available at his website.} This should be compared to a 1% level of significance. Apart from a few stocks, the rejection rate is typically close to the nominal level, showing that noise is not a major concern. Meanwhile, lowering the sampling frequency further raises the rejection rate materially (unreported, but available at request) and is not recommendable, unless a noise-robust approach is adopted.\footnote{One option is to pre-average the available high-frequency data, see, e.g., jacod-li-mykland-podolskij-vetter:09a, podolskij-vetter:09a, podolskij-vetter:09b. While this facilitates an increase in sampling frequency, one should be aware that noise-robust estimators converge at a very slow rate and may be less efficient than noise-free estimators in practice if the data are at the margin of being noisy. Still, pre-averaging can potentially improve the power of the test statistic, but we leave this extension for future research.}

The diurnal pattern in correlation

In Panel A of Figure (ref), we plot a representative example of the diurnal covariance pattern inherent in our data. We follow christensen-hounyo-podolskij:18a and compute it as the 0.5% trimmed mean realized covariance estimate (after jump-truncation) at a fixed 60-second time-of-day slot, where the average is taken across the $T = 3{,}325$ days in the sample and $\displaystyle \frac{d(d-1)}{2} = 465$ pairwise combinations of the number of included equities, $d$. We contrast this to the geometric mean of the idiosyncratic diurnal variance, $\hat{ \sigma}_{u,t}^{X} \hat{ \sigma}_{u,t}^{Y}$ (everything is normalized as in Assumption (C2) to be comparable). Since $\sigma_{u,t}^{XY} = \sigma_{u,t}^{X} \sigma_{u,t}^{Y} k_{u,t}$, the latter can be interpreted as the imputed diurnal covariance pattern present with no seasonality in the intraday correlation (i.e., $k_{u,t} = 1$). In agreement with prior literature andersen-bollerslev:97b, bibinger-hautsch-malec-reiss:19a, christensen-hounyo-podolskij:18a, $\hat{ \sigma}_{u,t}^{X} \hat{ \sigma}_{u,t}^{Y}$ resembles a “tilted J.” In contrast, we observe the actual diurnal covariance, $\hat{ \sigma}_{u,t}^{XY}$, is almost symmetric and much closer to U-shaped. This is anecdotal evidence that $k_{u,t}$ is not always equal to one.\footnote{Interestingly, there also appears to be a subperiodic structure in the diurnal covariance pattern at the whole- and half-hourly horizon.}

Next, we map each 60-second pairwise realized covariance matrix into a correlation estimate and repeat the above averaging procedure. The ensuing time-of-day correlation measure---portrayed in Panel B of Figure (ref)---should be randomly distributed around one under the null of no diurnal variation. Instead, we observe a pronounced upward-sloping and almost piecewise linear curve. There is notably lower (on average less positive) correlation in the morning than in the afternoon, which is in accord with allez-bouchaud:11a and hansen-luo:23a. These findings are further corroborated by estimating the equation $k_{u,t} = a + bt$ in (ref) from the empirical high-frequency data. The OLS parameter estimates, subject to the maintained restriction $b = 2(1-a)$, are $\hat{a} = 0.5969$ and $\hat{b} = 0.8062$ with the fitted regression line inserted into the figure as a reference point. In practice, of course, $k_{u,t}$ evolves in a much more nonlinear and discontinuous fashion. We notice a positive jump at 10:00am, arguably caused by the publication of macroeconomic information. There is another upsurge around 2:00pm, corresponding to the release of minutes from Federal Open Market Committee (FOMC) meetings.

The right-hand side of Table (ref) has further descriptive statistics on diurnal correlation. It also reports the outcome of our testing procedure. We proceed as in Section (ref) in terms of tuning parameters, i.e. for $n = 390$ we take $k_{n} = 130$. We calculate the test statistic each month (of which there are 160 in total) with a Parzen kernel and lag length $H_{T_{m}} = [T_{m}^{1/3}]$, where $T_{m}$ is the number of days in month $m$ (with $T_{m} = 21$ on average). The analysis is then divided in two: We correlate individual members of the DJIA index against the SPY (“versus SPY”) and summarize with the interquartile range the results of pairing each stock---including the SPY---against all the thirty remaining ones (“versus rest”).

Gauging at the “versus SPY” part, several interesting findings emerge. First, every asset in our sample is positively related with the stock market portfolio exhibiting a typical level of correlation $\bar{ \rho} = 0.556$. Second, on an individual stock basis the estimated $a$ and $b$ parameters are broadly in line with the aggregate figures reported above and remarkably consistent over the cross-section of equities. In the end, it translates into an average rejection rate of around two out of three with our proposed test statistic. Apart from a few instances, the latter are remarkably close for the vast majority of the assets.

Switching to the “versus rest” part, single names display a weaker association with each other than with the market. This is further reflected in the tendency for the intraday correlation to exhibit a more upward-sloping linear association with $a$ being lower and $b$ being higher. Interestingly, there is a somewhat larger discrepancy between the rejection rates of the test statistic for individual assets tested against each other, which is notably lower than our findings for the market index, but it remains far above the nominal level.

Overall, our results suggest diurnal variation in the correlation process is a nontrivial effect, which is present most of the months in our sample.

Conditioning information

To delve deeper into our empirical results, we follow the guidance from Section (ref) and extend the previous analysis by investigating whether and how conditioning information helps to determine the functional form of the intraday correlation curve.

First, we gauge the impact of macroeconomic news in the form of monetary policy decisions made by the Federal Open Market Committee (FOMC), which in the majority of our sample are released at 2:00pm followed by a press conference at 2:30pm.\footnote{Earlier, the FOMC statements were released at the conclusion of the meeting, which gradually converged toward 2:15pm. The current format was adopted beginning in 2011 and, hence, covers nearly our entire sample.} There are eight regularly scheduled meetings during the year. We acquired historical announcement dates from the Federal Reserve Board's website. Secondly, we analyze the influence of quarterly earnings announcements (QEA) issued by the individual companies in our stock universe. Here, the historical announcement dates were extracted from the Center for Research in Security Prices (CRSP) database. We only include earnings announcements released either in the after-hours session on the previous day or during pre-market trading on the same day, such that the earliest opportunity to react on the news for the general public is at the commencement of the exchange trading at 9:30am. Thus, whereas the former application centers around market-wide systematic announcements released during active trading that are likely to affect the stock market in its entirety, the latter concerns largely idiosyncratic news---at least within the domain of the equities we look at---that are released prior to the opening of the stock exchange.\footnote{Fiscal information from a company can trigger price changes in related firms and the broader market patton-verardo:12a,savor-wilson:16a. However, as shown by christensen-timmermann-veliyev:25a, the spillover effect is often small in magnitude.}

The outcome of this analysis is presented in Figure (ref). In Panel A, we show the results for the macroeconomic news announcements, while Panel B reports the associated results for earnings releases. The “no” curve refers to the contraindicator based on the no announcement sample. In both cases, the latter is very close to the unconditional curve from Panel B in Figure (ref), although the jump at 2:00pm is slightly smaller in Panel A of Figure (ref) than previously. Furthermore, we should note that since the announcement sample is much smaller than the no announcement sample, the reported point estimates are subject to considerable measurement error. However, the overall evolution can still be deciphered.

The results are compelling. In particular, the typical FOMC announcement leads to a distinct positive jump in the diurnal correlation pattern, which is much larger than above. As anticipated, the influx of a market-wide news component leads to a systematic response in the prices of most equities, which temporarily reinforces their intraday return correlation, before it starts to recede and taper off toward to no announcement curve at the closing of the stock exchange at 4:00pm. Turning our attention to Panel B for the earnings announcements, the results are also rather intuitive. Specifically, an earnings announcement causes the security price of the issuing company to be largely uncorrelated with the market during the early phases of trading while the price discovery process is being completed and portfolio holdings being updated, before the intraday correlation curve reconnects with the no announcement sample around noon.\footnote{In unreported results, we also examined whether stock characteristics can help to explain the pattern in the diurnal correlation process. In particular, we studied the influence of liquidity and industry connectedness. First, we sorted our stocks based on liquidity, as defined by the “$N$” column in Table (ref). We selected the ten most liquid and least liquid companies, while leaving out the middle portion of the sample, and calculated a separate intraday correlation curve for each subsample. However, there was no discernible difference between them. This is possibly because we are only considering large-cap stocks that are highly liquid in absolute terms, even if some are relatively illiquid. Second, we split the stocks based on industry proximity, as defined by the “closeness” of their SIC codes christensen-timmermann-veliyev:25a, wang-zajac:07a. This showed that more distant companies are less correlated in the morning. A finding that parallels our results for the QEA. Intuitively, when a company announces its fiscal results, its security price also trades relatively “distant” to the market, being driven mainly by the idiosyncratic contents of the announcement in the short-term. The details are available at request.}

Implications for risk management

In the closing, we highlight the importance of incorporating diurnal variation in the correlation process as exemplified via the operations of a trading desk. We suppose a dealer is long one stock from the DJIA index. The risk is offset with a dynamic short position in the market index (SPY in our context). We assume the trader employs a conventional five-minute frequency and updates the hedge at the end of each time interval---based on available information---in order to minimize the expected variance of the combined portfolio during the next five-minute window. The minimum variance hedge ratio, denoted $\phi_{i \mid i-1}^{n}$, is an adapted discrete-time stochastic process that is selected at the beginning of the $i$th interval $[(i-1)/n,i/n]$ via the following optimization problem:

The solution is given by:

for $i = 1, \ldots, n$, where $\Delta_{i}^{n} X$ is the subsequent five-minute log-return on the underlying asset and $\Delta_{i}^{n} Y$ is the associated SPY log-return (note that in this subsection we set $n = 78$ to represent a five-minute frequency for notational convenience).

The trading policy depends on the conditional covariance matrix:

In practice, $\Sigma_{i \mid i-1}^{n}$ is not known in advance and has to be modeled. However, we do not pursue this approach here. Instead, we assume that an estimator of $\Sigma_{i \mid i-1}^{n}$ is accessible via the 5-minute ex-post realized covariance matrix of $X$ and $Y$ (calculated from the 60-second high-frequency data extracted above).

$\phi_{i \mid i-1}^{n}$ is then selected as:

with $\hat{ \sigma}_{[i-1,i]}^{X}$ and $\hat{ \sigma}_{[i-1,i]}^{Y}$ being the square-root realized variance of $X$ and $Y$ on the $i$th interval, whereas $\hat{ \rho}_{[i-1,i]}$ is the realized correlation.

In other words, $\phi_{i \mid i-1}^{n}$ is the ex-post minimum variance hedge ratio, conditional on knowing the subsequent realized covariance matrix over that window. It follows that $(\phi_{i \mid i-1}^{n})_{i=1}^{n}$ adapts to intraday seasonality in both the variance and correlation processes. Suppose that the stochastic correlation component is constant within a day, i.e. $\rho_{t} = \rho_{sc, \lfloor t \rfloor} k_{u,t}$, where $\rho_{sc, \lfloor t \rfloor}$ is determined at the start of day $t$. This assumption is common in the discrete-time multivariate stochastic volatility literature, and it is a decent approximation to the dynamic of the stochastic correlation process in view of its persistence. In this case, the high-frequency correlation estimate can be decomposed as $\hat{ \rho}_{[i-1,i]} = \hat{k}_{u, [i-1,i]} \hat{ \rho}_{sc}$, where $\hat{ \rho}_{sc} = n^{-1} \sum_{i=1}^{n} \hat{ \rho}_{[i-1,i]}$ is the average realized correlation over the whole day and $\hat{k}_{u,[i-1,i]}$ is the diurnal coefficient. This further implies that

where ${\bar{ \phi}}_{i \mid i-1}^{n}$ is the optimal ex-post hedge ratio, when the local correlation estimate is replaced by an average for the entire day, all else equal. Hence, ${\bar{ \phi}}_{i \mid i-1}^{n}$ adapts to diurnal variation in the variance but not the correlation.

We compare $\phi_{i \mid i-1}^{n}$ and $\bar{ \phi}_{i \mid i-1}^{n}$ to illustrate the effect on risk management. The minimum variance hedge ratio is computed as described above across the components of the DJIA index and for each 5-minute interval in the sample. Figure (ref) reports the results. In Panel A, we plot the intraday profile of $\phi_{i \mid i-1}^{n}$ and $\bar{ \phi}_{i \mid i-1}^{n}$. The optimal $\phi_{i \mid i-1}^{n}$ is around 0.7 -- 0.8. In contrast, there is pronounced variation in $\bar{ \phi}_{i \mid i-1}^{n}$. The latter fails to acknowledge that lower correlation in the morning has a detrimental impact on the diversification effect, causing a reduced hedge ratio (and vice versa in the afternoon). Interestingly, this means there are fewer transaction costs associated with managing a portfolio based on $\phi_{i \mid i-1}^{n}$. In Panel B, we see the unconditional distribution of $\phi_{i \mid i-1}^{n}$ is more symmetric and has mass below zero, as it automatically adapts to brief lapses of low-to-negative correlation. In contrast, the histogram of $\bar{ \phi}_{i \mid i-1}^{n}$ is floored at zero, because the daily correlation with the stock index tends to be positive.

The variance ratio of the full sample ex-post portfolio return:

suggesting it is possible to achieve a highly nontrivial reduction in risk exposure of about 17.6% in a risk management model that controls for diurnal variation in correlation.

Conclusion

We develop a nonparametric test of the hypothesis that there is no diurnal variation in a correlation process. The proposed test statistic has a known distribution under the null, whereas it diverges under an alternative with deterministic variation in the correlation. In a simulation study, the testing procedure aligns closely with the theoretical predictions and it attains a good rejection rate for moderate sample sizes and realistic shapes in the diurnal correlation process. In our empirical application, we document pervasiveness in the intraday correlation dynamics in the US equity market. As consistent with allez-bouchaud:11a, bibinger-hautsch-malec-reiss:19a, and hansen-luo:23a, we find that correlations are low in the morning and rise systematically during the trading session. We further show how conditioning information about macroeconomic news and corporate earnings announcements affects the evolution of the intraday correlation curve.

andersen-thyrsgaard-todorov:19a test whether the intraday volatility curve is changing over time andersen-su-todorov-zhang:24a. They exploit an assumed stationarity of the stochastic volatility and compare the unconditional distribution of different time-of-the-day high-frequency returns. As in this paper, their results are derived based on a combination of infill and long-span analysis. It may be possible to adapt that setting to our framework by feeding their test statistic with devolatized high-frequency returns. We leave this idea for inspiration.