# The Epps effect under alternative sampling schemes

Patrick Chang<EMAIL_ADDRESS>Etienne Pienaar
<EMAIL_ADDRESS>Tim Gebbie<EMAIL_ADDRESS>Department of
Statistical Sciences, University of Cape Town, Rondebosch 7700, South Africa

###### Abstract

Time and the choice of measurement time scales is fundamental to how we choose
to represent information and data in finance. This choice implies both the
units and the aggregation scales for the resulting statistical measurables
used to describe a financial system. It also defines how we measure the
relationship between different traded instruments. As we move from high-
frequency time scales, when individual trade and quote events occur, to the
mesoscales when correlations emerge in ways that can conform to various latent
models; it remains unclear what choice of time and sampling rates are
appropriate to faithfully capture system dynamics and asset correlations for
decision making. The Epps effect is the key phenomenology that couples the
emergence of correlations to the choice of sampling time scales. Here we
consider and compare the Epps effect under different sampling schemes in order
to contrast three choices of time: calendar time, volume time and trade time.
Using a toy model based on a Hawkes process, we are able to achieve simulation
results that conform well with empirical dynamics. Concretely, we find that
the Epps effect is present under all three definitions of time and that
correlations emerge faster under trade time compared to calendar time, whereas
correlations emerge linearly under volume time.

###### keywords:

Epps effect, Hawkes process, Volume time, Trade time, Calendar time

##  1 Introduction

The Epps effect [18] is key phenomenology relating to high-frequency
correlation dynamics in financial markets. Central to the effect is the
relationship between correlations and the time scale of the sampling scheme.
The Epps effect is prevalent and has been observed in a range of financial
markets ranging from stock markets to foreign exchange markets (see
Mastromatteo et al. [29] and references therein).

The Epps effect is a long-studied phenomenon that still lacks a satisfactory
and comprehensive explanation. The main sources that contribute towards this
effect include: (i) asynchrony, (ii) lead-lag, and (iii) tick-size
(discretisation of financial prices). We highlight a few studies by various
scholars over the last few decades. Renò [35] explored the effect of
asynchrony under the presence of lead-lag. Precup and Iori [34] demonstrated
that different levels of asynchrony resulted in different behaviours of the
Epps effect. Tóth and Kertész [40] derived an analytical expression that
characterised the Epps effect as a function of the rate parameter from Poisson
sampling. They extended this further by decomposing the correlation at a
certain time scale $\Delta t$ as a function of the correlation at smaller time
scales $\Delta t_{0}$ [41]. The analytic expression was then further extended
by Mastromatteo et al. [29] who separated the effect of asynchrony and that of
lead-lag. Finally, Münnix et al. [30] investigated the direct impact of tick-
size on the Epps effect.

Another strand of literature around the Epps effect focuses on estimators
correcting for this decay in correlation. The canonical estimator that
addresses this is the Hayashi–Yoshida estimator [24] which corrected the
effect of asynchrony by allowing for multiple contributions using a cumulative
covariance estimator. Buccheri et al. [7] characterised the correction through
the probability of flat trading whereas Münnix et al. [31] provided a
correction that compensated for both the statistical cause of tick-size and
asynchrony. Other notable estimators that address this include: the quasi-
maximum likelihood estimator from Aït-Sahalia et al. [1] and the multi-asset
lagged adjustment estimator from Buccheri et al. [6].

Part of the reason why the phenomenon still lacks a satisfactory explanation
is because these causes do not account for the entirety of the Epps effect
[29, 31, 41]. The Epps effect is commonly understood as a bias that requires a
correction [7, 43]. However, Chang et al. [15] speculated whether the Epps
effect is better understood as a fundamental property arising from the
discrete nature of high-frequency financial markets, and questioned whether
diffusion processes are an appropriate underlying model representation.
Nonetheless, all the aforementioned studies investigated the Epps effect under
calendar time sampling. Chang et al. [11] deviated from this by providing a
preliminary investigation of the effect under volume time sampling.

Time and the choice of time scales is fundamental to how we choose to
represent information and data. This implies both the units and the
aggregation scales for the resulting statistical measurables used to describe
financial systems [9]. At low-frequency the measurement time scales conform
well with those convenient to human operators and researchers intuition —
calendar time. However, as one moves into the world of heterogeneous inter-
related individual events, before meaningful correlations have emerged, it
becomes increasingly difficult to uniquely provide global measures of event
synchronisation to the system.

The first real attempt at linking the dynamics under different definitions of
time was by Derman [16]. Derman was able to derive the relationship between
different definitions of time by assuming that each stock has its own
intrinsic time that has a linear map to calendar time based on the stock’s
trading frequency. Under these assumptions, Derman was able to show that
correlations are the same in calendar and intrinsic time. This linear
assumption is problematic because the inter-arrivals of empirical events
(market orders) are not homogeneous, but rather follows a Weibull distribution
[9].

The goal of this paper is to explore the Epps effect under alternative
sampling schemes. This is similar to what has been done for the Realised
Volatility (RV) [19, 22, 33]. We explore this by using the Malliavin–Mancino
(MM) Fourier estimator [27, 28] to directly probe the relationship between
measured correlations and time scale represented as frequencies: (i) the
number of Fourier coefficients $N$ will define the sampling frequency, and
(ii) the reconstruction frequency $M$. Here we are using integrated estimates
and thus do not need to concern ourselves with the choice of $M$ [10]. We will
contrast this with the well understood Hayashi–Yoshida (HY) estimator. This
comparison is informative because both are designed to compute estimates in
calendar time. However, the HY is based on the a priori assumption that the
process has some underlying Brownian motion contribution and is constructed to
be unbiased in this sense. The MM estimator has no such reliance on the
specification of bias with respect to the underlying process.

Many estimation methods do not allow the precise extraction of the
correlations as a direct function of the sampling frequency independent of
assumptions relating to the underlying stochastic processes. Fourier methods
provide the technology to achieve this. In addition, the Fourier method allows
direct extraction of correlations at a particular sampling time scale. This
will enable us to efficiently probe correlations at different high-frequency
time scales.

There are important caveats, one of which is captured by volume time. Volume
time is an example of when we cannot explicitly use sampling to map a
definition of time aggregated from local events into some global calendar time
that preserves event synchronisation in both representations. This is because
the rate of trading of each underlying stock is not well-defined in calendar
time which can lead to cases where trading assets may be synchronised in
volume time, but only intermittently in calendar time.

To simulate the order book events, in particular transactions, we need a time
labelled 3-tuple with time, price, and volume: $(t,p,v)$. Using the event
times defined by the Hawkes process we are free to choose the distributional
properties of the prices and volumes associated with events. We know
empirically that we can relate price changes $\Delta p_{t}$ of transactions at
some time $t$ to the volumes at that time $v_{t}$ using a price impact
function: $\Delta p_{t}\propto f(v_{t})$. We could exploit the price impact
function to simulate a realistic order-book of transaction events [9]. Here we
argue that one does not need to link the volumes to the price changes through
price impact to recover realistic Epps curves under the various sampling
schemes considered. Perhaps surprisingly, the fine-to-coarse model introduced
by Bacry et al. [2] conforms well with reality when combined with a power-law
distribution for the volumes [9].

The paper is structured as follows. Section 2 introduces the different
temporal metrics. Section 3 presents the estimators and their relationship to
each other. Section 4 uses a multivariate Hawkes process to simulate an event-
based price process which is used to investigate and compare calendar time
sampling, transaction time sampling and a proxy for volume time sampling.
Section 5 investigates and compares the various sampling methods for banking
securities on the Johannesburg Stock Exchange using trade-and-quote data.
Finally, Section 6 we conclude with some closing remarks. The key contribution
made is to demonstrate that the Epps effect emerges with different dynamics
depending on the choice of time. In particular, volume time rescales the
exponential emergence of correlation into a linear scale.

##  2 Temporal metrics

In order to investigate the Epps effect under alternative sampling schemes, we
need to evaluate the various definitions of time [9, 22, 33, 39]. First we
have calendar time also known as physical time. This is the most natural way
to perceive time as it is the way humans experience time at low-frequency and
it is common to all investors and markets. It is also the most common approach
used to compute statistical properties of financial time series because it can
be defined without reference to the system generating the data. Even though
the passage of time is most natural to human operators under this definition,
it is not without challenges. These problems include: intraday seasonality,
asynchrony across stocks, different and often overlapping market calendars and
time zones. However, calendar time can allow coordinated sampling in a unique
way independent of the data generation process.

There are other methods to measure the passage of time, approaches that arise
from the mechanics of the data generating processes, and the technologies used
to manage and make decisions in financial systems. We can count events and use
the event arrivals to increment a time. The most general being event time. For
example, if we consider the full order book then one may increment event time
by one unit when an order book event occurs. This may be a trade, a
cancellation, an order modification or any order book event that changes the
order book in some way. This is related to the idea of decision time, but here
we will use event time and decision time interchangeably because practically
any event in the order book is the result of a decision. Some prudence should
be exercised, more generally in an Economic system this need not be the case,
as many events can be truly random and exogenous relative to the strategic
agents making decisions and taking actions that generate events. An outcome of
using event time can be the “smoothing” of the data which is caused by bursts
of activity being stretched out in the time series — as we have one event per
time unit [9]. This can lead to the effect that under calendar time there is
significant clustering, while under different choices of event time the
clustering effect can be smoothed away.

There can be different types of event times. For example, when considering
transaction prices, one may use trade time or transaction time. Using this
count, time is increased by one unit each time a unique transaction occurs.
The advantage of this count is that limit orders do not affect the flow of
time. Additionally, aggregational normality is more clear in trade time [9].
However, this choice of time can exclude the many cascades and burst of order
book changes that can in turn lead to the actual trading events.

One can also use tick time. Using this count, time is increased by one unit
each time the price changes. Griffin and Oomen [22] found that for the
realised volatility, tick time sampling is superior to transaction time
sampling, which are both an improvement on calendar time sampling [33], at
least with respect to using a performance measure such as the mean squared
error. There are many such possible choices for performance measures.

Finally, one may use volume time [17] to incorporate the volume of
transactions. Using this count, time is increased by one unit for each unit of
volume transacted each time a single share or instrument is exchanged. The
advantage of this count is that stocks can naturally be synchronised,
irrespective of their liquidity, directly in terms of sequence of comparable
volumes traded. This is a practical data-science construct which is useful for
the aggregation of real-time relative risk measures such as VPIN [17].

However, mapping event times to calendar times can be problematic. For
example, mapping volume time to a calendar or external clock does not always
make sense because the mapping is highly nonlinear, intermittent and system
dependent. In finance, an example where this can make sense could be the
aggregation of transactions from trade time to bar data, so called OHLCV bars.
These are typically aggregated at 1 minute, 5 minutes, and so on; it should be
expected to require increments that may be correlated.111It should be noted
that mappings from trade events to bar data are still nonlinear mappings, but
ones that may well be possible to faithfully represent using approximations
that combine stochastic time and fractional process [21]. However, these are
still defined using continuous time limit representations without explicit
coarse-grained sampling, and so may not be natural to a system that is
fundamentally discrete at high-frequency [15]. Moreover, trade time can also
be problematic when one mixes data generating processes. For example, the
aggregation of intraday transactions in trade time to low-frequency
transactions sampled below one day. Here the data generating processes can be
vastly incommensurate; intraday transactions represent prices arising from
continuous-time double auctions and traders operating on these time scales.
Whereas, market close prices are from an entirely different market structure,
specifically closing auctions. These can be populated by different trading
agents with different decision horizons and liquidity expectations. Sampling
daily closing auction prices to even lower frequencies, such as weekly,
monthly or even yearly, can also be problematic if key properties of the power
spectrum are not preserved.

Here we restrict ourselves to high-frequency and intraday day time scales,
i.e., to the high-frequency and mesoscale. Here the intermediate mesoscale is
defined by time scales at which correlations are still emergent. For the
purposes of our discussion, low-frequency finance has well-defined
correlations that can be well represented by traditional latent models. Here
we conceptually associate low-frequency with anything sampled from daily
closing auction data where agent decisions are synchronised by the mechanism
of the auction process.

Aside from what time scales can be appropriate for a given event process and
the associated decisions being made from the resulting data, the question of
whether these can be uniquely aggregated or mapped into various notions of
global system time depends on how the correlations between these event
processes emerge in the representations and the implicit filtering that takes
place on the underlying data. Any choice of time scale implies bounds on what
can be measured and represented by those measures being computed from the
resulting filtered and sampled data. The choice of time carries implicit
implications for the uncertainty relations that link various measurables,
particularly under averaging and sampling — the cardinal theorems of
interpolation [26, 32, 36, 37, 38]. However, here it is the Whittaker-Shannon
interpolation theorems and the link to the Gabor limit theorems that is
probably of most interest. We may need to be particularly careful of the
bandwidth versus time-limiting constraints [20] when making choices relating
the sampling time scales near to the emergence of events themselves. The
importance of this will not be discussed any further here, however, it is
something that we would like to urge readers to remain cognisant of as it may
provide constraints on the correlations and their dynamics as they emerge.

The fundamental question remains unanswered: What choice of time is
appropriate when we have chosen a particular sampling time scale when
computing cross-correlations between different assets with vastly different
event generation processes? Correlations have yet to emerge at small time
scales near individual events. Whereas, at larger time scales, the aggregation
of events impacts the nature of cross-correlations and how one measures the
cross-impacts and relationships between event generation processes.

##  3 Estimators

###  3.1 Malliavin–Mancino

Malliavin and Mancino [27, 28] introduced a Fourier estimator that expresses
the Fourier coefficients of the volatility process using the Fourier
coefficients of the price process $X^{i}_{t}=\ln(p^{i}_{t})$, where
$p^{i}_{t}$ is the generic asset price at time $t$ for asset $i$.

Following Malliavin and Mancino [28], by rescaling the transactions times
between $[0,T]$ to $[0,2\pi]$ and using the Bohr convolution product, we have
that for all $k\in\mathbb{Z}$ and $N$ samples:

$\displaystyle\mathcal{F}(\Sigma^{ij})(k)=\lim_{N\rightarrow\infty}\frac{2\pi}{2N+1}\sum_{|s|\leq
N}\mathcal{F}(dX^{i})(s)\mathcal{F}(dX^{j})(k-s),$ (1)

where $\mathcal{F}(\ast)(\star)$ is the $\star$th Fourier coefficient of the
$\ast$ process.

Using previous tick interpolation and a simple function approximation for the
Fourier coefficients (see [28]), we obtain the Dirichlet representation of the
integrated volatility/co-volatility estimator:222Note that $\mathrm{i}\mkern
1.0mu\in\mathbb{C}$ is such that $\operatorname{Re}(\mathrm{i}\mkern 1.0mu)=0$
and $\operatorname{Im}(\mathrm{i}\mkern 1.0mu)=1$. It should not be confused
with integer indices $i$, for example on the times $t^{i}_{h}$.

$\hat{\Sigma}^{ij}_{n,N}=\frac{1}{2N+1}\sum_{\begin{subarray}{c}|s|\leq N\\\
h=0,\ell=0\end{subarray}}^{n_{i}-1,n_{j}-1}e^{\mathrm{i}\mkern
1.0mus(t^{j}_{\ell}-t^{i}_{h})}\delta_{i}(I_{h})\delta_{j}(I_{\ell}),$ (2)

where $U^{i}=\\{t_{h}^{i}\\}_{h=0}^{n_{i}}$ and
$U^{j}=\\{t_{\ell}^{j}\\}_{\ell=0}^{n_{j}}$ are the asynchronous transactions
times observed between $[0,T]$ for each asset, and the price fluctuations are:

$\delta_{i}(I_{h})=X^{i}_{t_{h+1}^{i}}-X^{i}_{t_{h}^{i}},\quad\delta_{j}(I_{\ell})=X^{j}_{t_{\ell+1}^{j}}-X^{j}_{t_{\ell}^{j}}$

for asset $i$ and $j$ respectively, and $n_{i}$ is the sample dimension for
price $p^{i}$ and $n_{j}$ that of the price $p^{j}$, which a priori can be
different.

The advantage behind the estimator is its ability to investigate different
time scales through the choice of $N$ [12]. The conversion is given as:

$N=\left\lfloor\frac{1}{2}\left(\frac{T}{\Delta t}-1\right)\right\rfloor.$ (3)

This means that there is no need to re-sample the raw observations onto a
homogeneous grid using previous tick interpolation, such as in the case when
using the RV estimator.

###  3.2 Hayashi–Yoshida

Hayashi and Yoshida [24] introduced a cumulative covariance estimator defined
as:

$\displaystyle\hat{\Sigma}^{ij}_{T}$
$\displaystyle=\sum_{h=1}^{\\#U^{i}}\sum_{\ell=1}^{\\#U^{j}}\left(X_{t_{h}^{i}}^{i}-X_{t_{h-1}^{i}}^{i}\right)\left(X_{t_{\ell}^{j}}^{j}-X_{t_{\ell-1}^{j}}^{j}\right)w_{h\ell},$
(4)

where

$w_{h\ell}=\left\\{\begin{array}[]{ll}1&\text{ if
}\left(t_{h-1}^{i},t_{h}^{i}\right]\cap\left(t_{\ell-1}^{j},t_{\ell}^{j}\right]\neq\emptyset,\\\
0&\text{ otherwise. }\end{array}\right.$ (5)

Here $U^{i}=\\{t_{h}^{i}\\}_{h\in\mathbb{Z}}$ is the set of asynchronous
transactions observed between $\left[0,T\right]$ for asset $i$ and
$\\#U^{i}=n_{i}$ denotes the cardinality of set $U^{i}$.

The advantage behind the estimator is its ability to ameliorate the
statistical cause of the Epps effect arising from asynchrony. The weakness of
this estimator is that it is unable to investigate different time scales,
which is particularly problematic because asynchrony is only one of the
various sources of the Epps effect [15].

###  3.3 Realised Volatility

Both the Malliavin–Mancino and Hayashi–Yoshida estimator were designed to
compute estimates in calendar time, while dealing with the issue of
asynchronous arrival of transactions. Both estimators overcome the need to re-
sample the process onto a synchronous and homogeneous grid, for example, by
using previous tick interpolation. However, given that we plan to re-sample
the process under different definitions of time, it becomes useful to consider
how the two estimators relate to the Realised Volatility (RV) estimator.

In the case of the Hayashi–Yoshida estimator, when $t_{h}^{i}$ and
$t_{\ell}^{j}$ are synchronous and homogeneously spaced, we see that eq. 4
reduces to the RV estimator given as:

$\displaystyle\hat{\Sigma}^{ij}_{T}$
$\displaystyle=\sum_{h=1}^{n}\left(X_{t_{h}^{i}}^{i}-X_{t_{h-1}^{i}}^{i}\right)\left(X_{t_{h}^{j}}^{j}-X_{t_{h-1}^{j}}^{j}\right).$
(6)

In the case of the Malliavin–Mancino estimator, when $t_{h}^{i}$ and
$t_{\ell}^{j}$ are synchronous, homogeneously spaced and $N=n/2$ (the Nyquist
frequency in the synchronous case, where $n=n_{i}=n_{j}$), then it was found
in [11] that the estimate were numerically the same as the RV estimate. This
will also be demonstrated and discussed in Section 4.4.

##  4 Experiments

Here we are only interested in transactions, for this reason the remainder of
the paper we will be using event time and transaction time interchangeably.

We will be comparing two types of sampling for calendar and event time. First,
when using the RV estimator, we will down sample a synchronous and homogeneous
grid (obtained through previous tick interpolation) by skipping observations
to investigate different time scales. For example, if we have 28,200 samples
that correspond to one second intervals in calendar time. This means we would
drop half the samples by sampling every second observation to achieve two
second intervals in calendar time. Second, when using the Malliavin–Mancino
estimator, we will be sampling different time scales through the Fourier
domain by choice of $N$ for the particular time definition.

The Malliavin–Mancino estimator presents several advantages. First, it does
not need to drop observations to investigate different time scales. For
example, suppose we have a total of 100 events. To investigate two unit time
scale in event time, we would need to drop half the events when re-sampling
for the RV estimator; whereas with the Malliavin–Mancino estimator, we would
use the full set of samples, but compute fewer Fourier coefficients to achieve
the two unit time scale using the relation in eq. 3.

Second, it has the ability to deal with asynchrony both in calendar and event
time (see Section 4.3 demonstrating why event time can present asynchrony).
Therefore, it does not require previous tick interpolation to synchronise the
observations. One must however be mindful that as a consequence of the
sampling theorem that the Malliavin–Mancino estimator can only investigate
time scales larger than the smallest interval between two transactions (in
both event and calendar time) to prevent aliasing.

Third, because it can deal with asynchrony, it uses the same fixed
observations when investigating different time scales. This leads to more
stable estimates at different time scales [10]. On the other hand,
observations are dropped when sampling at different time scales for the RV
estimator, therefore the time series changes slightly each time through the
loss in resolution. These changes cause larger changes to the correlation
estimate at each time scale, making the estimates less stable [10].

With regards to the Hayashi–Yoshida estimator, we include it as a baseline for
calendar and event time. This is because the estimator corrects for the
statistical cause of the Epps effect arising from asynchrony. Therefore, it is
of interest to see the baseline correlation level corrected for asynchrony
under each definition of time.

In the case of volume time, sampling in the Fourier domain with the
Malliavin–Mancino estimator is not possible. This is because the different
time scales in volume time are determined by the number of samples obtained,
which depends on the size of the volume bucket, which determines how many
shares will be aggregated to obtain a sample. This aggregation changes the
time series at different time scales — it is not just a loss in resolution
from skipping observations.

###  4.1 Hawkes process

To generate a price process based on events, we will be using the class of
multivariate point processes introduced by Hawkes [23] known as a Hawkes
process. Concretely, we will use the fine-to-coarse model introduced by Bacry
et al. [2]. Let the bivariate log-price be:

$\displaystyle X_{t}^{1}=X_{0}^{1}+N_{1}(t)-N_{2}(t),$ (7) $\displaystyle
X_{t}^{2}=X_{0}^{2}+N_{3}(t)-N_{4}(t),$

where $\\{N_{m}(t)\\}_{m=1}^{M}$ is a 4-dimensional ($M=4$) mutually exciting
Hawkes process with the associated intensity
$\lambda(t)=\\{\lambda^{m}(t)\\}_{m=1}^{M}$ taking the form:

$\lambda^{m}(t)=\mu+\sum_{n=1}^{M}\int_{-\infty}^{t}\phi^{mn}(t-s)dN_{s}^{n}.$
(8)

The counting processes are coupled through:

$\boldsymbol{\Phi}=\begin{pmatrix}0&\phi^{(r)}&\phi^{(c)}&0\\\
\phi^{(r)}&0&0&\phi^{(c)}\\\ \phi^{(c)}&0&0&\phi^{(r)}\\\
0&\phi^{(c)}&\phi^{(r)}&0\end{pmatrix},$ (9)

where $\phi^{(r)}=\alpha^{(r)}e^{-\beta t}\mathbbm{1}_{t\in\mathbb{R}^{+}}$
and $\phi^{(c)}=\alpha^{(c)}e^{-\beta t}\mathbbm{1}_{t\in\mathbb{R}^{+}}$. The
parameters are
$(\mu,\alpha^{(r)},\alpha^{(c)},\beta)=(0.015,0.023,0.05,0.11)$, borrowed from
Bacry et al. [2].

The interpretation of eq. 9 is as follows: $\phi^{(r)}$ achieves mean
reversion, because an uptick in $X^{1}$ by $N_{1}$ will lead to an increased
intensity in the down tick $N_{2}$ — allowing the price level to revert
(similarly for $X^{2}$ through $N_{3}$ and $N_{4}$). While $\phi^{(c)}$
induces a correlation between the prices by connecting the two prices, since
an uptick in $X^{1}$ by $N_{1}$ will lead to an increased intensity in the
uptick of $X^{2}$ through $N_{3}$ (similarly for down ticks through $N_{2}$
and $N_{4}$).333The specification of eq. 9 in the model is fully symmetric and
thus there are no lead-lag effects (see [2] and Remark 9 of [3]).

To ensure stability with stationary increments, the spectral radius of the
branching matrix $\boldsymbol{\Gamma}$ must be strictly less than 1, i.e., the
magnitude of the largest eigenvalue must be less than 1. For our exponential
kernel, we have
$\boldsymbol{\Gamma}=\\{{\alpha^{mn}}/{\beta^{mn}}\\}_{m,n=1}^{M}$.

A key insight from Bacry et al. [2] is their formulation of the covariance
matrix for the model under calendar time sampling with increment size $\Delta
t$:

$\displaystyle\frac{C_{\Delta t}^{11}}{\Delta t}=$
$\displaystyle\Lambda+\frac{RC_{1}}{2G_{1}}+\frac{RC_{2}}{2G_{2}}$
$\displaystyle+R\frac{C_{2}G_{1}^{2}\mathrm{e}^{-\Delta
tG_{2}}-C_{1}G_{2}^{2}+Q_{1}G_{2}^{2}\mathrm{e}^{-\Delta
tG_{1}}-C_{2}G_{1}^{2}}{2G_{2}^{2}G_{1}^{2}\Delta t},$

and

$\displaystyle\frac{C_{\Delta t}^{12}}{\Delta t}=$
$\displaystyle\frac{-RC_{1}}{2G_{1}}+\frac{RC_{2}}{2G_{2}}$
$\displaystyle+\frac{R\left(C_{1}G_{2}^{2}-C_{2}G_{1}^{2}-C_{1}G_{2}^{2}\mathrm{e}^{-G_{1}\Delta
t}+C_{2}G_{1}^{2}\mathrm{e}^{-G_{2}\Delta t}\right)}{2G_{2}^{2}G_{1}^{2}\Delta
t}.$

Here the parameters are:

$\displaystyle\Lambda$ $\displaystyle=\frac{\mu}{1-\Gamma_{12}-\Gamma_{13}},$
$\displaystyle R$ $\displaystyle=\frac{\beta\mu}{\Gamma_{12}+\Gamma_{13}-1},$
$\displaystyle C_{1}$
$\displaystyle=\frac{\left(2+\Gamma_{12}+\Gamma_{13}\right)\left(\Gamma_{12}+\Gamma_{13}\right)}{1+\Gamma_{12}+\Gamma_{13}},$
$\displaystyle C_{2}$
$\displaystyle=\frac{\left(2+\Gamma_{12}-\Gamma_{13}\right)\left(\Gamma_{12}-\Gamma_{13}\right)}{1+\Gamma_{12}-\Gamma_{13}},$
$\displaystyle Q_{1}=Q_{4}$
$\displaystyle=\frac{-\mu\left(\Gamma_{12}^{2}+\Gamma_{12}-\Gamma_{13}^{2}\right)}{\left(\left(\Gamma_{12}+1\right)^{2}-\Gamma_{13}^{2}\right)\left(1-\Gamma_{12}-\Gamma_{13}\right)},$
$\displaystyle Q_{2}=Q_{3}$
$\displaystyle=\frac{-\mu\Gamma_{13}}{\left(\left(\Gamma_{12}+1\right)^{2}-\Gamma_{13}^{2}\right)\left(1-\Gamma_{12}-\Gamma_{13}\right)},$

and

$G_{1}=\beta\left(1+\Gamma_{12}+\Gamma_{13}\right),\quad
G_{2}=\beta\left(1+\Gamma_{12}-\Gamma_{13}\right).$

The correlation is then given by:

$\rho_{\Delta t}^{12}=\frac{C_{\Delta t}^{12}}{C_{\Delta t}^{11}},$ (10)

and the Epps effect is present. Moreover, in the limit we have that [2]:

$\lim_{\Delta t\to\infty}\rho_{\Delta
t}^{12}=\frac{2\Gamma_{13}\left(1+\Gamma_{12}\right)}{1+\Gamma_{13}^{2}+2\Gamma_{12}+\Gamma_{12}^{2}}.$
(11)

The experiments that will follow in the next few subsections are conducted by
simulating 100 pairs of eq. 7 for $T=72,000$ seconds. Each of the price pairs
will be re-sampled into the appropriate format and correlations will be
computed under each definition of time for sampling intervals between 1 to 100
units. The mean correlation estimate at each sampling interval will be plotted
along with error ribbons that contain 95% of the estimates at each sampling
interval. These are computed from the 100 replications using the student
$t$-distribution with 99 degrees of freedom and the standard deviation of the
estimates between the replications at each sampling interval.

Moreover, to demonstrate how the sampling is achieved under each definition of
time, we simulate a single pair of eq. 7 for $T=300$ seconds.

###  4.2 Calendar time

(a) Sampled for RV

(b) Sampled for MM and HY

Figure 1: The Hawkes price model in calendar time. The figure demonstrates the
two types of input samples (given as bubbles) for the different estimators.
(a) has synchronous and homogeneous samples with 10 unit intervals between
observations. The synchronisation is achieved using previous tick
interpolation. (b) has asynchronous samples based on when the events occurred.
Figure 2: The correlation estimate at different sampling intervals in calendar
time for the MM (blue) and RV (red) estimates. The HY (brown) estimate is
provided as a baseline. The theoretical Epps effect from eq. 10 and the
limiting correlation from eq. 11 are also provided.

The calendar time sampling is relatively straightforward. When sampling for
the RV estimator, we sample eq. 7 at equidistant and synchronous points
between $[0,T]$ using a particular sampling interval. As an example, Figure 1a
demonstrates the samples (given as bubbles) where the sampling the done using
10 unit intervals (corresponding to 10 seconds). For the Malliavin–Mancino
(MM) and Hayashi–Yoshida (HY) estimator, the samples are when the events (in
this case jumps) occurred. This is demonstrated in Figure 1b where the bubbles
are the samples. Notice that when the sampling interval increases for Figure
1a we lose more resolution into the price path which is present in Figure 1b.

Figure 2 plots the correlation for the RV and MM estimates in calendar time
measured at different sampling intervals with the HY estimates provided as a
baseline estimate.

We see that the RV estimator recovers the theoretical Epps effect which is
expected as this was the estimator used to derive the covariance matrix in
Bacry et al. [2]. The MM estimator does not quite recover the same theoretical
curve but it captures a more pronounced the Epps effect. This is because there
is the effect of asynchrony contributing towards the Epps effect in this case.
The HY estimates recover a rather strange estimate. This is because it was
found that the correction achieved here depends on the inter-arrivals of the
asynchronous events [15]. This fact was used by Chang et al. [15] to design
experiments to detect if the system has emerging correlations as in the case
of eq. 7 or if the system has correlations that exist at all sampling
intervals such as a geometric Brownian motion.

###  4.3 Event time

(a) Sampled for RV

(b) Sampled for MM and HY

Figure 3: The Hawkes price model in event time where the securities share the
same event clock. The figure demonstrates the two types of input samples
(given as bubbles) for the different estimators. (a) has synchronous and
homogeneous samples with one unit intervals between observations. The
synchronisation is achieved using previous tick interpolation. (b) has
asynchronous samples on a homogeneous grid based on the ordering of when the
events occurred. Figure 4: The correlation estimate at different sampling
intervals in event time for the MM (blue) and RV (red) estimates. The HY
(brown) estimate is provided as a baseline.

Event time is interesting because in theory, there is a one-to-one mapping
between the event count $h$ to calendar time $t_{h}$ when considering one
security.444This mapping can be difficult to find because there is often
measurement error associated with when events occur on an exchange. Event time
however becomes slightly problematic when we want to compute the correlations
between two securities. This is because each security will have a different
number of events, therefore the event clock between securities are not
comparable. Thus we need a method to unify the time between the securities.
This can be achieved if the securities share the same event clock, where an
event in either security will increase the time by one unit.

Under this construction, the time series will be asynchronous but homogeneous,
therefore we need to use previous tick interpolation to create the time series
in event time for the RV estimator. Figure 3a demonstrates the samples in
event time for the RV estimator where the sampling is done using one unit
intervals. Figure 3b demonstrates the samples in event time for the MM and HY
estimator where the samples are asynchronous but live on a homogeneous grid.
Comparing Figure 3 against Figure 1b, we see that event time indeed achieves a
smoothing effect as the bursts of activity in calendar time is stretched out.

Figure 4 plots the correlation for the RV and MM estimates in event time
measured at different sampling intervals with the HY estimates provided as a
baseline estimate.

We see that both the MM and RV estimates present an Epps effect in event time.
Moreover, we see that correlations seem to emerge faster with a more
pronounced concavity in the curves. The HY estimates in this case is
particularly interesting, it actually recovers the same estimates as in
calendar time as a result of how the event time is constructed (see a clearer
comparison in Figure 8). This is because even though the time series is
stretched out, the observations used in eq. 4 remain the same and because the
events are asynchronous, the intervals where $w_{h\ell}$ is activated remains
the same.

The relationship between the MM and RV estimates under the two definitions of
time remains unclear. Particularly for the MM estimator. This is because the
shifting of events through stretching out the time series will alter the power
spectrum. Thus the relationship is unclear. Related to this, the pronounced
concavity in event time could potentially be related to the shifting and
stretching of the time series which could induce potential lead-lag effects.
However, currently it is unclear how this mechanism could potentially explain
the observed dynamics. The difficulty lies in that lead-lag is usually induced
from a calendar time perspective through the shifting of observations (see the
model in Mastromatteo et al. [29]), whereas here the time series is also
stretched. Moreover, the shifting and stretching here is contingent on the
asynchronicity between the two time series. Nonetheless, the well understood
concave Epps effect remains present under both definitions of time.

###  4.4 Volume time

(a) 10 unit sampling interval in volume time

(b) 5 unit sampling interval in volume time

Figure 5: The Hawkes price model with volumes samples from an IID power-law
distribution given in eq. 14. The sampling interval was chosen to be the
equivalent of (a) 10 unit and (b) 5 unit sampling interval in calendar time
based on the number of samples obtained (relative to Figure 1). The samples in
volume time are synchronous, homogeneous and requires different sized volume
buckets (for averaging) to achieve. Figure 6: The correlation estimate at
different sampling intervals in volume time for the MM (blue), RV (red) and HY
(brown) estimates. Figure 7: The Epps effect in volume time for different
distributions generating the volume samples. The estimates are obtained using
the RV estimator.

The sampling of volume time is different to that of calendar and event time.
Although time in this definition is increased by one unit for each share
traded, sampling it requires the use of volume buckets which determine the
number of samples we get for each security. This method of sampling follows
Easley et al. [17] which they used to compute VPIN. The sampling overcomes the
problem of different volumes traded between securities. The problem however
with this sampling method and volume time in general is that there is no clear
mapping back to calendar time. This is because prices between transactions
will be aggregated and multiple volume times will map back to the same
calendar time. For example, suppose a transaction occurred with five shares
traded. These five volume time counts would map back to the same instant in
calendar time.

To obtain a notion of sampling interval in volume time, we will use the number
of samples obtained to get an idea of the equivalent sampling interval in
calendar time. For example, if a sampling interval of 10 units translates to
30 samples in calendar time, in order to obtain a 10 unit sampling interval in
volume time, we need to pick a volume bucket that allows us to obtain 30
samples.

The sampling in volume time follows partly from appendix A1 from Easley et al.
[17]. The transaction time series for a security $i$ is defined by the tuple:

$(t^{i}_{h},p^{i}_{t^{i}_{h}},v^{i}_{t^{i}_{h}}),$ (12)

where $t^{i}_{h}$ is the transaction time, $p^{i}_{t^{i}_{h}}$ is the
transaction price and $v^{i}_{t^{i}_{h}}$ is the transaction volume. Let $n$
be the number of samples we want from the volume time sampling, then the
volume bucket $V$ is given as:

$V=\left\lfloor V_{\text{tot}}/n\right\rfloor,$ (13)

where $V_{\text{tot}}=\sum_{h}v^{i}_{t^{i}_{h}}$ is the total number of shares
traded. Expand the number of observations by repeating each transaction
$p^{i}_{t^{i}_{h}}$ as many times as $v^{i}_{t^{i}_{h}}$, generating
$V_{\text{tot}}$ number of observations. Looping through the expanded prices,
create a sample in volume time after $V$ shares by averaging the expanded
prices, resulting in a total of $n$ samples. In other words, average
$p^{i}_{t^{i}_{h}}$ by its volume contribution to each bucket $V$ for each
volume time sample. Note that the last bucket is always incomplete or empty,
thus these observations are discarded [17]. Under this sampling, we will have
synchronous and homogeneous samples as each security will always have $n$
samples.

Crucially, Hawkes processes only model when the events occur; it does not
provide us with information about the states of the features associated with
these event times. Additional model assumptions are necessary to generate a
realistic tuple.

A realistic model for the time series of transaction requires the model to
account for the couplings between the tuples. First, the prices
$p^{i}_{t^{i}_{h}}$ and volumes $v^{i}_{t^{i}_{h}}$ are coupled through market
impact [5, 25].555Strictly speaking, market impact is the coupling between the
mid-price $m_{t}$ and volume. However, the transaction price $p_{t}$ is
related to the mid-price as: $p_{t}=m_{t}+s_{t}/2$ for buyer-initiated
transactions where $s_{t}$ is the spread. Therefore, prices and volumes are
coupled in how the transaction volume shifts the mid-prices through time.
Second, the volumes $v^{i}_{t^{i}_{h}}$ depend on previous volumes as a large
source of the order-flow auto-correlation is found to be from order splitting
[42]. Additionally, the price observations are fundamentally discrete events.
Whether a continuous time representation using stochastic differential
equations is sufficient remains an important open question [15].

There appears to be paucity of realistic models that are able to provide a
convincingly realistic 3-tuple
$(t^{i}_{h},p^{i}_{t^{i}_{h}},v^{i}_{t^{i}_{h}})$. Point processes like Hawkes
processes provide good models for the time of the events $t^{i}_{h}$.
Stochastic differential equations may provide good models for the prices
$p^{i}_{t^{i}_{h}}$ at low-frequency, and naturally provide long-memory [21].
There seems to be no consensus for a model specification for the volume of
market orders as the distributions varies widely with the product and market
[9].

Solving these issues are beyond the scope of the paper. Our objective here is
to highlight various concerns and what we do empirically know about the
behaviour of collections of traded asset events at high-frequency under
different choices of time. For the purposes of investigating the Epps effect,
the Hawkes model in eq. 7 provides a good basis for the simulation because it
deals with $t^{i}_{h}$ and $p^{i}_{t^{i}_{h}}$ in the tuple. Even though it
has unrealistic price changes, the model naturally recovers microscopic
properties such as the signature plot, mean reversion and the Epps effect, and
it can also naturally recover large scale diffusive properties [4].

To complete the tuple, we need a method of generating $v^{i}_{t^{i}_{h}}$.
Empirically, it was found that the unconditional distribution of market orders
follows a power-law behaviour [9]. Therefore, combining eq. 7 and a power-law
for the volume samples, we have a toy model to investigate the Epps effect in
volume time. Even though the model does not account for the empirical form of
measured price impact, we still find that it recovers the empirical dynamics
surprisingly well.

In order to perform the simulation work for volume time, we will assume that
the volume associated with each transaction is from an independent and
identically distributed (IID) power-law where the probability density function
is given as:

$f(x)=\begin{cases}\frac{\alpha x_{\mathrm{m}}^{\alpha}}{x^{\alpha+1}}&x\geq
x_{\mathrm{m}}\\\ 0&x<x_{\mathrm{m}}.\end{cases}$ (14)

For the toy model, we set $x_{\mathrm{m}}=20$ with exponent of $1+\alpha=2.7$
— the value found in Chakraborti et al. [9] for market orders. Note that the
volume samples are rounded to an integer.

Figure 5 demonstrates the volume time samples of the Hawkes price models with
volume samples from an IID power-law distribution. Figures 5a and 5b are 10
unit and 5 unit sampling intervals in volume time respectively. The sampling
interval was chosen to be the equivalent interval in calendar time based on
the number of samples. Each security has different sized volume buckets (for a
given sampling interval) to achieve the synchronous and homogeneous samples.
The samples here are no longer the same as those in calendar time since these
samples have gone through an averaging process. This is fundamentally
different to event time where the samples remain the same but smoothed.

Figure 6 plots the correlation for the RV, MM and HY estimates in volume time
measured at different sampling intervals. Here we see that all the estimators
recover the same estimates (see Section 3.3).

We see that the Epps effect is present in volume time. However, something
interesting happens here. The concavity of the Epps effect is lost, here the
correlations emerge linearly for larger sampling intervals and there is a
significant drop in the correlation achieved at larger intervals. This
behaviour is not just an anomaly under simulation. We see that this is also
the case with empirical data (see Figure 11).

To try and further understand this effect, Figure 7 computes the same Epps
curves in volume time as Figure 6, but for a range of distributions generating
the volume samples.

We consider the additional uniform, normal and beta distributions. The uniform
distribution samples transaction volumes between 1 to 100 with equal
probability. The normal distribution samples transaction volumes with a mean
of 50 and standard deviation of 5. We consider three beta distributions with
$\alpha=\beta$ but taking on values of 0.1, 0.2 and 2.666The beta distribution
samples values between 0 and 1. Therefore, the samples are multiplied by 100
before rounding to obtain a transaction volume. The distributions all have a
mean of roughly 50 shares for each transaction volume.

We see that regardless of the distribution, the Epps effect is linear in
volume time. The exact cause leading to a linear Epps curve is not clear.
However, from Figure 7 we see that the linear effect does not seem to depend
on the particular distribution generating the volume samples as all the
distributions present this linear effect.777Unravelling potential lead-lag
effects is particularly difficult for volume time because lead-lag cannot be
easily separated from shrinking and stretching as these are also mixed by the
averaging required to construct volume bins.

Figure 8: Comparison of the Epps curves from Figures 2, 4 and 6. We
investigate calendar time (CT), event time (ET) and volume time (VT) using the
RV, MM and HY estimator. The theoretical Epps effect eq. 10 and limiting
correlation eq. 11 from calendar time are also included.

Finally, to end this section, we compare all the Epps curves from Figures 2, 4
and 6 together in Figure 8. We see that the Epps effect is present under all
three definitions of time. For calendar and event time, sampling in the time
domain or Fourier domain both result in an Epps effect. Interestingly, the
Epps effect becomes linear under volume time.

##  5 Empirical

We investigate the Epps effect under different definitions of time using
transaction data from the Johannesburg Stock Exchange (JSE) for five trading
days ranging from 24/06/2019 to 28/06/2019. We consider Standard Bank Group
Ltd. (SBK), Nedbank Group Ltd. (NED), Absa Group Ltd. (ABG) and FirstRand Ltd.
(FSR). The data was extracted from Bloomberg Pro and is only reported up to an
accuracy of seconds (hence the data has been effectively re-sampled). Although
the transactions were in the correct order, without an exact millisecond or
nanosecond time stamp means that we cannot correctly order the events for
event time sampling.888Consider two securities which have 3 and 5 transactions
that all occurred in the same second. It is not possible to order these events
on a joint event count. Therefore, we process the data by aggregating
transactions (for each security) with the same time stamp using a volume
weighted average.

(a) MM

(b) RV

(c) HY

Figure 9: The Epps effect in calendar time for the six correlation pairs. The
sampling interval ranges from 1 to 300 units. Figures 9a, 9b and 9c we have
the MM, RV and HY estimates respectively.

(a) MM

(b) RV

(c) HY

Figure 10: The Epps effect in event time for the six correlation pairs. The
sampling interval ranges from 1 to 300 units. Figures 10a, 10b and 10c we have
the MM, RV and HY estimates respectively. Figure 11: The Epps effect in volume
time for the six correlation pairs. The sampling interval ranges from 1 to 300
units. The figure only reports the RV estimates because the three estimators
recover the same estimates under volume time (see Figure 6).

Correlations are computed separately for the trading days so that we do not
have to worry about overnight returns. Therefore the correlations reported in
the following subsections are the average at each sampling interval aggregated
across the five trading days.

###  5.1 Calendar time

The continuous trading session on the JSE is between 09:00 and 16:50, this
translates to a seven hour and 50 minute trading day or $T=28,200$ seconds. As
before, we sample a synchronous and homogeneous grid at different time scales
using previous tick interpolation for the RV estimator. The MM and HY
estimator use the samples from when the transactions occurred. The MM
investigates different time scales by sampling the Fourier domain through the
relation in eq. 3 while the HY provides a baseline estimate after correcting
for asynchrony.

Figure 9 plots the Epps effect in calendar time for the six correlation pairs
for sampling intervals ranging from 1 to 300 units for the MM, RV and HY
estimator in Figures 9a, 9b and 9c respectively.

First, we see that the MM and RV estimates have yet to reach the limiting
(asymptotic) correlation after 300 seconds. Second, we see that the RV
estimates are less stable moving from one sampling interval to the next. This
is because sampling different time scales using previous tick interpolation
changes the samples (through the loss in resolution) for the estimator. This
was not seen in Figure 2 because this instability was hidden away when
averaging 100 correlation estimates; whereas here we are only averaging over
five days of trading.

###  5.2 Event time

Lining up the events in event time with empirical data is slightly
problematic. The data set we have only reports time stamps up to the second,
this means that we can have concurrent events at the same time between a pair
of securities (even after aggregating events on the same second using a volume
weighted average). In theory, these should be two separate event counts when
sharing the same event clock, however since there is no way to determine the
ordering of these events, we allow them to happen simultaneously, i.e., they
share the same event count. This was not a problem under simulation because we
had a well-defined point process.

Figure 10 plots the Epps effect in event time for the six correlation pairs
for sampling intervals ranging from 1 to 300 units for the MM, RV and HY
estimator in Figures 10a, 10b and 10c respectively.

We see again that the Epps effect is present and correlations seem to emerge
faster in event time. Both the MM and RV estimator reaches the limiting
(asymptotic) correlation after around 100 units in event time. Again, we also
see that the HY estimate is exactly the same as the estimate in calendar time.
This is due to how the events are lined up. Even though the time series is
smoothed out in event time, the observations in eq. 4 remain the same and the
intervals where $w_{h\ell}$ is activated remains the same.

(a) SBK/FSR

(b) NED/ABG

Figure 12: The Epps curves compared under different definitions of time for
(a) the SBK/FSR and (b) the NED/ABG pair. Included are the calendar time (CT),
event time (ET) and volume time (VT) estimates for the range of 1 to 300 unit
sampling intervals using the RV and MM estimator. The HY estimator is provided
as a baseline.

###  5.3 Volume time

Sampling in volume time is done by aggregating transaction prices over a
certain number of shares determined by the volume bucket. As before, the
sampling interval here is defined by the number of samples obtained. The
number of samples obtained is compared against the number of sampled obtained
in calendar time to find an equivalent sampling interval in volume time.

Figure 11 plots the Epps effect in volume time for the six correlation pairs
for sampling intervals ranging from 1 to 300 units. Here we only report the RV
estimates because the three estimators recover the same estimates in volume
time (see Figure 6).

We see that as in the case of our simulations, the Epps effect is present in
volume time. Moreover, correlations emerge linearly rather than exponentially
for larger sampling intervals as in the case of calendar and event time; and
there is a significant drop in the correlations achieved for larger sampling
intervals compared calendar and event time.

Looking at the NED/FSR pair (purple line) in Figure 11, we see that the
correlation becomes negative in volume time. This behaviour is possible under
simulation with our toy model as the error bars in Figure 6 do go into the
negative region, meaning that this is possible. However, at this stage it
remains unclear as to why this is occurs.

###  5.4 Comparison

Figure 12 compares the various the Epps curves under different definitions of
time. This is done for two of the six correlations pairs SBK/FSR and NED/ABG
in Figures 12a and 12b respectively.

Comparing Figure 12 against Figure 8, we see that the empirical behaviours of
the various Epps curves line up with what we have seen using the toy model
under simulation. The event time correlations emerge faster than those in
calendar time, while the Epps effect is linear under volume time. Sampling in
the time domain or Fourier domain both result in an Epps effect under calendar
and event time. In A, we provide the same comparison between uncorrelated
asset pairs. The generalisation of these results is only confirmed to hold for
correlated asset pairs where the Epps effect behaves as expected.

Given that this relatively simple toy model captures the various empirical
dynamics, it is unlikely that a more realistic simulation of the 3-tuple
$(t^{i}_{h},p^{i}_{t^{i}_{h}},v^{i}_{t^{i}_{h}})$ would affect the correlation
estimates. In particular, intraday seasonality of the volatility,
transactions, and volumes that are present with empirical data [8] does not
alter the dynamics compared to the toy model which does not capture these
seasonality effects. A possible reason why the intraday seasonality effects do
not alter the dynamics is because the correlation estimates are integrated
quantities.999We speculate that intraday seasonality may alter the dynamic of
the correlations in volume time, particularly when two assets have
significantly different intraday volume curves. Perhaps these seasonality
effects may influence the dynamics differently if instantaneous correlation
quantities were computed instead.

##  6 Conclusion

In this paper, we compared the Epps effect under three definitions of time:
calendar time, trade time and volume time. We use a Hawkes process,
specifically, the fine-to-coarse model by Bacry et al. [2] as the basis for
the simulation of price paths. Combining this with a power-law distribution
for the volumes samples [9], we find that the Epps effect is present under all
three definitions of time.

Moreover, the simulations reveal different rates of correlation emergence.
First, we saw that the correlations emerge faster under trade time compared to
calendar time. Second, correlations emerge linearly under volume time and do
not seem to depend strongly on the distribution generating the volume samples.
Under this setup, where the volumes and prices are independent, we find
realistic results. These results are found despite using a model that does not
directly encode price impact or intraday seasonality of the volatility,
transactions, and volumes.

We then investigated the Epps effect under these three definitions of time
using transaction data from the Johannesburg Stock Exchange. We found that the
empirical Epps curves conform well with what we observed under simulation for
the various definitions of time. However, the results hold for the case of
sufficiently correlated asset pairs where the Epps effect behaves as expected.

This paper presents some anomalies that raise several unresolved concerns.
First, the underlying mechanisms that link the different temporal definitions
and sampling schemes that lead to the different correlation dynamics remains
unclear. Second, an analysis of the instantaneous correlation estimates could
be performed for the different definitions of time. This could potentially
lead to a better understanding of how intraday seasonality could impact the
correlation dynamics under different sampling schemes. Finally, to explore how
market activity affects the Epps effect with different time choices. This
particularly relates to the observation made by Tóth and Kertész [40] where
the Epps curves do not scale with market activity.

## Reproducibility of the Research

The dataset can be found at [13]. The Julia code and instructions for
replication can be found in our GitHub site [14].

## Acknowledgements

We would like to thank the two anonymous referees for thoughtful comments
which helped us improve this paper. PC would like to thank Roger Bukuru for
work done together and various discussions.

## Funding

PC was supported by the Manuel & Luby Washkansky Scholarship and the South
African Statistical Association [grant number 127931].

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##  Appendix A Uncorrelated assets

(a) NED/AGL

(b) NED/BTI

Figure 13: The Epps curves compared under different definitions of time for
(a) the NED/AGL and (b) the NED/BTI pair. Included are the calendar time (CT),
event time (ET) and volume time (VT) estimates for the range of 1 to 300 unit
sampling intervals using the RV and MM estimator. The HY estimator is provided
as a baseline.

We investigate whether the correlation dynamics found between the different
definitions of time hold in general for all asset pairs. To this end, Figure
13 compares the Epps curves under different sampling schemes for the asset
pairs (a) NED and Anglo American plc (AGL), and (b) NED and British American
Tobacco plc (BTI). The correlations are reported in the same way as Section 5
by computing the correlations separately for the five trading days and
reporting the average at each sampling interval.

From the figure it is suggested that the Epps effect is not prominent in the
case of uncorrelated assets — this is an unsurprising result. However, we
still see that correlations emerge linearly and that there is are drop in the
correlations achieved in volume time compared to calendar and event time. The
correlation dynamics between calendar and event time becomes ambiguous.

Here the Epps effect itself starts to behave in an unexpected manner. The
calendar time correlations for the NED/AGL pair start off positive and become
negative for larger sampling intervals. A similar dynamic of the Epps effect
in calendar time has been reported between the FSR/AGL pair in Chang et al.
[12]. However, to the best of our knowledge, this has not been reported in the
boarder literature surrounding the Epps effect. This is likely because the
literature generally focuses on asset pairs with higher correlation levels
where the Epps effect then behaves as expected. Therefore, we conclude that
our results only hold in the case of sufficiently correlated assets. We
speculate that this is due to the nature of different types of trading agent
activity on different time scales.