# A Comparison of Cryptocurrency Volatility-benchmarking New and Mature Asset
Classes

Alessio Brini111Corresponding author<EMAIL_ADDRESS>Jimmie Lenz
<EMAIL_ADDRESS>
(Digital Asset Research and Engineering Collaborative (DAREC) Lab, Pratt
School of Engineering, Duke University
305 Teer Engineering Building Box 90271, Durham, NC 27708, United States)

Keywords: cryptocurrency; volatility; market regimes; leverage effect

## Abstract

The paper analyzes the cryptocurrency ecosystem at both the aggregate and
individual levels to understand the factors that impact future volatility. The
study uses high-frequency panel data from 2020 to 2022 to examine the
relationship between several market volatility drivers, such as daily
leverage, signed volatility and jumps. Several known autoregressive model
specifications are estimated over different market regimes, and results are
compared to equity data as a reference benchmark of a more mature asset class.
The panel estimations show that the positive market returns at the high-
frequency level increase price volatility, contrary to what is expected from
the classical financial literature. We attributed this effect to the price
dynamics over the last year of the dataset (2022) by repeating the estimation
on different time spans. Moreover, the positive signed volatility and negative
daily leverage positively impact the cryptocurrencies’ future volatility,
unlike what emerges from the same study on a cross-section of stocks. This
result signals a structural difference in a nascent cryptocurrency market that
has to mature yet. Further individual-level analysis confirms the findings of
the panel analysis and highlights that these effects are statistically
significant and commonly shared among many components in the selected
universe.

##  1 Introduction

Cryptocurrencies have continuously gained popularity since Bitcoin (BTC), the
first and most well-known cryptocurrency, was created in 2009, right after the
2008 financial crisis, as a response to traditional financial institutions’
perceived lack of trust. Since its advent, many other cryptocurrencies have
emerged, with around a 400% growth in the number of traded coins and tokens on
multiple exchanges since the beginning of
2020222https://www.statista.com/statistics/863917/number-crypto-coins-tokens/.
The rise of digital currencies can be attributed to several factors due to the
heterogeneous nature and use cases each has. One reason behind this sharp
increase in interest is their potential for enhancing financial freedom and
removing intermediaries from financial transactions. Individuals can make
peer-to-peer transactions with cryptocurrencies without a central authority or
intermediaries. This characteristic has made them particularly appealing to
those concerned about government control and censorship in traditional
financial systems. Another relevant factor contributing to the popularity of
cryptocurrencies is their growing acceptance and use as a form of payment by
merchants and businesses in less-developed countries where only some have
access to a traditional bank account. Furthermore, the development of
decentralized finance (DeFi) platforms, such as decentralized exchanges or
lending protocols, has provided additional motivation to enter the
cryptocurrency ecosystem for those who may have previously been excluded from
traditional financial systems. However, beyond those premises, the
cryptocurrency market is still relatively immature and subjected to
substantial uncertainty and sudden failures. In that regard, the
cryptocurrency market historically showed high volatility traits (Bouri,
Shahzad and Roubaud, 2019; Bouri, Lucey and Roubaud, 2020; Bouri, Roubaud and
Shahzad, 2020) and potential for significant returns, resulting in being
attractive for retail investors. Therefore, despite its growing popularity,
the cryptocurrency market, if viewed in the context of price versus more
mature asset classes, is notoriously unstable, with frequent and substantial
fluctuations in value.

A known stylized fact of traditional financial market is the asymmetric impact
of financial returns on the future volatility of the assets, referred to as
the leverage effect (Black, 1976), in which negative returns increase the
asset’s volatility more than positive returns (Bollerslev et al., 2006;
Bollerslev and Zhou, 2006; Carr and Wu, 2017). The presence of the leverage
effect in a financial return series is often motivated by the market
participants’ response to adverse shocks. The increased volatility due to
negative returns can start a self-reinforcing mechanism that leads to sudden
asset firesales, exacerbating the uncertainty even further. Given the novelty
of cryptocurrencies as an asset class, it is essential to determine if the
asymmetric effect of volatility, commonly seen in mature markets, is also
present here. The unclear classification of cryptocurrencies on whether they
align with established asset classes or stand apart Baur et al. (2018); Corbet
et al. (2018), compounded by the evolving regulatory landscape Shanaev et al.
(2020), makes this investigation particularly crucial to provide a starting
point for a sound and data-driven comparison that can reflect how the market
perceives cryptocurrencies. Some previous work tested the asymmetric effect of
cryptocurrency returns at daily frequency (Phillip et al., 2018; Baur and
Dimpfl, 2018; Brini and Lenz, 2022) on a more or less comprehensive cross-
section of cryptocurrencies. The results at the daily level detect the absence
of a statistically significant leverage effect in such a market, showing an
inversion of the asymmetric effect of returns. In this case, investors exploit
market crashes as a buying opportunity to enter at a perceived discount. This
absence of panicking behaviors among market participants indifferent to the
sign of the returns in terms of reaction size let them enter the market
without understanding the asset class, for the most part, investing money
because this is considered a new investment with many potential opportunities.

This paper extends the analysis of the price volatility inherent in the
cryptocurrency market at a higher frequency level, exploring its dynamics and
explaining its main drivers. The developing interest in cryptocurrencies from
regulatory bodies partially stems from the lack of clarity regarding their
classification as an asset class (Corbet et al., 2019). A detailed study of
market behaviors, such as the asymmetric effects on volatility, can contribute
to a deeper understanding of the fundamental characteristics of
cryptocurrencies. By comparing these behaviors with those observed in
established asset classes, we can provide insights that may assist in
developing appropriate regulatory measures other than better-informed
investment choices. Such insights are particularly valuable given the
cryptocurrency markets’ rapid growth and the increasing number of market
participants. Therefore, the outcomes of such analysis can inform the
direction of regulatory policies, ensuring they are based on empirical
evidence of the asset class’s inherent nature. Our analysis aims to understand
the cryptocurrency’s volatility dynamics that emerge from high-frequency data
due to different market participants since retail investors usually operate on
daily if not lower frequencies (Auer et al., 2022). Previous works (Jha and
Baur, 2020; Ftiti et al., 2021) have already studied the volatility dynamics
of cryptocurrency at a frequency higher than daily. However, these analyses
are restricted to a single or few cryptocurrencies, and the insights are
detected and described only within the realm of the crypto ecosystem. We
extend the analysis in three directions to further elaborate on these aspects.
First, we consider historical high-frequency cryptocurrency data represented
by a large cross-section of coins and tokens to understand the volatility
dynamics of the global cryptocurrency market. We significantly extend the
analysis of cryptocurrency volatility by enlarging the analyzed universe,
which includes 87 cryptocurrencies. Secondly, we provide a basis for
comparison by performing the same volatility analysis on a cross-section of
stocks, representing a proxy of a more established and mature asset class. The
selected stocks come from the NASDAQ technology index (NDXT), whose returns
are positively correlated with the return of cryptocurrencies in the past
years (Goodell and Goutte, 2021). By focusing the volatility analysis on
multiple and larger cross-sections, we can fulfill a gap of interpretability
available in the literature that often overlooks a thoughtful comparison with
traditional financial markets, even knowing that cryptocurrencies often
exhibit a certain level of correlation with them. Lastly, the period covered
by our dataset, from 2020 to 2022, allows us to appreciate the significance of
our selected volatility drives under the latest bullish and bearish market
regimes that characterize the cryptocurrency market. Such a period has been
crucial for developing the cryptocurrency industry, which welcomed more
institutional investors among the active participants and experienced
astounding growth followed by sudden drawdowns. Previous works focus on the
analysis of periods pre-2020 when the market dynamics and cryptocurrency
adoption were less developed than in the subsequent years. We consider
combining these three elements essential to disentangle the peculiar and
common traits of cryptocurrencies as a whole with respect to more traditional
assets.

To this end, we adopt the framework of (Patton and Sheppard, 2015) by
obtaining volatility estimators from minute-level price data and fitting a set
of renowned interpretable econometric models to describe the volatility
patterns over time. Estimating volatility from historical data is crucial in
financial econometrics, as it helps assess risk and make informed investment
decisions. Various methods have been developed to estimate volatility,
including historical, implied, and realized volatility. Historical volatility
is often indicated as a measure of the standard deviation of an asset’s
returns over a specified period, usually calculated daily, while implied
volatility is derived from the prices of options on the underlying asset. The
latter is a forward-looking measure incorporating market expectations about
future volatility by assuming market efficiency and exact price discovery for
the options. In our work, we adopt the realized volatility as an estimator for
historical volatility, which employs intraday data. Unlike the volatility
calculated daily, realized volatility allows us to capture the past volatility
dynamics more accurately and can account for sudden changes in market
conditions. Being calculated on a larger amount of data points, it is also
less sensitive to outliers and price gaps, and it is also not affected by
assumptions on the market as the implied volatility. The choice is further
motivated by the possibility of accounting for the effects of market
microstructure noise, which can distort the volatility estimation. This is
particularly important in highly volatile markets such as cryptocurrency,
where small deviations from the true volatility can significantly affect
investment decisions.

Studying the volatility dynamics of the cryptocurrency market is important for
several scopes, among which managing risk effectively and hedging losses
(Almeida and Gonçalves, 2022), enhancing market efficiency through trading
(Fang et al., 2022) and designing derivatives (Söylemez, 2019; Akyildirim et
al., 2020; Soska et al., 2021). In particular, the presence of derivatives in
the market modulates the volatility transmissions among centralized
cryptocurrency exchanges (Alexander et al., 2022; Badenhorst et al., 2018;
Beneki et al., 2019). The cryptocurrency market is relatively new and
unregulated, meaning price discovery can be highly volatile and erratic. Under
this view, it is a known fact that the cryptocurrency market still contains a
lot of inefficiencies compared to more established financial markets (see Al-
Yahyaee et al. (2020) for an analysis of how a high volatility period harms
the market efficiency for cryptocurrencies). Studying the volatility dynamics
of cryptocurrencies can help traders identify such inefficiencies and improve
the amount of information reflected by the prices when they take advantage of
them. Hence, it makes the study of volatility patterns crucial for market
stability. Moreover, the cryptocurrency market’s relatively young age, the
lack of regulation, and the high level of speculation make a proper
understanding of the volatility dynamics a pressing problem for the industry
aiming to gain market stability.

The rest of the paper is organized so that Section 2 illustrates the related
literature and Section 3 introduces the methodologies to estimate the
volatility, provides details regarding the structure of the analyzed data, and
explains the employed autoregressive models in detail. Section 4 includes the
results of the empirical analysis, from the model estimation using panel data
to the robustness check by repeatedly fitting the same model specification
over a different time window. In addition, we carry out the individual
estimation of the same model for every entity in the cryptocurrency cross-
section to disentangle the aggregate results and validate them. Then Section 6
discusses the key findings and the implications of our results for the
cryptocurrency ecosystem and Section 7 provides the takeaways from the
volatility analysis.

##  2 Related Literature

The literature relevant to this study bifurcates into two distinct yet
interconnected domains. The first is the foundational body of work on
volatility modeling in finance, which has been a key topic in financial
econometrics. This part of the literature particularly explores modeling the
volatility of equity returns through variants of the autoregressive
conditional heteroskedasticity (ARCH) model (Engle et al., 1987) and its
generalized (GARCH) version (Bollerslev, 1987) (see also Bollerslev et al.
(1994); Andersen et al. (2006) for a comprehensive overview). The use of high-
frequency data substantiates the discoveries regarding volatility modeling,
among which the impact of signed returns on future volatility, referred to as
the leverage effect, represents a well-established empirical regularity in
financial market data. Bollerslev et al. (2006); Barndorff-Nielsen et al.
(2008); Chen and Ghysels (2011) highlights the effect of negative equity
returns on increasing future volatility.

The second strand of literature explores how these studies analyze equity
volatility patterns by computing high-frequency volatility estimators such as
the realized variance (Andersen et al., 2001) and the bipower variation
Barndorff-Nielsen and Shephard (2006). Alternative volatility estimators like
realized semivariance (Barndorff-Nielsen et al., 2008) dissect the realized
variance measure, isolating the elements attributable solely to positive and
negative high-frequency returns. The significance of these estimators lies in
their ability to retain the critical information encoded in the sign of
returns, which is otherwise lost when volatility is measured through the sum
of squared or absolute value returns.

The econometrics literature has shown how high-frequency price data can
improve the estimation and predictability of the volatility for a cross-
section of equities (Patton and Sheppard, 2015; Bollerslev, Li and Zhao,
2020). These studies leverage the heterogeneous autoregressive (HAR) model
proposed by Corsi (2009) and elaborate on its variants to account for the
direction of the volatility movements. All the models use realized variance
estimators and their variants obtained from high-frequency data.

The choice of the realized variance as an estimator is motivated by
efficiency, being an estimator with low variance among all the unbiased ones,
and by consistency, approaching the true value of the integrated variance
process when the number of observations increases. In addition, it is easy to
compute, as it just requires price data sampled at an intraday frequency. It
is also flexible because it can be generalized to other volatility measures
that reflect specific aspects of the volatility process, such as the realized
semivariance to assess the downside risk of an asset (see Aït-Sahalia and
Jacod (2014) for more details). Other volatility estimators do not consider
the market microstructure dynamics (daily historical variance) or are too
complex and not easily adaptable to high-frequency data (GARCH-like models).
Those choices, such as the Parkinson estimator (Parkinson, 1980) and its
extension Garman and Klass (1980), systematically underestimate the magnitude
of the volatility, calculating it on daily data and assuming the price process
as a continuous process without jump components.

The analysis of volatility in the cryptocurrency market has attracted
significant attention from researchers and practitioners due to the highly
volatile nature of this market. The use of those volatility estimators is
relatively new to the field of cryptocurrencies, where the realized variance
estimator is calculated and modeled on just BTC (Hu et al., 2019; Yu, 2019;
Shen et al., 2020). Common findings are the presence of an inverse leverage
effect that impacts the estimation of future volatility and the role of jumps
in shaping the Bitcoin volatility dynamics (Chaim and Laurini, 2018; Charles
et al., 2019). Previous studies have explored various aspects of
cryptocurrency market volatility daily, including its determinants, spillover
effects, and forecasting accuracy. For instance, Baur and Dimpfl (2018)
examined the volatility of several cryptocurrencies and found that different
factors, such as market capitalization and trading volume, drive their
volatility. Zhang et al. (2018) explores the stylized facts of the daily
returns of eight cryptocurrencies, finding heavy-tailed distributions. In
contrast, Phillip et al. (2018) study the stylized facts of a larger cross-
section of cryptocurrency daily returns. In particular, the literature
analyzed the time-varying relationship between financial returns and their
volatility at the daily level is usually investigated through a GARCH
framework. Many works (Liu and Serletis, 2019; Fakhfekh and Jeribi, 2020;
Wajdi et al., 2020) retrieve an inverted leverage effect in the cryptocurrency
market using several variants of the original GARCH model on daily data. Chu
et al. (2017) apply GARCH-like models to fit the best volatility model for
prediction, while Mostafa et al. (2021) use asymmetric GARCH models for risk
management assessment. Alqaralleh et al. (2020) compares the predictive power
of nonlinear GARCH-type models for a small subset of cryptocurrency assets
similar to López-Martín et al. (2022), which retrieves an inverse leverage
effect by examining the return distribution. Chaim and Laurini (2019) employ a
multivariate stochastic volatility model with discontinuous jumps to mean
returns and volatility and signals larger transitory mean jumps starting from
2017 as an effect of shifts in cryptocurrencies return dynamics. Dyhrberg
(2016) identifies Bitcoin as useful in risk management and ideal for risk-
averse investors in anticipation of negative shocks to the market through an
asymmetric GARCH model. Gradojevic and Tsiakas (2021) analyze the volatility
cascades for some highly capitalized cryptocurrencies. Ji et al. (2019)
explores the volatility connectedness, retrieving an asymmetric effect
depending on the direction of the returns.

Specifically, Jha and Baur (2020); Ftiti et al. (2021) explores the analysis
of cryptocurrency volatility by using intraday data of a small number of coins
and tokens. They discovered the inverse leverage effect at high frequency
based on data before 2020. Similarly, Naeem et al. (2022); Yousaf and Ali
(2020) study spillover effect on the volatility using generalized vector
autoregressive model versions. Mensi et al. (2021) focuses on the portfolio
management implication of high-frequency volatility patterns. Katsiampa et al.
(2019a) explores the volatility co-movements of eight cryptocurrencies on
hourly data, while Sensoy et al. (2021); Ampountolas (2022) explore the
spillover effects on the volatility of high-frequency cryptocurrency returns.
Ji et al. (2021); Zhang et al. (2019) investigate the stylized fact of high-
frequency return in terms of the Hurst Exponent. Katsiampa (2019) find
asymmetric effects between good and bad news among cryptocurrencies.

Baur and Dimpfl (2018); Brini and Lenz (2022) analyze the inversion of
leverage effect for the cryptocurrency market at the daily level by linking
the increased excitement and speculative behavior when the market is rising
with behavioral traits of the average uninformed investors, such as the fear
of missing out (FoMO) (McGinnis, 2004). Wang et al. (2023) explore the FoMo
idea specifically on BTC finding positive asymmetric volatility behavior in
the Bitcoin market. Huang et al. (2022) detail the contribution of jumps in
cryptocurrency price series to the inverse leverage effect. Panagiotidis et
al. (2022) retrieves the inversion of the leverage effect for a wide cross-
section of cryptocurrencies using a GARCH-like framework. Ardia et al. (2019)
find evidence of regime changes in the GARCH volatility process for BTC,
highlighting the temporary and transient nature of such volatility dynamics.
Katsiampa et al. (2019b) retrieve volatility spillover effect for various
cryptocurrencies using a BEKK-MGARCH model.

##  3 Methodology and Data

This Section outlines the steps needed to perform the empirical analysis in
the following Section. We first describe the methodology for obtaining the
daily volatility estimators from high-frequency data. Then, we present the
collected data outlining descriptive statistics of the universe of the two
asset classes, cryptocurrency, and equity. The final part of the Section
describes the model specifications employed and the estimation techniques to
gain insights into the drivers of future volatility.

###  3.1 Volatility Estimator

All the estimated econometric models in our empirical analysis are based on
high-frequency volatility estimators, making it essential to start by
describing the underlying assumptions and the choices to compute such
estimators. As outlined in Section 1, volatility estimators based on high-
frequency data are more expressive of the true variation of the cryptocurrency
returns and are less sensitive to the presence of outliers, making it a
suitable choice to study a market subjected to sudden and irrational
movements. Hence, we approach modeling the future volatility by first
describing the employed volatility estimators. Let $p_{t}$ a continuous-time
stochastic process of log prices

$p_{t}=\int_{0}^{t}\mu_{s}\mathrm{\leavevmode\nobreak\
d}s+\int_{0}^{t}\sigma_{s}\mathrm{\leavevmode\nobreak\ d}W_{s}+J_{t}$ (1)

including two continuous components and a jump component. We call $\mu$ a
locally bounded drift process, $\sigma$ a strictly positive cadlag process,
and $J$ a jump process. The quadratic variation of the process in Eq. Equation
1 is given by

$[p,p]=\int_{0}^{t}\sigma_{s}^{2}\mathrm{\leavevmode\nobreak\
d}s+\sum_{0<s\leq t}\left(\Delta p_{s}\right)^{2}$ (2)

where $\Delta p_{s}=p_{s}-p_{s-}$ captures the jump component.

The realized variance (RV) is an estimator of Eq. Equation 2 such that

$RV_{t}=\sum_{i=1}^{n}r_{t,i}^{2}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}[p,p]\text{, as }n\longrightarrow\infty$ (3)

where $r_{t,i}=p_{t}-p_{t-1}$ is the return calculated as the log price
difference. The estimator has been shown to converge in probability to the
quadratic variation as the time interval between each equally spaced
observation becomes small (Andersen et al., 2003). The RV represents the
quadratic variation of a process as the sum of squared returns sampled at a
high frequency. For instance, the $n$-sample RV measure is obtained from $n+1$
equally spaced sample of the price process. In general, using higher frequency
data to calculate realized variance can result in more accurate estimates of
volatility, as it captures more of the price fluctuations that occur within
the time period being analyzed. However, there are trade-offs to consider when
using higher-frequency data since a large number of data points can lead to
more noise in the data and make it more difficult to identify the true
volatility of the asset. Hence, the optimal sampling frequency to compute
realized variance is a trade-off between bias and efficiency (McAleer and
Medeiros, 2008). That optimal has been identified as the 5-minute frequency to
provide accuracy in estimation and avoid microstructure noises such as bid-ask
bounce and price discreteness (Andersen and Bollerslev, 1998).

Another powerful estimator of the variance process is the bipower variation
(BV), which, unlike the RV, converges in the limit to just the continuous
component of the quadratic variation, called integrated variance (IV)

$BV=\frac{\pi}{2}\sum_{i=2}^{n}\left|r_{i}\right|\left|r_{i-1}\right|\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\int_{0}^{t}\sigma_{s}^{2}ds.$ (4)

Due to the nature of these two estimators, their difference allows capturing
the jump component of the quadratic variation $RV-BV\stackrel{{\scriptstyle
p}}{{\rightarrow}}\sum_{0\leq s\leq t}\Delta p_{s}^{2}$

The RV estimator can be split into two signed realized semivariances,
$RV=RV^{+}+RV^{-}$, that respectively capture variation due to positive and
negative returns

$\displaystyle RV^{+}=\sum
r_{i}^{2}I\left[r_{i}>0\right]\stackrel{{\scriptstyle
p}}{{\longrightarrow}}1/2\int_{0}^{t}\sigma_{s}^{2}ds+\sum_{0<s<=t}\left(\Delta
p_{s}^{2}\right)I\left[\Delta p_{s}>0\right]$ $\displaystyle RV^{-}=\sum
r_{i}^{2}I\left[r_{i}<0\right]\stackrel{{\scriptstyle
p}}{{\longrightarrow}}1/2\int_{0}^{t}\sigma_{s}^{2}ds+\sum_{0<s<=t}\left(\Delta
p_{s}^{2}\right)I\left[\Delta p_{s}<0\right].$ (5)

These measures have been shown to include variations of both the continuous
and the jump part of Eq. Equation 2. The first term is equal for both
semivariances and not decomposable into a signed measure. It follows then that
jump variation is defined as

$\Delta J^{2}\equiv RV^{+}-RV^{-}\stackrel{{\scriptstyle
p}}{{\rightarrow}}\sum_{0\leq s\leq l}\Delta p_{s}^{2}I\left\\{\Delta
p_{s}>0\right\\}-\sum_{0\leq s\leq l}\Delta p_{s}^{2}I\left\\{\Delta
p_{s}<0\right\\}$ (6)

The difference eliminates the common integrated variance term and is positive
when positive shocks are prevalent in a day and negative if adverse shocks
dominate a day. Using the signed jumps component instead of the whole jump
component, RV - BV, gives the advantage of separately studying the impact of
significant positive and negative price movements on future volatility. Merton
(1976) first introduce the idea of a jump component as a jump-diffusion
process where large variations of prices occur at a discrete time together
with small continuous movements.

Each of these variance estimators is obtained from a series of equally-spaced
observed prices $p_{0},p_{1},\ldots,p_{N}$ that span through the time horizon
of the analysis. Setting $N$ as the number of intraday observations available
daily, the RV estimator is the sum of intraday squared returns following Eq.
Equation 3, as well as realized semivariances, $RV^{+}$and $RV^{-}$, follows
Eq. Section 3.1.

The estimator for the bipower variation BV in Eq. Equation 4 is obtained by
averaging multiple skip versions of the same estimator defined as

$BV=\frac{\pi}{2}\sum_{i=q+2}^{n}\left|r_{i}\right|\left|r_{i-1-q}\right|$ (7)

where the usual $BV$ estimator is obtained when $q=0$. We obtain the BV
estimator by averaging skip-0 through skip-4 of the same estimator as an
important correction for the bipower variation estimator.

The purpose of computing the estimators described in this Section is to obtain
a daily proxy for the quadratic variation of the log-return prices process
from high-frequency data. By disentangling the different components of the
quadratic variation into their relative estimators, we can empirically analyze
the effect of different volatility drivers in the cryptocurrency market, such
as the signed part of the realized variance and the jump components.

###  3.2 Data Collection

We obtained the 5-minute level prices of the 100 most capitalized
cryptocurrencies333The list is available here. from the Binance API
(Application Programming Interface) 444https://www.binance.com/en/binance-api
at the time of writing, such that our sample covers more than 95% of the
cryptocurrency market capitalization. From the 100 cryptocurrencies, we
exclude stablecoins555Stablecoins are cryptocurrencies designed to maintain a
stable value relative to a specific asset, such as the US dollar or a basket
of assets. The value of stablecoins is backed by these assets, which are held
in reserve, and they aim to minimize price volatility compared to other
cryptocurrencies like Bitcoin and Ethereum (Fiedler and Ante, 2023). from our
analysis since we are interested in volatility fluctuations that are not
expected within that category of digital assets, remaining with 87 entities.
Then, we queried from Polygon666https://polygon.io/ the stock prices of the 42
companies included in the Nasdaq technology index with the same granularity.
To compare the two asset classes, we consider the historical period between
the year 2020 and year 2022 (up to October). Many of the most highly
capitalized cryptocurrencies at the time of this analysis began trading after
2020, making the sample representative of the current state of the market.

The amount of 5-minute level observation daily varies among cryptocurrencies,
continuously traded throughout the day, and stock, which follows the market
hours. This results in a different amount of high-frequency observation each
day in our dataset, 87 for cryptocurrencies and 79 for stocks. We compute the
estimators using a different number of observations to properly reflect the
realized variance within a given day for both asset classes. The entities in
the cryptocurrency cross-section have around four times the available
observation compared to equities. They are, therefore, subject to be more
volatile by construction since each entity has more recorded fluctuations.
However, this difference does not impact how we construct $n$-days volatility
estimators that follow the same process for both asset classes.

Table 1 contains the average values, the standard deviations, and some
quantiles of the distribution of the computed estimators. The table is divided
to provide the same information for the cross-sections of entities included in
our analysis. The included estimators, realized variance (RV ), bipower
variation (BV ), positive and negative semivariance (RV- and RV+), jump
variation (SJV), and signed jump variation (SJV- and SJV+), are those involved
in the autoregressive models outlined in the next Section. The descriptive
statistics show a higher order of magnitude for the cryptocurrency estimators
than those related to equities. This effect indeed points to the continuous
trading under which cryptocurrencies are subjected due to how the estimator is
constructed. However, the difference in scale is also due to the more
significant volatility shocks that are known to affect such a market.

Table 2 contains the two correlation matrices for the computed estimators. For
both the cross-sections, the RV estimator is highly correlated to its signed
components and the continuous volatility estimator BV, as expected by
construction. On the contrary, the SJV estimators and their signed components
are not strongly correlated with the other estimators for the equity cross-
section. At the same time, such a correlation tends to be higher in the case
of cryptocurrencies. Such results potentially signal a stronger influence of
the jump part to explain future cryptocurrency volatility.

|  | Mean | Std | 5% | 25% | 50% | 75% | 95%
---|---|---|---|---|---|---|---|---
Crypto | RV | 0.459 | 2.881 | 0.036 | 0.112 | 0.222 | 0.448 | 1.397
BV | 0.404 | 2.510 | 0.030 | 0.095 | 0.191 | 0.391 | 1.244
RV- | 0.228 | 1.806 | 0.017 | 0.055 | 0.110 | 0.222 | 0.679
RV+ | 0.230 | 1.124 | 0.017 | 0.053 | 0.106 | 0.223 | 0.731
SJV | 0.002 | 0.869 | -0.124 | -0.024 | -0.001 | 0.023 | 0.164
SJV- | -0.037 | 0.824 | -0.124 | -0.024 | -0.001 | 0.0002 | 0.0005
SJV+ | 0.039 | 0.270 | -0.012 | 0.001 | 0.003 | 0.023 | 0.164
Equity | RV | 0.069 | 0.284 | 0.002 | 0.008 | 0.010 | 0.053 | 0.284
BV | 0.025 | 0.070 | 0.022 | 0.061 | 0.001 | 0.025 | 0.110
RV- | 0.035 | 0.157 | 0.016 | 0.023 | 0.003 | 0.024 | 0.141
RV+ | 0.035 | 0.153 | 0.012 | 0.026 | 0.003 | 0.024 | 0.147
SJV | 0.001 | 0.123 | -0.046 | -0.001 | 0.0005 | 0.001 | 0.053
SJV- | -0.012 | 0.090 | -0.046 | -0.001 | 0.0004 | 0.001 | 0.003
SJV+ | 0.012 | 0.083 | 0.031 | 0.001 | 0.0002 | 0.004 | 0.053
Table 1: Descriptive statistics expressed in percentage of the constructed variance estimators for cryptocurrencies and equities. |  | RV | BV | RV- | RV+ | SJV | SJV- | SJV+
---|---|---|---|---|---|---|---|---
Crypto | RV | 1.000 | 0.995 | 0.989 | 0.972 | -0.798 | -0.901 | 0.179
BV | 0.995 | - | 0.984 | 0.968 | -0.793 | -0.878 | 0.128
RV- | 0.989 | 0.984 | - | 0.928 | -0.877 | -0.947 | 0.067
RV+ | 0.972 | 0.968 | 0.928 | - | -0.637 | -0.787 | 0.352
SJV | -0.798 | -0.793 | -0.877 | -0.637 | - | 0.950 | 0.317
SJV- | -0.901 | -0.878 | -0.947 | -0.787 | 0.950 | - | 0.006
SJV+ | 0.179 | 0.128 | 0.067 | 0.352 | 0.317 | 0.006 | -
Equity | RV | - | 0.819 | 0.919 | 0.914 | -0.036 | -0.539 | 0.527
BV | 0.819 | - | 0.784 | 0.719 | -0.106 | -0.424 | 0.298
RV- | 0.919 | 0.784 | - | 0.684 | -0.426 | -0.778 | 0.206
RV+ | 0.914 | 0.719 | 0.682 | - | 0.370 | -0.203 | 0.767
SJV | -0.036 | -0.106 | -0.426 | 0.370 | - | 0.738 | 0.688
SJV- | -0.539 | -0.424 | -0.778 | -0.203 | 0.738 | - | 0.019
SJV+ | 0.527 | 0.298 | 0.206 | 0.767 | 0.688 | 0.01930 | -
Table 2: Linear correlation matrices of the constructed variance estimators
for cryptocurrencies and equities.

###  3.3 Models

To study the driving factor of the volatility within the two asset classes, we
consider a set of different model specifications that estimate future
volatility using the information the computed volatility estimators provided.
All the model specifications are estimated in-sample since our scope is to
obtain insights regarding the driving factor of the volatility process. The
forecasting problem is not treated in this study, although the results can be
beneficial to improve modeling for that purpose.

The reference model specification is the heterogeneous autoregression (HAR)
specification (Corsi, 2009), which is a high-order autoregressive model
specified as

$RV_{h,t+h}=\mu+\phi_{d}RV_{t}+\phi_{w}\left(\frac{1}{4}\sum_{i=1}^{4}RV_{t-i}\right)+\phi_{m}\left(\frac{1}{17}\sum_{i=5}^{21}RV_{t-i}\right)+\epsilon_{t+h}.$
(8)

The model regresses the future realized variance estimator $t+h$ on three past
terms of the same estimator: the first lag, the average between the second and
the fifth lag, and the average between the sixth and the twenty-second lag.
These three regressors intuitively correspond to an estimation of the
volatility on the previous day and the average volatility over the past
trading week and trading month. The model is said to be rotated since each
regressor has no overlapped observation, i.e., the average of the past trading
week does not include the observation of the previous day since another
regressor already represents it. We estimate the reference specification and
each of its modifications at four different horizons, corresponding to one
day, a trading week (5 days ahead), a trading month (22 days ahead), and three
trading months (66 days ahead) in the future. This model specification looks
for the effects of three representative market agents operating at those three
different frequencies.

We compute the realized variance estimator described in Section 3.1 using the
5-minute level data in Section 3.2 for each of the cryptocurrencies. Once we
estimate these volatility proxies, it is straightforward to derive the
regressor of the reference model in Eq. Equation 8 by lagging or averaging
over the information according to the specification. The HAR model’s main
benefit is that it allows for incorporating multiple time scales, which is
particularly important because volatility exhibits specific patterns at
different frequencies. The HAR model is an extension of the Autoregressive
Conditional Heteroskedasticity (ARCH) model, which assumes that the variance
of a time series is a function of its past squared residuals and can capture
the long-memory properties of financial volatility. Lastly, it is a relatively
easy-to-interpret model, which makes it suitable for descriptive analyses of
linear volatility patterns. However, the reduced complexity can cause some
information regarding nonlinear volatility behaviors to be missing.

Starting from the reference model and following the framework of Patton and
Sheppard (2015), we estimate various HAR specifications, each accounting for
specific traits and stylized facts of the volatility dynamics. Each model
specification analyzes the effect of signed information, such as the positive
and negative part of the realized variance and the jump components of the
price process, in improving modeling and explainability of the global
cryptocurrency market volatility. All the model specifications tested are
derived from the reference model HAR-RV, which we rewrite from Equation 8 in a
compact form as

$RV_{h,t+h}=\mu+\phi_{d}RV_{t}+\phi_{w}RV_{w,t}+\phi_{m}RV_{m,t}+\epsilon_{t+h}$
(9)

Recalling the definition of RV as the sum of squared returns, one test for the
effect of signed returns through the realized semivariances in Equation 10.
The HAR-semiRV specification corresponds to

$RV_{h,t+h}=\mu+\phi^{+}_{d}RV^{+}_{t}+\phi^{-}_{d}RV^{-}_{t}+\phi_{w}RV_{w,t}+\phi_{m}RV_{m,t}+\epsilon_{t+h}$
(10)

when the most recent information is decomposed into the signed realized
semivariances. The Table 6 in the appendix also estimates a similar model
specification when the three regressors are all decomposed into the signed
components. Such an autoregressive model detects the effect of the market’s
direction on future volatility, describing the entity of the leverage effect
at the intraday level. As previously discussed, the granularity of the data
collected to obtain the volatility estimators defines the amount of
information contained in the regressor for the model specification. Therefore,
the leverage effect detected daily could differ from that detected on a more
fine-grained scale.

To also account for a daily leverage effect, in addition to the intraday
leverage effect captured by the semivariances, we also estimate the model
specification in Eq. Equation 9 and Equation 10 by including a supplementary
interaction term $RV_{t}I_{r_{t}<0}$:

$\displaystyle RV_{h,t+h}$ $\displaystyle=\mu+\phi_{d}RV_{t}+\gamma
RV_{t}I_{r_{t}<0}+\phi_{w}RV_{w,t}+\phi_{m}RV_{m,t}+\epsilon_{t+h}$ (11)
$\displaystyle RV_{h,t+h}$
$\displaystyle=\mu+\phi^{+}_{d}RV^{+}_{t}+\phi^{-}_{d}RV^{-}_{t}+\gamma
RV_{t}I_{r_{t}<0}+\phi_{w}RV_{w,t}+\phi_{m}RV_{m,t}+\epsilon_{t+h}.$ (12)

which is nonzero only when the daily return of the asset under study is
negative. This term reflects the definition of the leverage effect from the
traditional financial literature, where a negative price variation increases
the volatility more than a positive one. Since the interaction term is based
on daily information, while realized variance and semivariances are based on
5-minute level prices, we expect to find a different effect in the results.

An additional couple of tested model specifications disentangle the most
recent information into the continuous and the jump part of the volatility. We
obtain two variants, where the second one also decomposes the jump part to
account for its signed effect.

$\displaystyle RV_{h,t+h}$ $\displaystyle=\mu+\phi_{j}\Delta
J^{2}_{t}+\phi_{c}BV_{t}+\phi_{w}RV_{w,t}+\phi_{m}RV_{m,t}+\epsilon_{t+h}$
(13) $\displaystyle RV_{h,t+h}$ $\displaystyle=\mu+\phi_{j}+\Delta
J^{2+}_{t}+\phi_{j}-\Delta
J^{2-}_{t}+\phi_{c}BV_{t}+\phi_{w}RV_{w,t}+\phi_{m}RV_{m,t}+\epsilon_{t+h}.$
(14)

It is possible to calculate the jump parts of the volatility estimators from
Eq. Equation 3 and Equation 4 because the BV estimators converge in the limit
to the continuous component of the quadratic variation of the price process.
Including the jump component into a model specification is extremely important
in markets characterized by a lack of regulation and liquidity, as is the case
for cryptocurrencies. Table 1 highlights the different scale of magnitude of
the estimated realized variance for the cryptocurrency cross-section compared
to the equity one as a signal of more frequent large shifts in the price
variation.

For each model specification described in this section, we estimate a panel
regression to capture the average contribution of each entity in a specific
asset class and explain the volatility dynamics for the considered group.
Therefore, the panel regression is carried out separately for cryptocurrencies
and equities for a thoughtful comparison. Each panel HAR is estimated through
a weighted least squares (WLS) method to account for frequent regime shifting
retrieved in the dynamics of our time series. The WLS estimation is based on a
preliminary Ordinary Least Squares (OLS) estimation, which is necessary to
construct weights as the inverse of the fitted residual from such an OLS
model. As a further analysis, the same estimation sequence is repeated on
different time windows to capture the effect of different market regimes and
separately on each component of the cross-sections to disentangle the
aggregated effect at the entity level and validate the findings. When
estimating the panel regressions, we adopt the Newey-West Heteroskedasticity
and Autocorrelation Consistent (HAC) estimators. This approach is particularly
useful for addressing the potential issues of autocorrelation and
heteroskedasticity in panel data, which are common in financial time series
data. The Newey-West HAC estimators allow us to produce standard errors that
are robust to these issues, thereby ensuring that our inference is more
reliable.

##  4 Empirical Results

This Section provides the different results of our empirical analysis where we
comment on the estimation results of various model specifications described in
Section 3.3. The estimation procedure remains the same as described, even when
we restrict the time window of observation used for the estimation and when we
estimate the model specifications individually over each entity of a specific
cross-section instead of as a panel.

###  4.1 Impact of Signed Returns

As a first step in the empirical analysis of cryptocurrency volatility, we
estimate the reference specification that accounts for three different lags of
the estimated volatility observed in the past. The results on the reference
model described in Eq. Equation 9 are instrumental in describing past signed
returns’ impact on future volatility.

Table 3 presents the results for the reference specification of the panel HAR
and its variant where the first lag of the volatility is decomposed into its
signed components. Both model specifications are separately estimated for the
cryptocurrency and the equity entities, as shown in the table.

The first result to highlight is the relative amount of volatility
persistence, defined as $\phi_{d}+\phi_{w}+\phi_{m}=1$ in case of highest
persistence, for the first lag of the reference specification of
cryptocurrency cross-section at the horizon $t+1$ and $t+5$. This effect is
remarkably missing in crypto, where the relative contribution to the
volatility persistence is low for all the estimation horizons. This result
aligns with previous findings on cryptocurrency volatility persistence until
2018 (Katsiampa et al., 2019a; Charfeddine and Maouchi, 2019; Abakah et al.,
2020; Caporale et al., 2018). Higher volatility persistence suggests that the
market’s reaction to new information entering the financial system is not
instantaneous but occurs progressively over time. Consequently, historical
price fluctuations can serve as indicators for forecasting future price
movements, offering a counterpoint against the Efficient Market Hypothesis
(EMH), indicating that the cryptocurrency markets may not always reflect all
available information immediately and efficiently (Cross et al., 2021). This
preliminary observation serves as a starting point for a more comprehensive
discussion on the Adaptive Market Hypothesis (AMH) (Lo, 2004), which we
explore in detail in Section 5. The AMH provides a framework for understanding
the evolving efficiency of the cryptocurrency market, highlighting how its
unique ecosystem characteristics and the interplay of various market
participants contribute to its dynamic volatility patterns.

The role of recent information, as measured by $\phi_{d}$, is prevalent in the
crypto cross-section, while it tends to be more spread across different lags
for equity. This result signals the temporary nature of the volatility in the
cryptocurrency market, whose future values are not largely correlated with
distant observations in the past. A similar effect regarding the importance of
recent information to estimate volatility is captured by looking at the
diminishing role of $\phi_{d}$ as the horizon increases. Although more
pronounced for cryptocurrencies, such an effect is shared among the two asset
classes for the time period considered.

We observe that the decomposition of the most recent information on volatility
into its signed counterparts adds information to the analysis since
$\phi^{+}_{d}\neq\phi^{-}_{d}\neq\phi_{d}$ for both asset classes. The effect
of the positive returns is always significant and dominates the impact of the
negative returns. It is important to notice that, when statistically
significant, the negative part of the volatility has the opposite effect on
cryptocurrencies compared to what is observed for equities. In general, we
observe that the role of the negative volatility is negligible for
cryptocurrencies as a signal of the missing impact of negative shocks of the
market on future volatility, contrary to what is usual for other asset
classes. Such a result implies the absence of the leverage effect for
cryptocurrencies when using high-frequency data. The decreasing impact of
recent information is also present when using the signed past returns for the
most recent lag. Those effects are present when the RV estimator is decomposed
into the signed components at various lags. The Table 6 in the appendix
provides the estimated coefficients for such model specifications.

The results obtained at a high-frequency level for the cryptocurrency cross-
section highlight an absence of the leverage effect in the traditional sense,
where a negative return impacts future volatility more than a positive one.
Such behavior challenges the notion of market efficiency and the traded price
as an aggregator of all the available information. The hypothesis of rational
expectation in front of a negative return falls in the case of cryptocurrency
since it is a positive movement of the market that increases the volatility.
The market participants are not necessarily rational in the cryptocurrency
market, as the massive presence of retail investors exacerbates idiosyncratic
volatility in unstable market conditions (Low and Marsh, 2019; Ozdamar et al.,
2022; Wen et al., 2019). The next subsection elaborates more on the leverage
effect aspect by comparing the high-frequency effect with the daily one.

Another motivation for retrieving unexpected volatility patterns in the
cryptocurrency market compared to the equity one is the different mechanisms
and set of technologies the ecosystem builds upon (see Xu et al. (2019) for a
review of the Blockchain technology). Even though the level of on-chain
activity777 On-chain market activity refers to transactions recorded on the
blockchain, while off-chain market activity refers to transactions that occur
outside of the blockchain, typically facilitated by third-party services such
as centralized exchanges or wallets. is still a fraction of the volume of off-
chain trades that happens on centralized exchanges, it is in the process of
growing and represents an element of novelty compared to traditional finance.
Our analysis focuses on off-chain data for comparison with the equity market,
but substantial activity on a decentralized platform makes the market less
reactive and fractionated (Aspris et al., 2021). Indeed, traders can often
find arbitrages between centralized and decentralized exchanges. Hence, the
cryptocurrency market is slower than the equity one at transferring
information into the quoted prices due to these two concurrent layers of
activity.

$RV_{h,t+h}=\mu+\phi_{d}RV_{t}+\phi^{-}_{d}RV^{-}_{t}+\phi^{+}_{d}RV^{+}_{t}+\phi_{w}RV_{w,t}+\phi_{m}RV_{m,t}+\epsilon_{t+h}$ |  | $\mathbf{\phi_{d}}$ | $\mathbf{\phi^{-}_{d}}$ | $\mathbf{\phi^{+}_{d}}$ | $\mathbf{\phi_{w}}$ | $\mathbf{\phi_{m}}$
---|---|---|---|---|---|---
Crypto | t+1 |  0.554∗∗∗ | - | - | 0.006 |  0.056∗∗∗
(6.726) | - | - | (1.573) | (4.025)
- |  -0.142∗ |  1.262∗∗∗ |  0.007∗ |  0.036∗∗∗
- | (-1.694) | (10.183) | (1.774) | (4.764)
t+5 |  0.271∗∗∗ | - | - |  0.063∗∗∗ |  0.051∗∗∗
(4.834) | - | - | (2.773) | (3.585)
- | 0.078 |  0.543∗∗∗ |  0.054∗∗∗ |  0.032∗∗∗
- | (0.258) | (2.313) | (2.4) | (1.536)
t+22 |  0.076∗∗∗ | - | - |  0.031∗∗∗ |  0.049∗∗∗
(3.551) | - | - | (2.681) | (4.089)
- | -0.065 |  0.24∗∗∗ |  0.028∗∗∗ |  0.043∗∗∗
- | (-0.946) | (4.33) | (2.592) | (3.764)
t+66 |  0.025∗∗ | - | - | 0.006 |  -0.012∗∗∗
(2.335) | - | - | (1.293) | (-2.676)
- |  -0.069∗∗∗ |  0.145∗∗∗ |  0.006∗ |  -0.016∗∗∗
- | (-4.7734) | (8.392) | (1.174) | (-3.814)
Equity | t+1 |  0.187∗∗∗ | - | - |  0.182∗∗∗ |  0.165∗∗∗
(7.553) | - | - | (7.419) | (3.081)
- |  0.075∗∗ |  0.305∗∗∗ |  0.182∗∗∗ |  0.163∗∗∗
- | (2.276) | (5.339) | (7.527) | (3.051)
t+5 |  0.089∗∗∗ | - | - |  0.183∗∗∗ |  0.162∗∗∗
(7.95) | - | - | (12.1) | (6.211)
- |  0.055∗∗ |  0.125∗∗∗ |  0.184∗∗∗ |  0.162∗∗∗
- | (2.45) | (5.011) | (12.086) | (6.197)
t+22 |  0.048∗∗∗ | - | - |  0.108∗∗∗ |  0.169∗∗∗
(6.42) | - | - | (10.359) | (12.428)
- |  0.01∗∗∗ |  0.088∗∗∗ |  0.108∗∗∗ |  0.169∗∗∗
- | (2.811) | (5.513) | (10.393) | (12.428)
t+66 |  0.028∗∗∗ | - | - |  0.08∗∗∗ |  0.217∗∗∗
(5.967) | - | - | (11.21) | (23.229)
- |  0.01∗∗ |  0.047∗∗∗ |  0.08∗∗∗ |  0.216∗∗∗
- | (2.036) | (4.819) | (11.233) | (23.254)
***$p<0.01$, **$p<0.05$, *$p<0.1$ |  |  |  |
Table 3: Estimation results for HAR-RV and HAR-semiRV model specifications.
Results are displayed for the cryptocurrency and the equity panel over the
four different horizons. T-stats are in parentheses. The model equation on top
of the table reflects all the parameters common to the specification reported
in this table. The total number of observations available considering all the
entities used for the panel estimate is 47787 for crypto and 17111 for equity

###  4.2 Daily Leverage Effect

In this section, we introduce a daily leverage effect into the reference
specification estimated in the previous section. The leverage effect is a
known stylized fact of financial returns that exhibits an asymmetric impact of
financial returns on the future volatility of the assets. The financial
econometrics literature has extensively investigated such relationships at a
daily frequency, usually through Generalized AutoRegressive Conditional
Heteroskedasticity, GARCH-like, (Bollerslev, 1987) models.

We add a daily leverage effect component, $\gamma RV_{t}I_{r_{t}<0}$,
according to Eq. Equation 11. This model specification allows us to evaluate
the contribution of the decomposition of the realized variance at a high-
frequency level, adding value beyond the usual daily effect. Note that the
daily leverage effect is modeled using a lagged squared return interacted with
an indicator for negative returns, similar to an asymmetric GARCH-like model
as the GJR-GARCH (Glosten et al., 1993). Table 4 shows the estimation results
with the same structure proposed in the previous section. If the addition of
volatility estimator obtained from high-frequency data did not add additional
information, we would expect the estimation results to show
$\phi^{+}_{d}=\phi^{-}_{d}=\phi_{d}$ and the daily leverage parameter $\gamma$
as statistically significant. The results show that the positive part of the
most recent realized variance lag is always statistically significant at any
time horizon for both asset classes. In contrast, the negative part does not
have enough statistical evidence. The sign of the daily leverage effect
interaction is opposite for the cryptocurrency panel compared to the equity
panel, being negative for the former and positive for the latter. The
inversion of the sign also holds when the daily leverage component is included
in the reference specification. These results are consistent with the leverage
effect inversion in crypto due to the diminishing effect of a negative shock
on the realized variance.

This lack of statistical significance serves as an indicator of an inversion
in the leverage effect within the cryptocurrency market. The analysis
demonstrates that the expected negative influence typically associated with
downturns fails to materialize. Instead, this anomaly underscores the unique
dynamics, suggesting that the traditional leverage effect, as observed in
other financial markets, does not apply in the same manner to
cryptocurrencies. While the results of Section 4.1 show statistically
insignificant measures for the negative part of the realized variance only for
some horizons, the results hold even more when the daily leverage component is
included. This inversion is a critical insight, highlighting the distinct
market responses to positive versus negative shocks in the realm of
cryptocurrencies. The daily leverage aggregation provided by the $\gamma
RV_{t}I_{r_{t}<0}$ factor cancels out the effects captured by the high-
frequency semivariance estimator, especially if the leverage effect operates
differently within shorter time frames and is more prevalent at the daily
level as already explored by Baur and Dimpfl (2018); Brini and Lenz (2022).

The reversed asymmetric effect of high-frequency cryptocurrency returns
indicates that investors exploit market crashes as a buying opportunity to
enter at a discount. Such behavior connects to FoMo as a motivator for
investment behavior in the cryptocurrency market. Gerrans et al. (2023)
retrieves a larger association between FoMO and crypto ownership than equity
ownership for young age groups more likely to trade crypto assets. The concept
of FoMo among retail investors has been explored by Argan et al. (2023) to
observe the relationship with investment engagement and by Park et al. (2023)
to analyze the relationship with market stability. Kaur et al. (2023) observes
the FoMo effect in crypto investment decisions as a driving factor for the
herding effect in investment decisions, while Delfabbro et al. (2021) delve
into the psychological traits of cryptocurrency trading, among which FoMo
represents one of the factors. King and Koutmos (2021) elaborates on the
effect of herding behavior on the trading activity for cryptocurrencies. We
argue that the fear of missing out on potential profits when others are
succeeding is a potential cause for the leverage effect inversion retrieved by
the econometric analysis. Specifically, the market is relatively more active,
hence more volatile, when returns are positive and spark excitement in the
crowd that wants to enter this new investment opportunity, rather than when
returns are negative and sell-offs happen. Psychology-driven behaviors can
cause retail investors to enter the market without clearly understanding the
asset class, viewing it as a new opportunity with high potential (Pursiainen
and Toczynski, 2022). Indeed, it is known that the retail investors’ presence
is larger in crypto than equity due to the nascent nature of the former and
other peculiar characteristics such as the relatively low entry barrier, the
potential to disrupt the conventional financial system and the underlying
Blockchain technology that powers the ecosystem.

$RV_{h,t+h}=\mu+\phi_{d}RV_{t}+\phi^{-}_{d}RV^{-}_{t}+\phi^{+}_{d}RV^{+}_{t}+\gamma RV_{t}I_{r_{t}<0}+\phi_{w}RV_{w,t}+\phi_{m}RV_{m,t}+\epsilon_{t+h}$ |  | $\mathbf{\phi_{d}}$ | $\mathbf{\phi^{-}_{d}}$ | $\mathbf{\phi^{+}_{d}}$ | $\mathbf{\gamma}$ | $\mathbf{\phi_{w}}$ | $\mathbf{\phi_{m}}$
---|---|---|---|---|---|---|---
Crypto | t+1 |  0.632∗∗∗ | - | - | -0.126 | 0.006 |  0.054∗∗∗
(16.698) | - | - | (-1.444) | (1.455) | (4.163)
- | 0.298 |  0.661∗∗∗ |  -0.261∗∗∗ | 0.026 |  0.127∗∗∗
- | (1.436) | (3.355) | (-2.755) | (1.551) | (4.627)
t+5 |  0.32∗∗∗ | - | - | -0.053 |  0.056∗∗∗ |  0.047∗∗∗
(12.709) | - | - | (-0.929) | (2.645) | (3.445)
- | 0.314 | 0.151 | 0.026 |  0.084∗∗∗ |  0.036∗∗∗
- | (0.712) | (0.0363) | (0.247) | (3.072) | (1.485)
t+22 |  0.145∗∗∗ | - | - |  -0.083∗∗∗ |  0.026∗∗ |  0.043∗∗∗
(13.745) | - | - | (-4.183) | (2.534) | (3.777)
- | 0.07 |  0.196∗∗∗ |  -0.076∗∗∗ |  0.029∗∗∗ |  0.043∗∗∗
- | (0.822) | (3.254) | (-3.852) | (2.618) | (3.815)
t+66 |  0.084∗∗∗ | - | - |  -0.068∗∗∗ | 0.004 |  -0.016∗∗∗
(13.376) | - | - | (-7.32) | (0.917) | (-4.033)
- | 0.298 |  0.661∗∗∗ |  -0.261∗∗∗ | 0.026 |  0.127∗∗∗
- | (1.436) | (3.355) | (-2.755) | (1.551) | (4.627)
Equity | t+1 |  0.138∗∗∗ | - | - |  0.09∗∗ |  0.183∗∗∗ |  0.166∗∗∗
(4.403) | - | - | (1.988) | (7.471) | (3.137)
- | -0.012 |  0.266∗∗∗ |  0.115∗∗∗ |  0.184∗∗∗ |  0.165∗∗∗
- | (-0.319) | (4.578) | (2.721) | (7.635) | (3.13)
t+5 |  0.064∗∗∗ | - | - |  0.047∗∗ |  0.183∗∗∗ |  0.161∗∗∗
(5.443) | - | - | (2.417) | (12.113) | (6.197)
- | 0.012 |  0.107∗∗∗ |  0.055∗∗ |  0.184∗∗∗ |  0.161∗∗∗
- | (0.526) | (4.139) | (2.193) | (12.105) | (6.183)
t+22 |  0.036∗∗∗ | - | - |  0.021∗ |  0.108∗∗∗ |  0.169∗∗∗
(4.674) | - | - | (1.668) | (10.367) | (12.425)
- | -0.011 |  0.079∗∗∗ |  0.027∗∗ |  0.108∗∗∗ |  0.169∗∗∗
- | (-0.632) | (4.988) | (2.044) | (10.415) | (12.434)
t+66 |  0.027∗∗∗ | - | - | 0.002 |  0.08∗∗∗ |  0.217∗∗∗
(5.335) | - | - | (0.259) | (11.214) | (23.221)
- | 0.005 |  0.045∗∗∗ | 0.006 |  0.087∗∗∗ |  0.216∗∗∗
- | (0.475) | (4.581) | (0.703) | (11.242) | (23.242)
***$p<0.01$, **$p<0.05$, *$p<0.1$ |  |  |  |  |
Table 4: Estimation results for HAR-RV and HAR-semiRV model specifications
with daily leverage effect component. Results are displayed for the
cryptocurrency and the equity panel over the four different horizons. T-stats
are in parentheses. The model equation on top of the table reflects all the
parameters common to the specification reported in this table. The total
number of observations available considering all the entities used for the
panel estimate is 47787 for crypto and 17111 for equity

###  4.3 The Role of Jumps

The model specifications estimated in the previous subsections indicate that a
positive realized variance has a greater impact on estimating future
volatility than a negative realized variance. According to Barndorff-Nielsen
et al. (2008), this is due to differences in jump variation in a given time
series. To account for this effect, we insert the signed jump variation
measure, $\Delta J_{t}^{2}=RV^{+}-RV^{-}$, to identify information from signed
jumps. Such difference eliminates the integrated variance term and is positive
when a day has a prevalence of upward jumps or negative when downward jumps
are prevalent. Table 5 shows the estimation of the related model
specification, including the jump variation term. The coefficient is always
positive and statistically significant, except for the horizon $h=5$, for both
asset classes. The sign of $\phi_{j}$ means that days dominated by negative
jumps lead to lower future volatility, while days with positive jumps lead to
higher future volatility. The jump variation estimated coefficients for the
cryptocurrency cross-section are also larger than the equity cross-section,
remarking on the more frequent presence of extreme market movements in the
notoriously unstable and immature cryptocurrency market.

Table 5 also shows the results of the signed jump specification, which
replaces the jump variation component with two signed components regarding the
dominance of a positive or a negative jump, respectively. The coefficients for
the positive jump variations are significant, except at $h=5$, and negative in
sign, differently from the case of equity cross-section. Hence, this result
says that a day dominated by a positive jump decreases future volatility,
while a day dominated by a negative jump does the opposite. We conclude that
the two signed jump components do not have the same effect. When isolated in
two different factors, they provide additional information to interpret the
effect of a jump. In this case, a negative jump is associated with an increase
in future volatility, while a positive jump is with a decrease, only in the
cryptocurrency case. Although this result seems to conflict with the inversion
of the leverage effect in the cryptocurrency cross-section, the jump variation
gives a perspective on the volatility movement at the intraday level. In
contrast, the inverted leverage effect retrieved in the literature and the
previous model specification accounts for a daily effect. These results
highlight how the discrete component, i.e., the jump one, of the quadratic
variation exhibits the asymmetric effect of return on future volatility,
while, under the light of the previous section’s results, the continuous
component signals an inversion of such effect. The Table 7 in the appendix
provides the estimated coefficients for model specifications that include only
the continuous part of the volatility for the most recent lags, without any
jumps.

The presence of a leverage effect guaranteed by the discrete part of the
quadratic variation points to different categories of operators in the
cryptocurrency market. As outlined in the previous section, the inverse
leverage effect is caused by uninformed investors that permeate the
cryptocurrency market. Indeed, this is a market that, unlike more traditional
markets, has seen a huge exposure to retail investors, while the institutional
and professional operators came after. Hence, we explain the different
behaviors of the volatility drivers by associating them with a specific
category of market operators. The activity of high-frequency professional
traders regulates the volatility dynamics according to the renowned leverage
effect because the price variation associated with jumps is transient, so only
they can exploit such information to make an arbitrage. On the other hand,
some less-informed retail investors operate under the FoMO guide and inverse
the leverage direction.

$RV_{h,t+h}=\mu+\phi_{j}\Delta J^{2}_{t}+\phi_{j}-\Delta J^{2-}_{t}+\phi_{j}+\Delta J^{2+}_{t}+\phi_{c}BV_{t}+\phi_{w}RV_{w,t}+\phi_{m}RV_{m,t}+\epsilon_{t+h}$ |  | $\mathbf{\phi_{j}}$ | $\mathbf{\phi_{j}^{-}}$ | $\mathbf{\phi_{j}^{+}}$ | $\mathbf{\phi_{c}}$ | $\mathbf{\phi_{w}}$ | $\mathbf{\phi_{m}}$
---|---|---|---|---|---|---|---
Crypto | t+1 |  0.554∗∗∗ | - | - |  0.668∗∗∗ |  0.008∗ |  0.045∗∗∗
(6.867) | - | - | (12.469) | (1.783) | (5.798)
- |  0.643∗∗∗ |  -0.13∗∗ |  0.341∗∗∗ |  0.011∗∗∗ |  0.039∗∗∗
- | (4.565) | (-2.298) | (3.593) | (3.061) | (4.149)
t+5 | -0.039 | - | - |  0.359∗∗∗ |  0.042∗ | 0.025
(-0.085) | - | - | (3.809) | (1.982) | (0.964)
- | -0.022 | -0.083 | 0.186 |  0.051∗∗ | 0.024
- | (-0.037) | (-0.93) | (1.187) | (2.304) | (0.843)
t+22 |  0.144∗∗ | - | - |  0.099∗∗∗ |  0.026∗∗ |  0.042∗∗∗
(2.294) | - | - | (6.684) | (2.495) | (3.697)
- |  0.243∗∗ |  -0.044∗ |  0.112∗∗∗ |  0.025∗∗ |  0.039∗∗∗
- | (2.533) | (-1.828) | (4.408) | (2.41) | (3.461)
t+66 |  0.107∗∗∗ | - | - |  0.044∗∗∗ | 0.005 |  -0.016∗∗∗
(7.78) | - | - | (10.685) | (1.072) | (-3.994)
- |  0.19∗∗∗ |  -0.029∗∗ |  0.07∗∗∗ | 0.004 |  -0.017∗∗∗
- | (9.454) | (-2.53) | (9.073) | (0.939) | (-4.407)
Equity | t+1 |  0.061∗∗ | - | - |  1.076∗∗∗ |  0.129∗∗∗ |  0.129∗∗∗
(2.384) | - | - | (9.221) | (5.398) | (2.602)
- | 0.003 |  0.154∗∗ |  1.002∗∗∗ |  0.132∗∗∗ |  0.131∗∗∗
- | (0.099) | (2.197) | (8.061) | (5.472) | (2.661)
t+5 | 0.017 | - | - |  0.575∗∗∗ |  0.148∗∗∗ |  0.142∗∗∗
(0.937) | - | - | (11.106) | (10.182) | (5.53)
- | 0.055 |  0.07∗∗ |  0.552∗∗∗ |  0.148∗∗∗ |  0.14∗∗∗
- | (0.24) | (2.212) | (10.33) | (10.174) | (5.476)
t+22 |  0.035∗∗∗ | - | - |  0.335∗∗∗ |  0.088∗∗∗ |  0.154∗∗∗
(8.769) | - | - | (9.745) | (8.664) | (11.539)
- |  0.021∗∗∗ |  0.067∗∗∗ |  0.307∗∗∗ |  0.088∗∗∗ |  0.154∗∗∗
- | (2.279) | (2.791) | (8.634) | (8.722) | (11.553)
t+66 |  0.021∗∗∗ | - | - |  0.21∗∗∗ |  0.067∗∗∗ |  0.205∗∗∗
(3.627) | - | - | (9.01) | (9.55) | (22.214)
- |  0.018∗∗∗ |  0.028∗∗ |  0.205∗∗∗ |  0.067∗∗∗ |  0.205∗∗∗
- | (2.744) | (2.093) | (8.528) | (9.563) | (22.213)
***$p<0.01$, **$p<0.05$, *$p<0.1$ |  |  |  |  |
Table 5: Estimation results for HAR-JV and HAR-SJV model specifications.
Results are displayed for the cryptocurrency and the equity panel over the
four different horizons. T-stats are in parentheses. The model equation on top
of the table reflects all the parameters common to the specification reported
in this table. The total number of observations available considering all the
entities used for the panel estimate is 47787 for crypto and 17111 for equity

###  4.4 Behavior under different market trends

The cryptocurrency market underwent several changes in the regime due to the
developing nature of this ecosystem and the underlying global economic
condition. This aspect is especially true when considering the market run from
2020 to mid-year 2021 when many new protocols had been released, and terms
like DeFi protocol or Non-fungible Token (NFT) have been popularized among a
larger audience. Using BTC, the largest capitalized cryptocurrency, as a proxy
for the whole market dynamics, one can observe from Figure 1 different market
trends: a sharp increase in 2020, two major drawdowns in 2021, and a steep
decrease in 2022. We aim to estimate the same volatility model specifications
of the previous sections under those different market regimes to validate the
robustness of the previous sections’ findings. Hence, in this Section, we
provide the estimates considering just the windows 2020-2021 and 2021-2022 and
the whole period covered by the data, 2020-2022, which is available in the
previous Sections. The results of the estimated coefficients provided in this
Section are available in the table format in the appendix, where statistical
significance can be assessed by looking at the resulting t-statistics.
Considering all the entities used for the panel estimate, the total number of
observations available is 29005 for crypto and 11472 for equity in the window
2020-2021 and 45191 for crypto and 15968 for equity in the window 2021-2022.
The number of observations for the whole dataset (2020-2022) is the same as
provided in the previous subsection.

Figure 1: BTC price dynamics over the historical period under analyses. The
shades of blue separate the time series year by year, highlighting the
different market regimes in the cryptocurrency market.

The Figures of this Section comprise four subplots, one for each horizon. The
one-day ahead (first plot) and the five-day ahead (second plot) horizons are
in the first row, while the one-month ahead (first plot) and the three-month
ahead (second plot) are in the second row. Figure 2 shows the estimation for
the panel HAR model when the first regressor is decomposed into its signed
components. The first remarkable effect relies on the volatility’s more
considerable persistence in the cryptocurrency market for the two years
2020-2021, where the weekly component of the model plays a role in the
volatility estimation at any of the time horizons tested. The monthly
components in that period are less significant in magnitude. Still, it is
significant and more prominent than the other estimation windows, where the
estimated coefficient $\phi_{m}$ is more marginal. In addition, by looking at
the parameter $\phi^{+}_{d}$, we observe that the impact of the previous-day
positive volatility is less pronounced with respect to the whole estimation
period or the subsequent window that spans 2021-2022. The reason for this
effect can be retrieved in the extremely bullish market regime (see Figure 1)
when no sudden market drawdown occurred. That two-year period experienced a
majority of positive returns, making it more difficult to visualize the effect
compared to the adjacent period we have considered in the analyses. Given the
similarity of the estimation results for those two estimation windows, when
compared to the 2020-2021 window, we conclude that a large amount of the
effect of the positive part of the recent volatility is due to the most recent
period in 2022. This result indicates that the bear market regime in 2022
signaled market activity in response to upward movements at the intraday
level. Recalling the results from the previous subsection, this effect is
similar to the equity cross-section but larger in magnitude. Such results are
opposite to what was retrieved from results in previous works that employed
the realized semivariances as volatility proxies (Fang et al., 2017;
Bollerslev, Patton and Quaedvlieg, 2020), where the negative part of the
intraday volatility provides a relevant contribution to future volatility.
Under this view, cryptocurrencies show different traits in the volatility
pattern compared to other, more established asset classes due to a clear
separation of behaviors among informed traders and uninformed retail investors
that participate in the market.

Figure 2: Barplots of estimated coefficient for the panel HAR-semiRV model
specification over three different time windows. The number in the legend
reflects the year of analysis and is inclusive. Each subplot differs for the
future horizon of the model specifications: 1 day (top left), 5 days (top
right), 22 days (bottom left), and 66 (bottom right).

Moving from the intraday effect to the daily leverage effect, Figure 3 shows
the decomposed first lag specification estimation results when the latter
effect is added. The $\gamma$ parameter is consistently significant and with
the same magnitude for all the estimation windows tested, except for the
horizon equal to five days ahead. The coefficient is always negative,
reflecting the inversion of the leverage effect retrieved daily. As previously
studied in Baur and Dimpfl (2018); Brini and Lenz (2022), negative returns’
impact tends to lower the cryptocurrency volatility instead of increasing it,
as is commonly expected from the empirical literature. The effect is robust,
even though more pronounced during the overall bull market of the 2020-2021
period. Also, in this model specification, the volatility appears more
persistent in the two years, 2020-2021, than in the other estimation period.

Figure 3: Barplots of estimated coefficients for the panel HAR-semiRV model
specification with daily leverage effect over three different time windows.
The number in the legend reflects the year of analysis and is inclusive. Each
subplot differs for the future horizon of the model specifications: 1 day (top
left), 5 days (top right), 22 days (bottom left), and 66 (bottom right).

Figure 4 provided the result of the role of the most recent signed jump
components over the different estimation windows. As for the whole signed
volatility components, the jump one presents a less influential impact of the
$\phi_{j}^{-}$ when estimated over 2020-2021, except for the horizon equal to
five. Negative jumps happen to be more influential due to the bear market of
2022, while they are more negligible during the upward trend of 2020. Also,
the signed jump specification shows a different level of volatility
persistence among the selected estimation windows. In the appendix, Figure 11
and Figure 12 show the estimated coefficients for a fully decomposed HAR-RV
model into its signed components and a HAR-SJV that includes only the
continuous part of the volatility for the most recent lags, respectively.

Figure 4: Barplots of estimated coefficient for the panel HAR-JV and panel
HAR-SJV model specification over three different time windows. The number in
the legend reflects the year of analysis and is inclusive. Each subplot
differs for the future horizon of the model specifications: 1 day (top left),
5 days (top right), 22 days (bottom left), and 66 (bottom right).

The results of this section show the cryptocurrency ecosystem’s evolution from
a quantitative perspective. The structural change in the cryptocurrency
ecosystem from the beginning of 2020 is one reason for the decrease in
volatility persistence. The proliferation of new coins and tokens, the
emergence of different blockchains to challenge the most established Ethereum
blockchain, in terms of fees and validation speed, and the entrance of
institutional investors in the market contributed significantly to the
observed ecosystem growth (Białkowski, 2020). Such an effect is particularly
evident in the traded volume and the number of active participants in the
space (Bouri, Lau, Lucey and Roubaud, 2019; Bains et al., 2022).

###  4.5 Disentangling the effects at the individual level

After analyzing the cryptocurrency ecosystem at the macro level, we analyze
the data in our universe at the micro level by individually estimating each
model’s specifications for each entity. Since the panel estimation provides an
average effect of the whole ecosystem, we are also interested in evaluating
the tested effects at the individual level to understand which results in the
previous Sections are commonly shared among many universe components. Similar
to the previous section, each figure presents a group of subplots, each
referring to a different estimation horizon and following the same order. We
present the signed volatility component results, the daily leverage effect,
and the signed jump components. For each bar plot, the cryptocurrencies are
ranked by the magnitude of the respective coefficient, and the bars are either
blue if the estimation is statistically significant at least at 5%, or red if
they are not. For some cryptocurrencies, the individual estimation of the HAR-
like model does not converge; hence, these cryptocurrencies are excluded from
the graphic representation.

We acknowledge the challenges and limitations of applying OLS estimation to
individual time series within the panel. In our methodological approach, we
recognize that the properties of the error term can differ between panel and
time series data. In panel data, the error term may have a complex structure,
including both individual-specific and time-specific effects. This complexity
can sometimes be averaged out in panel estimations but might cause problems in
individual time series estimations. Moreover, when dealing with individual
time series, especially with short-time dimensions, OLS can be biased due to
the presence of lagged dependent variables as regressors, of which the HAR-
like specifications are an example. This bias might not be as pronounced when
considering a panel estimation, and it is one of the reasons for not having
all the individual estimations converge. As a third motivation for the OLS
convergence problem of certain individual model specifications, we note that
in a panel dataset, the OLS method benefits from the larger sample size, as it
combines cross-sectional and time series data. This increased sample size can
lead to more stable and reliable estimates. In contrast, individual time
series may have fewer observations, leading to less stable estimates and
potential convergence issues. Under this view, panel data often captures
heterogeneous units (different cryptocurrencies in our case) over time. When
you estimate the model for the entire panel, the OLS method captures the
average effect across all units. However, when you analyze individual time
series, the specific characteristics of each unit come into play. If a
particular time series has unique features that are not well-captured by the
OLS assumptions (like non-linearity, or structural breaks), the estimation
might fail to converge. Moreover, it allows us to identify any outliers or
influential observations that can drive the panel regression results and
contribute to a more granular understanding of the volatility pattern for each
cryptocurrency considered rather than treating them as a homogeneous group.

Despite this, the individual estimation procedure can provide additional
insights into the heterogeneity of the entities and the robustness of the
panel regression results. The cryptocurrency market is distinguished by its
pronounced volatility but also by the heterogeneity of its constituents. Some
of them are established cryptocurrencies such as BTC and ETH, and others have
only relatively recent history as they did their Initial Coin Offering (ICO)
(Ante et al., 2018; Ashta and Biot-Paquerot, 2018) or started being traded on
Binance (the data source) shortly before or after 2020. The idea of
heterogeneity is often overlooked when one thinks about the high level of
correlation among cryptocurrencies since the most capitalized one usually
drags the market dynamics, resulting in a correlation heatmap for the daily
returns over the period analyzed as in Figure 5.

Figure 5: Cryptocurrency daily return correlation over the 2020-2022 period.

However, each cryptocurrency is usually backed by a different project idea or
use case, which makes it interesting to disentangle the aggregated effect
provided by the pooled panel analysis. Our decision to undertake individual
estimations for each cryptocurrency is driven by the intention to investigate
the distinct nature, if any, of the cryptocurrency market, characterized by
its diversity and the unique backing of each currency by different project
ideas or use cases. Although we recognize the limitations inherent in using
price data alone, where the assumption that prices immediately reflect
underlying economic, technological, and regulatory developments may not hold
due to the known latency in market reactions, our decision to perform
individual estimations for each cryptocurrency is primarily driven by the
objective to investigate the distinct price patterns. This methodological
choice allows us to explore the nuances of price movements and volatility
patterns across different cryptocurrencies, acknowledging that these price
dynamics may serve as proxies for underlying determinants in the absence of
more detailed data888There is no established consensus for one or the other
approach, and they can be used concurrently depending on the use case (Bou and
Satorra, 2018). This comparison has been further analyzed within the
literature at the intersection of statistics and social sciences (Allison and
Bollen, 1997; Teachman et al., 2001; Ejrnæs and Holm, 2006; Allison, 2009)..

Figure 6 and Figure 7 show the individual estimates for the realized
semivariances parameters, respectively, the positive and negative part at any
of the previously mentioned future horizons. The significant estimated
parameters $\phi^{+}_{d}$ have the same sign for most of the sample under
study, with few exceptions at any horizon with negative signs. On the
contrary, the significant estimated parameters $\phi^{-}_{d}$ are less in
number, and the individual results are contrasting, with some of the
cryptocurrencies in the universe having a positive impact for the negative
realized semivariance and some others having a negative one. Those results
validate the estimation of the panel models in Section 4.1 where the effect of
$\phi^{+}_{d}$ impacts more than $\phi^{-}_{d}$ on future volatilities.

Figure 6: Barplots of $\phi^{+}_{d}$ coefficients for the individual HAR-
semiRV model specification over three different time windows. Each subplot
differs for the future horizon of the model specifications: 1 day (top left),
5 days (top right), 22 days (bottom left), and 66 (bottom right). Blue bars
identify statistically significant parameters at 5% confidence level, while
red bars are for statistically insignificant ones.

Figure 7: Barplots of $\phi^{-}_{d}$ coefficients for the individual HAR-
semiRV model specification over three different time windows. Each subplot
differs for the future horizon of the model specifications: 1 day (top left),
5 days (top right), 22 days (bottom left), and 66 (bottom right). Blue bars
identify statistically significant parameters at 5% confidence level, while
red bars are for statistically insignificant ones.

Differently from $\phi^{+}_{d}$ and $\phi^{-}_{d}$, where the percentage of
significant estimates in the selected cryptocurrency universe does not change
with the model horizons, the daily leverage effect parameter $\gamma$ tends to
be significant for the majority of the entities when the time horizon is
larger, one-month or three-month ahead. Figure 8 showcases these results,
highlighting that the significant parameters $\gamma$ are always negative with
few exceptions.

Figure 8: Barplots of $\gamma$ coefficients for the individual HAR-semiRV
model specification with daily leverage effect over three different time
windows. Each subplot differs for the future horizon of the model
specifications: 1 day (top left), 5 days (top right), 22 days (bottom left),
and 66 (bottom right). Blue bars identify statistically significant parameters
at 5% confidence level, while red bars are for statistically insignificant
ones.

Figure 9 and Figure 10 present the individual estimates for the positive and
the negative jump components, whose results are coherent with the panel
estimation of the same model specification. Negative jumps exhibit a strong
positive effect on future volatility. In contrast, positive jumps are
associated with significant estimates mixed among effects of opposite signs in
the universe of cryptocurrency analyzed.

Figure 9: Barplots of $\phi_{j}^{+}$ coefficients for the individual HAR-SJV
model specification over three different time windows. Each subplot differs
for the future horizon of the model specifications: 1 day (top left), 5 days
(top right), 22 days (bottom left), and 66 (bottom right). Blue bars identify
statistically significant parameters at 5% confidence level, while red bars
are for statistically insignificant ones.

Figure 10: Barplots of $\phi_{j}^{-}$ coefficients for the individual HAR-SJV
model specification over three different time windows. Each subplot differs
for the future horizon of the model specifications: 1 day (top left), 5 days
(top right), 22 days (bottom left), and 66 (bottom right). Blue bars identify
statistically significant parameters at 5% confidence level, while red bars
are for statistically insignificant ones.

##  5 Connection to Economic Theories

The empirical analysis presented in our study reveals significant insights
into the dynamics of cryptocurrency volatility, challenging traditional
financial theories with empirical evidence from high-frequency data.
Particularly, our findings on the inversion of the traditional leverage
effect, where positive returns correspond to increased volatility, suggest a
deviation from classical financial theory. This anomaly can be connected to
the presence of momentum effects in cryptocurrencies, as reported in Yang
(2019). Established behavioral finance theories, such as those addressing
investor overconfidence, herd behavior, and the disposition effect (De Long et
al., 1990; Barberis et al., 1998; Hong and Stein, 1999; Daniel et al., 1998),
can explain these unique volatility patterns. Cryptocurrencies exhibit a
distinct sensitivity to positive returns, possibly driven by the fear of
missing out (FOMO), as described in Section 4.2 and speculative trading, which
are characteristic of overconfident and herd-like investor behavior, following
the notion of noise traders in the Glosten-Milgrom model (Glosten and Milgrom,
1985). This divergence underscores the importance of considering psychological
and behavioral factors in further analysis of cryptocurrency asset volatility,
offering a new perspective on investor behavior in these markets, as done with
regards to herding behavior in da Gama Silva et al. (2019); Gurdgiev and
O’Loughlin (2020); Vidal-Tomás et al. (2019); Stavroyiannis and Babalos
(2019).

The findings from our study on cryptocurrency volatility dynamics align with
the principles of market microstructure theory, particularly when viewed
through the lens of the AMH theory (Lo, 2004) as explored by Khuntia and
Pattanayak (2018). The cryptocurrency market’s continuous trading cycle over
the whole day (Demiralay and Golitsis, 2021) and the prevalence of speculators
and uninformed traders (Grobys and Junttila, 2021) contribute to its dynamic
nature, contrasting with more traditional asset classes. In this context, the
AMH’s emphasis on market adaptation with temporary inefficiency and unusual
patterns in price volatility is particularly relevant, as the pace of
technological innovation and regulatory developments continually shape
cryptocurrency market dynamics. Specifically, the reliance on blockchain
technology as the underlying framework for this new asset class introduces new
microstructure variables to consider, such as the number of network addresses
related to a particular token and the processing power needed to maintain the
corresponding blockchain network in place (see Koutmos (2018) for such an
analysis on BTC and Fanning and Centers (2016) for an evaluation of the
technology impact on the financial services industry). The cryptocurrency
markets show notable distinctions from traditional markets that can motivate
the observed volatility patterns in this work. Silantyev (2019) observe those
as coming primarily from shallower average depths of the order book, leading
to various disparities in how order books handle order flow. Trade flow
imbalances appear effective in explaining price changes in cryptocurrencies.

##  6 Discussion

This section evaluates the implications of our findings on the cryptocurrency
market’s unique volatility landscape, drawing on high-frequency volatility
estimators. The econometrics analysis aims to offer valuable insights for
market participants and regulators, enriching the understanding of this
emerging asset class’s volatility patterns and stakeholder roles.

### Key Findings

Our empirical results reveal the cryptocurrency market’s volatility dynamics
as distinctly marked by speculative behaviors and psychological influences. A
primary observation is the inversion of the traditional leverage effect, where
positive returns are correlated with future volatility, suggesting a market
propelled by speculative enthusiasm and the psychological phenomenon of Fear
of Missing Out (FoMo). This inversion signifies a heightened market activity
in response to positive gains, attracting investors who anticipate joining a
new investment opportunity.

FoMo, particularly pronounced among retail investors, emerges as a crucial
determinant of investment behavior, encouraging entry into the market on
market downturns as opportunities for purchasing at perceived discounts. This
behavior underscores a herding effect in investment decisions, influenced by a
collective eagerness to participate in perceived lucrative opportunities
without fully grasping the market’s complexities. When separately estimating
the selected model specification over the three years that compose our
dataset, we find that the bearish period in 2022 is the one that is actually
driving the inversion of the leverage effect more than the two previous years.

The distinction between investor types, retail versus institutional, further
elucidates market dynamics. Retail investors, driven by FoMo and speculative
motives, often diverge from traditional market analysis, leading to an
inversion in leverage effect dynamics. Conversely, institutional investors
adhere to a more conventional understanding of market movements, leveraging
price variations for arbitrage and strategic advantage, highlighting their
role in sustaining market stability through informed decision-making. In this
regard, we document the asymmetric effect of returns when accounting for the
intraday jump dynamics. This distinction between investor behaviors underpins
the varied volatility drivers in the market. On one side, there are retail
investors driven by Fear of Missing Out (FoMO), causing an inversion of the
leverage effect at the daily level, and on the other, informed traders
exploiting price jumps for arbitrage purposes, causing the usual asymmetric
leverage effect at the intraday level.

Moreover, we observe less persistence and volatility memory in cryptocurrency
than in traditional equity markets. This finding motivates a deeper
investigation into the underlying factors that contribute to these distinctive
volatility patterns. The cryptocurrency market, characterized by its
innovative technologies and trading mechanisms, diverges significantly from
equity markets. The increase in on-chain activity, though still less
predominant than off-chain trades, introduces novel financial dynamics not
present in traditional finance. Additionally, the cryptocurrency market’s
structure, which includes both centralized and decentralized platforms,
creates unique arbitrage opportunities. These factors lead to a slower
assimilation of information into cryptocurrency prices as the market navigates
through these dual layers of activity. This complex and evolving landscape
underlines the importance of understanding the unique aspects of the
cryptocurrency market, particularly its reduced volatility persistence, and
memory, shedding light on its distinct behavior and challenges.

### Implications and Further Advancements

The inversion of the traditional leverage effect and the significant influence
of FoMo necessitate reconsidering the investment approach to this asset class.
This market’s distinct behavior, marked by inefficiencies in information
transmission and fragmentation across various trading platforms, suggests that
the Adaptive Market Hypothesis (AMH) may offer a more accurate framework than
the Efficient Market Hypothesis (EMH) for understanding cryptocurrency
dynamics. This insight prompts a more cautious and informed approach,
acknowledging the psychological underpinnings and the potential for sudden
market shifts due to the emergent and still unstable nature of the ecosystem.

The insights derived from our analysis of cryptocurrency market volatility
underscore the potential benefits of developing more sophisticated pricing
models for cryptocurrency derivatives. While our study does not delve into the
exploration of cryptocurrency derivatives directly, we suggest that an in-
depth understanding of the unique volatility patterns observed in the
cryptocurrency market is crucial for devising accurate risk management
strategies. This recognition highlights the importance of derivatives in
providing hedging capabilities, necessitating pricing models that effectively
account for the cryptocurrency market’s distinct volatility characteristics.
Consequently, this research points towards the need for innovative derivative
pricing mechanisms that can adapt to the peculiar aspects of volatility
inherent in cryptocurrencies, thereby aiding in mitigating unexpected
volatility patterns. Some preliminary results on cryptocurrency derivatives
come from Alexander and Imeraj (2023), which analyze the dynamic delta hedging
of Bitcoin options, revealing that Bitcoin’s implied volatility curves behave
very differently from those of equity index options, similarly to the
comparison we provide in this article from the historical volatility
perspective.

Regulatory efforts should enhance transparency through mandatory disclosure of
trading data by exchanges and wallet providers, strengthening financial
literacy with targeted educational initiatives on cryptocurrency risks and
management. These measures aim to stabilize the market and protect investors
against the volatility and speculative risks inherent in cryptocurrency
trading. The challenge relies on tailoring oversight mechanisms that recognize
and effectively manage the inherent volatility and the complex behaviors of
cryptocurrency market participants without harming the increased liquidity and
market participation in the cryptocurrency ecosystem. For instance, Chokor and
Alfieri (2021) has analyzed the effect of regulation on trader activity,
finding that investors reacted less negatively for most illiquid
cryptocurrencies and those with higher information asymmetry. However,
Feinstein and Werbach (2021) notes that such an effect is unclear.

As a recommendation for further advancements in the cryptocurrency ecosystem,
we acknowledge that volatility is a crucial component of any derivative
pricing model and can benefit the activities of both retail investors and
traders. Also, cryptocurrency markets are relatively new and less efficient
than traditional financial markets. Accurate volatility estimation can help
improve market efficiency by providing more reliable information about
cryptocurrencies’ risk and return characteristics and allowing portfolio
managers to use these assets as an alternative asset class. Furthermore,
prompt analysis of the volatility dynamics and a solid comparison with
traditional asset classes can guide the regulators in shaping policies for
this nascent asset class, which has been the object of an ongoing debate,
especially in the last two years. The increasing attention investment in
financial technology (Kou et al., 2021) further augments the importance of
having a framework of rules in place.

##  7 Conclusions

### Contribution

We provide an in-depth analysis of price volatility in the cryptocurrency
ecosystem at both the aggregate and the individual levels. At the aggregate
level, we analyzed the effects of signed volatility, daily leverage effect,
and signed jumps on the ecosystem as a whole. In addition, we compared the
results obtained on the cryptocurrencies cross-section to a cross-section of
stocks, a more mature asset class. The results showed that positive signed
volatility, negative daily leverage, and negative signed jumps positively
impact the ecosystem’s future volatility. The first two results contrast with
common stylized facts of financial volatility in more traditional asset
classes, signaling a structural difference in the relatively immature
cryptocurrency ecosystem. Then, at the individual level, we analyzed the
abovementioned effects for each cryptocurrency in the selected sample,
retrieving that most of the cryptocurrencies positively impacted the future
volatility for positive signed volatility, with a few exceptions. The results
also showed that the daily leverage effect tends to be negative for most
cryptocurrencies and statistically significant for most cross-section
components when the time horizon is larger. We also found that the dataset’s
most recent year (2022) contributes to highlighting the effect of positively
realized semivariances on future volatility. Such an effect contrasts the
notion of the asymmetric impact of returns on the realized variance and points
to an inverted asymmetry inherent in the cryptocurrency data.

Overall, the analysis of cryptocurrency volatility by using high-frequency
data sheds light on specific traits that this nascent market has had since
2020, going from astounding growth to significant drawdowns and disbelief.
Identifying the volatility dynamics and comparing it to a more renowned asset
class is a crucial aspect to consider for the cryptocurrency ecosystem as a
whole. A better assessment of the cryptocurrency market’s volatility could
stabilize the market by producing a more liquid and reliable derivative
market, which nowadays accounts for a few large centralized exchanges such as
Binance and various residual exchanges.

### Further Research Endeavors

An interesting improvement of the volatility analysis could be generated using
sophisticated on-chain data, accounting for the blockchain user activity and
network size measure where agents are operating. Indeed, the cryptocurrency
ecosystem lives on blockchains that produce a lot of data and do not follow
the same logic as the traditional financial market. Therefore, it could be
that the information is not fully reflected by the price of coins and tokens
on centralized exchanges, as there are two alternative and interoperable
layers of activities. Moreover, the autoregressive models we employed in our
empirical study are all linear in the parameters, neglecting any possible
nonlinear relationship in the volatility pattern. This aspect can be further
analyzed by extending the model specifications to regime-switching models or
employing machine learning techniques that account for nonlinearities as
discussed in Sebastião and Godinho (2021). We reserve these points as a study
for further analysis since we believe it is possible to deepen the
understanding of the cryptocurrency volatility following those lines.

## Acknowledgements

The DAREC initiative is funded by donor Teymour H. Farman-Farmaian.

## List of Abbreviations

BTC |  Bitcoin
---|---
DeFi |  Decentralized Finance
NDTX |  NASDAQ Technology Index
RV |  Realized Variance
BV |  Bipower Variation
IV |  Integrated Variance
$RV^{+}$ |  Positive Realized Semivariance
$RV^{-}$ |  Negative Realized Semivariance
API |  Application Programming Interface
SJV |  Jump Variation
$SJV^{+}$ |  Positive Signed Jump Variation
$SJV^{-}$ |  Negative Signed Jump Variation
OLS |  Ordinary Least Squares
HAR |  Heterogeneuous Autoregression
WLS |  Weighted Least Squares
GARCH |  Generalized AutoRegressive Conditional Heteroskedasticity
GJR-GARCH |  Glosten-Jagannathan-Runkle GARCH
NFT |  Non-Fungible Token
FoMO |  Fear-of-missing-out
ICO |  Initial Coin offering

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##  Appendix A Additional Estimation Results

$RV_{h,t+h}=\mu+\phi_{d}RV_{t}+\phi^{-}_{d}RV^{-}_{t}+\phi^{+}_{d}RV^{+}_{t}+\phi^{-}_{w}RV^{-}_{t}+\phi^{+}_{w}RV^{+}_{t}+\phi^{-}_{m}RV^{-}_{t}+\phi^{+}_{m}RV^{+}_{t}+\epsilon_{t+h}$ |  | $\mathbf{\phi^{-}_{d}}$ | $\mathbf{\phi^{+}_{d}}$ | $\mathbf{\phi^{-}_{w}}$ | $\mathbf{\phi^{+}_{w}}$ | $\mathbf{\phi^{-}_{m}}$ | $\mathbf{\phi^{+}_{m}}$
---|---|---|---|---|---|---|---
Crypto | t+1 | -0.108 |  1.224∗∗∗ | -0.001 |  0.152∗∗∗ |  -0.245∗∗∗ |  0.472∗∗∗
(-1.311) | (9.895) | (-0.037) | (2.607) | (-3.473) | (4.218)
t+5 | 0.157 | 0.433 | -0.037 |  0.229∗∗∗ |  -0.316∗∗∗ |  0.479∗∗∗
(0.425) | (1.473) | (-0.921) | (3.448) | (-5.615) | (9.886)
t+22 | -0.048 |  0.196∗∗∗ |  0.042∗∗ |  0.11∗∗∗ |  -0.48∗∗∗ |  0.729∗∗∗
(-0.711) | (3.56) | (2.442) | (4.493) | (-21.146) | (21.917)
t+66 |  -0.054∗∗∗ |  0.108∗∗∗ |  0.02∗∗ |  0.07∗∗∗ |  -0.37∗∗∗ |  0.528∗∗∗
(-4.333) | (6.968) | (2.144) | (5.252) | (-23.174) | (23.225)
Equity | t+1 |  0.073∗∗ |  0.3∗∗∗ |  0.282∗∗∗ |  0.154∗∗∗ |  -0.136∗ |  0.477∗∗∗
(2.277) | (5.347) | (3.94) | (2.321) | (-1.971) | (5.373)
t+5 |  0.054∗∗ |  0.12∗∗∗ |  0.184∗∗∗ |  0.258∗∗∗ |  -0.167∗∗∗ |  0.504∗∗∗
(2.416) | (4.945) | (5.735) | (6.884) | (-4.252) | (10.034)
t+22 | 0.009 |  0.085∗∗∗ |  0.096∗∗∗ |  0.198∗∗∗ |  -0.081∗∗ |  0.423∗∗∗
(0.77) | (5.498) | (4.546) | (7.799) | (-2.336) | (11.013)
t+66 | 0.01 |  0.046∗∗∗ |  0.101∗∗∗ |  0.161∗∗∗ |  0.123∗∗∗ |  0.313∗∗∗
(1.045) | (4.755) | (5.706) | (9.176) | (4.011) | (10.867)
***$p<0.01$, **$p<0.05$, *$p<0.1$ |  |  |  |  |
Table 6: Estimation results for the full HAR-semiRV model specifications.
Results are displayed for the cryptocurrency and the equity panel over the
four different horizons. T-stats are in parentheses. The model equation on top
of the table reflects all the parameters common to the specification reported
in this table. The total number of observations available considering all the
entities used for the panel estimate is 47787 for crypto and 17111 for equity

Figure 11: Barplots of estimated coefficient for the full panel HAR-semiRV model specification over three different time windows. Each subplot differs for the future horizon of the model specifications: 1 day (top left), 5 days (top right), 22 days (bottom left), and 66 (bottom right). $RV_{h,t+h}=\mu+\phi_{c}BV_{t}+\phi_{w}RV_{w,t}+\phi_{m}RV_{m,t}+\epsilon_{t+h}$ |  | $\mathbf{\phi_{c}}$ | $\mathbf{\phi_{w}}$ | $\mathbf{\phi_{m}}$
---|---|---|---|---
Crypto | t+1 |  0.634∗∗∗ | 0.005 |  0.053∗∗∗
(7.65) | (1.35) | (4.26)
t+5 |  0.318∗∗∗ |  0.051∗∗ |  0.05∗∗∗
(5.445) | (2.525) | (3.661)
t+22 |  0.09∗∗∗ |  0.028∗∗∗ |  0.048∗∗∗
(4.021) | (2.58) | (4.084)
t+66 |  0.03∗∗∗ |  0.006∗ |  -0.012∗∗∗
(2.651) | (1.188) | (-2.721)
Equity | t+1 |  1.088∗∗∗ |  0.126∗∗∗ |  0.129∗∗∗
(9.098) | (5.202) | (2.588)
t+5 |  0.582∗∗∗ |  0.148∗∗∗ |  0.14∗∗∗
(11.173) | (10.17) | (5.491)
t+22 |  0.335∗∗∗ |  0.087∗∗∗ |  0.154∗∗∗
(9.397) | (8.592) | (11.512)
t+66 |  0.209∗∗∗ |  0.067∗∗∗ |  0.205∗∗∗
(8.755) | (9.528) | (22.162)
***$p<0.01$, **$p<0.05$, *$p<0.1$ |  |
Table 7: Estimation results for HAR-SJV model specifications with no jump
components at the first lag. Results are displayed for the cryptocurrency and
the equity panel over the four different horizons. T-stats are in parentheses.
The model equation on top of the table reflects all the parameters common to
the specification reported in this table. The total number of observations
available considering all the entities used for the panel estimate is 47787
for crypto and 17111 for equity

Figure 12: Barplots of estimated coefficient for the panel for HAR-SJV model
specifications with no jump components over three different time windows. The
number in the legend reflects the year of analysis and is inclusive. Each
subplot differs for the future horizon of the model specifications: 1 day (top
left), 5 days (top right), 22 days (bottom left), and 66 (bottom right).