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Archives of Business Research – Vol. 11, No. 12
Publication Date: December 25, 2023
DOI:10.14738/abr.1112.16086.
Džafić, J., & Hečimović, E. (2023). (Un)Stable Cryptocurrency Markets: Insights from Volatility Modeling. Archives of Business
Research, 11(12). 103-119.
Services for Science and Education – United Kingdom
(Un)Stable Cryptocurrency Markets: Insights From Volatility
Modeling
Jasmina Džafić
University of Zenica, Faculty of economics,
Fakultetska bb, 72000 Zenica, Bosnia & Herzegovina
Emir Hečimović
Prague University of Economics and Business
Faculty of Finance and Accounting
Winston Churchill Sq 1938/4 CZ-13067 Prague 3 – Žižkov
ABSTRACT
The cryptocurrency market has attracted considerable attention from investors
and researchers alike. This paper examines the volatility patterns of two major
cryptocurrencies utilizing GARCH modeling: Bitcoin, based on a proof-of-work
mechanism, and Cardano, operating on a proof-of-stake mechanism. Our findings
reveal differences in the volatility structures of the two cryptocurrencies, with
Cardano demonstrating a reduced long-term volatility compared to Bitcoin. This
study suggests that transitioning from proof-of-work to proof-of-stake mechanisms
might lead to a decrease in market volatility.
Keywords: Financial Innovation, Cryptocurrency, Bitcoin, Cardano, Jel Classification: G1,
G11, G15, F31
INTRODUCTION
Digital money existed long before Bitcoin's emergence (January 2009). Thus, investor's
enthusiasm towards cryptocurrencies primarily refers to the promise of the underlying
technology: blockchain. Conventional currencies and payment systems always require some
central authority that must be trusted when two parties want to make a transaction.
Cryptocurrencies circumvent this issue by using blockchain technology. Blockchain technology
can potentially serve a much broader purpose than just settling cryptocurrencies transactions.
Thus, under the umbrella term of decentralized finance emerged numerous blockchain
utilizations: smart contracts (blockchain contracts that can be enforced without human
interaction), non-fungible tokens (NFTs), video games based on blockchain, various
applications in financial services including tokenization of stocks, and other. However, with the
exception of cryptocurrencies, none of these applications has become mainstream so far. With
more than 18,000 cryptocurrencies in existence1, there are substantial technical, functional,
and conceptual differences between various cryptocurrencies. However, cryptocurrency
research has been generally focused on Bitcoin and very little attention has been given to the
understanding of cryptocurrency differences and the impact these characteristics might have
1
See https://coinmarketcap.com/
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on the markets. This paper aims to shed light on one such difference. There are two main
consensus mechanisms behind blockchain: proof-of-work and proof-of-stake. The proof-of- work is a common consensus mechanism used by most popular cryptocurrencies including
Bitcoin. However, the proof-of- work mechanism consumes substantial amounts of electricity.
Thus, proof-of-stake cryptocurrencies emerged as a potential solution to rising electricity
consumption. Historically high volatility of cryptocurrency markets might be a specific feature
of proof-of-work cryptocurrencies due to their heavy reliance on energy and prevalence among
cryptocurrencies. Prices of basic energy (natural gas, electricity, heating oil) are generally more
volatile than prices of other commodities. Thus, the relationship between proof-of-work
cryptocurrencies and electricity consumption might contribute to volatility of cryptocurrency
markets. In this essay, we thus hypothesize that the blockchain mechanism upon which
cryptocurrencies are based might influence volatility. Therefore, we use GARCH to model
volatility of two major cryptocurrencies for each consensus mechanism: Bitcoin (proof-of- work) and Cardano (proof-of-stake), and compare the conditional volatility of both.
THEORETICAL FRAMEWORK AND LITERATURE REVIEW
Despite a large number of cryptocurrencies actively trading on the market, the existing
literature focuses predominantly on Bitcoin. Considering Bitcoin's popularity and market
capitalization, this is somewhat expected. However, cryptocurrencies differ significantly in
terms of their characteristics. Thus, not only the majority of cryptocurrencies have been
underrepresented in literature, there was very little effort to understand how these differences
might impact the volatility and also drive various cryptocurrencies to respond differently to
external shocks. When it comes to volatility, researchers have usually attempted to model the
volatility of specific cryptocurrencies. Understanding the volatility dynamics received
particular interest in academic literature provided that cryptocurrencies have been
characterized with extreme price movements. Briere et al. (2013) show that Bitcoin has
substantially higher volatility compared with other assets classes but also higher average
return. Yermack (2013) compared the volatility of Bitcoin to several major fiat currencies
including the Japanese Yen, Swiss Franc, and Euro. This study contributed to the early
understanding that Bitcoin’s high volatility (compared to traditional currencies) prevents it
from serving as a means of payment. Similarly, Sapuric & Kokkinaki (2014) compared the
volatility of the Bitcoin exchange rate against several major currencies (Euro, Swiss Franc,
Russian Ruble, and Japanese Yen) finding that the volatility of Bitcoin is substantially higher.
Glaser et al. (2014) were among the first to employ the GARCH model for such a purpose. Since
GARCH family models have been used extensively in modeling cryptocurrencies volatility.
Gronwald (2014) used an autoregressive jump-intensity GARCH model to model the extreme
price swings of Bitcoin. GARCH models on Bitcoin time series data eventually concluding that
the Asymmetric Component GARCH provides the best performance. Stavroyiannis (2018)
utilized a GJR-GARCH model to determine whether Bitcoin violates the VaR more than other
speculative assets including gold. His conclusion underlines the high volatility of Bitcoin and
the stronger tendency to violate VaR as compared to gold. Ardia et al. (2019) used MSGARCH
to successfully detect regime changes in the volatility of Bitcoin. Furthermore, Bayesian
estimations for GARCH family models have been proposed by some authors including Bauwens
et al. (2010) who utilized the Bayesian MCMC algorithm to estimate MSGARCH. As opposed to
solely focusing on Bitcoing, Chu et al. (2017) provided a GARCH modeling on the seven most
popular cryptocurrencies by fitting 12 different types of GARCH models and using information
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URL: http://doi.org/10.14738/abr.1112.16086
criteria to evaluate the models. Their conclusion indicates that IGARCH and GJR-GARCH models
provide the best fit in-sample for most of the cryptocurrencies. Burnie (2018) focused on
correlations between various cryptocurrencies finding strong tendencies. Some attempts to
employ high-frequency data in modeling Bitcoin volatility were also made. Most notably, Baur
and Dimpfl (2018) modeled a Bitcoin realized volatility showing that the volatility of Bitcoin is
excessive when compared to fiat currencies. Since external shocks are within the area of our
interest, it is worth mentioning several papers investigating the impact of external shocks on
cryptocurrencies volatility. Aysan et al. (2019) analyze the effects of geopolitical risks on
bitcoin returns and volatility. Their findings suggest that Bitcoin volatility is increasing with
increased geopolitical risks. Wang et al. (2020) find that Bitcoin volatility increases with the
increase in economic policy uncertainty while Lyócsa and Molnár (2020) point to increased
volatility on days of cryptocurrency-related hacking attacks. Finally, there are several
influential studies explaining the relationship between public sentiment and financial markets.
Brown & Cliff (2005) argue that mispricing of stock valuation can be explained by sentiment.
Engelberg (2008) investigates earning announcements news for approximately 5000
companies over a period between 1999 and 2005. He finds that earning announcements
published in news articles contain predictive power over future returns. Liao et al. (2019)
discuss whether news affects mergers and acquisitions. Their findings indicate that the more
optimistic sentiment embedded in the news leads to a higher chance of reaching successful
acquisitions. However, they also speculate that if the acquirer receives high media coverage, it
will on average experience negative post-acquisition returns. Zhang et al. (2011) were pioneers
of investigating a potential correlation between Twitter sentiment and financial markets
(specifically: Dow Jones, NASDAQ, and S&P 500). However, they did not detect the correlation
between either positive or negative emotions embedded within tweets with stock markets.
Using the supervised machine learning approach, Liu (2017) reaches the same conclusion.
Bollen et al. (2011) measured six dimensions of mood (i.e., calm, alert, sure, vital, kind, and
happy) and binary sentiment polarity (positive or negative) in tweets finding the correlation of
sentiment with stock markets. Similarly, Mittal & Goel (2012) found a correlation between
happiness and calmness with Dow Jones.
SUBJECT OF THE RESEARCH
Broadly speaking, there are two main consensus mechanisms behind blockchain: proof-of- work and proof-of-stake. Blockchain is a decentralized peer-to-peer system with no central
authority. Thus, without central authority serving as an intermediator, a consensus mechanism
is needed to settle transactions between different parties. The proof-of-work is a common
consensus mechanism used by most popular cryptocurrencies including Bitcoin, Litecoin,
Monero, and others. According to Cryptoslate exchange 2 , the 10 largest proof-of-work
cryptocurrencies measured by market capitalization account alone for more than 66% of the
cryptocurrency market as of September 2023. However, the proof-of-work mechanism
consumes substantial amounts of electricity. In fact, the estimated power demand of the Bitcoin
network alone is currently approximately 117 TWh annually3. As such, the Bitcoin network
2 https://cryptoslate.com/cryptos/proof-of-work/
3
See https://cbeci.org/
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alone consumes approximately 4.2 times more electricity than the whole of Slovakia, 3.3 times
more than Denmark, and about the same as Norway4.
Figure 1: Daily Bitcoing network power demand 2011 – 2023
Source: https://ccaf.io/cbeci/index
Obviously, the near-exponential jump in electricity consumption by Bitcoin and other proof- of- work cryptocurrencies has been severely criticized by Gallersdörfer et al. (2020), Li et al.
(2019), de Vries (2021), Mora et al. (2018), Dittmar et al. (2018), and others. Proof-of-stake
cryptocurrencies emerged as a potential solution to rising electricity consumption. Platt et al.
(2021) found that the electricity consumption of proof-of-work-based Bitcoin is three times
higher than that of the highest consuming proof-of-stake cryptocurrency. However, in terms of
market capitalization proof-of-stake cryptocurrencies are still marginal compared to proof-of- work cryptocurrencies. According to Coinmarketcap exchange 5 top 10 proof-of-stake
cryptocurrencies in terms of market capitalization amount to approximately 12.3% of Bitcoin's
market capitalization alone, as of September 20236. In the previous section, we have discussed
that high volatility has been well-established for Bitcoin and other major cryptocurrencies.
However, literature has not devoted enough attention to the distinction between proof-of-work
and proof-of-stake cryptocurrencies. Thus, high volatility might be a specific feature of proof- of-work cryptocurrencies due to their reliance on energy. Prices of basic energy (natural gas,
electricity, heating oil) are generally more volatile than prices of other commodities. Thus, this
relationship between proof-of-work cryptocurrencies and electricity consumption might
contribute to volatility. We propose modeling and comparing volatilities between proof-of- work and proof-of-stake cryptocurrency classes to detect and understand these potential
differences. Why is this important? Although proof-of-stake is still marginal in terms of market
capitalization, this might change. The second-largest cryptocurrency by market capitalization
Ethereum has recently fully transitioned to a proof-of-stake mechanism7.
. Dogecoin8 (currently
the 7th largest cryptocurrency by market capitalization) is also planning the transition and
4
See https://www.iea.org/countries/norway
5 https://coinmarketcap.com/
6 This figure is not taking into account market capitalization of Ethereum which has just recently transitioned to a proof- of-stake. With Ethereum added, top 10 biggest proof-of-stake cryptocurrencies would amount to just over 50% of
Bitcoin's market capitalization.
7 https://www.technologyreview.com/2023/02/28/1069190/ethereum-moved-to-proof-of-stake-why-cant-bitcoin/
8 https://www.binance.com/en/news/top/6912062
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other cryptocurrencies might follow. In other words, proof-of-stake will take a much larger
share of the overall cryptocurrency market in the following years. If proof-of-stake
cryptocurrencies are less volatile, transitioning to a new consensus mechanism will contribute
to a more stable cryptocurrency market overall. Cryptocurrencies were meant to serve
primarily as money. However, there is a rather strong consensus in the literature that so far,
cryptocurrencies are in reality serving mainly a role of speculative assets (Yermack (2015),
Ciaian et al. (2016), Baur et al. (2018), Shahzad et al. (2019)). Transitioning to less volatile
consensus mechanisms might lead to increased adoption of cryptocurrencies as money.
Furthermore, this might yield several important policy outcomes too. Less volatile
cryptocurrency markets might nudge governments into issuing their own CBDC (central bank
digital coins). At the moment, China has confirmed the development of a government-back
digital coin9 so-called eCNY. According to the Atlantic Council10 11 nations have already issued
CBDCs, 21 are enrolled into a pilot program, 33 are developing digital coins, while 46 are
officially researching and considering (including USA and EU). For most developed nations high
electricity consumption of traditional cryptocurrencies is a major stumbling block when it
comes to larger adoption of this asset as well as potential issuance of CBDCs. This is because
proof-of-work cryptocurrencies are simply not aligned with global warming initiatives as such.
Government-backed digital coins that would significantly increase the nation’s ecological
footprint would be disregarded. However, a proof-of-stake mechanism would allow
governments to issue CBDC without compromising global warming policies.
RESEARCH METHODOLOGY
Both proof-of-work and proof-of-stake cryptocurrency prices are publicly available. Data for
cryptocurrencies are collected via Bitstamp exchange (https://www.bitstamp.net/). Bitstamp
is one of the oldest cryptocurrency exchanges operating since 2011 as well as the first fully EU- licensed and regulated bitcoin exchange in Europe. In this paper, we have chosen Bitcoin as a
representative for proof-of-work cryptocurrencies and Cardano as a representative for proof- of-stake cryptocurrencies. Bitcoin, the pioneer in the cryptocurrency space, operates on a
proof-of-work consensus mechanism, which requires miners to solve complex mathematical
problems to validate transactions and create new blocks. This mechanism, while energy- intensive, has stood the test of time, offering robust security and decentralized control. On the
other hand, Cardano, a newer generation cryptocurrency, utilizes a proof- of-stake consensus
mechanism, which is considered to be more energy-efficient and scalable. In this mechanism,
validators are chosen to create a new block based on the number of coins they hold and are
willing to "stake" or lock up as collateral. Although Ethereum is arguable the most well-know
proof-of-stake cryptocurrency, it was not selected as a representative for proof-of-stake
cryptocurrencies in this paper since it has just recently transitioned from a proof-of-work to a
proof-of-stake consensus mechanism. This means that Ethereum has operated on a hybrid
model for the past several years, incorporating features of both proof-of-work and proof-of- stake mechanisms. This transitional phase brought in considerable variability and uncertainty
in the data, potentially affecting the accuracy and reliability of volatility modeling. The selection
of Bitcoin and Cardano as representatives for their respective consensus mechanisms was
9 https://www.reuters.com/world/china/china-will-advance-cbank-digital-currency-improve-its-design-governor-says- 2021-11-09/
10 https://www.atlanticcouncil.org/cbdctracker/
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based on several criteria, including market capitalization and transaction volume. Bitcoin, being
the first and the most established cryptocurrency, has a high market capitalization, indicating
its wide acceptance and trust among the community. Its substantial transaction volume also
showcases the liquidity and activity in the Bitcoin network, making it a fitting choice for
representing proof-of-work cryptocurrencies. Cardano, on the other hand, has rapidly emerged
as a significant player in the cryptocurrency market, backed by a strong academic foundation
and a growing community. It holds a substantial market capitalization, reflecting its growing
influence and adoption in the cryptocurrency space. Its transaction volume also indicates a
vibrant and active network, making it a suitable choice for representing proof-of-stake
cryptocurrencies in the study. The empirical analysis in this paper was conducted using Eviews
13 and supplemented with Python, a programming language well-suited for data analysis and
modeling. Libraries such as Pandas and NumPy were utilized for data manipulation and
analysis, offering powerful tools for handling datasets efficiently. Visualization libraries like
Matplotlib and Seaborn were employed to create graphs and plots, aiding in the interpretation
of the data. As with regards to methodology, GARCH models variance as the weighted average
of long-run variance (α0), new information that is represented by current period’s variance (αi)
and the variance predicted for this period (βj). GARCH (m, s) can be written in a following form:
(1)
where {εt} represent independent and identically distributed random variables (iid) with zero
mean and variance of 1. Stationarity condition dictates that ∑max (m, s) (αi + βi) < 1. We refer
i=1 to αi as ARCH parameter while βi is GARCH parameter. Thus, if s = 0 GARCH equation will
reduce to ARCH equation.
RESULTS OF EMPIRICAL RESEARCH AND DISCUSSION
In this section of the paper, we focus on examining the volatility patterns found in two
cryptocurrencies: Bitcoin, representing the proof-of-work mechanism, and Cardano,
representing the proof-of-stake mechanism. The ARMA-GARCH modeling of both time series
was conducted on a range between January 1, 2018, to September 7, 2023. We opted for this
timeframe because Cardano launched in late 2017, and to mitigate the impact of high volatility
in its early days, we focused on data from 2018 onward. In aligning with our approach, we also
examined the Bitcoin time series for the corresponding period. Exploratory analysis was
extended to a longer time series, encompassing data from 2015, specifically for Bitcoin.
Bitcoin Modeling
Before diving into the core of the volatility modeling, it is important to note that we have
converted Bitcoing prices into returns. Working with returns instead of prices is a vital step in
producing a reliable and precise analysis. One of the main advantages of using returns is that
they are generally stationary, meaning their properties do not change over time, which helps in
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URL: http://doi.org/10.14738/abr.1112.16086
avoiding misleading outcomes in the analysis. Moreover, returns are more likely to have a
consistent spread of values over time, making the volatility modeling process simpler and more
effective.
Rt = ln(
Pt
Pt−1
) (2)
Here, Rt is the return at time t, while Pt and Pt−1 represent the prices at time t and t−1
respectively. Plotting Bitcoing daily prices and returns yields the following Figure 2.
Figure 2: Bitcoin Daily Prices and Returns
We can observe that the price of Bitcoin has experienced significant growth, especially in the
recent years, mixed with periods of sharp declines. The average daily percentage gain (on days
with positive returns) for Bitcoin is approximately 2.51% while the average daily percentage
loss (on days with negative returns) for Bitcoin is approximately 2.41%. However, as can be
seen from the Figure 2, daily gains/losses did go as high as 30% on certain days. Next, we
proceed with the ARMA modeling for Bitcoin. The ARMA model is a popular time series model
that combines the Autoregressive (AR) and Moving Average (MA) models. Before fitting an
ARMA model, it is essential to ensure that the time series is stationary. A common method to
test for stationarity is the Augmented Dickey-Fuller (ADF) test. The ADF test on the initial time
series data before differencing resulted in a p-value of 0.7492, indicating non-stationarity and
the presence of a unit root. To address this, a log transformation and differencing were applied.
After these adjustments, the ADF test was repeated, yielding a p-value of less than 0.01,
confirming the stationarity of the transformed time series. Detailed test statistics can be found
in Appendix A for reference. This implies that the log transformation and differencing
effectively addressed the non-stationarity observed in the initial analysis, providing a more
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suitable basis for subsequent econometric modeling and analysis. Next, we need to identify the
best-fitting parameters for the ARMA model. We will rely on the minimization of the AIC
(Akaike Information Criterion) to identify these. The parameters are: p - The number of lag
observations included in the model (lag order); q- The size of the moving average window
(order of moving average). A lower AIC value indicates a better model. After expanding the
range of parameters p and q to include values from 0 to 3 and attempting the grid search to find
the optimal parameters for the ARMA model, we find ARMA (2,2) to be optimal. This model
includes two autoregressive terms, and two moving average terms. While ARMA (2,3) and
ARMA (3,3) exhibited slightly lower AIC values, it's noteworthy that the evaluation of these
models revealed coefficients that did not achieve statistical significance.
Table 1: Akaike Information Criteria for ARMA Models (Bitcoin)
MA (0) MA (1) MA (2) MA (3)
AR (0) - -3.735631 -3.736479 -3.735712
AR (1) -3.735396 -3.735730 -3.737439 -3.736501
AR (2) -3.735809 -3.737634 -3.751557 -3.751768
AR (3) -3.741304 -3.740373 -3.741153 -3.752590
After evaluating the model, we get the following parameter values.
Table 2: Summary of ARMA (2,2) Model Estimation (Bitcoin)
Parameter/Statistic Value
AR (1) 1.226030
AR (2) -0.897201
MA (1) -1.250466
MA (2) 0.929703
Log-Likelihood 3890.613
AIC -3.751557
The AR (1) term shows a positive coefficient, indicating a positive correlation with the previous
day’s return. In contrast, the AR (2) term has a larger negative coefficient, suggesting a stronger
negative correlation with the second lag of the dependent variable.
The MA (1) term is significantly negative, implying a substantial negative moving average
component associated with the first lag of the error term. The MA (2) term is positive, indicating
a positive moving average effect linked to the second lag of the error term. The Durbin-Watson
statistic of 2.061700 indicates some positive autocorrelation in the residuals. Next, we can plot
the residuals to visualize any patterns and conduct further diagnostic checks to ensure no
information is left for volatility modeling.
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URL: http://doi.org/10.14738/abr.1112.16086
Figure 3: ARMA (2,2) Residual Density Plot (Bitcoin)
When it comes to the density plot (Figure 3), ideally, we want to see a normal distribution
centered around zero. The plot indicates that the residuals are somewhat normally distributed,
albeit with heavy tails, which is also reflected in the high kurtosis value we observed earlier.
Figure 4: ARMA (2,2) Autocorrelation Function of Residuals (Bitcoin)
This plot shows the autocorrelations of the residuals at different lags. We would like to see no
significant autocorrelations, indicating that the model has captured all the information in the
data. It seems there are no significant autocorrelations in the residuals, which is a positive
indication. Additionally, an ARCH (Autoregressive Conditional Heteroskedasticity) test was
conducted to examine the presence of ARCH effects in the residuals. The obtained p-value from
the ARCH test is 0.0377. This p-value indicates that there is some evidence to reject the null
hypothesis of no ARCH effects. In other words, the data suggests the presence of
heteroskedasticity in the residuals, implying that the variance of the error term is not constant
across observations. The significance of this result implies that there may be patterns or
clusters of volatility in the data that the model has not fully captured. It's worth exploring
whether a model that accounts for conditional heteroskedasticity, such GARCH models, might
offer improvements in capturing the time-varying volatility in the series. Thus, we proceed to
the second stage of the analysis: fitting a GARCH (Generalized Autoregressive Conditional
Heteroskedasticity) model to the data. The GARCH (p, q) model is defined by the order
parameters p and q, which represent the number of lagged variance terms and lagged squared
return terms to include in the model, respectively. The GARCH (2,2) model was selected based
on the criteria mentioned before.
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Table 3: Summary of GARCH (2,2) Model Estimation (Bitcoin)
Parameter/Statistic Value
Constant 7.41E-05
ARCH 0.083769
GARCH 0.865768
Log-Likelihood 3982.587
AIC -3.837439
The coefficient for the constant term represents the baseline or average volatility. In this case,
the small magnitude suggests a low baseline volatility in the absence of shocks or previous
volatility. The ARCH coefficient indicates the impact of past shocks on the current volatility. In
this context, a higher coefficient would suggest that past shocks have a substantial influence on
the current level of volatility. The value of of 0.083769 indicates a moderate impact of past
shocks on the current volatility. The GARCH coefficient reflects the persistence of volatility over
time. A higher coefficient indicates a strong and persistent influence of past volatility on the
current level of volatility. The coefficient of 0.865768 suggests a strong and persistent influence
of past volatility on the current level of volatility. We have conducted a repeat of the ARCH test,
and the resulting p-value is 0.8978. This p-value suggests that we do not have sufficient
evidence to reject the null hypothesis of no ARCH effects. In other words, there is no significant
evidence of conditional heteroskedasticity in the residuals. This suggests that the GARCH (2,2)
model adequately captures and explains the conditional variance patterns in the data, and there
is no compelling evidence to suggest that additional ARCH effects need to be considered. Let us
proceed with plotting conditional volatility.
Figure 5: GARCH (2,2) Conditional Volatility (Bitcoin)
As we can observe, the volatility seems to exhibit the characteristic volatility clustering effect,
where periods of high volatility are followed by additional periods of high volatility, and
similarly for low volatility.
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URL: http://doi.org/10.14738/abr.1112.16086
Cardano Modeling
Let us start by visualizing the prices and daily returns for Cardano.
Figure 6: Cardano Daily Prices and Returns
Similar to Bitcoin, the ADF test on the initial time series data before differencing resulted in a
p-value of 0.5820, indicating non-stationarity and the presence of a unit root. To address this,
a log transformation and differencing were applied. After these adjustments, the ADF test was
repeated, yielding a p-value of less than 0.01, confirming the stationarity of the transformed
time series. We proceed with plotting the Autocorrelation Function (ACF) and Partial
Autocorrelation Function (PACF) of the Cardano returns. This will help in identifying the
potential orders of the AR and MA components for the ARMA model.
Figure 7: ACF and PACF of Cardano Daily Returns
To determine the best-fitting ARMA model, we consider Akaike Information Criteria.
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Table 4: Akaike Information Criteria for ARMA Models (Cardano)
MA (0) MA (1) MA (2) MA (3)
AR (0) - -2.930669 -2.932148 -2.931372
AR (1) -2.931226 -2.935783 -2.939977 -2.939036
AR (2) -2.945119 -2.945305 -2.945417 -2.944454
AR (3) -2.945450 -2.947059 -2.944546 -2.945674
Based on AIC minimization, we will evaluate ARMA (3,1) mode. The summary statistics can be
seen below.
Table 5: Summary of ARMA (3,1) Model Estimation (Cardano)
Parameter Value
AR (1) -1.002511
AR (2) 0.008186
AR (3) 0.050452
MA (1) 0.964038
Log-Likelihood 3057.154
AIC -2.947059
The AR (1) coefficient, standing at -1.002511, reveals a negative correlation between the
current return and its preceding day's return, emphasizing a robust persistence in returns over
time. This underscores the considerable influence of the previous day's return on the current
day's performance. In contrast, the AR (2) coefficient of 0.008186, with a non-significant p- value of 0.7917, indicates a minimal impact of the second lag on the current return. It suggests
that the second lag might not significantly contribute to the overall model, given its proximity
to zero and lack of statistical significance. Moving on to the AR (3) coefficient, which stands at
0.050452 with a statistically significant p-value of 0.0207, it implies a discernible positive
correlation. This suggests that the third lag contributes meaningfully to the current return,
contributing to the overall persistence in returns over time. The positive MA (1) coefficient of
0.964038 highlights the model's significant adjustment in response to a shock in the previous
day's return. This adjustment effectively diminishes residuals in the following periods,
showcasing the model's adaptability and ability to mitigate the impact of unexpected
fluctuations. The higher Log-Likelihood value in this context signals that the model fits the data
well, maximizing the likelihood of observing the given data.
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URL: http://doi.org/10.14738/abr.1112.16086
Figure 8: ARMA (3,1) Residual Density Plot (Cardano)
The density plot of the residuals shows that the residuals are approximately normally
distributed, which is a desired property for the residuals from a well-fitted model.
Figure 9: ARMA (3,1) Autocorrelation Function of Residuals (Cardano)
The ACF plot showed that most of the autocorrelations for the lagged residuals were within the
confidence bands (shaded area). This suggests that there is no significant autocorrelation in the
residuals, indicating a good fit of the model. We ran an ARCH test to check for heteroskedasticity
in the residuals. Results indicate significant ARCH effects. Thus, we can proceed with GARCH
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modeling. We fit a GARCH (3,1) model to the residuals from the ARMA model to analyze the
conditional volatility of Cardano returns.
Table 6: Summary of GARCH (3,1) Model Estimation (Cardano)
Parameter/Statistic Value
Constant 0.000175
ARCH 0.112732
GARCH 0.834591
Log-Likelihood 3183.505
AIC -3.065159
The ARCH coefficient is 0.112732 with a highly significant p-value. This implies that the squared
residuals from the previous day significantly impact the current variance. The coefficient
suggests a strong persistence in the volatility over time, highlighting the influence of past
volatility on the current day's volatility. The GARCH coefficient is 0.834591. This indicates a
substantial impact of the lagged conditional variance on the current variance. The persistence
in the GARCH term suggests that past volatility plays a crucial role in determining current
volatility, reinforcing the presence of long-term volatility patterns.
Figure 10: GARCH (3,1) Conditional Volatility (Cardano)
Conditional volatility seems to capture the volatility clustering effect commonly observed in
financial time series data, where periods of high volatility are followed by periods of high
volatility and vice versa.
Comparative Analysis
Bitcoin, characterized by a higher GARCH coefficient of 0.865768, suggests a more persistent
volatility over time. The GARCH parameter represents the impact of past squared residuals on
current volatility. With a higher GARCH coefficient, Bitcoin's volatility is likely to exhibit a
clustered pattern, where periods of high volatility tend to be followed by similarly high
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119
URL: http://doi.org/10.14738/abr.1112.16086
volatility periods. This indicates that Bitcoin may experience sustained and prolonged periods
of elevated volatility. Conversely, Cardano displays a higher ARCH coefficient of 0.112732 in its
GARCH (3,1) model. The ARCH coefficient reflects the responsiveness of volatility to recent
market shocks. A higher ARCH coefficient suggests that Cardano is more reactive to recent
shocks, and these shocks have a more pronounced effect on its volatility. Additionally,
Cardano's higher constant coefficient of 0.000175 suggests a higher inherent volatility in the
series, not influenced solely by past shocks or volatilities. In comparing the two
cryptocurrencies, Bitcoin's higher GARCH coefficient implies a more persistent and clustered
volatility pattern, although the difference is relatively subtle, potentially leading to more
prolonged periods of elevated volatility. On the other hand, Cardano, with a higher ARCH and
constant coefficients, seems to be more reactive to recent market shocks, and it exhibits a
higher inherent volatility that is not solely influenced by past shocks. In conclusion, both
cryptocurrencies exhibit significant volatility, but their patterns differ. Bitcoin's volatility tends
to persist over time, while Cardano is more responsive to recent market shocks. These
differences in volatility patterns should be taken into account when assessing the risk and
potential returns associated with each cryptocurrency. Thus, our analysis indicates that
Cardano, a cryptocurrency operating on a proof-of-stake mechanism, tends to have lower long- term volatility compared to Bitcoin. This finding can be important in hypothesizing that the
transition from proof-of-work to proof-of-stake mechanisms can potentially normalize the
volatility in the cryptocurrency market. Proof-of-stake mechanisms, like the one Cardano
employs, are considered more energy-efficient and secure, reducing the chances of price
manipulation and sudden market fluctuations. Furthermore, the staking element of proof-of- stake cryptocurrencies encourages users to hold onto their assets, potentially reducing sell-off
events and stabilizing prices.
The introduction of government-backed digital tokens, or Central Bank Digital Currencies
(CBDCs), can potentially further contribute to reducing market volatility. Governments and
central banks are exploring the possibility of introducing CBDCs as a digital form of their fiat
currencies. Leveraging the proof-of-stake mechanism in CBDCs can further enhance market
stability. By doing so, central banks can potentially ensure lower transaction costs, faster
transactions, and enhanced security features, thus creating a more stable digital financial
ecosystem. Moreover, the backing of central banks provides a level of trust and stability that is
generally not associated with cryptocurrencies, which can further mitigate market volatility.
CONCLUSION
The rapid rise of cryptocurrencies has reshaped the financial landscape, bringing with it both
opportunities and challenges. Understanding the volatility patterns of cryptocurrencies is vital
in navigating this evolving market. This paper analyzed the volatility structures of two major
cryptocurrencies, Bitcoin and Cardano, using GARCH modeling. The analysis reveals that
Bitcoin, which operates on a proof-of-work mechanism, tends to exhibit prolonged periods of
high volatility but the dfference is subtle. On the other hand, Cardano, built on a proof-of-stake
mechanism, shows signs of lesser long-term volatility, potentially making it a more attractive
option for investors seeking stability alongside returns. This study suggests that a shift towards
proof-of-stake cryptocurrencies could be a step towards reducing the overall volatility of the
cryptocurrency market. Furthermore, the potential introduction of Central Bank Digital
Currencies (CBDCs), particularly those grounded in proof-of-stake mechanisms, could further
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stabilize the market by introducing assets backed by central bank guarantees, thereby instilling
a greater degree of trust and stability in the digital currency sphere. Looking forward, this
research recommends an extension of this study to include a broader range of cryptocurrencies
representing both proof-of-work and proof-of-stake mechanisms. This approach would provide
a more comprehensive view of the market dynamics and potentially lead to more informed
policy and investment strategies.
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