# Price Stability of Cryptocurrencies as a Medium of Exchange

Tatsuru Kikuchi  Faculty of Economics, The University of Tokyo Toranosuke
Onishi  Tokio Marine & Nichido Fire Insurance Co., Ltd.  Kenichi Ueda  Faculty
of Economics, The University of Tokyo

((November, 2021))

###### Abstract

We present positive evidence of price stability of cryptocurrencies as a
medium of exchange. For the sample years from 2016 to 2020, the prices of
major cryptocurrencies are found to be stable, relative to major financial
assets. Specifically, after filtering out the less-than-one-month cycles, we
investigate the daily returns in US dollars of the major cryptocurrencies
(i.e., Bitcoin, Ethereum, and Ripple) as well as their comparators (i.e.,
major legal tenders, the Euro and Japanese yen, and the major stock indexes,
S&P 500 and MSCI World Index). We examine the stability of the filtered daily
returns using three different measures. First, the Pearson correlations
increased in later years in our sample. Second, based on the dynamic time-
warping method that allows lags and leads in relations, the similarities in
the daily returns of cryptocurrencies with their comparators have been present
even since 2016. Third, we check whether the cumulative sum of errors to
predict cryptocurrency prices, assuming stable relations with comparators’
daily returns, does not exceeds the bounds implied by the Black-Scholes model.
This test, in other words, does not reject the efficient market hypothesis.

##  1 Introduction

A key question on cryptocurrencies is whether they can be used as money, a
medium of exchange. Many argue they cannot. It is said to be primarily because
they are not backed by any valuable goods and because their price movements
are too volatile to use as money.

This paper presents positive evidence of price stability of cryptocurrencies
as a medium of exchange. For this purpose, we examine price stability after
filtering out the high frequency components. For example, the Euro or Japanese
yen are known to be quite volatile against the US dollar but still they are
used in the daily lives by people in the Euro area countries or in Japan
because the high-frequency exchange rate volatility does not matter much for
daily real-goods transactions, as long as their average values are stable
against the US dollar.

Specifically, we apply the frequency-filtering technique developed by Müller
and Watson  (2018), abbreviated as the MW filter hereafter, to the daily
returns of major cryptocurrencies, legal tenders, and stock indexes. We
essentially get rid of the less-than-one-month (i.e., 20 business days)
movements in their daily returns. Then, we compare the stability in the
filtered daily returns of cryptocurrencies against those of legal tenders and
stock indexes. For cryptocurrencies, we pick the three major ones, that is,
Bitcoin (BTC), Ethereum (ETH), and Ripple (XRP). For major legal tenders to
compare, we pick the Euro (EUR) and Japanese yen (JPY), and we pick the S&P
500 (S&P500) and MSCI World Index (MSCI) for the major stock indexes. All
values are defined in terms of US dollars. The daily data are taken from the
beginning of 2016 to the end of 2020.

To see the stability of these prices based on the MW-filtered daily returns,
we use three different measures. The first one is the Pearson correlation as
is also used by Müller and Watson  (2018) in a different context. We find that
the MW filter makes cryptocurrency returns closer to the other financial asset
returns. Moreover, the correlations in daily returns between cryptocurrencies
and other assets increased in later years in our sample.

The second one is a measure based on the dynamic time warping (DTW) algorithm.
In general, DTW is a method that calculates an optimal match between two time
series data. For the Pearson correlation, we compare, for example, BTC returns
with S&P 500 returns in each same day. However, in DTW, the data sequences are
warped non-linearly in the time dimension to see a similar pattern allowing
for lags and leads. DTW is a widely applied algorithm in non-economic fields,
such as speech pattern recognition, though some studies have already used DTW
in economics, for example, Augustyński and Laskoś-Grabowski  (2018), Franses
and Wiemann  (2020), and Sidi  (2020). Based on the DTW-based similarity
measure, we find that the similarity in daily returns of cryptocurrencies with
other assets has been present even since 2016, the beginning of our sample,
and not much has changed throughout our sample period. Note that the MW
filtering also does the job by removing erratic ups and downs in the
similarity over the years for some pairs of cryptocurrencies and other assets.

The third one is the cumulative sum (CUSUM) test, a test of the structural
stability under the efficient market hypothesis. It is simply a two-sample
Kolmogorov-Smirnov test in statistics. It was first developed based on the
recursive residuals by Brown, Durbin, and Evans  (1975), and the method was
refined using the OLS residuals by Ploberger and Krämer  (1992). Some
discussions on the boundaries were made by Zeileis  (2004)111R package was
also developed by Zeileis, Leisch, Hornik, and Kleiber  (2002).. The CUSUM
test is essentially based on the Black-Scholes model with the assumption of
stable relationships with other financial assets. In economics literature, for
example, the stability of money-demand models has been examined by the CUSUM
test (see Stock and Watson  (1996), Piehl, Cooper, Braga, and Kennedy  (2003)
and Elliott and Müller  (2006)).

We find that cryptocurrency daily returns, based on the whole sample period,
do pass the CUSUM test after the MW filtering, though they do not pass before
the MW filtering. As for the other assets (EUR, JPY, S&P500, and MSCI), all of
them essentially pass the test either before or after the MW filtering. In
other words, cryptocurrency price movements have been enjoying stable
relations with other major assets over the whole sample period, from 2016 to
2020. We also conduct the CUSUM tests for later years only, that is, samples
starting from 2017, 2018, or 2019. BTC starting 2018, and JPY starting 2018 do
not pass the test before the MW filtering. However, after the MW filtering,
only ETH starting 2018 fails the test.

There have been a few studies that examined the price stability of
cryptocurrencies, such as Vejačka  (2014). A typical finding is a high price
volatility of cryptocurrency compared with traditional financial assets. Some
similarity between Bitcoin and gold has been discussed in Dyhrberg  (2016). A
more comprehensive study has been done in Sovbetov  (2018), who analyzed the
underlying factors that influence prices of cryptocurrencies, stocks, and gold
in both the short- and long-term. Also, Kim, Kim, and Kim  (2020) show a
somewhat stable relationship across prices of Bitcoin, gold, and S&P 500 based
on a GARCH model. Moreover, Shams  (2019) shows that cryptocurrency prices co-
move a lot, perhaps based on SNS-based demand effects. Most of these existing
studies, however, methodologically requires some sort of stationarity or
parametric stability, for which Müller and Watson  (2018) renounce for key
economic and financial time series data and instead advocate to use
nonparametric or semi-parametric models, such as the Pearson correlation after
the MW filtering. Note that DTW is also nonparametric, and the CUSUM test is
the test for a stable relationship itself, not taking it as given.

A few studies on cryptocurrency price movements have shown that the high
frequency movements may be contaminated by inefficient market forces. This is
also a key reason to get rid of such high frequency components of price
movements. For example, Aoyagi and Hattori  (2018) find that a technical cost
issue (i.e., hash rate) matters for determining the daily raw returns of
Bitcoin. Moreover, Li, Shin, and Wang  (2020) show that pump-and-dump schemes
have been prevalent in cryptocurrency markets. Such a scheme is an illegal act
in the US stock market as it is backed by a small group of people who buy up a
target cryptocurrency up to a target price level and then sell it, all under
close coordination, to obtain huge profits.

A fact that a cryptocurrency is not backed by real assets cannot be used as a
reason to dismiss a cryptocurrency’s role as a medium of exchange. The theory-
of-money literature has shown the existence of a fiat money that is not backed
by any real assets. Indeed, money has a role when financial contracts are not
completely enforceable due to costs (especially for small transactions), and
such an environment is often considered as trading with strangers for whom
contract enforcement is difficult (Bewley  (1979), Townsend  (1980), and
Kiyotaki and Wright  (1989)). Moreover, intrinsically valueless money has a
value only when people think it is accepted by other people, who think it is
accepted by yet other people, who think…(repeated infinitely). This means that
money is in essence a bubble and has to be common knowledge (Chwe  (1999)).
Given these formal theoretical definitions in economics, cryptocurrencies (or
sea shells in old times) are legitimate candidates of money. What has remained
is an empirical question as to whether an acceptance by the people is stable
or not.

The rest of the paper is organized as follows. In Section 2, we give a
detailed description of the data for the cryptocurrencies and other financial
assets. In Section 3, we perform a standard spectrum analysis to see the
frequency distributions. Section 4 gives a brief explanation of the MW filter.
In Section 5, we show the Pearson correlation before and after the MW
filtering. Section 6 explains the DTW method and results. In Section 7, we
discuss the structural stability based on the CUSUM test. Section 8 concludes.

##  2 Data

We use daily time series data from January 1, 2016, to December 31, 2020, for
three major cryptocurrencies, Bitcoin (BTC), Ethereum (ETH), and Ripple (XRP).
Those data are obtained through the API provided by the Cryptocompare
(https://min-api.cryptocompare.com/). We evaluate the stability of
cryptocurrency values compared with the other financial data, such as S&P500,
gold, MSCI World Index (MSCI), Japanese Yen (JPY), and the Euro (EUR). While
the historical data of gold prices and the MSCI World Index are obtained from
Investing.com (https://investing.com/), the historical data of S&P500 is
obtained from Yahoo finance (https://finance.yahoo.com/), and the historical
data of JPY and EUR are obtained from the API provided by Cryptocompare
(https://min-api.cryptocompare.com/).

The prices of the cryptocurrencies and other financial assets are all defined
in US dollars. We let $P_{t}$ denote the price of a financial asset at time
$t$ and $P_{t-1}$ the price of the asset at time $t-1$, and $r_{t}$ the
(logarithmic) returns of the asset at $t$ by

$r_{t}=\log\frac{P_{t}}{P_{t-1}}\times 365\;.$ (2.1)

The descriptive statistics of the daily returns of cryptoprices and the key
financial assets for the whole sample period are described in Table 1222The
skewness measures symmetricity of the distribution. The kurtosis tells us
whether the data is heavy-tailed or light-tailed, relative to a normal
distribution. Both skewness and kurtosis of normal distribution are equal to
$0$.. Figure 1 shows the 50-day moving averages of the daily returns of the
cryptocurrencies and other financial assets.

##  3 Short-Run and Long-Run Movements

The short-run and long-run price stability appear a bit differently in the
return data. The daily movements of cryptocurrencies are in general very
large, relative to the traditional financial assets. This means that when we
make a Fourier transformation from the time domain to the frequency domain, it
is expected to see a large amount of high-frequency component in their
spectrum.

A power spectrum analysis is a useful tool to see the frequency distribution
of some financial assets data. Let us consider the daily return of asset data
$r_{t}$ of length $T=n\,\Delta t=1.258\times 10^{5}$, where $n=1258$ is the
number of daily data and $\Delta t=100$ is taken as the sampling interval, and
the frequency is defined by $f=1/T=1/(n\,\Delta t)=7.949\times 10^{-6}$. Then,
the discretized frequency and the discretized time are given by
$f_{k}=k/(n\,\Delta t)~{}(k=1,\cdots,n)$ and $t_{j}=j\,\Delta
t~{}(j=1,\cdots,n)$, respectively. With this setup, the discrete Fourier
transformation (DFT) of the return of asset data $r_{j}$ is defined by

$Q_{k}=\sum_{j=1}^{n}r_{j}\exp(-i\,2\pi j\,k/n)\;,$ (3.2)

where $i=\sqrt{-1}$ is the imaginary unit. The discretization of time with the
sampling period $\Delta t$ implies a limitation of frequency $f$ to the band
$f\in[-\frac{1}{2\,\Delta t},~{}\frac{1}{2\;\Delta t}]$ as frequencies outside
the range are folded inside by the finite sampling (see, for example, Hamming
(2012)). This boundary frequency is called the Nyquist frequency,
$f_{\text{NQ}}=1/(2\,\Delta t)$.

The power spectrum density $I_{\text{PSD}}(f_{k})$ for the return of asset
data $r_{t}$ is then calculated by

$I_{\text{PSD}}(f_{k})=\frac{2\,\Delta t^{2}}{n}Q_{k}\,Q_{k}^{*}\;,$ (3.3)

where $*$ stands for the complex conjugate. In practice, we use the Fast
Fourier Transformation (FFT) to calculate the Fourier transformation of the
return of asset data, and take the sampling period as $\Delta t=100$, hence,
the Nyquist frequency is $f_{\text{NQ}}=0.005$.

The power spectrum density for the returns of cryptocurrencies and key
financial assets are shown in Figures 2, 3, 4, 5, 6, 7, 8, and 9. These
figures typically show that returns of cryptocurrencies and key financial
assets vary intensively with frequency. In the frequency region higher than
$f_{\text{NQ}}=0.005$, the noises seem dominant over the structurally
important signals. On the other hand, the structural signals are quite clear
in the frequency region lower than $f_{\text{NQ}}=0.005$ (i.e., $5\times
10^{-3}$).

##  4 Frequency Filtering

Now we know that the short-run movements of cryptocurrencies are somewhat
strange, we focus on the rest by filtering out the short-run movements. A
proper way of using a high-frequency filtering method for noisy non-stationary
data has been proposed in Müller and Watson  (2018), who use an orthogonal
projection of time series data $y_{j}(j=1,\cdots,n)$ onto the space spanned by
the cosine function based on low-frequency periodic vectors. Their approach is
an extension of the discrete cosine transformation (DCT) for the time series
$y_{j}$,

$\displaystyle\hat{y}_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{n}}\sum_{j=1}^{n}y_{j}\;,$
$\displaystyle\hat{y}_{k}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{n}}\sum_{j=1}^{n}\cos\left[(j-1)\left(k-\frac{1}{n}\right)\frac{\pi}{n}\right]y_{j}\;,~{}\text{for}~{}k=2,\cdots,n\;.$
(4.4)

The long-run projection of Müller-Watson is made by using the following
transformation, truncating the approximation with $q<n$,

$\displaystyle\hat{y}_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{n}}\sum_{j=1}^{q}y_{j}\;,$
$\displaystyle\hat{y}_{k}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{n}}\sum_{j=1}^{q}\cos\left[(j-1)\left(k-\frac{1}{2}\right)\frac{\pi}{n}\right]y_{j}\;,~{}\text{for}~{}k=2,\cdots,q\;.$
(4.5)

Here, up to the frequency that is represented by parameter $q$ are extracted.
More specifically, $\hat{y}_{q}$ is the data that is filtered to extract only
frequencies lower than $\hat{f}=1/\hat{T}=q/(2n)$ or $\hat{T}=2n/q$ period. We
use the following condition to decide the value of parameter $q$.

$q=\left[\frac{n}{10}\right]+3\;,$ (4.6)

where $\left[\frac{n}{10}\right]$ is the integer part of $\frac{n}{10}$. For
example, in the case of BTC, we set these parameters as $n=1420~{}({\it
i.e.},q=145),~{}\hat{T}=19,~{}\hat{f}=0.0526$. This roughly means that we take
out short-run cycles occurring for less than $20$ business days, which is
essentially one month in the calendar333In the literature, a similar method of
high-frequency filtering has been proposed and named as the separating
information maximum likelihood (SIML) method in Kunitomo and Sato  (2013). We
use an R script provided by Sato, for whom we are grateful..

In other words, the MW filtering smooths the higher frequency movements in
more than one month (20 business days) as shown in the power spectrum density
in Figures 2, 3, 4, 5, 6, 7, 8, and 9. Note that the MW filtering also
somewhat smooths the low frequencies.

##  5 Similarity to the Major Financial Assets: The Müller-Watson (2018)
Approach

Recall that our objective in this study is to see the stability of prices or
returns of cryptocurrencies relative to key financial assets. The time series
similarity could be measured by the degree of comovements across two or more
variables. How to evaluate comovements is discussed extensively in Müller and
Watson  (2018). They renounce the cointegtation approach and instead propose
to use the Pearson correlation after filtering out high-frequency movements in
the data.

Recall that the Pearson correlation is defined for two time series of data
$x_{t}$ and $y_{t}$ as

$P(x_{t},y_{t})=\frac{\text{cov}\left(x_{t},y_{t}\right)}{\sigma_{x}~{}\sigma_{y}}=\frac{\mathbb{E}\left[\left(x_{t}-\mu_{x}\right)\left(y_{t}-\mu_{y}\right)\right]}{\sigma_{x}~{}\sigma_{y}}\;.$
(5.7)

In order to show the importance of the Müller-Watson filtering, we show the
correlation matrices for both before and after filtering. The correlations
before the Müller-Watson filtering are shown in Tables 2, 3, 4, 5, and 6 for
2016, 2017, 2018, 2019, and 2020, respectively. The correlations after the
Müller-Watson filtering are shown in Tables 7, 8, 9, 10, and 11 for 2016,
2017, 2018, 2019, and 2020, respectively.

The correlation matrices indicate the long-run stability of cryptocurrencies
relative to the major financial assets. More specifically, we find three
patterns. First, the correlations between the returns of cryptocurrencies and
key financial assets becomes more significant after the Müller-Watson
filtering for each year. Second, the returns of cryptocurrencies and key
financial assets become more and more correlated to each other in later years.
Third, the correlations among cryptocurrencies are getting stronger over time.

##  6 Dynamic Time Warping

Next, we investigate the similarities of cryptocurrency returns to key
financial asset returns by looking at the Dynamic Time Warping (DTW). This is
often used in pattern recognition, as developed by Itakura  (1975) and Sakoe
and Chiba  (1978) (for a review, see Müller  (2007)), which allows leads and
lags of data sequences over time when measuring the similarity.

A set of time series, which are the returns of assets in our case,
$x=\\{x_{t}\\}=(x_{1},\cdots,x_{n})$ and $y=\\{y_{t}\\}=(y_{1},\cdots,y_{m})$,
can be expressed in terms of the warping path, $\Lambda=(w_{1},\cdots,w_{K})$.
The warping path is a contiguous set of matrix elements that define a mapping
between $x_{t}$ and $y_{t}$. A typical element $w_{\ell}$ is represented by
$(i,j)$, that selects $x_{i}$ from $x$ and $y_{j}$ from $y$.

Formally, the DTW distance between two given sequences, $x$ and $y$, can be
calculated by

$D(x,y)=\min_{\Lambda^{*}}~{}\sum_{(i,j)\in\Lambda^{*}}d(x_{i},y_{j})\;,$
(6.8)

where $\Lambda^{*}$ is the warping path that minimizes the cumulative distance
of all mapped point-pairs on the path. Although the element distance function
$d(x_{i},y_{j})$ could take one of several forms, normally and here, it is
given by the Euclidean distance, that is,

$d(x_{i},y_{j})=\sqrt{\left(x_{i}-y_{j}\right)^{2}}\;.$ (6.9)

The best warping path $\Lambda^{*}$ is found using a dynamic programming
approach to align two sequences. Going through all possible paths is
combinatorially explosive, as pointed out by Berndt and Clifford  (1994),
hence, the possible warping paths need to be restricted. When we apply the DTW
method to two economic data series, we think it natural to constrain the
possible warping paths by following three conditions.

The first constraint is the boundary condition. This requires the warping path
to start and finish in diagonally opposite corner cells (i.e., the starting
date and the ending date of the sample) of the warping path matrix,

$w_{1}=(1,1)~{}\text{and}~{}w_{K}=(n,m)\;.$ (6.10)

The second constraint is the continuity condition. This constraint limits the
path transitions to adjacent points in time,

$w_{\ell}-w_{\ell-1}\in\\{(1,0)\;,(0,1)\;,(1,1)\\}\;.$ (6.11)

The third constraint is the monotonicity condition. This constraint preserves
the time-order of points.

$\text{If}~{}w_{\ell}=(i,j)~{}\text{and}~{}w_{\ell+1}=(i^{\prime},j^{\prime})\;,~{}\text{then}~{}i\leq
i^{\prime}~{}\text{and}~{}j\leq j^{\prime}\;.$ (6.12)

In practice, the DTW distance is calculate in a recursive way. First, we
construct two subsequences $\tilde{x}_{i}=(x_{1},\cdots,x_{i})$ for
$i=1,\cdots,n$ and $\tilde{y}_{j}=(y_{1},\cdots,y_{j})$ for $j=1,\cdots,m$
from an assumed optimal path sequence. Then, we define the cost function for
each subsequence as

$\gamma(i,j)=D(\tilde{x}_{i},\tilde{y}_{j})~{}\text{for}~{}i=1,\cdots,n~{}\text{and}~{}j=1,\cdots,m\;.$
(6.13)

The cost function $\gamma(i,j)$ specifies the total cost of an assumed optimal
warping path starting from $w_{1}=(1,1)$ and ending at $w_{\ell}=(i,j)$.

Second, we find the cost function $\gamma(i,j)$ in the recursive way. Namely,
it is computed iteratively using a nested loop according to the following
formula,

$\hat{\gamma}(i,j)=d(x_{i},y_{j})+\min\\{\gamma(i-1,j),\,\gamma(i,j-1),\,\gamma(i-1,j-1)\\}\;,$
(6.14)

and

$\hat{\gamma}(i,j)=\gamma(i,j)\;,$ (6.15)

where $i=1,\cdots,n$ and $j=1,\cdots,m$, with appropriate initial values444The
python code is available upon request.. Once the cost function is found, it is
the DTW distance, $D(x,y)=\gamma(n,m)$, and the optimal warping path
$\Lambda^{*}$ is identified. For example, the paired data (i.e., the DTW
optimal warping path $\Lambda^{*}$) between BTC and GOLD in 2020 is shown in
Figure 10.

Note that the Pearson correlation always compares the same period data
$(x_{t},y_{t})$. For the DTW distance, the periods to compare are related
$(x_{i},y_{j})$, though $i$ and $j$ are not necessary the same time $t$. For
example, if $x=\\{x_{t}\\}$ and $y=\\{y_{t}\\}$ follow the simple sine and
cosine functions, respectively, in the time dimension, then they are
orthogonal, and thus their Pearson correlation is zero. However, they look the
same when one of their phases is shifted by $\pi$, thus, their DTW distance is
$0$, which means their shapes are exactly the same.

It is useful to normalize the DTW distance relative to some base value. Let us
use the maximum of the DTW distance between a pair of key financial assets
(i.e., JPY, EUR, GOLD, S&P500, and MSCI), $D_{\text{max}}$, as the base value
for all other DTW distances. That is, we normalize the DTW distance as
follows,

$\bar{D}(x,y)=\frac{D(x,y)}{D_{\text{max}}}\;.$ (6.16)

We show the normalized DTW distance $\bar{D}(x,y)$ before the Müller-Watson
filtering in Tables 12, 13, 14, 15, and 16 for each sample year. The
corresponding results after the Müller-Watson filtering are shown in Tables
17, 18, 19, 20, and 21.

There are several salient features which appear in these Tables. First, the
strengthening trend in similarities among returns of cryptocurrencies and key
financial assets seem less clear than in the Pearson correlations (Tables 2
and 3). Before the MW filtering, they, especially BTC, seem to behave
similarly even in 2016, that is the initial period of our sample. These
similarities dwindled in 2017 until 2019. In 2020, it recovered especially for
BTC. However, after the MW filtering, similarities are less fluctuating and
relatively stable, especially for BTC, in almost all the sample years.

Second, frequency filtering does make some differences. For example, Table 13
shows quite a different movement only in XRP in 2017 compared with the other
assets. This result also seems to be supported in Figure 1. In particular, it
is shown in Figure 1 that the returns of XRP dramatically rises in 2017.
However, after the MW filtering, the returns of XRP behave in a similar way,
as shown in Table 18. The following year, 2018, also show almost the same case
for the returns of XRP before the MW filtering (see Table 14) and after the
frequency filtering (see Table 19). This is interesting as there was the well-
known 2018 cryptocurrency crash (also known as the great crypto crash). Note,
however, that even after the MW filtering, XRP seems to move somewhat
differently compared to other assets in 2020.

##  7 Stability from the Viewpoint of the Black-Scholes Model

Under a simple, efficient market hypothesis, can we assume the price stability
of cryptocurrencies? Since the traditional financial asset returns can be
considered as more or less following the Brownian motion, we test whether the
cryptocurrency returns also follows a simple form of the Black-Scholes formula
under stable relationships with key financial asset returns.

Following Elliott and Müller  (2006), we consider the tests of the null
hypothesis of a stable coefficient $\bar{\beta}$ in

$z_{t}={\bf X}^{T}_{t}\bar{\beta}+{\bf
Z}^{T}_{t}\bar{\gamma}+\bar{\epsilon}_{t}$ (7.17)

against the alternative of variable coefficient $\beta_{t}$ in

$z_{t}={\bf X}^{T}_{t}\beta_{t}+{\bf Z}^{T}_{t}\gamma+\epsilon_{t}\;.$ (7.18)

The null hypothesis of stable $\bar{\beta}$ is rejected, if the cumulative sum
(CUSUM) of residual with constant $\bar{\beta}$ exceed the limit that is
consistent with the Brownian bridge (see, Appendix A).

First, we consider the structural stability based on the OLS-CUSUM test by
using all the data. Here, $\bar{\beta}$ is obtained once by OLS. As is clearly
explained in Krämer, Ploberger and Schlüter  (1991), the sum of the OLS
residual always starts from $0$ and ends with $0$. In between, it has some
bounds.

Second, we also employ the recursive CUSUM (Rec-CUSUM) test, which can be
useful to monitor price stability every day. Here, the residuals are obtained
as one-step ahead forecast errors. In other words, $\bar{\beta}$ is obtained
using historical data at each date. The null hypothesis of the structural
stability depends on the choice of the sampling periods. We divide our data
into four sets of data samples as follows: (i) all the data; (ii) a sample
data from 2017 to 2020; (iii) a sample data from 2018 to 2020; and (iv) a
sample data from 2019 to 2020. We then proceed with the Rec-CUSUM test for
each sample set.

We consider two types of boundaries for the CUSUM tests555The crucial
difference between the Rec-CUSUM test and the OLS-CUSUM test is in the
limiting process. In the former case, the limiting process is the Wiener
process (Brownian motion), but in the latter case, the limiting process is the
Brownian bridge.. We use the following two types of boundaries for the OLS-
CUSUM test. One is a constant in time and another is proportional to the
standard deviation function of the corresponding theoretical process, as
proposed in Zeileis  (2004),

$\displaystyle b(t)$ $\displaystyle=$ $\displaystyle\nu$ (7.19) $\displaystyle
b(t)$ $\displaystyle=$ $\displaystyle 2\nu\,\sqrt{t(1-t)}$ (7.20)

for the OLS-CUSUM path. We take $\nu=1.358$ at the 95% confidence interval.

When we apply the Rec-CUSUM test, we consider the following two types of
boundaries as proposed in Brown, Durbin, and Evans  (1975) and Zeileis
(2004),

$\displaystyle b(t)$ $\displaystyle=$ $\displaystyle\lambda\,(2\,t-1)$ (7.21)
$\displaystyle b(t)$ $\displaystyle=$ $\displaystyle 2\nu\,\sqrt{t}$ (7.22)

with $\lambda$ takes the value $\lambda=0.948$ at the 95% confidence
interval666There are several constructions of the boundaries of the stochastic
process. In the monitoring context, a nearly linear boundary was considered in
Chu, Stinchcombe and White  (1996) and Leisch, Hornik, and Kuan  (2000), which
is written as follows:
$b(t)=\left[t(t-1)\left(a^{2}+\ln\left(\frac{t}{t-1}\right)\right)\right]^{1/2}\;,$
where $a$ only depends on the confidence level $\alpha$..

Note that there is no need to change conditions in the test of the structural
stability even if we apply frequency filtering. This is known as pattern
recovery in Mehrizi and Chenouri  (2020) and Mehrizi and Chenouri  (2021).
Similar studies can be found, for example, in Kim, Koh, Boyd, and Gorinevsky
(2009) and Phillips and Jin  (2021).

We perform the test of this structural stability, based on the following
regression for the daily returns of each cryptocurrency or key financial asset
on the returns of other financial assets. For example, for Bitcoin,

$\displaystyle\tt{BTC}_{t}$ $\displaystyle=$
$\displaystyle\bar{\beta}_{0}+\tt{BTC}_{t-1}~{}\bar{\beta}_{1}+\tt{JPY}_{t-1}~{}\bar{\beta}_{2}+\tt{EUR}_{t-1}~{}\bar{\beta}_{3}+\tt{GOLD}_{t-1}~{}\bar{\beta}_{4}+\tt{S\&P500}_{t-1}~{}\bar{\beta}_{5}$
(7.23) $\displaystyle+$
$\displaystyle\tt{MSCI}_{t-1}~{}\bar{\beta}_{7}+\bar{\epsilon}_{t}~{},$

where the symbolic notations stand for the daily returns, for example,
$\tt{BTC}_{t}$ for the BTC daily return at time $t$.

First, we show the results of the OLS-CUSUM test for both before and after the
MW filtering is applied. Figures 11, 12, 13, 14, 15, 16, 17 and 18 illustrate
the results, that is, whether the daily returns exceed the boundary consistent
with the Black-Scholes formula under the assumption of a stable relationship
with other financial assets before the Müller-Watson filtering.

Based on the OLS-CUSUM test, there are two findings from the structural
stability tests. We find that before the MW filtering, only ETH and XRP
apparently exceed the boundaries of the Brownian bridge. Moreover, BTC almost
reaches the boundaries of the Brownian bridge as well.

However, this is not the case after the MW filtering. Also, all the other
assets do not exceed the boundaries either, before or after the MW filtering
(see Figures 19, 20, 21, 22, 23, 24, 25 and 26). In other words,
cryptocurrencies are stable enough within the boundaries of the Brownian
bridge after the MW filtering is applied, as is the case for the key financial
assets.

Second, we conduct the Rec-CUSUM test. Figures 27, 31, 35, 39, 43, 47, 51 and
55 illustrate the results using all the data, that is, whether the daily
returns exceed the boundary consistent with the Black-Scholes formula before
the MW filtering.

Similar to the OLS-CUSUM test, we find that before the MW filtering, only ETH
(and perhaps GOLD) exceeds the boundaries of the Brownian motion in the Rec-
CUSUM test when we use all the data from 2016 to 2020.

However, once again, this is not the case after the MW filtering. All assets
(perhaps except for GOLD) do not exceed the boundaries of the Brownian motion
after the MW filtering (see Figures 59, 63, 67, 71, 75, 79, 83 and 87). Once
again, these results show that cryptocurrencies are stable enough within the
boundaries of the Brownian motion after the MW filtering is applied, as is the
case for the key financial assets.

In addition to that, we also conduct the Rec-CUSUM tests using the latter
parts of data before the MW filtering, that is, (i) a sample data from 2017 to
2020; (ii) a sample data from 2018 to 2020; and (iii) a sample data from 2019
to 2020. The results before the MW filtering can be seen in Figures 28, 29,
and 30 for BTC, Figures 32, 33, and 34 for ETH, Figures 36, 37, and 38 for
XRP, Figures 40, 41, and 42 for JPY, Figures 44, 45, and 46 for EUR, Figures
48, 49, and 50 for GOLD, Figures 52, 53, and 54 for S&P500, and Figures 56,
57, and 58 for MSCI.

For the sample data from 2018 to 2020, BTC and JPY before the MW filtering
exceed the boundaries of the Brownian motion. For other subsample years, all
the return movements are within the boundaries.

However, the results change in the Rec-CUSUM tests after the MW filtering (see
Figures 60, 61, and 62 for BTC, Figures 64, 65, and 66 for ETH, Figures 68,
69, and 70 for XRP, Figures 72, 73, and 74 for JPY, Figures 76, 77, and 78 for
EUR, Figures 80, 81, and 82 for GOLD, Figures 84, 85, and 86 for S&P500, and
Figures 88, 89, and 90 for MSCI). Even for the sample data from 2018 to 2020,
all the assets, except for ETH, after the MW filtering do not exceed the
boundaries of the Brownian motion.

In summary, we confirm that the returns on cryptocurrencies after the MW
filtering are essentially stable enough within the boundaries of the Brownian
bridge or the Brownian motion, as is the case for the key financial assets.

##  8 Conclusion

A key question on cryptocurrencies is whether they can be used as money, a
medium of exchange. Many argue they cannot. A major reason is that their price
movements are too volatile to use as money.

We have presented positive evidences of price stability of cryptocurrencies.
We focus on the daily returns after filtering out the high frequency
components, which are contaminated by technical forces. Also, for the daily
transaction uses, people do not seem to care about the high frequency
movements of money, say, the Euro or Japanese yen against the US dollar.

Specifically, we apply the filter developed by Müller and Watson  (2018) to
the daily return data of major cryptocurrencies (i.e., Bitcoin (BTC), Ethereum
(ETH), and Ripple (XRP)) as well as their comparators (i.e., major legal
tenders, the Euro (EUR) and Japanese yen (JPY), and major stock indexes, the
S&P 500 and the MSCI World Index (MSCI)). By doing so, we essentially get rid
of the less-than-one-month cycles of their price movements.

We then investigate the stability of the filtered daily returns using three
different measures. First, we find that the Pearson correlations in the daily
returns between cryptocurrencies and other assets increased in latter years in
our sample from the beginning of 2016 to the end of 2020. Second, however,
based on the DTW method that allow lags and leads, we find that the similarity
in the daily returns of cryptocurrencies with other assets has been present
even since 2016, the beginning of our sample, and not much changed throughout
our sample period. Third, we test the stability of the relationships between
daily returns of cryptocurrencies and those of their comparators by checking
if the cumulative sum of errors, under the assumption of stable coefficients
on comparators’ daily returns, does not exceeds the statistical bounds. This
CUSUM test is based on the efficient market hypothesis and the results assure
the market efficiency and structural price stability of cryptocurrencies, as
is also the case for other major financial assets.

In summary, interestingly, for the years from 2016 to 2020, the prices of
major cryptocurrencies are found to be stable. This conclusion is not well
shown without filtering out the high frequency movements or without conducting
deeper investigations than simple correlations. Still, apparently, an
empirical question remains if such stability can be continued for the
foreseeable future.

## Acknowledgement

The views expressed in this paper are those of the authors and should not be
attributed to the Tokio Marine & Nichido Fire Insurance Co., Ltd., or any
institutions that the authors have been affiliated with. This work is
supported by the Digital Economy Project at the University of Tokyo, funded by
the Silicon Valley Community Foundation. We would like to thank these
organizations. We are grateful for the helpful comments from participants of
The University Blockchain Research Initiative (UBRI) Conference, hosted
(online) by the University of London, the Digital Currency and Finance
Workshop, hosted (online) by the University of Tokyo, and the Blockchain in
Kyoto 2021 Conference, hosted (online) by the Kyoto University.

## Appendix

##  Appendix A CUSUM Test

Let $\\{z_{t}\\}$, for $t=1,\cdots,n$, be the observed time series and we
write it as.

$z_{t}=\mu_{t}+\epsilon_{t}\;,$ (A.24)

where $\mu_{t}={\bf X}^{T}_{t}{\bf\beta}_{t}$ is a trend component in the
linear regression model and $\epsilon_{t}$ is the i.i.d. disturbances, which
are assumed to be stationary and ergodic with the following condition:

$\mathbb{E}\left[\epsilon_{t}\right]=0~{}\text{and}~{}\mathbb{V}\left[\epsilon_{t}\right]=\sigma^{2}\;.$
(A.25)

In the standard linear regression model, the coefficients $\beta_{t}$ is
estimated as $\hat{\beta}_{t}$ by using the ordinary least squares (OLS)
method. Then, the OLS residual $\hat{\epsilon}_{t}$, can be also estimated
with

$\mathbb{E}[\hat{\epsilon}_{t}]=0~{}\text{and}~{}\mathbb{V}[\hat{\epsilon}_{t}]=\hat{\sigma}^{2}\;.$
(A.26)

The key is to evaluate the fluctuations of the cumulative sum (CUSUM) of
residuals. The null hypothesis is that $\hat{\sigma}^{2}$ is not explosive.
See discussion by Brown, Durbin, and Evans  (1975).

It is useful to define a continuous time stochastic process (sometimes known
as the empirical fluctuation process) of the sum of residuals as

$W_{n}(\tau)=\frac{1}{\hat{\sigma}\sqrt{n}}\sum_{i=1}^{n}{\bf 1}_{\\{\tau\leq
t\\}}\hat{\epsilon}_{i}\;.$ (A.27)

In the recursive CUSUM (Rec-CUSUM) test, the residual $\hat{\epsilon}_{t}$ is
essentially the one-step-ahead forecast error and given by

$\hat{\epsilon}_{t}=\frac{z_{t}-{\bf X}^{T}_{t}\hat{\beta}_{t-1}}{\sqrt{1+{\bf
X}^{T}_{t}\left(\sum_{i=1}^{t-1}{\bf X}_{i}{\bf X}^{T}_{i}\right)^{-1}{\bf
X}_{t}}}\;.$ (A.28)

Here, $\hat{\beta}_{t-1}$ is estimated by the OLS using the data up to the
previous period $t-1$. In this case, the limiting process becomes the Brownian
motion (i.e., Wiener process).

In fact, according to the functional central limit theorem or the Donsker’s
theorem, $W_{n}(\tau)$ converges to the Wiener process $W(\tau)$:

$W(\tau)=\lim_{n\to\infty}\|W_{n}(\tau)\|\;,~{}\text{for}~{}0\leq\tau\leq
1\;,$ (A.29)

where the convergence means the weak convergence of the associated probability
measures. The Wiener process (i.e., the Brownian motion) has the following
properties:

$\mathbb{E}\left[W(\tau)\right]=0~{}\text{and}~{}\mathbb{V}\left[W(\tau)\right]=\tau\;.$
(A.30)

Based on this property, the Rec-CUSUM test bounds are obtained.

In the case of the OLS-CUSUM tests, $\hat{\beta}$ based on all samples is used
instead of $\hat{\beta}_{t-1}$. As the OLS residuals are correlated to each
other and their sum is zero by definition, the limiting process of the OLS-
CUSUM is no longer a Brownian motion. Instead, the limiting process is known
to become a Brownian bridge:

$B(\tau)=W(\tau)-\tau\,W(1)\;,~{}\text{for}~{}0\leq\tau\leq 1\;.$ (A.31)

An alternative representation of the Brownian bridge is given by the
stochastic differential equation,

$dB(\tau)=dW(\tau)-\frac{B(\tau)}{1-\tau}d\tau\;,~{}\text{for}~{}0\leq\tau\leq
1\;,$ (A.32)

whose solution is given by

$B(\tau)=\int_{0}^{\tau}\frac{1-\tau}{1-t}\;dW(t)\;,~{}\text{for}~{}0\leq\tau\leq
1\;.$ (A.33)

The Brownian bridge has the following properties:

$\mathbb{E}\left[B(\tau)\right]=0~{}\text{and}~{}\mathbb{V}\left[B(\tau)\right]=\tau(1-\tau)\;.$
(A.34)

Here, the OLS-CUSUM test uses the following quantity:

$M_{n}(\tau)=\max_{0\leq s\leq\tau}\|W_{n}(s)\|\;.$ (A.35)

The distribution $M_{n}(\tau)$ converges to the Kolmogorov distribution (or
the Markov process) as follows:

$M(\tau)=\lim_{n\to\infty}\|M_{n}(\tau)\|\;,$ (A.36)

then, it satisfies

$M(\tau)=\sup_{0\leq\tau\leq 1}\|B(\tau)\|\;.$ (A.37)

In general, the cumulative distribution function of the Kolmogorov
distribution $M(\tau)$ is given by

$\mathbb{P}\left(M(\tau)\leq
x\right)=1-2\sum_{k=1}^{\infty}(-1)^{k-1}\exp(-2k^{2}x^{2})\;.$ (A.38)

The goodness-of-fit test or the Kolmogorov–Smirnov test can be constructed by
using the critical values of the Kolmogorov distribution,

$\mathbb{P}\left(\sup_{0\leq\tau\leq 1}\|B(\tau)\|\leq
b(\tau)\right)=1-\alpha\;,$ (A.39)

where $1-\alpha$ represents the confidence interval.

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Figure 1: 50 days moving average returns (annualized, %) of cryptocurrencies
and key financial assets

(a) 50 days moving average returns (annualized, %) of cryptocurrencies (BTC,
ETH, and XRP)

(b) 50 days moving average returns (annualized, %) of JPY, EUR, and GOLD

(c) 50 days moving average returns (annualized, %) of S&P500 and MSCI

Figure 2: The power spectrum density (PSD) of BTC

(a) The power spectrum density (PSD) of BTC (before the Müller-Watson
filtering)

(b) The power spectrum density (PSD) of BTC (after the Müller-Watson
filtering)

Figure 3: The power spectrum density (PSD) of ETH

(a) The power spectrum density (PSD) of ETH (before the Müller-Watson
filtering)

(b) The power spectrum density (PSD) of ETH (after the Müller-Watson
filtering)

Figure 4: The power spectrum density (PSD) of XRP

(a) The power spectrum density (PSD) of XRP (before the Müller-Watson
filtering)

(b) The power spectrum density (PSD) of XRP (after the Müller-Watson
filtering)

Figure 5: The power spectrum density (PSD) of JPY

(a) The power spectrum density (PSD) of JPY (before the Müller-Watson
filtering)

(b) The power spectrum density (PSD) of JPY (after the Müller-Watson
filtering)

Figure 6: The power spectrum density (PSD) of EUR

(a) The power spectrum density (PSD) of EUR (before the Müller-Watson
filtering)

(b) The power spectrum density (PSD) of EUR (after the Müller-Watson
filtering)

Figure 7: The power spectrum density (PSD) of GOLD

(a) The power spectrum density (PSD) of GOLD (before the Müller-Watson
filtering)

(b) The power spectrum density (PSD) of GOLD (after the Müller-Watson
filtering)

Figure 8: The power spectrum density (PSD) of S&P500

(a) The power spectrum density (PSD) of S&P500 (before the Müller-Watson
filtering)

(b) The power spectrum density (PSD) of S&P500 (after the Müller-Watson
filtering)

Figure 9: The power spectrum density (PSD) of MSCI

(a) The power spectrum density (PSD) of MSCI (before the Müller-Watson
filtering)

(b) The power spectrum density (PSD) of MSCI (after the Müller-Watson
filtering)

Figure 10: DTW alignment between BTC and GOLD in 2020 Figure 11: The OLS-CUSUM test of BTC (before the Müller-Watson filtering) Figure 12: The OLS-CUSUM test of ETH (before the Müller-Watson filtering) Figure 13: The OLS-CUSUM test of XRP (before the Müller-Watson filtering) Figure 14: The OLS-CUSUM test of JPY (before the Müller-Watson filtering) Figure 15: The OLS-CUSUM test of EUR (before the Müller-Watson filtering) Figure 16: The OLS-CUSUM test of GOLD (before the Müller-Watson filtering) Figure 17: The OLS-CUSUM test of S&P500 (before the Müller-Watson filtering) Figure 18: The OLS-CUSUM test of MSCI (before the Müller-Watson filtering) Figure 19: The OLS-CUSUM test of BTC (after the Müller-Watson filtering) Figure 20: The OLS-CUSUM test of ETH (after the Müller-Watson filtering) Figure 21: The OLS-CUSUM test of XRP (after the Müller-Watson filtering) Figure 22: The OLS-CUSUM test of JPY (after the Müller-Watson filtering) Figure 23: The OLS-CUSUM test of EUR (after the Müller-Watson filtering) Figure 24: The OLS-CUSUM test of GOLD (after the Müller-Watson filtering) Figure 25: The OLS-CUSUM test of S&P500 (after the Müller-Watson filtering) Figure 26: The OLS-CUSUM test of MSCI (after the Müller-Watson filtering) Figure 27: The Rec-CUSUM test of BTC [2016-2020] (before the Müller-Watson filtering) Figure 28: The Rec-CUSUM test of BTC [2017-2020] (before the Müller-Watson filtering) Figure 29: The Rec-CUSUM test of BTC [2018-2020] (before the Müller-Watson filtering) Figure 30: The Rec-CUSUM test of BTC [2019-2020] (before the Müller-Watson filtering) Figure 31: The Rec-CUSUM test of ETH [2016-2020] (before the Müller-Watson filtering) Figure 32: The Rec-CUSUM test of ETH [2017-2020] (before the Müller-Watson filtering) Figure 33: The Rec-CUSUM test of ETH [2018-2020] (before the Müller-Watson filtering) Figure 34: The Rec-CUSUM test of ETH [2019-2020] (before the Müller-Watson filtering) Figure 35: The Rec-CUSUM test of XRP [2016-2020] (before the Müller-Watson filtering) Figure 36: The Rec-CUSUM test of XRP [2017-2020] (before the Müller-Watson filtering) Figure 37: The Rec-CUSUM test of XRP [2018-2020] (before the Müller-Watson filtering) Figure 38: The Rec-CUSUM test of XRP [2019-2020] (before the Müller-Watson filtering) Figure 39: The Rec-CUSUM test of JPY [2016-2020] (before the Müller-Watson filtering) Figure 40: The Rec-CUSUM test of JPY [2017-2020] (before the Müller-Watson filtering) Figure 41: The Rec-CUSUM test of JPY [2018-2020] (before the Müller-Watson filtering) Figure 42: The Rec-CUSUM test of JPY [2019-2020] (before the Müller-Watson filtering) Figure 43: The Rec-CUSUM test of EUR [2016-2020] (before the Müller-Watson filtering) Figure 44: The Rec-CUSUM test of EUR [2017-2020] (before the Müller-Watson filtering) Figure 45: The Rec-CUSUM test of EUR [2018-2020] (before the Müller-Watson filtering) Figure 46: The Rec-CUSUM test of EUR [2019-2020] (before the Müller-Watson filtering) Figure 47: The Rec-CUSUM test of GOLD [2016-2020] (before the Müller-Watson filtering) Figure 48: The Rec-CUSUM test of GOLD [2017-2020] (before the Müller-Watson filtering) Figure 49: The Rec-CUSUM test of GOLD [2018-2020] (before the Müller-Watson filtering) Figure 50: The Rec-CUSUM test of GOLD [2019-2020] (before the Müller-Watson filtering) Figure 51: The Rec-CUSUM test of S&P500 [2016-2020] (before the Müller-Watson filtering) Figure 52: The Rec-CUSUM test of S&P500 [2017-2020] (before the Müller-Watson filtering) Figure 53: The Rec-CUSUM test of S&P500 [2018-2020] (before the Müller-Watson filtering) Figure 54: The Rec-CUSUM test of S&P500 [2019-2020] (before the Müller-Watson filtering) Figure 55: The Rec-CUSUM test of MSCI [2016-2020] (before the Müller-Watson filtering) Figure 56: The Rec-CUSUM test of MSCI [2017-2020] (before the Müller-Watson filtering) Figure 57: The Rec-CUSUM test of MSCI [2018-2020] (before the Müller-Watson filtering) Figure 58: The Rec-CUSUM test of MSCI [2019-2020] (before the Müller-Watson filtering) Figure 59: The Rec-CUSUM test of BTC [2016-2020] (after the Müller-Watson filtering) Figure 60: The Rec-CUSUM test of BTC [2017-2020] (after the Müller-Watson filtering) Figure 61: The Rec-CUSUM test of BTC [2018-2020] (after the Müller-Watson filtering) Figure 62: The Rec-CUSUM test of BTC [2019-2020] (after the Müller-Watson filtering) Figure 63: The Rec-CUSUM test of ETH [2016-2020] (after the Müller-Watson filtering) Figure 64: The Rec-CUSUM test of ETH [2017-2020] (after the Müller-Watson filtering) Figure 65: The Rec-CUSUM test of ETH [2018-2020] (after the Müller-Watson filtering) Figure 66: The Rec-CUSUM test of ETH [2019-2020] (after the Müller-Watson filtering) Figure 67: The Rec-CUSUM test of XRP [2016-2020] (after the Müller-Watson filtering) Figure 68: The Rec-CUSUM test of XRP [2017-2020] (after the Müller-Watson filtering) Figure 69: The Rec-CUSUM test of XRP [2018-2020] (after the Müller-Watson filtering) Figure 70: The Rec-CUSUM test of XRP [2019-2020] (after the Müller-Watson filtering) Figure 71: The Rec-CUSUM test of JPY [2016-2020] (after the Müller-Watson filtering) Figure 72: The Rec-CUSUM test of JPY [2017-2020] (after the Müller-Watson filtering) Figure 73: The Rec-CUSUM test of JPY [2018-2020] (after the Müller-Watson filtering) Figure 74: The Rec-CUSUM test of JPY [2019-2020] (after the Müller-Watson filtering) Figure 75: The Rec-CUSUM test of EUR [2016-2020] (after the Müller-Watson filtering) Figure 76: The Rec-CUSUM test of EUR [2017-2020] (after the Müller-Watson filtering) Figure 77: The Rec-CUSUM test of EUR [2018-2020] (after the Müller-Watson filtering) Figure 78: The Rec-CUSUM test of EUR [2019-2020] (after the Müller-Watson filtering) Figure 79: The Rec-CUSUM test of GOLD [2016-2020] (after the Müller-Watson filtering) Figure 80: The Rec-CUSUM test of GOLD [2017-2020] (after the Müller-Watson filtering) Figure 81: The Rec-CUSUM test of GOLD [2018-2020] (after the Müller-Watson filtering) Figure 82: The Rec-CUSUM test of GOLD [2019-2020] (after the Müller-Watson filtering) Figure 83: The Rec-CUSUM test of S&P500 [2016-2020] (after the Müller-Watson filtering) Figure 84: The Rec-CUSUM test of S&P500 [2017-2020] (after the Müller-Watson filtering) Figure 85: The Rec-CUSUM test of S&P500 [2018-2020] (after the Müller-Watson filtering) Figure 86: The Rec-CUSUM test of S&P500 [2019-2020] (after the Müller-Watson filtering) Figure 87: The Rec-CUSUM test of MSCI [2016-2020] (after the Müller-Watson filtering) Figure 88: The Rec-CUSUM test of MSCI [2017-2020] (after the Müller-Watson filtering) Figure 89: The Rec-CUSUM test of MSCI [2018-2020] (after the Müller-Watson filtering) Figure 90: The Rec-CUSUM test of MSCI [2019-2020] (after the Müller-Watson filtering) Table 1: Descriptive statistics of daily returns (annualized, %) for the whole sample period Statistics | BTC | ETH | XRP | JPY | EUR | GOLD | S&P500 | MSCI
---|---|---|---|---|---|---|---|---
${\bf N}$ | 1258 | 1258 | 1258 | 1249 | 1249 | 1249 | 1249 | 1258
${\bf Mean~{}(\%)}$ | 0.446 | 0.804 | 0.648 | 0.022 | 0.012 | 0.049 | 0.057 | 0.045
${\bf SD~{}(\%)}$ | 4.695 | 7.567 | 9.159 | 1.400 | 0.647 | 0.872 | 1.211 | 1.023
${\bf Min~{}(\%)}$ | -38.118 | -43.420 | -43.069 | -12.702 | -4.160 | -5.723 | -11.984 | -9.915
${\bf Max~{}(\%)}$ | 25.561 | 65.995 | 109.760 | 15.346 | 6.071 | 4.805 | 9.383 | 8.770
${\bf Skewness}$ | -0.059 | 1.281 | 3.750 | 0.823 | 0.278 | -0.200 | -0.731 | -1.234
${\bf Kurtosis}$ | 7.096 | 9.353 | 33.954 | 36.265 | 10.922 | 4.652 | 20.552 | 22.681
Table 2: Correlation matrix in 2016 (before the Müller-Watson filtering) Table
3: Correlation matrix in 2017 (before the Müller-Watson filtering) Table 4:
Correlation matrix in 2018 (before the Müller-Watson filtering) Table 5:
Correlation matrix in 2019 (before the Müller-Watson filtering) Table 6:
Correlation matrix in 2020 (before the Müller-Watson filtering) Table 7:
Correlation matrix in 2016 (after the Müller-Watson filtering) Table 8:
Correlation matrix in 2017 (after the Müller-Watson filtering) Table 9:
Correlation matrix in 2018 (after the Müller-Watson filtering) Table 10:
Correlation matrix in 2019 (after the Müller-Watson filtering) Table 11:
Correlation matrix in 2020 (after the Müller-Watson filtering) Table 12: DTW
similarity matrix in 2016 (before the Müller-Watson filtering) Table 13: DTW
similarity matrix in 2017 (before the Müller-Watson filtering) Table 14: DTW
similarity matrix in 2018 (before the Müller-Watson filtering) Table 15: DTW
similarity matrix in 2019 (before the Müller-Watson filtering) Table 16: DTW
similarity matrix in 2020 (before the Müller-Watson filtering) Table 17: DTW
similarity matrix in 2016 (after the Müller-Watson filtering) Table 18: DTW
similarity matrix in 2017 (after the Müller-Watson filtering) Table 19: DTW
similarity matrix in 2018 (after the Müller-Watson filtering) Table 20: DTW
similarity matrix in 2019 (after the Müller-Watson filtering) Table 21: DTW
similarity matrix in 2020 (after the Müller-Watson filtering)


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