Bachelor’s Thesis, 15 ECTS Economics C100:2
Spring 2021 Supervisor: Niklas Hanes
FEDERAL FUNDS RATE ON BITCOIN VOLATILITY
Using the symmetric GARCH and asymmetric EGARCH models
Elias Atmander
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i
Abstract
This thesis examines the volatility of Bitcoin during four years from 2014-04-01 until 2020-04-01. The main objective of the thesis was to answer the research question: “Is the return volatility of Bitcoin affected by interest rate change announcements by the FOMC?” and given Bitcoin’s decentralized characteristics, the hypothesis to this was that Bitcoin should not be affected by such changes. The GARCH (1,1) and EGARCH (1,1) models were used to analyze the transformed logarithmic returns of Bitcoin. The number of observations sum to 1462 observations (days). Additionally, 13 observations of change announcements in the federal funds rate were used with a dummy variable approach to analyze for effects on Bitcoin volatility. The main findings of this thesis indicate that Bitcoin is not affected by announcements of a change in the federal funds rate, and thus, the hypothesis that Bitcoin is immune to changes in the federal funds rate is supported.
Keywords: ARCH, Bitcoin, Cryptocurrency, Dummy Variables, EGARCH (1,1), Federal Funds Rate, FOMC, GARCH (1,1), Time series, Volatility
ii Acknowledgements
I would like to thank my supervisor, Niklas Hanes, Senior Lecturer at the Department of Economics at Umeå University for his valuable suggestions and guidance throughout this thesis. I’d also like to extend my gratitude to the rest of the inspiring staff at the Department of Economics at Umeå University, for motivating me to pursue a degree in economics.
iii Table of Contents
1. INTRODUCTION ... 1
2. THEORETICAL FRAMEWORK ... 4
2.1CRYPTOCURRENCIES ... 4
2.1.1 Bitcoin ... 5
2.2BLOCKCHAIN ... 6
2.3FOMC AND MONETARY POLICY ... 7
2.4FIAT CURRENCY ... 8
2.5VOLATILITY ... 9
2.6REVIEW OF PREVIOUS STUDIES ... 10
3. METHOD ... 12
3.1GARCH(1,1)MODEL ... 12
3.1.1 Application ... 14
3.2EGARCH(1,1)MODEL ... 15
3.2.1 Application ... 16
3.3TESTS ... 17
3.3.1 Phillips-Perron ... 17
3.3.2 ARCH-LM ... 18
3.3.3 Ljung-Box Portmanteau White Noise ... 19
4. DATA ... 21
4.1COLLECTION OF DATA ... 21
4.1.1 Choice of time period ... 21
4.2TRANSFORMATION & VISUALIZATION OF DATA ... 22
4.2.1 Logarithmic returns ... 22
4.2.2 Volatility ... 24
4.3DESCRIPTIVE STATISTICS ... 25
5. RESULTS ... 28
5.1STATISTICAL TESTS ... 28
5.1.1 Phillips-Perron ... 28
5.1.2 ARCH-LM ... 28
5.2GARCH(1,1) ... 29
5.2.1 Regression ... 29
5.3EGARCH(1,1) ... 30
5.3.1 Regression ... 30
5.4PORTMANTEAU WHITE NOISE TEST ... 31
6. DISCUSSION ... 33
7. CONCLUSION ... 36
REFERENCES ... 38
APPENDIX ... 41
APPENDIX 1 ... 41
APPENDIX 2 ... 41
APPENDIX 3 ... 42
iv List of Figures
FIGURE 1:DEVELOPMENT OF BTC/USD FROM 2016-04-01 TO 2020-04-01. ... 6
FIGURE 2:HOW BLOCKCHAIN WORKS.SOURCE:CROSBY ET AL.(2016) ... 7
FIGURE 3:BTC/USD GRAPH WITH CHANGE ANNOUNCEMENTS. ... 22
FIGURE 4:LOGARITHMIC RETURNS. ... 23
FIGURE 5:LOGARITHMIC RETURNS AND CHANGE ANNOUNCEMENTS. ... 23
FIGURE 6:VOLATILITY (CONDITIONAL VARIANCE) OF BITCOIN. ... 24
FIGURE 7:VOLATILITY (CONDITIONAL VARIANCE) OF BITCOIN WITH ANNOUNCEMENTS. ... 25
FIGURE 8:HISTOGRAM OF LOGARITHMIC RETURNS OF BITCOIN. ... 26
List of Tables TABLE 1:LOGARITHMIC RETURN OF BITCOIN. ... 25
TABLE 2:95% CONFIDENCE INTERVAL OF THE MEAN. ... 26
TABLE 3:SHAPIRO-WILK TEST FOR NORMALITY ... 27
TABLE 4:OVERVIEW OF THE FEDERAL FUNDS RATE DURING THE PERIOD.SOURCE:FEDERAL RESERVE ... 27
TABLE 5:RESULTS PHILLIPS-PERRON ... 28
TABLE 6:RESULTS ARCH-LM. ... 29
TABLE 7:RESULTS GARCH(1,1). ... 29
TABLE 8:RESULTS EGARCH(1,1). ... 31
TABLE 9:RESULTS PORTMANTEAU WHITE NOISE ... 32
v List of Abbreviations
CPI Consumer Price Index
FED Federal Reserve
FOMC Federal Open Market Committee
PPI Producer Price Index
UIP Uncovered Interest Parity
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1. Introduction
During the last couple of years, cryptocurrencies have increased in influence in today's societies and have risen dramatically in popularity. From 2015 to 2020, Bitcoin’s total market capitalization has risen from approximately 4 billion USD to 356 billion USD in 2020 and continued to grow to over 1 trillion USD in March 2021 (Coinmarketcap, 2021). To put this into perspective, the market capitalization of Facebook Inc., as the 6th largest company globally, is, by Jan-01-2021, 778 billion USD. This means that the total market capitalization of Bitcoin is larger than that of Facebook (Value.today, 2021).
One of the drivers behind the rise of Bitcoin is that it is increasingly accepted as a means of payment and large corporations have started to invest in Bitcoin. This
increases Bitcoin’s influence as a means of payment and as a currency. On February 8, 2021, the electric automobile corporation Tesla invested 1.5 billion USD in Bitcoin (Kovach, 2021) and on March 24, 2021, Tesla’s CEO Elon Musk announced that it accepts Bitcoin as a payment method (Shead, 2021).
Bitcoin is a digital currency that is used as a form of money. However, Bitcoin is not backed up by any government or authority but by mathematics, trust, and adaptation (Bitcoin, 2021). The USD, conversely, has a direct relationship to the federal funds rate through the UIP condition. When the FOMC operates a monetary contraction, the USD exchange rate appreciates initially, but over time the exchange rate depreciates (Kim &
Roubini, 2000, p. 562). With this in mind, and with Bitcoin’s decentralized nature, one can expect that Bitcoin is not affected by changes in interest rates (Glaser et al., 2014, p.
5).
How Bitcoin is defined has been a topic on many researcher’s agendas. For example, Haubrich & Orr (2014) compared Bitcoin to the USD and obtained results that there are many differences between the two. For example, Bitcoin is mined compared to the USD where the Fed determines the amount of high-powered money in circulation. There are also differences in the total value and price stability of the two. The supply of Bitcoin is also limited to 21 million (Hayes, 2021a) compared to the USD, where there are no limitations on the supply other than that decided by the FOMC.
Whether Bitcoin defines as a currency or speculative investment has also been asked amongst researchers. Glaser et al. (2014, p. 13) argued that most new Bitcoin users treat it as a speculative investment rather than a currency. Additionally, Dyhrberg (2016) examined Bitcoin by comparing it to the USD and gold and concluded that Bitcoin has many similarities to the two. Dyhrberg also noted that Bitcoin reacts significantly to the federal funds rate and concluded that Bitcoin should be defined somewhere between a currency and a commodity (Dyhrberg, 2016, p. 10). However, Baur et al. (2018) replicated and extended this study and obtained contradictory results to Dyhrberg (2016). Baur et al. (2018) obtained significant results that Bitcoin is very different from gold and fiat currencies with unique risk characteristics, different volatility processes and that Bitcoin is uncorrelated to other assets (Baur et al., 2018, p. 109).
Corbet et al. (2014) studied the influence of four central bank’s monetary policy announcements on Bitcoin’s return volatility and found that all quantitative easing and interest rate adjustments from the central banks significantly affected Bitcoin’s return volatility. Corbet et al. (2014) argued in line with Dyhrberg (2016) that Bitcoin shares the characteristics of gold and fiat currencies and is somewhere between the two as an asset.
Pyo & Lee (2020) examined if FOMC macroeconomic announcements affected the price of Bitcoin. Pyo & Lee analyzed FOMC announcements and the specific announcements of the employment rate, PPI, and CPI. Pyo & Lee (2020) obtained significant results that the price of Bitcoin decreased approximately 1% on the
announcement day, compared to an increase of approximately 0.26% on a day without an announcement. However, they did not obtain significant results on the announcement of the three variables of employment rate, PPI, and CPI.
The diffusion amongst researchers on how to classify Bitcoin and the contradictory results of prior studies makes cryptocurrencies a fascinating and relevant topic to study.
The use and significance of cryptocurrencies, especially Bitcoin, have also risen during the last couple of years, making it even more relevant than ever to study. This thesis aims to examine if there is an impact of a change in the federal funds rate on the return volatility of Bitcoin. As Bitcoin classifies itself as a decentralized currency, independent from governments and policy interventions, interest rate change announcements from
central banks should not affect Bitcoin. This study investigates interest rate changes by the FOMC on the return volatility of Bitcoin during the period of 2016-04-01 to 2020- 04-01. During this period, the FOMC has changed the federal funds rate 13 times (Federal Reserve, 2020).
Bitcoin closing prices are transformed to daily returns data to examine if there is an effect by generating dummy variables on the days of a change announcement (as well as one day before and after) by the FOMC on the federal funds rate. The study uses the GARCH (1,1) and EGARCH (1,1) methodology, which is widely used in volatility analysis studies related to financial time series data.
The results of the GARCH model in this study indicate that there is a significant effect on the volatility of Bitcoin by changes in the federal funds rate. However, as this study demonstrates, the GARCH model results are unreliable due to an unsatisfiable model restriction. On the contrary, the results of the EGARCH model indicate that changes in the federal funds rate do not influence the volatility of Bitcoin. This supports the hypothesis of Bitcoin as a decentralized currency independent of governments.
The thesis is structured as follows; section 2 presents the theoretical framework and explains fundamental theories and previous studies. In section 3 the method is assessed where the regression models are explained further. Section 4 presents the data and explains it in more detail. The results are presented in section 5, followed by a discussion in section 6. In the last section, section 7, the conclusions are presented.
2. Theoretical Framework 2.1 Cryptocurrencies
In February 2021, data from Statista estimates that there exist more than 4500 cryptocurrencies (De Best, 2021). However, far from all of these are significant. The two largest, Bitcoin and Ethereum, constitute by March 2021, for over 70% of the total market capitalization and the top 10 cryptocurrencies for over 80% (Coinmarketcap, 2021).
Milutinovic (2018) studied the characteristics of cryptocurrencies. Her study concludes that a cryptocurrency is a form of digital or virtual currency protected by encryption technology. The underlying encryption technology makes the transaction secure and protects the information about the transaction and all the exchanges made on the digital market (Milutinovic, 2018, p. 120). The primary purpose of a cryptocurrency is to be decentralized and free from government intervention and independent of any central authority or institution. Most of the cryptocurrencies are decentralized and based on blockchain technology and managed by computer networks.
There are different characteristics of different types of cryptocurrencies, some are mineable, such as Ethereum and Bitcoin, and some have a supply cap, such as Bitcoin.
The mining process is where the underlying computer network rewards verifiers with coins when a transaction block is verified. This process expands the number of coins.
During this mining process, the marginal cost of mining equals the marginal benefit because of the high electricity costs required for the mining process (Harwick, 2016, p.
571). Furthermore, Milutinovic (2018, p. 120) acknowledged the discussion on how cryptocurrencies will affect the global economy, and she concluded that it is impossible to know how they will develop. Many believe that cryptocurrencies could replace fiat currencies, while others believe that governments could adapt to the technology and replace their fiat currencies with a digital currency. The Swedish Riksbank already has an ongoing pilot project with a digital currency, the e-krona (Sveriges Riksbank, 2021).
2.1.1 Bitcoin
Bitcoin is the largest and the first cryptocurrency. In April 2021, the total market capitalization of Bitcoin was more than 1 trillion USD, which represents approximately 59% of the market capitalization of all cryptocurrencies (Coinmarketcap, 2021). In 2008, Satoshi Nakamoto introduced it as the first decentralized peer-to-peer electronic cash system (Nakamoto, 2008). Satoshi Nakamoto is a pseudonym, and the real identity behind the pseudonym is still unknown, despite much speculation (De Filippi &
Loveluck, 2016, p. 8).
Bitcoin is a decentralized currency that is independent of any government, agency, or institution. Bitcoin is essentially electronic money that is used as a means of payment or a store of value. Nowadays there are many restrictions on using Bitcoin as a means of payment. However, it is becoming more accepted. On February 8, 2021, the electronic automobile corporation Tesla invested 1.5 billion USD in Bitcoin (Kovach, 2021), and on March 24, 2021, Elon Musk, CEO of Tesla, announced that it is accepting Bitcoin as a means of payment (Shead, 2021). Other corporations that accept Bitcoin as a means of payment include Microsoft, Starbucks, Paypal, and Coca-Cola (Haqqi, 2021).
Like any other (crypto)currency, Bitcoins can be exchanged for another currency such as USD or EUR. Supply and demand are the factors that determine the exchange rate for Bitcoin (Segendorf, 2014, p. 73). Compared to fiat currencies, Bitcoin has a supply cap of 21 million Bitcoins, estimated to be reached around 2140 (Bariviera et al., 2017, p.
3). What is essential to understand concerning this study is that Bitcoin does not have a centralized system, and no one can control it entirely. Compared to the traditional banking system, the central banks control the money supply (Milutinovic, 2018, p. 106).
Thus, it would be logical to view Bitcoin as immune to government policies such as monetary policy compared to fiat currencies.
Figure 1: Development of BTC/USD from 2016-04-01 to 2020-04-01.
2.2 Blockchain
Blockchain technology has a crucial role in cryptocurrencies. Crosby et al. (2016) describe a blockchain as a database of records. The database is a public digital ledger, and when a transaction occurs, participants in the system verifies the transaction. When it is verified, the information remains in the digital ledger forever. The blockchain contains verifiable information on every transaction made in the blockchain.
Yaga et al. (2019) describe the two general categories of a blockchain: (1)
permissionless and (2) permissioned. A permissionless blockchain is a blockchain where everyone can read and write to the blockchain without authorization. The two largest cryptocurrencies: Bitcoin and Ethereum are categorized as permissionless blockchains. Contrariwise, a permissioned blockchain is a blockchain that is limited to specific people or organizations and generally has stricter access requirements. An example of a cryptocurrency that is permissioned is Ripple.
Although the primary use of blockchains is related to cryptocurrencies, there are many fields in which blockchains can be used. For example, assets such as property can be registered in a blockchain, and in that way, it can be verified by ownership by insurance companies. Blockchains could also be a place where people could store legal
documents. In this case, there is no need for third parties (Crosby et al., 2016, p. 15).
These are just examples of industries where blockchain technology can be adapted, and Crosby et al. (2016) expects that there will be a significant adaptation of blockchain within a decade or two.
Figure 2: How Blockchain Works. Source: Crosby et al. (2016)
Figure 2 above illustrates how a blockchain works. (1) Someone wants to transfer money. (2) The transaction is represented online as a block. (3) The block is signaling to the rest of the network. (4) Participants in the network approve if the transaction is valid. (5) The block adds to the chain if verified, and the chain provides a record of the transaction. (6) The money transfers to the other party.
2.3 FOMC and Monetary policy
The FOMC is the federal reserve system branch consisting of twelve members. The members of the FOMC are seven members of the board governors, the president of the federal reserve bank of New York, and the four remaining members are on a rotating basis (Federal Reserve, 2021).
The FOMC aims to determine the direction of the monetary policy in the United States.
Monetary policy aims to influence the availability and cost of money and credit to support and aid the Fed to achieve national economic goals (Federal Reserve, 2021).
The Fed controls three tools of monetary policy: (1) open market operations, which is
the purchase or sale of securities, (2) the discount rate, which is the interest rate charged to commercial banks and other institutions when making loans, and (3), reserve
requirements, which is the amount of money commercial banks must have available on hand every day. The Fed will affect the supply and demand of the Fed’s bank custodial accounts through these three tools, thereby changing the federal funds rate.
The federal funds rate is the short-term interest rate at which banks can borrow from each other. When the federal funds rate is low, the FOMC has adapted an expansionary monetary policy. Expansionary policy often follows with relatively high inflation. The opposite is the case for a high federal funds rate. The federal funds rate directly relates to the US economy because it is the basis for the interest rates provided by financial institutions to businesses and consumers (Segal, 2020). The federal funds rate influences the USD, which section 2.4 explains in more detail below.
2.4 Fiat Currency
Fiat currencies are government-backed currencies that are not backed by a commodity, such as gold. With fiat currencies, the central banks have more control over the currency because they decide how much money to print. A problem with fiat currencies and their centralized nature are that they may result in high inflation when central banks print too much of them. Today, most currencies are fiat currencies, including the USD, EUR, and JPY (Chen, 2021).
There is usually a relationship between fiat currencies and corresponding interest rates in macroeconomic theories and valuation models (Glaser et al., 2014, p. 5). For
example, the USD directly relates to the federal funds rate through the UIP condition.
The UIP condition explains the relationship between foreign and domestic interest rates and currency exchanges. The UIP states that the price of goods should be equal
everywhere globally once interest rates and exchange rates are factored in (Hayes, 2021b).
To put this in context, for example, when the FOMC operates a monetary contraction (increase in the federal fund’s interest rate), the exchange rate appreciates initially, but over time the exchange rate depreciates (Kim & Roubini, 2000, p. 562). Additionally,
according to the study of Dominguez (1998), central bank intervention tends to increase the exchange rate volatility.
The European Central Bank (ECB), identifies money with the three following functions:
(1) as a medium of exchange, which means that money is used as an intermediary in trade to avoid the inconvenience of a barter system. (2) as a unit of account; money is presented as a numerical unit that corresponds to a value in trade. (3) A store of value;
money is something that can be saved and retrieved in the future (European Central Bank, 2012, p. 23). Bitcoin checks all these three functions to some extent. Thus, according to this definition, Bitcoin is categorized as money.
2.5 Volatility
Volatility usually refers to the degree of uncertainty or risk associated with the
fluctuations in a value of an economic variable (Aizenman & Pinto, 2005, p. 3). Higher volatility indicates large swings in a short period of time, either positive or negative, in the variable’s value. Low volatility indicates the opposite; small fluctuations in the value of a variable in a short period of time. One way to measure fluctuations in an asset's price is to quantify the past daily returns (percentage changes per day). This is known as historical volatility, which represents the degree of volatility of the returns of an asset. Variance is the distribution of returns around the average value of the entire asset, and volatility measures this limited to a specific period (Kuepper, 2021).
Return volatility is calculated using the standard deviation of an economic variable’s returns using continuous compounding (Hull, 2012, p. 201). The unit of time can either be yearly, monthly, or daily values of the standard deviation of the returns of the
variable. When using a sample, the standard deviation calculated is used to estimate the variability for the entire population (Altman & Bland, 2005, p. 903). The continuously compounded return, or the return volatility, can be calculated using the following formula:
𝑟! = 𝑙𝑛(""!
!"#) (1)
The numerator is the variable’s value at the end of the day (t), and the denominator denotes the variable’s value the day before (t-1). As mentioned in section 2.4, when
central banks operate interventions, the exchange rate volatility tends to increase (Dominguez, 1998, p. 186). Nevertheless, given Bitcoins decentralized characteristics, we expect the volatility of Bitcoin not to be influenced by central bank interventions.
2.6 Review of Previous Studies
Many previous studies on cryptocurrencies and particularly Bitcoin have examined how Bitcoin should be defined as an asset. Haubrich & Orr (2014) compared Bitcoin to the USD and found many differences between the two. They noted that Bitcoin is more volatile than the USD, indicating that Bitcoin may experience higher inflation and deflation rates than the USD. Conversely, the FOMC uses monetary measures to prevent high levels of inflation of the USD. Another difference is that there is a supply cap of Bitcoin, just like with a commodity such as gold. The supply of Bitcoin is limited to 21 million Bitcoins (Hayes, 2021a). The FOMC solely determines the supply and high-powered money in circulation, and there is no supply cap on the USD.
Glaser et al. (2014) argued that most of the new Bitcoin users treat it as a speculative investment rather than a currency and that few people who own Bitcoins intend to rely on it as a means of payment for goods and services in everyday life. These conclusions support the results of Haubrich & Orr (2014) that Bitcoin is different compared to fiat currencies. Furthermore, Dyhrberg (2016) compared Bitcoin to both the USD and gold and obtained contradictory results to Haubrich & Orr (2014). Dyhrberg (2016) found many similarities between Bitcoin, the USD and gold. Dyhrberg (2016) concluded that Bitcoin is acting like a currency and that Bitcoin also reacts significantly to the federal funds rate. Dyhrberg further concluded that Bitcoin is classified as an asset between a fiat currency and a commodity such as gold (Dyhrberg, 2016, p. 10). However, a replication and extension to this study were conducted by Baur et al. (2018), and they obtained contradictory results to that of Dyhrberg (2016) and results that support the conclusion made by Haubrich & Orr (2014) and Glaser et al. (2014). Baur et al. (2018) concluded that Bitcoin is very different from gold and fiat currencies and that Bitcoin has unique risk characteristics, different volatility processes and that Bitcoin is uncorrelated to other assets (Baur et al., 2018, p. 109).
The impact of monetary policy announcements on assets price and volatility has also
been a subject of research. Glick & Leduc (2012) studied QE announcements by the Fed and Bank of England and concluded that the long-term interest rate decreased on days when QE programs were announced, and the USD and Pound depreciated following this. Chulia et al. (2010) studied the effect of surprise changes in the federal funds rate on stock market volatility. Chulia et al. (2010) concluded that stock market volatility was substantially affected by these surprise changes in the federal funds rate. Corbet et al. (2014) conducted a similar study as above, but on Bitcoin return volatility. Corbet et al. (2014) studied the impact of four central banks: ECB, FOMC, Bank of Japan, and Bank of England. Corbet et al. (2014) concluded in line with Dyhrberg (2016) that all QE adjustment and the interest rate changes had a significant effect on the return volatility of Bitcoin and thus that Bitcoin shares similar characteristics to gold and fiat currencies and that Bitcoin is an asset somewhere in between fiat currencies and gold.
Further studies are done on policy announcements on the price and volatility of Bitcoin.
Pyo & Lee (2020) studied FOMC macroeconomic announcements and whether specific announcements of the employment rate, PPI, and CPI influenced the price of Bitcoin.
Pyo & Lee (2020) obtained significant results in line with that of Dyhrberg (2016) and Corbet et al. (2014) that the price of Bitcoin on an FOMC announcement day decreased by approximately 1% compared to an increase of approximately 0.26% on a day when there were no announcements. However, Pyo & Lee (2020) concluded, in contrast to general FOMC announcements, that Bitcoins price is insignificant to announcements of the three macroeconomic variables of employment rate, PPI, and CPI. These findings are different from that of the US stock market, where the announcement of employment rate, PPI, and CPI positively affected the stock market.
This study is related to the studies above, especially that of Dyhrberg (2016), Baur et al.
(2018), Corbet et al. (2014), and Pyo & Lee (2020). This study focuses specifically on announcements from FOMC of changes in the federal funds rate on Bitcoins return volatility. The hypothesis to be answered is that in line with Bitcoins decentralized characteristics contrary to fiat currencies, that the volatility of Bitcoin should be immune to federal funds rate change announcements. This study will answer the
research question with up-to-date data on Bitcoin and changes in the federal funds rate.
3. Method
In this section, the choice of method and approach to the method are described. The choice of the methodological approach is adapted from the methodology of previous studies of Dyhrberg (2016), Baur et al. (2018), Corbet et al. (2014), and Pyo & Lee (2020). These studies are based on an event study approach using the ARCH family models. This approach is described in more detail concerning this study below.
Furthermore, the statistical tests that are used in this study are explained in detail.
3.1 GARCH (1,1) Model
The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model is widely used in many event-driven financial studies that estimate volatility effects. The GARCH model is an extension of the ARCH model, which the Nobel laureate Robert F.
Engle developed in 1982 (Stock & Watson, 2015, p. 712). The GARCH model was developed independently by Bollerslev and Taylor in 1986 (Brooks, 2019, p. 512), and compared to the ARCH model, the GARCH model allows for a more flexible lag structure (Bollerslev, 1986, p. 308).
The GARCH model allows the conditional variance to depend on the previous lags (Brooks, 2019, p. 512). The fundamentals behind the model are that high volatility tends to be followed by high volatility, and low volatility tends to be followed by low
volatility. This is known as volatility clustering or heteroskedasticity and is common in financial time series data (Stock & Watson, 2015, pp. 710-711). These irregularities can be captured using Autoregressive Conditional Heteroskedasticity (ARCH) family models, which is why it is preferred to Ordinary Least square (OLS), which has assumptions of homoscedasticity and constant volatility.
Compared to the GARCH model, the ARCH model often requires many parameters to explain the volatility of an economic variable, making it a complex process (Tsay, 2010, p. 131). The GARCH model utilizes declining weights, but the weights never go
entirely to zero compared to the ARCH model. This, together with the characteristic that the GARCH model depends on past squared residuals, makes it a simple yet successful model in predicting conditional variance, even in its simplest form (Engle, 2001, p.
159).
The GARCH (1,1) model can be generalized to a GARCH (p, q) model, where the p and q allow for additional lag terms, which is often required when data over long periods are analyzed, such as over several decades (Engle, 2001, p. 166). Thus, the GARCH (1,1) model indicates that the variance is calculated from the most recent lag of the squared residuals and the most recent variance (Hull, 2012, p. 218). The coefficients in the GARCH model are estimated with the maximum likelihood (ML) method. Using the ML estimation method, the aim is to find the probability distribution that makes the observed data most likely to occur (Myung, 2003, p. 93).
The GARCH model consists of two equations, the conditional mean equation (equation 2) and the conditional variance equation (equation 5). The mean equation specifies the behavior of the daily returns and its error term: 𝜀! represents the conditional mean equations error process related to the conditional variance equation and is known as the market shock (Alexander, 2008, p. 136). The GARCH (1,1) model is the simplest but also the most robust of the ARCH family models (Engle, 2001, p. 166), and by far the most popular (Hull, 2012, p. 218).
The interpretation of the GARCH and ARCH parameters relation to shocks is explained by Alexander (2008, p. 137) as following:
• The ARCH error parameter, 𝛼#, measures the reaction of the volatility to market shocks. When 𝛼# is large (above 0.1), the volatility is sensitive to market shocks.
• The GARCH lag parameter, 𝜔, measures the persistence in the volatility independently to the market. A large 𝜔 (above 0.9) indicates that the volatility takes a long time to decline following a market shock.
• The sum of 𝛼#+ 𝜔 determines the rate of convergence of the volatility to the long-term average.
• The GARCH constant: 𝛼$ together with 𝛼#+ 𝜔 determines the long-term average volatility. A large value of (#'%%$
#( * ) indicates that the long-term volatility is relatively high.
3.1.1 Application
Now the GARCH (1,1) methodology will be applied to the study in question. The conditional mean model is presented below as follows:
𝑙𝑛 ,"!'#"! - = 𝛽$+ 𝜀! (2)
𝜀! = 𝜐!𝜎! (3)
The dependent variable in equation (2) is the logarithm of the price of Bitcoin at time t, divided by the price the previous day, also known as the return volatility. 𝛽$ is the intercept constant, which is assumed to be zero, which follows the theory that daily stock returns are unpredictable and follow a random walk (Stock & Watson, 2015, p.
713). The error term 𝜀! is part of the model related to the conditional variance equation and important for estimating volatility. This part is explained in more detail below. The error term is not expected to follow the normal distribution; it is simply a disturbance term from the regression model. To test for normality, it is plausible to construct the statistic:
𝜐!= ,-!
! (4)
The statistic is interpreted as the model’s disturbance at each point in time t, divided by the conditional standard deviation at point t. Thus, 𝜐! is assumed to be normally
distributed, and 𝜀!, on the other hand, is not assumed to be normally distributed. 𝜐! is known as a standardized residual and is tested for normality. However, even if the assumptions of normality are not fulfilled, it is not a complication if the mean and variance equation is correctly specified (Brooks, 2019, p. 520). Financial time-series data of returns are known to follow a leptokurtic distribution. To solve this issue, Bollerslev suggested in 1987 that the assumptions of a normal distribution regarding financial time series data to be exchanged for an assumption of a student t-distribution (Bollerslev, 2008, p. 15). In large samples, t statistics have a standard normal
distribution, and the estimators of the ARCH and GARCH coefficients follow the normal distribution in large samples (Stock & Watson, 2015, p. 713).
Equation (5) below is the variance of the errors that follow the GARCH (1,1) methodology.
𝜎!" = 𝛼#+ 𝛼$(𝜀!%$" ) + 𝜔𝜎!%$" + ∑𝑖=1𝑖=−1𝛽𝑖𝛿𝑖,𝑡𝐹𝐸𝐷 (5)
Where:
𝜎!" = Conditional variance. 𝛼# = Constant.
(𝜀!%$" ) = Error term, day t-1. 𝛼$ = ARCH parameter.
𝜔 = GARCH parameter.
𝛿7,!89: = Dummy variable that takes on the value 1 on the day there is a change announcement of the federal funds rate and 0 otherwise.
In equation (5), the error term from equations (2 & 3) is applied. The conditional variance 𝜎!; is in equation (5) dependent on its variance in the previous period, 𝜎!'#; , and the previous periods squared error; (𝜀!'#; ). The intercept is 𝛼$, 𝛼# is the ARCH, and 𝜔 is the GARCH. A good model follows that 𝛼#+ 𝜔 < 1 and 𝛼$ > 0 (Bollerslev, 1986, p. 310). 𝛿7,!89: takes on the value 1 when there is a change announcement in the federal funds rate, including the day before the change and the day after the change, to capture potential information asymmetry effects and delays in reactions to the change.
3.2 EGARCH (1,1) Model
The additional model used in this study is the EGARCH (1,1) model, also known as the
“Exponential” GARCH model. The EGARCH model is an extension of the GARCH model proposed by Nelson (1991) and is used to overcome the GARCH model’s weakness and allow for asymmetric effects between positive and negative returns of an asset (Tsay, 2010, p. 143).
The EGARCH model solves three drawbacks of the GARCH model. First, it allows for the volatility to respond asymmetrically to positive and negative asset returns. Secondly, it deals with the limitation of the GARCH model regarding nonnegative constraints on the coefficients. Thirdly, it deals with the drawback of the GARCH model’s estimation of the persistence of shocks to the conditional variance (Nelson, 1991, p. 349).
The EGARCH model solves these three issues by formulation the conditional variance equation in terms of the logarithm of the variance rather than the variance itself. The logarithm may still be negative, but the variance will always be positive (Alexander, 2008, p. 151). Additionally, the EGARCH model uses an asymmetric term to allow the model to respond asymmetrically to positive and negative return values (Tsay, 2010, p.
143).
The sign of the asymmetric term in the EGARCH model: 𝛾 (gamma) determines the asymmetric volatility size and if there is leverage. If leverage exists, there is a negative correlation between past returns and subsequent volatility (McAleer & Hafner, 2014, pp. 94-96). The term has the following characteristics:
• If 𝛾 = 0, there is symmetry.
• If 𝛾 < 0, negative shocks will increase the volatility more than positive.
• If 𝛾 > 0, positive shocks will increase the volatility more than negative shocks.
• If 𝛾 < 0 and 𝛾 < 𝛼# < −𝛾 leverage exists.
Equivalent to the GARCH model, the EGARCH model also consists of two equations:
the conditional mean equation and the conditional variance equation (Alexander, 2008, p. 152). The conditional mean equation can be obtained by the same procedure as used in the GARCH model (Tsay, 2010, p. 143). Additionally, similarly to GARCH, the parameters of the EGARCH model are estimated using the Maximum Likelihood (ML) method. The simplest of the EGARCH models is the EGARCH (1,1) model, but the model can be extended and generalized to the EGARCH (p, q) model and allow for additional lag terms.
3.2.1 Application
A difference from the GARCH (1,1) model is that the EGARCH, as assumed by Nelson (1991) is that the errors follow a Generalized Error Distribution (GED) structure
(Brooks, 2019, p. 522). The conditional mean equation of the EGARCH (1,1) model will be set up the same way as in the GARCH (1,1) case as following:
𝑙𝑛 ,"!'#"! - = 𝛽$+ 𝜀! (6)
𝜀! = 𝜐!𝜎! (7)
The conditional variance equation of the EGARCH (1,1) is vastly different from the GARCH (1,1) and is listed as equation (8) below.
ln(𝜎!;) = 𝛼$+ 𝛼#,-,!"#
!"#- + 𝜔ln(𝜎!'#; ) + 𝛾(:,-!"#
!"#: − ;;<) + ∑7=#7='#𝛽7𝛿7,!89: (8) Where:
ln(𝜎!;) = Logarithmic conditional variance. 𝛼$ = Constant.
(𝜀!'#; ) = Error term, day t-1. 𝛼# = ARCH parameter.
𝜔 = GARCH parameter. 𝛾 = Asymmetric term.
𝛿7,!89: = Dummy variable that takes on the value 1 on the day there is a change announcement of the federal funds rate and 0 otherwise.
The conditional variance equation of the EGARCH in equation (8) applies the error term from the conditional mean equation (equation 6). The logarithmic conditional variance 𝜎!; is in equation (8) dependent on its logarithmic variance in the previous period, 𝜎!'#; . The intercept is 𝛼$, 𝛼# is the ARCH, and 𝜔 is the GARCH. The 𝛾 is the asymmetric term that allows for asymmetries in the time series and is expected to be negative for a leverage effect (Tsay, 2010, p. 144). 𝛿7,!89: is the dummy variable, which has the same characteristics as the dummy variable used in the GARCH model in section 3.1.1.
3.3 Tests
3.3.1 Phillips-Perron
To test for stationarity of the time series a unit root test is used. A stationary time series is one with a constant mean, variance, and autocovariance for each given lag (Brooks, 2019, p. 437). The use of non-stationary data can provide biased estimates of the coefficients in the regression model. The Phillips-Perron (Phillips & Perron, 1988) is more robust than other unit-root tests because the test allows for autocorrelated residuals (Brooks, 2019, p. 451). The Phillips-Perron test is based on the following model:
𝑦! = 𝑐$ + 𝛿!+ 𝛼𝑦!'#+ 𝜖! (9)
The following hypotheses are then tested:
𝐻$: The time series is non-stationary. (10) 𝐻>: The time series is stationary. (11)
𝑦! represents the time series, 𝑐$ and 𝛿! represent the drift and trend coefficients
respectively. 𝐻$ restricts 𝛼 = 1 and the drift and trend coefficients 𝑐$ and 𝛿! = 0. If the test leads to a rejection of 𝐻$, the alternative hypothesis that the time series is stationary is accepted. The p-value to reject 𝐻$ is if P < 0.05 (Brooks, 2019, p. 454).
3.3.2 ARCH-LM
The ARCH-LM test verifies that there is heteroskedasticity in the time series and whether it is necessary to go further and use the GARCH and EGARCH models
(Bollerslev, 1986). To test for ARCH effects, the first step is following the methodology in Brooks (2019, p. 510) as follows:
Step 1: Run the linear regression that is going to be tested with the help of OLS. In this case, the linear regression is the same as the mean model in equation (2). The regression is set up:
𝑙𝑛 ,"!'#"! - = 𝛽$+ 𝜀! (12)
After the regression has run, step 2 predicts the residuals ê from the regression and squares the predicted residuals, and regress them on q (1) own lags to test for ARCH of order q (1):
ê! = 𝛾$+ 𝛾#ê!'#+ 𝜐! (13)
Where:
𝜐! = error term.
From this regression, we obtain the 𝑅;. The test statistic is defined as 𝑇𝑅; and follows a
𝜒;(𝑞) distribution. The test statistic represents the number of observations multiplied by the correlation coefficient from the regression of equation (13).
Step 3 is to test the null and alternative hypothesis respectively:
𝐻$: 𝛾# = 0 (14) 𝐻>: 𝛾# ≠ 0 (15)
𝐻$ is equivalent to that there is no ARCH-effect, and 𝐻> is equivalent to an ARCH effect. The test is also a test for autocorrelation in the residuals and tests the estimated model’s residuals (Brooks, 2019, p. 511).
3.3.3 Ljung-Box Portmanteau White Noise
The Portmanteau White Noise test is a test for nonlinear patterns and model
misspecifications. The Portmanteau White Noise test that the model has power over a broad range of different model structures (Brooks, 2019, p. 835). The test was originally developed by Box & Pierce (1970) but was further developed and refined by Ljung &
Box (1978). A white noise process should have the following characteristics (Brooks, 2019, p. 333):
𝐸(𝑦!) = 𝜇 (16)
𝑣𝑎𝑟(𝑦!) = 𝜎; (17)
𝑦!'? = K $ @!AB?C7DB-% 7E ! = ? L (18)
That is, a white noise process has a constant mean; equation (16), a constant variance;
equation (17), and zero autocovariance except at lag zero; equation (18). Another way to define equation (18) is to say that each observation should be uncorrelated to all other values in the sequence (Brooks, 2019, p. 333).
The test statistic that is tested for the white noise process is known as the Ljung-Box statistic:
Q = 𝑛(𝑛 + 2) ∑IH=#G'HF&% ∼ 𝜒I; (19)
Where:
𝑚 = Number of autocorrelations. Q = Ljung-Box test statistic.
𝜏H; = Estimate of the autocorrelation of lag k. 𝑛 = Sample size.
𝜒I; = 𝜒 ; distribution with m degrees of freedom.
The test is done on the estimated standardized residuals from equation (4). The hypotheses that are being tested are stated as follows:
𝐻$: 𝜐! follows a white noise process. (20) 𝐻>: 𝜐! does not follow a white noise process. (21)
A rejection of 𝐻$ means an acceptance of 𝐻>, indicating that the data is not independently distributed, and that the data is non-stationary. However, if it is not possible to reject the null hypothesis, the data is randomly distributed, and the model follows a white noise process. Thus, the conditions in equations (16, 17, and 18) are satisfied. Furthermore, a failure to reject the null hypothesis indicates that there is no serial correlation, which is also desirable.
4. Data
This section describes the data, how the data was collected, and why the specific time period used in this study was selected. This section also describes how the data was converted from closing price data to return data and visualizes the data. The section furthermore provides descriptive statistics of the data to give an idea of the properties of the data.
4.1 Collection of data
The data used in this study are two types of data: (1) daily closing prices of Bitcoin and (2) change announcement of the federal funds rate by the FOMC. What is unique to currencies such as Bitcoin is that it is tradable seven days a week, denoting that the data used include weekends. Bitcoin data is collected from Yahoo Finance (2020-04-09) in USD closing price from 2016-04-01 to 2020-04-01, which sum to 1462 observations (days).
The data regarding the change announcements in the federal funds rate by the FOMC is collected from the Federal Reserve’s website (Federal Reserve, 2020) on the 2021-03- 30. The number of change announcements of the federal funds rate in the time period of 2016-04-01 to 2020-04-01 sum to 13. Since this study focuses on the volatility, a rise and a decrease in the federal funds rate are not treated differently.
4.1.1 Choice of time period
The period used in this study is from 2016-04-01 to 2020-04-01. There are two main reasons why this period is used. The first reason is that there are frequent changes in the federal funds rate during this period (13 changes), and the second reason is that previous studies analyzed the periods prior to this, which brings originality to the study in
question.
Figure 3: BTC/USD graph with change announcements.
The graph in figure 3 above represents Bitcoin’s price development during the chosen period of 2016-04-01 to 2020-04-01, together with the 13 federal funds rate change announcements represented as vertical lines.
4.2 Transformation & visualization of data 4.2.1 Logarithmic returns
The Bitcoin closing price data is converted into daily return data. There are many good reasons why it is preferred to use return data to closing prices. In comparison to closing price data, return data makes it possible to observe volatility and as mentioned in section 2.5, volatility is the standard deviation of the return (Hull, 2012, p. 213).
Furthermore, modeling the volatility of a time series can improve the efficiency and the accuracy of the parameter estimations and interval forecasts (Tsay, 2010, p. 110).
In this study, the continuously compounded return is calculated in accordance to equation (1) as follows:
𝑟! = 𝑙𝑛(𝑃!) − 𝑙𝑛(𝑃!'#) = 𝑙𝑛(""!
!"#) (22)
Where:
𝑟! = Logarithmic (continuously compounded) return.
𝑙𝑛(𝑃!) = Natural logarithm of Bitcoin closing price at time t.
𝑙𝑛(𝑃!'#) = Natural logarithm of Bitcoin closing price at time t-1.
Figure 4: Logarithmic returns.
In Figure 4 above, the daily logarithmic return of Bitcoin is illustrated in the period of 2016-04-01 to 2020-04-01. By looking at it, it is possible to see signs of volatility clustering.
Figure 5: Logarithmic returns and change announcements.
In Figure 5, the daily logarithmic returns of Bitcoin from 2016-04-01 to 2020-04-01 is illustrated together with vertical lines representing the days of the change
announcements in the federal funds rate by the FOMC.
4.2.2 Volatility
Furthermore, the volatility (conditional variance) is estimated and plotted from the GARCH (1,1) and EGARCH (1,1) models.
Figure 6: Volatility (Conditional Variance) of Bitcoin.
In figure 6, it is easier to visualize the volatility clustering. It is possible to see that periods of high volatility are followed by high volatility, and periods of low volatility are followed by periods of low volatility. The figure also captures the asymmetry effect with the EGARCH model, which can be spotted by comparing the red and blue lines.
Furthermore, the volatility spikes seem to be slightly more significant in the GARCH model compared to the EGARCH model.
0.01.02.03.04
1/1/2016 1/1/2017 1/1/2018 1/1/2019 1/1/2020
Date
Conditional variance, EGARCH Conditional variance, GARCH
Figure 7: Volatility (Conditional Variance) of Bitcoin with Announcements.
Figure 7 above visualizes the volatility together with the federal funds rate change announcements by the FOMC. In this figure it is also easier to spot the volatility clustering. By looking at figure 7, it is suggested to analyze more in-depth regarding if there is an effect of a change announcement on the return volatility of Bitcoin with statistical inferences.
4.3 Descriptive Statistics
Table 1: Logarithmic Return of Bitcoin.
Variable Obs Mean Variance Std. Deviation Skewness Kurtosis
LnBitcoin 1462 0.00189 0.00174 0.04173 -0.09435 16.7106
Table 1 shows the descriptive statistics of the logarithmic return of Bitcoin. The daily mean return is estimated to be 0.189%, with a standard deviation of 4.173%. The value of the mean suggests that the assumption of a zero-mean is justified. The high value of kurtosis suggests that Bitcoin returns follow a leptokurtic distribution rather than the normal distribution.
0.01.02.03.04
1/1/2016 1/1/2017 1/1/2018 1/1/2019 1/1/2020
Date
Conditional variance, EGARCH Conditional variance, GARCH
To investigate the assumptions of a zero-mean, a 95% confidence interval was set up with the assumption that the daily return is equal to zero.
Table 2: 95% confidence interval of the mean.
Variable Obs Mean Std. Err. Std. Deviation [95% Conf. Interval]
LnBitcoin 1462 0.00189 0.00109 0.04173 -0.00025 0.00403
Table 2 provides a 95% confidence interval of the mean of the logarithmic returns of Bitcoin; the data suggests that the assumption of a mean approximately equal to zero is a reasonable assumption to make.
Figure 8: Histogram of logarithmic returns of Bitcoin.
The figure above confirms the assumptions that the logarithmic returns of Bitcoin do not follow a normal distribution but rather a leptokurtic distribution with fat tails and a sharp peak around the mean. Thus, alternative solutions of the assumptions of which distribution the GARCH (1,1) and EGARCH (1,1) models and the estimated residuals will follow is made. The assumption that the GARCH (1,1) follows a t-distribution instead of the normal distribution and that the EGARCH (1,1) the GED will make more sense.
Furthermore, the deviation from the normal distribution is confirmed by doing the Shapiro-Wilk test for normality. The results from the test are provided in table 3 below.
