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+Time-Varying Coefficient DAR Model and
+Stability Measures for Stablecoin Prices: An
+Application to Tether
+Antoine Djogbenou,∗ Emre Inan,† Joann Jasiak‡
+This Version: January 3, 2023
+Abstract
+This paper examines the dynamics of Tether, the stablecoin with the largest
+market capitalization. We show that the distributional and dynamic proper-
+ties of Tether/USD rates have been evolving from 2017 to 2021. We use local
+analysis methods to detect and describe the local patterns, such as short-lived
+trends, time-varying volatility and persistence. To accommodate these pat-
+terns, we consider a time varying parameter Double Autoregressive tvDAR(1)
+model under the assumption of local stationarity of Tether/USD rates. We es-
+timate the tvDAR model non-parametrically and test hypotheses on the func-
+tional parameters. In the application to Tether, the model provides a good fit
+and reliable out-of-sample forecasts at short horizons, while being robust to
+time-varying persistence and volatility. In addition, the model yields a simple
+plug-in measure of stability for Tether and other stablecoins for assessing and
+comparing their stability.
+Keywords:
+Stablecoins, Tether, Prices, DAR Model, Persistence, Time-
+Varying Parameters, Conditional Heteroskedasticity, Local Stationarity.
+JEL number: C58, C13.
+∗York University, Canada, e-mail: daa@yorku.ca
+†York University, Canada, e-mail: emreynan@yorku.ca
+‡York University, Canada, e-mail: jasiakj@yorku.ca.
+The authors thank C. Gourieroux and H. Kim and the participants of CMStatistics 2022 and Canadian
+Economic Association (CEA) 2022 meetings for helpful comments. This project was supported by the
+Digital Currency Research Clusters Initiative, the Natural Sciences and Engineering Research Council of
+Canada (NSERC), and the Social Sciences and Humanities Research Council of Canada (SSHRC).
+arXiv:2301.00509v1  [econ.EM]  2 Jan 2023
+
+THIS VERSION: January 3, 2023
+1
+1
+Introduction
+The total market capitalization of cryptocurrencies is currently over 1 trillion U.S. dollar,
+with the top three cryptocurrencies in terms of market capitalization being Bitcoin (BTC),
+Ethereum (ETH), and Tether (USDT). While Bitcoin and Euthereum are characterized
+by high price volatility, Tether is a stablecoin, i.e. a cryptocurrency designed to maintain
+a stable price compared to other cryptocurrencies such as Bitcoin and Ethereum. It is the
+first and by far the largest stablecoin in the market with the highest daily volume of over
+$100 billion. In order to achieve price stability, the value of Tether is pegged 1-to-1 with
+the U.S. dollar. There also exist other stablecoins with values to other currency or gold
+and managed by either a single authority (usually the service provider) or a network of
+participants (the whole protocol).
+Allen, Gu, and Jagtiani (2022) recently discussed how stable cryptocurrencies provide
+alternative financial instruments for market participants and how appropriately regulated
+crypto markets could allow increased public confidence and lead to growth and innova-
+tion.
+The November 2021 report by the US President’s Working Group on Financial
+Markets (PWG), the Federal Deposit Insurance Corporation (FDIC), and the Office of
+the Comptroller of the Currency (OCC) highlight various risks that need to be addressed.
+These include user protection and run risk, payment system risk, systemic risk and con-
+centration of economic power.
+They provided various recommendations, including the
+requirement for stablecoin issuers to be insured depository institutions. See President’s
+Working Group (2021) for more details. Furthermore, Li and Mayer (2021) show that
+collateralized stablecoins like Tether could create systemic risk if the issuer does not have
+enough reserve to maintain its stability. More recently, Chen, Qin, and Zhang (2022, page
+5) pointed out the important role of Tether in the trading volume of Bitcoin compared
+to US dollars since 2017 and noted the limited reserve of Tether according to anecdotal
+evidence.
+Despite the increased interest in stablecoins and the recommendations of more scrutiny
+by regulators, these crypto assets’ stability is still ineffective. For example, TerraUSD,
+an algorithmic stablecoin, collapsed in May 2022.
+This situation posits the need for
+predictability of stablecoin prices and easy tools for proactive assessment of stability.
+To address those issues, this paper made the following contributions. First, we analyze
+
+THIS VERSION: January 3, 2023
+2
+the local Tether price from historical data and pin down important features in its dynamic.
+These features include the local pattern of the mean and the conditional pattern of Tether
+price as well as the role of specific events in this dynamic. Second, we develop a time-
+varying model for Tether price that incorporates these specificities. Third, we propose,
+based on the model, measures that can be used to assess the stability of stablecoins and
+mitigate risks.
+More specifically, we examine the dynamics of Tether/USD rates and documents the
+time varying distributional properties of this series. We apply local analysis methods to
+reveal the time varying mean, volatility and persistence. In particular, we observe periods
+when Tether rates deviate from the peg, which are often combined with increased volatility.
+During those episodes, local persistence measures increase, suggesting unit root dynamics
+of Tether.
+Based on these findings, we consider an extension of the Double-Autoregressive (DAR)
+model, called the dynamic time-varying parameter Double-Autoregressive (tvDAR). The
+DAR model [Ling (2004)] accommodates the conditional heteroscedasticity and nests the
+ARCH and the autoregressive of order one AR(1) models, including the unit root model
+with the autoregressive coefficient equal to 1. More specifically, the DAR is a nonlinear
+Markov 1 process, which becomes a stationary martingale when the autoregressive coef-
+ficient is equal to 1 [Gourieroux, Jasiak (2019)]. The DAR model, unlike the traditional
+autoregressive AR(1)-ARCH process, provides valid inference and consistent parameter
+estimators for the autoregressive coefficient values including 1. The proposed extension to
+a deterministic time-varying parameters model relies on the assumption of local strict sta-
+tionarity of the process, following the approach of Dahlhaus (2000) and Dahlhaus, Richter,
+Wu (2019). Then, during the episodes of unit root dynamics, the process satisfies locally
+the stationary martingale condition. The time varying tvDAR model provides a good fit
+to the Tether/USD rates and gives reliable one step ahead out-of-sample predictions. To
+obtain the empirical results, we employed a rectangular kernel and an Epanechnikov ker-
+nel. The first is an asymmetric kernel, which permits the incorporation of past information
+in a pre-specified window and could be used for out-of-sample prediction. The second is a
+symmetric kernel that uses information around any time period and is more suitable for
+inference on the parameters in the model.
+Moreover, the tvDAR model provides a simple plug-in measure of stability for sta-
+
+THIS VERSION: January 3, 2023
+3
+blecoins, based on the Lyapunov exponent. This measure is commonly used to assess
+the stability of deterministic dynamical systems and to test for chaos [see, e.g., Sprott
+(2014)]. The Lyapunov exponent for the AR(1)-ARCH model has been determined by
+Borkovec and Kluppenberg (2001), and shown to be the condition of strict stationarity
+of that process [see also Borkovec (2000)]. It has been also considered by Nelson (1990)
+in the context of the IGARCH model and by Cline and Pu (2004) in a non-parametric
+framework. The Lyapunov exponent was also used as a stability measure in application
+to the Vector Autoregressive VAR model by Dechert and Gencay (1992). Those authors
+have introduced an alternative stability measure based on the noise-to-signal ratio for
+linear dynamic models [see also LeBaron (1994) for introduction to chaos].
+In this paper, the sample Lyapunov exponent is computed from the model parameter
+estimates and proposed as a measure of stability for stablecoins. A more conservative
+measure, based on the condition of second-order stationarity is also introduced.
+Both
+measures can be computed locally and used to assess the stability of a stablecoin over
+time, or to compare the stability of different stablecoins.
+The time-varying coefficient approach based on the assumption of local stationarity dis-
+tinguishes our approach from the literature that relies on the assumption of global strong
+stationarity of the series. For instance, Baum¨ohl and Vyrost (2022) use high frequency
+data to compute a spectral density-based quantile dependence measure under a strict
+stationarity condition, which does not seem to be satisfied by Tether. Bianchi, Rossini,
+and Iacopini (2022) estimate a Bayesian VAR with stochastic volatility and Student-t dis-
+tributed shocks (BVAR-SV-t). However, the conditional volatility equation is constrained
+to unit root dynamics, which is inconsistent with the empirical evidence provided in this
+paper.
+The paper is organized as follows. Section 2 describes the stablecoins. Section 3 dis-
+cusses the local dynamic analysis of the price of Tether. Section 4 discusses the modelling
+approach, estimation procedures and stability measures. Section 5 presents the empiri-
+cal results based on the estimation and inference on the DAR(1) and tvDAR(1) models,
+including the sample stability measures. Section 6 concludes. Appendix A contains the
+technical results. Simulation and additional empirical results on stability measures are
+relegated to Appendices B and C, respectively.
+
+THIS VERSION: January 3, 2023
+4
+2
+Stablecoins
+This section defines stablecoins and discusses their classification, issuance, and redemption
+mechanisms. In addition, we discuss how the market prices of stablecoins are determined.
+2.1
+Definition and Classification of Stablecoins
+Stablecoins are a type of cryptocurrency designed to maintain a stable price and reduced
+volatility, compared to other cryptocurrencies such as Bitcoin and Ethereum. Conven-
+tionally, stablecoin companies peg the value of their coins to that of a physical asset such
+as a fiat currency or gold with the assumption that the market price of their coins will
+eventually stabilize, establishing equivalency with the reference asset. The strategies used
+to achieve price stability of stablecoins are discussed below.
+There is currently no standard in place that private enterprises should comply with
+to qualify as a legitimate stablecoin company. This leaves stablecoin enterprises with un-
+limited design options to choose from to differentiate their business models. Currently,
+business models of stablecoin companies differ in their economic design, the quality of
+backing they maintain, stability assumptions they rely on, and legal protection they pro-
+vide for coin holders (Catalini and de Gortari, 2021).
+While the underlining business models may be diverse and complex, there is interest
+in the elements of such models to understand their economic implications. For example,
+one element of interest is the mechanism stabelcoins rely on to stabilize price and another
+is how the responsibilities are distributed over stablecoin protocols.
+There exist two alternative mechanisms used by stablecoin companies to achieve price
+stability. They either hold collaterals in their reserves to back the value of their coins
+or they adjust the supply of coins through software codes to restore the peg with the
+reference asset. When the market value of a cryptocurrency is backed by collaterals, the
+cryptocurrency is referred to as a collateralized stablecoin. Conventionally, collateralized
+stablecoins are split into two sub-categories including off-chain collateralized stablecoins
+and on-chain collateralized stablecoins. Off-chain collateralized stablecoins are backed by
+a set of collaterals that have an economic value outside of the blockchain. The reserves of
+this type of stablecoins usually consist of a fiat currency such as the US dollar for Tether
+(USDT) or a commodity such as gold for PAX Gold (PAXG). Stablecoins are labelled as
+
+THIS VERSION: January 3, 2023
+5
+on-chain collateralized if the underlying collaterals are composed of crypto assets that are
+created in a digital form and recorded on a distributed ledger. For instance, Dai (DAI),
+the largest on-chain stablecoin project, supports 18 collateral assets including not only
+cryptocurriencies such as Ethereum (ETH) and Chainlink (LINK) but also stablecoins
+such as Tether (USDT), USD Coin (USDC), TrueUSD (TUSD) and PAX dollar (USDP).
+Some projects opt for developing software codes to minimize price fluctuations instead
+of collateralizing their coins. This type of cryptocurrencies is called an algorithmic stable-
+coin as they try to stabilize their price around the peg by contracting or expanding the
+coin supply with the help of computer algorithms embedded in their design. TerraUSD
+(UST) was until May 2022 the only example of an algortihmic stablecoin that has a market
+capitalization over a billion US dollar.
+In terms of distribution of responsibilities, stablecoins can be categorized as centralized
+or decentralized. Centralized stablecoins rely on a single legal entity to maintain the price
+stability, to manage and protect the collaterals, and to fulfill its obligations to users. For
+instance, Tether Limited is the legal entity that has the authority as well as the respon-
+sibility over every Tether in the circulation. Unlike centralized stablecoins, decentralized
+stablecoins distribute these responsibilities within their network through smart contracts.
+This allows network participants to take an active role in determining the rules of the
+stablecoin protocol such as the set of eligible collaterals and the minimum collateral re-
+quirements. The decentralized stablecoin DAI grants users who hold its governance token
+Maker (MKR) the right to vote on the changes to its protocol.
+Li and Mayer (2021) noted that the introduction of stablecoins is comparable to “the
+unregulated creation of safe assets to meet agents’ transactional demands” known as
+shadow banking. Unlike stablecoin issuers, shadow banks must play the role of credit
+guarantees in the case of insolvency. However, as we will see later, the observed prices of
+stablecoins tend to deviate from the peg. In addition, these crypto assets face multiple
+risks including the risk of liquidation.
+For stablecoins designed to be on par with a fiat currency, the use of reserve allows
+the stablecoins issuers to sell or buy the currency to achieve its price stability.
+This
+mechanism helps stablecoin companies to underpin the market value of their coins and
+protect against the highly volatile nature of the cryptocurrency markets. It resembles fixed
+exchange rate regimes currently implemented in Panama, Qatar, and Saudi Arabia. In the
+
+THIS VERSION: January 3, 2023
+6
+fixed exchange rate regime, the central bank also uses its foreign reserves to buy or sell its
+domestic currency to maintain the fixed parity with the currency peg. When the reserve
+system fails, the domestic currency can be devaluated.
+Lyons and Viswanath-Natraj
+(2020) documented that contrary to central banks with some macroeconomic mandate,
+including keeping inflation around its target, stablecoin issuers do not have any policy
+functions. In addition, stablecoin companies cannot use the interest rate or devaluation
+policy to control the exchange rate. For our analysis, we focus on Tether, which is by
+far the primarily traded stablecoin in terms of market capitalization. As we will show
+later, Tether price tends to be noticeably affected by events in the crypto world, leading
+to deviations from the peg despite using reserves.
+Given the aforementioned significant risks for stablecoin holders, there is a need to
+develop appropriate tools to assess their predictability and stability. This paper develops
+a model based on the properties of Tether price and uses it to propose tests for its sta-
+bility. Before discussing the specificity of the cryptocurrency of interest and the modeling
+strategy, we provide further explanation on the issuance and redemption of stablecoins.
+2.2
+Issuance and Redemption
+Issuance and redemption are the two fundamental market activities that determine the
+equilibrium quantity of a stablecoin in the market.
+The equilibrium quantity goes up
+when new coins are issued, and it goes down when existing coins are redeemed. While the
+equilibrium quantity changes with issuance and redemption, the price of a stablecoin is
+held constant during these transactions by the service provider. This constant price policy
+is the result of the pegging strategy explained in the previous section.
+Issuance and redemption of stablecoins are presumably initiated by users.1 How these
+transactions are executed depends on whether stablecoin has a centralized or decentralized
+structure. Issuance takes place following the transfer of funds by user to the stablecoin
+enterprise. Depending on whether stablecoin is centralized or decentralized, these funds
+are deposited either into banking accounts of a custodian or into a cryptographical vault.
+For example, an individual or a business who wants to buy Tether should transfer the
+funds, specifically the US dollar, to Tether Limited’s accounts at Cathay Bank and Hwatai
+1See Griffin and Shams (2020) for further discussion on whether Tether issuances are supply-driven or
+demand-driven.
+
+THIS VERSION: January 3, 2023
+7
+Bank in Taiwan. The collection of these funds constitutes the reserves of the stablecoin
+enterprise, and they are meant to be kept as a collateral to back the value of every coin in
+the circulation. Once the funds are successfully deposited, stablecoins are issued through
+smart contracts and credited into the user’s wallet. For centralized stablecoins, it is the
+issuer or the agent that authorizes the issuance of the coins whereas it is done automatically
+by the blockchain technology for decentralized stablecoins.
+Redemption of stablecoins is also initiated by user but the difference is that the trans-
+actions take place in the reserve order. To redeem stablecoins, users place an order on the
+blockchain to exchange their stablecoins for the collateral. Upon the order, the stablecoin
+enterprise becomes obliged to withdraw the stablecoins from circulation and give the user
+the corresponding amount of collateral in return. The stablecoins that the user redeems
+are destroyed subsequently from the protocol. Stablecoin projects pledge in their whitepa-
+pers that their coins are 100% redeemable and redemptions can be performed any time
+users want at the predetermined price. Hence, one can argue that redeemability becomes
+the liability of stablecoin enterprises and plays a key role in the sustainability of their
+projects. It is the issuer that is liable to users for redemptions in centralized stablecoins.
+Decentralized stablecoins, on the other hand, have no single legal entity that shoulders the
+responsibility. It is the whole network that is responsible for undertaking redemptions.
+2.3
+Market Price of Stablecoins
+Stablecoin prices are usually fluctuating around the target value. While their value in
+terms of the reference asset is fixed during issuance and redemptions, they are often
+traded at a premium or a discount on the exchange platforms.2 This splits the market
+for stablecoins between the primary market and the secondary market, which can be
+considered analogous to the market for traditional securities. The primary market for
+stablecoins is where stablecoins are created (issuance) or destroyed (redemption) at the
+fixed exchange rate predetermined by the stablecoin initiative.
+The secondary market
+is where the users trade stablecoins within and across cryptocurrency exchanges such
+as Binance, Coinbase Exchange, Kraken and Bitfinex. The price of stablecoins in the
+secondary market is determined as a result of the market activities. According to Griffin
+2See Lyons and Viswanath-Natraj (2020) for the detailed analysis of premium and discount on stablecoin
+prices.
+
+THIS VERSION: January 3, 2023
+8
+and Shams (2020), the secondary market activities account for most of the aggregate
+Tether flow from 2014 to 2018. Hence, the price of Tether and other stablecoins in the
+secondary market can be considered as the effective rate at which individuals or businesses
+can buy and sell stablecoins on a day-to-day basis.
+At first glance, the design elements stands out as the primary mechanism through which
+stablecoin projects try to achieve the price stability. However, the price stabilization in
+the stablecoin market could be multifaceted. For instance, Lyons and Viswanath-Natraj
+(2020) argue that the price gap between the primary market and the secondary market,
+which corresponds to the deviation from the peg, can be mitigated also by arbitrage
+activities. As long as investors have access to the primary market, the price deviations in
+the secondary market creates an opportunity for them to make profit. When the price of
+a stablecoin in the secondary market is above the peg, the arbitrager can buy the coin at
+the target exchange rate from the primary market and sell it in the secondary market to
+make profit. This increases the supply of the stablecoin in the secondary market, so it puts
+downward pressure on its price. Similarly, when the price of a stablecoin in the secondary
+market is below the parity, an arbitrager can buy the coin from the secondary market
+and redeem it at the peg ratio in the primary market. In this case, the demand from the
+arbitrager puts upward pressure on the secondary market price. Hence, one could expect
+that the price of stablecoins across cryptocurrency exchanges would stabilize around the
+peg through arbitrage activities. For example, introduction of Tether to the Ethereum
+blockchain in April 2019, which is associated with increased direct access of investors to
+the primary market, is found to have a stabilizing effect on the price of Tether in the
+secondary market (Lyons and Viswanath-Natraj, 2020).
+Bullman et al. (2019) provides the list of alternative tools that each type of stablecoins
+can adopt to maintain the peg. The list consists of fees, redemption limits, and penalty
+fees to name a few. Fees and redemption limits could be used by collateralized stablecoins
+to limit the users’ transactions and prevent sudden liquidations while penalty fees can help
+maintain the minimum level of collateralization. For example, Tether Limited imposes the
+minimum amount of 100,000 USD required for a fiat withdrawal or deposit and charges
+the greater of $1,000 or 0.1% fee per fiat withdrawal and per fiat deposit.3
+The stabilization strategies of Tether have not always been fully successful. The next
+3https://tether.to/fees/
+
+THIS VERSION: January 3, 2023
+9
+section presents empirical evidence based on the dynamic analysis of the data.
+3
+Tether Dynamics
+This section examines the patterns in Tether dynamics in the sample of T = 1361 daily
+closing prices recorded between November 9, 2017, and July 31, 2021.
+Figure 1 displays the evolution of daily closing rates of the Tether against US Dollar.
+We observe the episodes of explosive dynamics mixed with more stable periods as well as
+the convergence of the process at the end of the trajectory to a smooth process taking
+values close to 1. The convergence of Tether towards the peg and its reduced range after
+2021 are associated with increased volume displayed in Figure 2.
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+0.96
+0.98
+1
+1.02
+1.04
+1.06
+1.08
+Daily Price
+Figure 1: Tether/USD daily closing rates
+During the sampling period, the lowest and highest price were 0.9666 and 1.0779,
+respectively. Although 0.0334 and 0.0779 deviations from the one US dollar parity may
+look small, they can provide important arbitrage opportunities if the investor is holding
+a large position in the crypto asset. While the mean over this period is 1.022 and close
+to one, as expected, the volatility around the mean is 0.066. The evolution of the price
+shows an alternate of relatively large and small deviation in the stablecoin price due to
+changes in its demand and lags in intervention by Tether to maintain price stability.
+
+THIS VERSION: January 3, 2023
+10
+The fluctuation of the Tether price around the one-dollar peg can be connected with
+the European snake in the tunnel currency system created in April 1972 by an agreement.
+To increase the convergence among the different currencies in the European Economic
+Community (EEC), the agreement objective was to create a single currency band within
+which all the EEC currencies could fluctuate and not deviate too much from a peg. The
+peg was defined using first gold and, later on, the US dollar. More details on the system
+can be found in Day (1976). To achieve stability around the peg, central banks had to use
+their reserve to intervene by buying or selling local currencies. The system was difficult
+to sustain as several currencies left the agreement. Although stablecoin issuers do not
+have a macroeconomic policy function as central banks, the difficulties in maintaining the
+snake currency system also speak to the challenge of maintaining stablecoin prices around
+its peg using the reserve system discussed above. To understand the price movements of
+Tether, we first discuss factors that affect its demand during the sampling period.
+Figure 2: Time varying volume of Tether
+The daily volume series in Figure 2 exhibits larger fluctuations during the year 2021
+of the sampling period. Although the overall trend was positive, the daily volume varied
+
+Volume in million USD (USDT)
+2.5
+1.5
+0.5
+Jui 2017
+Jan 2018
+ Jul 2018
+ Jan 2019
+ Jul 2019
+ Jan 2020
+ Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022THIS VERSION: January 3, 2023
+11
+roughly between $15.4 billion and $279 billion. The highest daily volume of $279 billion
+was achieved on May 19, 2021, when the Chinese government cracked down on its domestic
+market for cryptocurrencies. Later, the daily volume plunged to as low as $33.7 billion in
+July 2021.
+The high levels of daily volume in 2020 and 2021 could be explained by an increasing
+interest from investors as the market for cryptocurrencies grew substantially during the
+initial stages of the Covid-19 pandemic. Tether’s daily volume increased drastically from
+less than $10 billion in 2018 to as high as $279 billion in 2021. While the daily volume
+is observed to be increasing almost steadily in 2018 and 2019, it exhibits rather a volatile
+pattern in 2020 and 2021. For instance, in early 2021, the daily volume of Tether more
+than doubled in a matter of a few months and reached a peak of 99.3 billion USD on the
+March 13th, a day after the infamous “Crypto Black Thursday”. However, the pattern
+in Figure 2 indicates that the daily volume of Tether decreased between May and July of
+2021 and returned to its pre-pandemic level.
+The convergence to reduced range and small variation around the constant value of 1
+occurs first in Tether in May 2018 and is interrupted by the end of September 2018. In
+the environnement of Tether, the convergence is observed simultaneously for other mostly
+traded traded stablecoins such as USD Coin, Binance USD, True USD, and Pax Dollar
+starting from July 2020 and in Dai starting from December 2020. During this period,
+these stablecoins displayed a period of improved stability towards the end of the sampling
+period. Also, the Bitcoin and Etheureum prices in US Dollars have increased. This period
+of stability overlaps with the period of bullish run in the cryptocurrency market. The
+cryptocurrency market indices such as the S&P Cryptocurrency Broad Digital Market
+(BDM) Index recorded more than fivefold increase between September 2020 and May
+2021.4 The strong demand for cryptocurrencies also benefited the stablecoin companies
+as the total market capitalization of the top 10 stablecoins went up from approximately
+$20 billion in September 2020 to slightly over $100 billion in July 2021.5
+The stability is interrupted again for all stablecoins when the cryptocurrency market
+shrunk by over $300 billion on April 17, 2021 in less than 24 hours.6 While the waves
+4https://www.spglobal.com/spdji/en/indices/digital-assets/sp-cryptocurrency-broad-digital-market-
+index/#overview
+5https://www.statista.com/statistics/1255835/stablecoin-market-capitalization/
+6https://www.forbes.com/sites/jonathanponciano/2021/04/18/crypto-flash-crash-wiped-out-300-
+
+THIS VERSION: January 3, 2023
+12
+of sell-off caused the price of Bitcoin to plummet by 10.5%, the price of all stablecoins
+increased simultaneously. This can be evidence in favor of the previous studies which
+suggest that stablecoins could provide hedging opportunities for cryptocurrency investors
+against Bitcoin’s volatility, e.g., Wang and Wu (2020). However, one could also argue that
+the risk mitigating properties of stablecoins, which are closely linked to the comovements
+between the price of stablecoins and that of Bitcoin, could be changing locally. For exam-
+ple, stablecoins showed resilience against even a much stronger market crash in mid-May
+2021. On May 19, 2021, the Chinese government announced that the banks in China are
+banned from providing cryptocurrency services to their clients.7 The market reacted to
+this news almost immediately as Bitcoin shed 30% of its value over the course of the day.
+During the crash, all stablecoins except for Terra USD managed to keep their price stable
+around the one-dollar peg.
+3.1
+Local Analysis of Tether Price
+This subsection analyzes the local dynamics of Tether price series. We examine its local
+means, variances, and autocorrelations.
+(a)
+Local mean and variance
+This section studies the evolution of Tether/USD rates and identifies local patterns in this
+series by considering time varying descriptive statistics computed by rolling over a window
+of 50 days.
+billion-in-less-than-24-hours-spurring-massive-bitcoin-liquidations/?sh=7d60735b2c89
+7https://www.theguardian.com/technology/2021/may/19/bitcoin-falls-30-after-china-crackdown
+
+THIS VERSION: January 3, 2023
+13
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+0.98
+0.99
+1
+1.01
+1.02
+Local Mean
+=1
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+0
+1
+2
+3
+Local Variance
+10
+-4
+Figure 3: Local mean and variance for the price of Tether
+The locally estimated marginal mean µt and variance σ2
+t are displayed in panels a) and
+b) of Figure 3.8
+The figure’s top panel reports the local mean of the price series and shows its evolution
+over the sampling period. We observe that: the local mean of Tether varies across sub-
+periods, and it displays local trends. Especially, in the first half of the sampling period,
+a strong local trend is observed, which is interrupted by a return of the series to values
+close to 1. For example, in August 2019, the local mean increases, which is akin to the
+pattern of financial bubbles observed in stock prices (rational stochastic bubbles as in
+Blanchard and Watson (1982) or intrinsic bubbles as in Froot and Obstfeld (1991)). See
+Kortian (1995) for more details. These patterns, however, disappear towards the end of
+the sampling period and the local mean becomes more stable and close to the target value
+of 1. Moreover, there are periods where the target value of 1 falls within the confidence
+intervals of local means: April 19, 2018 to May 5, 2018, May 9, 2018 to June 16, 2018,
+8The lower and upper bounds of the confidence interval at the 95% level are calculated under the iid
+assumption as ˆµt ∓ 1.96
+�
+ˆσt
+n for each window of n days.
+
+THIS VERSION: January 3, 2023
+14
+October 10, 2018 to October 17, 2018, January 2, 2019 to January 3, 2019, and April
+2, 2020 to May 1, 2020. The local variance of the price series is plotted in the bottom
+panel of the figure. It varies over time, and its variation is much higher in the first half
+of the sampling period. For example, in 2018, Tether had periods of high volatility from
+January to March as well as periods of low volatility such as from April to November.
+On the other hand, Tether is less volatile during the second half of the sampling period
+as the local variance takes smaller values except for a short period of increased volatility
+between mid-March and early May of 2020.
+Although the rolling window approach helps detect local trends, it needs to be inter-
+preted with caution. For a window size of n days, the first n-1 observations in the dataset
+are eliminated due to rolling. In addition, using a longer rolling window (e.g., 100 days)
+may over-smooth the changes in the mean and variance as compared to a shorter window
+(e.g., 50 days). Therefore, we use the window of 50 days for further computations.9
+The distributional changes in Tether also concern the range and quantiles of the series.
+Overall, we observe that:
+1. The local mean is changing over time and is close to 1 between April and June 2018,
+and after January 2021.
+2. The variance is time varying and diminishes over time.
+In Section 3.2, we identify a series of events that are closely related to Tether, and
+provide a detailed explanation of the reason why those events could be the driving force
+behind the changes we observe in the local statistics of Tether.
+3.2
+Event Analysis
+The dynamics of Tether are strongly influenced by events, which can be used to distinguish
+the episodes of distinct patterns in the local mean and variance.
+Figure 4 shows the evolution of the local mean and the local variance of Tether along
+with the series of events that can be important for the dynamics of Tether, which can
+9Also, note that n=50 is large enough to estimate parameters within each window consistently. Fur-
+thermore, n=50 divided by the sample size of T=1361 is the bandwidth bT = n/T = 0.0367 in our local
+analysis, which will be discussed later. In the literature, an optimal choice should satisfy Tb3
+T = o(1). In
+our case, we have Tb3
+T = 0.0675, which is relatively small. See Dahlhaus, Richter, Wu (2019, page 1039)
+for more details.
+
+THIS VERSION: January 3, 2023
+15
+help explain the trend reversals in its local statistics. Total of 11 such events are identified
+including 7 events for the local mean and 4 events for the local variance.
+In 2018, the local mean shows a downward trend for the most part of the year, except
+for a brief period of recovery between early May and Late September. This should come as
+no surprise because 2018 was a very tumultuous year for Tether Limited and its business
+partners. Tether Limited was being scrutinized by the media and scholars for the quality of
+its reserves and its close ties to the cryptocurrency exchange Bitfinex. More specifically,
+Tether was publicly accused for not holding enough reserves to back all of its coins in
+circulation and for manipulating the price of Bitcoin by pumping unbacked supply of
+Tether into the market through Bitfinex to buy Bitcoins. Amid these controversies, the
+local mean of Tether is found to be decreasing for the most part of the year, which could
+be linked directly or indirectly to a shift in investor sentiment towards Tether.
+In 2018, there is also a short period of a slight upward trend in the local mean roughly
+between early May and late September.
+In early May, the owners of Tether Limited
+made their first significant attempt to show their willingness to address the investors‘
+concerns about the accountability of their business and that of Bitfinex.
+On May 7,
+2018, the cryptocurrency exchange Bitfinex officially announced that Peter Warrack, who
+worked previously at RBC Royal Bank for 20 years as an anti-money laundering specialist,
+joins their team as the Chief Compliance Officer.
+Upon this news, the local mean of
+Tether enjoys a period of recovery and hovers above the one-dollar peg. Nevertheless, the
+local mean of Tether starts to decrease once again in September and reaches its lowest
+level in November. The downfall of Tether during this period could be triggered by the
+introduction of USD Coin (USDC) on September 26, 2018.
+USD Coin, which is also
+designed to be on par with the US dollar, relies on the business principles similar to that
+of Tether but it claims to offer its users an improved transparency in its business activities.
+
+THIS VERSION: January 3, 2023
+16
+Figure 4: Important events for the local mean and the local variance of Tether
+Note: This note provides a description of the events.
+Peter Warrack: Peter Warrack was hired by Bitfinex as the Chief Compliance Officer on May 7,
+2018.
+USDC launched: USD Coin was launched on September 26, 2018.
+Partial Backing: Bloomberg suggested on December 12, 2018 that Tether could be fully backed.
+Partial Backing 2: On March 14, 2019, Tether made changes to its backing policy on its official
+website.
+Tether wins appellate: Bitfinex won a motion in the New York Supreme Court to delay sub-
+mission of its business documents.
+WHO Covid: WHO made an announcement on Twitter on December 31, 2019 to acknowledge
+the cases of pneumonia in Wuhan, China.
+Coinbase Outage: Bitcoin shed over 10% of its value in a matter of minutes on May 9, 2020,
+which was followed by an outage in the cryptocurrency exchange Coinbase.
+Wire Deposits: Bitfinex “temporarily paused” EUR, USD, JPY, and GBP wire deposits on
+October 11, 2018.
+TRON: Tether went live on the Tron network on April 17, 2019.
+Black Thursday: Black Thursday: Bitcoin’s price reduced by around 50% in less than a day on
+March 12, 2020.
+Chain Swap: On June 22, 2020, Tether announced on Twitter that they would implement a
+chain swap for a sizable amount of USDT from Tron TRC20 to ERC20 protocol on June 29th.
+
+Local Mean (USDT)
+1.02 
+Warrack
+PIAOS OHM
+6
+uedde
+1.015
+no
+ieie.
+1.01
+1.005
+0.995
+0.99
+0.985
+ Jui 2017
+ Jan 2018
+ Jul 2018
+ Jan 2019
+ Jul 2019
+Jan 2020
+Jul 2020
+ Jan 2021
+ Jul 2021
+Jan 2022
+×10-4
+Local Variance(USDT)
+TRON.
+Deposits
+2.5
+1.5
+0.5
+Jui 2017
+Jan 2018
+Jul 2018
+ Jan 2019
+ Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+ Jul 2021
+Jan 2022THIS VERSION: January 3, 2023
+17
+Tether makes up for the loses quickly as the local mean increases remarkably from
+its lowest level in November 2018 to its highest level in January 2019 just under two
+months. The surge in the mean price of Tether could be a sign of positive reaction from
+the investors as the bank statements obtained by Bloomberg showed that on the contrary
+to the allegations, Tether Limited could be holding enough reserves to back its coins in the
+circulation.10 However, this positive attitude towards Tether does not seem to last long as
+the local mean diminishes once again starting in January 2019. The downward pressure
+on the local mean was so strong during this period that it persisted up until August
+2019 despite Tether’s effort to be more transparent about the composition of its reserves.
+According to coindesk.com,11 Tether changed the terms on its website in Mid-February
+implying that while every Tether is always 100% backed by their reserves, the reserves are
+not necessarily composed of only fiat currency as they claimed before but also includes
+cash equivalents, other assets, and receivables from loans. This update, however, did not
+stop the New York State Attorney General (NYSAG) sue Bitfinex and Tether Limited on
+April 24, 2019 on the basis of an ongoing fraud.12
+As the downward trend in the local mean dies out in August 2019, another episode
+of increasing local trend is observed subsequently. The trend becomes more noticeable
+around September 24, 2019 when iFinex Inc, the parent company of Bitfinex and Tether
+Limited, won a motion in the court against NYSAG meaning that the company does not
+have to hand over the documents related to its business activities until further notice.13
+Following this small victory in the court, the local mean of Tether diverges from the peg
+and keeps increasing until the end of the year.
+On December 31, 2019, the World Health Organization (WHO) announced via its
+official Twitter account that they were informed of cases of pneumonia of unknown cause
+in Wuhan City, China.14 With this announcement, WHO acknowledged the problem of
+an epidemic in China, which would soon turn into a global health and economic crisis of
+10https://www.bloomberg.com/news/articles/2018-12-18/crypto-mystery-clues-suggest-tether-has-the-
+billions-it-promised
+11https://www.coindesk.com/markets/2019/03/14/tether-says-its-usdt-stablecoin-may-not-be-backed-
+by-fiat-alone/
+12https://ag.ny.gov/press-release/2019/attorney-general-james-announces-court-order-against-crypto-
+currency-company
+13https://www.forbes.com/sites/michaeldelcastillo/2019/09/24/bitfinex-and-tether-win-appeal-from-
+new-york-supreme-court-in-900-million-case/?sh=7c8bf41132bc
+14see https://twitter.com/who/status/1213795226072109058?lang=en for the original tweet from WHO
+
+THIS VERSION: January 3, 2023
+18
+Figure 5: Autocorrelation functions of Tether over different periods
+COVID-19 pandemic. Subsequently, the local mean of Tether starts to decline cancelling
+out the gains from the last quarter of 2019.
+3.3
+Persistence in Tether Series
+Let us now examine the serial correlation in Tether. The autocorrelation function (ACF)
+of Tether is computed from the entire sampling period 2017-2021 as well as from each
+calendar year separately. Figure 5 presents the computed ACF functions: the ACF over
+the entire period (panel (a)), and in years 2017-2021 in panels (b) to (e), consecutively.
+The ACF calculated from the entire sample exhibits a long range persistence. However,
+the subperiod analysis reveals that the persistence in the ACF of Tether is strong up to
+and including 2019,15 whereas the series has a short memory in 2020 and 2021.
+When combined with the results from the local statistics, it can be inferred that the
+period of long-range persistence in Tether coincides with the period of level shifts and high
+volatility as documented in Figure 1. Likewise, when the variation is small and the local
+mean stabilizes around the one-dollar peg as in 2020 and 2021, Tether displays a short
+memory.
+15In 2017, there are only 53 observations, which could be the reason for the weak evidence for the
+persistence.
+
+Panel (a): 2017-2021
+Panel (b): 2017
+Panel (c): 2018
+0.8
+0.6
+0.6
+ 0.4
+0.2
+0.2
+0.2
+-0.2 
+20
+10
+15 
+20
+10
+15
+10
+Lag
+Lag
+Lag
+Panel (d): 2019
+Panel (e): 2020
+Panel (f): 2021
+1
+0.8
+0.8
+0.6
+4 0.4
+00.4
+0.2
+0.2
+0.2
+0.2
+-0.2
+29
+10
+15
+20
+5
+10
+15
+10
+15
+Lag
+Lag
+LagTHIS VERSION: January 3, 2023
+19
+In brief, the empirical results show that the analysis of Tether based on global statistics
+would provide unreliable results, especially concerning the serial correlation of the series.
+For example, different values and range of serial correlation are obtained in year 2021, as
+compared to years 2018-2019.
+Let us now focus on the autocorrelation values. More specifically, the autocorrela-
+tion at lag one of the series can be estimated from the autoregressive coefficient of an
+autoregressive of order 1 (AR(1)) model. We first consider the autoregressive coefficient
+estimated from the AR(1) model fitted to the whole sample of demeaned Tether prices
+xt = yt − µ
+xt = ρxt−1 + σeet,
+(3.1)
+where et is assumed to be a white noise with mean 0 and variance 1. The AR(1) process
+is stationary when |ρ| < 1 and nonstationary and explosive when ρ = 1. Model (3.1) is
+estimated globally by the OLS, or equivalently by maximizing the Gaussian Maximum
+Likelihood as follows:
+ˆθT = Argmaxθ
+T
+�
+t=1
+l(xt|xt−1; θ) = Argmaxθ
+T
+�
+t=2
+−1
+2
+�
+log
+�
+2πσ2
+e
+�
++ (xt − ρxt−1)2
+σ2e
+�
+where θ = (ρ, σ2
+e).
+The estimated parameter values are ˆρT = 0.674 and ˆσ2
+e,T = 0.545. Next, model (3.1) is
+fitted locally and estimated by rolling with a window of length 50 and displayed in Figure
+6.
+The top panel of Figure 6 shows the autoregressive coefficient/correlation at lag 1
+estimates. We observe that the autoregressive coefficient is close to 1 during the explosive
+episodes in 2018 and 2019, which violates the stationary condition of the AR(1) process.
+The unit root dynamics of Tether resembles the stock prices and exchange rates.
+It
+suggests that Tether is then locally efficient in financial terms. The autocorrelation at
+lag 1 is close to 0.5 at the end of the sampling period. In comparison with the results
+presented in Section 3.1, estimating the autocorrelation at lag one based on the AR(1)
+model gives us greater flexibility to assess the change in the persistence of Tether as we
+are able to examine its evolution at a daily frequency rather than on a yearly basis. For
+
+THIS VERSION: January 3, 2023
+20
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+0.012
+0.014
+Figure 6: AR(1) parameter estimates and conditional volatility for xt
+example, the estimated values of the AR(1) coefficient ρ suggest that the autocorrelation
+at lag one ranges between -0.20 and 1.02 in 2018 whereas in Section 3.1, the ACF at lag
+1 was estimated to be slightly over 0.76 for the entire year.
+The bottom panel shows the conditional volatility ˆσe(t) =
+�
+1
+50
+�t
+τ=t−49(xτ − ˆρ(τ) xτ−1)2
+of the price of Tether under the AR(1) assumption. The figure shows that this price ex-
+hibits periods of low volatility and high volatility. Especially from October 2020 onwards,
+the conditional volatility decreases remarkably and becomes very close to 0. This result is
+consistent with the period of stability we observed in the price series of Tether during the
+cryptocurrency bull market explained in Section 3. If the volatility was computed over
+the full sample we would have a constant estimate which does not capture the changes in
+volatility over time. To accommodate this feature, we consider the time-varying volatil-
+ity model which also allows us to have valid inference when the estimated correlation
+parameter is close to one unlike the AR(1) model.
+
+THIS VERSION: January 3, 2023
+21
+3.4
+Properties of rolling estimators
+Let us now examine the properties of the rolling estimators of time varying parameters
+written as deterministic functions of time. First, we estimated by rolling the time varying
+marginal mean and variance functions, hoping to approximate m(t) and σ2(t) in a simple
+model
+yt = m(t) + σ(t)ut,
+under the simplifying assumption of Normally distributed i.i.d. process ut with mean 0
+and variance 1. Then, in finite sample
+ˆm(t) = 1
+50
+t
+�
+τ=t−49
+yτ ∼ N
+�m(t) + · · · + m(t − 49)
+50
+, σ2(t) + · · · + σ2(t − 49)
+250
+�
+We see that ˆm(t) is biased of m(t) towards an integrated mean. By the same argument it
+can be shown that ˆσ2(t) is biased of both σ2(t) and the integrated variance.
+Let us now consider the time varying parameters θ(t) = (ρ(t), σ2
+e(t)) of a time varying
+parameter AR(1) model:
+xt = ρ(t)xt−1 + σe(t)et,
+where xt = yt −m(t). Then, the normality-based MLE estimator ˆθ∗(t) obtained by rolling
+can be written as the following kernel MLE estimator (see Fan et al. (1998) for the method
+of local kernel-weighted likelihood estimation using local polynomial fitting).
+ˆθ∗
+T (t)
+=
+Argmaxθ
+1
+50
+t
+�
+τ=t−49
+l(xτ|xτ−1; θ)
+=
+Argmaxθ
+1
+50
+T
+�
+τ=1
+1t−49≤τ≤t l(xτ|xτ−1; θ)
+=
+Argmaxθ
+T
+�
+τ=1
+� 1
+501−49≤τ−t≤0 l(xτ|xτ−1; θ)
+�
+=
+Argmaxθ
+T
+�
+τ=1
+1
+50 1−49/50≤(τ−t)/50≤0 l(xτ|xτ−1; θ)
+=
+Argmaxθ
+T
+�
+τ=1
+1
+50 K
+�τ − t
+50
+�
+l(xτ|xτ−1; θ), t = 1, ..., T
+
+THIS VERSION: January 3, 2023
+22
+with the kernel K(u) = 1[−1,0](u).
+It is easy to see that the dimension of the parameter of interest [θ(1), ..., θ(T)] depends on
+the number of observations T. To circumvent this difficulty, we can replace the functional
+parameter [θ∗(1), ..., θ∗(T)] with t ∈ N by an alternative functional parameter θ(c), c ∈
+(0, 1) on [0,1], such that θ∗(t) = θ(t/T). The rolling MLE is such that
+ˆθ∗
+T (t) = ˆθT (t/T) = Argmaxθ
+T
+�
+τ=1
+T
+50 K
+�τ/T − t/T
+50/T
+�
+l(xτ|xτ−1; θ), t = 1, ..., T.
+This formula can be extended to any value of argument c ∈ [0, 1]:
+ˆθT (c) = Argmaxθ
+T
+�
+τ=1
+T
+50 K
+�τ/T − c
+50/T
+�
+l(xτ|xτ−1; θ), c ∈ [0, 1].
+Dahlhaus (2000) and Dahlhaus, Richter, Wu (2019) show that under regularity conditions
+the functional parameters θ(c) of a locally stationary process can be consistently estimated.
+Instead of considering the time varying parameters in calendar time, we have now defined
+the functional parameters in a deformed time t → t/T that depends on T. The functional
+parameter is now independent of the observations, while the effect of T is introduced by
+considering a triangular array approach. This leads to a sequence of models indexed by
+T:
+xt,T = ρ(t/T)xt−1,T + σe(t/T)et,T ,
+where xt,T = yt,T − m(t/T).
+This approach motivates our modelling approach presented in the next section.
+4
+DAR(1) Model for Tether Price
+To account for time-varying conditional mean and volatility, we introduce the Double
+Autoregressive (tvDAR) process of order 1 with time varying parameters. The first part
+of this section recalls the constant parameter DAR model. Next, the estimation of both
+type of models is discussed. The last part of this section presents the stability measures
+and their estimators.
+
+THIS VERSION: January 3, 2023
+23
+4.1
+Model with Constant Parameters
+We consider the DAR process of order 1 for the demeaned Tether price series:
+xt = φxt−1 + ηt
+�
+ω + αx2
+t−1,
+(4.2)
+where ω > 0, α > 0, and ηt, t = 1, ..., T is an independent and identically distributed (i.i.d.)
+sequence with mean 0 and variance 1. The parameter φ captures the conditional mean
+dependence. Parameter α represents the past dependence in the conditional variance. The
+model is semi-parametric and conditionally heteroskedastic.16 Borkovec and Kluppenberg
+(2001), Ling (2004) show that there exists a unique strictly stationary and ergodic solution
+to model (4.2) when the following assumptions hold:
+Assumption A.1: ηt has a symmetric and continuous density with mean 0 and variance
+1.
+Assumption A.2: The parameter space is Θ = {θ = (φ, ω, α) : E(ln|φ+ηt
+√α|) < 0 with
+|φ| ≤ ˜φ, ω ≤ ω ≤ ˜ω, α ≤ α ≤ ˜α where ˜φ, ω, ˜ω, α, ˜α are positive constants.
+Assumption A.1 is not a stringent assumption. It is satisfied in particular if ηt, t =
+1, ..., T are normally distributed.
+Assumption A.2 is the existence and negativity of
+the Lyapunov exponent ensuring the existence and uniqueness of a stationary solution
+[Borkovec and Kluppenberg (2001)]. The region of φ, α that satisfy the negativity condi-
+tion is displayed in Figure 1, p. 64 Ling (2004) and Figure 1, p. 191 Chen et al. (2014).
+It includes cases when φ ≥ 1 as well as E(x2
+t ) = ∞. The processes xt that satisfy As-
+sumption 2 are strictly stationary. Some of these processes are also weakly (second-order)
+stationary and satisfy additionally the condition φ2 + α < 1 ensuring that E(x2
+t ) < ∞.
+Thus, the marginal variance of those processes is finite.
+When φ = 1, and E(ln |1 + ηt
+√α|) < 0, the process xt is a strictly stationary martin-
+gale process with volatility induced “mean-reversion” [Gourieroux, Jasiak (2019)]. Model
+(4.2) is non-stationary when the Lyapunov exponent is non-negative. In particular, it is
+nonstationary at the boundary points (φ, α) = (±1, 0) and nests the standard unit root
+models at these two points. When φ = 0, the process is an ARCH(1) model. Moreover,
+process (xt) is strictly stationary when x0 is drawn from a stationary distribution.
+16More on the models with conditional heteroscedasticity and their applications in finance can be found
+in Gourieroux (1997). Zakoian (1994) also proposed maximum likelihood and least squares estimators for
+conditionally heteroscedastic model with threshold.
+
+THIS VERSION: January 3, 2023
+24
+Under assumptions A.1 and A.2, the parameter space Θ is compact and there exists
+a unique strictly stationary solution of the model for any θ ∈ Θ. In addition, we assume
+that:
+Assumption A.3: The model is well-specified, i.e. the process satisfies equation (4.2)
+for the true value of parameter θ0 = (φ0, w0, α0) and the true density ψ0 of η. The true
+parameter value θ0 is an interior point in Θ.
+Assumption A.4: The observed process is the unique, strictly stationary solution asso-
+ciated with (θ0, ψ0).
+These two conditions are introduced for the identification of the model and parameter
+estimation.
+4.2
+Model with Time-Varying Parameters
+The DAR(1) model can be extended to a time-varying parameter model by using the
+triangular array approach for locally stationary processes [Dahlhaus (2000), Dahlhaus,
+Richter, Wu (2019)]. The time varying tvDAR(1) model is written for locally demeaned
+observations xt,T , t = 1, ..., T indexed by t and T (triangular array) and defined by:
+xt,T = φ(t/T)xt−1,T + ηt,T
+�
+ω(t/T) + α(t/T)x2
+t−1,T ,
+(4.3)
+where for each time T, (ηt,T ) is a strong (i.i.d) white noise with mean zero, unit variance
+and a symmetric distribution invariant in T.
+φ(c), ω(c) > 0, α(c) > 0, c ∈ [0, 1] are
+deterministic functions. We assume that these functions are smooth.
+Assumption a.1: The functions φ(.), ω(.), α(.), are positive, deterministic and twice
+differentiable on [0, 1].
+Moreover, the trajectories of the process have to be little responsive to small changes of
+the parameters, which is ensured by a Lipschitz condition. More precisely, let us consider
+process xt(c) defined by:
+xt(c) = φ(c)xt−1(c) + ηt(c)
+�
+ω(c) + α(c)xt−1(c)2,
+(4.4)
+We assume that the following condition holds:
+Assumption a.2:
+
+THIS VERSION: January 3, 2023
+25
+Let the Lq norm for q > 0 be denoted by ||.||q. Then,
+i) For each c ∈ (0, 1), process {xt(c)} is stationary and ergodic.
+ii) c → xt(c) is continuous for any t and ||supcxt(c)||q < ∞.
+iii) There exists α, 1 ≥ α > 0 and CB > 0, such that
+||xt(c) − xt(c′)||q < CB|c − c′|α uniformly in t and c, c′ ∈ (0, 1).
+Under Assumptions a.1 and a.2, if T is large and t/T in a small interval (c−ϵ, c), then
+the parameters are almost constant over that interval and locally model (4.4) is close to
+model (4.3) with φ = φ(c), ω = ω(c), α = α(c). This explains the local stationarity.
+When all the observations xt,T , t = 1, ..., T are considered, the variation of the pa-
+rameters prevents the DAR process from being globally stationary. However, it is locally
+stationary, if Assumption A.2 is locally satisfied, i.e. E[ln|φ(c) + ηt(c)α(c)|] < 0 for any c.
+This is the condition on the negativity of the local Lyapunov exponent.
+4.3
+Estimation
+4.3.1 Estimation of the Model with Constant Parameters
+The parameter estimates of model (4.2) are obtained by maximizing the quasi-maximum
+likelihood (QML) objective function, i.e.
+the log-likelihood function for normally dis-
+tributed ηt.
+LT (θ) = −1
+2
+T
+�
+t=2
+ln
+�
+ω + αx2
+t−1
+�
+− 1
+2
+T
+�
+t=2
+(xt − φxt−1)2
+�
+ω + αx2
+t−1
+� ,
+(4.5)
+where θ = [φ, ω, α]′. The QML estimators of Model (4.2):
+ˆθT = Argmaxθ∈ΘLT (θ)
+are consistent under Assumptions A.1 to A.4, and the vector of QMLE estimators ˆθT =
+[ˆφT ˆωT ˆαT ]′ → θ0 in probability, where θ0 = [φ0 ω0 α0]′ [Ling (2004,2007)]. Moreover, if
+the following assumption:
+Assumption A.5: E(η4
+t ) < ∞
+is satisfied, Li and Ling (2008) and Chen, Li and Ling (2014) show that the Quasi Maxi-
+mum Likelihood estimators (QMLE) of θ are also asymptotically normal when |φ| ≥ 1 17
+as well as E(x2
+t ) = ∞.
+17The ML/OLS estimators of φ from a linear autoregressive AR(1) model with constant parameters are
+not asymptotically normal when φ = 1.
+
+THIS VERSION: January 3, 2023
+26
+√
+T(ˆθT − θ0) → N(0, diag(Σ−1, κΩ−1))
+where this convergence is in distribution, Σ = E0[x2
+t−1/(ω0 + α0x2
+t−1)]
+Ω = E0
+�
+1
+(ω0 + α0x2
+t−1)2
+�
+1
+x2
+t−1
+x2
+t−1
+x4
+t−1
+��
+,
+diag(Σ−1, κΩ−1) denotes the block-diagonal matrix with Σ−1 as the upper left block and
+κΩ−1 as the bottom right block and κ is the kurtosis less 1 of the distribution of η. In
+particular, κ = 2 when ηt is normal.
+These asymptotic results are valid for any true
+distribution of η, not necessarily a Gaussian distribution. The consistent estimators of Σ
+and Ω are
+ˆΣT =
+1
+T − 1
+T
+�
+t=2
+[x2
+t−1/(ˆωT + ˆαT x2
+t−1)],
+ˆΩT =
+1
+T − 1
+T
+�
+t=2
+1
+(ˆωT + ˆαT x2
+t−1)2
+�
+1
+x2
+t−1
+x2
+t−1
+x4
+t−1
+�
+.
+The model residuals are defined as:
+ˆηt,T = (xt − ˆφT xt−1)/
+�
+ˆωT + ˆαT x2
+t−1.
+The model residuals ˆηt,T , t = 1, ..., T allow us to estimate non-parametrically the error
+density to verify ex-post the symmetry assumption. The parameter κ is estimated by
+ˆκT =
+1
+T−1
+�T
+t=2 ˆη2
+t,T − 1 =
+1
+T−1
+�T
+t=2
+(xt−ˆφxt−1)
+4
+(ˆω+ˆαx2
+t−1)
+2 − 1 allowing us to accommodate the
+heavy tailed distribution of the stablecoin prices.
+4.3.2 Estimation of the Model with Time-Varying Parameters
+Let us consider the locally stationary tvDAR model.
+The dynamic model (4.3) of
+triangular arrays xt,T , t = 1, ..., T is non-parametric and depends on the functional param-
+eters φ(c), ω(c) > 0, α(c) > 0, c ∈ [0, 1] and on the density function of the noise ηt,T . The
+estimation of φ(.), ω(.), α(.) can be done by the local-in-time QML estimators. We con-
+sider a kernel K defined on [−1/2, 1/2] and bandwith bT , bT > 0 (following the notation
+
+THIS VERSION: January 3, 2023
+27
+used in Dahlhaus, Richter, Wu (2019), p. 1035). The local negative log-conditional quasi
+likelihood is
+LT,b(c, φ, ω, α) =
+1
+TbT
+T
+�
+t=2
+K
+�t/T − c
+bT
+� �
+�−1
+2 ln
+�
+ω + αx2
+t−1,T
+�
+− 1
+2
+(xt,T − φxt−1,T )2
+�
+ω + αx2
+t−1,T
+�
+�
+� ,
+(4.6)
+Then, the local negative QML estimator of θ(c) = [φ(c), ω(c), α(c)] is:
+ˆθT,b(c) = ArgmaxθLT,b(c, φ, ω, α).
+(4.7)
+Under suitable regularity conditions given in [Dahlhaus, Richter, Wu (2019)], this
+functional QML estimator ˆθT,bT (c), c ∈ [0, 1] is consistent of θ(c), c ∈ [0, 1] and its limiting
+distribution is normal:
+�
+TbT (ˆθb(c) − θ0(c)) → N(0,
+�
+K(y)2dy J(c)−1I(c)J(c)−1),
+where → denotes the weak convergence of processes indexed by c, J(c) is the Hessian
+matrix and I(c) is the outer product of scores, both evaluated at c. Under the symmetry
+assumption on ηt and for a strictly stationary and ergodic xt, the information matrix I(c)
+simplifies to a block diagonal matrix [Ling (2004), Remark 1].
+The regularity conditions concern the functional parameter, the distribution of noise
+ηt, the kernel K and the bandwidth bT . They can be found in Dahlhaus, Richter, Wu
+(2019), since the DAR model is a nonlinear autoregressive model discussed in Dahlhaus,
+Richter, Wu (2019), Example 5.5, p. 1039. In particular, the bandwidth has to satisfy the
+conditions bT → 0, TbT → ∞, Tb3
+T → 0 when T tends to infinity.
+4.4
+Stability measures
+The Lyapunov exponent measures the average logarithmic rate of separation or conver-
+gence of initially close trajectories in chaotic systems, and the sensitivity to initial con-
+ditions, in general. A negative value of the Lyapunov exponent indicates the stability
+of the dynamical system, while a positive value indicates chaos. The more negative the
+Lyapunov exponent, the more stable the system. Therefore, it is used for testing for chaos
+
+THIS VERSION: January 3, 2023
+28
+[Sprott (2003)]. In this section, the Lyapunov exponent is proposed as a measure of stabil-
+ity of Tether and other stable coins. In the framework of the DAR model, the Lyapunov
+exponent is:
+λ = E(ln |φ + η√α|).
+The more negative the Lyapunov exponent, the less explosive the process18 and more likely
+its marginal variance is finite.
+The behavior of E(ln |φ+ηt
+√α|) as a function of φ, α can be examined analytically for
+selected densities of η, and/or simulated and illustrated graphically [see, Liu et al. (2018)
+for graphical illustration]. Appendix A.1 shows the analytical formula of λ for a uniformly
+distributed sequence {ηt}. More precisely:
+Proposition 1: If η ∼ U[−1,1], the Lyapunov exponent is given by:
+λ(φ, α) =
+1
+2√α
+�
+(|φ| + √α) ln(|φ| + √α) − (|φ| + √α) − ||φ| − √α| ln ||φ| − √α| + ||φ| − √α|
+�
+Proof: See Appendix A.1.
+This example clarifies that the Lyapunov exponent is a continuous function of φ, α,
+although with points of non-differentiability.
+Moreover, it is easy to show that that
+E(ln |φ + ηt
+√α|) is always an even function of φ, i.e. it takes the same value for φ and
+−φ. To see that, consider a symmetric density function ψ(η). Then,
+E(ln | − φ + ηt
+√α|) =
+�
+(ln | − φ + ηt
+√α|)ψ(η)dη.
+Because ψ(η) is symmetric, we can change the variable η → −η:
+E(ln|−φ+ηt
+√α|) =
+�
+(ln |−φ−ηt
+√α|)ψ(−η)dη =
+�
+(ln |φ+ηt
+√α|)ψ(η)dη = E(ln |φ+ηt
+√α|).
+This proves that the Lyapunov exponent is even in φ. The Lyapunov exponent can be
+computed by plug-in from the parameter estimates. Let us first consider the constant
+parameter DAR model. The following estimators can be considered:
+a) Suppose that the true density function ψ = ψ0 is known. Then, the estimator ˆλ1,T
+of the Lyapunov exponent is:
+ˆλ1,T =
+�
+ln |ˆφT + η
+�
+ˆαT | ψ0(η) dη.
+18A strictly stationary process can have infinite moments.
+
+THIS VERSION: January 3, 2023
+29
+Proposition 2:
+When the density ψ(η) = ψ0(η) is known, then under assumptions A.1 to A.4 and the
+following condition:
+(A.6)
+∃δ > 0, such that
+�
+sup φ0 − δ < φ < φ0 + δ
+α0 − δ < α < α0 + δ
+| ln |φ + η√α||ψ0(η)dη < ∞
+the estimator ˆλ1,T converges in probability to the true value λ0 =
+�
+ln |φ0 +η√α0|ψ0(η)dη
+of the Lyapunov exponent when T → ∞.
+Proof: See Appendix A.2
+b) When the density ψ(η) is unknown, the model is semi-parametric. The Lyapunov
+exponent can be estimated by the estimator ˆλ2,T such that:
+ˆλ2,T = 1
+T
+T
+�
+t=1
+ln |ˆφT + ˆηt,T
+�
+ˆαT |.
+Therefore this estimator is equal to :
+ˆλ2,T = 1
+T
+T
+�
+t=1
+ln
+������
+ˆφT +
+xt − ˆφT xt−1
+�
+ˆωT + ˆαT x2
+t−1
+�
+ˆαT
+������
+= 1
+T
+T
+�
+t=1
+g(xt, xt−1; ˆθT )
+where g(xt, xt−1; θ) = ln
+����φ +
+xt−φxt−1
+√
+ω+αx2
+t−1
+√α
+����. We also introduce the notation GT (θ) =
+1
+T
+�T
+t=1 g(xt, xt−1; θ).
+Proposition 3:
+Let us introduce the additional conditions:
+(A.7) Eθ0g(xt, xt−1; θ) < ∞, ∀θ ∈ Θ.
+(A.8) Sufficient Lipschitz condition for stochastic equicontinuity: There exists a stochas-
+tic sequence BT with BT = Op(1) and an increasing function h from [0, ∞) to [0, ∞),
+continuous at 0 with h(0) = 0, such that for all ˜θ, θ ∈ Θ, |GT (˜θ) − GT (θ)| ≤ BT h(d(˜θ, θ)).
+Then, under assumptions A.1-A.4 and conditions A.7, A.8 the estimator ˆλ2,T = GT (ˆθT ) →
+G(θ0) = λ0 in probability.
+Proof: See Appendix A.3.
+The asymptotic distributions of estimators ˆλ1,T , ˆλ2,T cannot be obtained asymptot-
+ically from the Taylor series expansion because function φ, α →
+�
+ln |φ + η√α|ψ(η)dη
+
+THIS VERSION: January 3, 2023
+30
+does not satisfy the necessary differentiability assumption, as pointed out in Proposition
+1. However, the distribution of ˆλ1,T , ˆλ2,T can be determined by simulations and used for
+hypothesis testing.
+c) An alternative stability measure is ξ = φ2+α, which depicts the region of parameter
+space ξ < 1 where the marginal variance of xt remains finite, so that the process is both
+strictly and weakly stationary. In that sense ξ is a more conservative measure of stability
+than the Lyapunov exponent because there is a region of parameter values φ, α where
+the condition ξ < 1 no longer holds, while the condition λ < 0 remains satisfied. The
+estimator ˆξT of ξ is:
+ˆξT = ˆφ2
+T + ˆαT .
+Proposition 4:
+Under Assumptions A.1-A.5, the estimator ˆξT converges in probability to ξ0 when
+T → ∞ and it is asymptotically Normally distributed:
+√
+T(ˆξT − ξ0) A∼ N(0, Vξ),
+where the formula of the asymptotic variance Vξ is given in Appendix A.4.
+The asymptotic variance provides the asymptotically valid standard errors that can be
+used to test the null hypothesis ξ < ξ0 using a Wald test statistic. The interpretation of
+measure ξ is similar to that of λ: the smaller ξ, the more stable xt.
+4.4.1 Model with time varying parameters
+The Lyapunov exponent λ2(c) can be estimated locally by computing ˆλ2,T (c) from the
+plugged in parameter estimates and residual values of the tvDAR model. For example,
+the Lyapunov exponent can be estimated from a kernel-weighted formula
+ˆλ2,T (c) =
+1
+TbT
+T
+�
+t=1
+K
+�t/T − c
+bT
+�
+(ln|ˆφ(t/T) + ˆηt,T
+�
+ˆα(t/T)|),
+with the Epanechnikov kernel K(c) = 3
+2(1−(2c)2) for c ∈ [−1/2, 1/2] and K(c) = 0 other-
+wise, which satisfies the regularity condition for localizing kernel [see Dahlhaus, Richter,
+Wu (2019, Assumption 2.6)].
+
+THIS VERSION: January 3, 2023
+31
+A similar approach can be used to estimate locally the measure ξ(c) = φ2(c) + α(c).
+The local plug-in estimator ˆλ2,T (c) is illustrated in the next section.
+5
+Empirical Results
+This section presents the parameter estimates for the constant parameter DAR model and
+the time varying parametr tvDAR model. The time varying parameters DAR model is
+estimated first by rolling, which is equivalent to the use of an asymmetric rectangular
+kernel, as shown in the previous section. Next, the model is estimated by using a kernel
+which assigns higher weights to the observations close to the estimation date, providing
+consistent and asymptotically normally distributed estimates, which are used for testing
+hypotheses on the constancy of parameters. The estimators of stability measures are also
+computed and illustrated graphically.
+The estimation of the DAR(1) process with constant parameters is straightforward.
+First, we demean the price series of Tether by subtracting the total mean of 1.0022 and
+then fit the model 4.2 to the entire series of the demeaned prices, which gives the following
+result
+Table 1: Estimation of the DAR(1) model using the entire
+sample
+DAR(1) parameters
+φ
+ω
+α
+Estimates
+0.699
+9.102e−06
+0.484
+Standard deviation
+(0.034)
+(2.174e−06)
+(0.205)
+where the standard deviations of the parameters are obtained using the asymptotic dis-
+tribution described in Section 5.3.1.
+In the following sections, the model 4.3 is estimated by using two types of kernels. The
+first one is the asymmetric rectangular kernel, equivalent to the rolling estimation over the
+window of 50. This window ensures good properties of the estimators, while preventing
+the over-smoothing.
+
+THIS VERSION: January 3, 2023
+32
+The second approach relies on the Epanechnikov kernel and produces consistent and
+asymptotically normally distributed parameter estimates used for hypotheses testing.
+5.1
+Rectangular kernel
+The tvDAR(1) is estimated from the demeaned Tether price by rolling, which is a common
+practice in applied literature, and a window of length 50 days. This is equivalent to using
+an asymmetric rectangular kernel, K(u) = 1(−1,0)(u), bT = 50/T which is computationally
+simple, but does not satisfy the smoothness conditions ensuring locally the validity of
+asymptotic distribution of the QMLE. The rolling estimate and the confidence interval of
+the parameter of interest φ(t/T) is displayed in the first paned of Figure 7 below.
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+-1
+-0.5
+0
+0.5
+1
+1.5
+=1
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+0.012
+0.014
+Conditional Volatility
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+-6
+-5
+-4
+-3
+-2
+-1
+0
+Figure 7: tvDAR(1) parameter φ(t/T), conditional volatility and Lyapunov exponent
+λ2(t/T)
+
+THIS VERSION: January 3, 2023
+33
+The second panel shows the estimated conditional volatility
+�
+ˆω(t/T) + ˆα(t/T)x2
+t for
+Tether price. The third panel in Figure 7 presents the local estimates ˆλ2,T (t/T) of the
+Lyapunov exponent computed by plugging in the local parameter estimators.
+We observe that there are periods when the autoregressive coefficient is not signifi-
+cantly different from 1. For instance, this is the case between October and December 2018
+and between February and May 2019. These results from the tvDAR (1) model confirm
+our initial observation in Section 3.1 that the price series of Tether shows strong persis-
+tence in 2018 and 2019 whilst allowing us to pinpoint its exact timing. The estimated
+conditional volatility based on the estimated parameters has a pattern consistent with the
+local variance estimator in Figure 3. The results in the third panel suggests that Assump-
+tion 2 holds and the Lyapunov exponent remains negative even for the highest recorded
+persistence. The sample Lyapunov exponent varies across time becoming on average more
+negative before the end of the sampling period. This indicates that Tether achieves higher
+stability over that period.
+Full Sample
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+0.96
+0.98
+1
+1.02
+1.04
+1.06
+1.08
+y=1
+Predicted E(yt+1|yt)
+y
+October 2018 to October 2019
+Jul 2018
+Oct 2018
+Jan 2019
+Apr 2019
+Jul 2019
+Oct 2019
+Jan 2020
+0.96
+0.97
+0.98
+0.99
+1
+1.01
+1.02
+1.03
+1.04
+y=1
+Predicted E(yt+1|yt)
+y
+Figure 8: Tether prices compared to one-step ahead out-of-sample forecasts
+
+THIS VERSION: January 3, 2023
+34
+The tvDAR(1) model estimated with the asymmetric rectangular kernel can be used for
+forecasting at short horizons, under the assumption that the parameter functions remain
+constant and equal to the last estimated value. Figure 8 presents the observed Tether price
+and the estimate ˆyt+1 of the one-day-ahead conditional mean E(yt+1|yt, . . .) of the price of
+Tether using a rolling window. To get ˆyt+1, we add the local mean of Tether price to the
+estimated ˆφ using data over 50 days up to date t times xt. The figure shows a close match
+between Tether price and its best prediction based on the previous day price. In addition,
+the computed mean square prediction error is 1.7428 × 10−5 which is very small. Under
+the assumption of locally constant parameters, the 95 % asymptotically valid prediction
+intervals in Figure 9 are given by
+�
+ˆyt+1 − 1.96
+√
+T
+�
+x2
+t ˆΣ−1, ˆyt+1 + 1.96
+√
+T
+�
+x2
+t ˆΣ−1
+�
+.
+Full Sample
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+0.95
+0.96
+0.97
+0.98
+0.99
+1
+1.01
+1.02
+1.03
+1.04
+1.05
+y=1
+October 2018 to October 2019
+Jul 2018
+Oct 2018
+Jan 2019
+Apr 2019
+Jul 2019
+Oct 2019
+Jan 2020
+0.95
+0.96
+0.97
+0.98
+0.99
+1
+1.01
+1.02
+1.03
+1.04
+1.05
+y=1
+Figure 9: One-step ahead out-of-sample predicted Tether prices and prediction intervals
+Next, we conduct further investigations to analyze the goodness of fit of the tvDAR(1).
+For this analysis, we keep the estimation window of 50 and perform the Ljung-Box test of
+
+THIS VERSION: January 3, 2023
+35
+white noise on ˆηt and ˆη2
+t , while rolling the sample used for the test [see Li (1992), and Li
+and Mak (1994)]. Because we consider all subsamples of 50 consecutive dates, it is likely
+that at some dates the serial correlation is not fully captured by the tvDAR(1) model.
+Our results show that for most periods, residuals ˆηt and ˆη2
+t are serially uncorrelated. More
+precisely, we reject the null of no serial correlation for ˆηt only for 1.52% of the subsamples,
+while we reject the null of no serial correlation for ˆη2
+t only for 8.08% of the subsamples.
+The results suggest that the model captures most of the nonlinear serial correlation in
+Tether prices.
+5.2
+Epanechnikov kernel
+We now use a symmetric Epanechnikov kernel producing consistent and asymptotically
+normally distributed estimates for hypothesis testing.
+The first panel of Figure 10 plots the estimated time-varying autoregressive coefficient
+and its confidence band, while the second panel presents the time-varying estimates for
+of the Lyapunov exponent using the ˆλ2,T (t/T) estimator. For the bandwidth, we choose
+bT = 50/T using the same window as before.
+The results confirm that when the estimation is conducted locally over each period,
+the Lyapunov exponent displayed in the second panel of Figure 10 is negative. Hence the
+critical validity condition holds for all the dates. Moreover, the Lyapunov exponent is
+on average more negative at the end of the sampling period, confirming that Tether has
+achieved higher stability at the end of the sampling period.
+Furthermore, the estimated dynamic of estimated parameter φ for Tether price in the
+first panel of of Figure 10 remains similar to that of Figure 7. These findings confirm
+that Tether price has causal dynamics, as the autoregressive parameter and its confidence
+interval are mostly between −1 and 1. However, there are few dates when this estimate
+is not statistically different from 1, which suggests strong persistence in Tether prices.
+
+THIS VERSION: January 3, 2023
+36
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+=1
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+0
+Figure 10: Kernel-based parameter φ estimates and Lyapunov exponent λ(t/T)
+To detect the periods of strong persistence, we test the null hypothesis H0 : φ = 1
+using the series of ˆφ(t/T) and its 95% confidence intervals for each t = 1, ..., T.
+We
+conduct the hypothesis test using the series of ˆφ(t/T) obtained from the estimation with
+the Epanechnikov kernel and report the results in Table 2.
+The results suggest that although Tether price is predictable most of the time, there
+are intervals of time periods where this tends not to be the case. During these episodes
+presented in Table 2, we find high persistence in Tether price. However, as explained
+above, the proposed tvDAR approach remains valid and accommodates strong persistence
+in the price series. In light of the results in Figure 4, we further inspect the identified
+periods.
+We observed that the identified episodes in 2018 and 2019 overlap with the
+periods of high volatility observed in Figure 4, which ended in February 2020, but started
+around the introduction in September 2018 of USD Coin, another stablecoin designed to
+maintain price equivalence to the U.S. dollar. Moreover, the episodes in 2020 match with
+a small rise in Tether price volatility by the end of July 2020, while the episodes in 2021
+
+THIS VERSION: January 3, 2023
+37
+can be associated with the period of increased volatility at the end of our sample.
+Table 2: Episodes of high persistence in Tether characterized by φ = 1.
+Year
+From
+To
+2018
+September 24
+October 29
+December 3
+October 5
+2019
+January 11
+January 30
+February 9
+February 11
+February 16
+February 21
+2020
+July 25
+August 8
+August 16
+August 19
+2021
+May 14
+June 17
+5.3
+Test for Conditional Homoscedasticity
+Let us now consider a simple test of model specification of the tvDAR(1) model with
+time-varying parameters. It is based on testing for the constancy of the variance function
+in model 4.3 which is given by the following expression
+σ(t/T) =
+�
+ω(t/T) + α(t/T)x2
+t−1,T ,
+t = 1, ..., T.
+(5.8)
+To test the null hypothesis H0 : σ(t/T) = σ0 ∀t, we consider the below test statistics
+proposed by Chandler and Polonik (2017) for time-varying autoregressive processes
+CPT = sup
+α∈[0,1]
+�
+T
+(γ(1 − γ)| ˆGT,γ(α) − αγ|,
+(5.9)
+where
+• ˆGT,γ(α) = 1
+T
+�[αT]
+t=1 1(ˆϵ2
+t ≥ ˆq2
+γ),
+
+THIS VERSION: January 3, 2023
+38
+• ˆq2
+γ = min
+�
+q2 ≥ 0 : 1
+T
+�
+t∈[aT,bT] 1(ˆϵ2
+t > q2) ≤ γ
+�
+,
+• ˆϵt = xt − ˆφ( t
+T )xt−1.
+The process ˆGT,γ(α) counts the number of squared residuals within the first (100×α)%
+of the observations that are larger than the empirical quantile of the squared residuals
+denoted by ˆq2
+γ. The series of squared residuals is constructed by computing ˆϵt = xt −
+ˆφ( t
+T )xt−1 for each period t where ˆφ( t
+T ) in our case is the corresponding DAR(1) estimate
+obtained in the previous section.
+Chandler and Polonik (2017) shows that the test statistics CPT in equation (5.9) under
+the null hypothesis converges asymptotically to the supremum of a Brownian bridge. This
+asymptotic result is found to be still valid when the time-varying functions of the model
+parameters are estimated nonparametrically (see Chandler and Polonik (2012)).
+When computing the test statistics CPT , we consider different alternatives for the em-
+pirical upper γ-quantile for comparison, particularly γ = (0.7, 0.8, 0.9) following Chandler
+and Polonik (2017). Given the choice of γ, we then calculate the expression
+�
+T
+(γ(1−γ)| ˆGT,γ(α)−
+αγ| for different values of α and report the results in table 3.
+Table 3: The calculated values of
+�
+T
+(γ(1−γ)| ˆGT,γ(α) − αγ| for the given pairs of γ
+and α.
+α
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+0.7
+0.922
+1.302
+2.466
+3.328
+4.251
+5.957
+6.759
+6.837
+3.539
+γ
+0.8
+0.912
+1.478
+2.390
+3.233
+3.937
+5.471
+6.106
+6.327
+3.509
+0.9
+0.562
+1.031
+1.593
+2.155
+2.993
+3.923
+4.485
+5.047
+3.490
+The table shows that regardless of the choice of γ, the largest value is achieved when
+α = 0.8. By the definition of supremum, the test statistics CPT should satisfy CPT ≥
+�
+T
+(γ(1−γ)| ˆGT,γ(α)−αγ| for all α ∈ [0, 1]. Hence, it is safe to say that we have CPT ≥ 5.047
+in the worst case scenario, i.e when γ = 0.9 is chosen.19
+Compared with the critical
+19Alternatively, we could fix our choice of γ and find the exact value of α ∈ [0, 1] at which the expression
+�
+T
+(γ(1−γ)| ˆGT,γ(α) − αγ| attains its maximum on this fixed interval.
+This could slightly improve the
+
+THIS VERSION: January 3, 2023
+39
+values from the asymptotic distribution of the test statistics,20 this result leads us to the
+conclusion that we can reject the null hypothesis of constant variance function even at
+the 99% confidence level. In other words, we have a statistically significant evidence in
+favor of the alternative that the variance function is varying over time, which consequently
+justifies our strategy to estimate the model with time-varying parameters.
+6
+Concluding Remarks
+We show that the distributional and the dynamic properties of stablecoins have been
+evolving over the sampling period. We implement local analysis to detect and describe
+local explosive patterns, time-varying volatility and persistence. We model the dynamic of
+the most important stablecoin which is Tether, and provide evidence that the tvDAR(1)
+model with time varying coefficients provides locally a good fit and reliable short-term
+predictions of Tether prices. Our modelling strategy enables us to have valid inference
+even when the tvDAR(1) coefficient φt of Tether price is not locally different from 1.
+The sample Lyapunov exponent computed from the parameter estimates of the model
+provides a measure of stability. It confirms that at the end of the sampling period Tether
+becomes relatively more stable and allows for comparing the stability of Tether with other
+stablecoins.
+Appendix A: Technical Results
+This Appendix contains the proofs of Propositions 1, 2, 3 and 4.
+A.1
+Proposition 1
+Because of the symmetry of the density of η, the Lyapunov exponent is an even function
+of φ. Hence we can suppose that φ > 0 to find the expression of the Lyapunov exponent,
+and then replace φ by |φ|.
+For φ > 0 we have:
+precision of the lower bound for the test statistics. For example, when γ = 0.9, the expression attains its
+maximum value of 5.106 at α∗ = 0.8012.
+20see https://homepages.ecs.vuw.ac.nz/ ray/Brownian/ for the distribution of the supremum of a Brow-
+nian Bridge.
+
+THIS VERSION: January 3, 2023
+40
+λ(φ, α) = E ln |φ + η√α| =
+�
+ln |φ + η√α|ψ(α)dα.
+Let us assume that η ∼ U[−1,1]. Then, its density is ψ(η) = 1
+21η∈[−1,1].
+We have:
+λ(φ, α) = 1
+2
+� 1
+−1 ln |φ + η√α|dα.
+We observe that:
+φ + η√α > 0
+⇐⇒ η > −φ/√α,
+φ + η√α < 0
+⇐⇒ η < −φ/√α.
+Hence,
+λ(φ, α) = 1
+2
+� 1
+−1
+1η>−φ/√α ln(φ + η√α)dη + 1
+2
+� 1
+−1
+1η<−φ/√α ln(−φ − η√α)dη
+Let us now examine the two cases:
+a) If φ/√α > 1 ⇐⇒ −φ/√α < −1, we get:
+λ(φ, α)
+= 1
+2
+� 1
+−1 ln(φ + η√α)dη + 1
+20 = 1
+2
+1
+√α
+� 1
+−1 ln(φ + η√α)d(√αη)
+=
+1
+2√α
+� φ+√α
+φ−√α ln(u)du with change of variable u = φ + η√α
+=
+1
+2√αu ln(u) − u|φ+√α
+φ−√α
+=
+1
+2√α [(φ + √α) ln(φ + √α) − (φ + √α) − (φ − √α) ln(φ − √α) + (φ − √α)]
+b) If φ/√α < 1 ⇐⇒ −φ/√α > −1, we get:
+λ(φ, α) = 1
+2
+� 1
+−φ/√α ln(φ + η√α)dη + 1
+2
+� −φ/√α
+−1
+ln(−φ − η√α)dη
+= 1
+2
+1
+√α
+� φ+√α
+0
+ln(u)du + 1
+2
+� 1
+φ/√α ln(−φ + η√α)dη with the change of variable u = φ + η√α
+=
+1
+2√α
+� φ+√α
+0
+ln(u)du +
+1
+2√α
+� −φ+√α
+0
+ln(u)du
+=
+1
+2√α [(φ + √α) ln(φ + √α) − (φ + √α) − [(−φ + √α) ln(−φ + √α)] + (−φ + √α)]
+By putting the two expressions in a) and b) together we get for φ > 0:
+λ(φ, α) =
+1
+2√α
+�
+(φ + √α) ln(φ + √α) − (φ + √α) − |φ − √α| ln |φ − √α)| + |φ − √α|
+�
+The general expression of the Lyapunov Exponent without the sign constraint on φ is:
+λ(φ, α) =
+1
+2√α
+�
+(|φ| + √α) ln(|φ| + √α) − (|φ| + √α) − ||φ| − √α| ln ||φ| − √α)| + ||φ| − √α|
+�
+We observe a non-differentiability in φ = ±√α. Moreover, for φ = 0, we get a zero value
+of the Lyapunov exponent:
+
+THIS VERSION: January 3, 2023
+41
+λ(φ, α) =
+1
+2√α
+�√α ln √α − √α ln √α + √α
+�
+= 0
+A.2
+Proof of Proposition 2
+We need to show that when T → ∞, then
+�
+ln[ˆφT +η√ˆαT ]ψ0(η)dη →
+�
+ln(φ0+η√α0)ψ0(η)dη
+in probability if (ˆφT , ˆαT ) → (φ0, α0) in probability, and if condition A.6 of Proposition 2:
+∃δ > 0, such that
+�
+sup φ0 − δ < φ < φ0 + δ
+α0 − δ < α < α0 + δ
+| ln |φ + η√α||ψ0(η)dη < ∞
+(a.1)
+holds.
+Proof of convergence:
+If (ˆφT , ˆαT ) → (φ0, α0) in probability, then they also converge in distribution. It follows
+from the Skorokhod theorem that up to a change of probability space, we can assume that
+the almost sure (a.s.) convergence also holds [Billingsley (1999)]. Therefore, if g is a con-
+tinuous function of (φ, α), we have g(ˆφT , ˆαT ) → g(φ0, α0) a.s. and in distribution in that
+new space. Then, g(ˆφT , ˆαT ) d→ g(φ0, α0) in the initial space and also in probability because
+the limit is constant, we get the ”in probability” version of the continuous mapping theo-
+rem. Therefore, we need only the condition ensuring that g(φ, α) =
+�
+ln |φ+η√α|ψ0(η)dη
+is continuous. Condition (A.6) ensures the continuity of integral function g, which follows
+from the dominated convergence theorem.
+A.3
+Proof of Proposition 3
+We first prove a general lemma, which is next applied to the DAR model and Lyapunov
+estimator λ2,T .
+Lemma
+Let us consider a sequence GT (θ) of stochastic functions of θ, θ ∈ Θ, and a sequence
+of estimators ˆθT . We assume that:
+i) Θ is compact and θ0 is in the interior of Θ.
+ii) ˆθT tends in probability to θ0.
+iii) GT (θ) tends in probability to a limit G(θ), ∀θ ∈ Θ.
+iv) Sufficient Lipschitz condition for stochastic equicontinuity:
+
+THIS VERSION: January 3, 2023
+42
+There exists a stochastic sequence BT with BT = Op(1) and an increasing function
+h : [0, ∞) → [0, ∞) continuous at zero, with h(0) = 0 and such that for all ˜θ, θ ∈ Θ,
+|GT (˜θ) − GT (θ)| ≤ BT h(d(˜θ, θ)).
+Then, ˆGT (ˆθT ) tends in probability to G(θ0).
+Proof:
+We have :
+|GT (ˆθT ) − G(θ0)|
+= |GT (ˆθT ) − GT (θ0) + GT (θ0) − G(θ0)|
+≤ |GT (ˆθT ) − G(θ0)| + |GT (θ0) − G(θ0)|
+≤ BT h[d(ˆθT , θ0)] + |GT (θ0) − G(θ0)|.
+We know that if XT
+P→ 0, YT
+P→ 0 => XT + YT
+P→ 0, i.e. the sum of op(1) is op(1).
+Under condition iii) |GT (θ0) − G(θ0)| = op(1). It remains to be shown that BT h[d(ˆθT , θ0)]
+is op(1).
+We have:
+[BT < M and h[d(ˆθT , θ0)] < ϵ/M] => [BT h[d(ˆθT , θ0)]] < ϵ]
+⇐⇒
+�
+(BT < M) ∩ [h[d(ˆθT , θ0)] < ϵ/M]
+�
+⊂ [BT h[d(ˆθT , θ0)] < ϵ]
+Consider the complement: (A ∩ B)c = Ac ∪ Bc. We get:
+((BT > M) ∪ (h[d(ˆθT , θ0)] > ϵ/M) ⊃ [BT h[d(ˆθT , θ0)] > √ϵ]
+It follows that:
+P[BT h[d(ˆθT , θ0)] > ϵ]
+≤ P[(BT > M) ∪ (h[d(ˆθT , θ0)] > ϵ/M)]
+≤ P[BT > M] + P[h[d(ˆθT , θ0)] > h−1(ϵ/M)],
+because P[A ∪ B] ≤ P(A) + P(B).
+Then, for any ϵ, we can choose a value of M and a number of observations T sufficiently
+large to get P[BT h[d(ˆθT , θ0)] > ϵ] arbitrarily small. Therefore, BT h[d(ˆθT , θ0)] tends to
+zero in probability.
+QED
+Then, the lemma can be applied with GT (θ) = 1
+T
+�T
+t=2 g(xt, xt−1; θ) and g(xt, xt−1; θ) =
+ln |φ+ xt−φxt−1
+√
+ω+αx2
+t−1
+√α|. Under assumptions A.1, A.4, A.7, conditions i), ii), iii) are satisfied.
+For example:
+GT (θ) P→ E0g(xt, xt−1; θ),
+by the weak law of large numbers applied to the transformation g(xt, xt−1; θ) of the ergodic
+stationary process (xt). Assumption A.8 corresponds to condition iv) of the lemma.
+
+THIS VERSION: January 3, 2023
+43
+A.4
+Proof of Proposition 4
+a) Proof of convergence
+We need to show that ˆφ2
+T + ˆαT → φ2
+0 + α0 in probability if (ˆφ, ˆα) → (φ0, α0) in
+probability when T → ∞.
+Thi is a consequence of the ”in probability” version of the continuous mapping theorem
+given in Appendix A.2
+b) Proof of Normality
+The Taylor series expansion pre-multiplied by
+√
+T implies:
+√
+T
+�
+(ˆφ2
+T + ˆαT ) − (φ2
+0 + α0)
+�
+=
+� 2φ0
+1
+�′ √
+T
+� ˆφT − φ0
+ˆαT − α0
+�
++ op(1)
+= A′√
+T
+� ˆφT − φ0
+ˆαT − α0
+�
++ op(1)
+where A′ = [2φ0 1]. We get the asymptotic normal distribution of ˆξT :
+√
+T(ˆξT − ξ0) ∼ N(0, Vξ),
+where Vξ = A′Ω∗A. The matrix Ω∗ is: Ω∗ = diag(Σ−1, V (ˆα)) where Σ = E0(y2/(ω0+α0y2)
+given in Section 4.3.1.
+Matrix V (ˆα) = (E0
+1
+(ω0+α0y2)2 )/ ˜V0(y2) and ˜V0(y2) = ˜E0(y4) −
+( ˜E0y2)2. In this formula, ˜E0 denotes the expectation of variables y4
+1
+(ω0+α0y2)2 /E0
+1
+(ω0+α0y2)2
+and y2
+1
+(ω0+α0y2)2 /E0
+1
+(ω0+α0y2)2 [see, section 4.3.1 and Ling (2004) for the variance estima-
+tor formula].
+Appendix B: Simulation Results
+The purpose of this section is to illustrate the derived results in Appendix A using simu-
+lation experiments. We distinguish the case where the distribution of the innovation η is
+known and the case it is not.
+First, we use the result in Proposition 1 and plot the Lyapunov exponent λ = E(ln(|φ+
+√αη|)) for different values of the parameter φ and α. To do so, we assume η ∼ U[−1, 1].
+Figure B1 shows that the Lyapunov exponent λ remains lower than zero as as φ varies in
+{−1, −0.8, −0.6, . . . , 0.6, 0.8, 1} and α varies in {0, 0.1, 0.2, 0.3, . . . , 0.8, 0.9, 1}.
+Second, we assume η ∼ N(0, 1), set the true parameters to φ0 = 0.7, α0 = 0.5,
+ω0 = 0.01. Note that the parameters are chosen to be close to their estimated value from
+the entire data in our application. The estimated densities are based on 4, 000 simulations
+
+THIS VERSION: January 3, 2023
+44
+and obtained via kernel density estimation. Figure B2 plots the estimated density for
+the Lyapunov exponent ˆλ2,T and the stability measure ˆξT when the three parameters are
+estimated from a sample of size T and plugged in. The results on panel (a) of the figure
+show that the mostly frequent estimated value is below zero for T = 50 or T = 100,
+implying valid inference. In addition, panel (b) of Figure B2 shows that the density of the
+estimated alternative stability measure ˆξT = ˆφ2
+T + ˆαT has its mode around the true value
+of ξ, which is ξ0 = φ2
+0 + α0 = 0.99.
+As a by-product, we present Figure B3, which shows the estimated density for ˆφT , ˆαT
+and ˆωT for T = 50 and T = 100. The three panels in the figure provide evidence that the
+three parameters are fairly accurately estimated. More specifically, the estimated values
+have modes close to their true unknown parameters. The accuracy improves as the sample
+size increases from T = 50 to T = 100.
+Figure B1: Lyapunov exponent λ = E(ln(|φ + √αη|)) in terms of φ and α when η ∼
+U[−1, 1]
+
+0
+-2
+-4 ~
+-6
+1
+0.8
+0.6
+0.4
+0
+0.2
+-0.5
+0
+-11
+0.5THIS VERSION: January 3, 2023
+45
+-3.5
+-3
+-2.5
+-2
+-1.5
+-1
+-0.5
+0
+0.5
+0
+0.5
+1
+1.5
+2
+2.5
+Density
+T=50
+T=100
+(a) Density of the Lyapunov exponent ˆλ2,T
+-0.5
+0
+0.5
+1
+1.5
+2
+2.5
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Density
+T=50
+T=100
+(b) Density of the alternative stability measure ˆξT
+Figure B2: Densities for stability measures based on estimated parameters
+
+THIS VERSION: January 3, 2023
+46
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+Density
+T=50
+T=100
+0
+(a) Density of ˆφT
+-0.005
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Density
+T=50
+T=100
+0
+(b) Density of ˆωT
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+0
+0.5
+1
+1.5
+2
+2.5
+3
+Density
+T=50
+T=100
+0
+(c) Density of ˆαT
+Figure B3: Densities of estimators for φ, α and ω
+
+THIS VERSION: January 3, 2023
+47
+Appendix C: More Empirical Results
+Given that the proposed stability measure can be used as a mechanical tool to detect
+periods of instability in stablecoins, we use the results of Proposition 4 in Appendix A to
+construct an interval for ξ employing the same rolling window approach as before. Figure
+C1 presents the results and contains, in its first panel, the estimated coefficient DAR model
+using the rolling windows approach, in its second panel, the conditional heteroskedasticity,
+and in the third panel, the Lyapunov exponent over time. In addition to the episodes of
+high persistence mentioned above, we observed, around September 2020, an important
+instability that is not due to high persistence in Tether price, but more frequent changes
+in the conditional heteroskedasticity, which can be seen in the second panel. This period
+can also be linked to higher local volatility in the observed data in Figure 4. There is no
+specific event we can associate with this movement, as is sometimes the case in crypto
+markets. However, the proposed model allows capturing those changes.
+As explained above, the measure of stability ξ plotted in Figure C1 is more conservative
+than the Lyapunov exponent λ. Because we can have ξ ≥ 1 while λ < 0 so that valid
+inference is still possible, the rejection of ξ < 1 should be interpreted as the need for
+investors, regulators or stablecoin issuers to be cautious when predicting Tether future
+price around the tested periods.
+
+THIS VERSION: January 3, 2023
+48
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+-1
+-0.5
+0
+0.5
+1
+1.5
+=1
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+0.012
+0.014
+Conditional Volatility
+Jul 2017
+Jan 2018
+Jul 2018
+Jan 2019
+Jul 2019
+Jan 2020
+Jul 2020
+Jan 2021
+Jul 2021
+Jan 2022
+-4
+-2
+0
+2
+4
+6
+Figure C1: tvDAR(1) parameter φ(t/T) and Lyapunov exponent ξ(t/T)
+
+THIS VERSION: January 3, 2023
+49
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