Time-Varying Coefficient DAR Model and Stability Measures for Stablecoin
Prices: An Application to Tether

Antoine Djogbenou,111York University, Canada, e-mail<EMAIL_ADDRESS>Emre
Inan,222York University, Canada, e-mail<EMAIL_ADDRESS>Joann Jasiak333York
University, Canada, e-mail<EMAIL_ADDRESS>
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).

This Version:

Abstract

This paper examines the dynamics of Tether, the stablecoin with the largest
market capitalization. We show that the distributional and dynamic properties
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
patterns, we consider a time varying parameter Double Autoregressive tvDAR(1)
model under the assumption of local stationarity of Tether/USD rates. We
estimate the tvDAR model non-parametrically and test hypotheses on the
functional 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.

##  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 innovation. 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
concentration 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 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
coefficient 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 stationarity 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 kernel. 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
stablecoins, 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 distinguishes our approach from the literature that relies on the
assumption of global strong stationarity of the series. For instance, Baumöhl
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 distributed
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 discusses the local dynamic analysis of the price of Tether. Section
4 discusses the modelling approach, estimation procedures and stability
measures. Section 5 presents the empirical 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.

##  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. Conventionally, 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 unlimited 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 provide 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 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 stablecoin 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
responsibility 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
requirements. 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 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 stability. 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.111See Griffin and Shams (2020) for further discussion on whether Tether
issuances are supply-driven or demand-driven. 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 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 transactions 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
whitepapers 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.222See Lyons and Viswanath-Natraj (2020) for the detailed analysis
of premium and discount on stablecoin prices. 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 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.333https://tether.to/fees/

The stabilization strategies of Tether have not always been fully successful.
The next 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.

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.

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 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
2021.444https://www.spglobal.com/spdji/en/indices/digital-assets/sp-
cryptocurrency-broad-digital-market-index/#overview 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
2021.555https://www.statista.com/statistics/1255835/stablecoin-market-
capitalization/

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.666https://www.forbes.com/sites/jonathanponciano/2021/04/18/crypto-
flash-crash-wiped-out-300-billion-in-less-than-24-hours-spurring-massive-
bitcoin-liquidations/?sh=7d60735b2c89 While the waves 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 example,
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.777https://www.theguardian.com/technology/2021/may/19/bitcoin-
falls-30-after-china-crackdown 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.

Figure 3: Local mean and variance for the price of Tether

The locally estimated marginal mean $\mu_{t}$ and variance $\sigma_{t}^{2}$
are displayed in panels a) and b) of Figure 3.888 The lower and upper bounds
of the confidence interval at the 95% level are calculated under the iid
assumption as $\hat{\mu}_{t}\mp 1.96\sqrt{\frac{\hat{\sigma}_{t}}{n}}$ for
each window of n days.

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, 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
interpreted 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.999Also, note
that n=50 is large enough to estimate parameters within each window
consistently. Furthermore, n=50 divided by the sample size of T=1361 is the
bandwidth $b_{T}=n/T=0.0367$ in our local analysis, which will be discussed
later. In the literature, an optimal choice should satisfy $Tb_{T}^{3}=o(1)$.
In our case, we have $Tb^{3}_{T}=0.0675$, which is relatively small. See
Dahlhaus, Richter, Wu (2019, page 1039) for more details.

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 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.

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 submission 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.

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.101010https://www.bloomberg.com/news/articles/2018-12-18/crypto-
mystery-clues-suggest-tether-has-the-billions-it-promised 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,111111https://www.coindesk.com/markets/2019/03/14/tether-says-
its-usdt-stablecoin-may-not-be-backed-by-fiat-alone/ 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.121212https://ag.ny.gov/press-
release/2019/attorney-general-james-announces-court-order-against-crypto-
currency-company

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.131313https://www.forbes.com/sites/michaeldelcastillo/2019/09/24/bitfinex-
and-tether-win-appeal-from-new-york-supreme-court-in-900-million-
case/?sh=7c8bf41132bc 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.141414see
https://twitter.com/who/status/1213795226072109058?lang=en for the original
tweet from WHO With this announcement, WHO acknowledged the problem of an
epidemic in China, which would soon turn into a global health and economic
crisis of 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.

Figure 5: Autocorrelation functions of Tether over different periods

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 2019,151515In 2017, there are only 53
observations, which could be the reason for the weak evidence for the
persistence. 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.

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
autocorrelation 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 $x_{t}=y_{t}-\mu$

$x_{t}=\rho x_{t-1}+\sigma_{e}e_{t},$ (3.1)

where $e_{t}$ is assumed to be a white noise with mean 0 and variance 1. The
AR(1) process is stationary when $|\rho|<1$ and nonstationary and explosive
when $\rho=1$. Model (3.1) is estimated globally by the OLS, or equivalently
by maximizing the Gaussian Maximum Likelihood as follows:

$\hat{\theta}_{T}=Argmax_{\theta}\sum_{t=1}^{T}l(x_{t}|x_{t-1};\theta)=Argmax_{\theta}\sum_{t=2}^{T}-\frac{1}{2}\left(\log\left(2\pi\sigma_{e}^{2}\right)+\frac{\left(x_{t}-\rho
x_{t-1}\right)^{2}}{\sigma_{e}^{2}}\right)\,$

where $\theta=(\rho,\sigma_{e}^{2})$.

The estimated parameter values are $\hat{\rho}_{T}=0.674$ and
$\hat{\sigma}_{e,T}^{2}=0.545$. Next, model (3.1) is fitted locally and
estimated by rolling with a window of length 50 and displayed in Figure 6.

Figure 6: AR(1) parameter estimates and conditional volatility for $x_{t}$

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 example, the estimated values of the AR(1)
coefficient $\rho$ 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
$\hat{\sigma}_{e}(t)=\sqrt{\frac{1}{50}\sum_{\tau=t-49}^{t}(x_{\tau}-\hat{\rho}(\tau)\,x_{\tau-1})^{2}}$
of the price of Tether under the AR(1) assumption. The figure shows that this
price exhibits 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 volatility model which also allows us to have valid inference
when the estimated correlation parameter is close to one unlike the AR(1)
model.

###  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 $\sigma^{2}(t)$ in a simple model

$y_{t}=m(t)+\sigma(t)u_{t},$

under the simplifying assumption of Normally distributed i.i.d. process
$u_{t}$ with mean 0 and variance 1. Then, in finite sample

$\hat{m}(t)=\frac{1}{50}\sum_{\tau=t-49}^{t}y_{\tau}\sim
N\left(\frac{m(t)+\cdots+m(t-49)}{50},\frac{\sigma^{2}(t)+\cdots+\sigma^{2}(t-49)}{250}\right)$

We see that $\hat{m}(t)$ is biased of m(t) towards an integrated mean. By the
same argument it can be shown that $\hat{\sigma}^{2}(t)$ is biased of both
$\sigma^{2}(t)$ and the integrated variance.

Let us now consider the time varying parameters
$\theta(t)=(\rho(t),\sigma^{2}_{e}(t))$ of a time varying parameter AR(1)
model:

$x_{t}=\rho(t)x_{t-1}+\sigma_{e}(t)e_{t},$

where $x_{t}=y_{t}-m(t)$. Then, the normality-based MLE estimator
$\hat{\theta}^{*}(t)$ 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).

$\displaystyle\hat{\theta}^{*}_{T}(t)$ $\displaystyle=$ $\displaystyle
Argmax_{\theta}\frac{1}{50}\sum_{\tau=t-49}^{t}\,l(x_{\tau}|x_{\tau-1};\theta)$
$\displaystyle=$ $\displaystyle
Argmax_{\theta}\frac{1}{50}\sum_{\tau=1}^{T}1_{t-49\leq\tau\leq
t}\,l(x_{\tau}|x_{\tau-1};\theta)$ $\displaystyle=$ $\displaystyle
Argmax_{\theta}\sum_{\tau=1}^{T}\left[\frac{1}{50}1_{-49\leq\tau-t\leq
0}\,l(x_{\tau}|x_{\tau-1};\theta)\right]$ $\displaystyle=$ $\displaystyle
Argmax_{\theta}\sum_{\tau=1}^{T}\frac{1}{50}\,1_{-49/50\leq(\tau-t)/50\leq
0}\,l(x_{\tau}|x_{\tau-1};\theta)$ $\displaystyle=$ $\displaystyle
Argmax_{\theta}\sum_{\tau=1}^{T}\frac{1}{50}\,K\left(\frac{\tau-t}{50}\right)\,l(x_{\tau}|x_{\tau-1};\theta),\;t=1,...,T$

with the kernel $K(u)=1_{[-1,0]}(u)$.

It is easy to see that the dimension of the parameter of interest
$[\theta(1),...,\theta(T)]$ depends on the number of observations T. To
circumvent this difficulty, we can replace the functional parameter
$[\theta^{*}(1),...,\theta^{*}(T)]$ with $t\in N$ by an alternative functional
parameter $\theta(c),c\in(0,1)$ on [0,1], such that
$\theta^{*}(t)=\theta(t/T)$. The rolling MLE is such that

$\hat{\theta}_{T}^{*}(t)=\hat{\theta}_{T}(t/T)=Argmax_{\theta}\sum_{\tau=1}^{T}\frac{T}{50}\,K\left(\frac{\tau/T-t/T}{50/T}\right)\,l(x_{\tau}|x_{\tau-1};\theta),\;t=1,...,T.$

This formula can be extended to any value of argument $c\in[0,1]$:

$\hat{\theta}_{T}(c)=Argmax_{\theta}\sum_{\tau=1}^{T}\frac{T}{50}\,K\left(\frac{\tau/T-c}{50/T}\right)\,l(x_{\tau}|x_{\tau-1};\theta),\;c\in[0,1].$

Dahlhaus (2000) and Dahlhaus, Richter, Wu (2019) show that under regularity
conditions the functional parameters $\theta(c)$ 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 $t\rightarrow t/T$ 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:

$x_{t,T}=\rho(t/T)x_{t-1,T}+\sigma_{e}(t/T)e_{t,T},$

where $x_{t,T}=y_{t,T}-m(t/T)$.

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.

###  4.1 Model with Constant Parameters

We consider the DAR process of order 1 for the demeaned Tether price series:

$x_{t}=\phi x_{t-1}+\eta_{t}\sqrt{\omega+\alpha x_{t-1}^{2}},$ (4.2)

where $\omega>0,\alpha>0$, and $\eta_{t},t=1,...,T$ is an independent and
identically distributed (i.i.d.) sequence with mean 0 and variance 1. The
parameter $\phi$ captures the conditional mean dependence. Parameter $\alpha$
represents the past dependence in the conditional variance. The model is semi-
parametric and conditionally heteroskedastic.161616More 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. 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: $\eta_{t}$ has a symmetric and continuous density with mean 0
and variance 1.

Assumption A.2: The parameter space is
$\Theta=\\{\theta=(\phi,\omega,\alpha):E(ln|\phi+\eta_{t}\sqrt{\alpha}|)<0$
with
$|\phi|\leq\tilde{\phi},\underline{\omega}\leq\omega\leq\tilde{\omega},\underline{\alpha}\leq\alpha\leq\tilde{\alpha}$
where
$\tilde{\phi},\underline{\omega},\tilde{\omega},\underline{\alpha},\tilde{\alpha}$
are positive constants.

Assumption A.1 is not a stringent assumption. It is satisfied in particular if
$\eta_{t},t=1,...,T$ 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
$\phi,\alpha$ that satisfy the negativity condition is displayed in Figure 1,
p. 64 Ling (2004) and Figure 1, p. 191 Chen et al. (2014). It includes cases
when $\phi\geq 1$ as well as $E(x_{t}^{2})=\infty$. The processes $x_{t}$ that
satisfy Assumption 2 are strictly stationary. Some of these processes are also
weakly (second-order) stationary and satisfy additionally the condition
$\phi^{2}+\alpha<1$ ensuring that $E(x_{t}^{2})<\infty$. Thus, the marginal
variance of those processes is finite.

When $\phi=1$, and $E(\ln|1+\eta_{t}\sqrt{\alpha}|)<0$, the process $x_{t}$ is
a strictly stationary martingale 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 $(\phi,\alpha)=(\pm 1,0)$ and nests the standard unit root
models at these two points. When $\phi=0$, the process is an ARCH(1) model.
Moreover, process $(x_{t})$ is strictly stationary when $x_{0}$ is drawn from
a stationary distribution.

Under assumptions A.1 and A.2, the parameter space $\Theta$ is compact and
there exists a unique strictly stationary solution of the model for any
$\theta\in\Theta$. 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
$\theta_{0}=(\phi_{0},w_{0},\alpha_{0})$ and the true density $\psi_{0}$ of
$\eta$. The true parameter value $\theta_{0}$ is an interior point in
$\Theta$.

Assumption A.4: The observed process is the unique, strictly stationary
solution associated with $(\theta_{0},\psi_{0})$.

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 $x_{t,T},t=1,...,T$ indexed by $t$
and $T$ (triangular array) and defined by:

$x_{t,T}=\phi(t/T)x_{t-1,T}+\eta_{t,T}\sqrt{\omega(t/T)+\alpha(t/T)x_{t-1,T}^{2}},$
(4.3)

where for each time $T$, $(\eta_{t,T})$ is a strong (i.i.d) white noise with
mean zero, unit variance and a symmetric distribution invariant in $T$.
$\phi(c),\omega(c)>0,\alpha(c)>0$, $c\in[0,1]$ are deterministic functions. We
assume that these functions are smooth.

Assumption a.1: The functions $\phi(.),\omega(.),\alpha(.)$, 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 $x_{t}(c)$ defined by:

$x_{t}(c)=\phi(c)x_{t-1}(c)+\eta_{t}(c)\sqrt{\omega(c)+\alpha(c)x_{t-1}(c)^{2}},$
(4.4)

We assume that the following condition holds:

Assumption a.2:

Let the $L_{q}$ norm for $q>0$ be denoted by $||.||_{q}$. Then,

i) For each $c\in(0,1)$, process $\\{x_{t}(c)\\}$ is stationary and ergodic.

ii) $c\rightarrow x_{t}(c)$ is continuous for any $t$ and
$||sup_{c}x_{t}(c)||_{q}<\infty$.

iii) There exists $\alpha,1\geq\alpha>0$ and $C_{B}>0$, such that

$||x_{t}(c)-x_{t}(c^{\prime})||_{q}<C_{B}|c-c^{\prime}|^{\alpha}$ uniformly in
$t$ and $c,c^{\prime}\in(0,1)$.

Under Assumptions a.1 and a.2, if $T$ is large and $t/T$ in a small interval
$(c-\epsilon,c)$, then the parameters are almost constant over that interval
and locally model (4.4) is close to model (4.3) with
$\phi=\phi(c),\omega=\omega(c),\alpha=\alpha(c)$. This explains the local
stationarity.

When all the observations $x_{t,T},t=1,...,T$ are considered, the variation of
the parameters prevents the DAR process from being globally stationary.
However, it is locally stationary, if Assumption A.2 is locally satisfied,
i.e. $E[ln|\phi(c)+\eta_{t}(c)\alpha(c)|]<0$ 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 distributed $\eta_{t}$.

$L_{T}(\theta)=-\frac{1}{2}\sum_{t=2}^{T}\ln\left(\omega+\alpha
x_{t-1}^{2}\right)-\frac{1}{2}\sum_{t=2}^{T}\frac{\left(x_{t}-\phi
x_{t-1}\right)^{2}}{\left(\omega+\alpha x_{t-1}^{2}\right)},$ (4.5)

where $\theta=[\phi,\ \omega,\ \alpha]^{\prime}$. The QML estimators of Model
(4.2):

$\hat{\theta}_{T}=Argmax_{\theta\in\Theta}L_{T}(\theta)$

are consistent under Assumptions A.1 to A.4, and the vector of QMLE estimators
$\hat{\theta}_{T}=[\hat{\phi}_{T}\ \hat{\omega}_{T}\
\hat{\alpha}_{T}]^{\prime}\rightarrow\theta_{0}$ in probability, where
$\theta_{0}=[\phi_{0}\ \omega_{0}\ \alpha_{0}]^{\prime}$ [Ling (2004,2007)].
Moreover, if the following assumption:

Assumption A.5: $E(\eta_{t}^{4})<\infty$

is satisfied, Li and Ling (2008) and Chen, Li and Ling (2014) show that the
Quasi Maximum Likelihood estimators (QMLE) of $\theta$ are also asymptotically
normal when $|\phi|\geq 1$ 171717The ML/OLS estimators of $\phi$ from a linear
autoregressive AR(1) model with constant parameters are not asymptotically
normal when $\phi=1$. as well as $E(x_{t}^{2})=\infty$.

$\sqrt{T}(\hat{\theta}_{T}-\theta_{0})\rightarrow
N(0,diag(\Sigma^{-1},\kappa\Omega^{-1}))$

where this convergence is in distribution,
$\Sigma=E_{0}[x_{t-1}^{2}/(\omega_{0}+\alpha_{0}x_{t-1}^{2})]$

$\Omega=E_{0}\left(\frac{1}{(\omega_{0}+\alpha_{0}x_{t-1}^{2})^{2}}\left[\begin{array}[]{cc}1&x_{t-1}^{2}\\\
x_{t-1}^{2}&x_{t-1}^{4}\end{array}\right]\right),$

$diag(\Sigma^{-1},\kappa\Omega^{-1})$ denotes the block-diagonal matrix with
$\Sigma^{-1}$ as the upper left block and $\kappa\Omega^{-1}$ as the bottom
right block and $\kappa$ is the kurtosis less 1 of the distribution of $\eta$.
In particular, $\kappa=2$ when $\eta_{t}$ is normal. These asymptotic results
are valid for any true distribution of $\eta$, not necessarily a Gaussian
distribution. The consistent estimators of $\Sigma$ and $\Omega$ are

$\hat{\Sigma}_{T}=\frac{1}{T-1}\sum_{t=2}^{T}[x_{t-1}^{2}/(\hat{\omega}_{T}+\hat{\alpha}_{T}x_{t-1}^{2})],$

$\hat{\Omega}_{T}=\frac{1}{T-1}\sum_{t=2}^{T}\frac{1}{(\hat{\omega}_{T}+\hat{\alpha}_{T}x_{t-1}^{2})^{2}}\left[\begin{array}[]{cc}1&x_{t-1}^{2}\\\
x_{t-1}^{2}&x_{t-1}^{4}\end{array}\right].$

The model residuals are defined as:

$\hat{\eta}_{t,T}=(x_{t}-\hat{\phi}_{T}x_{t-1})/\sqrt{\hat{\omega}_{T}+\hat{\alpha}_{T}x_{t-1}^{2}}.$

The model residuals $\hat{\eta}_{t,T},t=1,...,T$ allow us to estimate non-
parametrically the error density to verify ex-post the symmetry assumption.
The parameter $\kappa$ is estimated by
$\hat{\kappa}_{T}=\frac{1}{T-1}\sum_{t=2}^{T}\hat{\eta}_{t,T}^{2}-1$
$=\frac{1}{T-1}\sum_{t=2}^{T}\frac{\left(x_{t}-\hat{\phi}x_{t-1}\right)^{4}}{\left(\hat{\omega}+\hat{\alpha}x_{t-1}^{2}\right)^{2}}-1$
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 $x_{t,T},t=1,...,T$ is non-parametric and depends on the
functional parameters $\phi(c),\omega(c)>0,\alpha(c)>0,c\in[0,1]$ and on the
density function of the noise $\eta_{t,T}$. The estimation of
$\phi(.),\omega(.),\alpha(.)$ can be done by the local-in-time QML estimators.
We consider a kernel $K$ defined on $[-1/2,1/2]$ and bandwith
$b_{T},\,b_{T}>0$ (following the notation used in Dahlhaus, Richter, Wu
(2019), p. 1035). The local negative log-conditional quasi likelihood is

$L_{T,b}(c,\phi,\omega,\alpha)=\frac{1}{Tb_{T}}\sum_{t=2}^{T}K\left(\frac{t/T-c}{b_{T}}\right)\left[-\frac{1}{2}\ln\left(\omega+\alpha
x_{t-1,T}^{2}\right)-\frac{1}{2}\frac{\left(x_{t,T}-\phi
x_{t-1,T}\right)^{2}}{\left(\omega+\alpha x_{t-1,T}^{2}\right)}\right],$ (4.6)

Then, the local negative QML estimator of
$\theta(c)=[\phi(c),\omega(c),\alpha(c)]$ is:

$\hat{\theta}_{T,b}(c)=Argmax_{\theta}L_{T,b}(c,\phi,\omega,\alpha).$ (4.7)

Under suitable regularity conditions given in [Dahlhaus, Richter, Wu (2019)],
this functional QML estimator $\hat{\theta}_{T,b_{T}}(c),\;c\in[0,1]$ is
consistent of $\theta(c),c\in[0,1]$ and its limiting distribution is normal:

$\sqrt{Tb_{T}}(\hat{\theta}_{b}(c)-\theta_{0}(c))\rightarrow N(0,\int
K(y)^{2}dy\,J(c)^{-1}I(c)J(c)^{-1}),$

where $\rightarrow$ 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 $\eta_{t}$ and for a
strictly stationary and ergodic $x_{t}$, 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 $\eta_{t}$, the kernel $K$ and the bandwidth $b_{T}$. 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
$b_{T}\rightarrow 0,Tb_{T}\rightarrow\infty,Tb_{T}^{3}\rightarrow 0$ when $T$
tends to infinity.

###  4.4 Stability measures

The Lyapunov exponent measures the average logarithmic rate of separation or
convergence of initially close trajectories in chaotic systems, and the
sensitivity to initial conditions, 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 [Sprott
(2003)]. In this section, the Lyapunov exponent is proposed as a measure of
stability of Tether and other stable coins. In the framework of the DAR model,
the Lyapunov exponent is:

$\lambda=E(\ln|\phi+\eta\sqrt{\alpha}|).$

The more negative the Lyapunov exponent, the less explosive the process181818A
strictly stationary process can have infinite moments. and more likely its
marginal variance is finite.

The behavior of $E(\ln|\phi+\eta_{t}\sqrt{\alpha}|)$ as a function of
$\phi,\alpha$ can be examined analytically for selected densities of $\eta$,
and/or simulated and illustrated graphically [see, Liu et al. (2018) for
graphical illustration]. Appendix A.1 shows the analytical formula of
$\lambda$ for a uniformly distributed sequence $\\{\eta_{t}\\}$. More
precisely:

Proposition 1: If $\eta\sim U_{[-1,1]}$, the Lyapunov exponent is given by:

$\lambda(\phi,\alpha)=\frac{1}{2\sqrt{\alpha}}\left[(|\phi|+\sqrt{\alpha})\ln(|\phi|+\sqrt{\alpha})-(|\phi|+\sqrt{\alpha})-||\phi|-\sqrt{\alpha}|\ln||\phi|-\sqrt{\alpha}|+||\phi|-\sqrt{\alpha}|\right]$

Proof: See Appendix A.1.

This example clarifies that the Lyapunov exponent is a continuous function of
$\phi,\alpha$, although with points of non-differentiability. Moreover, it is
easy to show that that $E(\ln|\phi+\eta_{t}\sqrt{\alpha}|)$ is always an even
function of $\phi$, i.e. it takes the same value for $\phi$ and $-\phi$. To
see that, consider a symmetric density function $\psi(\eta)$. Then,

$E(\ln|-\phi+\eta_{t}\sqrt{\alpha}|)=\int(\ln|-\phi+\eta_{t}\sqrt{\alpha}|)\psi(\eta)d\eta.$

Because $\psi(\eta)$ is symmetric, we can change the variable
$\eta\rightarrow-\eta$:

$E(ln|-\phi+\eta_{t}\sqrt{\alpha}|)=\int(\ln|-\phi-\eta_{t}\sqrt{\alpha}|)\psi(-\eta)d\eta=\int(\ln|\phi+\eta_{t}\sqrt{\alpha}|)\psi(\eta)d\eta=E(\ln|\phi+\eta_{t}\sqrt{\alpha}|).$

This proves that the Lyapunov exponent is even in $\phi$. 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 $\psi=\psi_{0}$ is known. Then, the
estimator $\hat{\lambda}_{1,T}$ of the Lyapunov exponent is:

$\hat{\lambda}_{1,T}=\int\ln|\hat{\phi}_{T}+\eta\sqrt{\hat{\alpha}_{T}}|\,\psi_{0}(\eta)\,d\eta.$

Proposition 2:

When the density $\psi(\eta)=\psi_{0}(\eta)$ is known, then under assumptions
A.1 to A.4 and the following condition:

(A.6) $\;\;\;\exists\delta>0,\mbox{such that}\int
sup_{\begin{array}[]{c}\phi_{0}-\delta<\phi<\phi_{0}+\delta\\\
\alpha_{0}-\delta<\alpha<\alpha_{0}+\delta\end{array}}|\ln|\phi+\eta\sqrt{\alpha}||\psi_{0}(\eta)d\eta<\infty\;\;\;$

the estimator $\hat{\lambda}_{1,T}$ converges in probability to the true value
$\lambda_{0}=\int\ln|\phi_{0}+\eta\sqrt{\alpha}_{0}|\psi_{0}(\eta)d\eta$ of
the Lyapunov exponent when $T\rightarrow\infty$.

Proof: See Appendix A.2

b) When the density $\psi(\eta)$ is unknown, the model is semi-parametric. The
Lyapunov exponent can be estimated by the estimator $\hat{\lambda}_{2,T}$ such
that:

$\hat{\lambda}_{2,T}=\frac{1}{T}\sum_{t=1}^{T}\ln|\hat{\phi}_{T}+\hat{\eta}_{t,T}\sqrt{\hat{\alpha}_{T}}|.$

Therefore this estimator is equal to :

$\hat{\lambda}_{2,T}=\frac{1}{T}\sum_{t=1}^{T}\ln\left|\hat{\phi}_{T}+\frac{x_{t}-\hat{\phi}_{T}x_{t-1}}{\sqrt{\hat{\omega}_{T}+\hat{\alpha}_{T}x_{t-1}^{2}}}\sqrt{\hat{\alpha}_{T}}\right|=\frac{1}{T}\sum_{t=1}^{T}g(x_{t},x_{t-1};\hat{\theta}_{T})$

where $g(x_{t},x_{t-1};\theta)=ln\left|\phi+\frac{x_{t}-\phi
x_{t-1}}{\sqrt{\omega+\alpha x_{t-1}^{2}}}\sqrt{{\alpha}}\right|$. We also
introduce the notation
$G_{T}(\theta)=\frac{1}{T}\sum_{t=1}^{T}g(x_{t},x_{t-1};\theta)$.

Proposition 3:

Let us introduce the additional conditions:

(A.7) $E_{\theta_{0}}g(x_{t},x_{t-1};\theta)<\infty,\;\forall\theta\in\Theta$.

(A.8) Sufficient Lipschitz condition for stochastic equicontinuity: There
exists a stochastic sequence $B_{T}$ with $B_{T}=O_{p}(1)$ and an increasing
function $h$ from $[0,\infty)$ to $[0,\infty)$, continuous at 0 with $h(0)=0$,
such that for all $\tilde{\theta}$, $\theta\in\Theta$,
$|G_{T}(\tilde{\theta})-G_{T}(\theta)|\leq B_{T}h(d(\tilde{\theta},\theta))$.

Then, under assumptions A.1-A.4 and conditions A.7, A.8 the estimator
$\hat{\lambda}_{2,T}=G_{T}(\hat{\theta}_{T})\rightarrow
G(\theta_{0})=\lambda_{0}$ in probability.

Proof: See Appendix A.3.

The asymptotic distributions of estimators $\hat{\lambda}_{1,T}$,
$\hat{\lambda}_{2,T}$ cannot be obtained asymptotically from the Taylor series
expansion because function
$\phi,\alpha\rightarrow\int\ln|\phi+\eta\sqrt{\alpha}|\psi(\eta)d\eta$ does
not satisfy the necessary differentiability assumption, as pointed out in
Proposition 1. However, the distribution of $\hat{\lambda}_{1,T}$,
$\hat{\lambda}_{2,T}$ can be determined by simulations and used for hypothesis
testing.

c) An alternative stability measure is $\xi=\phi^{2}+\alpha$, which depicts
the region of parameter space $\xi<1$ where the marginal variance of $x_{t}$
remains finite, so that the process is both strictly and weakly stationary. In
that sense $\xi$ is a more conservative measure of stability than the Lyapunov
exponent because there is a region of parameter values $\phi,\alpha$ where the
condition $\xi<1$ no longer holds, while the condition $\lambda<0$ remains
satisfied. The estimator $\hat{\xi}_{T}$ of $\xi$ is:

$\hat{\xi}_{T}=\hat{\phi}_{T}^{2}+\hat{\alpha}_{T}.$

Proposition 4:

Under Assumptions A.1-A.5, the estimator $\hat{\xi}_{T}$ converges in
probability to $\xi_{0}$ when $T\rightarrow\infty$ and it is asymptotically
Normally distributed:

$\sqrt{T}(\hat{\xi}_{T}-\xi_{0})\stackrel{{\scriptstyle\rm
A}}{{\rm\sim}}N(0,V_{\xi}),$

where the formula of the asymptotic variance $V_{\xi}$ is given in Appendix
A.4.

The asymptotic variance provides the asymptotically valid standard errors that
can be used to test the null hypothesis $\xi<\xi_{0}$ using a Wald test
statistic. The interpretation of measure $\xi$ is similar to that of
$\lambda$: the smaller $\xi$, the more stable $x_{t}$.

4.4.1 Model with time varying parameters

The Lyapunov exponent $\lambda_{2}(c)$ can be estimated locally by computing
$\hat{\lambda}_{2,T}(c)$ 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

$\hat{\lambda}_{2,T}(c)=\frac{1}{Tb_{T}}\sum_{t=1}^{T}K\left(\frac{t/T-c}{b_{T}}\right)(ln|\hat{\phi}(t/T)+\hat{\eta}_{t,T}\sqrt{\hat{\alpha}(t/T)}|),$

with the Epanechnikov kernel $K(c)=\frac{3}{2}(1-(2c)^{2})$ for
$c\in[-1/2,1/2]$ and $K(c)=0$ otherwise, which satisfies the regularity
condition for localizing kernel [see Dahlhaus, Richter, Wu (2019, Assumption
2.6)].

A similar approach can be used to estimate locally the measure
$\xi(c)=\phi^{2}(c)+\alpha(c)$. The local plug-in estimator
$\hat{\lambda}_{2,T}(c)$ 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

| DAR(1) parameters
---|---
| $\phi$ |  | $\omega$ |  | $\alpha$
Estimates | 0.699 |  | $9.102e^{-06}$ |  | 0.484
Standard deviation | (0.034) |  | ($2.174e^{-06}$) |  | (0.205)

Table 1: Estimation of the DAR(1) model using the entire sample

where the standard deviations of the parameters are obtained using the
asymptotic distribution 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.

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,
$K(u)=1_{(-1,0)}(u),\;b_{T}=50/T$ 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 $\phi(t/T)$ is displayed in the first
paned of Figure 7 below.

Figure 7: tvDAR(1) parameter $\phi(t/T)$, conditional volatility and Lyapunov
exponent $\lambda_{2}(t/T)$

The second panel shows the estimated conditional volatility
$\sqrt{\hat{\omega}(t/T)+\hat{\alpha}(t/T)x_{t}^{2}}$ for Tether price. The
third panel in Figure 7 presents the local estimates
$\hat{\lambda}_{2,T}(t/T)$ of the Lyapunov exponent computed by plugging in
the local parameter estimators.

We observe that there are periods when the autoregressive coefficient is not
significantly 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 persistence 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 Assumption
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

October 2018 to October 2019

Figure 8: Tether prices compared to one-step ahead out-of-sample forecasts

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 $\hat{y}_{t+1}$
of the one-day-ahead conditional mean $E(y_{t+1}|y_{t},\ldots)$ of the price
of Tether using a rolling window. To get $\hat{y}_{t+1}$, we add the local
mean of Tether price to the estimated $\hat{\phi}$ using data over $50$ days
up to date $t$ times $x_{t}$. 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 $1.7428\times 10^{-5}$ which is
very small. Under the assumption of locally constant parameters, the 95 %
asymptotically valid prediction intervals in Figure 9 are given by
$\left[\hat{y}_{t+1}-\frac{1.96}{\sqrt{T}}\sqrt{x_{t}^{2}\hat{\Sigma}^{-1}},\
\hat{y}_{t+1}+\frac{1.96}{\sqrt{T}}\sqrt{x_{t}^{2}\hat{\Sigma}^{-1}}\right]$.

Full Sample

October 2018 to October 2019

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 white noise on $\hat{\eta}_{t}$ and
$\hat{\eta}_{t}^{2}$, 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 $\hat{\eta}_{t}$ and $\hat{\eta}_{t}^{2}$ are serially
uncorrelated. More precisely, we reject the null of no serial correlation for
$\hat{\eta}_{t}$ only for $1.52\%$ of the subsamples, while we reject the null
of no serial correlation for $\hat{\eta}_{t}^{2}$ 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
$\hat{\lambda}_{2,T}(t/T)$ estimator. For the bandwidth, we choose
$b_{T}=50/T$ 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 $\phi$ 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.

Figure 10: Kernel-based parameter $\phi$ estimates and Lyapunov exponent
$\lambda(t/T)$

To detect the periods of strong persistence, we test the null hypothesis
$H_{0}:\phi=1$ using the series of $\hat{\phi}(t/T)$ and its 95% confidence
intervals for each $t=1,...,T$. We conduct the hypothesis test using the
series of $\hat{\phi}(t/T)$ 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 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 $\phi=1$.  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

$\sigma(t/T)=\sqrt{\omega(t/T)+\alpha(t/T)x_{t-1,T}^{2}},\quad t=1,...,T.$
(5.8)

To test the null hypothesis $H_{0}:\sigma(t/T)=\sigma_{0}\>\forall t$, we
consider the below test statistics proposed by Chandler and Polonik (2017) for
time-varying autoregressive processes

$CP_{T}=\sup_{\alpha\in[0,1]}\sqrt{\frac{T}{(\gamma(1-\gamma)}}\lvert\hat{G}_{T,\gamma}(\alpha)-\alpha\gamma\rvert,$
(5.9)

where

  * •

$\hat{G}_{T,\gamma}(\alpha)=\frac{1}{T}\sum_{t=1}^{[\alpha
T]}\mathbbm{1}(\hat{\epsilon}^{2}_{t}\geq\hat{q}^{2}_{\gamma}),$

  * •

$\hat{q}^{2}_{\gamma}=\min\left(q^{2}\geq
0:\frac{1}{T}\sum_{t\in[aT,bT]}\mathbbm{1}(\hat{\epsilon}^{2}_{t}>q^{2})\leq\gamma\right),$

  * •

$\hat{\epsilon}_{t}=x_{t}-\hat{\phi}(\frac{t}{T})x_{t-1}.$

The process $\hat{G}_{T,\gamma}(\alpha)$ counts the number of squared
residuals within the first $(100\times\alpha)\%$ of the observations that are
larger than the empirical quantile of the squared residuals denoted by
$\hat{q}_{\gamma}^{2}$. The series of squared residuals is constructed by
computing $\hat{\epsilon}_{t}=x_{t}-\hat{\phi}(\frac{t}{T})x_{t-1}$ for each
period t where $\hat{\phi}(\frac{t}{T})$ in our case is the corresponding
DAR(1) estimate obtained in the previous section.

Chandler and Polonik (2017) shows that the test statistics $CP_{T}$ 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 $CP_{T}$, we consider different
alternatives for the empirical upper $\gamma$-quantile for comparison,
particularly $\gamma=(0.7,0.8,0.9)$ following Chandler and Polonik (2017).
Given the choice of $\gamma$, we then calculate the expression
$\sqrt{\frac{T}{(\gamma(1-\gamma)}}\lvert\hat{G}_{T,\gamma}(\alpha)-\alpha\gamma\rvert$
for different values of $\alpha$ and report the results in table 3.

|  | $\alpha$
---|---|---
|  | 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
$\gamma$ | 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

Table 3: The calculated values of
$\sqrt{\frac{T}{(\gamma(1-\gamma)}}\lvert\hat{G}_{T,\gamma}(\alpha)-\alpha\gamma\rvert$
for the given pairs of $\gamma$ and $\alpha$.

The table shows that regardless of the choice of $\gamma$, the largest value
is achieved when $\alpha=0.8$. By the definition of supremum, the test
statistics $CP_{T}$ should satisfy
$CP_{T}\geq\sqrt{\frac{T}{(\gamma(1-\gamma)}}\lvert\hat{G}_{T,\gamma}(\alpha)-\alpha\gamma\rvert$
for all $\alpha\in[0,1]$. Hence, it is safe to say that we have $CP_{T}\geq
5.047$ in the worst case scenario, i.e when $\gamma=0.9$ is
chosen.191919Alternatively, we could fix our choice of $\gamma$ and find the
exact value of $\alpha\in[0,1]$ at which the expression
$\sqrt{\frac{T}{(\gamma(1-\gamma)}}\lvert\hat{G}_{T,\gamma}(\alpha)-\alpha\gamma\rvert$
attains its maximum on this fixed interval. This could slightly improve the
precision of the lower bound for the test statistics. For example, when
$\gamma=0.9$, the expression attains its maximum value of 5.106 at
$\alpha^{*}=0.8012$. Compared with the critical values from the asymptotic
distribution of the test statistics,202020see https://homepages.ecs.vuw.ac.nz/
ray/Brownian/ for the distribution of the supremum of a Brownian Bridge. 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 $\phi_{t}$ 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 $\eta$, the Lyapunov exponent is an
even function of $\phi$. Hence we can suppose that $\phi>0$ to find the
expression of the Lyapunov exponent, and then replace $\phi$ by $|\phi|$.

For $\phi>0$ we have:

$\lambda(\phi,\alpha)=E\ln|\phi+\eta\sqrt{\alpha}|=\int\ln|\phi+\eta\sqrt{\alpha}|\psi(\alpha)d\alpha.$

Let us assume that $\eta\sim U_{[-1,1]}$. Then, its density is
$\psi(\eta)=\frac{1}{2}1_{\eta\in[-1,1]}$.

We have:

$\lambda(\phi,\alpha)=\frac{1}{2}\int_{-1}^{1}\ln|\phi+\eta\sqrt{\alpha}|d\alpha$.

We observe that:

$\begin{array}[]{ll}\phi+\eta\sqrt{\alpha}>0&\iff\eta>-\phi/\sqrt{\alpha},\\\
\phi+\eta\sqrt{\alpha}<0&\iff\eta<-\phi/\sqrt{\alpha}.\end{array}$

Hence,

$\lambda(\phi,\alpha)=\frac{1}{2}\int_{-1}^{1}1_{\eta>-\phi/\sqrt{\alpha}}\ln(\phi+\eta\sqrt{\alpha})d\eta+\frac{1}{2}\int_{-1}^{1}1_{\eta<-\phi/\sqrt{\alpha}}\ln(-\phi-\eta\sqrt{\alpha})d\eta$

Let us now examine the two cases:

a) If $\phi/\sqrt{\alpha}>1\iff-\phi/\sqrt{\alpha}<-1$, we get:

$\begin{array}[]{ll}\lambda(\phi,\alpha)&=\frac{1}{2}\int_{-1}^{1}\ln(\phi+\eta\sqrt{\alpha})d\eta+\frac{1}{2}0=\frac{1}{2}\frac{1}{\sqrt{\alpha}}\int_{-1}^{1}\ln(\phi+\eta\sqrt{\alpha})d(\sqrt{\alpha}\eta)\\\
&=\frac{1}{2\sqrt{\alpha}}\int_{\phi-\sqrt{\alpha}}^{\phi+\sqrt{\alpha}}\ln(u)du\mbox{
with change of variable}\;\;u=\phi+\eta\sqrt{\alpha}\\\
&=\frac{1}{2\sqrt{\alpha}}u\ln(u)-u|_{\phi-\sqrt{\alpha}}^{\phi+\sqrt{\alpha}}\end{array}$

$=\frac{1}{2\sqrt{\alpha}}\left[(\phi+\sqrt{\alpha})\ln(\phi+\sqrt{\alpha})-(\phi+\sqrt{\alpha})-(\phi-\sqrt{\alpha})\ln(\phi-\sqrt{\alpha})+(\phi-\sqrt{\alpha})\right]$

b) If $\phi/\sqrt{\alpha}<1\iff-\phi/\sqrt{\alpha}>-1$, we get:

$\begin{array}[]{l}\lambda(\phi,\alpha)=\frac{1}{2}\int_{-\phi/\sqrt{\alpha}}^{1}\ln(\phi+\eta\sqrt{\alpha})d\eta+\frac{1}{2}\int_{-1}^{-\phi/\sqrt{\alpha}}\ln(-\phi-\eta\sqrt{\alpha})d\eta\\\
=\frac{1}{2}\frac{1}{\sqrt{\alpha}}\int_{0}^{\phi+\sqrt{\alpha}}\ln(u)du+\frac{1}{2}\int_{\phi/\sqrt{\alpha}}^{1}\ln(-\phi+\eta\sqrt{\alpha})d\eta\mbox{
with the change of variable}\;u=\phi+\eta\sqrt{\alpha}\\\
=\frac{1}{2\sqrt{\alpha}}\int_{0}^{\phi+\sqrt{\alpha}}\ln(u)du+\frac{1}{2\sqrt{\alpha}}\int_{0}^{-\phi+\sqrt{\alpha}}\ln(u)du\end{array}$

$=\frac{1}{2\sqrt{\alpha}}\left[(\phi+\sqrt{\alpha})\ln(\phi+\sqrt{\alpha})-(\phi+\sqrt{\alpha})-[(-\phi+\sqrt{\alpha})\ln(-\phi+\sqrt{\alpha})]+(-\phi+\sqrt{\alpha})\right]$

By putting the two expressions in a) and b) together we get for $\phi>0$:

$\lambda(\phi,\alpha)=\frac{1}{2\sqrt{\alpha}}\left[(\phi+\sqrt{\alpha})\ln(\phi+\sqrt{\alpha})-(\phi+\sqrt{\alpha})-|\phi-\sqrt{\alpha}|\ln|\phi-\sqrt{\alpha})|+|\phi-\sqrt{\alpha}|\right]$

The general expression of the Lyapunov Exponent without the sign constraint on
$\phi$ is:

$\lambda(\phi,\alpha)=\frac{1}{2\sqrt{\alpha}}\left[(|\phi|+\sqrt{\alpha})\ln(|\phi|+\sqrt{\alpha})-(|\phi|+\sqrt{\alpha})-||\phi|-\sqrt{\alpha}|\ln||\phi|-\sqrt{\alpha})|+||\phi|-\sqrt{\alpha}|\right]$

We observe a non-differentiability in $\phi=\pm\sqrt{\alpha}$. Moreover, for
$\phi=0$, we get a zero value of the Lyapunov exponent:

$\lambda(\phi,\alpha)=\frac{1}{2\sqrt{\alpha}}\left[\sqrt{\alpha}\ln\sqrt{\alpha}-\sqrt{\alpha}\ln\sqrt{\alpha}+\sqrt{\alpha}\right]=0$

###  A.2 Proof of Proposition 2

We need to show that when $T\rightarrow\infty$, then
$\int\ln[\hat{\phi}_{T}+\eta\sqrt{\hat{\alpha}_{T}}]\psi_{0}(\eta)d\eta\rightarrow\int\ln(\phi_{0}+\eta\sqrt{\alpha_{0}})\psi_{0}(\eta)d\eta$
in probability if
$(\hat{\phi}_{T},\hat{\alpha}_{T})\rightarrow(\phi_{0},\alpha_{0})$ in
probability, and if condition A.6 of Proposition 2:

$\exists\delta>0,\mbox{such that}\int
sup_{\begin{array}[]{c}\phi_{0}-\delta<\phi<\phi_{0}+\delta\\\
\alpha_{0}-\delta<\alpha<\alpha_{0}+\delta\end{array}}|\ln|\phi+\eta\sqrt{\alpha}||\psi_{0}(\eta)d\eta<\infty\;\;\;(\mbox{a}.1)$

holds.

Proof of convergence:

If $(\hat{\phi}_{T},\hat{\alpha}_{T})\rightarrow(\phi_{0},\alpha_{0})$ 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 continuous function of $(\phi,\alpha)$, we have
$g(\hat{\phi}_{T},\hat{\alpha}_{T})\rightarrow g(\phi_{0},\alpha_{0})$ a.s.
and in distribution in that new space. Then,
$g(\hat{\phi}_{T},\hat{\alpha}_{T})\stackrel{{\scriptstyle\rm
d}}{{\rm\rightarrow}}g(\phi_{0},\alpha_{0})$ in the initial space and also in
probability because the limit is constant, we get the ”in probability” version
of the continuous mapping theorem. Therefore, we need only the condition
ensuring that
$g(\phi,\alpha)=\int\ln|\phi+\eta\sqrt{\alpha}|\psi_{0}(\eta)d\eta$ 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 $\lambda_{2,T}$.

Lemma

Let us consider a sequence $G_{T}(\theta)$ of stochastic functions of
$\theta$, $\theta\in\Theta$, and a sequence of estimators $\hat{\theta}_{T}$.
We assume that:

i) $\Theta$ is compact and $\theta_{0}$ is in the interior of $\Theta$.

ii) $\hat{\theta}_{T}$ tends in probability to $\theta_{0}$.

iii) $G_{T}(\theta)$ tends in probability to a limit
$G(\theta),\;\forall\theta\in\Theta$.

iv) Sufficient Lipschitz condition for stochastic equicontinuity:

There exists a stochastic sequence $B_{T}$ with $B_{T}=O_{p}(1)$ and an
increasing function $h:$ $[0,\infty)\rightarrow[0,\infty)$ continuous at zero,
with $h(0)=0$ and such that for all $\tilde{\theta}$, $\theta\in\Theta$,
$|G_{T}(\tilde{\theta})-G_{T}(\theta)|\leq B_{T}h(d(\tilde{\theta},\theta))$.

Then, $\hat{G}_{T}(\hat{\theta}_{T})$ tends in probability to $G(\theta_{0})$.

Proof:

We have :

$\begin{array}[]{ll}|G_{T}(\hat{\theta}_{T})-G(\theta_{0})|&=|G_{T}(\hat{\theta}_{T})-G_{T}(\theta_{0})+G_{T}(\theta_{0})-G(\theta_{0})|\\\
&\leq|G_{T}(\hat{\theta}_{T})-G(\theta_{0})|+|G_{T}(\theta_{0})-G(\theta_{0})|\\\
&\leq
B_{T}h[d(\hat{\theta}_{T},\theta_{0})]+|G_{T}(\theta_{0})-G(\theta_{0})|.\end{array}$

We know that if $X_{T}\stackrel{{\scriptstyle\rm P}}{{\rm\rightarrow}}0$,
$Y_{T}\stackrel{{\scriptstyle\rm P}}{{\rm\rightarrow}}0$
$=>X_{T}+Y_{T}\stackrel{{\scriptstyle\rm P}}{{\rm\rightarrow}}0$, i.e. the sum
of $o_{p}(1)$ is $o_{p}(1)$.

Under condition iii) $|G_{T}(\theta_{0})-G(\theta_{0})|=o_{p}(1)$. It remains
to be shown that $B_{T}h[d(\hat{\theta}_{T},\theta_{0})]$ is $o_{p}(1)$.

We have:

$[B_{T}<M\;\mbox{and}\;h[d(\hat{\theta}_{T},\theta_{0})]<\epsilon/M]=>[B_{T}h[d(\hat{\theta}_{T},\theta_{0})]]<\epsilon]$
$\iff\left((B_{T}<M)\cap[h[d(\hat{\theta}_{T},\theta_{0})]<\epsilon/M]\right)\subset[B_{T}h[d(\hat{\theta}_{T},\theta_{0})]<\epsilon]$

Consider the complement: $(A\cap B)^{c}=A^{c}\cup B^{c}$. We get:

$((B_{T}>M)\cup(h[d(\hat{\theta}_{T},\theta_{0})]>\epsilon/M)\supset[B_{T}h[d(\hat{\theta}_{T},\theta_{0})]>\sqrt{\epsilon}]$

It follows that:

$\begin{array}[]{ll}P[B_{T}h[d(\hat{\theta}_{T},\theta_{0})]>\epsilon]&\leq
P[(B_{T}>M)\;\cup\;(h[d(\hat{\theta}_{T},\theta_{0})]>\epsilon/M)]\\\ &\leq
P[B_{T}>M]+P[h[d(\hat{\theta}_{T},\theta_{0})]>h^{-1}(\epsilon/M)],\end{array}$

because $P[A\cup B]\leq P(A)+P(B)$.

Then, for any $\epsilon$, we can choose a value of $M$ and a number of
observations $T$ sufficiently large to get
$P[B_{T}h[d(\hat{\theta}_{T},\theta_{0})]>\epsilon]$ arbitrarily small.
Therefore, $B_{T}h[d(\hat{\theta}_{T},\theta_{0})]$ tends to zero in
probability. QED

Then, the lemma can be applied with
$G_{T}(\theta)=\frac{1}{T}\sum_{t=2}^{T}g(x_{t},x_{t-1};\theta)$ and
$g(x_{t},x_{t-1};\theta)=\ln|\phi+\frac{x_{t}-\phi
x_{t-1}}{\sqrt{\omega+\alpha x_{t-1}^{2}}}\sqrt{\alpha}|$. Under assumptions
A.1, A.4, A.7, conditions i), ii), iii) are satisfied. For example:

$G_{T}(\theta)\stackrel{{\scriptstyle\rm
P}}{{\rm\rightarrow}}E_{0}g(x_{t},x_{t-1};\theta),$

by the weak law of large numbers applied to the transformation
$g(x_{t},x_{t-1};\theta)$ of the ergodic stationary process $(x_{t})$.
Assumption A.8 corresponds to condition iv) of the lemma.

###  A.4 Proof of Proposition 4

a) Proof of convergence

We need to show that
$\hat{\phi}_{T}^{2}+\hat{\alpha}_{T}\rightarrow\phi_{0}^{2}+\alpha_{0}$ in
probability if $(\hat{\phi},\hat{\alpha})\rightarrow(\phi_{0},\alpha_{0})$ in
probability when $T\rightarrow\infty$.

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 $\sqrt{T}$ implies:

$\sqrt{T}\left[(\hat{\phi}_{T}^{2}+\hat{\alpha}_{T})-(\phi_{0}^{2}+\alpha_{0})\right]=\
\left(\begin{array}[]{c}2\phi_{0}\\\
1\end{array}\right)^{\prime}\sqrt{T}\left(\begin{array}[]{c}\hat{\phi}_{T}-\phi_{0}\\\
\hat{\alpha}_{T}-\alpha_{0}\end{array}\right)+o_{p}(1)$

$=A^{\prime}\sqrt{T}\left(\begin{array}[]{c}\hat{\phi}_{T}-\phi_{0}\\\
\hat{\alpha}_{T}-\alpha_{0}\end{array}\right)+o_{p}(1)$

where $A^{\prime}=[2\phi_{0}\;\;1]$. We get the asymptotic normal distribution
of $\hat{\xi}_{T}$:

$\sqrt{T}(\hat{\xi}_{T}-\xi_{0})\sim N(0,V_{\xi}),$

where $V_{\xi}=A^{\prime}\Omega^{*}A$. The matrix $\Omega^{*}$ is:
$\Omega^{*}=diag(\Sigma^{-1},V(\hat{\alpha}))$ where
$\Sigma=E_{0}(y^{2}/(\omega_{0}+\alpha_{0}y^{2})$ given in Section 4.3.1.
Matrix
$V(\hat{\alpha})=(E_{0}\frac{1}{(\omega_{0}+\alpha_{0}y^{2})^{2}})/\tilde{V}_{0}(y^{2})$
and $\tilde{V}_{0}(y^{2})=\tilde{E}_{0}(y^{4})-(\tilde{E}_{0}y^{2})^{2}$. In
this formula, $\tilde{E}_{0}$ denotes the expectation of variables
$y^{4}\frac{1}{(\omega_{0}+\alpha_{0}y^{2})^{2}}/E_{0}\frac{1}{(\omega_{0}+\alpha_{0}y^{2})^{2}}$
and
$y^{2}\frac{1}{(\omega_{0}+\alpha_{0}y^{2})^{2}}/E_{0}\frac{1}{(\omega_{0}+\alpha_{0}y^{2})^{2}}$
[see, section 4.3.1 and Ling (2004) for the variance estimator formula].

## Appendix B: Simulation Results

The purpose of this section is to illustrate the derived results in Appendix A
using simulation experiments. We distinguish the case where the distribution
of the innovation $\eta$ is known and the case it is not.

First, we use the result in Proposition 1 and plot the Lyapunov exponent
$\lambda=E(ln(|\phi+\sqrt{\alpha}\eta|))$ for different values of the
parameter $\phi$ and $\alpha$. To do so, we assume $\eta\sim U[-1,1]$. Figure
B1 shows that the Lyapunov exponent $\lambda$ remains lower than zero as as
$\phi$ varies in $\\{-1,-0.8,-0.6,\ldots,0.6,0.8,1\\}$ and $\alpha$ varies in
$\\{0,0.1,0.2,0.3,\ldots,0.8,0.9,1\\}$.

Second, we assume $\eta\sim N(0,1)$, set the true parameters to
$\phi_{0}=0.7$, $\alpha_{0}=0.5$, $\omega_{0}=0.01$. 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 and
obtained via kernel density estimation. Figure B2 plots the estimated density
for the Lyapunov exponent $\hat{\lambda}_{2,T}$ and the stability measure
$\hat{\xi}_{T}$ 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
$\hat{\xi}_{T}=\hat{\phi}^{2}_{T}+\hat{\alpha}_{T}$ has its mode around the
true value of $\xi$, which is $\xi_{0}=\phi_{0}^{2}+\alpha_{0}=0.99$.

As a by-product, we present Figure B3, which shows the estimated density for
$\hat{\phi}_{T}$, $\hat{\alpha}_{T}$ and $\hat{\omega}_{T}$ 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 $\lambda=E(ln(|\phi+\sqrt{\alpha}\eta|))$ in
terms of $\phi$ and $\alpha$ when $\eta\sim U[-1,1]$

(a) Density of the Lyapunov exponent $\hat{\lambda}_{2,T}$

(b) Density of the alternative stability measure $\hat{\xi}_{T}$

Figure B2: Densities for stability measures based on estimated parameters

(a) Density of $\hat{\phi}_{T}$

(b) Density of $\hat{\omega}_{T}$

(c) Density of $\hat{\alpha}_{T}$

Figure B3: Densities of estimators for $\phi$, $\alpha$ and $\omega$

## 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 $\xi$ 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 $\xi$ plotted in Figure C1 is
more conservative than the Lyapunov exponent $\lambda$. Because we can have
$\xi\geq 1$ while $\lambda<0$ so that valid inference is still possible, the
rejection of $\xi<1$ should be interpreted as the need for investors,
regulators or stablecoin issuers to be cautious when predicting Tether future
price around the tested periods.

Figure C1: tvDAR(1) parameter $\phi(t/T)$ and Lyapunov exponent $\xi(t/T)$

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