A Peek into the Unobservable: Hidden States and Bayesian Inference for the Bitcoin and Ether Price Series
AA Peek into the Unobservable: Hidden States and BayesianInference for the Bitcoin and Ether Price Series
Constandina Koki , Stefanos Leonardos , and Georgios Piliouras Athens University of Economics and Business, 76 Patission Str. GR-10434 Athens, Greece { kokiconst,vrontos } @aueb.gr Singapore University of Technology and Design, 8 Somapah Rd. 487372 Singapore, Singapore { stefanos leonardos,georgios } @sutd.edu.sg Abstract.
Conventional financial models fail to explain the economic and monetary propertiesof cryptocurrencies due to the latter’s dual nature: their usage as financial assets on the oneside and their tight connection to the underlying blockchain structure on the other. In an effort toexamine both components via a unified approach, we apply a recently developed Non-HomogeneousHidden Markov (NHHM) model with an extended set of financial and blockchain specific covariateson the Bitcoin (BTC) and Ether (ETH) price data. Based on the observable series, the NHHMmodel offers a novel perspective on the underlying microstructure of the cryptocurrency marketand provides insight on unobservable parameters such as the behavior of investors, traders andminers. The algorithm identifies two alternating periods (hidden states) of inherently differentactivity – fundamental versus uninformed or noise traders – in the Bitcoin ecosystem and unveilsdifferences in both the short/long run dynamics and in the financial characteristics of the twostates, such as significant explanatory variables, extreme events and varying series autocorrelation.In a somewhat unexpected result, the Bitcoin and Ether markets are found to be influenced bymarkedly distinct indicators despite their perceived correlation. The current approach backs earlierfindings that cryptocurrencies are unlike any conventional financial asset and makes a first steptowards understanding cryptocurrency markets via a more comprehensive lens.
Keywords:
Cryptocurrencies · Blockchain · Bitcoin · Ethereum · Non Homogeneous HiddenMarkov · Bayesian Inference
The present study is motivated by the still limited understanding of the economic and financial propertiesof cryptocurrencies. Sheding light on such properties constitutes a necessary step for their wider publicadoption and is fundamental for blockchain stakeholders, investors, interested authorities and regulators([25,71]). More importantly, it may provide hints about market manipulation and fraud detection.Unfortunately, existing financial models that are used to study fiat currency exchange rates fail tocapture the convoluted nature of cryptocurrencies ([19]). The additional challenge that they face is thetight connection between cryptocurrency prices and the underlying blockchain technology which drivesthe dynamics of the observable market. To some extent, this is expressed via the particular marketmicrostructure of cryptocurrencies: the market depth which depends on the exchange and the marketmaker, the functionality of exchanges as custodians (unique property among financial assets) and theabsence of stocks, equities or other financial investment instruments (with the exception of Bitcoinfutures, [53]) which render acquiring and/or trading the cryptocurrency the main way of investing in thisnew technology ([61]). The miners and/or stakers emerge as the main actors who drive the creation anddistribution of the currency whereas the cheap and immediate transactions essentially obviate the needfor conventional brokers. All these features (among many others), starkly distinguish cryptocurrenciesfrom conventional financial assets or fiat money. However, a precise understanding of their definingfinancial and economic properties is still elusive ([15,31,83]). With this in mind, the concrete researchquestions that we set out to understand are the following: – How do cryptocurrencies compare – in terms of their economic and financial properties – to wellunderstood financial assets like commodities, precious metals, equities and fiat currencies ([64,6,52])?How do they relate to traditional financial markets and global macroeconomic indicators? – What are the defining microstructure characteristics of the cryptocurrency market and which are thedistinguishing features (if any) between different coins ([3,56])? a r X i v : . [ ec on . E M ] S e p o address these questions, we use a recently developed instance of Non-Homogeneous Hidden Markov(NHHM) modeling, namely the Non-Homogeneous P´olya Gamma Hidden Markov model (NHPG) of [59],which has been shown to outperform similar models in conventional financial data ([67]). Using financialand blockchain specific covariates on the Bitcoin ([70]) and Ether ([16,17]) log-return series (henceforthBTC and ETH, respectively), the NHHM methodology aims not only to capture dynamic patterns andstatistical properties of the observable data but more importantly, to shed some light on the unobservablefinancial characteristics of the series, such as the activity of investors, traders and miners.The present model falls into the Markov-switching or regime-switching literature with two possiblestates that is the benchmark for predicting exchange rates ([37,63,39,7,45,86]). This linear model wasfirst introduced by [46] as an alternative approach to model non-linear and non-stationary data. Itinvolves switches between multiple structures (equations) that can characterize the time series behaviorin different regimes (states). The switching mechanism is governed by an unobservable state variablethat follows a first-order Markov chain . Therefore, the NHMM is suitable for describing correlatedand heteroskedastic data with distinct dynamic patterns during different time periods, as are preciselycryptocurrency prices ([1,13,56]).Although standard in financial applications ([65]), Hidden Markov models have only been applied inthe cryptocurrency context by [77] as state space models, by [61] to capture the liquitity uncertainty and[74] in the context of price bubbles. Yet, their more extensive use is supported by the specific character-istics of cryptocurrency data that have been identified by earlier research. [4,49] and [33] demonstratethe non-stationarity of BTC prices and volume and underline the importance of modeling non-linearitiesin Bitcoin prediction models. This is further elaborated by [7,75,72] and [89] who suggest that modelselection and the use of averaging criteria are necessary to avoid poor forecasting results in view ofthe cryptocurrencies’ extreme and non-constant volatility. Along these lines, [24] show that the Bitcoinprice series exhibits structural breaks and suggest that significant price predictors may vary over time.Additional motivation for the analysis of cryptocurrency data with regime-switching models as the oneemployed here, is provided by [54] who demonstrate the heteroskedasticity of BTC prices and [5] whoidentify periods of different trading activity. Our main findings can be summarized as follows – The NHPG algorithm identifies two hidden states with frequent alternations for the BTC log-returnseries, cf. Figure 2. State 1 corresponds to periods with higher volatility and returns and accounts forroughly one third of the sample period (2014-2019). By contrast, state 2 marks periods with lowervolatility, series autocorrelation (long memory), trend stationarity and random walk properties, cf.Table 2. At the more variable state 1, the BTC data series is influenced by miners’ activity and morevolatile covariates (stock indices) in comparison to more stable indicators (exchange rates) in state 2,cf. Table 3. – The results for the hidden process are the same for both the long run (2014-2019) and the short run(2017-2019) BTC data, cf. Figures 2 and 3a. However, differences in the significant predictors indicatemore speculative activity in the short run compared to more fundamental investor behavior in thelong run, cf. Table 3. In sum, speculative activity (noise traders) is identified in the less frequent state1 and in the short run whereas increased activity of fundamental investors is seen in state 2 and inthe long run. – The algorithm does not mark a well defined hidden process with clear transitions for the ETH series,cf. Figure 3b. This is further supported by the low number and the small values of significant predictorsfrom the current set, cf. Table 4. These results imply that ETH prices are still driven by variablesbeyond the currenlty selected set of predictors, showing characteristics of an emerging market that ismore isolated than BTC from global financial and macroeconomic indicators.More details are presented in Section 3. Overall, the outcome of the NHPG model can be useful forinvestors and blockchain stakeholders by providing hints on periods of differentiating activities andeffects in the cryptocurrency markets. From a theoretical perspective, it backs earlier findings thatcryptocurrencies are unlike any other financial asset and suggests that their understanding requires notonly the integration of existing financial tools but also a more refined framework to account for theirbundled technological and financial features ([79]). For example, in the seminal paper of [46], the author used the underlying hidden process to define the businesscycles (recession periods). More recent examples and a comprehensive theory about NHHM in finance can befound in [65]. .2 Related Literature
The literature on the financial properties of cryptocurrencies is expanding at an exponential rate and anexhaustive review is not possible (see [29] and references therein for a more comprehensive reference list).More relevant to the current context is the scarcity (to the best of our knowledge) of papers that addressthe bundled nature of cryptocurrencies as both blockchain applications and financial assets. Existingstudies focus either on the underlying blockchain technology/consensus mechanism or on the observablefinancial market but not on both. By contrast, the current NHPG model parses the observable financialinformation to recover the underlying structure of cryptocurrency markets and hence makes a first steptowards a unified approach to fill this gap. Its limitations are discussed in Section 4. In the remainingpart of this section, we provide a (non-exhaustive) list of studies that focus on the financial part.Early research, mainly focusing on BTC has provided mixed insights on the properties of cryptocur-rencies. [58] claim that BTC is fundamentally different from valuable metals like gold due to its shortagein stable hedging capabilities. Along with [22], [24] also argue that standard economic theories cannotexplain BTC price formation and using data up to 2015, they provide evidence that BTC lacks the nec-essary qualities to be qualified as money. However, [36] demonstrate that BTC has similarities to bothgold and the US dollar (USD) and somewhat surprisingly, that it may be ideal for risk-averse investors.[12,10] and [14] also explore BTC’s characteristics as a financial asset and find that while BTC is use-ful to diversify financial portfolios – due to its negative correlation to the US implied volatility index(VIX) – it otherwise has limited safe haven properties. Using data from a longer period (between 2010and 2017), [33] conclude the opposite, namely that BTC may indeed serve as a hedging tool, due to itsrelationship to the Economic Policy Uncertainty Index (EUI). In comparative studies, [40,30] provideempirical evidence of bubbles in both BTC and ETH and [44] suggest that BTC is less risky than ETH,i.e., that it exhibits less fat tailed behavior. [73] confirm that Bitcoin exhibits long memory and het-eroskedasticity and argue that cryptocurrencies display mild leverage effects, predictable patterns withmostly oscillating persistence, varied kurtosis and volatility clustering. Comparing BTC with ETH, theyargue that kurtosis is lower for ETH being easier to transact than BTC. Along this line, the findings of[68] and [56] further motivate the use of non-homogeneous and regime-switching modeling for both theBTC and ETH log-returns series.The differences between cryptocurrencies and conventional financial markets are further elaboratedby [54,47,72]. High volatility, speculative forces and large dependence on social sentiment at least duringits earlier stages are shown by some as the main determinants of BTC prices ([42,43,26,88]). Yet, alarge amount of price variability remains unaccounted for ([48,66,49]). Moreover, the proliferation ofcryptocurrencies on different blockchain technologies suggests that their current correlation may bediscontinued in the near future and calls for comparative studies as the one conducted here ([9,85]).
The rest of the paper is structured as follows. In Section 2, we describe the NHPG model and simulationscheme and present the set of variables that have been used (some preliminary descriptive statisticsand tests about this data are relegated to Appendix A). Section 3 contains the main results and theiranalysis. In the first part (Sections 3.1 to 3.3), we present the outcome of the algorithm and discuss thestatistical findings for the hidden states and the generated subseries. In the second part, Section 3.4, wefocus on the significant explanatory variables for the BTC data series in both the short and long run andthe ETH data series. We conclude the paper with a discussion of the limitations of the present modeland directions for future work in Section 4.
Given a time horizon T ≥ t = 1 , , . . . , T , we consider an observedrandom process { Y t } t ≤ T and a hidden underlying process { Z t } t ≤ T . The hidden process { Z t } is assumedto be a two-state non-homogeneous discrete-time Markov chain that determines the states ( s ) of theobserved process. In our setting, the observed process is either the BTC or the ETH log-return series.Importantly, the description of the hidden states is not pre-determined and is subject to the outcome ofthe algorithm and interpretation of the results.Let y t and z t be the realizations of the random processes { Y t } and { Z t } , respectively. We assume thatat time t, t = 1 , . . . , T , y t depends on the current state z t and not on the previous states. Consider also aset of r − { X t } with realization x t = (1 , x t , . . . , x r − t ) at time t . The explanatoryariables (covariates) { X t } that are used in the present analysis are described in Table 1. The effectof the covariates on the cryptocurrency price series { Y t } is twofold: first, linear, on the mean equationand second non-linear, on the dynamics of the time-varying transition probabilities, i.e., the probabilitiesof moving from hidden state s = 1 to the hidden state s = 2 and vice versa. Given the above, thecryptocurrency price series { Y t } can be modeled as Y t | Z t = s ∼ N ( x t − B s , σ s ) , s = 1 , , where B s = ( b s , b s , . . . , b r − s ) (cid:48) are the regression coefficients and N ( µ, σ ) denotes the normal distri-bution with mean µ and variance σ . The dynamics of the unobserved process { Z t } can be describedby the time-varying (non-homogeneous) transition probabilities, which depend on the predictors and aregiven by the following relationship P ( Z t +1 = j | Z t = i ) = p ( t ) ij = exp( x t β ij ) (cid:80) j =1 exp( x t β ij ) , i, j = 1 , , where β ij = ( β ,ij , β ,ij , . . . , β r − ,ij ) (cid:48) is the vector of the logistic regression coefficients to be estimated.Note that for identifiability reasons, we adopt the convention of setting, for each row of the transitionmatrix, one of the β ij to be a vector of zeros. Without loss of generality, we set β ij = β ji = for i, j = 1 , , i (cid:54) = j . Hence, for β i := β ii , i = 1 ,
2, the probabilities can be written in a simpler form p ( t ) ii = exp( x t β i )1 + exp( x t β i ) and p ( t ) ij = 1 − p ( t ) ii , i, j = 1 , , i (cid:54) = j. To make inference on the hidden process, we use the smoothed marginal probabilities P ( Z t = i | Y T , z t +1 , θ ) which are the probabilities of the hidden state conditional on the full observed processas derived from the Forward-Backward algorithm ([46]). In the rest of the paper, we use the notation P ( Z t = i ) for convenience. Algorithm 1
MCMC Sampling Scheme for Inference on Model Specification and Parameters % After each procedure the parameters and model space are updated conditionally on the previousquantities procedure Scaled Forward Backward (( z T )) %Simulation of a realization of the hidden states z t for t = 1 , . . . , T and i = 1 , do π t ( i | θ ) ← α t ( s ) (cid:80) j =1 α t ( j ) = P ( z t = i | θ, y t ) ( (cid:46) ) Simulation of the scaled forward variables for t = T, T − , . . . , do z t ← P (cid:0) z t | y T , z t +1 (cid:1) = p iz t +1 π t ( i | θ ) (cid:80) mj =1 p jz t +1 π t ( j | θ ) ( (cid:46) ) Backwards simulation of z t using the smoothed probabilities procedure Mean Regres Param ( B s , σ s , s = 1 , %Simulation of the mean regression parameters for s = 1 , do ( (cid:46) ) Conjugate analysis with Gibbs sampler B | σ ∼ f B , σ ∼ IG ( (cid:46) ) f B ≡ Normal and
IG ≡
Inverse-Gamma procedure
Log Regres Coef (( β s , ω s )) %Simulation of the logistic regression coefficients for s = 1 , do ( (cid:46) ) P´olya-Gamma data augmentation scheme → augment the model space with ω s ( (cid:46) ) Conjugate analysis on the augmented space → sample from β s ∼ f β s | ω and ω s | β s ∼ PG → posteriors f β s | ω ≡ Normal and
PG ≡
P´olya-Gamma .1 Simulation Scheme
The unknown quantities of the NHPG are (cid:8) θ s = (cid:0) B s , σ s (cid:1) , β s , s = 1 , (cid:9) , i.e., the parameters in the meanpredictive regression equation and the parameters in the logistic regression equation for the transitionprobabilities. We follow the methodology of [59]. In brief, the authors propose the following MCMCsampling scheme for joint inference on model specification and model parameters.1. Given the model’s parameters, the hidden states are simulated using the Scaled Forward-Backwardof algorithm of [78].2. The posterior mean regression parameters are simulated using the standard conjugate analysis, viaa Gibbs sampler method.3. The logistic regression coefficients are simulated using the P´olya-Gamma data augmentation scheme[76], as a better and more accurate sampling methodology compared to the existing schemes.The steps 1-3 of the MCMC algorithm are detailed in Algorithm 1. We assess the ability of 11 financial–macroeconomic and 3 cryptocurrency specific variables, outlined inTable 1, in explaining and forecasting the prices of BTC and ETH via the NHPG model. In the relatedcryptocurrency literature these indices are commonly studied under various settings ([84,87,14,38,75,48],[49] and [77]). The findings of the descriptive statistics and preliminary stationarity tests, cf. Appendix A,indicate that the logarithmic return (log-return), i.e., the change in log price, r t = log ( y t ) − log ( y t − ),series of BTC and ETH exhibit trend non-stationarity, non-linearities, rich (i.e., non-random) underlyinginformation structure and non-normalities. Based on these properties, the NHPG model seems appro-priate for the study of the log-return data series. Accordingly, we apply the NHPG algorithm on dailylog-returns of BTC and ETH, with normalized explanatory variables. We perform two experiments overtwo different time frames: in the first, we study the BTC series between 1/2014 and 8/2019 and in thesecond, we study both the BTC and ETH series between 1/2017 and 8/2019. The second time framehas been selected to allow reasonable comparisons between the BTC and ETH prices after eliminatingan initial period following the launch of the ETH currency. It is further motivated by the outcome of atest-run of the NHPG model on BTC prices, cf. Figure 1, which indicates a transition point to a differentperiod for the BTC price series in January 2017.Fig. 1: Application of the NHPG model on the BTC price series. The algorithm essentially identifiestwo periods, the first from 2014 (start of the dataset) to 2017 and the second from 2017 to date. Thismotivates separate analysis of the BTC for the latter period and comparison with the ETH price seriesover the same period.xplanatory VariablesDescription Symbol Retrieved fromUS dollars to Euros exchange rate USD/EUR investing.comUS dollars to GBP exchange rate USD/GBP investing.comUS dollars to Japanese Yen exchange rate USD/JPY investing.comUS dollars to Chinese Yuan exchange rate USD/CNY investing.comStandard & Poor’s 500 index SP500 finance.yahoo.comNASDAQ Composite index NASDAQ finance.yahoo.comSilver Futures price Silver investing.comGold Futures price Gold investing.comCrude Oil Futures price Oil investing.comCBOE Volatility index VIX finance.yahoo.comEquity market related Economic Uncertainty index EUI fred.stlouisfed.orgDaily Block counts Blocks coinmetrics.ioHash Rate Hash quandl.com, etherscan.ioTransfers of native units Tx-Units coinmetrics.ioTable 1: List of variables and online resources. The Hash Rate (Hash) has been retrieved from quandl.comfor Bitcoin (BTC) and from etherscan.io for Ether (ETH). In this section, we discuss the findings from the NHPG model on the BTC and ETH log-return series. Wefirst present the graphics with the output of the algorithm for the whole 2014-2019 period on BTC log-returns (Section 3.1) and the shorter 2017-2019 period on both BTC and ETH log-returns (Section 3.2).Then, we interpret the results and compare the statistical properties and the significant covariates be-tween the two hidden states of both the BTC and ETH series and between the short and long run BTCseries (Sections 3.3 and 3.4).
Figure 2 displays the BTC log-return series (blue line) along with the smoothed marginal probabilities(gray bars) of the hidden process being at state 1. Using as a threshold the probability P ( Z t = 1) > . Figure 3 shows the results of the NHPG model for both the BTC (left panel) and ETH (right panel)log-return series over the shorter 1/2017-8/2019 period. The algorithm has again identified two states inthe BTC series, Figure 3a, as indicated by the clear distinction between high-low marginal probabilitiesof state 1, i.e., P ( Z t = 1), that are given by the gray bars. Moreover, a comparison with the same periodig. 2: BTC logarithmic-return series (blue line – right axis) for the period 1/2014-8/2019 with the meansmoothed marginal probabilities of state 1, i.e., P r ( Z t = 1) (gray bars – left axis).in Figure 2 demonstrates that the NHPG has produced the same result (zoom in) – in terms of statisticalquality – even over this smaller period, i.e., the algorithm has converged and returns essentially the sameprobabilities for the underlying process. However, as we will see below, cf. Section 3.4, the statisticalanalysis unveils differences in the significant predictors and financial properties between the short andlong run.The picture is different for the ETH series, cf. Figure 3b. Here, the hidden process is not well definedsince the probabilities of state 1 at each time period are mostly close to 0 .
5. This indicates high degree ofrandomness in the transitions of the algorithm and along with the low number of significant covariatesthat have been identified for ETH (cf. Table 4 below), it suggests that ETH prices are still influenced byforces which are beyond the current set of financial and blockchain indicators ([55,72,73]). This impliesthat ETH – when viewed as a financial asset – shows characteristics of an evolving, non-static and stillemerging market. However, the relative isolation of ETH from other financial assets agrees with earlierfindings in the literature ([72,31]).Our next task is to provide additional insight on the structural financial and economic attributes thatdifferentiate these two states for all experiments. Based on the similarities between the short and longrun BTC time frames and the poor convergence of the algorithm for the ETH series, we focus on thelong-run BTC series.
The results of both the descriptive statistics and the relevant statistical tests are summarized in Table 2.Each entry – BTC price, log-price and log-return series – consists of two rows that correspond to the (a) BTC: 1/2017-8/2019 (b) ETH: 1/2017-8/2019
Fig. 3: BTC (left panel) and ETH (right panel) logarithmic-returns series (blue lines – right axis) for theperiod 1/2017-8/2019 with the mean smoothed marginal probabilities of state 1, i.e.,
P r ( Z t = 1), (graybars – left axis).ubseries of state 1 (upper row) and state 2 (lower row), respectively. The first two columns of Table 2verify that the estimated hidden process segments the series into two subseries with high/low mean andvariance values for all the examined data series. Log-returns exhibit increased kurtosis in comparison tothe initial estimates, cf. Table 5, for both subseries (in particular for state 2). Similarly, the skewnessof both subseries has increased and has turned positive with the skewness of the second subseries beingagain much higher than that of the first (cf. [82]). These distributional properties lead to rejection ofnormality for either subseries and suggest the presence of heavy-tailed data (phenomena in which exremeevents are likely, [92]) .The identification of two subchains with different kurtosis and skewness can be a useful tool to in-vestors ([51,60,34]). As risk measures, kurtosis and skewness cause major changes to the constructionof the optimal portofolio ([23,27]), especially in emerging and highly volatile markets ([18]). The asym- Descriptive statistics Tests
Mean Variance Kurtosis Skewness DF LBQ KPSS VR JB
BTC
Price 4920 2 . × . × Table 2: Descriptive statistics (left panels) and p-values for the time series statistical tests (right panels)for the two (2) BTC price, log-price and log-return subseries – first and second line of each entry– which correspond to the two hidden states that were identified by the NHPG model for the whole1/2014-8/2019 time period.metry on the distributions and the difference of volatility between the two subchains can be related tothe activity of informed or fundamental vs uninformed, noise or non-fundamental investors (or traders).Intuitively, the activity of uninformed investors leads to periods with higher volatility (cf. [5] and refer-ences therein). This is true for state 1 and refines the findings of [90,5] who attribute the informationalinefficiency of BTC not only to its endogenous factors of an emerging, non-mature market but also tothe non-existence of fundamental traders.The differences between the two states are further explained by the statistical tests. While the p-values of the Dickey-Fuller (DF) and Jarque-Bera (JB) remain the same as for the combined data series,cf. Table 5, the results for the Ljung-Box-Q (LBQ), KPSS and Variance Ratio (VR) tests unveil differentcharacteristics of the two subseries. In state 2 of the log-return series, the zero hypothesis is rejected forthe LBQ test but not for the KPSS and VR tests. This suggests that the subseries defined by state 2is a random walk with trend stationarity and long memory. These findings are related to (and to someextent refine) the results of [50,62,57,68,91] by determining periods with (state 2) and without (state1) permanent effects (long memory). The subchain of state 1 stills exhibits richer structure which canbe potentially attributed to the combined activity and herding behavior of the non-fundamental traders([11,80,81]).
The second functionality of the NHPG model is to identify the significant explanatory variables fromthe set of available predictors that affect the underlying series both linearly, i.e., in the mean equation(observable process), and non-linearly, i.e., in the non-stationary transition probabilities (unobservableprocess). The algorithm also distinguishes between the variables that are significant in each state. Thecorresponding results for the BTC log-return series over both the 2014-2019 and 2017-2019 time periodsare given in Table 3 and the results for the ETH log-return series over the 2017-2019 time period aregiven in Table 4. We use B i to denote the posterior mean equation coefficients and β i the posterior meanlogistic regression coefficients for states i = 1 ,
2, as described in Section 2. The predictors that have beenfound significant at the 0.05 level are marked with bold font and ∗ . The main findings are the following: Inclusion of a third hidden state could potentially lead to smoothing of these measurements, cf. Section 4. stimations BTC2014-2019 2017-2019Variables B B β β B B β β USD/EUR 0.00 0.00 0 . . ∗ -0.01 0.00 0 . . ∗ USD/GBP . ∗ ≈ . ∗ - . ∗ -0.01 0.00 -1.82 . ∗ USD/JPY 0.00 0.00 0.52 -0.77 0 .
00 -0.00 -0.53 -0.77USD/CNY -0.01 0.00 0.90 0.57 ≈ .
98 1 . . ∗ -0.01 3.90 -1.87 0 .
04 0.01 . ∗ . . ∗ -2.04Silver 0.00 ≈ .
01 -0.00 -0.42 1 . ≈ . ∗ -0.18 0.00 0.00 . ∗ -0.35Oil ≈ ≈ .
00 0 .
00 0.00 . ∗ . ∗ VIX ≈ ≈ .
47 -0.18 ≈ ≈ .
53 0 . ≈ ≈ ≈ ≈ . ≈ . . ∗ . ∗ -0.03 . ∗ .
01 - . ∗ . ≈ .
51 -0.13 ≈ .
53 -0.50Table 3: Posterior mean estimations for the BTC log-return series in the 2014-2019 (left) and 2017-2019(right) time periods. B , B are the mean equation coefficients and β , β are the logistic regressioncoefficients for states 1,2. Statistically significant coefficients (at the 0.05 level) are marked with ∗ . BTC: state 1 vs state 2.
The significant predictors (covariates) that dominate both the observableand the unobservable processes in the more volatile state 1 (cf. Section 3.3), correspond to more volatilefinancial instruments such as stock markets (S&P500 and NASDAQ). By contrast, state 2 is mostlyinfluenced by the more stable exchange rates, cf. Figure 4. These findings suggest increased speculativeactivity in state 1 in comparison to fundamental investors in state 2.
BTC: short vs long run.
While the algorithm has identified essentially the same hidden process forboth the short and long run windows, cf. Figures 2 and 3a, the significant predictors that affect boththe observable and unobservable processes are remarkably different: more volatile for the short runversus more fundamental (monetary) for the long run. In line with [32], these findings provide evidencefor increased speculative behavior in the short run. They also extend BTC’s financial and safen havenproperties to more recent windows ([77,4,10]). Additionally, they refine the results of [28] and [20] whoargue about the differences in the short and long run BTC markets and the hedging properties of BTCagainst volatile stock indices in time varying periods, respectively.
ETH vs BTC: short run.
The lower number of significant predictors in the ETH log-return seriesreflects the inability of the NHPG model to parse the underlying process, cf. Figure 3b. This differen-tiates the ETH from the BTC market and provides evidence that ETH is still at its infancy, evolvingindependently from established economic indicators and fundamentals. Yet, the main – and somewhatunexpected – conclusion is that, despite the evident correlation between the prices of BTC and ETH(Pearsons serial correlation 0.62), the two cryptocurrencies are affected by different fundamental finan-cial and macroeconomic indicators over the same time period.Finally, an observation that applies to all series is that the current set of predictors cannot fullyexplain the data volatility. Excluding the miners’ activity (as expressed by the Hash Rate) which ap-pears significant in state 1 for all series (both for the observable and the unobservable processes), thisobservation follows from the small values of the predictors in the mean equation of state 1 (cf. columns B in Table 3) and the absence of predictors in the mean equation (observable process) of state 2 (cf.columns B in Table 3).ig. 4: Comparison of the USD/EUR exchange rate (blue line), S&P500 (green line) and Crude Oil FuturePrices (gray line) as a percentage of price changes from the initial period. The USD/EUR exchange rateis less volatile than the other predictors. Estimations ETH 2017-2019Variables B B β β Variables B B β β USD/EUR -0.01 -0.00 -0.36 -0.38 USD/GBP 0.01 -0.06 0.21 0.59USD/JPY -0.01 -0.00 -0.69 0.68 USD/CNY -0.02 0.01 -0.14 -0.01SP500 . ∗ -0.00 2.70 0.13 VIX 0.01 0.00 -0.19 0.26NASDAQ - . ∗ . ∗ ≈ ≈ . ∗ Silver -0.01 0.01 -0.60 0.07 Blocks -0.01 ≈ . ∗ . ∗ -1.41Oil -0.01 -0.01 -0.64 -1.92 Tx-Units 0.00 0.00 1.03 1.16 Table 4: Posterior mean estimations for the ETH log-return series in the 2017-2019 time period. Statis-tically significant coefficients (at the 0.05 level) are marked with ∗ . The application of NHHM modeling in cryptocurrency markets comes with its own limitations. Froma methodogical perspective, the main concerns stem from the decision rule for each state which isprobabilistic and the exogenously given number of hidden states. In the present study, we used thethreshold of 0 . gray zone for time periods in which the algorithm returns probabilities around 0.5 for both states. Thiswill allow for the identification of periods with high uncertainty about the underlying process and hence,will lead to more scarce, yet more uniform (in terms of distributional properties) subseries.From a contextual perspective, the present approach does not account for qualitative attributes ofthe predictive variables. For instance, it does not measure centralization of the transactions or allegedfake volumes among different exchanges ([41,13]). Coupling the present approach with transaction graphanalysis, [35], and user metrics to capture potential market manipulation and the purpose of usagesuch as speculative trading or exchange of goods and services ([22,8,6]) will lead to improved results.Lastly, as more blockchains transition to alternative consensus mechanisms such as Proof of Stake,urther iterations of the current model should also include the underlying technology (e.g., staking versusmining) as a determining factor. At the current stage, such a comparative study is not possible froma statistical perspective, since the market capitalization and trading volume of “conventional” Proof ofWork cryptocurrencies is still not comparable to that of coins with alternative consensus mechanisms.The long-anticipated transition of the Ethereum blockchain to Proof of Stake consensus may define suchan opportunity in the near future.Along these lines, extensions of the current model may enrich the set of covariates (explanatoryvariables) to capture technological features and/or advancements of various cyrptocurrencies, refine theNHPG model with potentially three hidden states and finally, couple the statistical/economic findingswith transaction graph analysis. The expected outcome is a more detailed understanding of the financialproperties of cryptocurrencies and the assembly of a model with improved explanatory and predictiveability for cryptocurrency markets. Acknowledgements
Stefanos Leonardos and Georgios Piliouras were supported in part by the National Research Foundation(NRF), Prime Minister’s Office, Singapore, under its National Cybersecurity R&D Programme (AwardNo. NRF2016NCR-NCR002-028) and administered by the National Cybersecurity R&D Directorate.Georgios Piliouras acknowledges SUTD grant SRG ESD 2015 097, MOE AcRF Tier 2 Grant 2016-T2-1-170 and NRF 2018 Fellowship NRF-NRFF2018-07.
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A.1 Data: Descriptive Statistics and Stationarity Tests
In Table 5, we summarize the descriptive statistics for the BTC and ETH data series, log-prices and thep-values of the necessary preliminary statistical tests that assess the properties of the given data seriesprior to the application of the NHPG model. In detail:
Descriptive statisticsMean & variance:
We report the mean and variance of prices, log-prices and log-returns of BTC andETH. As expected, all series exhibit very high (to extreme) volatility.
Kurtosis:
Based on the kurtosis values, the distributions of all series – except the log-price BTC series– are leptokurtic, i.e., they exhibit tail data exceeding the tails of the normal distribution (values above3), indicating the large number of outliers (extreme values).
Skewness:
Additionally, we report the skewness values, as measure of the asymmetry of the data aroundthe sample mean. If skewness is negative, the data are spread out more to the left of the mean andthe opposite if skewness is positive. We observe that the price series are highly right skewed, whereasthe skewness of the log-returns for both coins are close to 0, indicating an approximately symmetrical,around the mean, series. escriptive statistics Tests
Mean Variance Kurtosis Skewness DF LBQ KPSS VR JB
BTC
Price 3057 1 . × ETH
Price 311 6 . × Table 5: Descriptive statistics (left panels) and p-values of the time series statistical tests (right panels)for the BTC and ETH price, log-price and log-return series. DF denotes the Dickey-Fuller test, KPSSthe Kwiatkowski-Phillips-Schmidt-Shin test, LBQ is the Ljung-Box Q test, VR the variance ratio testand JB the Jarque-Bera test.
Statistical tests
Stationarity captures the intuitive idea that certain properties of a (data generating)process are unchanging. This means that if the process does not change at all over time, it does not matterwhich sample portion of observations we use to estimate the parameters of the process, cf. Sections 3.1and 3.2.
DF-ADF:
First, we report the p-values of the Dickey-Fuller (DF) unit root test . This test assesses thenull hypothesis of a unit root using the model y t = φy t − + (cid:15) t . The null hypothesis is H : φ = 1 underthe alternative H : φ <
1. The H was rejected only in the log-return series. The existence of the unitroot is one of the common causes of non-stationarity. Intuitively, if a series is unit root nonstationarythen the impact of the previous shock (cid:15) t − on the series has a permanent effect on the series. LBQ:
To test for serial autocorrelation on the long-run, i.e., to detect if the observations are ran-dom and independent over time, we used the Ljung-Box-Q (LBQ) test which assesses the presenceof autocorrelations ( ρ ) at lags p in one hypothesis, jointly. The null hypothesis of the LBQ test is H : ρ = · · · = ρ p = 0, under every possible alternative. The null hypothesis was not rejected onlyfor the log-return series and for lags up to p BT C = 10 and p ET H = 6, for BTC and ETH respectively.However, when p BT C >
10 and p ET H > KPSS:
The next column presents the p-values of the Kwiatkowsi, Phillips Schimdt, Shin (KPSS) test.The KPSS test assesses the null hypothesis that a univariate time series is trend stationary againstthe alternative that it is a non stationary unit root process. The test uses the structural model: y t = c t + δ t + u t , c t = c t − + u t where δ t is the trend coefficient at time t , u t is a stationary process and u t is an independent and identically distributed process with mean 0 and variance σ . The null hypothesisis that σ = 0, which implies that the random walk term ( c t ) is constant and acts as the model intercept.The alternative hypothesis is that σ >
0, which introduces the unit root in the random walk. Basedon the p-values, we reject all the hypothesis of trend stationarity of the series.
VR:
Additionally, we report the p-values of the Variance Ratio (VR) test which assesses the hypothesisof a random walk. The random walk hypothesis provides a mean to test the weak-form efficiency andhence, non-predictability of financial markets, and to measure the long run effects of shocks on thepath of real output in macroeconomics, see [21] and references therein. The model under the H is y t = c + y t − + (cid:15) t , where c is a drift constant and (cid:15) t are uncorrelated innovations with zero mean. Therandom walk hypothesis is rejected only in the log-return series for both coins. Essentially, the rejectionof the random walk hypothesis shows that there exists information that can be used in explaining themovement of the returns. JB:
Lastly, we report the Jarque-Bera (JB) test, as a normality test. Based on these results, all theseries are not normally distributed. We also performed the Augmented Dickey-Fuller test with drift c , which assesses the null hypothesis of a unitroot using the model y t = c + φy t − + β ∆y t − + · · · + β p ∆y t − p + (cid:15) t where ∆y t = y t − y t − and lagged operator pp