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Home , Economics of bitcoin, Equation of exchange, Quantity theory of money, Velocity of money

and the Velocity of

Ingolf G. A. Pernice Georg Gentzen Hermann Elendner Weizenbaum Institute Weizenbaum Institute Independent Humboldt-Universität zu Berlin Humboldt-Universität zu Berlin

Abstract to quantify the for cryptocurrencies, and propose a novel one. Velocity of money is central to the , Until recently, meaningful measures for velocity of cryp- which relates it to the general . While the theory tocurrencies did not exist, and most studies resorted to proxy motivated countless empirical studies to include velocity as variables.2 Recent years saw first advances to measure— price determinant, few find a significant relationship in the instead of approximate—the velocity of money. In [2] and short or medium run. Since the velocity of money is gener- [3] the quantity equation of money has first been considered ally unobservable, these studies were limited to use proxy to measure velocity as the ratio of transaction volume and variables, leaving it unclear whether the lacking relationship . In [2], however, this this approach is mod- refutes the theory or the proxies. Cryptocurrencies on pub- ified to create a measure handling the change transactions lic , however, visibly record all transactions, and in systems. While [2] and later [4] focused thus allow to measure—rather than approximate—velocity. on adjusting the transaction volume in the above ratio, we This paper evaluates most suggested proxies for velocity and complement their approach by adjusting the money supply. also proposes a novel measurement approach. We introduce Money in effective circulation should be differentiated from velocity measures for UTXO-based cryptocurrencies focused money held for long-term investment or speculation. Not only on the subset of the money supply effectively in use for the does the total monetary aggregate contain technically dysfunc- processing of transactions. Our approach thus explicitly ad- tional money (burnt ), a major portion of cryptocurrency dresses the hybrid use of cryptocurrencies as media of ex- is stored unused over long time periods (compare [5] or [4]). change and as stores of , a major distinction in recently like [1], [6] or [7] have argued to exclude such proposed theoretical pricing models. We show that each of the funds and focus on money in circulation. To our knowledge, velocity estimators is approximated best by the simple ratio [3] and [2] were first to apply this distinction in theoretical of on-chain transaction volume to total supply. Moreover, cryptocurrency pricing models. Both link feedback effects “coin days destroyed”, if used as an approximation for velocity, from speculation and price levels to a reduction of coins in shows considerable discrepancy from the other approaches. effective circulation. In [3] velocity of money is explicitly defined as based on the component of coin supply in effective 1 Introduction circulation. In this paper, we operationalize this definition for velocity measurement. Velocity of money plays a key role in traditional monetary In implementing this concept, we make common implicit since having been popularized by [1] over a cen- assumptions explicit. For example, the separation of money tury ago. Broadly speaking, velocity of money denotes the into hoarded or circulating depends on the choice of a time average number of transactions per monetary unit within a window. Tokens can be defined as circulating if moved within certain time period.1 In the quantity theory of money, velocity the last day, month, year or any other period. The choice of is related to the price level. While empirical studies frequently [2] and [4], defining money in circulation as the total coin apply this concept to cryptocurrencies, surprisingly few find a supply, implies an infinite time window. The other extreme significant relationship between velocity and prices. We take 2We use measure and estimator interchangeably but contrast them to the this discrepancy as occasion to evaluate current approaches terms proxy variable or approximating variable. The former quantify the addressed concept directly (e.g. length with a yardstick), while the latter rely 1We refer to its definition arising from the “transaction form” of the on the quantification of a distinct concept which is assumed to be correlated quantity theory of money. with the one sought (e.g. wealth via horsepower of owned cars).

1 might be a very restrictive definition requiring coins to be Theoretically, [3] use velocity as central building block of moved within the period for which velocity is measured. As their pricing model. They decompose the velocity of money the optimal time-window might depend on the respective use into a part for monetary units used as media of exchange and case, we operationalize a velocity measure for UTXO-based3 a part for those used as long-term investment. The paper does cryptocurrencies as a function of the respective time-window. not specify, however, how this decomposition could be imple- Subsequently, we apply our approach to and com- mented. Our paper is the first to offer an operationalization pare a variety of potential proxy variables to measures char- for UTXO-based cryptocurrencies. acterizing the two extremes of the design space. Measuring In [2] a measure of velocity of Bitcoin acknowledging the goodness of fit from a variety of perspectives, we show the need for adjustments addressing the transaction volume that the most common proxy-variable, coin days destroyed generated by change transactions is presented. [4] adopt the (CDD)4 in the vast majority of tests shows higher approxi- same concept but provide deeper insights into its technical mation errors than the simple ratio of unadjusted, on-chain configuration as a byproduct of introducing a new transaction volume and total coin supply as shown by a series parser for UTXO-based cryptocurrencies. of Model Confidence Set (MCS) tests. As the majority of Recognizing the need for a more precise method, [18] pro- research opted for CDD, our results might suggest a reason posed a cryptocurrency’s turnover as derivation of CDD. We for the unexpectedly missing relation between velocity and compare this approach to the other methods and show that, prices in most studies. compared to CDD, the measure is indeed closer to the velocity Our implementation is based on the open-source estimates in many tests. blockchain parser BlockSci5. The codebase to calculate the evaluated velocity measures for UTXO-based cryptocurren- 3 Theoretical concepts of measures for the ve- cies is re-usable and will be openly available after publication. In summary, we offer three contributions to research on cryp- locity of money tocurrencies: At its core, velocity of money refers to the average number • a review of approaches to quantify velocity,6 of turnovers per monetary unit within a period of time. This definition stems from the transaction form of the quantity • novel measures based on money in circulation, and theory of money as formalized by [1].7 The central equation • an evaluation of common approximation methods. of the theory equates the flows of real transactions, given by the scalar product hPp,Tpi of prices Pp and transaction volumes Tp, to total money flows, equal to the product of the 2 Literature review money supply Mp and its velocity Vp, where p denotes the time period considered. With n ∈ N this amounts to [1, 8] spawned an extensive literature in n based on what he called the , in which ve- MpVp = hPp,Tpi with Mp,Vp ∈ R, and Pp,Tp ∈ R . (1) locity played a crucial role. Since this literature does not relate to account for the information blockchains make available, The scalar product on the right-hand side is referred to as the and is reviewed at length by [9] we restrict our review to the price sum. In this product, Pp = (Pp1,Pp2,··· ,Ppn) denotes a literature on the velocity of cryptocurrencies. Both theoretical vector of prices Ppt of transacted goods and services in trans- pricing models and empirical studies of price determinants action t during period p. Transaction volumes Tp are given in have addressed velocity. units of goods and services. They are conceptualized as the Empirically, CDD is commonly used as proxy variable in vector Tp = (Tp1,Tp2,··· ,Tpn) with volume Tpt in transaction regressions of cryptocurrency return patterns. Based on the t in period p. On the left-hand side, Mp stands for the number quantity equation, these studies expect a significant positive of all units of money supply available in period p. Vp denotes 8 relationship between prices and their chosen proxy. While the velocity of money. While Tp, Vp and Pp are measured [10] and [11] confirm the hypothesis, more often it is rejected 7The transaction form highlights the use of money as a medium of ex- [12, 13, 14, 15, 16]. Following [1], [2] estimate velocity as change; other forms stress different characteristics. All forms, however, relate the ratio of adjusted on-chain transaction volume to the total money flows to real transactions. While the transaction form is an identity Bitcoin supply when modeling the bitcoin price. Additionally and thus by definition correct, it is nowadays mostly considered impractical for use in -making [cf. 9]. General criticism was spurred by they employ the ratio of off-chain transaction volume and coin [19, 20, 21], who reasoned that an unstable, endogenously forming velocity supply (both denominated in USD) as a velocity estimator. of money would render the equations useless for steering inflation. The re- lation of monetary aggregates and price levels is disputed until this day [cf. 3Descendants of Bitcoin’s approach to build a transaction graph are known 22, 23, 24, 25, 26]. as UTXO-based cryptocurrencies. See Section4 for details. 8 To see this equation less abstractly, imagine an economy with only 2 4The variable is discussed in Section 9.1.1. gold coins, Alice, Bob and sheep Eve. Alice owns the 2 gold coins and Bob 5https://citp.github.io/BlockSci/index.html the sheep Eve. In 2020, Alice buys Eve from Bob for 1 gold coin. Later in the 6Refer to Appendix 11 for a condensed summary of all approaches. year, Bob regrets his decision and buys Eve back for the same price—Alice

2 Figure 1: Blockchain and Transactions (adapted from [17]).

over a time period, Mp is a point-in-time measure. To simplify, 4 UTXO-based cryptocurrencies we assume the money supply is fixed during period p and record it at the period’s beginning pstart. Bitcoin builds its transaction graph chaining transaction out- puts. This approach is followed by many altcoins, referred To develop intuition for velocity Vp, it can be viewed as weighted average number of turnovers of all monetary units to as UTXO-based cryptocurrencies. UTXO refers to the 10 [cf. 9]. The weights are derived from sorting the units into “coins” that can be spent—unspent transaction outputs. Thus, UTXO-based cryptocurrencies record only transactions, groups g ∈ Gp with respect to their number of turnovers vp g in contrast to account-based cryptocurrencies which store a during period p. Velocity Vp then is balance for each address on their blockchain. As an exten- sive exposition is provided by [17], we restrict ourselves to  Npg  Vp = ∑ vpg · , (2) features relevant for calculating velocity measures. ∑ Np g∈Gp g∈Gp g UTXO-based cryptocurrency protocols ensure an or- dered transaction history using a linked chain of so-called with Npg monetary units in group g in period p. Velocity thus blocks (compare Figure1). These blocks contain a hash is the sum of turnover numbers vpg, weighted by their re- (BlockHash) fingerprinting the information of all transac- spective fractions.9 While this definition is intuitive, it cannot tions recorded in the block, a timestamp (BlockTime) and be used to measure velocity in practice. Turnover numbers the hash of the previous block (PrevBlockHash). This con- per monetary unit are neither recorded for fiat , nor stellation of hashes and timestamps establishes pointers that can they be inferred unambiguously for UTXO-based cryp- are determining the order of blocks. The process of creating tocurrencies (compare Section4). In practice, the velocity of new blocks, called mining, creates new monetary units in so money is thus backed out of Equation (1): Vp = hPp,Tpi/Mp. called transactions. Creating blocks usually involves For cryptocurrencies, this appears a simple task, using on- solving a computationally demanding puzzle (proof-of-work) chain transaction volume and total coin supply. However, or proving stake in the existing coin supply (proof-of-stake). due to their technical implementation, transaction volumes Generally, transactions contain a hash as identifier recorded on-chain are distorted. The next section therefore (TxHash) as well as inputs and outputs. Coinbase transac- discusses the relevant subtleties of UTXO-based cryptocur- tions are exceptions as they include an output but no input. rency systems. This effectively increases the amount of spendable outputs, and thus money in the system. Inputs are recorded as links back to outputs of a previous transaction identified by an receives her coin back. The quantity equation states the following: coins !   index (PrevTxHash). Outputs can be sent to addresses 1 sheep 1sheep 2coins · V2020 = h coins , i = 2coins. (Address) corresponding to public keys of an appropri- 1 sheep 1sheep Now the velocity V2020 of the economy’s money can be backed out as 1. ately generated public–private key pair. Unspent previous 9Returning to the example in footnote8, one coin was turned over twice— outputs can be used as inputs only upon proving ownership 1coins 1coins and one not at all. Thus V2020 = 0 · 2coins + 2 · 2coins . This shows how the quantity equation is an identity and an implicit definition of velocity (rather 10For a detailed explanation of UTXO-based and account-based cryptocur- than a testable statement, compare [9]). rency systems see [27].

3 (Signature). Generating this proof requires the private key Note that in practice we only observe the above transaction belonging to the public key which received the output. Im- volume in terms of monetary units rather than the full vector portantly, transaction inputs can only be spent as a whole. If of prices and transacted units. a fraction of the input is to be retained, an additional output must be included which links to an address belonging to the spender. Such addresses are known as change addresses and 5.2 Adjustment heuristics to deflate transac- we refer to the respective outputs as change outputs. tion volumes We refer to all outputs sent to an address controlled by the As discussed in Section4, only addresses but no identities are sender, following [4], as self-churn. Since the concept of user recorded in the transaction . This complicates the classi- identities does not exist in UTXO-based cryptocurrencies, fications of self-churn, needed to calculate adjusted (deflated) there is no direct way to clearly separate self-churn from out- transaction volume. However, statistical properties have been puts transferred to third parties [cf. 28]. Moreover, no relation used to classify outputs as likely belonging to the same indi- between elements of the set of inputs and those in the set vidual user as the transactions inputs (compare [28]). While of outputs of a transaction is determined. In other words, no heuristics, such procedures are now commonly employed to technical link between individual outputs and inputs exists; un- create user clusters of addresses. Outputs are classified as spent outputs are fully fungible. Therefore, velocity measures self-churn if the cluster of their destination address equals the cannot be calculated by following “coins” or UTXOs through cluster of their input addresses. their transactions: due to splitting and rejoining within blocks As our empirical analysis builds on a blockchain parser there exists no “path” the money took. proposed by [4], we follow their choice of heuristics. They employ one heuristic first proposed by [28] and one account- 5 Self-churn and Clustering ing for peeling chains. Peeling chains are transaction patterns where large unspent transaction outputs are split into smaller As discussed in the last section, by construction many cryp- amounts in a chain of transactions. Upon manual inspection tocurrency transactions contain outputs sending a fraction of [4] concluded that outputs created and spent within a rela- the value back to the sender. Such change outputs as well as tively short time period often belong to the same user cluster. other self-churn ought to be excluded from transaction vol- The heuristics used are thus: ume: “What is desired is the rate at which money is used for purchasing goods, not for making change ” ([8]). 1. All inputs in a transaction stem presumably from one person. 12 5.1 Transaction volume inflated by self-churn 2. Outputs created and spent within 4 blocks are classified as self-churn transactions. While the transaction volume in principle can be calculated ac- cumulating the output values valOut(o)11 of all transactions t recorded within period p this yields an inflated aggregate 6 Velocity based on total money supply 0 0 hPp ,Tp i when defined as The quantity equation (1) requires a measure of the money 0 0 supply M . To our knowledge, all prior work has employed hPp ,Tp i = ∑ ∑ valOut(o), (3) p t∈Tp o∈Ot the sum total of ever-mined coins as its measure. We denote this money-supply measure Mtotal p and calculate it at the 13 with o ∈ Opt the set of all outputs of transaction t in period p beginning of each period p. In all common UTXO-based and Tp the set of all transactions t recorded within period p. cryptocurrencies it is a deterministic function of block height. selfchurn Thus, this volume needs to be adjusted. Defining O as Technically, Mtotal p can be calculated as the aggregate of the set of all self-churn outputs, the accumulated transaction outputs from the set of coinbase transactions coinbase be- TP volume from these outputs Op can be obtained by summing longing to the set P of all periods with a maximum block time selfchurn the individual self-churn outputs c ∈ O as smaller pstart and thus O = (o). p ∑ ∑ valOut (4) Mtotal p = valOut(o). (5) selfchurn ∑ t∈Tp o∈O t∈ coinbase TP

Hence a corrected transaction volume can be calculated as 12Note that [4] added the additional restriction that this heuristic is not 0 0 hPp,Tpi = hPp ,Tp i − Op. applied to coinjoin transactions which are used to obfuscate transaction paths. Coinjoin transaction are classified as in [29]. 11To simplify notation, we denote many variables as functions of transac- 13As discussed in Section3, we simplify by assuming a fixed money supply tions, outputs or inputs. For example, function valOut(o) extracts the output over period p, using the amount at the beginning of the period. Neither [4] value of output o of transaction t in period p. nor [2] clearly specify their adopted choice.

4 Based on the quantity equation (1), the simplest measure of 0 0 velocity arises from dividing total transaction volume hPp Tp i Npg by total coin supply Mtotal p, which was described in [3] and Vcirc p = ∑ vpg , (10) 0 ∑g∈ 0 Np adopted by [30]. Formally, g∈Gp Gp g 0 0 msr hPp ,Tp i Hence, the measure can be interpreted as the average number Vtriv p = . (6) Mtotal p of turnovers of units effectively circulating in period p. Drop- ping con-circulating money in Equation (10) amounts to an V msr offers the advantages of providing a theoretically triv p adjustment of the money supply Mp in the quantity equation sound interpretation and extremely simple calculation. More- (1). msr over, data for calculating Vtriv p (raw on-chain transaction volume and total coin supply) are widely available. 14 How- ever, the result is biased: Self-churn transactions lead to an 7.2 Theoretical basis and practical considera- overestimation of transaction volume. tions [2] and [4] propose a similar velocity measure. However, An advantage of basing velocity on money in effective cir- they clean the price sum from self-churn and get culation is higher information content. To begin with, it is msr msr hPp,Tpi questionable whether Vtriv and Vtotal capture more informa- msr 0 0 Vtotal = . (7) tion than transaction volumes hPp,Tpi, respectively hPp ,Tp i. Mtotal p After all, for most UTXO-based cryptocurrencies money sup- In line with Section3, both measures can interpreted as the ply is just a simple function of block height.15 Thus, the two turnover of coins during period p averaged over the total coin former measures appear very close to merely scaled versions supply. of their price-sums. msr msr Moreover, the total coin supply used in Vtriv and Vtotal includes money that is technical dysfunctional (burnt coins). 7 Velocity of money in effective circulation Yet also coins held unused as storage of wealth16 or specula- tion17 do not fulfill one of the key functions of money, use as While these measures advanced quantifying the velocity of . cryptocurrencies, they suffer from inaccuracy as money sup- Furthermore, the amount of money frozen in speculative ply is defined as “the aggregate of all monetary units ever investments might not be neutral to money flows or prices. M issued” ( total p). We therefore propose a measure based on Since the beginnings of monetary economics, spec- the component of money that circulates effectively. Denoting ulation has been associated with patterns in price levels. In M this circulating amount circ p, our measure yields [36] and [37], the illiquid component is considered a reservoir for neutralizing demand shocks and excluded from the money msr hPp,Tpi Vcirc p = . (8) supply. For [7] and [6] hoarded money, as destroyed money, Mcirc p is leakage which must be compensated to stabilise the price level. [1] associates the rise in market prices for one of the 7.1 Formal derivation early fiat U.S. bank notes with a relation of speculation and circulating money as well: “speculation acted as a regulator To see how excluding hoarded money relates to the quantity of the quantity of money.” theory, expand the sum in Equation (2) and differentiate the Our proposed measure captures precisely this velocity of set of all monetary units treated as investment, h, from its money in effective circulation. This perspective is already complement 0 := \{h}: Gp Gp present in current theoretical research on cryptocurrency prices. For instance, the model of [3] distinguishes demand msr Nph Vtotal p = vph  (9) for transactions from demand for rational speculation. When N + 0 N ph ∑g∈Gp pg using the quantity equation, they deduct coins bought and Npg held as speculative investment. Their modified quantity equa- + ∑ vpg . tion relates the exchange rate between fiat and cryptocurrency 0 Np + ∑ 0 Np g∈Gp h g∈Gp g Sp to the velocity of cryptocurrency in effective circulation

By definition, h encompasses only non-transferred units, and 15Difficulty adjustments for mining lead to close-to-constant spans be- thus vph = 0 in period p. Consequently tween block creation times [17]. 16Compare [31] for a detailed discussion of the discrepancy arising from 14Time-series data can be downloaded for free from www.blockchair. money as storage of wealth. com, www.blockchain.com, www.btc.com. Also www.blockwatch.cc and 17Most studies conclude cryptocurrencies are currently used more for long- many other data brokers provide the data. term speculation rather than as media of exchange (see e.g. [32, 33, 34, 35]).

5 TxA TxF

Input Output TxC Input Output

Out1 Input Output Inp1 Out1

Inp1 Out2 Out1 Inp Chng Chng Inp1 2 Chng

TxB Tx Input Output TxD E Input Output Input Output

Inp1 Out1 Inp1 Out1 Out Out1 2 Inp2 Chng

p w

Figure 2: An example of a transaction chain.

∗ Vcirc p and the volume of transactions Tp denominated in which monetary units are to be classified as circulating or units of cryptocurrency as not. Money is referred to as circulating if it has been moved

∗ economically within the last day, month, year or generally Tp /Vcirc p w [w ,w ] Sp = . (11) any time period covering start end . Mcirc p To identify the fraction of the money supply which circu- lated within w, we step trough every transaction recorded in The speculators in the model purchase cryptocurrency units period w. Transactions spending outputs generated before w if traded prices lie below their risk-adjusted, discounted ex- as well as outputs from coinbase transactions (seignorage) pected future price. With increasing aggregate speculative have the interpretation of bringing an amount into circula- positions, the risk of marginal speculative investments rises, tion that corresponds to the value of spent outputs. All inputs lowering the price speculators are willing to pay. On the other referring to UTXOs generated within period w, on the other hand, higher speculative positions imply reduced M and circ p hand, re-spend money which has already been counted as increase current price according to their modified quantity circulating.18 equation. A similar relation between money in circulation and Note that we define the time window w within which spent speculation is modelled in [2] as well. coins are considered as circulating distinct from the period Hence, implicitly or explicitly theoretical research already p for which the velocity measure is calculated, [p , p ]. employs velocity based on circulating money rather than the start end This distinction can be parametrized via a maximum length total money supply. We close the gap in empirical research by ` of the look-back window w, where w = p − ` and operationalizing the circulation-based velocity measure for start start w = p . UTXO-based cryptocurrencies. end end The approach as characterized so far, however, does not account for two technical properties of UTXO-based cryp- 8 Novel measures of velocity of money in effec- tocurrencies: First, transactions always spend prior transaction tive circulation outputs in full. Second, there exists no attribution of individ- ual outputs to the inputs of a transaction; thus it remains Based on the concept of velocity of money in effective circu- undefined which input corresponds to which output(s). lation as clarified last section, we now propose algorithms to 18 calculate velocity for UTXO-based cryptocurrencies. To develop an intuition, remember the example of sheep Eve being sold and re-bought by Bob in 2020. This time Bitcoin was used and Alice held unspent outputs of 2 bitcoin from transactions 5 years ago. The transaction formed by Alice to buy sheep Eve, Tx1, spends her five-year-old output and 8.1 Determining money in circulation generates a new one in 2020 which is controlled by Bob. When Bob buys Eve Conceptually, a monetary unit has circulated in a period if back, he pays with the output of 1 bitcoin from Tx1 in a second transaction (Tx2). For w = p = 2020, the value of Tx1 (1 bitcoin) originated prior to and only if it has been used as a medium of exchange. There- 2020 and is thus money brought into circulation. Tx2, however, spent an fore, first a time window must be specified with respect to output generated in 2020—an already counted monetary unit.

6 The first property raises the question how to deal with McircWba p and simply calculated as transactions which send back change: Should the sum of all inputs be considered in circulation, as technically all was msr hPp,Tpi VcircWba p[`] = msr . (12) transferred—or should only the fraction sent to third parties McircWba p[`] be considered? We analyze both choices, naming the first whole-bill approach (WB approach) and the second moved- VcircWba p is recommended if a conservative measurement of coin approach (MC approach). They are visualized in Fig- coin turnover is sought or additional assumptions about how to ure2. The moved-coin approach considers only output Out1 link inputs to outputs should be avoided. The second and third of transaction TxC as circulating, not the change output. This measures are based on McircMcaFifo p[`] and McircMcaLifo p[`], approach captures the net economic value transferred to a respectively: third party. The WB approach classifies the whole input of hP ,T i transaction Tx as circulating. This approach captures the msr p p C VcircMcaFifo p[`] = msr , (13) amount of money that has been moved technically; it can also McircMcaFifo p[`] be interpreted as revealed to be available for transactions. msr hPp,Tpi VcircMcaLifo p[`] = msr . (14) In the MC approach, the sum of inputs is counted as circu- McircMcaLifo p[`] lating only net of change outputs. However, due to the second msr msr property of UTXO blockchains ambiguous constellations can VcircMcaFifo p and VcircMcaLifo p are more stringent on the defi- occur: If for a given transaction one input was generated nition of money in circulation. Their monetary aggregates do within and one before w, it remains unspecified which one not count “touched” funds, but only amounts transferred to corresponds to the change output. somebody other than the sender. However, this comes at the cost of the additional assumption with respect to the assign- Transaction TxF in Figure2 illustrates the point. It has ment rule between transaction inputs and outputs. two inputs: Inp1 originated before period w and Inp2 origi- nated within w. If only amounts sent to third parties matter, it remains unclear which of the two inputs funded the change 8.3 Algorithmic implementation output. For Inp1, the transaction would not increase money in Having defined three velocity measures, we now detail our circulation. In contrast, if Inp2 funded the change output, the technical approach and the implementation. transaction would increase the amount in circulation by Inp1. Money in circulation under the WB approach is measured To resolve the ambiguity, an assignment rule between trans- as in Algorithm1. For every period w, we loop over all trans- action inputs and outputs is required. We consider both end- actions t ∈ Tw and add their inputs to circulating money points on the spectrum of the age of the input assigned to if they either reference outputs from coinbase transactions, the change transaction and thus differentiate between "last-in- denoted by genByCoinbase(i), or outputs with timestamps first-out" (LIFO), where oldest inputs get assigned to outputs dateGen(i) before the first timestamp wstart of period w. first, and "first-in-first-out" (FIFO), where it is the other way Measuring money in circulation under the MC approach around. is depicted in Algorithm2. As in Algorithm1, for time win- Naturally, with the WB approach this differentiation is dow w we loop over all transactions t ∈ Tp and add inputs void. Hence, we have three definitions of money in cir- based on the same core condition (compare lines2-15 and culation, each a function of the activity window length `: 1-6). This time, however, inputs are added step by step— Money in circulation for period w adopting the WB approach only until the amount sent to third parties is reached. The order of inputs to consider is determined by the LIFO or (McircWba p[`]), and both the MC approach with the LIFO rule FIFO principle. For every transaction t in time window w, the (McircMcaLifo p[`]) and the FIFO rule (McircMcaFifo p[`]). amount sent to third parties is determined net of self-churn toOthers 0 as valOut (t) = 0 valOut(o) where denotes ∑o∈Ot Ot non-self-churn outputs of transaction t. If all outputs are iden- 8.2 Definition of velocity measures tified as self-churn, valOuttoOthers(t) = 0 and the algorithm continues with the next transaction. If valOuttoOthers(t) > 0, the algorithm collects input values in a vector sort; they are Based on the above definitions of money in circulation, three It sorted in either ascending (LIFO) or descending (FIFO) or- variations of velocity can be calculated in accordance with der w.r.t. to the timestamp when the UTXOs were generated. Equation (8)—one per money aggregate. All measures cap- Then, looping over inputs i, input values valInp(i) are added ture the average number of peer-to-peer coin turnovers of ef- to M (t) if they meet the core condition (compare fectively circulating monetary units in period p. They concur circMca p[`] 19 in capturing on-chain liquidity; they differ w.r.t. to definitions line 17) introduced in line 6 of Algorithm1. However, one of circulating monetary units and assignment rules linking 19 Summands McircMca p[`](t) can be interpreted as money drawn into ef- transaction inputs to outputs. The first measure is based on fective circulation by transaction t.

7 additional condition applies: If the last added input would 9 Benchmarking popular velocity proxies on increase the summand McircMca p[`](t) beyond the value of the Bitcoin blockchain outputs sent to third parties valOuttoOthers(t), we only add up to the latter amount. Equipped with our proposed velocity measures that are cal- culated on a level of detail of each output in each transaction in each block on the blockchain, we can now use these mea- 8.4 Manipulation potential sures to empirically assess the quality of popular proxies for velocity. One important concern with any measure of economic activity To this end, we first review the most common proxies in asks, how amenable is it to manipulation? In the case of veloc- Section 9.1, then implement the proposed estimators to calcu- ity, the question is specifically if the measure can be inflated late velocity measures for Bitcoin in Section 9.2, and finally by a single agent (or a small group) of limited means. After run tests in Section 9.3 to evaluate the goodness of fit of the all, one proxy variable for velocity has been designed as a proxies w.r.t. the measures. manipulation-proof alternative to turnover (see Section 9.1.1). How easy would it be to create fake velocity in order to inflate our measures? 9.1 Popular proxy variables for velocity of Indeed, no direct technical impossibility prevents gener- UTXO-based cryptocurrencies ating transactions affecting Equations (12)–(14). Nonethe- less, there exist reasonably tight limits to the manipulation 9.1.1 Coin Days Destroyed potential of our measures, in particular compared to trading volumes on exchanges. Since turnover can be gamed via repeated transfers of an 20 First and foremost, calculating our measure on on-chain agent to herself, the CDD measure was introduced as .org transactions puts an upper limit to fake transfers. First, a fake a manipulation-proof alternative in in 21 coin days transaction needs to be committed to the blockchain, and 2011. For each input, refer to the product of its thus can be repeated only once per block time. As long as monetary value and the number of days “since it was last the manipulator is not the miner confirming the next block spent,” i.e. how many days the funding output had remained (a random and unlikely outcome even for large pools), the a UTXO. CDD then is defined as the sum of coin days over p on-chain settlement is also costly, incurring fees. all transactions within a period : Second, the manipulator must fully fund her fake transac- V app = cdd(t), (15) tions: she cannot send more than she owns once per block. cdd p ∑ t∈Tp The question then turns to how she should structure the fake transfers. Send the entire amount in a single transaction, or where split it up into as many as can fit into a block? In the latter case, the fees rise. In the former, should she minimize or max- cdd(t) = ∑ ∆(i) · valInp(i) (16) imize the number of outputs? If she maximizes, the fees rise i∈It again, and she quickly fragments her wealth across a diverg- ing multitude of wallets, reducing the funds available in each with ∆(i) denoting the number of days since the respective for the next round (block) of manipulation. input originated as output (or coinbase transaction) in a prior In this context it is critical that we follow the literature in block, and valInp(i) the value of input i of transaction t in clustering user addresses: As long as the clustering works, period p. the manipulator cannot combine her funds ever again, lest she The CDD measure puts largeer weight on the reactivation would be deconspired as a single agent and all her fake trans- of long-dormant coins compared to ones frequently spun, a actions disregarded. It follows that her strategy minimizes feature clearly differing from the concept of velocity. the number of outputs, generating a chain of fresh addresses across which a large sum, tied up in the manipulation, tra- 9.1.2 Turnover based on dormancy verses indefinitely to make-believe “newcomers.” This, how- ever, matches peeling transactions, which we also exclude. In [18] a proxy for velocity, aiming at “the average number of She would thus need to break frequently enough in order to times the actively used [coins] can be expected to turn over” escape this classification, slowing her manipulation further. was proposed. While named “turnover” in [18], this is not In sum, manipulation attempts are both technically elab- 20 orate and relatively straightforward to detect and exclude in Primarily, allegations centered on exchanges inflating trading volumes to pretend to be more liquid. the future. We thus do not consider manipulation a serious 21https://bitcointalk.org/index.php?topic=6172.msg90789# concern for our current results, nor for our measures. msg90789

8 Algorithm 1: Whole-bill-approach: Measurement of McircWba within period p with look-back window w. Data: wstart /* Beginn of look-back window w */ Data: Tp /* Set of transactions in period p */ Result: McircWba /* Money supply to be estimated within period p */ 1 McircWba ← 0 2 foreach t ∈ Tp /* Loop through transactions t of period p */ 3 do 4 foreach i ∈ It /* Loop through inputs i of transaction t */ 5 do 6 if dateGen(i) < wstart or genByCoinbase(i) /* Check, whether i stems from a coinbase transaction generated before wstart */ 7 then 8 McircWba = McircWba + valInp(i) 9 end 10 end 11 end 12 return McircWba /* Return estimated money in circulation for WB approach */

Algorithm 2: Moved-coin-approach: Measurement McircMca for type (FIFO, LIFO) within period p, window w. Data: wstart /* Beginn of look-back window w */ Data: Tp /* Set of transactions in period p */ Data: Oselfchurn /* Set of self-churning outputs */ Result: McircMca /* Money supply to be estimated in period p with window w */ 1 McircMca ← 0 2 foreach t ∈ Tp do 3 McircMca(t) ← 0 /* Set summand of McircMca(t) for every transaction t */ 0 selfchurn 4 Ot ← Ot \ O /* Determine the set of outputs of t excluding self-churning outputs */ toOthers 5 valOut (t) ← 0 valOut(o) /* Amount of money sent to third parties */ ∑o∈Ot toOthers 6 if valOut (t) = 0 then 7 continue /* Skip current transaction and go to the next */ 8 end 9 if type = LIFO then sort 10 It ← It sorted ascending w.r.t. generation time /* Here, It is the set of all inputs of t */ 11 else sort 12 It ← It sorted descending w.r.t. generation time 13 end sort 14 foreach i ∈ It do 15 if dateGen(i) < wstart or genByCoinbase(i) /* Check, whether i stems from a coinbase transaction generated before wstart */ 16 then 17 McircMca(t) ← McircMca(t) + valInp(i) toOthers 18 if McircMca(t) > valOut (t) then toOthers toOthers 19 McircMca(t) ← valOut (t) /* Use valOut (t) as upper cap for McircMca(t) */ 20 end 21 break /* Break foreach-loop on line 14 and proceed with line 24*/ 22 end 23 end 24 McircMca ← McircMca + McircMca(t) 25 end 26 return McircMca /* Return estimated money in circulation for MC approach */

9 Table 1: Descriptives for velocity proxies and measures with p = ` = 1 day for 2013-06-01 to 2019-06-01.

Obs. Mean Med. Min. Max. Std. Dev. Kurtosis app Vcdd 2174.00 9.78 5.72 0.87 361.99 16.22 137.39 app Vturn 2174.00 98.66 80.74 1.63 2635.37 93.81 255.22 msr VcircMcaFifo 2174.00 4.20 3.95 1.36 22.42 1.41 23.92 msr VcircMcaLifo 2174.00 4.20 3.95 1.36 22.42 1.41 23.92 msr VcircWba 2174.00 2.53 2.36 0.77 15.70 0.94 36.57 msr Vtriv 2174.00 0.11 0.08 0.02 4.46 0.13 577.05 msr Vtotal 2174.00 0.04 0.03 0.01 0.46 0.02 89.18

to be confused with the classical turnover that refers to the [4] (see Section 5.2), in order to mitigate the amount and simple sum total of transacted amounts within a period p: effect of false positives we exclude one unreasonably large cluster with over 297 million addresses. 24 app Vturnover p = ∑ ∑ valInp(i). (17) We calculate our velocity measures based on a time window t∈ i∈ Tp It for money in active circulation ` equal to period p = 1 day.25 The measure in [18] in contrast is constructed as the inverse Therefore, a coin is in circulation if it is transferred at least of average dormancy dormp multiplied by the time period, once within the day for which velocity is computed. This daily where dormancy amounts to CDD scaled by the sum of inputs measure can be interpreted as average turnover of monetary spent during the period p [cf. 18]: units which are part of the daily circulating money supply. While our choice for the window ` within which moved coins app 1 are considered actively in circulation is short, this allows us Vturn p = · δ, (18) dormp to flesh out the difference between velocity based on the total money supply and our proposed measures based on circulating with supply most clearly. Moreover, since [4] provide results with app an implicitly26 infinite `, we provide evidence on the opposite Vcdd p dormp = . (19) endpoint on the spectrum. Naturally, neither the BlockSci ∑ ∑ valInp(i) t∈Tp i∈It parser nor our approach are restricted to these choices and Here δ refers to the length of the period for which turnover is can be extended both to other time windows and other UTXO- calculated. To illustrate the concept, assume the coins spent based cryptocurrencies. today stayed unused on average for 6 hours before their trans- Table1 shows descriptive statistics for the proxies and the app app 24h measures. V denotes CDD in million coin days, while Vturn action. Hence, circulating coins are turned over 6h = 4 times cdd per day on average. Dormancy-based turnover, however, re- denotes active turnover in expected on-chain coin transfers. mains an approximation depending on transactions distributed For both proxy variables, means exceed medians, suggesting msr homogeneously over time. outliers in the skewed distribution. According to Vtriv, a mon- etary unit of the total coin supply is turned over 0.11 times on msr average, while the more sophisticated measure Vtotal results 9.2 Data sources and calculations in turnover of 0.04. The difference stems from the deflated msr We collect a dataset spanning from June 2013 until June 2019, transaction volume used by Vtotal being strictly lower then starting with the rise of the first cryptocurrency exchanges the inflated one (see Section5). According to the measure msr and thus reliable trading data. VcircWba, which is based on the WB approach, coins in ef- For the proxy variables, we tap existing sources as far as fective circulation during the day reach turnover of around possible. CDD data is gathered via API from Blockwatch,22 2.5. Assuming the clustering heuristics work well, coin trans- 23 msr trading data taken from CoinMarketCap. fers correspond to peer-to-peer hops. Accordingly, VcircWba To calculate the velocity measures, however, access to estimates that monetary units in circulation change owners msr msr the atomic units of cryptocurrency transactions is needed. 2.5 times per day, while VcircMcaFifo and VcircMcaLifo give an We rely on the open-source blockchain parser BlockSci in- estimate of around 4.2 peer-to-peer hops. The reason for the troduced by [4]. We also provide the code of our paper higher turnover estimate is the more conservative operational- at https://github.com/wiberlin/cryptocurrencies_ and_velocity/. In replicating the clustering approach of 24Only 13 clusters contain more than 20,000 addresses. 25We define days based on transaction block times and UTC timezone. 22https://blockwatch.cc 26[4] equate money in circulation with total coin supply, implying an 23https://coinmarketcap.com infinite time window.

10 15 Vol 0.33

10 Mtotal 0.46 0.64 BTC (in m) 5 msr VcircWba 0.35 0.69 0.33

0 msr Vtotal 0.66 0.3 0.98 0.23 2014 2016 2018 Date

BTC Vol / msr total Vol.(deflated) Mtotal M USD V circWba P

McircWba

(a) Components of velocity measures. (b) Correlations between the measures and price.

Figure 3: Descriptives for velocity proxies and measures with p = ` = 1 day for 2013-06-01 to 2019-06-01.

being in circulation stand X−µ(X) ization of the concept of (see Section 8.3). is based on Z-scores X = σ(X) , with mean µ(·) and The different levels can be disentangled by looking at the standard deviation σ(·). Both are sensitive to outliers [cf. 38], components of the velocity measures. Figure 3a exemplarily so we truncate at 10 standard deviations around the mean.28 msr msr shows this for the components of Vtotal and VcircWba. While Figure4 shows the time series of proxy variables for velocity. Mtotal increases steadily over time, the subset of coins trans- The scaling leads to a visible difference, relevant when com- acted at least once per day, with an average 1.5 % of the total paring proxies to measurement results. A first indication of supply, is minuscule but volatile in comparison.27 Deflated the quality of the proxy variables is their diversity. Not only on-chain transaction volume varies widely—clearly always spikes but also general trends vary across methods. Figure5 below total supply, yet above the supply in circulation. While depicts the different velocity measures. Differences across msr the in the transaction volume feeds fully into Vtotal, measures are smaller, highs and lows correspond more. The msr for VcircWba the relation is less obvious. In Figure 3b, the com- next section provides quantitative evidence. ponents’ co-variation with bitcoin price indicates that not only deflated on-chain transaction volume Vol, but also the mone- tary aggregates are positively correlated with price changes. 9.3 Assessing the goodness of fit of the proxies While this provides only anecdotal evidence, it is consistent We now turn to evaluate the proxy variables for velocity based with the hypothesis that transaction volumes as well as the on our estimated measures. In so doing, we include the trivial liquid component of the money supply rise with price, and msr velocity measure Vtriv in the set of proxies, as it is calculated in consequence lower the correlation of the velocity measure on very high-level inputs (raw on-chain transaction volume V msr circWba with price. This hypothesis is related to the broader and total coin supply) that are similarly easy to obtain as CDD question how far speculation might act as “regulator of the or turnover. quantity of money” [1]. In order to evaluate the quality of proxy variables perfectly, Comparing maxima and minima across time series, as Ta- the “true” velocity ought to be known. At the same time, we ble1 shows, requires scaling. We use two methods: normaliza- have proposed three novel measures, in addition to those sug- tion and standardization. Normalization is based on the usual gested by [2] and [4]. We do not view this a contradiction: norm X = (X − Xmin)/(Xmax − Xmin) ∈ [0,1]. Standardization after all, the measures capture different concepts of velocity. Ours, for example, address the velocity of money in circula- 27Compare appendix5,6,7 and8 for additional descriptive statistics on the dataset. 28This procedure truncated a maximum of 3 values across time series.

11 12.5 1.00

10.0 0.75 7.5

0.50 5.0 Misc.

2.5 0.25 Stnd. est. avg. turnovers 0.0 0.00 2014 2016 2018 2014 2016 2018 Date Date

app app Vcdd Vcdd

app app Vturn Vturn

(a) Normalized variables of volatility approximation. (b) Standardized variables of volatility approximation.

Figure 4: Time series plots for proxy-variables. tion. Therefore, we do not horserace all proxies against one 9.3.2 Model confidence set test benchmark, but rather evaluate the goodness of fit of the proxy To understand if the approximation methods indeed differ variables with any of the measures. We do, however, take the significantly, we perform Model Confidence Set (MCS) tests. position that the measures are more precise estimators of “true [39] introduced MCS tests to measure the performance of velocity” (its respective concepts) than the proxies, and thus forecasting methods, but they are applicable more generally. evaluate the latter in terms of their fit to the former. The method uses equivalence tests and an elimination pro- Perhaps surprisingly, the results do not vary qualitatively cedure to determine which set of models significantly out- much across different measurement approaches. performs the rest of the models. For an intuitive understand- ing of the test one might think of a set M of models (here the different approximation methods). The models are in- 9.3.1 Approximation errors dexed using i and j ∈ 1,..,m. When comparing model i to a benchmark, for each period p model i leads to loss func- A common approach assesses approximation errors by com- msr app msr tions Lip = L(V ,V ), where V denotes the benchmark paring mean squared errors or mean absolute errors. MSE, p ip p velocity measure and V app the approximation method i. by squaring them, punishes large deviations more rigorously ip To compare the performance of the models in M, relative than the linear MAE. We provide both, not only to the stan- performance metrics d are defined as dardized and normalized time series, but also on their first i jp differences. This is motivated, as in econometric studies like di jp = Lip − L jp those in Section2, by doubts about the stationarity of the time series. for all i, j ∈ M where i 6= j. We use squared errors for the loss Table2 shows that the above transformations differ in their function L, so assessment of the goodness of fit of approximation methods. msr app2 msr app2 di jp = V −V − V −V . When judging by the normalized but undifferenced dataset, p ip p jp app turnover Vturn (as alternatively defined by [18]) achieves the The relative performance of model i compared to all other lowest error in approximating velocity measures based on models then is effectively circulating money supply. However, for almost 1 all other constellations we find that the trivial measure V msr di· = di j, triv m − 1 ∑ provides the closest fit to all ways to measure velocity. j∈M\i

12 15 1.00

0.75 10

0.50 5

0.25 0 Stnd. est. avg. turnovers Norm. est. avg. turnovers 0.00 2014 2016 2018 2014 2016 2018 Date Date

msr msr msr msr Vtriv Vtotal Vtriv Vtotal

msr msr msr msr VcircWba VcircMcaLifo VcircWba VcircMcaLifo

msr msr VcircMcaFifo VcircMcaFifo

(a) Normalized variables of volatility measurement. (b) Standardized variables of volatility approximation.

Figure 5: Time series plots for volatility measures with p = ` = 1 day.

msr with i = 1,...,m. The null hypothesis states the trivial measure Vtriv. Only in cases with time trends and less focus on outliers than on smaller changes, turnover might app H0M : E(di·) = 0, ∀i ∈ M. be the better choice. Again, Vcdd , the most common proxy, is significantly outperformed in most constellations. If it can be rejected for set M, there exist models in the set that are significantly outperforming the remaining ones. The above equivalence test is then re-iterated after elimination 9.3.3 Mincer-Zarnowitz regressions of the worst-performing model(s) until the null hypothesis The Mincer-Zarnowitz (MZ) approach regresses estimates on cannot be rejected. For details, see [39]. the benchmark in simple ordinary-least-squares regressions Table2 displays the MCS tests’ results, denoting signif- [40]. In order to avoid spurious regression results, we only icance at the 1 % level with †. Results for the differenced use the first differences of the standardized and normalized dataset are clear: the MCS tests select even at the tight 1 % series. The regression structure for proxy i can be expressed significance levels a single winning approximation method as: for each constellation. This evidence confirms Section 9.3.1 msr at high significance levels. V is the single best element of app triv ∆V msr = α + β ∆V + u . the 1 %-MCS for all constellations. For certain constellations ip i i ip ip of the raw dataset, differences in approximation errors are An ideal proxy would yield an intercept of 0 and a β equal too small for the MCS test to determine a unique winner.29 to 1 with an adjusted R2 of 1. The results of the MZ regres- While for the normalized undifferenced dataset the turnover sions in Table3 confirm the simple ratio of inflated on-chain V app is the single best proxy for velocity based on circulating turn transaction volume to total coin supply V msr as superior. This money for all but one of the constellations, for V msr the triv circWba simple ratio yields an adjusted R2 between 0.04 and 0.14; trivial measure remains in the model confidence set as well. other proxies’ R2 is close to zero for all measurement methods If velocity is measured as V msr , however, the trivial measure total of velocity. While for none of the approximations a signifi- V msr again performs significantly better than all other proxies. triv cant intercept indicates bias, the slope coefficients β for most For the standardized undifferenced data, MCS tests remain constellations are significant and positive. Furthermore, with mostly inconclusive. In summary, MCS tests mostly support msr respect to the β coefficients, Vtriv performs best. The latter 29While we show the results only for the 1 % significance level, MCS tests shows coefficients between 0.35 and 0.51 for normalized and for the 5 % or 10 % levels yield the same conclusions. between 0.28 and 0.42 for standardized data.

13 Table 2: Mean absolute error for normalized data of approximation methods compared to measurement methods with p = ` = 1 day. Proxy-variables in 1 % Model Confidence Set marked by †.

Raw Data First Differences standardized normalized standardized normalized Approximation Estimator MAE MSE MAE MSE MAE MSE MAE MSE app msr Vcdd Vtotal 2510.61 2510.61 27.33 27.33 3376.81 3376.81 23.47 23.47 app msr † † Vcdd VcircWba 3790.03 3790.03 50.40 50.40 4203.97 4203.97 29.37 29.37 app msr † † Vcdd VcircMcaLifo 4089.56 4089.56 57.60 57.60 4616.46 4616.46 32.26 32.26 app msr † † Vcdd VcircMcaFifo 4089.56 4089.56 57.60 57.60 4616.46 4616.46 32.26 32.26 app msr Vturn Vtotal 4274.63 4274.63 27.59 27.59 3205.86 3205.86 19.24 19.24 app msr † † Vturn VcircWba 4247.09 4247.09 35.38 35.38 3619.50 3619.50 22.42 22.42 app msr † † † † Vturn VcircMcaLifo 4110.31 4110.31 37.76 37.76 3770.22 3770.22 23.55 23.55 app msr † † † † Vturn VcircMcaFifo 4110.31 4110.31 37.76 37.76 3770.22 3770.22 23.55 23.55 app msr † † † † † † † † Vtriv Vtotal 2336.82 2336.82 20.06 20.06 1409.80 1409.80 7.81 7.81 app msr † † † † † † † Vtriv VcircWba 3827.77 3827.77 40.65 40.65 2119.77 2119.77 12.54 12.54 app msr † † † † † † Vtriv VcircMcaLifo 3897.17 3897.17 45.66 45.66 2384.19 2384.19 14.33 14.33 app msr † † † † † † Vtriv VcircMcaFifo 3897.17 3897.17 45.66 45.66 2384.19 2384.19 14.33 14.33

Table 3: Model Confidence Set regressions for standardized and normalized approximation and measurement data with p = ` = 1 day.

Normalized Standardized 2 α β F−Test 2 α β F−Test Approximation Estimator Rad j α p β p p Rad j α p β p p app msr ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∆ Vcdd ∆ Vtotal 0.03 0.00 0.97 0.12 0.00 0.00 0.03 0.00 0.97 0.12 0.00 0.00 app msr ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∆ Vcdd ∆ VcircWba 0.01 0.00 0.97 0.06 0.00 0.00 0.01 0.00 0.97 0.06 0.00 0.00 app msr ∗∗∗ ∗∗∗ ∆ Vcdd ∆ VcircMcaLifo 0.00 0.00 0.98 0.03 0.14 0.00 0.00 0.00 0.98 0.03 0.14 0.00 app msr ∗∗∗ ∗∗∗ ∆ Vcdd ∆ VcircMcaFifo 0.00 0.00 0.98 0.03 0.14 0.00 0.00 0.00 0.98 0.03 0.14 0.00 app msr ∗ ∗∗∗ ∗ ∗∗∗ ∆ Vturn ∆ Vtotal 0.00 0.00 0.97 0.03 0.07 0.00 0.00 0.00 0.97 0.03 0.07 0.00 app msr ∗∗ ∗∗∗ ∗∗ ∗∗∗ ∆ Vturn ∆ VcircWba 0.00 0.00 0.97 0.05 0.02 0.00 0.00 0.00 0.97 0.04 0.02 0.00 app msr ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∆ Vturn ∆ VcircMcaLifo 0.00 0.00 0.98 0.07 0.00 0.00 0.00 0.00 0.98 0.06 0.00 0.00 app msr ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∆ Vturn ∆ VcircMcaFifo 0.00 0.00 0.98 0.07 0.00 0.00 0.00 0.00 0.98 0.06 0.00 0.00 app msr ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∆ Vtriv ∆ Vtotal 0.14 0.00 0.97 0.50 0.00 0.00 0.14 0.00 0.97 0.42 0.00 0.00 app msr ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∆ Vtriv ∆ VcircWba 0.05 0.00 0.98 0.38 0.00 0.00 0.05 0.00 0.98 0.30 0.00 0.00 app msr ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∆ Vtriv ∆ VcircMcaLifo 0.04 0.00 0.99 0.35 0.00 0.00 0.04 0.00 0.99 0.28 0.00 0.00 app msr ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∆ Vtriv ∆ VcircMcaFifo 0.04 0.00 0.99 0.35 0.00 0.00 0.04 0.00 0.99 0.28 0.00 0.00

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16 11 Summary of on-chain velocity measures and proxy-variables

Table 4: Summary of on-chain velocity measures and proxy-variables.

Proxy-Variables Measures

Type non-segregated money supply segregated money supply

app app msr msr msr msr msr Notation Vcdd p Vturn p Vtriv p Vtotal VcircWba p[`] VcircMcaLifo p[`] VcircMcaFifo p[`]

Description Coin days destroyed Turnover Trivial velocity Velocity measure Based on money in Based on money in Based on money in measure based on total money effective circulation effective circulation effective circulation supply using the Whole Bill using the Moved using the Moved Approach Coin Approach Coin Approach (FIFO) (LIFO)

0 0 1 hPp ,Tp i hPp,Tpi hPp,Tpi hPp,Tpi hPp,Tpi Formula ∑t∈ cdd(t) · δ msr msr msr Tp dormp Mtotal p Mtotal p McircWba p[`] McircMcaLifo p[`] McircMcaFifo p[`] Related Section 9.1.1 9.1.266 8.2 8.2 8.2

Sources Bitcointalk.org, 2011 [18][3][2,4] novel novel novel

Considered in [10, 11, 12, 13, 14, 15, 16] [18][3][2,4] 17

t ... an individual transaction

Tp ... the set of all transactions in period p cdd(t) ... coin days destroyed for transaction t as defined in Section 9.1

dormp ... dormancy in period p as defined in Section 9.1.2 δ ... the period for which an approximated turnover per coin in circulation is calculated 0 0 hPp ,Tp i ... the inflated price-sum

hPp,Tpi ... the deflated price-sum

Mtotal p ... the complete, ever mined coin supply msr McircWba p[`] ... money in effective circulation using the Whole Bill Approach msr McircMcaFifo p[`] ... money in effective circulation using the Moved Coin Approach adopting first-in-first-out assignment between inputs and outputs msr McircMcaLifo p[`] ... money in effective circulation using the Moved Coin Approach adopting last-in-first-out assignment between inputs and outputs 12 Descriptives of the entirety of raw data

Table 5: Basic descriptive statistics based on Bitcoin data from 01.06.2013 to 01.06.2019 with p = ` = 1 day. Variables marked by "∗" are scaled down by factor 1,000,000.

Obs. Mean Med. Min. Max. Std. Dev. Kurtosis

McircMcaFifo 2174.00 0.14 0.13 0.03 0.52 0.06 2.96

McircMcaLifo 2174.00 0.14 0.13 0.03 0.52 0.06 2.96

McircWba 2174.00 0.23 0.21 0.06 0.66 0.09 0.77

Mtotal 2174.00 15.12 15.63 11.23 17.73 1.88 -1.04 PUSD/BTC 2174.00 2525.95 645.16 67.81 19535.70 3387.80 3.17 Vol.(deflated) 2174.00 0.58 0.52 0.10 4.25 0.33 18.07 Vol.(inflated) 2174.00 1.64 1.28 0.26 21.84 1.45 60.21 app Vcdd 2174.00 9.67 5.72 0.87 171.96 14.48 52.80 app Vturn 2174.00 97.92 80.74 1.63 1036.72 78.87 29.82 msr VcircMcaFifo 2174.00 4.20 3.95 1.36 18.34 1.39 17.07 msr VcircMcaLifo 2174.00 4.20 3.95 1.36 18.34 1.39 17.07 msr VcircWba 2174.00 2.52 2.36 0.77 11.93 0.91 22.66 msr Vtriv 2174.00 0.11 0.08 0.02 1.42 0.09 66.46 msr Vtotal 2174.00 0.04 0.03 0.01 0.26 0.02 18.44

13 Descriptives of the normalized proxy-variables and measurements

Table 6: Basic descriptive statistics based on normalized velocity measurement and approximation data from 01.06.2013 to 01.06.2019 with p = ` = 1 day.

Obs. Mean Med. Min. Max. Std. Dev. Kurtosis app Vcdd 2174.00 0.02 0.01 0.00 1.00 0.04 137.39 app Vturn 2174.00 0.04 0.03 0.00 1.00 0.04 255.22 msr VcircMcaFifo 2174.00 0.14 0.12 0.00 1.00 0.07 23.92 msr VcircMcaLifo 2174.00 0.14 0.12 0.00 1.00 0.07 23.92 msr VcircWba 2174.00 0.12 0.11 0.00 1.00 0.06 36.57 msr Vtriv 2174.00 0.02 0.01 0.00 1.00 0.03 577.05 msr Vtotal 2174.00 0.07 0.06 0.00 1.00 0.05 89.18

18 14 Descriptives of the standardized proxy-variables and measurements

Table 7: Basic descriptive statistics based on standardized velocity measurement and approximation data from 01.06.2013 to 01.06.2019 with p = ` = 1 day.

Obs. Mean Med. Min. Max. Std. Dev. Kurtosis app Vcdd 2174.00 0.00 -0.27 -0.61 11.20 1.00 52.80 app Vturn 2174.00 0.00 -0.22 -1.22 11.90 1.00 29.82 msr VcircMcaFifo 2174.00 -0.00 -0.18 -2.04 10.16 1.00 17.07 msr VcircMcaLifo 2174.00 -0.00 -0.18 -2.04 10.16 1.00 17.07 msr VcircWba 2174.00 -0.00 -0.18 -1.92 10.32 1.00 22.66 msr Vtriv 2174.00 0.00 -0.24 -0.94 14.24 1.00 66.46 msr Vtotal 2174.00 0.00 -0.19 -1.51 11.05 1.00 18.44

15 Correlations of the entirety of raw data

Table 8: Correlations based on Bitcoin data from 01.06.2013 to 01.06.2019 with p = ` = 1 day. ) ) inflated deflated BTC ( ( / . . circMcaFifo total circMcaLifo circWba msr msr app msr msr msr app app USD total triv circMcaLifo circMcaFifo turn triv circWba cdd M M M M Vol Vol P V V V V V V V V msr Vtriv 1 0.46 0.12 0.1 0.1 0.19 0.54 1 0.15 0.47 0.47 0.52 0.99 0.44 -0.07 msr Vtotal 0.46 1 0.66 0.58 0.58 0.42 0.02 0.46 0.3 0.74 0.74 0.72 0.49 0.98 0.23 msr VcircWba 0.12 0.66 1 0.95 0.95 0.13 0.02 0.12 0.35 0.13 0.13 0.05 0.15 0.69 0.33 msr VcircMcaLifo 0.1 0.58 0.95 1 1 0.06 0.05 0.1 0.31 -0.02 -0.02 -0.01 0.13 0.6 0.29 msr VcircMcaFifo 0.1 0.58 0.95 1 1 0.06 0.05 0.1 0.31 -0.02 -0.02 -0.01 0.13 0.6 0.29 app Vcdd 0.19 0.42 0.13 0.06 0.06 1 -0.3 0.19 0.17 0.48 0.48 0.44 0.21 0.43 0.17 app Vturn 0.54 0.02 0.02 0.05 0.05 -0.3 1 0.54 0 -0.01 -0.01 0.02 0.52 0.01 -0.24 app Vtriv 1 0.46 0.12 0.1 0.1 0.19 0.54 1 0.15 0.47 0.47 0.52 0.99 0.44 -0.07 Mtotal 0.15 0.3 0.35 0.31 0.31 0.17 0 0.15 1 0.38 0.38 0.38 0.24 0.46 0.64

McircMcaLifo 0.47 0.74 0.13 -0.02 -0.02 0.48 -0.01 0.47 0.38 1 1 0.95 0.5 0.75 0.2

McircMcaFifo 0.47 0.74 0.13 -0.02 -0.02 0.48 -0.01 0.47 0.38 1 1 0.95 0.5 0.75 0.2

McircWba 0.52 0.72 0.05 -0.01 -0.01 0.44 0.02 0.52 0.38 0.95 0.95 1 0.54 0.72 0.18 Vol.(inflated) 0.99 0.49 0.15 0.13 0.13 0.21 0.52 0.99 0.24 0.5 0.5 0.54 1 0.49 0 Vol.(deflated) 0.44 0.98 0.69 0.6 0.6 0.43 0.01 0.44 0.46 0.75 0.75 0.72 0.49 1 0.33 PUSD/BTC -0.07 0.23 0.33 0.29 0.29 0.17 -0.24 -0.07 0.64 0.2 0.2 0.18 0 0.33 1

19