Velocity

Reuben Youngblom

To this end, we first review the most common proxies in , then implement the proposed estimators to calculate velocity measures for Bitcoin in , and finally run tests in to evaluate the goodness of fit of the proxies w.r.t. the measures.

${M_{\mathtt{circWba}}}\gets {0}$ return ${M_{\mathtt{circWba}}}$

${M_{\mathtt{circMca}}}\gets {0}$ ${M_{\mathtt{circMca}}}(t) \gets {0}$ ${\mathtt{valOut}}^\mathtt{break}(t) \gets {0}$ ${\mathbb{O}}'_{t} \gets{{\mathbb{O}}_{t}\setminus{}O^{{\mathtt{selfchurn}}}}$ ${\mathtt{valOut}}^{\mathtt{toOthers}}(t) \gets{\sum_{o\in{}{\mathbb{O}}'_{t}} {\mathtt{valOut}}(o)}$ ${\mathbb{I}^{\mathtt{sort}}_{t}}(\mathtt{type}) \gets{{\mathbb{I}_{\mathtt{t}}}\ \text{sorted ascending (\acs*{fifo})/descending (\acs*{lifo}) according to $\mathtt{type}$ and thus generation time of inputs}}$
${M_{\mathtt{circMca}}}\gets{{M_{\mathtt{circMca}}}+ {M_{\mathtt{circMca}}}(t)}$ [algo:code_mcirc_mc_break]

return ${M_{\mathtt{circMca}}}$

1 Mcirc <- 0

2 foreach t E Tp

The measure puts larger weight on the reactivation of long-dormant coins compared to ones frequently spun, a feature clearly differing from the concept of velocity.

	Obs.	Mean	Med.	Min.	Max.	Std. Dev.	Kurtosis
$V^{\mathtt{app}}_{\mathtt{cdd}}$	2174.00	9.78	5.72	0.87	361.99	16.22	137.39
$V^{\mathtt{app}}_{\mathtt{turn}}$	2174.00	98.66	80.74	1.63	2635.37	93.81	255.22
$V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	2174.00	6.78	6.09	1.52	57.10	3.49	41.05
$V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	2174.00	6.78	6.09	1.52	57.10	3.49	41.05
$V^{\mathtt{msr}}_{\mathtt{circWba}}$	2174.00	5.83	5.22	1.49	48.99	3.05	40.44
$V^{\mathtt{msr}}_{\mathtt{triv}}$	2174.00	0.11	0.08	0.02	4.46	0.13	577.05
$V^{\mathtt{msr}}_{\mathtt{total}}$	2174.00	0.09	0.07	0.02	1.03	0.07	33.66

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

The mz regressions are in line with the evidence in prior sections. Again, the trivial measure approximates the more sophisticated velocity measures better than the two commonly used proxy variables.

Approximation	Estimator	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE
$V^{\mathtt{app}}_{\mathtt{cdd}}$	$V^{\mathtt{msr}}_{\mathtt{total}}$	1405.30	1405.30	152.42	30.15	1356.97	1356.97	116.79	28.06
$V^{\mathtt{app}}_{\mathtt{cdd}}$	$V^{\mathtt{msr}}_{\mathtt{circWba}}$	1611.26	1611.26	210.77	41.95	1600.15	1600.15	135.49	33.55
$V^{\mathtt{app}}_{\mathtt{cdd}}$	$V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	1607.60	1607.60	221.03	43.47	1623.72	1623.72	136.76	33.48
$V^{\mathtt{app}}_{\mathtt{cdd}}$	$V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	1607.61	1607.61	221.03	43.47	1623.72	1623.72	136.76	33.48
$V^{\mathtt{app}}_{\mathtt{turn}}$	$V^{\mathtt{msr}}_{\mathtt{total}}$	1570.56	1570.56	129.35	16.85	1470.08	1470.08	116.05	14.50
$V^{\mathtt{app}}_{\mathtt{turn}}$	$V^{\mathtt{msr}}_{\mathtt{circWba}}$	1328.73	1328.73	$^{\dag}$ 120.34	$^{\dag}$ 14.95	1542.02	1542.02	121.99	16.33
$V^{\mathtt{app}}_{\mathtt{turn}}$	$V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	1373.43	1373.43	$^{\dag}$ 129.71	$^{\dag}$ 16.09	1549.60	1549.60	122.28	16.01
$V^{\mathtt{app}}_{\mathtt{turn}}$	$V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	1373.40	1373.40	$^{\dag}$ 129.71	$^{\dag}$ 16.09	1549.61	1549.61	122.28	16.01
$V^{\mathtt{app}}_{\mathtt{triv}}$	$V^{\mathtt{msr}}_{\mathtt{total}}$	$^{\dag}$ 381.20	$^{\dag}$ 381.20	$^{\dag}$ 85.24	$^{\dag}$ 7.32	$^{\dag}$ 306.69	$^{\dag}$ 306.69	$^{\dag}$ 35.09	$^{\dag}$ 4.60
$V^{\mathtt{app}}_{\mathtt{triv}}$	$V^{\mathtt{msr}}_{\mathtt{circWba}}$	$^{\dag}$ 892.30	$^{\dag}$ 892.30	143.68	15.71	$^{\dag}$ 838.69	$^{\dag}$ 838.69	$^{\dag}$ 69.11	$^{\dag}$ 8.65
$V^{\mathtt{app}}_{\mathtt{triv}}$	$V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	$^{\dag}$ 912.54	$^{\dag}$ 912.54	156.52	17.52	$^{\dag}$ 864.98	$^{\dag}$ 864.98	$^{\dag}$ 70.60	$^{\dag}$ 8.55
$V^{\mathtt{app}}_{\mathtt{triv}}$	$V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	$^{\dag}$ 912.55	$^{\dag}$ 912.55	156.52	17.52	$^{\dag}$ 864.99	$^{\dag}$ 864.99	$^{\dag}$ 70.60	$^{\dag}$ 8.55

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

Approximation	Estimator	$R^{2}_{adj}$	$p^{\alpha}$	$\beta$	$p^{\beta}$	$p^{F-Test}$	$R^{2}_{adj}$	$p^{\alpha}$	$\beta$	$p^{\beta}$	$p^{F-Test}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{cdd}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{total}}$	0.00	0.99	0.05	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00	0.99	0.05	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{cdd}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{circWba}}$	0.00	0.99	0.00	0.97	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00	0.99	0.00	0.97	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{cdd}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	0.00	0.99	-0.01	0.67	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00	0.99	-0.01	0.67	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{cdd}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	0.00	0.99	-0.01	0.67	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00	0.99	-0.01	0.67	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{turn}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{total}}$	0.11	0.99	0.30	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.11	0.99	0.26	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{turn}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{circWba}}$	0.13	0.99	0.37	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.13	0.99	0.34	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{turn}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	0.13	0.99	0.37	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.13	0.99	0.34	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{turn}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	0.13	0.99	0.37	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.13	0.99	0.34	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{triv}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{total}}$	0.53	1.00	1.07	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.53	1.00	0.80	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{triv}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{circWba}}$	0.35	1.00	1.01	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.90	0.35	1.00	0.79	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{triv}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	0.34	1.00	0.99	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.89	0.34	1.00	0.78	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$
$\Delta$ $V^{\mathtt{app}}_{\mathtt{triv}}$	$\Delta$ $V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	0.34	1.00	0.99	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.89	0.34	1.00	0.78	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$	0.00 $^{\ast}$ $^{\ast}$ $^{\ast}$

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

Summary of on-chain velocity measures and proxy-variables

L0.1C0.15C0.15C0.12C0.12C0.12C0.12C0.12 & &
Type & & &
Notation & ${{V^{{\mathtt{app}}}_{\mathtt{cdd}}}_{\!{p}}}$ & ${{V^{{\mathtt{app}}}_{\mathtt{turn}}}_{{p}}}$ & ${{V_\mathtt{triv}^{\mathtt{msr}}}_{p}}$ & ${V_\mathtt{total}^{\mathtt{msr}}}$ & ${{V_{\mathtt{circWba}}^{\mathtt{msr}}}_{{p}[{\ell}]}}$ & ${{V_{\mathtt{circMcaLifo}}^{\mathtt{msr}}}_{{p}[{\ell}]}}$ & ${{V_{\mathtt{circMcaFifo}}^{\mathtt{msr}}}_{{p}[{\ell}]}}$
Description & Coin days destroyed& Turnover& Trivial velocity measure& Velocity measure based on total money supply& Based on money in effective circulation using the Whole Bill Approach& Based on money in effective circulation using the Moved Coin Approach (FIFO)& Based on money in effective circulation using the Moved Coin Approach (LIFO)
Formula & $\sum_{t\in{\mathbb{T}_{p}}} \mathtt{cdd}(t)$ & $\frac{1}{\mathtt{dorm}_{p}} \cdot \delta$ & $\frac{\langle{P_{p}}',{T_{p}}'\rangle}{{{M_\mathtt{total}}_{p}}}$ & $\frac{\langle{P_{p}},{T_{p}}\rangle}{{{M_\mathtt{total}}_{p}}}$ & $\frac{\langle{P_{p}},{T_{p}}\rangle}{{{M_{\mathtt{circWba}}^{\mathtt{msr}}}_{{p}[{\ell}]}}}$ & $\frac{\langle{P_{p}},{T_{p}}\rangle}{{{M_{\mathtt{circMcaLifo}}^{\mathtt{msr}}}_{{p}[{\ell}]}}}$ & $\frac{\langle{P_{p}},{T_{p}}\rangle}{{{M_{\mathtt{circMcaFifo}}^{\mathtt{msr}}}_{{p}[{\ell}]}}}$
Related Section& 9.1.1& 9.1.2& 6& 6& 8.2& 8.2& 8.2
Sources & Bitcointalk.org, 2011& [1]& [2]& [3][4]& novel& novel& novel
Considered in & [5][6][7][8][9][10][11]& [1]& [2]& [3][4]& & &


$t$	$\hdots{}$	an individual transaction
${\mathbb{T}_{p}}$	$\hdots{}$	the set of all transactions in period ${p}$
$\mathtt{cdd}(t)$	$\hdots{}$	coin days destroyed for transaction $t$ as defined in Section 9.1
$\mathtt{dorm}_{p}$	$\hdots{}$	dormancy in period ${p}$ as defined in Section 9.1.2
$\delta$	$\hdots{}$	the period for which an approximated turnover per coin in circulation is calculated
$\langle{P_{p}}',{T_{p}}'\rangle$	$\hdots{}$	the inflated price-sum
$\langle{P_{p}},{T_{p}}\rangle$	$\hdots{}$	the deflated price-sum
${{M_\mathtt{total}}_{p}}$	$\hdots{}$	the complete, ever mined coin supply
${{M_{\mathtt{circWba}}^{\mathtt{msr}}}_{{p}[{\ell}]}}$	$\hdots{}$	money in effective circulation using the Whole Bill Approach
${{M_{\mathtt{circMcaFifo}}^{\mathtt{msr}}}_{{p}[{\ell}]}}$	$\hdots{}$	money in effective circulation using the Moved Coin Approach adopting first-in-first-out assignment between inputs and outputs
${{M_{\mathtt{circMcaLifo}}^{\mathtt{msr}}}_{{p}[{\ell}]}}$	$\hdots{}$	money in effective circulation using the Moved Coin Approach adopting last-in-first-out assignment between inputs and outputs

Descriptives of the entirety of raw data

	Obs.	Mean	Med.	Min.	Max.	Std. Dev.	Kurtosis
$^{\ast} M_{\mathtt{circMcaFifo}}$	2174.00	0.20	0.18	0.05	0.60	0.08	1.35
$^{\ast} M_{\mathtt{circMcaLifo}}$	2174.00	0.20	0.18	0.05	0.60	0.08	1.35
$^{\ast} M_{\mathtt{circWba}}$	2174.00	0.23	0.21	0.06	0.66	0.09	0.77
$^{\ast} M_{\mathtt{total}}$	2174.00	15.12	15.63	11.23	17.73	1.88	-1.04
$P^{\mathtt{USD}/\mathtt{BTC}}$	2174.00	2525.95	645.16	67.81	19535.70	3387.80	3.17
$^{\ast} \mathtt{Vol.(deflated)}$	2174.00	1.37	1.10	0.21	12.03	1.05	20.32
$^{\ast} \mathtt{Vol.(inflated)}$	2174.00	1.64	1.28	0.26	21.84	1.45	60.21
$V^{\mathtt{app}}_{\mathtt{cdd}}$	2174.00	9.67	5.72	0.87	171.96	14.48	52.80
$V^{\mathtt{app}}_{\mathtt{turn}}$	2174.00	97.92	80.74	1.63	1036.72	78.87	29.82
$V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	2174.00	6.76	6.09	1.52	41.67	3.36	24.24
$V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	2174.00	6.77	6.09	1.52	41.67	3.36	24.24
$V^{\mathtt{msr}}_{\mathtt{circWba}}$	2174.00	5.82	5.22	1.49	36.34	2.95	26.05
$V^{\mathtt{msr}}_{\mathtt{triv}}$	2174.00	0.11	0.08	0.02	1.42	0.09	66.46
$V^{\mathtt{msr}}_{\mathtt{total}}$	2174.00	0.09	0.07	0.02	0.76	0.07	22.90

[tbl:appendix_desc_other]

Descriptives of the normalized proxy-variables and measurements

	Obs.	Mean	Med.	Max.	Std. Dev.	Kurtosis
$V^{\mathtt{app}}_{\mathtt{cdd}}$	2174.00	0.02	0.01	1.00	0.04	137.39
$V^{\mathtt{app}}_{\mathtt{turn}}$	2174.00	0.04	0.03	1.00	0.04	255.22
$V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	2174.00	0.09	0.08	1.00	0.06	41.05
$V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	2174.00	0.09	0.08	1.00	0.06	41.05
$V^{\mathtt{msr}}_{\mathtt{circWba}}$	2174.00	0.09	0.08	1.00	0.06	40.44
$V^{\mathtt{msr}}_{\mathtt{triv}}$	2174.00	0.02	0.01	1.00	0.03	577.05
$V^{\mathtt{msr}}_{\mathtt{total}}$	2174.00	0.07	0.06	1.00	0.07	33.66

[tbl:appendix_desc_other_norm]

Descriptives of the standardized proxy-variables and measurements

	Obs.	Mean	Med.	Min.	Max.	Std. Dev.	Kurtosis
$V^{\mathtt{app}}_{\mathtt{cdd}}$	2174.00	0.00	-0.27	-0.61	11.20	1.00	52.80
$V^{\mathtt{app}}_{\mathtt{turn}}$	2174.00	0.00	-0.22	-1.22	11.90	1.00	29.82
$V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	2174.00	0.00	-0.20	-1.56	10.38	1.00	24.24
$V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	2174.00	0.00	-0.20	-1.56	10.38	1.00	24.24
$V^{\mathtt{msr}}_{\mathtt{circWba}}$	2174.00	0.00	-0.20	-1.47	10.34	1.00	26.05
$V^{\mathtt{msr}}_{\mathtt{triv}}$	2174.00	0.00	-0.24	-0.94	14.24	1.00	66.46
$V^{\mathtt{msr}}_{\mathtt{total}}$	2174.00	-0.00	-0.25	-1.11	10.23	1.00	22.90

[tbl:appendix_desc_other_stand]

Correlations of the entirety of raw data


$V^{\mathtt{msr}}_{\mathtt{triv}}$	1	0.86	0.73	0.72	0.72	0.19	0.54	1	0.15	0.52	0.52	0.52	0.99	0.85	-0.07
$V^{\mathtt{msr}}_{\mathtt{total}}$	0.86	1	0.84	0.83	0.83	0.22	0.44	0.86	0.18	0.62	0.62	0.61	0.87	0.99	-0.05
$V^{\mathtt{msr}}_{\mathtt{circWba}}$	0.73	0.84	1	0.99	0.99	0.02	0.55	0.73	0.18	0.2	0.2	0.18	0.74	0.84	-0.07
$V^{\mathtt{msr}}_{\mathtt{circMcaLifo}}$	0.72	0.83	0.99	1	1	0.03	0.53	0.72	0.24	0.19	0.19	0.2	0.74	0.84	0.01
$V^{\mathtt{msr}}_{\mathtt{circMcaFifo}}$	0.72	0.83	0.99	1	1	0.03	0.54	0.72	0.24	0.19	0.19	0.2	0.74	0.84	0.01
$V^{\mathtt{app}}_{\mathtt{cdd}}$	0.19	0.22	0.02	0.03	0.03	1	-0.3	0.19	0.17	0.43	0.43	0.44	0.21	0.24	0.17
$V^{\mathtt{app}}_{\mathtt{turn}}$	0.54	0.44	0.55	0.53	0.54	-0.3	1	0.54	0	0.04	0.04	0.02	0.52	0.41	-0.24
$V^{\mathtt{app}}_{\mathtt{triv}}$	1	0.86	0.73	0.72	0.72	0.19	0.54	1	0.15	0.52	0.52	0.52	0.99	0.85	-0.07
$M_{\mathtt{total}}$	0.15	0.18	0.18	0.24	0.24	0.17	0	0.15	1	0.3	0.3	0.38	0.24	0.29	0.64
$M_{\mathtt{circMcaLifo}}$	0.52	0.62	0.2	0.19	0.19	0.43	0.04	0.52	0.3	1	1	0.99	0.54	0.64	0.09
$M_{\mathtt{circMcaFifo}}$	0.52	0.62	0.2	0.19	0.19	0.43	0.04	0.52	0.3	1	1	0.99	0.54	0.64	0.09
$M_{\mathtt{circWba}}$	0.52	0.61	0.18	0.2	0.2	0.44	0.02	0.52	0.38	0.99	0.99	1	0.54	0.63	0.18
$\mathtt{Vol.(inflated)}$	0.99	0.87	0.74	0.74	0.74	0.21	0.52	0.99	0.24	0.54	0.54	0.54	1	0.87	0
$\mathtt{Vol.(deflated)}$	0.85	0.99	0.84	0.84	0.84	0.24	0.41	0.85	0.29	0.64	0.64	0.63	0.87	1	0.03
$P^{\mathtt{USD}/\mathtt{BTC}}$	-0.07	-0.05	-0.07	0.01	0.01	0.17	-0.24	-0.07	0.64	0.09	0.09	0.18	0	0.03	1

[tbl:corrtable_all]