# A Hypothesis Test Method for Detecting Multifractal Scaling, Applied to Bitcoin Prices

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Monofractal vs. Multifractal Processes

**Definition**

**1.**

**Theorem**

**1.**

**.**If $\left\{X\right(t),t\phantom{\rule{3.33333pt}{0ex}}\ge 0\}$ is self-similar and stochastically continuous at t = 0, then there exists a unique $H>0$ such that for all $a>0$,

#### 1.2. Examination of Multifractality

#### 1.3. Simulation of Multifractal Processes

**Definition**

**2.**

**.**Brownian motion in multifractal time is defined as

- In Stage 1, divide the time interval $[0,1]$ into $b>1$ non-overlapping subintervals with equal length $1/b$. Assign multipliers ${M}_{1,\beta}\phantom{\rule{4pt}{0ex}}(0\le \beta \le b-1)$ to each subinterval, where the ${M}_{1,\beta}$ are random variables with distributions that are not necessarily discrete. For computational convenience, we assume the ${M}_{1,\beta}$ to be identically distributed with a common distribution M.
- In Stage 2, each of the b intervals is further divided into b subintervals of length $1/{b}^{2}$. Again, we assign multipliers ${M}_{2,\beta}\phantom{\rule{4pt}{0ex}}(0\le \beta \le b-1)$ to each subinterval. The ${M}_{2,\beta}$ are assumed to be identically distributed with distribution M. Thus, after the second stage, the mass on an interval, for example $[0,1/{b}^{2}]$, will be ${\mu}_{2}[0,1/{b}^{2}]={m}_{0}{m}_{1}$ if ${M}_{1,0}={m}_{0}$ and ${M}_{2,0}={m}_{1}$ with probability$$\mathbb{P}({\mu}_{2}[0,1/{b}^{2}]={m}_{0}{m}_{1})=\mathbb{P}(M={m}_{0})P(M={m}_{1}),$$
- Repetition of this scheme generates a sequence of measures ${\left({\mu}_{k}\right)}_{k\in \mathbb{N}}$ which converges to our desired multiplicative measure $\mu $ as $k\to \infty $.

**Remark**

**1.**

**Theorem**

**2.**

**.**Define $p\left(\alpha \right)$ to be the continuous density of $V=-{log}_{b}M$, and ${p}_{k}\left(\alpha \right)$ to be the density of the k-th convolution product of p. The scaling function of a multiplicative measure satisfies

#### 1.4. Multifractality and Heavy Tails

**Theorem**

**3.**

**.**Let X be a random variable with $\mathbb{E}\left(X\right)=0$ such that the distribution function of $\left|X\right|$ has a regularly varying tail of order $-\alpha $ where $\alpha >2$; that is,

**Definition**

**3**

**.**A stochastic process $\left\{X\right(t),t\ge 0\}$ is said to be of type $\mathcal{E}$ if $Y\left(t\right)=X\left(t\right)-X(t-1)$, $t\in \mathbb{N}$, is a strictly stationary sequence having heavy-tailed marginal distribution with index α, satisfying the strong mixing property with an exponentially decaying rate and such that $\mathbb{E}Y\left(t\right)=0$ when $\alpha >1$.

**Theorem**

**4**

**.**Suppose $\left\{X\right(t),0\le t\le T\}$ is of type $\mathcal{E}$ and suppose $\Delta {t}_{i}$ is of the form ${T}^{\frac{i}{N}}$ for $i=1,\dots ,N$. Then, for every $q>0$ and $s\in (0,1)$

## 2. Methodology

#### 2.1. Measure of Concavity

**Definition**

**4**

**.**Assume that

- 1.
- there is an iid sample ${\left\{({x}_{i},{\u03f5}_{i})\right\}}_{i=1}^{n}$ drawn from the joint distribution of the random variables $(x,\u03f5)$, where ϵ is symmetrically distributed about 0 (conditional on x), so that $\mathbb{E}\left({\u03f5}_{i}\right|{x}_{i})=0,i\phantom{\rule{3.33333pt}{0ex}}=1,\dots ,n$;
- 2.
- the observed sample is ${\left\{({y}_{i},{x}_{i})\right\}}_{i=1}^{n}$, where ${y}_{i}$ is generated by ${y}_{i}=f\left({x}_{i}\right)+{\u03f5}_{i},\phantom{\rule{4.pt}{0ex}}i=1,\dots ,n,$ and the functional form of f is left unspecified.

**Definition**

**5.**

#### 2.2. Simulation Results

#### 2.3. Look-Up Table

## 3. Results

#### 3.1. Application to Bitcoin

- the daily Bitcoin open price, daily data (in USD) from 28 April 2013 to 3 September 2019 with 2,320 observations, retrieved from CoinMarketCap (2019); and

#### 3.1.1. Daily Bitcoin Price Data

- ${X}_{1}\left(t\right)$ from 28 April 2013 to 16 July 2017 with 1541 observations; and,
- ${X}_{2}\left(t\right)$ from 17 July 2017 to 3 September 2019 with 779 observations.

#### 3.1.2. High-Frequency Bitcoin Price Data

#### 3.2. Bitcoin Compared to Other Financial Assets

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Look-Up Table

Localised Concavity Measure | Global Concavity Measure | ||
---|---|---|---|

Tail Index | Critical Value | Tail Index | Critical Value |

0.50 | −0.8889 | 0.50 | −0.9979 |

0.75 | −0.8889 | 0.75 | −0.9987 |

1.00 | −0.8333 | 1.00 | −0.9987 |

1.25 | −0.7778 | 1.25 | −0.9986 |

1.50 | −0.6667 | 1.50 | −0.9963 |

1.75 | −0.5556 | 1.75 | −0.8543 |

2.00 | −0.3889 | 2.00 | −0.7503 |

2.25 | −0.3333 | 2.25 | −0.5482 |

2.50 | −0.2222 | 2.50 | −0.5264 |

2.75 | −0.1111 | 2.75 | −0.4992 |

3.00 | −0.0556 | 3.00 | −0.6231 |

3.25 | −0.1111 | 3.25 | −0.7011 |

3.50 | −0.1667 | 3.50 | −0.7562 |

3.75 | −0.2222 | 3.75 | −0.8330 |

4.00 | −0.2778 | 4.00 | −0.8791 |

4.25 | −0.3333 | 4.25 | −0.9274 |

4.50 | −0.3889 | 4.50 | −0.9291 |

4.75 | −0.5000 | 4.75 | −0.9228 |

5.00 | −0.5556 | 5.00 | −0.9259 |

5.25 | −0.5556 | 5.25 | −0.9149 |

5.50 | −0.5556 | 5.50 | −0.9133 |

5.75 | −0.5556 | 5.75 | −0.9086 |

6.00 | −0.5083 | 6.00 | −0.9006 |

6.25 | −0.5000 | 6.25 | −0.8835 |

6.50 | −0.4444 | 6.50 | −0.8548 |

6.75 | −0.4444 | 6.75 | −0.8233 |

7.00 | −0.4444 | 7.00 | −0.8047 |

7.25 | −0.3333 | 7.25 | −0.7321 |

7.50 | −0.3333 | 7.50 | −0.7018 |

7.75 | −0.2528 | 7.75 | −0.6804 |

8.00 | −0.2222 | 8.00 | −0.6545 |

8.25 | −0.2222 | 8.25 | −0.6203 |

8.50 | −0.2528 | 8.50 | −0.5517 |

8.75 | −0.2222 | 8.75 | −0.4959 |

9.00 | −0.2222 | 9.00 | −0.4449 |

9.25 | −0.2222 | 9.25 | −0.4065 |

9.50 | −0.2222 | 9.50 | −0.3928 |

9.75 | −0.1167 | 9.75 | −0.3969 |

10.00 | −0.1111 | 10.00 | −0.3293 |

## Appendix B. Multifractality Test Results

#### Appendix B.1. Hypothesis Test Results on Daily Bitcoin Open Price Data—Annual Breakdown

Time Period | Tail Index | Localised Measure | Sample Size | Test Result |
---|---|---|---|---|

28/04/2013–31/12/2013 | 2.0886 | −0.5556 | 2026 | Multifractal |

01/01/2014–31/12/2014 | 1.4785 | −0.3333 | 1904 | Non-Multifractal |

01/01/2015–31/12/2015 | 0.9464 | 0.2222 | 980 | Non-Multifractal |

01/01/2016–31/12/2016 | 1.7584 | −0.7778 | 2036 | Multifractal |

01/01/2017–31/12/2017 | 2.2187 | −0.6111 | 1997 | Multifractal |

01/01/2018–31/12/2018 | 1.6145 | 0.1667 | 1977 | Non-Multifractal |

01/01/2019–03/09/2019 | 1.4411 | 0.6111 | 1867 | Non-Multifractal |

Time Period | Tail Index | Global Measure | Sample Size | Test Result |
---|---|---|---|---|

28/04/2013–31/12/2013 | 2.0886 | −0.9260 | 72 | Multifractal |

01/01/2014–31/12/2014 | 1.4785 | −0.7300 | 77 | Non-Multifractal |

01/01/2015–31/12/2015 | 0.9464 | −0.0511 | 50 | Non-Multifractal |

01/01/2016–31/12/2016 | 1.7584 | −0.9473 | 80 | Multifractal |

01/01/2017–31/12/2017 | 2.2187 | −0.9551 | 70 | Multifractal |

01/01/2018–31/12/2018 | 1.6145 | 0.2706 | 74 | Non-Multifractal |

01/01/2019–03/09/2019 | 1.4411 | 0.8439 | 80 | Non-Multifractal |

#### Appendix B.2. Hypothesis Test Results on High Frequency Bitcoin Price Data

Time Period | Tail Index | Localised | Sample Size | Test Result |
---|---|---|---|---|

22/05/2018 14:01–08/07/2018 23:23 | 2.1226 | 0.0000 | 2015 | Non-Multifractal |

08/07/2018 23:24–24/08/2018 23:24 | 2.9599 | −0.3333 | 1572 | Multifractal |

24/08/2018 23:25–11/10/2018 00:25 | 2.5649 | −0.4444 | 1889 | Multifractal |

11/10/2018 00:26–27/11/2018 10:56 | 2.5984 | −0.6667 | 1847 | Multifractal |

27/11/2018 10:57–13/01/2019 10:57 | 2.8381 | −0.2778 | 1654 | Multifractal |

13/01/2019 10:58–01/03/2019 10:58 | 2.6245 | −0.5000 | 1845 | Multifractal |

Time Period | Tail Index | Global | Sample Size | Test Result |
---|---|---|---|---|

22/05/2018 14:01–08/07/2018 23:23 | 2.1226 | −0.4811 | 72 | Non-Multifractal |

08/07/2018 23:24–24/08/2018 23:24 | 2.9599 | −0.3872 | 96 | Non-Multifractal |

24/08/2018 23:25–11/10/2018 00:25 | 2.5649 | −0.5587 | 74 | Multifractal |

11/10/2018 00:26–27/11/2018 10:56 | 2.5984 | −0.8287 | 74 | Multifractal |

27/11/2018 10:57–13/01/2019 10:57 | 2.8381 | −0.6690 | 88 | Multifractal |

13/01/2019 10:58–01/03/2019 10:58 | 2.6245 | −0.4125 | 75 | Non-Multifractal |

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1. | “False positive” corresponds to the scenario that we mistakenly detect multifractality when the underlying process does not possess the multifractal property. |

2. | For convenience, in the rest of this paper, we call uniscaling multifractal processes monofractal processes while referring to multiscaling multifractal processes as multifractal processes. |

3. | We compared the scaling functions of the Bitcoin and other financial assets with the ones of BMMTs simulated using multiplicative cascades with Poisson distribution, Gamma distribution and Normal distribution. The scaling function of the BMMT simulated through log-Normal multiplicative cascade displays the most similar behaviour. |

4. | The choice of Student’s t-distribution with 4 degrees of freedom is suggested by the findings of Platen and Rendek (2008). |

5. | Knots were chosen so that the function is evaluated on 18 equal sub-intervals. This gave the best approximation considering computational efficiency. |

6. | For example, if Hill’s estimator takes value ${h}^{\mathrm{Hill}}=2$, we generate simulated student t-distributed processes with 2 degrees of freedom, select those with tail indices in the interval $[1.5,2.5]$, and construct the null distribution using the empirical distribution of their concavity measures. |

7. | Retrieved from Yahoo Finance (2019) for the period 30 December 1927 to 26 February 2020 with 23,147 observations. |

8. | Retrieved from Yahoo Finance (2020) for the period 5 February 1971 to 25 February 2020 with 12,372 observations. |

9. | Retrieved from investing.com Australia (2020b) for the period 4 March 1988 to 28 February 2020 with 8337 observations. |

10. | Retrieved from investing.com Australia (2020a) for the period 27 December 1979 to 28 February 2020 with 10,190 observations. |

**Figure 1.**Asymptotic scaling functions for various values of $\alpha $. The grey line is the $q/2$ reference line.

**Figure 2.**

**Left**—Scaling Functions of the original S&P 500 open price data (black solid line), the S&P 500 open price data after removing 19 October 1987 (dotted line), the S&P 500 open price data after truncating any return larger than 4 standard deviations from the mean (grey solid line).

**Right**—Scaling Functions of the Student t-distributed processes before (black lines) and after truncation (grey lines). For comparison, the reference lines $q/2$ (black dashed line) are included.

**Figure 3.**Distribution of global simplex statistics for Brownian motion (

**left**) and fractional Brownian motion (

**right**).

**Figure 4.**Distributions of global simplex statistics for Student t-distributed process (black) and the simulated BMMT with log-normal multiplicative cascade (grey) around tail index 3.06. The two vertical lines represent the 95th percentile for the simulated BMMTs (

**left**) and 5th percentiles for the Student t-distributed processes, respectively (

**right**).

**Figure 5.**Daily mean-centered log Bitcoin open price process (

**top**) and mean-centered log returns process (

**bottom**).

**Figure 6.**

**Left**—Autocorrelations of mean-centered log-returns (black) and absolute returns (grey) for daily Bitcoin open price (28 April 2013–3 April 2019).

**Right**—Scaling function for daily Bitcoin open price (28 April 2013–3 September 2019). The dashed line is the $q/2$ reference line (same for all the scaling function figures).

**Figure 7.**Scaling functions for Daily Bitcoin open price from 28 April 2013 to 16 July 2017 (

**left**) vs. from 17 July 2017 to 3 September 2019 (

**right**).

**Figure 8.**High frequency mean-centered log Bitcoin open price process (

**top**) and mean-centered log returns process (

**bottom**).

**Figure 9.**

**Left**—Autocorrelations of mean-centered log returns (black) and absolute returns (grey) for high frequency Bitcoin open prices (22 May 2018–1 March 2019).

**Right**—Scaling function for high frequency Bitcoin open prices (22 May 2018–01 March 2019).

**Figure 10.**Scaling functions for S&P500, NASDAQ Index, USD/JPY Exchange and Gold Futures Prices the period 28 April 2013 to 3 September 2019.

**Figure 11.**Scaling functions for S&P500, NASDAQ Index, USD/JPY Exchange and Gold Futures Prices during the dot-com bubble (3 January 1994–8 October 2004). The time period from 3 January 1994 to 8 October 2004 corresponds to the dot-com bubble and the dot-com crash.

**Figure 12.**Scaling functions for S&P500, NASDAQ Index, USD/JPY Exchange and Gold Futures Prices the period 4 March 1988 to 3 September 2019.

Tail Index | 3.0630 | |||
---|---|---|---|---|

Sample Size ^{1} | Test Statistic | Rejection Region | Test Result | |

Localised Test | 1479 | $-0.2222$ | $[-1,-0.0556)$ | Multifractal |

Global Test | 99 | $-0.6256$ | $[-1,-0.6148)$ | Multifractal |

^{1}Sample size refers to the number of student t-distributed processes used in estimating the distribution of concavity measures under ${H}_{0}$. In the case of Bitcoin open prices from 28/04/2013 to 03/09/2019, there are 1479 student t-distributed processes with tail indices ranging in the interval [2.5630, 3.5630] when constructing the distribution of localised concavity measures. 99 student t-distributed processes are used in the construction of the distributions of global simplex statistics.

Tail Index | 3.2791 | |||
---|---|---|---|---|

Sample Size | Test Statistic | Rejection Region | Test Result | |

Localised Test | 1403 | $-0.4444$ | $[-1,-0.1111)$ | Multifractal |

Global Test | 98 | $-0.5378$ | $[-1,-0.7014)$ | Non-Multifractal |

Tail Index | 1.6528 | |||
---|---|---|---|---|

Sample Size | Test Statistic | Rejection Region | Test Result | |

Localised Test | 1992 | $-0.8333$ | $[-1,-0.6111)$ | Multifractal |

Global Test | 75 | $-0.9950$ | $[-1,-0.8703)$ | Multifractal |

Tail Index | 2.7759 | |||
---|---|---|---|---|

Sample Size | Test Statistic | Rejection Region | Test Result | |

Localised Test | 1682 | $0.2778$ | $[-1,-0.0556)$ | Non-multifractal |

Global Test | 83 | $-0.2714$ | $[-1,-0.5049)$ | Non-multifractal |

Financial Asset | Tail Index | Test | Test Statistics | Rejection Region | Test Result (Local) |
---|---|---|---|---|---|

BTC Daily | 3.06 | Local | −0.22 | [−1, −0.06) | Multifractal |

Global | −0.63 | [−1, −0.61) | Multifractal | ||

S&P500 | 3.13 | Local | −0.11 | [−1, −0.11) | Non-Multifractal |

Global | −0.14 | [−1, −0.66) | Non-Multifractal | ||

NASDAQ | 3.19 | Local | 0.11 | [−1, −0.11) | Non-Multifractal |

Global | −0.44 | [−1, −0.66) | Non-Multifractal | ||

USD/JPY | 2.69 | Local | −0.28 | [−1, −0.17) | Multifractal |

Global | −0.86 | [−1, −0.49) | Multifractal | ||

Gold Futures | 1.35 | Local | −0.67 | [−1, −0.72) | Non-Multifractal |

Global | −0.89 | [−1, −0.9979) | Non-Multifractal |

**Table 6.**Test results among different financial assets during the dot-com bubble (3 January 1994–8 October 2004).

Financial Asset | Tail Index | Test | Test Statistics | Rejection Region | Test Result (Local) |
---|---|---|---|---|---|

S&P500 | 3.41 | Local | −0.22 | [−1, −0.17) | Multifractal |

Global | −0.40 | [−1, −0.71) | Non-Multifractal | ||

NASDAQ | 2.81 | Local | −0.44 | [−1, −0.06) | Multifractal |

Global | −0.79 | [−1, −0.52) | Multifractal | ||

USD/JPY | 3.50 | Local | −0.33 | [−1, −0.17) | Multifractal |

Global | −0.80 | [−1, −0.76) | Multifractal | ||

Gold Futures | 3.48 | Local | 0.22 | [−1, −0.17) | Non-Multifractal |

Global | −0.48 | [−1, −0.71) | Non-Multifractal |

Financial Asset | Tail Index | Test | Test Statistics | Rejection Region | Test Result (Local) |
---|---|---|---|---|---|

S&P500 | 3.20 | Local | 0.78 | [−1, −0.11) | Non-Multifractal |

Global | 0.93 | [−1, −0.66) | Non-Multifractal | ||

NASDAQ | 3.84 | Local | −0.61 | [−1, −0.22) | Multifractal |

Global | −0.93 | [−1, −0.84) | Multifractal | ||

USD/JPY | 3.33 | Local | −0.28 | [−1, −0.17) | Multifractal |

Global | −0.72 | [−1, −0.70) | Multifractal | ||

Gold Futures | 1.36 | Local | −0.11 | [−1, −0.72) | Non-Multifractal |

Global | 0.11 | [−1, −0.9979) | Non-Multifractal |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jiang, C.; Dev, P.; Maller, R.A.
A Hypothesis Test Method for Detecting Multifractal Scaling, Applied to Bitcoin Prices. *J. Risk Financial Manag.* **2020**, *13*, 104.
https://doi.org/10.3390/jrfm13050104

**AMA Style**

Jiang C, Dev P, Maller RA.
A Hypothesis Test Method for Detecting Multifractal Scaling, Applied to Bitcoin Prices. *Journal of Risk and Financial Management*. 2020; 13(5):104.
https://doi.org/10.3390/jrfm13050104

**Chicago/Turabian Style**

Jiang, Chuxuan, Priya Dev, and Ross A. Maller.
2020. "A Hypothesis Test Method for Detecting Multifractal Scaling, Applied to Bitcoin Prices" *Journal of Risk and Financial Management* 13, no. 5: 104.
https://doi.org/10.3390/jrfm13050104