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

#### **Chuxuan Jiang, Priya Dev and Ross A. Maller \***

School of Finance & Applied Statistics, Australian National University, Canberra ACT 0200, Australia; chuxuan.jiang@anu.edu.au (C.J.); priya.dev@anu.edu.au (P.D.)

**\*** Correspondence: ross.maller@anu.edu

Received: 2 March 2020; Accepted: 14 May 2020; Published: 20 May 2020

**Abstract:** Multifractal processes reproduce some of the stylised features observed in financial time series, namely heavy tails found in asset returns distributions, and long-memory found in volatility. Multifractal scaling cannot be assumed, it should be established; however, this is not a straightforward task, particularly in the presence of heavy tails. We develop an empirical hypothesis test to identify whether a time series is likely to exhibit multifractal scaling in the presence of heavy tails. The test is constructed by comparing estimated scaling functions of financial time series to simulated scaling functions of both an iid Student *t*-distributed process and a Brownian Motion in Multifractal Time (BMMT), a multifractal processes constructed in Mandelbrot et al. (1997). Concavity measures of the respective scaling functions are estimated, and it is observed that the concavity measures form different distributions which allow us to construct a hypothesis test. We apply this method to test for multifractal scaling across several financial time series including Bitcoin. We observe that multifractal scaling cannot be ruled out for Bitcoin or the Nasdaq Composite Index, both technology driven assets.

**Keywords:** multifractal processes; fractal scaling; heavy tails; long range dependence; financial models; Bitcoin
