**4. Conclusions**

Multifractal scaling cannot be assumed to exist a priori, but instead should be established after accounting for the heavy tail effect on scaling functions. In this paper, we demonstrate how one can develop a multifractal scaling hypothesis test that accounts for the heavy tail effect on scaling functions. The test outlined in this paper doesn't incorporate the possibility of monofractal scaling, but it can be incorporated into the test in the future and we will undertake such an approach in further research. The hypothesis test presented in this paper distinguishes between the heavy tail effect that distorts a linear scaling function to look concave, and true multifractal scaling. To implement the test, a look-up table is employed to simplify the hypothesis test procedure. This makes it easy to implement the test on various time series with marginal distributions of varying tail indexes. Our test results are of course contingent upon the validity of the underlying distributional assumptions. While a sound statistical theory awaits further development, a thorough examination of the test properties by means of Monte Carlo simulation can be carried out straightforwardly, in future research.

We apply this hypothesis test to Bitcoin prices and reveal that Bitcoin exhibits scaling behaviour more similar to a multifractal model than to a heavy tail process. We then extend the test to a set of other financial assets to ascertain whether Bitcoin prices are likely to share multifractal scaling relationships akin to other financial time series. Our results show that Bitcoin, USD/JPY exchange rates, and the technology heavy NASDAQ could all share multifractal scaling properties, after accounting for heavy tails. The findings of this paper are that, while Bitcoin prices span a relatively short period, the hypothesis test indicates that multifractal scaling is plausible and such scaling could be a feature of foreign exchange markets and technology stocks as well. This indicates some helpful methodology for model selection in risk analysis. Furthermore, the research suggests that financial time series may be classified by their statistical scaling properties in addition to the asset class they belong to. This additional type of classification could allow practitioners to construct statistically diverse portfolios based on assets grouped by their scaling dynamics.

**Author Contributions:** Research and analysis, C.J.; Supervision, P.D. and R.A.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially funded by ARC gran<sup>t</sup> DP160104737.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
