Statistical tests of distributional scaling properties for financial return series

Abstract

Existing empirical evidence of distributional scaling in financial returns has helped motivate the use of multifractal processes for modelling return processes. However, this evidence has relied on informal tests that may be unable to reliably distinguish multifractal processes from other related classes. The current paper develops a formal statistical testing procedure for determining which class of fractal process is most consistent with the distributional scaling properties in a given sample of data. Our testing methodology consists of a set of test statistics, together with a model-based bootstrap resampling scheme to obtain sample p-values. We demonstrate in Monte Carlo exercises that the proposed testing methodology performs well in a wide range of testing environments relevant for financial applications. Finally, the methodology is applied to study the scaling properties of a data-set of intraday equity index and exchange rate returns. The empirical results suggest that the scaling properties of these return series may be inconsistent with purely multifractal processes.

Keywords:

• Multifractal
• Unifractal
• Bootstrap inference
• Hypothesis testing
• Financial returns

Acknowledgements

The authors would also like to thank participants at the 7th CSDA International Conference on Computational and Financial Econometrics and anonymous referees for constructive feedback and comments

Notes

No potential conflict of interest was reported by the authors.

1 On the more financial and econometric side of the literature, wavelet-based methods have been widely applied to develop statistical tests in many other contexts, such as for serial correlation (Gençay and Signori 2015) and unit roots (Fan and Gençay 2010), in addition to more empirical applications, such as hedging (Conlon et al. 2016).

2 More detailed treatments of these topics can be found in Mandelbrot et al. (1997), Calvet and Fisher (2002) or Kantelhardt (2009).

3 The stationarity requirement for the increments does not have to hold for the original time series and unifractal or multifractal time series may be non-stationary. However, in this case an appropriate transformation must first be applied that results in stationary increments, before equation (2.1(2.1) ) will hold. The majority of estimators for and employed in the literature, including those used in the current work, automatically perform local detrending to remove non-stationarities, which is typically sufficient for the case of financial returns.

4 The term self-affinity index is also widely employed in the literature on unifractal processes; however, the Hurst exponent terminology will be employed here to remain consistent with the multifractal case.

5 This is possible because for the majority of theoretical unifractal processes, the parameter controls not only the distributional scaling properties, but also the smoothness and volatility of the process.

6 In the sense that both multifractional and multifractal processes nest the simplest unifractal class as a special case, when in all time periods or for all moment orders respectively.

7 In principle estimates of the generalized Hurst exponent could also be used, but in practice previous work exclusively employs the scaling function representation.

8 See Lashermes et al. (2005) and Schumann and Kantelhardt (2011) for some informal exploration of this topic using simulated data.

9 Despite this theoretical limitation, such a testing approach was nonetheless briefly explored but was also found to perform poorly in practice and so was not pursued further.

10 Note again that in both cases rejection only occurs for negative sample values, so that only deviations consistent with a multifractal process for a given are considered.

11 As with the statistics based on above, the alternative hypotheses of and technically correspond to no unifractality rather than multifractality and should be combined with the same type of additional check described above.

12 At this stage, it is not necessary to assume a specific process a priori, since the various estimators available for the simple Hurst exponent are valid for any unifractal process.

13 Note that the fractional Gaussian noise process is generally characterized by two parameters; and the increment standard deviation . The latter however is not relevant in the current testing environment, since changes to leave the scaling properties unchanged, and can thus be set to any arbitrary value such as .

14 For the fGn, the same parameter controls both the scaling and dependence properties of the process. This is a characteristic of the subclass of ‘self-affine’ processes, of which the fBm and fGn are members; however, unifractal processes more generally may possess independent scaling and dependence properties. The construction of such generalized unifractal processes is more complex and so few examples exist in the literature, with a notable example being that of Gneiting and Schlather (2004).

15 Indeed, a secondary benefit of using the fGn is that the methods available for simulating sample paths include a number of very efficient algorithms, thus reducing computational requirements compared to alternative unifractal processes.

16 The additional parameters for the integral timescale, , and sampling interval were set to and , though these are of less direct interest and are only defined relative to the sampling frequency of the data, which is arbitrary when dealing with simulated series.

17 Values of in the range are typical in empirical studies of multifractality in financial data, such as Muzy et al. (2001), Di Matteo (2007) or Bacry et al. (2008).

18 To check robustness to alternative choices of estimator for , the MF-DFA estimator was also substituted for the multifractal centred moving average (MF-CMA) estimator of Schumann and Kantelhardt (2011). The empirical size and power obtained for the testing methodology employing MF-CMA was very similar to that employing the MF-DFA estimator, with results available upon request.

19 Note that for figure 4 a logarithmic scale has been employed for the horizontal axis to improve legibility; however, the vales of marked on the axis correspond to the actual values of and not their logarithms.

20 Additional theoretical multifractal processes have been developed in other fields of study, such as the physics literature. However, given that our interest lies specifically in financial applications we focus on processes developed for this context.

21 Although as previously discussed, the MRW process used for testing power under the alternative will exhibit non-linear serial dependence, particularly in the variance, even when linearly independent.

22 This problem is also encountered in the literature on realized volatility, where the 5-min sampling interval has generally been found to be a good compromise between these two factors (see e.g. Andersen et al. 2001).

23 For assets that are very liquid, such as those considered here, even higher sampling frequencies, such as 1-min data, could arguably be used without encountering serious issues with market microstructure effects.

25 If desired, to avoid discarding observations steps (1) and (2) can be repeated starting from the end of the original series , producing segments of length . However, the effects on the resulting estimates of will typically be small unless is large relative to , which should typically be avoided.

26 It is also possible to use higher order polynomial detrending, allowing more complex forms of nonstationarity to be removed from the series. However, it is standard in the finance literature to employ simple linear detrending due to the computational demands of higher order detrending.

27 Values in this range were used as a starting point for the Monte Carlo studies of section 5 and were found to produce tests with good performance in the current context.

28 Because for any unifractal process for any , an estimate of can in principle be obtained from the MF-DFA estimator using any , but the value of simplifies the expression in Step 4 and so is used in the unifractal case.

29 The time discretization step could be set to value of without loss of generality by simply normalizing the units of time used for measurement.

Additional information

Funding

Mark Hallam would like to thank Science Foundation Ireland (under grant number 08/SRC/FMC1389) and The Scientific and Technological Research Council of Turkey (under the BİDEB 2216 Research Fellowship Programme for International Researchers, undertaken at Koç University) for financial support.