To read the full version of this content please select one of the options below:

Time scale and fractionality in financial time series

Thomas W. Sproul (Department of Environmental & Natural Resource Economics, University of Rhode Island, Kingston, Rhode Island, USA)

Agricultural Finance Review

ISSN: 0002-1466

Article publication date: 3 May 2016



Turvey (2007, Physica A) introduced a scaled variance ratio procedure for testing the random walk hypothesis (RWH) for financial time series by estimating Hurst coefficients for a fractional Brownian motion model of asset prices. The purpose of this paper is to extend his work by making the estimation procedure robust to heteroskedasticity and by addressing the multiple hypothesis testing problem.


Unbiased, heteroskedasticity consistent, variance ratio estimates are calculated for end of day price data for eight time lags over 12 agricultural commodity futures (front month) and 40 US equities from 2000-2014. A bootstrapped stepdown procedure is used to obtain appropriate statistical confidence for the multiplicity of hypothesis tests. The variance ratio approach is compared against regression-based testing for fractionality.


Failing to account for bias, heteroskedasticity, and multiplicity of testing can lead to large numbers of erroneous rejections of the null hypothesis of efficient markets following an independent random walk. Even with these adjustments, a few futures contracts significantly violate independence for short lags at the 99 percent level, and a number of equities/lags violate independence at the 95 percent level. When testing at the asset level, futures prices are found not to contain fractional properties, while some equities do.

Research limitations/implications

Only a subsample of futures and equities, and only a limited number of lags, are evaluated. It is possible that multiplicity adjustments for larger numbers of tests would result in fewer rejections of independence.


This paper provides empirical evidence that violations of the RWH for financial time series are likely to exist, but are perhaps less common than previously thought.



JEL Classification — Q14, G13, G12

The author would like to thank the editor, Calum Turvey, and two anonymous referees for insightful comments and suggestions leading to an improved final draft, and thanks the participants of the 2015 International Agricultural Risk, Finance and Insurance Conference (IARFIC) meeting sessions for feedback on an earlier draft. Dr Sproul is grateful for financial support from USDA National Institute of Food and Agriculture Hatch Project No. RI00H-108, Accession No. 229284, and USDA ERS Cooperative Research Agreement No. 58-3000-2-0063.


Sproul, T.W. (2016), "Time scale and fractionality in financial time series", Agricultural Finance Review, Vol. 76 No. 1, pp. 76-93.



Emerald Group Publishing Limited

Copyright © 2016, Emerald Group Publishing Limited