Statistical Papers

, Volume 60, Issue 3, pp 687–716 | Cite as

A Gini-based time series analysis and test for reversibility

  • Amit ShelefEmail author
  • Edna Schechtman
Regular Article


Time reversibility is a fundamental hypothesis in time series. In this paper, Gini-based equivalents for time series concepts that enable to construct a Gini-based test for time reversibility under merely first-order moment assumptions are developed. The key idea is that the relationship between two variables using Gini (as measured by Gini autocorrelations and partial autocorrelations) can be measured in two directions, which are not necessarily equal. This implies a built-in capability to discriminate between looking at forward and backward directions in time series. The Gini creates two bi-directional Gini autocorrelations (and partial autocorrelations), looking forward and backward in time, which are not necessarily equal. The difference between them may assist in identifying models with underlying heavy-tailed and non-normal innovations. Gini-based test and Gini-based correlograms, which serve as visual tools to examine departures from the symmetry assumption, are constructed. Simulations are used to illustrate the suggested Gini-based framework and to validate the statistical test. An application to a real data set is presented.


Autocorrelation Autoregression Gini correlation Gini regression Moving block bootstrap Time reversibility 

Mathematics Subject Classification

62-07 62G08 



The authors thank Shlomo Yitzhaki, Robert Serfling, Yisrael Parmet, Jeff Hart and Gideon Schechtman for their helpful comments on a previous version of this paper. The authors would also like to thank the anonymous reviewers for their thorough review and highly appreciate their comments and suggestions, which significantly contributed to improving the quality of the paper.


  1. Andrews B, Calder M, Davis RA (2009) Maximum likelihood estimation for \(\alpha \)-stable autoregressive processes. Ann Stat 37(4):1946–1982MathSciNetCrossRefzbMATHGoogle Scholar
  2. Andrews B, Davis RA (2013) Model identification for infinite variance autoregressive processes. J Econom 172(2):222–234MathSciNetCrossRefzbMATHGoogle Scholar
  3. Box GEP, Jenkins GM, Reinsel GC (1994) Time series analysis: forecasting and control, 3rd edn. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  4. Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2nd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
  5. Carcea M (2014) Contributions to time series modeling under first order moment assumptions. PhD Dissertation, University of Texas at DallasGoogle Scholar
  6. Carcea M, Serfling R (2015) A Gini autocovariance function for time series modeling. J Time Ser Anal 36(6):817–838CrossRefzbMATHGoogle Scholar
  7. Chen YT, Chou RY, Kuan CM (2000) Testing time reversibility without moment restrictions. J Econom 95(1):199–218CrossRefzbMATHGoogle Scholar
  8. Davis RA, Resnick S (1985) More limit theory for the sample correlation function of moving averages. Stoch Process Their Appl 20(2):257–279MathSciNetCrossRefzbMATHGoogle Scholar
  9. Davis RA, Resnick S (1986) Limit theory for the sample covariance and correlation functions of moving averages. Ann Stat 14:533–558MathSciNetCrossRefzbMATHGoogle Scholar
  10. Feigin PD, Resnick SI (1999) Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes 1(4):391–422CrossRefzbMATHGoogle Scholar
  11. Franses PH, Van Dijk D (2000) Non-linear time series models in empirical finance. Cambridge University Press, CambridgeGoogle Scholar
  12. Gini C (1914) On the measurement of concentration and variability of characters. Reprinted in Metron 63(2005):3–38Google Scholar
  13. Kunsch HR (1989) The jackknife and the bootstrap for general stationary observations. Ann Stat 17(3):1217–1241MathSciNetCrossRefzbMATHGoogle Scholar
  14. Lerman RI, Yitzhaki S (1984) A note on the calculation and interpretation of the Gini index. Econ Lett 15(3):363–368CrossRefGoogle Scholar
  15. Liu RY, Singh K (1992) Moving blocks jackknife and bootstrap capture weak dependence. In: Lepage R, Billard L (eds)Exploring the limits of bootstrap. Wiley, New York, p 225–248Google Scholar
  16. Olkin I, Yitzhaki S (1992) Gini regression analysis. Int Stat Rev 60:185–196CrossRefzbMATHGoogle Scholar
  17. Paulaauskas V, Rachev ST (2003) Maximum likelihood estimators in regression models with infinite variance innovations. Stat Pap 44(1):47–65MathSciNetCrossRefzbMATHGoogle Scholar
  18. Psaradakis Z (2008) Assessing time-reversibility under minimal assumptions. J Time Ser Anal 29(5):881–905MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ramsey JB, Rothman P (1996) Time irreversibility and business cycle asymmetry. J Money Credit Bank 28(1):1–21CrossRefGoogle Scholar
  20. Rubinstein ME (1973) The fundamental theorem of parameter-preference security valuation. J Financ Quant Anal 8:61–69CrossRefGoogle Scholar
  21. Schechtman E, Yitzhaki S (1987) A measure of association based on Gini mean difference. Commun Stat Theory Methods 16(1):207–231MathSciNetCrossRefzbMATHGoogle Scholar
  22. Schechtman E, Yitzhaki S (1999) On the proper bounds of the Gini correlation. Econ Lett 63(2):133–138MathSciNetCrossRefzbMATHGoogle Scholar
  23. Schechtman E, Yitzhaki S, Artsev Y (2008) Who does not respond in the household expenditure survey: an exercise in extended Gini regressions. J Bus Econ Stat 26(3):329–344MathSciNetCrossRefGoogle Scholar
  24. Serfling R (2010) Fitting autoregressive models via Yule-Walker equations allowing heavy tail innovations (preprint)Google Scholar
  25. Serfling R, Xiao P (2007) A contribution to multivariate L-moments: L-comoment matrices. J Multivar Anal 98(9):1765–1781MathSciNetCrossRefzbMATHGoogle Scholar
  26. Shelef A (2013) Statistical analyses based on Gini for time series data. PhD Dissertation, Ben-Gurion University of the NegevGoogle Scholar
  27. Shelef A (2014) A Gini-based unit root test. Comput Stat Data Anal. doi: 10.1016/j.csda.2014.08.012
  28. Shelef A, Schechtman E (2011) A Gini-based methodology for identifying and analyzing time series with non-normal innovations (preprint)Google Scholar
  29. Stuart A, Ord JK (1987) Kendall’s advanced theory of statistics, 5 edn, vol 1. Oxford University Press, New YorkGoogle Scholar
  30. Wei WWS (1993) Time series analysis. Addison-Wesley, New YorkGoogle Scholar
  31. Weiss G (1975) Time-reversibility of linear stochastic processes. J Appl Probab 12:831–836MathSciNetCrossRefzbMATHGoogle Scholar
  32. Wodon Q, Yitzhaki S (2006) Convergence forward and backward? Econ Lett 92(1):47–51MathSciNetCrossRefzbMATHGoogle Scholar
  33. Yitzhaki S (2003) Gini’s mean difference: a superior measure of variability for non-normal distributions. Metron 61(2):285–316MathSciNetzbMATHGoogle Scholar
  34. Yitzhaki S, Schechtman E (2013) The Gini methodology. Springer, New YorkCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Logistics, Sapir Academic CollegeD.N. Hof AshkelonSderotIsrael
  2. 2.Department of Industrial Engineering and ManagementBen-Gurion University of the NegevBeer-ShebaIsrael

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