The Effect of Tick Size on Testing for Nonlinearity in Financial Markets Data


The discrete nature of financial markets time-series data may prejudice the BDS and Close Returns test for nonlinearity. Our estimation results suggest that a tick/volatility ratio threshold exists, beyond which the test results are biased. Further, tick/volatility ratios that exceed these thresholds are frequently observed in financial markets data, which suggests that the results of the BDS and CR test must be interpreted with caution.

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H. Mitchell and M. McKenzie, "The Effect of Tick Size on Testing for Nonlinearity in Financial Markets Data," Journal of Mathematical Finance, Vol. 1 No. 1, 2011, pp. 1-7. doi: 10.4236/jmf.2011.11001.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] V. R. Anshuman and A. Kalay, “Market Making with Discrete Prices,” The Review of Financial Studies, Vol. 11, No. 1, 1998, pp. 81-109. doi:10.1093/rfs/11.1.81
[2] C. Vorlow, “Stock Price Clustering and Discreteness: The ‘Compass Rose’ and Complex Dynamics,” Working Paper, University of Durham, Durham, 2004.
[3] A. Antoniou and C. E. Vorlow, “Price Clustering and Discreteness: Is There Chaos behind the Noise,” Physica A, Vol. 348, 2005, pp. 389-403. doi:10.1016/j.physa.2004.09.006
[4] C. Lee, K. Gleason and I. Mathur, “The Tick/Volatility Ratio as a Determinant of the Compass Rose Pattern,” European Journal of Finance, Taylor and Francis Journals, Vol. 11, No. 2, April 2005, pp. 93-109.
[5] H. Amilon, “GARCH Estimation and Discrete Stock Pri- ces: An Application to Low-Priced Australian Stocks,” Economics Letters, Vol. 81, No. 2, 2003, pp. 215-222. doi:10.1016/S0165-1765(03)00172-1
[6] G. Gottleib and A. Kalay, “Implications of the Discreteness of Observed Stock Prices,” Journal of Finance, Vol. 40, No. 1, 1985, pp. 135-154. doi:10.2307/2328052
[7] R. Koppl and S. Tuluca, “Random Walk Hypothesis Test- ing and the Compass Rose,” Finance Letters, Vol. 2, No. 1, 2002, pp. 14-17.
[8] Y. Fang, “The Compass Rose and Random Walk Tests,” Computational Sta-tistics and Data Analysis, Vol. 39, No. 3, 2002, pp. 299-310. doi:10.1016/S0167-9473(01)00063-9
[9] T. F. Crack and O. Ledoit, “Robust Structure without Predictability: The Compass Rose Pattern of the Stock Market,” Journal of Finance, Vol. 51, No. 2, 1996, pp. 751-762. doi:10.2307/2329379
[10] W. Kramer and R. Runde, “Chaos and the Compass Rose,” Eco-nomics Letters, Vol. 54, No. 2, 1997, pp. 113-118. doi:10.1016/S0165-1765(97)00020-7
[11] W. Brock, W. De-chert and J. Scheinkman, “A Test for Independence Based on the Correlation Dimension,” Working Paper, University of Wisconsin, Madison, 1987.
[12] W. D. Dechert, “Testing Time Series for Nonlinearities: The BDS Approach,” In: W. A. Kirman,et al., Eds. Non- linear Dynamics and Economics: Proceedings of the 10th International Symposium in Economic Theory and Eco- nometrics, Cambridge University Press, Cambridge, 1996
[13] P. Grassberger and I. Procacia, “Meas-uring the Strangeness of Strange Attractors,” Physica D: Nonlinear Phenomena, Vol. 9, No. 1-2, 1983, pp. 189-208.
[14] Y.-T. Chen and C.-M. Kuan, “Time Irreversibil-ity and EGARCH Effects in US Stock Index Returns,” Journal of Applied Econometrics, Vol. 17, No. 5, 2002, pp. 565-578. doi:10.1002/jae.692
[15] R. Ostermark, J. Aaltonen, H. Saxen and K. Soderlund, “Nonlinear Modelling of the Finnihs Banking and Finance Branch Index,” European Journal of Finance, Vol. 10, No. 4, 2004, pp. 277-289. doi:10.1080/13518470210124641
[16] R. T. Baillie, A. A. Cecen and Y.-W. Han, “High Frequency Deutschemark-US Dollar Returns: FIGARCH Representation and Non Linearities,” Multinational Finance Journal, Vol. 4, No. 3-4, 2000, pp. 247-268.
[17] C. G. Gilmore, “An Examination of Nonlinear Dependence in Exchange Rates Using Recent Methods from Chaos Theory,” Global Finance Journal, Vol. 12, No. 1, 2001, pp. 139-151. doi:10.1016/S1044-0283(01)00018-7
[18] V. Chwee, “Chaos in Natural Gas Futures?” The Energy Journal, Vol. 19, No. 2, 1998, pp. 149-164. doi:10.5547/ISSN0195-6574-EJ-Vol19-No2-10
[19] M. Frank and T. Stengos, “Chaotic Dynamics in Economic Time Series,” Journal of Economic Surveys, Vol. 2, No. 2, 1988, pp. 103-133. doi:10.1111/j.1467-6419.1988.tb00039.x
[20] N. Kohzadi and M. S. Boyd, “Testing for Chaos and Nonlinear Dynamics in Cattle Prices,” Canadian Journal of Agricultural Economics, Vol. 43, No. 3, 1995, pp. 475- 484. doi:10.1111/j.1744-7976.1995.tb00136.x
[21] P. M. Bodman, “Steepness and Deepness in the Australian Macroeconomy,” Applied Economics, Vol. 33, No. 3, 2001, pp. 375-382. doi:10.1080/00036840122115
[22] C. G. Gilmore, “An Ex-amination of Nonlinear Dependence in Exchange Rates Using Recent Methods from Chaos Theory,” Global Finance Journal, Vol. 12, No. 1, 2001, pp. 139-151. doi:10.1016/S1044-0283(01)00018-7
[23] C. G. Gilmore, “A New Test for Chaos,” Journal of Eco- nomic Behaviour and Organisation, Vol. 22, No. 2, 1993, pp. 209-237. doi:10.1016/0167-2681(93)90064-V
[24] C. G. Gilmore, “A New Approach to Testing for Chaos, with Applications in Fi-nance and Economics,” International Journal of Bifurcation and Chaos, Vol. 3, No. 3, 1993, pp. 583-587. doi:10.1142/S0218127493000477
[25] C. G. Gilmore, “De-tecting Linear and Nonlinear Dependence in Stock Returns: New Methods Derived from Chaos Theory,” Journal of Business and Finance Accounting, Vol. 23, No. 9-10, 1996, pp. 1357-1377. doi:10.1111/1468-5957.00084
[26] M. D. McKenzie, “Chaotic Behaviour in National Stock Market Indices: New Evidence from the Close Returns Test,” Global Finance Journal, Vol. 12, No. 1, 2001, pp. 35-53. doi:10.1016/S1044-0283(01)00024-2
[27] D. E. Knuth, “The Art of Computer Programming,” 3rd Edition, Addison-Wesley Publishing Company, London. 1997.
[28] W. A. Brock, D. A. Hsieh and B. LeBaron, “Nonlinear Dynamics, Chaos and In-stability: Statistical Theory and Economic Evidence,” MIT Press, London, 1991.
[29] P. L’Ecuyer, “Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators,” Operations Research, Vol. 47, No. 1, 1999, pp. 159-164. doi:10.1287/opre.47.1.159
[30] M. Ma-tsumoto and T. Nishimura, “Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudo-Ran- dom Number Generator,” ACM Transactions on Modeling and Computer Simulation, Vol. 8, No. 1, 1998, pp. 3- 30. doi:10.1145/272991.272995
[31] C. I. Lee, K. C. Gleason and I. Mathur, “A Comprehensive Examination of the Compass Rose Pattern in Futures Markets,” Journal of Futures Markets, Vol. 19, No. 5, 1999, pp. 541-564. doi:10.1002/(SICI)1096-9934(199908)19:5<541::AID-FUT3>3.0.CO;2-7
[32] E. Wang, R. Hudson and K. Keasey, “Tick Size and the Compass Rose: Further Insights,” Economics Letters, Vol. 68, No. 2, 2000, pp. 119-125. doi:10.1016/S0165-1765(00)00237-8
[33] D. Ruelle, “Deter-ministic Chaos: The Science and the Fiction,” Proceedings of the Royal Society of London, Vol. 427, No. 1873, 1991, pp. 241-248.
[34] J. B. Ramsay and H. J. Yuan, “Bias and Error Bars in Dimension Calculations and Their Evaluation in Some Simple Models,” Physics Letters A, Vol. 134, No. 5, 1989, pp. 287-297. doi:10.1016/0375-9601(89)90638-5
[35] J. B. Ram-say and H. J. Yuan, “The Statistical Properties of Dimension Calculations Using Small Data Sets,” Non- linearity, Vol. 3, No. 1, 1990, pp. 155-175. doi:10.1088/0951-7715/3/1/009
[36] K. C. Gleason, C. I. Lee and I. Mathur, “An Explanation for the Compass Rose Pattern,” Economics Letters, Vol. 68, No. 2, 2000, pp. 127-133. doi:10.1016/S0165-1765(00)00252-4
[37] M. D. McKenzie and A. Frino, “The Tick/Volatility Ratio as a Determinant of the Compass Rose: Empirical evidence from decimalisation on the NYSE,” Accounting and Finance, Vol. 43, No. 3, 2003, pp. 331-344. doi:10.1111/j.1467-629x.2003.00094.x

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