An Independence Test Based on Joint Recurrences

Teresa Aparicio, Eduardo F. Pozo^*, Dulce Saura

Departamento de Análisis Económico, Universidad de Zaragoza, Zaragoza, Spain.

DOI: 10.4236/me.2015.68085                

Abstract

We propose in this paper a test procedure to determine whether two series proceed from independent systems or not. Our starting point is a multivariate extension of the methodology called Recurrence Quantification Analysis (RQA). We derive the test procedure from the probability distribution of the number of joint recurrences of both series under the null hypothesis of independence. The behavior of the test is evaluated by means of a large set of simulations, carried out with different types of dynamical systems: random, deterministic chaotic, deterministic non-chaotic, systems affected by noise and coupled systems. We obtain satisfactory results in all cases. Finally, the methodology is used to study two questions, on which the bulk of the existing economic literature agrees: 1) the relationship between the nominal interest rate and the inflation rate; and 2) the relationship between the gross domestic product and the employment. The results suggest that our test can be a suitable tool for detecting linear and nonlinear dependence between real series.

Keywords

Independence Test, Joint Recurrence Plot, Recurrence Quantification Analysis, Dynamical Systems

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Szekely, G., Rizzo, M. and Bakirov, N. (2007) Measuring and Testing Independence by Correlation of Distances. Annals of Statistics, 35, 2769-2794. http://dx.doi.org/10.1214/009053607000000505
[2]	Reshef, D.N., Reshef, Y.A., Finucane, H.K., Grossman, S.R., McVean, G., Turnbaugh, P.J., Lander, E.S., Mitzenmacher, M. and Sabeti, P.C. (2011) Detecting Novel Associations in Large Data Sets. Science, 334, 1518-1524. http://dx.doi.org/10.1126/science.1205438
[3]	Heller, Y., Heller, R. and Gorfine, M. (2013) A Consistent Multivariate Test of Association Based on Ranks of Distances. Biometrika, 100, 495-502. http://dx.doi.org/10.1093/biomet/ass070
[4]	Sklar, A. (1959) Fonctions de Répartition à n Dimensions et Leurs Marges. Vol. 8, Institut Statistique de l’Université de Paris, Paris, 229-231.
[5]	Siqueira, S., Takahashi, D.Y., Nakata, A. and Fujita, A. (2014) A Comparative Study of Statistical Methods Used to Identify Dependencies between Gene Expression Signals. Briefings in Bioinformatics, 15, 906-918. http://dx.doi.org/10.1093/bib/bbt051
[6]	Eckmann, J.P., Kamphorst, S. and Ruelle, D. (1987) Recurrence Plots of Dynamical Systems. Europhysics Letters, 4, 973-977. http://dx.doi.org/10.1209/0295-5075/4/9/004
[7]	Zbilut, J.P. and Webber Jr., C.L. (1992) Embeddings and Delays as Derived Quantification of Recurrence Plots. Physics Letters A, 171, 199-203. http://dx.doi.org/10.1016/0375-9601(92)90426-M
[8]	Webber Jr., C.L. and Zbilut, J.P. (1994) Dynamical Assessment of Physiological Systems and States Using Recurrence Plot Strategies. Journal of Applied Physiology, 76, 965-973.
[9]	Holyst, J.A., Zebrowska, M. and Urbanowicz, K. (2001) Observations of Deterministic Chaos in Financial Time Series by Recurrence Plots, Can One Control Chaotic Economy? The European Physical Journal B, 20, 531-535. http://dx.doi.org/10.1007/PL00011109
[10]	Belaire-Franch, J. (2004) Testing for Non-Linearity in an Artificial Financial Market: A Recurrence Quantification Approach. Journal of Economic Behavior & Organization, 54, 483-494. http://dx.doi.org/10.1016/j.jebo.2003.05.001
[11]	Kyrtsou, C. and Vorlow, C.E. (2005) Complex Dynamics in Macroeconomics: A Novel Approach. In: Diebolt, C. and Kyrtsou, C., Eds, New Trends in Macroeconomics, Springer-Verlag, Berlin, 223-238. http://dx.doi.org/10.1007/3-540-28556-3_11
[12]	Zbilut, J.P. (2005) Use of Recurrence Quantification Analysis in Economic Time Series. In: Salzano, M. and Kirman, A., Eds, Economics: Complex Windows, Springer-Verlag, Milan, 91-104. http://dx.doi.org/10.1007/88-470-0344-x_5
[13]	Barkoulas, J.T. (2008) Testing for Deterministic Monetary Chaos: Metric and Topological Diagnostics. Chaos, Solitons and Fractals, 38, 1013-1024. http://dx.doi.org/10.1016/j.chaos.2007.01.065
[14]	Aparicio, T., Pozo, E. and Saura, D. (2009) Detecting Determinism Using Recurrence Quantification Analysis: Three Test Procedures. Journal of Economic Behavior & Organization, 65, 768-787. http://dx.doi.org/10.1016/j.jebo.2006.03.005
[15]	Aparicio, T., Pozo, E. and Saura, D. (2011) Detecting Determinism Using Recurrence Quantification Analysis: A Solution to the Problem of Embedding. Studies in Nonlinear Dynamics and Econometrics, 15, 1-10.
[16]	Aparicio, T., Pozo, E. and Saura, D. (2013) Do Exchange Rate Series Present General Dependence? Some Results Using Recurrence Quantification Analysis. Journal of Economics and Behavioral Studies, 5, 678-686.
[17]	Kyrtsou, C. and Terraza, M. (2010) Seasonal Mackey-Glass-GARCH Process and Short-Term Dynamics. Empirical Economics, 38, 325-345. http://dx.doi.org/10.1007/s00181-009-0268-8
[18]	Guhathakurta, K., Bhattarcharya, B. and Roy Chowdhury, A. (2010) Using Recurrence Plot Analysis to Distinguish between Endogenous and Exogenous Stock Market Crashes. Physica A: Statistical Mechanics and Its Applications, 389, 1874-1882. http://dx.doi.org/10.1016/j.physa.2009.12.061
[19]	Marwan, N. (2008) A Historical Review of Recurrence Plots. European Physical Journal—Special Topics, 164, 3-12. http://dx.doi.org/10.1140/epjst/e2008-00829-1
[20]	Schinkel, S., Dimigen, O. and Marwan, N. (2008) Selection of Recurrence Threshold for Signal Detection. The European Physical Journal—Special Topics, 164, 45-53. http://dx.doi.org/10.1140/epjst/e2008-00833-5
[21]	Zbilut, J.P., Giuliani, A. and Webber Jr., C.L. (1998) Detecting Deterministic Signals in Exceptionally Noisy Environments Using Cross-Recurrence Quantification. Physics Letters A, 246, 122-128. http://dx.doi.org/10.1016/S0375-9601(98)00457-5
[22]	Romano, M.C. (2004) Synchronization Analysis by Means of Recurrences in Phase Space. Ph.D. Dissertation, University of Potsdam, Potsdam.
[23]	Marwan, N., Romano, M.C., Thiel, M. and Kurths, J. (2007) Recurrence Plots for the Analysis of Complex Systems. Physics Reports, 438, 237-329. http://dx.doi.org/10.1016/j.physrep.2006.11.001
[24]	Goswami, B., Ambika, G., Marwan, N. and Kurths, J. (2012) On Interrelations of Recurrences and Connectivity Trends between Stocks Indices. Physica A: Statistical Mechanics and Its Applications, 391, 4364-4376. http://dx.doi.org/10.1016/j.physa.2012.04.018
[25]	Goswami, B., Marwan, N., Feulner, G., Kurths, J. (2013) How Do Global Temperature Drivers Influence Each Other? The European Physical Journal—Special Topics, 222, 861-873. http://dx.doi.org/10.1140/epjst/e2013-01889-8
[26]	Grebogi, C., Ott, E., Pelikan, S. and Yorke, L. (1984) Strange Attractors That Are Non Chaotic. Physica D: Nonlinear Phenomena, 13, 261-268. http://dx.doi.org/10.1016/0167-2789(84)90282-3
[27]	Packard, N., Crutchfield, J.P., Farmer, J.D. and Shaw, R.S. (1980) Geometry from a Time Series. Physical Review Letters, 45, 712-716. http://dx.doi.org/10.1103/PhysRevLett.45.712
[28]	Takens, F. (1981) Detecting Strange Attractors in Turbulence. In: Rand, D. and Young, L., Eds, Dynamical Systems and Turbulence, Springer-Verlag, Berlin, 366-381. http://dx.doi.org/10.1007/bfb0091924