# Nonparametric Tests for Independence

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## Definition of the Subject

One of the central goals of data analysis is to measure and model the statistical dependence among random variables. Not surprisingly, therefore,the question whether two or more random variables are statistically independent can be encountered in a wide range of contexts. Although this articlewill focus on tests for independence among time series data, its relevance is not limited to the time series context only. In fact many of the dependencemeasures discussed could be utilized for testing independence between random variables in other statistical settings (e.?g. cross-sectionaldependence in spatial statistics).

When working with time series data that are noisy by nature, such as financial returns data, testing for serial independence is oftena preliminary step carried out before modeling the data generating process or implementing a prediction algorithm for futureobservations. A straightforward application in finance consists of testing the random walk...

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## Notes

1. 1.

It is exactly degenerate for i.i.d. data from the uniform distribution on the circle,i.?e. the interval $${ [0,a] }$$ withthe endpoints identified [16]. Theiler [102]simulated variance of $$S=C_{m,n}(\varepsilon)-(C_{i,n}(\varepsilon))^m$$ in the degenerate case, andfound that it converges to 0 at the rate $${ n^{-2} }$$ instead of the usual rate $${ n^{-1} }$$.

## Abbreviations

Hypothesis :

A hypothesis is a statement concerning the (joint) distribution underlying the observed data.

Nonparametric test :

In contrast to a parametric test, a nonparametric test does not presume a particular parametric structure concerning the data generating process.

Serial dependence :

Statistical dependence among time series observations.

Time series :

A sequence of observed values of some variable over time, such as a historical temperature record, a sequence of closing prices of a stock, etc.

## Bibliography

1. Aparicio FM, Escribano A (1998) Information-theoretic analysis of serial dependence and cointegration. Stud nonlinear dyn econ 3:119–140

2. Ashley RA, Patterson DM (1986) A nonparametric, distribution-free test for serial independence in stock returns. J Financial Quant Anal 21:221–227

3. Ashley RA, Patterson DM (1989) Linear versus nonlinear macroeconomics: A statistical test. Int Econ Rev 30:165–187

4. Ashley RA, Patterson DM, Hinich MN (1986) A diagnostic check for nonlinear serial dependence in time series fitting errors. J Time Ser Anal 7:165–187

5. Barnard GA (1963) Discussion of Professor Bartlett's paper. J Royal Stat Soc Ser B 25:294

6. Bartels R (1982) The rank version of von Neumann's ratio test for randomness. J Am Stat Assoc 77:40–46

7. Benghabrit Y, Hallin M (1992) Optimal rank-based tests against 1st-order superdiagonal bilinear dependence. J Stat Plan Inference 32:45–61

8. Bera AK, Robinson PM (1989) Tests for serial dependence and other specification analysis in models of markets in equilibrium. J Bus Econ Stat 7:343–352

9. Beran J (1992) A goodness-of-fit test for time-series with long-range dependence. J Royal Stat Soc Ser B 54:749–760

10. Blum JR, Kiefer J, Rosenblatt M (1961) Distribution free tests of independence based on sample distribution functions. Ann Math Stat 32:485–498

11. Bollerslev T (1986) Generalized autoregressive heteroskedasticity. J Econometrics 31:307–327

12. Booth GG, Martikainen T (1994) Nonlinear dependence in Finnish stock returns. Eur J Oper Res 74:273–283

13. Box GEP, Pierce DA (1970) Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. J Am Stat Assoc 332:1509–1526

14. Bradley R (1986) Basic properties of strong mixing conditions. In: Eberlein E, Taqqu MS (eds) Dependence in Probability and Statistics. Birkäuser, Basel

15. Brock WA, Dechert WD, Scheinkman JA (1987) A test for independence based on the correlation dimension. Working paper 8702. University of Wisconsin, Madison

16. Brock WA, Dechert WD, Scheinkman JA, LeBaron B (1996) A test for independence based on the correlation dimension. Econometric Rev 15:197–235

17. Brockett PL, Hinich MD, Patterson D (1988) Bispectral based tests for the detection of Gaussianity and linearity in time series. J Am Stat Assoc 83:657–664

18. Brooks C, Heravi SM (1999) The effect of (mis-specified) GARCH filters on the filite sample distribution of the BDS test. Comput Econ 13:147–162

19. Brooks C, Hinich MJ (1999) Cross-correlations and cross-bicorrelations in Sterling exchange rates. J Empir Finance 6:385–404

20. Brooks C, Hinich MJ (2001) Bicorrelations and cross-bicorrelations as non-linearity tests and tools for exchange rate forecasting. J Forecast 20:181–196

21. Caporale GM, Ntantamis C, Pantelidis T, Pittis N (2005) The BDS test as a test for the adequacy of a GARCH(1,1) specification: A Monte Carlo study. J Financial Econometric 3:282–309

22. Carlstein E (1984) The use of sub-series methods for estimating the variance of a general statistic from a stationary time series. Ann Stat 14:1171–1179

23. Carlstein E (1988) Degenerate U-statistics based on non-independent observations. Calcutta Stat Assoc Bull 37:55–65

24. Chan NH, Tran LT (1992) Nonparametric tests for serial independence. J Time Ser Anal 13:19–28

25. Corrado CJ, Schatzberg J (1990) A nonparametric, distribution-free test for serial independence in stock returns: A correction. J Financial Quant Anal 25:411–415

26. Csörgo S (1985) Testing for independence by the empirical characteristic function. J Multivar Anal 16:290–299

27. Debnath L, Mikusinski P (2005) Introduction to Hilbert Spaces With Applications, 3rd edn. Elsevier Academic Press, Burlington

28. Delgado M, Mora J (2000) A nonparametric test for serial independence of regression errors. Biometrika 87:228–234

29. Delgado MA (1996) Testing serial independence using the sample distribution function. J Time Ser Anal 11:271–285

30. Denker M, Keller G (1983) On U-statistics and v. Mises' statistics for weakly dependent processes. Z Wahrscheinlichkeitstheorie verwandte Geb 64:505–522

31. Denker M, Keller G (1986) Rigorous statistical procedures for data from dynamical systems. J Stat Phys 44:67–93

32. Diks C, Panchenko V (2007) Nonparametric tests for serial independence based on quadratic forms. Statistica Sin 17:81–97

33. Diks C, Panchenko V (2008) Rank-based entropy tests for serial independence. Stud Nonlinear Dyn Econom 12(1)art.2:0–19

34. Diks C, Tong H (1999) A test for symmetries of multivariate probability distributions. Biometrika 86:605–614

35. Dionísio A, Menezes R, Mendes DA (2006) Entropy-based independence test. Nonlinear Dyn 44:351–357

36. Dufour JM (1981) Rank tests for serial dependence. J Time Ser Anal 2:117–128

37. Dufour JM (2006) Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics. J Econom 133:443–477

38. Durbin J, Watson GS (1950) Testing for serial correlation in least-squares regression, I. Biometrika 37:409–428

39. Durbin J, Watson GS (1951) Testing for serial correlation in least-squares regression, II. Biometrika 38:159–177

40. Durbin J, Watson GS (1971) Testing for serial correlation in least-squares regression, III. Biometrika 58:1–19

41. Durlauf S (1991) Spectral based testing of the martingale hypothesis. J Econometrics 50:355–376

42. Engle R (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50:987–1007

43. Ferguson TS, Genest C, Hallin M (2000) Kendall's tau for serial dependence. Can J Stat 28:587–604

44. Fernandes M, Neri B (2008) Nonparametric entropy-based tests of independence between stochastic processes. EconometricReviews; Forthcoming

45. Genest C, Quessy JF, Rémillard B (2002) Tests of serial independence based on Kendall's process. Can J Stat 30:1–21

46. Genest C, Rémillard B (2004) Tests of independence and randomness based on the empirical copula process. Test 13:335–369

47. Genest C, Verret F (2005) Locally most powerful rank tests of independence for copula models. Nonparametric Stat 17:521–539

48. Genest C, Ghoudi K, Rémillard B (2007) Rank-based extensions of the Brock, Dechert, and Scheinkman test. J Am Stat Assoc 102:1363–1376

49. Ghoudi K, Kulperger RJ, Rémillard B (2001) A nonparametric test of serial independence for time series and residuals. J Multivar Anal 79:191–218

50. Granger C, Lin JL (2001) Using the mutual information coefficient to identify lags in nonlinear models. J Time Ser Anal 15:371–384

51. Granger CW, Maasoumi E, Racine J (2004) A dependence metric for possibly nonlinear processes. J Time Ser Anal 25:649–669

52. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189–208

53. Grassberger P, Schreiber T, Schaffrath C (1991) Nonlinear time sequence analysis. Int J Bifurc Chaos 1:521–547

54. Hallin M, Ingenbleek J-F, Puri ML (1985) Linear serial rank tests for randomness against ARMA alternatives. Ann Stat 13:1156–1181

55. Hallin M, Mélard G (1988) Rank-based tests for randomness against first-order serial dependence. J Am Stat Assoc 83:1117–1128

56. Hannan EJ (1957) Testing for serial correlation in least squares regression. Biometrika 44:57–66

57. Hinich M, Patterson D (1985) Evidence of nonlinearity in stock returns. J Bus Econ Stat 3:69–77

58. Hinich MJ (1982) Testing for Gaussianity and linearity of a stationary time series. J Time Ser Anal 3:169–176

59. Hinich MJ (1996) Testing for dependence in the input to a linear time series model. J Nonparametric Stat 8:205–221

60. Hjellvik V, Tjøstheim D (1995) Nonparametric tests of linearity for time series. Biometrika 82:351–368

61. Hjellvik V, Yao Q, Tjøstheim D (1998) Linearity testing using polynomial approximation. J Stat Plan Inference 68:295–321

62. Hoeffding W (1948) A non-parametric test of independence. Ann Math Stat 19:546–557

63. Hong Y (1999) Hypothesis testing in time series via the empirical characteristic function: a generalized spectral density approach. J Am Stat Assoc 94:1201–1220

64. Hong Y (2000) Generalized spectral tests for serial dependence. J Royal Stat Soc Ser B 62:557–574

65. Hong Y, White H (2005) Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73:837–901

66. Horowitz JL (2001) The bootstrap. In: Heckman JJ, Leamer EE (eds) Handbook of Econometrics, vol 5. Elsevier, Amsterdam, pp 3159–3228

67. Horowitz JL, Spokoiny VG (2001) An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69:599–631

68. Joe H (1989) Relative entropy measures of multivariate dependence. J Am Stat Assoc 84:157–164

69. Joe H (1990) Multivariate concordance. J Multivar Anal 35:12–30

70. Johnson D, McLelland R (1997) Nonparametric tests for the independence of regressors and disturbances as specification tests. Rev Econ Stat 79:335–340

71. Johnson D, McLelland R (1998) A general dependence test and applications. J Appl Econometrics 13:627–644

72. Kallenberg WCM, Ledwina T (1999) Data driven rank tests for independence. J Am Stat Assoc 94:285–301

73. Kendall MG (1938) A new measure of rank correlation. Biometrika 30:81–93

74. Kocenda E, Briatka L (2005) Optimal range for the IID test based on integration across the correlation integral. Econometric Rev 24:265–296

75. Kulperger RJ, Lockhart RA (1998) Tests of independence in time series. J Time Ser Anal 1998:165–185

76. Künsch HR (1989) The jackknife and the bootstrap for general stationary observations. Ann Stat 17:1217–1241

77. Lim KP, Hinich MJ, Liew VKS (2005) Statistical inadequacy of GARCH models for Asian stock markets: Evidence and implications. J Emerg Mark Finance 4:263–279

78. Lima P De (1996) Nuisance parameter free properties of correlation integral based statistics. Econometric Rev 15:237–259

79. Ljung GM, Box GEP (1978) On a measure of lack of fit in time series models. Biometrika 65:297–202

80. Lo AW (2000) Finance: A selective survey. J Am Stat Assoc 95:629–635

81. Maasoumi E (2002) Entropy and predictability of stock market returns. J Econometrics 107:291–312

82. McLeod AI, Li WK (1983) Diagnostic checking ARMA time series models using squared-residual autocorrelations. J Time Ser Anal 4:269–273

83. Von Neumann J (1941) Distribution of the ratio of the mean square successive difference to the variance. Ann Math Stat 12:367–395

84. Pinkse J (1998) A consistent nonparametric test for serial independence. J Econometrics 84:205–231

85. Politis DN, Romano JP (1994) The stationary bootstrap. J Am Stat Assoc 89:1303–1313

86. Racine J, Maasoumi E (2007) A versatile and robust metric entropy test for time-irreversibility, and other hypotheses. J Econometrics 138:547–567

87. Ramsey JB, Rothman P (1990) Time irreversibility of stationary time series: estimators and test statistics. Unpublished manuscript, Department of Economics, New York University and University of Delaware

88. Robinson PM (1991) Consistent nonparametric entropy-based testing. Rev Econ Stud 58:437–453

89. Rosenblatt M (1975) A quadratic measure of deviation of two-dimensional density estimates and a test of independence. Ann Stat 3:1–14

90. Rosenblatt M, Wahlen BE (1992) A nonparametric measure of independence under a hypothesis of independent components. Stat Probab Lett 15:245–252

91. Rothman P (1992) The comparative power of the TR test against simple threshold models. J Appl Econometrics 7:S187–S195

92. Scaillet O (2005) A Kolmogorov–Smirnov type test for positive quadrant dependence. Can J Stat 33:415–427

93. Serfling RJ (1980) Approximation Theorems of Mathematical Statistics. Wiley, New York

94. Silverman BW (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York

95. Skaug HJ, Tjøstheim D (1993a) A nonparametric test for serial independence based on the empirical distribution function. Biometrika 80:591–602

96. Skaug HJ, Tjøstheim D (1993b) Nonparametric tests of serial independence. In: Subba Rao T (ed) Developments in Time Series Analysis: the M. B. Priestley Birthday Volume. Wiley, New York

97. Spearman C (1904) The proof and measurement of association between two things. Am J Psychol 15:72–101

98. Subba Rao T, Gabr MM (1980) A test for linearity of stationary time series. J Time Ser Anal 1:145–158

99. Takens F (1981) Detecting strange attractors in turbulence. In: Rand DA, Young LS (eds) Dynamical Systems and Turbulence, Warwick 1980. (Lecture Notes in Mathematics), vol 898. Springer, Berlin, pp 366–381

100. Terdik G, Máth J (1998) A new test of linearity of time series based on the bispectrum. J Time Ser Anal 19:737–753

101. Theil H, Nagar AL (1961) Testing the independence of regression disturbances. J Am Stat Assoc 56:793–806

102. Theiler J (1990) Statistical precision of dimension estimators. Phys Rev A 41:3038–3051

103. Tjøstheim D (1996) Measures of dependence and tests of independence. Statistics 28:249–284

104. Tong H (1990) Non-linear Time Series: A Dynamical Systems Approach. Clarendon Press, Oxford

105. Tsallis C (1998) Generalized entropy-based criterion for consistent testing. Phys Rev E 58:1442–1445

106. Wolff RC (1994) Independence in time series: another look at the BDS test. Philos Trans Royal Soc Ser A 348:383–395

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### Cite this entry

Diks, C. (2009). Nonparametric Tests for Independence. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_369