×

zbMATH — the first resource for mathematics

A consistent nonparametric test for serial independence. (English) Zbl 0924.62041
Summary: We propose a nonparametric test for serial independence against serial dependence of fixed order that is consistent against all such alternatives. The conditions required are weak, the asymptotic distribution under the null is \(\chi^2_1\), and the test works regardless of the underlying distribution. Also included are a nuisance parameter result, Monte Carlo simulations, a theoretical efficiency study, an empirical example, and a review of possible extensions. In addition, we derive a similar consistent test for lack of serial dependence of order one against serial dependence of order one.

MSC:
62G10 Nonparametric hypothesis testing
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P20 Applications of statistics to economics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Andrews, D. W. K.: Asymptotic normality of series estimators for nonparametric and semiparametric regression models. Econometrica 49, 307-346 (1991) · Zbl 0727.62047
[2] Bank of England, Quarterly Bulletin, Selected Issues, Bank of England, London.
[3] Box, G. E. P.; Pierce, D. A.: Distribution of residual autocorrelations in autoregressive integrated moving average time series models. Journal of the American statistical association 65, 1509-1526 (1970) · Zbl 0224.62041
[4] Brock, W. A.; Dechert, W. D.; Scheinkman, J. A.: A test for independence based on the correlation dimension. (1987) · Zbl 0893.62034
[5] Brock, W. A.; Dechert, W. D.; Scheinkman, J. A.; Lebaron, B. D.: A test for independence based upon the correlation dimension. (1987) · Zbl 0893.62034
[6] Chan, N. H.; Tran, L. T.: Nonparametric tests for serial dependence. Journal of time series analysis 13, 19-28 (1992) · Zbl 0745.62044
[7] Csörgö, S.: Testing for independence by the empirical characteristic function. Journal of multivariate analysis 16, 290-299 (1985) · Zbl 0585.62097
[8] Denker, M.; Keller, G.: On U-statistics and von Mises statistics for weakly dependent processes. Zeitschrift für wahrscheinlichkeitstheorie und verwandte gebiete 64, 502-522 (1983) · Zbl 0519.60028
[9] Drost, F. C.; Werker, B. J. M.: A note on Robinson’s test of independence. (1993)
[10] Durbin, J.; Watson, G. S.: Testing for serial correlation in least squares regression I. Biometrika 37, 409-428 (1950) · Zbl 0039.35803
[11] Dufour, J. -M.: Rank tests for serial dependence. Journal of time series analysis 2, 117-127 (1981) · Zbl 0514.62054
[12] Dufour, J. -M.; Lepage, Y.; Zeidan, H.: Nonparametric testing for time series: a bibliography. The canadian journal of statistics 10, 1-38 (1982) · Zbl 0483.62034
[13] Engle, R. F.: Autoregressive conditional heteroskedasticity with estimates of the variance of united kingdom inflations. Econometrica 50, 987-1008 (1982) · Zbl 0491.62099
[14] Epanechnikov, V.: Nonparametric estimates of a multivariate probability density. Theory of probability and its applications 14, 153-158 (1969) · Zbl 0175.17101
[15] Godfrey, L. G.: Misspecification tests in econometrics: the Lagrange multiplier principle and other approaches. (1988) · Zbl 0721.62111
[16] Gorodetskii, V. V.: On the strong mixing property for linear sequences. Theory of probability and its applications 22, 411-413 (1977) · Zbl 0377.60046
[17] Guerre, E.: Estimating Kullback contrast by the kernel method: asymptotic distribution. (1991) · Zbl 0735.62039
[18] Hallin, M.; Ingenbleek, J. -F.; Puri, M. L.: Linear serial rank test for randomness against ARMA alternatives. Annals of statistics 12, 1156-1181 (1985) · Zbl 0584.62064
[19] Hallin, M.; Mélard, G.: Optimal rank based tests for randomness against first-order serial dependence. Journal of the American statistical association 83, 1117-1122 (1988)
[20] Hallin, M.; Puri, M.: Rank tests for time series analysis: a survey. New directions in time series analysis (1992) · Zbl 0768.62075
[21] Härdle, W.: Resistant smoothing using the fast Fourier transform, AS 222. Applied statistics 36, 104-111 (1987) · Zbl 0613.62053
[22] Härdle, W.: Applied nonparametric regression. (1990) · Zbl 0714.62030
[23] Hidalgo, F. J.: Adaptive semiparametric estimation in the presence of autocorrelation of unknown form. Journal of time series analysis 13, 47-78 (1992) · Zbl 0769.62066
[24] Ibragimov, I.; Linnik, Y.: Independent and stationary sequences of random variables. (1971) · Zbl 0219.60027
[25] Joe, H.: Relative entropy measures of multivariate dependence. Journal of the American statistical association 84, 157-163 (1989) · Zbl 0677.62054
[26] King, M. L.: Testing for autocorrelation in linear regression models: a survey. Specification analysis in the linear regression model (1987)
[27] Knoke, J. D.: Testing for randomness against autocorrelation: alternative tests. Biometrika 64, 523-529 (1977) · Zbl 0382.62037
[28] Kullback, S.: Information theory and statistics. (1959) · Zbl 0088.10406
[29] Kullback, S.; Leibler, R. A.: On information and sufficiency. Annals of mathematical statistics 22, 79-86 (1951) · Zbl 0042.38403
[30] Ljung, G. M.; Box, G. E.: On a measure of lack of fit in time series models. Biometrika 65, 297-303 (1978) · Zbl 0386.62079
[31] Lukacs, E.: Characteristic functions. (1970) · Zbl 0201.20404
[32] Mokkadem, A.: Sur une modéle autorègressif non-linéaire. Ergodicité et ergodicité géométrique. Journal of time series analysis 8, 195-204 (1987) · Zbl 0621.60076
[33] Noether, G. E.: On a theorem of Pitman. Annals of mathematical statistics 26, 64-68 (1955) · Zbl 0066.12001
[34] Pham, T. D.; Tran, L. T.: Some mixing properties of time series models. Stochastic processes and their applications 19, 297-303 (1985) · Zbl 0564.62068
[35] Pinkse, C. A. P.: Nonparametric and semiparametric estimation and testing. Ph.d. thesis (1994)
[36] Pitman, E. J. G.: Nonparametric statistical inference. Lecture notes (1948)
[37] Robinson, P. M.: Kernel estimation and interpolation for time series containing missing observations. Annals of the institute of mathematical statistics 36, 401-412 (1984) · Zbl 0573.62089
[38] Robinson, P. M.: Testing for serial correlation in regression with missing observations. Journal of the royal statistical society B 47, No. 3, 429-437 (1985) · Zbl 0587.62167
[39] Robinson, P. M.: Root-N-consistent semiparametric regression. Econometrica 56, 931-954 (1988) · Zbl 0647.62100
[40] Robinson, P. M.: Hypothesis testing in semiparametric and nonparametric models for econometric time series. Review of economic studies 56, 511-534 (1989) · Zbl 0681.62101
[41] Robinson, P. M.: Consistent nonparametric entropy based testing. Review of economic studies 58, 437-453 (1991) · Zbl 0719.62055
[42] Rosenblatt, M.: A quadratic measure of deviation of two-dimensional density estimates and a test of independence. Annals of statistics 3, 1-14 (1975) · Zbl 0325.62030
[43] Rosenblatt, M.; Wahlen, B. E.: A nonparametric measure of independence under a hypothesis of independent components. Statistics and probability letters (1992) · Zbl 0770.62039
[44] Serfling, R.: Approximation theorems of mathematical statistics. (1980) · Zbl 0538.62002
[45] Silverman, B. W.: Kernel density estimation using the fast Fourier transform. Statistical algorithm AS170, applied statistics 31, 93-97 (1982) · Zbl 0483.62032
[46] Skaug, H. J.; Tjøstheim, D.: Nonparametric tests for serial independence. The M.B. Priestley birthday volume (1992) · Zbl 0880.62052
[47] Skaug, H. J.; Tjøstheim, D.: A nonparametric test for serial independence based on the empirical distribution function. (1992) · Zbl 0790.62044
[48] Von Neumann, J.: Distribution of the ratio of mean square successive difference to the variance. Annals of mathematical statistics 12, 367-395 (1941) · Zbl 0060.29911
[49] Von Neumann, J.: A further remark concerning the distribution of the ratio of mean square successive difference to the variance. Annals of mathematical statistics 13, 86-88 (1942) · Zbl 0060.29912
[50] Wahlen, B. E.: A nonparametric measure of independence. Ph.d. thesis (1991)
[51] Whistler, D. E. N.: Semi-parametric models of daily and intra-daily exchange rate volatility. Ph.d. thesis (1990)
[52] White, K.: The durbin-Watson test for autocorrelation in nonlinear models. Review of economics and statistics, 370-373 (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.