Abstract
This paper considers the independence test for two stationary infinite order autoregressive processes. For a test, we follow the empirical process method and construct the Cramér-von Mises type test statistics based on the least squares residuals. It is shown that the proposed test statistics behave asymptotically the same as those based on true errors. Simulation results are provided for illustration.
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References
Bai, J. (1994). Weak convergence of the sequential empirical processes of residuals in ARMA models,Annals of Statistics,22, 2051–2061.
Berk, K. (1974). Consistent autoregressive spectral estimates,Annals of Statistics,2, 489–502.
Billingsley, P. (1968).Convergence of Probability Measure, Wiley, New York.
Blum, J. R., Kiefer, J. and Rosenblatt, M. (1961). Distribution free tests of independence based on the sample distribution function,Annals of Mathematical Statistics,32, 485–498.
Brockwell, P. J. and Davis, R. A. (1990).Time Series: Theory and Models, 2nd ed., Springer-Verlag, New York.
Carlstein, E. (1988). Degenerate U-statistic based on non-independent observations,Calcutta Statistical Association Bulletin,37, 55–65.
Delgado, M. A. (1996). Testing serial independence using the sample distribution function,Journal of Time Series Analysis,17, 271–286.
Delgado, M. A. and Mora, J. (2000) A nonparametric test for serial independence of regression errors,Biometrika,87, 228–234.
Dunford, N. and Schwartz, J. T. (1963).Linear Operators, Wiley, New York.
Geweke, J. (1981). A comparison of tests of the independence of two covariance-stationary time series,Journal of the American Statistical Association,76, 363–373.
Haugh, L. D. (1976). Checking the independence of two covariance-stationary time series: A univariate residual cross-correlation approach.Journal of the American Statistical Association,71, 378–385.
Hoeffding, W. (1948). A nonparametric test of independence,Annals of Mathematical Statistics,26, 189–211.
Hong, Y. (1996). Testing for independence between two covariance stationary time series,Biometrika,83, 615–625.
Hong, Y. (1998). Testing for pairwise serial independence via the empirical distribution function,Journal of the Royal Statistical Society, Series B,60, 429–453.
Lee, S. and Karagrigoriou, A. (2001). An asymptotically optimal selection of the order of a linear processes,Sankhyā A,63, 93–106.
Lee, S. and Wei, C. Z. (1999). On residual empirical processes of stochastic regression models with applications to time series,Annals of Statistics,27, 237–261.
Neuhaus, G. (1971). On weak convergence of stochastic processes with multidimensional time parameter,Annals of Mathematical Statistics,42, 1285–1295.
Pierce, D. A. (1977). Relationships- and the lack thereof-between economic time series, with special reference to money and interest rates,Journal of the American Statistical Association,72, 11–22.
Priestley, M. B. (1981).Spectral Analysis and Time Series, Vol. 1, Academic Press, London.
Serfling, R. (1980).Approximation Theorems of Mathematical Statistics, Wiley, New York.
Shibata, R. (1980). Asymptotically efficient selection of the order of the model for estimating parameters of a linear processes,Annals of Statistics,8, 147–164.
Shorack, G. R. and Wellner, J. A. (1986).Empirical Processes with Applications to Statistics, Wiley, New York.
Skaug, H. J. (1993). The limit distribution of the Hoeffding statistic for test of serial independence (unpublished).
Skaug, H. J. and Tjøstheim, D. (1993). A nonparametic test of serial; independence based on the empirical distribution function,Biometrika,80, 591–602.
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Kim, E., Lee, S. A test for independence of two stationary infinite order autoregressive processes. Ann Inst Stat Math 57, 105–127 (2005). https://doi.org/10.1007/BF02506882
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DOI: https://doi.org/10.1007/BF02506882