×
Boldin, M. V.; Erlikh, I. G.

Testing hypotheses on the “drift” of parameters in ARMA and ARCH models. (English) Zbl 1282.62114

Math. Methods Stat. 18, No. 1, 1-20 (2009).
Summary: For an ARMA model, we test the hypothesis that the coefficients of this model remain constant in time and satisfy the stationarity condition against the alternative that the coefficients change (“drift”) in time. We propose asymptotically distribution free tests for such hypothesis based on sequential residual processes. A similar problem is solved for the ARCH model.

MSC:

62G10 Nonparametric hypothesis testing
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G20 Asymptotic properties of nonparametric inference

Keywords:

sequential processes; asymptotically distribution free tests; ARMA model; ARCH model; \(M\)-estimates
Cite Review PDF
Full Text: DOI

References:

[1] T.W. Anderson, The Statistical Analysis of Time Series (Wiley, New York, 1971). · Zbl 0225.62108
[2] J. Bai, ”Weak Convergence of the Sequential Empirical Processes of Residuals in ARMA Models”, Ann. Statist. 22, 2051–2061 (1994). · Zbl 0826.60016 · doi:10.1214/aos/1176325771
[3] P. J. Bickel and M. J. Wichura, ”Convergence for Multiparameter Stochastic Processes and Some Applications”, Ann. Statist. 42, 1656–1670 (1971). · Zbl 0265.60011 · doi:10.1214/aoms/1177693164
[4] P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968). · Zbl 0172.21201
[5] M. V. Boldin, ”On Median Estimates and Tests in Autoregressive Models”, Math. Methods Statist. 3, 114–129 (1994). · Zbl 0824.62039
[6] M.V. Boldin, ”On Sequential Residual Empirical Processes in Heteroscedastic Time Series”, Math. Methods Statist. 11, 453–464 (2002).
[7] M. V. Boldin and W. Stute, ”On Sign Tests in ARMA Models with Possibly Infinite Error Variance”, Theory Probab. Appl. 49(3), 392–413 (2005). · Zbl 1090.62091 · doi:10.1137/S0040585X97981160
[8] P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods (Springer, New York, 1987). · Zbl 0604.62083
[9] P. Doukhan, Mixing: Properties and Examples, Lecture Notes in Statistics, Vol. 85 (Springer, New York, 1994). · Zbl 0801.60027
[10] R. F. Engle, ”Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation”, Econometrica 5, 987–1008 (1982). · Zbl 0491.62099 · doi:10.2307/1912773
[11] P. Hall and C. C. Heyde, Martingale Limit Theory and its Applications (Academic Press, New York, 1980). · Zbl 0462.60045
[12] H. L. Koul, ”Asymptotics of Some Estimators and Sequential Residual Empiricals in Nonlinear Time Series”, Ann. Statist. 24(1), 380–404 (1996). · Zbl 0865.62065 · doi:10.1214/aos/1033066216
[13] S. Ling, ”Weak Convergence of the Sequential Empirical Processes of Residuals in Nonstationary Autoregressive Models”, Ann. Statist. 26, 741–754 (1998). · Zbl 0932.62064 · doi:10.1214/aos/1028144857
[14] J.W. Milnor, Topology from the Differentiable Viewpoint (Princeton Univ. Press, NJ, 1965). · Zbl 0136.20402
[15] A. Mokkadem, ”Mixing Properties of ARMA Processes”, Stochastic Processes Applications 29, 309–315 (1988). · Zbl 0647.60042 · doi:10.1016/0304-4149(88)90045-2
[16] A. M. Pardzhanadze, ”Functional Limit Theorems of a Posteriori Detection of Disorder”, Theory Probab. Appl. 31, 355–358 (1986). · Zbl 0626.62084 · doi:10.1137/1131047
[17] D. Picard, ”Testing and Estimating Change-Point in Time Series”, Adv. in Appl. Probab. 7, 841–867 (1985). · Zbl 0585.62151 · doi:10.1017/S0001867800015433
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.