zbMATH — the first resource for mathematics

Empirical process of the squared residuals of an ARCH sequence. (English) Zbl 1012.62053
Summary: We derive the asymptotic distribution of the sequential empirical process of the squared residuals of an ARCH(p) sequence. Unlike the residuals of an ARMA process, these residuals do not behave in this context like asymptotically independent random variables, and the asymptotic distribution involves a term depending on the parameters of the model. We show that in certain applications, including the detection of changes in the distribution of the unobservable innovations, our result leads to asymptotically distribution free statistics.

62G30 Order statistics; empirical distribution functions
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G20 Asymptotic properties of nonparametric inference
60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
Full Text: DOI
[1] Amemiya, T. (1985). Advanced Econometrics. Harvard Univ. Press.
[2] Bai, J. (1994). Weak convergence ofthe sequential empirical processes ofresiduals in ARMA models. Ann. Statist. 22 2051-2061. · Zbl 0826.60016
[3] Blum, J. R., Kiefer, J. and Rossenblatt, M. (1961). Distribution free tests of independence based on the sample distribution function. Ann. Math. Statist. 32 485-498. · Zbl 0139.36301
[4] Boldin, M. V. (1998). On residual empirical distribution functions in ARCH models with applications to testing and estimation. Mitt. Math. Sem. Giessen 235 49-66. · Zbl 0921.62063
[5] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods. Springer, New York. · Zbl 0709.62080
[6] Burke, M. D., Cs örg o, M., Cs örg o, S. and Révész, P. (1979). Approximations ofthe empirical process when parameters are estimated. Ann. Probab. 7 790-810. · Zbl 0433.62017
[7] Chu, C.-S. J. (1995). Detecting parameter shift in GARCH models. Econometric Rev. 14 241-266. · Zbl 0832.62099
[8] Cotteril, D. S. and Cs örg o, M. (1985). On the limiting distribution ofand the critical values for the Hoeffding, Blum, Kiefer and Rosenblatt independence criterion. Statist. Decisions 3 1-48. · Zbl 0569.62014
[9] Cs örg o, M. and Horváth, L. (1997). Limit Theorems in Change-Point Analysis. Wiley, New York. · Zbl 0884.62023
[10] Cs örg o, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York. · Zbl 0539.60029
[11] Cs örg o, S. (1983). Kernel-transformed empirical processes. Multivariate Anal. 13 511-533. · Zbl 0559.62012
[12] Davis, R. A. and Mikosch, T. (1998). The sample autocorrelations ofheavy-tailed processes with applications to ARCH. Ann. Statist. 26 2049-2080. Durbin, J. (1973a). Weak convergence ofthe sample distribution function when parameters are estimated. Ann. Statist. 1 279-290. Durbin, J. (1973b). Distribution Theory for Tests Based on the Sample Distribution Function. · Zbl 0929.62092
[13] SIAM, Philadelphia.
[14] Frances, P. H. and van Dijk, D. (2000). Outlier detection in GARCH models. Erasmus Univ. Econometric Institute Research Report, EI-9926/rv.
[15] Giraitis, L., Kokoszka, P. and Leipus, R. (2000). Testing for long memory in the presence of a general trend. Preprint. Available at http://math.usu.edu/ piotr/research.html URL: · Zbl 1140.62341
[16] Gouriéroux, C. (1997). ARCH Models and Financial Applications. Springer, New York. · Zbl 0880.62107
[17] Guegan, D. and Diebolt, J. (1994). Probabilistic properties of -arch model. Statist. Sinica 4 71-87. · Zbl 0826.60060
[18] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Applications. Academic Press, New York. · Zbl 0462.60045
[19] He, C. and Teräsvirta, T. (1999). Fourth moment structure ofthe GARCH(p q) process. Econometric Theory 15 824-846. · Zbl 0961.62077
[20] Horváth, L. (1985). Empirical kernel transforms of parameter-estimated empirical processes. Acta Sci. Math. 48 201-213. · Zbl 0586.62072
[21] Horváth, L. and Kokoszka, P. (2001). Large sample distribution ofARCH (p) squared residual correlations. Econometric Theory. · Zbl 0973.62074
[22] Horváth, L., Kokoszka, P. and Teyssiére, G. (2000). Bootstrap specification tests for ARCH. · Zbl 1060.62097
[23] Khamaladze, E. V. (1981). Martingale approach in the theory ofgoodness-of-fit tests. Theory Probab. Appl. 26 240-257. · Zbl 0481.60055
[24] Koul, H. L. and Stute, W. (1999). Nonparametric model checks for time series. Ann. Statist. 27 204-236. · Zbl 0955.62089
[25] Li, W. K. and Mak, T. K. (1994). On the squared residual autocorrelations in nonlinear time series with conditional heteroskedasticity. J. Time Ser. Anal. 15 627-636. · Zbl 0807.62070
[26] Lu, Z. and Cheng, P. (1997). Distribution-free strong consistency for nonparametric kernel regression involving nonlinear time series. J. Statist. Plann. Inference 65 67-86. · Zbl 0907.62053
[27] Lundbergh, S. and Teräsvirta, T. (1998). Evaluating GARCH models. Working paper 292, Stockholm School ofEconomics. · Zbl 1040.62078
[28] Martynov, G. V. (1992). Statistical tests based on empirical processes and related questions. J. Soviet Math. 61 2195-2273.
[29] Mikosch, T. and St aric a, C. (1999). Change ofstructure in financial time series, long range dependence and the GARCH model. Preprint. Available at http://www.cs.nl/ eke/ iwi/preprints. URL:
[30] Picard, D. (1977). Testing and estimating change-point in time series. Adv. in Appl. Probab. 17 841-867. JSTOR: · Zbl 0585.62151
[31] Seber, G. A. F. (1977). Linear Regression Analysis. Wiley, New York. · Zbl 0354.62055
[32] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365
[33] Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York. · Zbl 0321.60022
[34] Tjøstheim, D. (1999). Nonparametric specification procedures for time series. In Asymptotics, Nonparametrics and Time Series (S. Ghosh, ed.) 149-200. Dekker, New York. · Zbl 1069.62547
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.