## The Annals of Statistics

### High moment partial sum processes of residuals in GARCH models and their applications

#### Abstract

In this paper we construct high moment partial sum processes based on residuals of a GARCH model when the mean is known to be 0. We consider partial sums of kth powers of residuals, CUSUM processes and self-normalized partial sum processes. The kth power partial sum process converges to a Brownian process plus a correction term, where the correction term depends on the kth moment μk of the innovation sequence. If μk=0, then the correction term is 0 and, thus, the kth power partial sum process converges weakly to the same Gaussian process as does the kth power partial sum of the i.i.d. innovations sequence. In particular, since μ1=0, this holds for the first moment partial sum process, but fails for the second moment partial sum process. We also consider the CUSUM and the self-normalized processes, that is, standardized by the residual sample variance. These behave as if the residuals were asymptotically i.i.d. We also study the joint distribution of the kth and (k+1)st self-normalized partial sum processes. Applications to change-point problems and goodness-of-fit are considered, in particular, CUSUM statistics for testing GARCH model structure change and the Jarque–Bera omnibus statistic for testing normality of the unobservable innovation distribution of a GARCH model. The use of residuals for constructing a kernel density function estimation of the innovation distribution is discussed.

#### Article information

Source
Ann. Statist., Volume 33, Number 5 (2005), 2395-2422.

Dates
First available in Project Euclid: 25 November 2005

https://projecteuclid.org/euclid.aos/1132936567

Digital Object Identifier
doi:10.1214/009053605000000534

Mathematical Reviews number (MathSciNet)
MR2211090

Zentralblatt MATH identifier
1086.62100

#### Citation

Kulperger, Reg; Yu, Hao. High moment partial sum processes of residuals in GARCH models and their applications. Ann. Statist. 33 (2005), no. 5, 2395--2422. doi:10.1214/009053605000000534. https://projecteuclid.org/euclid.aos/1132936567

#### References

• Berkes, I. and Horváth, L. (2003). The rate of consistency of the quasi-maximum likelihood estimator. Statist. Probab. Lett. 61 133--143.
• Berkes, I. and Horváth, L. (2004). The efficiency of the estimatiors of the parameters in GARCH processes. Ann. Statist. 32 633--655.
• Berkes, I., Horváth, L. and Kokoszka, P. (2003). GARCH processes: Structure and estimation. Bernoulli 9 201--227.
• Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
• Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307--327.
• Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714--1730.
• Bougerol, P. and Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52 115--127.
• Bowman, K. O. and Shenton, L. R. (1975). Omnibus test contours for departures from normality based on $\sqrtb_1$ and $b_2$. Biometrika 62 243--250.
• Brown, R. L., Durbin, J. and Evans, J. M. (1975). Techniques for testing the constancy of regression relationships over time (with discussion). J. Roy. Statist. Soc. Ser. B 37 149--192.
• Campbell, J. Y., Lo, A. W. and MacKinlay, A. C. (1997). The Econometrics of Financial Markets. Princeton Univ. Press.
• Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. Chapman and Hall, New York.
• Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987--1007.
• Gasser, T. (1975). Goodness-of-fit tests for correlated data. Biometrika 62 563--570.
• Hall, P. and Yao, Q. (2003). Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71 285--317.
• Jarque, C. M. and Bera, A. K. (1987). A test for normality of observations and regression residuals. Internat. Statist. Rev. 55 163--172.
• Kilian, L. and Demiroglu, U. (2000). Residual based tests for normality in autoregressions: Asymptotic theory and simulation evidence. J. Bus. Econom. Statist. 18 40--50.
• Kim, S., Cho, S. and Lee, S. (2000). On the CUSUM test for parameter changes in GARCH($1,1$) models. Comm. Statist. Theory Methods 29 445--462.
• Kokoszka, P. and Leipus, R. (2000). Change-point estimation in ARCH models. Bernoulli 6 513--539.
• Lu, P. G. (2001). Identification of distribution for ARCH innovation. Master's project, Univ. Western Ontario.
• Rossi, P., ed. (1996). Modelling Stock Market Volatility: Bridging the Gap to Continuous Time. Academic Press, Toronto.
• Silverman, B. W. (1998). Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York.
• Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York.
• Yu, H. (2004). Analyzing residual processes of (G)ARCH time series models. In Asymptotic Methods in Stochastics (L. Horváth and B. Szyszkowicz, eds.) 469--486. Amer. Math. Soc., Providence, RI.