×

On diagnostics in conditionally heteroskedastic time series models under elliptical distributions. (English) Zbl 1049.62100

Summary: In statistical diagnostics and sensitivity analysis, the local influence method plays an important role. We use this method to study financial time series data and conditionally heteroskedastic models under elliptical distributions. We start with a likelihood displacement, and consider data- and model-perturbation schemes. We obtain the corresponding matrices of derivatives, and measures of slope and normal curvature, and then discuss the assessment of local influence.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62J20 Diagnostics, and linear inference and regression
62J05 Linear regression; mixed models

Software:

itsmr
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bera, A. K. and Higgins, M. L. (1995). On ARCH models: properties, estimation and testing. In Surveys in Econometrics , eds L. Oxley, D. A. R. George, C. J. Roberts and S. Sayer, Blackwell, Oxford, pp. 215–272.
[2] Chatterjee, S. and Hadi, A. S. (1988). Sensitivity Analysis in Linear Regression . John Wiley, New York. · Zbl 0648.62066
[3] Billor, N. and Loynes, R. M. (1993). Local influence: a new approach. Commun . Statist. Theory Meth. 22 , 1595–1611. · Zbl 0792.62060 · doi:10.1080/03610929308831105
[4] Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. J . Econometrics 31 , 307–327. · Zbl 0865.62085 · doi:10.1016/S0304-4076(95)01749-6
[5] Brockwell, P. J. (2000). Heavy-tailed and non-linear continuous-time ARMA models for financial time series. In Statistics and Finance : an Interface, eds W. S. Chan, W. K. Li and H. Tong, Imperial College Press, London, pp. 3–22.
[6] Brockwell, P. J. and Davis, R. A. (2002). Introduction to Time Series and Forecasting , 2nd edn. Springer, New York. · Zbl 0994.62085 · doi:10.1007/b97391
[7] Cook, R. D. (1986). Assessment of local influence (with discussion). J . R. Statist. Soc. B 48 , 133–169. · Zbl 0608.62041
[8] Cook, R. D. (1997). Local influence. In Encyclopedia of Statistical Sciences , Vol. 1 Update, eds S. Kotz, C. B. Read and D. L. Banks, John Wiley, New York, pp. 380–385.
[9] Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 , 987–1007. · Zbl 0491.62099 · doi:10.2307/1912773
[10] Fang, K. T. and Zhang, Y. T. (1990). Generalized Multivariate Analysis . Springer, Berlin. · Zbl 0724.62054
[11] Farebrother, R. W. (1992). Relative local influence and the condition number. Commun . Statist. Simul. Comput. 21 , 707–710.
[12] Galea, M., Paula, G. A. and Bolfarine, H. (1997). Local influence in elliptical linear regression models. Statistician 46 , 71–79.
[13] Gourieroux, C. and Jasiak, J. (2001). Financial Econometrics . Problems, Models, and Methods. Princeton University Press. · Zbl 1028.62083
[14] Heyde, C. C. (1999). A risky asset model with strong dependence through fractal activity time. J . Appl. Prob. 36 , 1234–1239. · Zbl 1102.62345 · doi:10.1239/jap/1032374769
[15] Heyde, C. C. and Kou, S. G. (2002). On the controversy over tailweight of distributions. · Zbl 1075.62039
[16] Heyde, C. C. and Liu, S. (2001). Empirical realities for a minimal description risky asset model. The need for fractal features. J . Korean Math. Soc. 38 , 1047–1059. · Zbl 0999.91070
[17] Heyde, C. C., Liu, S. and Gay, R. (2001). Fractal scaling and Black–Scholes: the full story. A new view of long-range dependence in stock prices. J. Austral. Soc. Security Anal. 1 , 29–32.
[18] Jung, K. M., Kim, M. G. and Kim, B. C. (1997). Second order local influence in linear discriminant analysis. J . Japan. Soc. Comput. Statist. 10 , 1–11. · Zbl 0901.62080
[19] Liu, S. (2000). On local influence in elliptical linear regression models. Statist . Papers 41 , 211–224. · Zbl 0948.62054 · doi:10.1007/BF02926104
[20] Liu, S. (2002a). Local influence in multivariate elliptical linear regression models. Linear Algebra Appl . 354 , 159–174. · Zbl 1009.62060 · doi:10.1016/S0024-3795(01)00585-7
[21] Liu, S. (2002b). On regression diagnostics in multivariate models under elliptical distributions. Research Division, National Center for University Entrance Examinations, Tokyo, November 2002.
[22] Liu, S. and Heyde, C. C. (2002). On estimation in conditionally heteroskedastic time series models under non-normal distributions (talk). 16th Austral. Statist. Conf., Canberra, July 2002.
[23] Magnus, J. R. and Neudecker, H. (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics . John Wiley, Chichester. · Zbl 0912.15003
[24] Pan, J. X. and Fang, K. T. (2002). Growth Curve Models with Statistical Diagnostics . Springer, New York. · Zbl 1024.62025
[25] Peña, D. (2001). Outliers, influential observations, and missing data. In A Course in Time Series Analysis , eds D. Peña, G. C. Tiao and R. S. Tsay, John Wiley, New York, pp. 136–170.
[26] Poon, W. Y. and Poon, Y. S. (1999). Conformal normal curvature and assessment of local influence. J . R. Statist. Soc. B 61 , 51–61. · Zbl 0913.62062 · doi:10.1111/1467-9868.00162
[27] Praetz, P. D. (1972). The distribution of share price changes. J . Business 45 , 49–55.
[28] Rachev, S. T. and Mittnik, S. (2000). Stable Paretian Models in Finance . John Wiley, New York. · Zbl 0972.91060
[29] Schall, R. and Dunne, T. T. (1991). Diagnostics for regression-ARMA time series, in Directions in Robust Statistics and Diagnostics , Part II, eds W. Stahel and S. Weisberg, Springer, New York, pp. 205–221.
[30] Tsay, R. S. (2002). Analysis of Financial Time Series . John Wiley, New York. · Zbl 1037.91080
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.