Abstract
We consider the codifference and the normalized codifference function as dependence measures for stationary processes. Based on the empirical characteristic function, we propose estimators of the codifference and the normalized codifference function. We show consistency of the proposed estimators, where the underlying model is the ARMA with symmetric α-stable innovations, 0 < α ≤ 2. In addition, we derive their limiting distribution. We present a simulation study showing the dependence of the estimator on certain design parameters. Finally, we provide an empirical example using some stocks from Indonesia Stock Exchange.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adler RJ, Feldman RE, Gallagher CM (1998) Analyzing stable time series. In: Adler RJ, Feldman RE, Taqqu MS (eds) A practical guide to eavy tails: statistical techniques and applications. Birkhäuser, Boston, pp 133–158
Bhansali RJ (1983) Estimation for the order of MA model from autoregressive and windows estimates of the inverse correlation function. J Time Ser Anal 4: 137–162
Brockwell PJ, Davis RA (1987) Time series: theory and methods. Springer, New York
Chambers JM, Mallows CL, Stuck BW (1976) A method for simulating stable random variables. J Am Stat Assoc 71: 340–344
Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Springer, Berlin
Hesse CH (1990) Rates of convergence for the empirical distribution function and the empirical characteristic function of a broad class of linear process. J Multivar Anal 35: 186–202
Hong Y (1999) Hypothesis testing in time series via the empirical characteristic function: a generalized spectral density approach. J Am Stat Assoc 94(448): 1201–1214
Janicki A, Weron A (1994) Simulation and chaotic behavior of α-stable stochastic processes. Marcel Dekker, New York
Kogon SM, Williams DB (1998) Characteristic function based estimation of stable distribution parameters. In: Adler RJ, Feldman RE, Taqqu MS (eds) Practical guide to heavy tails: statistical techniques and applications. Birkhäuser, Boston, pp 311–335
Kokoszka PS, Taqqu MS (1994) Infinite variance stable ARMA processes. J Time Ser Anal 15: 203–220
Koutrouvelis IA (1980) A goodness-of-fit test of simple hypotheses based on the empirical characteristic function. Biometrika 67: 238–240
Lehmann E (1999) Elements of large-sample theory. Springer, New York
Nikias CL, Shao M (1995) Signal processing with α-stable distributions and applications. Wiley, New York
Nolan JP (1999a) Numerical calculation of stable densities and distribution functions. Technical report. Department of Mathematics and Statistics, American University
Nolan JP (1999b) Fitting data and assessing goodness-of-fit with stable distributions. Technical report. Department of Mathematics and Statistics, American University
Nowicka J (1997) Asymptotic behavior of the covariation and the codifference for ARMA models with stable innovations. Stoch Models 13: 673–686
Nowicka J, Weron A (1997) Measures of dependence for ARMA models with stable innovations. Annales UMCS Sect A (Mathematica) 51: 133–144
Paolella MS (2001) Testing the stable Paretian assumption. Math Comput Model 34: 1095–1112
Proakis JG, Manolakis DG (1996) Digital signal processing. Prentice Hall, London
R Development Core Team (2008) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. ISBN 3-900051-00-3
Rachev ST, Mittnik S (2000) Stable paretian models in finance. Wiley, New York
Rosadi D (2005) Asymptotic behavior of the codifference and the dynamical function for ARMA models with infinite variance. J Ind Math Soc 59–69
Rosadi D (2006) Order identification for gaussian moving averages using the codifference function. J Stat Comput Simul 76: 553–559
Rosadi D (2007) Identification of moving average process with infinite variance. Stat Probab Lett 77(14): 1490–1496
Samorodnitsky G, Taqqu MS (1994) Stable nonGaussian processes: stochastic models with infinite variance. Chapman and Hall, New York
Yang Q, Petropulu A, Pesquet JC (2001) Estimating long-range dependence in impulsive traffic flows. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp 3413–3416
Yu J (2004) Empirical characteristic function estimation and its applications. Econ Rev 23: 93–123
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rosadi, D., Deistler, M. Estimating the codifference function of linear time series models with infinite variance. Metrika 73, 395–429 (2011). https://doi.org/10.1007/s00184-009-0285-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-009-0285-9