Una nota sobre el delta-ésimo momento en modelos ARMA-APARCH con distribuciones condicionales estables y GEV

• Sousa, Thiago R. ^[2] ; G. Otiniano, Cira E. ^[2] ; C. Lopes, Silvia R. ^[1]
  1. [1] Universidade Federal do Rio Grande do Sul

     Universidade Federal do Rio Grande do Sul

     Brasil

     
  2. [2] Departamento de Estatística, UnB, 70910-900 Brasília, DF, Brazil
• Localización: Selecciones Matemáticas, ISSN-e 2411-1783, Vol. 5, Nº. 1 (Enero - Julio), 2018, págs. 7-16
• Idioma: español
• DOI: 10.17268/sel.mat.2018.01.02
• Títulos paralelos:
  • A note about the delta-moment in ARMA-APARCH models with stable conditional distributions and GEV
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    En un modelo de series temporales ARMA-APARCH con innovaciones Z, la condición de delta - estacionariedad del proceso APARCH envuelve el delta-ésimo momento de la diferencia entre el valor absoluto de las innovaciones con el producto del parámetro de asimetría y las innovaciones. Este momento permite calcular de forma mas eficiente las estimativas de máxima verosimilitud de los parámetros del modelo. En este artículo, son obtenidas expresiones explícitas de ese delta-ésimo momento onde Z tem distribución estable y GEV. Esos momentos se han implementado en nuestro paquete GEVStableGarch disponible en CRAN R-PROJECT desarrollado para estimar los parámetros de los modelos ARMA-GARCH / APARCH con innovaciones estables y GEV.

  • English

    In a ARMA-APARCH time series model with innovations Z, the delta-stationarity condition of the APARCH process involves the delta-th moment of the difference between the absolute value of the innovations with the product of the asymmetry parameter and the innovations. This moment allows calculating more efficiently the estimates of the parameters of the model by maximum likelihood. In this article, we obtain explicit expressions of this delta - th moment where Z has stable and GEV distribution. These moments have been implemented in our GEVStableGarch package available in CRAN R-PROJECT developed to estimate the parameters of ARMA-GARCH / APARCH models with stable innovations and GEV.

• Referencias bibliográficas
  • Bollerslev,T. and Ghysels, E. Autoregressive Conditional Heteroscedasticity, Journal of Business and Economic Statistics,14(1986), pp 139–151.
  • Bougerol, P. and Picard, P. Stationarity of GARCH Processes and of some Non-Negative Time Series, Journal of Econometrics, 52 (1992), pp 115–127.
  • Brockwell, P.J. Linear Prediction of ARMA Processes with Infinite Variance, Stochastic Processes and their Applications, 19 (1985), pp 281–296.
  • Brockwell, P.J. Davis, R.A. Time Series: Theory and methods, Springer (1991).
  • Brummelhuis, R. and Kaufmann, R. Time Scaling For GARCH(1,1) and AR(1)-GARCH(1,1) Processes, Journal of Risk, 2007, www.risk.net/journal-of-risk/technical-paper/2161017/time-scaling-value-risk-garch-ar-garch-processes.
  • Curto, J.D. and Tavares, A.N. and Tavares, G.N. Modelling Heavy Tails and Asymmetry Using ARCH-type Models with Stable Paretian Distributions,...
  • Ding, Z. and Granger, C.W.J. and Engle, R.F. A Long Memory Property of Stock Market Returns and a New Model , Journal of Empirical Finance...
  • Diongue, A.K. An investigation of Stable-Paretian Asymmetric Power GARCH Model , Journal des Sciences , 8 (2008), pp 15-26.
  • Engle, R.F. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica 50(1982),...
  • Frain, J.C. Studies on the Application of the Alpha-stable Distribution in Economics , Phd Thesis-Trinity College, http://www.tcd.ie/Economics/staff/frainj/Stable-Distribution/thesis-main-5.pdf,...
  • Jenkinson , A.F. The frequency distribution of the annual maximum (or minimum) values of meteorological elements, Quarterly Journal of the...
  • Jondeau, E.; Poon, S.H. and Rockinger, M. Financial Modeling under non-Gaussian Distributions , Springer Science & Business Media , 2007.
  • Ling, S. and McAleer, M. Necessary and Sufficient Moment Conditions for the GARCH(r,s) and Asymmetric Power GARCH(r,s) Models , Econometric...
  • Mathai, A.M.; Saxena, R.K.; Haubold, H.J. The H-function: Theory and Applications, Springer-Verlag New York, (2010).
  • Mittnik, S. and Panosrka, A.K. and Rachev, S. T. Stable GARCH Models for Financial Time Series, Applied Mathematical Letters, 8 (1995), pp...
  • Mittnik, S. amd Paolella, M.C. Conditional Density and Value-At-Risk Prediction of Asian Currency Exchange Rates , journal of Forecasting...
  • Mittnik, S. and Paolella, M.S. and Rachev, S.T. Stationarity of Stable Power-GARCH Processes , Journal of Econometrics , 106 (2002), pp 97–107.
  • Nakatsuma, T. Bayesian Analysis of ARMA-GARCH Models: A Markov Chain Sampling Approach, Journal of Econometrics, 95 (2000), pp 57–69.
  • Nelson, D. Stationarity and Persistence in the GARCH(1,1) Model , Econometric Theory, 6 (1990), pp 318–344.
  • Nolan, J.P. Numerical Calculations of Stable Densities and Distribution Functions , Communication Statistics. Stochastic Models, 13 (1997),...
  • Nolan, J.P. Stable Distributions Models for Heavy Tailed Data, Copyright, http://www.math.ucla.edu/ biskup/275b.1.13w/ PDFs/Nolan.pdf (2009).
  • Schneider, W.R. Stable distributions: Fox function representation and generalization, Lecture Notes in Physics, 262 (1986), pp 497–511.
  • Sousa, T.R.; Otiniano, C.E.G. and Lopes, S.R.C. GEVStableGarch: An R Package for ARMA-GARCH or ARMA-APARCH Estimation with Conditional Stable...
  • Wuertz,D.; Chalabi,Y. and Luksan, L. Estimation of ARMA Models with GARCH/APARCH Errors: An R and SPlus Software Implementation, (2009),Journal...
  • Zhao, X.; Scarrott, C.; Oxley, L.; Reale, M. Dependence in Extreme Value Models with Bayesian Inference, Mathematics and Computers in Simulation,...