Vitesse de convergence de l'estimateur des moindres carrés dans les ARCH périodiques
DOI :
https://doi.org/10.5269/bspm.66149Résumé
On étudie l'estimateur des moindres carrés d'un modèle ARCH(p) périodique (PARCH(p)). Cet estimateur est construit à partir de la représentation AR périodique (PAR) du modèle PARCH. Nous montrons que cet estimateur est asymptotiquement stable, fortement consistent et nous déterminons sa vitesse de convergence presque sûre (p:s:).
Références
1. A. Aknouche, A. Bibi, Quasi maximum likelihood estimation of periodic GARCH processes. J. Time Ser. Anal, 29(1), 19-45, (2009).
2. A. Aknouche, Causality conditions and autocovariance calculations in PVAR models. Journal of Statistical Computation and Simulation, 77, 769-780, (2007).
3. I. V. Basawa, R. Lund, Large sample properties of parameters estimates for periodic ARMA models. Journal of Time Series Analysis, 22, 1-13, (2001).
4. A. Bibi, A. Aknouche, On periodic GARCH processes: Stationarity, Existence of moments and geometric ergodicity. Math. methods of Statistics, 17(4), 305-316, (2008).
5. T. Bollerslev, E. Ghysels, Periodic autoregressive conditional heteroscedasticity. J. of Business & Economic Statistics, 14, 139-151, (1996).
6. P.E. Caines, Stochastic linear systems. Wiley & Sons, (1988).
7. G. Ciolek, P. Potorski, Bootstrapping periodically autoregressive models . ESAIM. Probability and Statistics, 21, 2157-2178, (2017).
8. R. F. Engle, T. Bollerslev, Modelling the persistence of conditional variances. Econometric Review, 5, 1-50, (1986).
9. D. Freedman, Another note on the Borel-Cantilli lemma and the strong law with the Poisson approximation as a by-product. Ann. Prob. 1, 910-925, (1973).
10. E.G. Gladyshev, Periodically correlated random sequences. Soviet Mathematics, 2, 385-388, (1961).
11. L. Glosten, R. Jegannathan, D. Runke, Relationship between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48, 1779-1801, (1993).
12. C. Gourieroux, A. Monfort, Qualitative threshold ARCH models. Journal of Econometrics, 52, 159-200, (1992).
13. D. Guegan, J. Diebolt, Probabilistic properties of the β − ARCH model. Statistica Sinica, 4, 71-87, (1994).
14. T.L. Lai, C.Z. Wei, Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Ann. Statist, 10(1), 154-166, (1982).
15. G.W. Schwert, Stock volatility and the crash of 87. Review of Financial Studies, 3, 77-102, (1990).
16. G. A. F. Seber, A matrix handbook for statisticians. Wiley & Sons, (2008).
17. T.A. Ula, A.A. Smadi, Periodic stationary conditions for periodic autoregressive moving average processes as eigenvalues problems. Water Resources Research, 33, 1929-1934, (1997).
18. S.O.A. Voffal, L’estimateur des moindres carres dans les modeles ARCH. C.R. Acad. Sci. Paris. t. 322, serie I, 971-974, (1996).
2. A. Aknouche, Causality conditions and autocovariance calculations in PVAR models. Journal of Statistical Computation and Simulation, 77, 769-780, (2007).
3. I. V. Basawa, R. Lund, Large sample properties of parameters estimates for periodic ARMA models. Journal of Time Series Analysis, 22, 1-13, (2001).
4. A. Bibi, A. Aknouche, On periodic GARCH processes: Stationarity, Existence of moments and geometric ergodicity. Math. methods of Statistics, 17(4), 305-316, (2008).
5. T. Bollerslev, E. Ghysels, Periodic autoregressive conditional heteroscedasticity. J. of Business & Economic Statistics, 14, 139-151, (1996).
6. P.E. Caines, Stochastic linear systems. Wiley & Sons, (1988).
7. G. Ciolek, P. Potorski, Bootstrapping periodically autoregressive models . ESAIM. Probability and Statistics, 21, 2157-2178, (2017).
8. R. F. Engle, T. Bollerslev, Modelling the persistence of conditional variances. Econometric Review, 5, 1-50, (1986).
9. D. Freedman, Another note on the Borel-Cantilli lemma and the strong law with the Poisson approximation as a by-product. Ann. Prob. 1, 910-925, (1973).
10. E.G. Gladyshev, Periodically correlated random sequences. Soviet Mathematics, 2, 385-388, (1961).
11. L. Glosten, R. Jegannathan, D. Runke, Relationship between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48, 1779-1801, (1993).
12. C. Gourieroux, A. Monfort, Qualitative threshold ARCH models. Journal of Econometrics, 52, 159-200, (1992).
13. D. Guegan, J. Diebolt, Probabilistic properties of the β − ARCH model. Statistica Sinica, 4, 71-87, (1994).
14. T.L. Lai, C.Z. Wei, Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Ann. Statist, 10(1), 154-166, (1982).
15. G.W. Schwert, Stock volatility and the crash of 87. Review of Financial Studies, 3, 77-102, (1990).
16. G. A. F. Seber, A matrix handbook for statisticians. Wiley & Sons, (2008).
17. T.A. Ula, A.A. Smadi, Periodic stationary conditions for periodic autoregressive moving average processes as eigenvalues problems. Water Resources Research, 33, 1929-1934, (1997).
18. S.O.A. Voffal, L’estimateur des moindres carres dans les modeles ARCH. C.R. Acad. Sci. Paris. t. 322, serie I, 971-974, (1996).
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Publié
2025-02-12
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Research Articles
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Comment citer
Bibi, A. (2025). Vitesse de convergence de l’estimateur des moindres carrés dans les ARCH périodiques. Boletim Da Sociedade Paranaense De Matemática, 43, 1-7. https://doi.org/10.5269/bspm.66149
