Numerical optimization of the likelihood function based on Kalman filter in the GARCH models

In this work, we propose a new estimate algorithm for the parameters of a $\mathrm{GARCH}(p,q)$ model.  This algorithm turns out to be very reliable in estimating the true parameter’s values of a given model.  It combines maximum likelihood method, Kalman filter algorithm and the simulated annealing (SA) method, without any assumptions about initial values.  Simulation results demonstrate that the algorithm is liable and promising.

GARCH models
maximum likelihood
фільтр Калмана
метод імітації відпалу
  1. Harvey A. C., Phillips G. D. A.  Maximum Likelihood Estimation of Regression Models With Autoregressive-Moving Average Disturbances.  Biometrika. 66 (1), 49–58 (1979).
  2. Pearlman J. G.  An algorithm for the exact likelihood of a high-order autoregressive-moving average process.  Biometrika. 67 (1), 232–233 (1980).
  3. Gardner G., Harvey A. C., Phillips G. D. A.  An Algorithm for Exact Maximum Likelihood Estimation of Autoregressive-Moving Average models by Means of Kalman Filtring.  Applied Statistics. 29 (3), 311–322 (1980).
  4. Bollerslev T.  Generalized autoregressive conditional heteroskedasticity.  Journal of Econometrics. 31 (3), 307–327 (1986).
  5. Engle R. E.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation.  Econometrica. 50 (4), 987–1007 (1982).
  6. Weiss A. A.  Asymptotic theory for ARCH models: estimation and testing.  Econometric Theory. 2 (1), 107–131 (1986).
  7. Lumsdaine R. L.  Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models.  Econometrica. 64 (3), 575–596 (1996).
  8. Lee S. W., Hansen B. E.  Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator.  Econometric Theory. 10 (1), 29–52 (1994).
  9. Berkes I., Horváth L., Kokoszka P. S.  GARCH processes: structure and estimation.  Bernoulli. 9 (2), 201–227 (2003).
  10. Berkes I., Horváth L.  The rate of consistency of the quasi-maximum likelihood estimator.  Statistics and Probability Letters. 61 (2), 133–143 (2003).
  11. Berkes I., Horváth L.  The efficiency of the estimators of the parameters in GARCH processes.  Annals of Statistics. 32 (2), 633–655 (2004).
  12. Boussama F.  Ergodicité, mélange et estimation dans les modles GARCH. Doctoral thesis, Université Paris 7 (1998).
  13. Boussama F.  Normalité asymptotique de l'estimateur du pseudo-maximum de vraisemblance d'un modèle GARCH.  Comptes Rendus de l'Académie des Sciences – Series I – Mathematics. 331 (1), 81–84 (2000).
  14. Francq C., Zakoïan J.-M.  Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes.  Bernoulli. 10 (4), 605–637 (2004).
  15. Kalman R. E.  A New Approach to Linear Filtering and Prediction Problems Transaction of the ASME.  Journal of Basic Engineering. 82 (1), 34–45 (1960).
  16. Kalman R. E.  New methods in Wiener filtering theory.  In: John L. Bogdanoff and Frank Kozin, eds. Proceeding of the first symposium of engineering applications of random function; Function theory and probability, New York, Wiley, 270–388 (1963).
  17. Corana A., Marchesi M., Martini C., Ridella S.  Minimizing Multimodal Functions of Continuous Variables With The "Simulated Annealing" Algorithm.  ACM Transactions on Mathematical Software. 13 (3), 262–280 (1987).
  18. Allal J., Benmoumen M.  Parameter estimation for GARCH(1,1) models based on Kalman filter.  Advances and Applications in Statistics. 25 (2), 115–130 (2011).
  19. Allal J., Benmoumen M.  Parameter estimation for first-order Random Coefficient Autoregressive (RCA) Models based on Kalman Filter.  Communications in Statistics – Simulation and Computation. 42 (8), 1750–1762 (2013).
  20. Allal J., Benmoumen M.  Parameter estimation for ARCH(1) models based on Kalman filter.  Applied Mathematical Sciences. 8 (56), 2783–2791 (2014).
  21. Benmoumen M.  Numerical optimization of the likelihood function based on Kalman filter in the ARCH models.  AIP Conference Proceedings. 2074, 020020 (2019).
  22. Benmoumen M., Allal J., Salhi I.  Parameter Estimation for p-Order Random Coefficient Autoregressive (RCA) Models Based on Kalman Filter.  Journal of Applied Mathematics. 2019, 8479086 (2019).
  23. Settar A., Fatmi N. I., Badaoui M.  On the computational estimation of high order GARCH model.  Mathematical Modeling and Computing. 8 (4), 797–806 (2021).
  24. Bougerol P., Picard N.  Stationarity of GARCH processes and of some nonnegative time series.  Journal of Econometrics. 52 (1–2), 115–127 (1992).
  25. Nelson D. B.  Stationarity and persistence in the GARCH(1,1) model.  Econometric Theory. 6 (3), 318–334 (1990).
  26. Klüppelberg C., Lindner A., Maller R.  A continuous time GARCH process driven by a Lévy process: stationarity and second order behaviour.  Journal of Applied Probability. 41 (3), 601–622 (2004).
  27. Robinson P. M.  Testing for strong correlation and dynamic conditional heteroskedasticity in multiple regression.  Journal of Econometrics. 47 (1), 67–84 (1991).
  28. Giraitis L., Kokoszka P., Leipus R.  Stationary ARCH models: dependence structure and central limit theorem.  Econometric Theory. 16 (1), 3–22 (2000).
  29. Giraitis L., Leipus R., Surgailis D.  ARCH($\infty$) and long memory properties.  In: Mikosch T., Kreiß J. P., Davis R., Andersen T. (eds) Handbook of Financial Time Series.  Springer, Berlin, Heidelberg (2009).
  30. Milhøj A.  The moment structure of ARCH processes.  Scandinanvian Journal of Statistics. 12 (4), 281–292 (1984).
  31. Karanasos M.  The second moment and the autocovariance function of the squared errors of the GARCH model.  Journal of Econometrics. 90 (1), 63–76 (1999).
  32. He C., Teräsvirta T.  Fourth-moment structure of the GARCH$(p,q)$ process.  Econometric Theory. 15 (6), 824–846 (1999).
  33. Ling S., McAleer M.  Necessary and sufficient moment conditions for the GARCH$(r,s)$ and asymmetric power GARCH$(r,s)$ models.  Econometric Theory. 18 (3), 722–729 (2002).
  34. Ling S., McAleer M.  Stationarity and the existence of moments of a family of GARCH processes.  Journal of Econometrics. 106 (1), 109–-117 (2002).
  35. Chen M., An H. Z.  A note on the stationarity and the existence of moments of the GARCH model.  Statistica Sinica. 8, 505–510 (1998).