Estymacja modeli GARCH : MNW i podejście bayesowskie

• Autorzy: Mateusz Pipień

Warianty tytułu

Estimation of GARCH Models : Maximum Likelihood and the Bayesian Approach

• Języki publikacji: PL

Abstrakty

W artykule omówiono estymację modeli GARCH(p,q), stosowanych w analizie finansowych szeregów czasowych. Przedstawiono metodę największej wiarygodności oraz wnioskowanie bayesowskie, a więc podejście, które w pełni wykorzystuje postać funkcji wiarygodności. Metody te zostały oprogramowane i zilustrowane dla modelu GARCH(1,1), zarówno na danych wygenerowanych sztucznie jak i na rzeczywistym szeregu finansowym.

EN

This paper discusses estimation of GARCH(p, q) models, which are used in financial time series analysis. We present those methods of estimation, which use fully the likelihood function, namely ML and Bayesian estimation. Both cases require advanced numerical methods: optimisation with conditions given by inequalities (ML) and Monte Carlo integration (Bayesian estimation). These methods were implemented and presented for the GRACH(1,1) process both for simulated and empirical data. The numerical algorithms that were used seem to converge well. Their numerical efficiency (even with a few iterations) makes them useful and convenient tools. Only small differences in posterior results corresponding to different priors are observed. Bigger differences can be noticed between the Bayesian and ML estimation of parameters. Since ML results can be treated as based on the asymptotic normal approximation of the posterior distribution, it seems reasonable to use exact Bayesian analysis with Monte Carlo numerical techniques. (original abstract)

Słowa kluczowe

PL

Model GARCH; Wnioskowanie bayesowskie; Metody ekonometryczne

EN

GARCH model; Bayesian inference; Econometric methodology

• Czasopismo: Przegląd Statystyczny
• Rocznik: 1999
• Tom: 46
• Numer: z. 2
• Strony: 239--255

Twórcy

autor

Mateusz Pipień

Bibliografia

• [1] Bartoszewicz J., Wykłady ze statystyki matematycznej, PWN 1996.
• [2] Bollerslev T., Generalized Autoregressive Conditional Heteroscedasticity, Journal of Econometrics 31, 307-327, 1986.
• [3] Bollerslev T., Engle R.F., Common persistence in conditional variances, Econometrica 61,167-186,1993.
• [4] Chen Ch.-F., On asymptotic normality of limiting density functions with Bayesian implications, Journal of the Royal Statistical Society, Series B, 47, 540-546, 1985.
• [5] DeGroot M.H., Optymalne decyzje statystyczne, PWN, Warszawa 1981.
• [6] Engle R.F., Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica 50, 987-1007, 1982.
• [7] Engle R.F., Bollerslev T., Modeling the Persistence of Conditional Variance, Econometric Reviews 5, 1-50, 1986.
• [8] Engle R.F., Estimates of the variance of U.S. inflation based on the ARCH model, Journal of Money Credit and Banking 15, 286-301, 1989.
• [9] Engle R.F., Kraft D., Multiperiod forecast error variances of inflation estimated from ARCH models, [w:] A. Zellner (red.), Applied Time Series Analysis of Economic Data (Bureau of The Census, Washington, DC) 293-302, 1993.
• [10] Geweke J., Exact Predictive Densitites for Linear Models with ARCH Disturbances, Journal of Econometrics 40, 63-86, 1989.
• [11] Granger C.W.J., Andersen ?., An Introduction to Bilinear Time-Series Models, Gottingen: Vandenhoeck and Ruprecht, 1978.
• [12] Greene W., Econometric Analysis, Macmillan, New York 1993.
• [13] Hartigan J.A., Bayes Theory, Springer-Verlag, New York 1983.
• [14] Mandelbrot B., The variation of Certain Speculative Prices [w:] Cootner P.H. (red.), The Random Character of Stock Market Prices, MIT Press, Cambridge, MA 1994.
• [15] McNees S.S., The forecasting record for the 1970's, New England Economic Review, Sept/Oct 1979, 33--53.
• [16] Nelson D., Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica 59, 347-370, 1991.
• [17] Patrykiejew A., Wprowadzenie do metody Monte Carlo, Uniwersytet M. Curie-Skłodowskiej, Lublin 1993.
• [18] Shephard N., Statistical Aspect of ARCH and Stochastic Volatility, [w:] "Likelihood, Time Series with Econometric and Other Applications", D.R. Cox, D.V. Hinkley, O.E. Barndorff-Nielsen (red.), Chapman & Hall, 1995.
• [19] Weiss A.A., ARMA models with ARCH errors, Journal of Time Series Analysis 5, 129-143, 1993.
• [20] Weiss A.A., Asymptotic Theory for ARCH Modeb: Estimation and Testing, Econometric Theory 2, 107-131, 1986.
• [21] Zellner A., An Introduction to Bayesian Inference in Econometrics, J. Wiley, New York 1971.