Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 31824
Numerical Optimization within Vector of Parameters Estimation in Volatility Models

Authors: J. Arneric, A. Rozga

Abstract:

In this paper usefulness of quasi-Newton iteration procedure in parameters estimation of the conditional variance equation within BHHH algorithm is presented. Analytical solution of maximization of the likelihood function using first and second derivatives is too complex when the variance is time-varying. The advantage of BHHH algorithm in comparison to the other optimization algorithms is that requires no third derivatives with assured convergence. To simplify optimization procedure BHHH algorithm uses the approximation of the matrix of second derivatives according to information identity. However, parameters estimation in a/symmetric GARCH(1,1) model assuming normal distribution of returns is not that simple, i.e. it is difficult to solve it analytically. Maximum of the likelihood function can be founded by iteration procedure until no further increase can be found. Because the solutions of the numerical optimization are very sensitive to the initial values, GARCH(1,1) model starting parameters are defined. The number of iterations can be reduced using starting values close to the global maximum. Optimization procedure will be illustrated in framework of modeling volatility on daily basis of the most liquid stocks on Croatian capital market: Podravka stocks (food industry), Petrokemija stocks (fertilizer industry) and Ericsson Nikola Tesla stocks (information-s-communications industry).

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1070761

References:

[1] C. Alexander, Market Models: A Guide to Financial Data Analysis, John Wiley and & Sons Ltd., New York, 2001.
[2] J. Arnerić, B. ┼ákrabić, and Z. Babić, "Maximization of the likelihood function in financial time series models", in Proceedings of the International Scientific Conference on Contemporary Challenges of Economic Theory and Practice, Belgrade, 2007, pp. 1-12.
[3] M. S. Bazarra, H. D. Sherali, and C. M. Shetty, Nonlinear Programming - Theory and Algorithms (second edition), John Wiley and & Sons Ltd., New York, 1993.
[4] E. Berndt, B. Hall, R. Hall, and J. Hausman, "Estimation and Inference in Nonlinear Structural Models", Annals of Social Measurement, Vol. 3, 1974, pp. 653-665.
[5] T. Bollerslev, "Generalized Autoregressive Conditional Heteroscedasticity", Journal of Econometrics, Vol. 31, 1986, pp. 307- 327.
[6] W. Enders, Applied Econometric Time Series (second edition), John Wiley and & Sons Ltd., New York, 2004.
[7] R. Engle, "The Use of ARCH/GARCH Models in Applied Econometrics", Journal of Economic Perspectives, Vol. 15, No. 4, 2001, pp. 157-168.
[8] W. Gould, J. Pitblado, and W. Sribney, Maximum Likelihood Estimation with Stata (third edition), College Station, StatCorp, 2006.
[9] C. Gourieroux, and J. Jasiak, Financial Econometrics: Problems, Models and Methods, Princeton University Press, 2001.
[10] L. Neralić, Uvod u Matemati─ìko programiranje 1, Element, Zagreb, 2003.
[11] J. Petrić, and S. Zlobec, Nelinearno programiranje, Nau─ìna knjiga, Beograd, 1983.
[12] P. Posedel, "Properties and Estimation of GARCH(1,1) Model", Metodolo┼íki zvezki, Vol. 2, No. 2, 2005, pp. 243-257.
[13] R. Schoenberg, "Optimization with the Quasi-Newton Method", Aptech Systems working paper, Walley WA, 2001, pp. 1-9.
[14] D. F. Shanno, "Conditioning of quasi Newton methods for function minimization", Mathematics of Computation, No. 24, 1970, pp. 145-160.