Numerical Optimization within Vector of Parameters Estimation in Volatility Models
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).
[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",
Metodološki 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.
[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",
Metodološki 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.
@article{"International Journal of Business, Human and Social Sciences:57448", author = "J. Arneric and A. Rozga", title = "Numerical Optimization within Vector of Parameters Estimation in Volatility Models", 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).", keywords = "Heteroscedasticity, Log-likelihood Maximization,
Quasi-Newton iteration procedure, Volatility.", volume = "3", number = "1", pages = "44-5", }