Non-parametric Estimation of GARCH (2, 2) Volatility model: A new Algorithm

Cassim, Lucius (2018): Non-parametric Estimation of GARCH (2, 2) Volatility model: A new Algorithm.

Abstract

The main objective of this paper is to provide an estimation approach for non-parametric GARCH (2, 2) volatility model. Specifically the paper, by combining the aspects of multivariate adaptive regression splines(MARS) model estimation algorithm proposed by Chung (2012) and an algorithm proposed by Buhlman and McNeil(200), develops an algorithm for non-parametrically estimating GARCH (2,2) volatility model. Just like the MARS algorithm, the algorithm that is developed in this paper takes a logarithmic transformation as a preliminary analysis to examine a nonparametric volatility model. The algorithm however differs from the MARS algorithm by assuming that the innovations are i.d.d. The algorithm developed follows similar steps to that of Buhlman and McNeil (200) but starts by semi parametric estimation of the GARCH model and not parametric while relaxing the dependency assumption of the innovations to avoid exposing the estimation procedure to risk of inconsistency in the event of misspecification errors.

• Item Type: MPRA Paper
• Language: English
• Keywords: GARCH (2,2), MARS, Algorithm, Parametric, Semi parametric, Nonparametric
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics

References

Bollerslev. (1987). A conditional Heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics,69, 542-547.

Bollerslev, T. (1986). Generalized Autoregressive Heteroskedasticity. Journal of Econometrics, 307-327.

Bollersslev, J., & Woodridge, J. (1992). Quasi maximum likelihood estimation and inference in dynamic models with time varying covariances. Econometric Reviews,11(2), 143-172.

Buhlman, P., & McNeil, A. J. (2000). Non-parametric GARCH Models. Zurich, Switzerland: Seminar Fur Statistik:CH-8092.

Cameron, C. A., & Trivedi, P. (2005). Microeconometrics:Methods and Applications. New York,USA: Cambrdge University Press.

Choi, E. J. (2004). Estimation of Stochastic Volatility Models by Simulated Maximum Likelihood Method. University of Waterloo.

Chung, S. S. (2012). A class of non-parametric volatility models:Application to financial time series. Journal of Econometrics.

Dahl, C. M., & Levine, M. (2010). Non-parametric estimation of volatility models under serially dependent innovations. Econometrica.

Davidson, J. (2000). Econometric Theory. Blackwel: Oxford University Press.

Davidson, R., & Mackinnon, J. (1993). Estimation and inference in Econometrics. London: Oxford University Press.

Drost, F. C., & Klasssen, C. (1996). Efficient estimation in Semi-parametric GARCH Models. Discussion paper;vol(1996-38),Tilburg.

Duan, J. (1997). Augmented GARCH(p,q) process and its diffusion limit. Journal of Econometrics,79(1), 97-127.

Engle, R. (1982). Autoregressive Conditional Heteroskedasticity with estimation of the variance of U.K inflation. Econometrica, 987-1008.

Engle, R. F., & Gonzale-Rivera, G. (October,1991). Semiparametric ARCH Models. Journal of Business and Economic Statistics, 9(4), 345-359.

Engle, R. F., & Ng, V. (1993). Measuring and testing the impact of news on volatility. Journal of finance,48, 1747-1778.

Fan, & Gijbels, I. (1995). Data-driven bandwidth selection in local polynomial fitting:variable bandwidth and spatial adaptation. Journal of Royal Statistical Society,B,57, 371-394.

Fan, J., & Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika,85, 645-660.

Gallant, A. R., & Hsieh, D. (1989). Fitting a Recalcitrant series:The Pound/Dollar Exchange Rate,1974-83. Econometrica.

Geweke, J. (1986). Modelling Persistence in Conditional Variances: A Comment. Econometric Review,5, 56-61.

Glosten, L. R., & Runkle, D. (1993). On the relation between the expected value and volatility of the nominal excess return of stocks. Journal of Finance,48, 1779-1801.

Gourieroux, C., & Trognon, A. (1984). Pseudo Maximum Likelihood Methods:Theory,52(1). Econometrica, 681-700.

Haafner, C. M. (2003). Analytical quasi maximum likelihood inference in BEKK-GARCH models. Econometric Institution,Erasmus University,Rotterdam.

Hafner, C. M., & Rombonts, J. (2002). Semiparametric multivariate GARCH models . Discussion paper,2002/XX,CORE.

Hansen, B. C. (2006). Econometrics. New York: Cambridge University Press.

Hansen, P., & Heyde, C. (1980). Martingale limit theory and its applications. New York: Academic Press.

Hansen, R. P., & Lunde, A. (2001). A comparison of volatility models:Does anything beat GARCH(1,1)? Centre for analytica finance:University of AARHUS.

Hentshel, L. (1995). All in the family: Nesting Symetric and Asymetric GARCH Models. Journal of Financial Economics,39, 71-104.

Herwartz, H. (2004). ConditionalHeteroskedasticity.In H. Lutkepoh, & M. Kratzig (Eds.) Themes in Modern Econometrics, pp. 197-220.

Higging, M. L., & Bera, A. (1992). A Class of non-linear ARCH Models. International Economic Review.

Holly. (2009). Modelling Risk using fourth order Pseudo Maximum Likelihood Methods. University of Lausanne, Institute of Healthy Economics.

Holly, A., & Montifort, A. (2010). Fourth Order Pseudo Maximum Likelihood Methods. Econometrica.