Semiparametric Efficient Adaptive Estimation of the PTTGARCH model

Ciccarelli, Nicola (2016): Semiparametric Efficient Adaptive Estimation of the PTTGARCH model.

Abstract

Financial data sets exhibit conditional heteroskedasticity and asymmetric volatility. In this paper we derive a semiparametric efficient adaptive estimator of a conditional heteroskedasticity and asymmetric volatility GARCH-type model (i.e., the PTTGARCH(1,1) model). Via kernel density estimation of the unknown density function of the innovation and via the Newton-Raphson technique applied on the root-n-consistent quasi-maximum likelihood estimator, we construct a more efficient estimator than the quasi-maximum likelihood estimator. Through Monte Carlo simulations, we show that the semiparametric estimator is adaptive for parameters in- cluded in the conditional variance of the model with respect to the unknown distribution of the innovation.

• Item Type: MPRA Paper
• Language: English
• Keywords: Semiparametric adaptive estimation; Power-transformed and threshold GARCH.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Awartani, B.M.A., and Corradi, V., 2005. “Predicting the volatility of the S&P-500 stock index via GARCH models: the role of asymmetries,” International Journal of Forecasting, 21(1), 167-183.

Bickel, P.J., 1982. “On adaptive estimation,” The Annals of Statistics, 10(3), 647-671.

Bickel, P.J., Klaassen, C.A.J., Ritov, Y., and Wellner, J.A., 1993. “Efficient and adaptive estimation for semiparametric models,” Johns Hopkins University Press, Baltimore.

Black, F., 1976. “Studies of stock price volatility changes,” in Proceedings from the American Statistical Association, Business and Economics Statistics Section, 177-181.

Bollerslev, T., 1986. “Generalized autoregressive conditional heteroskedasticity,” Journal of Econometrics, 31(3), 307-327.

Christie, A.A., 1982. “The stochastic behavior of common stock variances: value, leverage and interest rate effects,” Journal of Financial Economics, 10(4), 407-432.

Diaz-Emparanza, I., 2002. “Is a small Monte Carlo analysis a good analysis?” Statistical Papers, 43(4), 567-577.

Drost, F.C., and Klaassen, C.A.J., 1997. “Efficient estimation in semiparametric GARCH models,” Journal of Econometrics, 81(1), 193-221.

Drost, F.C., Klaassen, C. A. J., and Werker, B. J. M., 1997. “Adaptive estimation in time series models,” The annals of statistics, 25(2), 786-817.

Engle, R.F., 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, 50(4), 987-1007.

Engle, R.F., 1990. “Discussion: Stock Market Volatility and the Crash of 87,” Review of Financial Studies, 3(1), 103-106.

Engle, R.F., and Ng, V.K., 1993. “Measuring and Testing the Impact of News on Volatility,” The Journal of Finance, 48(5), 1749-1778.

Francq, C., and Zako¨ıan, J.M., 2010. “GARCH Models. Structure, statistical inference and financial applications,” John Wiley & Sons, Chichester, West Sussex, United Kingdom.

Hajek, J., 1970. “A characterization of limiting distributions of regular estimates,” Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 14(4), 323-330.

Hamadeh, T., and Zakoıan, J.M., 2011. “Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH processes,” Journal of Statistical Planning and Inference, 141(1), 488-507.

Hardy, G.H., 1991. “Divergent Series,” Chelsea Publishing Company, New York.

Hwang, S.Y., and Kim, T.Y., 2004. “Power transformation and threshold modeling for ARCH innovations with applications to tests for ARCH structure,” Stochastic Processes and their Applications 110(2), 295-314.

Kreiss, J.P., 1987. “On adaptive estimation in stationary ARMA processes,” The Annals of Statistics, 15(1), 112-133.

Le Cam, L., 1960. “Locally Asymptotically Normal Families of Distributions. Certain Approximations to Families of Distributions and Their Use in the Theory of Estimation and Testing Hypotheses,” Berkeley & Los Angeles: University of California Press.

Linton, O., 1993. “Adaptive Estimation in Arch Models”, Cowles Foundation Discussion Paper No. 1054, Yale University.

Liu, J.C., 2006. “On the tail behaviors of Box-Cox transformed threshold GARCH(1,1) process,” Statistics & Probability Letters, 76(13), 1323-1330.

Nelson, D.B., 1991. “Conditional Heteroskedasticity in Asset Returns: A New Approach,” Econometrica, 59(2), 347-370.

Newey, W.K., 1990. “Semiparametric efficient bounds,” Journal of Applied Econometrics, 5(2), 99-135.

Pakel, C., Shephard, N., and Sheppard, K., 2011. “Nuisance parameters, composite likelihoods and a panel of GARCH models,” Statistica Sinica, 21(1), 307-329.

Pan, J., Wang, H., and Tong, H., 2008. “Estimation and tests for power-transformed and threshold GARCH models,” Journal of Econometrics, 142(1), 352-378.

Roussas, G.G., 1979. “Asymptotic distribution of the log likelihood function for stochastic processes,” Zeitschrift Wahrscheinlichkeitstheorie und Verwandte Gebiete, 47(1), 31-46.

Schick, A., 1986. “On asymmetrically efficient estimation in semiparametric models,” The Annals of Statistics, 14(3), 1139-1151.

Schwert, G.W., 1989. “Why does stock market volatility change over time?”, Journal of Finance, 44(5), 1115- 1153.

Sun, Y., and Stengos, T., 2006. “Semiparametric efficient adaptive estimation of asymmetric GARCH models,” Journal of Econometrics, 133(1), 373-386.

Swensen, A.R., 1985. “The Asymptotic Distribution of the Likelihood Ratio for Autoregressive Time Series with a Regression Trend,” Journal of Multivariate Analysis, 16(1), 54-70.

Taylor, S.J., 1986. “Modelling Financial Time Series,” John Wiley & Sons, Ltd.

Wand, M.P., and Jones, M.C., 1995. “Kernel smoothing,” Chapman & Hall/CRC, London.

Wellner, J.A., Klaassen, C.A.J., and Ritov, Y., 2006. “Semiparametric Models: a Review of Progress since BKRW (1993),” in Fan, J. and Koul, H.L., 2006. “Frontiers in Statistics”, World Scientific.