Aknouche, Abdelhakim and Almohaimeed, Bader and Dimitrakopoulos, Stefanos
(2024):
*Noising the GARCH volatility: A random coefficient GARCH model.*

Preview |
PDF
MPRA_paper_120456.pdf Download (960kB) | Preview |

## Abstract

This paper proposes a noisy GARCH model with two volatility sequences (an unobserved and an observed one) and a stochastic time-varying conditional kurtosis. The unobserved volatility equation, equipped with random coefficients, is a linear function of the past squared observations and of the past observed volatility. The observed volatility is the conditional mean of the unobserved volatility, thus following the standard GARCH specification, where its coefficients are equal to the means of the random coefficients. The means and the variances of the random coefficients as well as the unobserved volatilities are estimated using a three-stage procedure. First, we estimate the means of the random coefficients, using the Gaussian quasi-maximum likelihood estimator (QMLE), then, the variances of the random coefficients, using a weighted least squares estimator (WLSE), and finally the latent volatilities through a filtering process, under the assumption that the random parameters follow an Inverse Gaussian distribution, with the innovation being normally distributed. Hence, the conditional distribution of the model is the Normal Inverse Gaussian (NIG), which entails a closed form expression for the posterior mean of the unobserved volatility. Consistency and asymptotic normality of the QMLE and WLSE are established under quite tractable assumptions. The proposed methodology is illustrated with various simulated and real examples.

Item Type: | MPRA Paper |
---|---|

Original Title: | Noising the GARCH volatility: A random coefficient GARCH model |

English Title: | Noising the GARCH volatility: A random coefficient GARCH model |

Language: | English |

Keywords: | Noised volatility GARCH, Randon coefficient GARCH, Markov switching GARCH, QMLE, Weighted least squares, filtering volatility, time-varying conditional kurtosis. |

Subjects: | C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: 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 C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics |

Item ID: | 120456 |

Depositing User: | Prof. Abdelhakim Aknouche |

Date Deposited: | 23 Mar 2024 10:32 |

Last Modified: | 23 Mar 2024 10:32 |

References: | Aknouche, A. (2015). Quadratic random coefficient autoregression with linear-in-parameters volatility. Statistical Inference for Stochastic Processes 18, 99-125. Aknouche, A., Almohaimeed, B. and Dimitrakopoulos, S. (2022a) Forecasting transaction counts with integer-valued GARCH models. Studies in Nonlinear Dynamics & Econometrics 26, 529-539. Aknouche, A., Almohaimeed, B. and Dimitrakopoulos, S. (2022b). Periodic autoregressive conditional duration. Journal of Time Series Analysis 43, 5--29. Aknouche, A. and Francq, C. (2021). Count and duration time series with equal conditional stochastic and mean orders. Econometric Theory 37, 248--280. Aknouche, A. and Francq, C. (2022). Stationarity and ergodicity of Markov-switching positive conditional mean models. Journal of Time Series Analysis 43, 436-459. Aknouche, A. and Francq, C. (2023). Two-stage weighted least squares estimator of the conditional mean of observation-driven time series models. Journal of Econometrics 237, 105-174. Aknouche, A. and Rabehi, N. (2010). On an independent and identically distributed mixture bilinear time series model. Journal of Time Series Analysis 31, 113-131. Ayala, A. and Blazsek, S. (2019). Score-driven currency exchange rate seasonality as applied to the Guatemalan Quetzal/US Dollar. SERIEs 10, 65--92. Barndorff-Nielsen, O.E. (1978). Hyperbolic Distributions and distributions on Hyperbolae. Scandinavian Journal of Statistics 5, 151-157. Barndorff-Nielsen, O.E. (1997). Normal Inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics 24, 1--13. Barndorff-Nielsen, O.E., Mikosch T. and Resnick, S.I. (2013). Lévy Processes: Theory and Applications. Birkhäuser. Barndorff-Nielsen, O.E. and Prause, K., (2001). Apparent scaling. Finance and Stochastics 5, 103--113. Billingsley, P. (2008). Probability and measure. John Wiley and Sons, second edition. Blazsek, S., Ho, H.-C. and Liu. S.-P. (2018). Score-Driven Markov-Switching EGARCH Models: An Application to Systematic Risk Analysis. Applied Economics 50, 6047--6060. Bollerslev, T. (1982). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 7-327. Bougerol, P. and Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115-127. Chen, M. and An, H.Z. (1998). A note on the stationarity and the existence of moments of the GARCH model. Statistica Sinica 8, 505-510. Cox, D.R. (1981). Statistical analysis of time series: Some recent developments. Scandinavian Journal of Statistics 8, 93-115. Engle, R.F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, 987-1007. Engle, R.F. (2002). New frontiers for ARCH models. Journal of Applied Econometrics 17, 425--446. Engle, R. and Russell, J. (1998). Autoregressive conditional duration: A new model for irregular spaced transaction data. Econometrica 66, 1127--1162. Francq, C. and Zakoian J.M. (2004). Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605-637. Francq, C. and Zakoian, J.-M. (2005). The L2-structures of standard and switching-regime GARCH models. Stochastic Processes and their Applications 115, 1557--1582. Francq, C. and Zakoian, J.-M. (2008). Deriving the autocovariances of powers of Markov-switching GARCH models, with applications to statistical inference. Computational Statistics and Data Analysis 52, 3027--3046. Francq, C. and Zakoian, J. -M. (2019). GARCH Models, Structure, Statistical inference and nancial applications. John Wiley & Sons, Ltd., Publication. Gray, S. (1996). Modeling the conditional distribution of interest rates as a regime-switching process. Journal of Financial Economics 42, 27--62. Haas, M., Mittnik, S. and Paolella, M. (2004). A new approach to Markov-switching GARCH models. Journal of Financial Econometrics 2, 493--530. Hamilton, J.D. and Susmel, R. (1994). Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics 64, 307--333. Karlis, D. (2002). An EM type algorithm for maximum likelihood estimation of the normal--inverse Gaussian distribution. Statistics & Probability Letters 57, 43--52. Kazakevicius, V., Leipus, R. and Viano, M.-C. (2004). Stability of random coefficient ARCH models and aggregation schemes, Journal of Econometrics 120, 139--158. Klaassen, F. (2002). Improving GARCH volatility forecasts with regime-switching GARCH. Empirical Economics 27, 363--94. Klivecka, A. (2004). Random coefficient GARCH(1,1) model with i.i.d. coefficients. Lithuanian Mathematical Journal 44, 374--384. Murphy, K.P. (2007). Conjugate Bayesian analysis of the Gaussian distribution. Preprint. Mozumder, S., Talukdar, B. Kabir M.H. and Li, B. (2024). Non-linear volatility with normal inverse Gaussian innovations: ad-hoc analytic option pricing. Review of Quantitative Finance and Accounting 62, 97--133. Nicholls, F. and Quinn, B.G. (1982). Random coefficient autoregressive models: An Introduction. Springer-Verlag, New York. Paolella, M.S. (2007). Intermediate Probability: A computational Approach. John Wiley & Sons. Rachev, S.T. (2003). Handbook of heavy tailed distributions in Finance, Volume 1: Handbooks in Finance, North Holland. Regis, M., Serra P. and van den Heuvel E.R. (2022). Random autoregressive models: A structured overview. Econometric Reviews 41, 207--230. Stentoft, L. (2008). American Option Pricing Using GARCH Models and the Normal Inverse Gaussian Distribution. Journal of Financial Econometrics 6, 540--582. Taylor, S.J. (1982). Financial returns modelled by the product of two stochastic processes- a study of daily sugar prices, 1961-1979. In "Time Series Analysis: Theory and Practice", Anderson O.D. (ed.). North-Holland, Amsterdam, 203-226. Taylor, S.J. (1986). Modelling Financial Time Series. Wiley, Chichester. Thavaneswaran, A., Appadoo, S.S. and Samanta, M. (2005). Random coefficient GARCH models. Mathematical and Computer Modelling 41, 723--733. Tsay, R.S. (2010). Analysis of financial time series: Financial Econometrics, 3rd edition, Wiley. Wee, D.C.H., Chen, F., Dunsmuir, W.T.M. (2022). Likelihood inference for Markov switching GARCH(1,1) models using sequential Monte Carlo. Econometrics and Statistics, 21, 50 -- 68. White, H., Kim, T.H. and Manganelli, S. (2010). Modeling Autoregressive Conditional Skewness and Kurtosis with Multi-Quantile CAViaR. In Volatility and Time Series Econometrics: Essays in Honor of Robert Engle. Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199549498.003.0012. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/120456 |