MODELLING OF OIL PRICE VOLATILITY USING ARIMA-GARCH MODELS F. Merabet, H. Zeghdoudi, R. H Yahia, and I. Saba

In this paper, the behavior of the oil price series named OIL is examined. The non-stationarity on average and variance, with the non-normality of the OIL series distribution, indicate the volatility of the series. The study is based on a combination of the Box-Jenkins methodology with the GARCH processes (Engle and Bollerslev). The first part models the lnOIL series in which, by applying the first difference the series becomes DlnOIL. Then the Box-Jenkins methodology is applied. The choice of the model was made on basis of minimization of criterion -Akaike (AIC), Shwarz (SIC)- and maximization of log likelihood (LL). Of the four models identified, ARMA (3.1) is retained. According to the statistical indicators of the ARMA model (3,1), the nature of the residuals and other tests, it is shown that the series of squares of the residuals follows a conditionally heteroscedastic ARCH model. The second part is devoted to a symmetrical and asymmetrical GARCH modelling. The model used for predicting volatility is the EGARCH model (1,2). The data available relates to 3652 daily values of the change in OIL, from 01/01/2019 to 12/31/2019. The forecast is made for the first three months of 2020; the result concludes that the predicted values and the current values are very close, and that the model ARIMA (3,1,1) + EGARCH (1,2) is the best forecast model.

CHARACTERIZATION OF THE TAIL BEHAVIOR OF A CLASS OF BEKK PROCESSES: A STOCHASTIC RECURRENCE EQUATION APPROACH

We consider conditions for strict stationarity and ergodicity of a class of multivariate BEKK processes $(X_t : t=1,2,\ldots )$ and study the tail behavior of the associated stationary distributions. Specifically, we consider a class of BEKK-ARCH processes where the innovations are assumed to be Gaussian and a finite number of lagged $X_t$ ’s may load into the conditional covariance matrix of $X_t$ . By exploiting that the processes have multivariate stochastic recurrence equation representations, we show the existence of strictly stationary solutions under mild conditions, where only a fractional moment of $X_t$ may be finite. Moreover, we show that each component of the BEKK processes is regularly varying with some tail index. In general, the tail index differs along the components, which contrasts with most of the existing literature on the tail behavior of multivariate GARCH processes. Lastly, in an empirical illustration of our theoretical results, we quantify the model-implied tail index of the daily returns on two cryptocurrencies.

ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches

This chapter considers estimation of autoregressive conditional heteroscedasticity (ARCH) and the generalized autoregressive conditional heteroscedasticity (GARCH) models using quasi-likelihood (QL) and asymptotic quasi-likelihood (AQL) approaches. The QL and AQL estimation methods for the estimation of unknown parameters in ARCH and GARCH models are developed. Distribution assumptions are not required of ARCH and GARCH processes by QL method. Nevertheless, the QL technique assumes knowing the first two moments of the process. However, the AQL estimation procedure is suggested when the conditional variance of process is unknown. The AQL estimation substitutes the variance and covariance by kernel estimation in QL. Reports of simulation outcomes, numerical cases, and applications of the methods to daily exchange rate series and weekly prices’ changes of crude oil are presented.