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Recently, the interest of researchers in the use of hybrid models in the process of analyzing model time series with fluctuations and forecasting fluctuations in financial time series has increased significantly. Hybrid ARMA-GARCH models were created for medium- and long-term forecasts of time series of financial market index prices: ARMA models are used to analyze their linear component, which is a combination of autoregressive models and moving average models, and GARCH models are used to analyze the nonlinear component. which are generalized autoregressive models that depend on the nonconstancy of variance models. Hybrid ARMA-GARCH models eliminate the weaknesses and gaps that exist in each group of models (ARMA and GARCH) separately, which increases their forecasting accuracy and reliability, so they have already been successfully applied to model and forecast daily stock returns for three standard indices in the USA. The purpose of this article is to investigate which of the hybrid ARMA-GARCH models is optimal for forecasting the return of the DAX index, which is the most important stock index of the German securities market. It is the German equivalent of the American Dow Jones Index, has been calculated since 1988 by Deutsche Börse AG and reflects the total return on capital of the largest stock companies listed on the Frankfurt Stock Exchange (currently 40; by 2021 – 30): calculated as a weighted average of capitalization of the value of Free Float share prices on the Xetra electronic exchange, and also takes into account dividends on shares, assuming that the dividend is reinvested in the share on which it was accrued. The database of this study consisted of the daily closing prices of the DAX index presented on the official website of the Frankfurt Stock Exchange during the period from 01.01.2018 to 09.30.2023 (altogether about 1,500 observations), the stability of the time series was assessed using Expanded Dickey Fuller Liquidity (ADF). The article proposes 7 hybrid models, from which the one that is best suited for modeling the volatility of the DAX index is selected. It is an ARMA (2,3)-EGARCH (1,1) model because it captures volatility and leverage effects on DAX returns and its expected returns more than other models. The selection of the best alternative from the developed array of hybrid models was carried out according to the following criteria: AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), H-QIC (Hannan-Quinn Information Criterion).
Khelifa et al. (Sun,) studied this question.