1. Introduction

Many papers deal with the departures from normality of asset return distributions. It is well known that the distributions of stock return exhibit negative skewness and excess kurtosis; see among others [2, 9, 14, 15]. The higher moments of the return specifically, excess kurtosis (the fourth moment of the distribution) makes extreme observations more likely than in the normal case, which means that the market gives higher probability to extreme observations than in

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

normal distribution. However, the existence of negative skewness (the third moment of the distribution) has the effect of accentuating the left-hand side of the distribution, which means that a higher probability of decreases given to asset pricing than increases in the market.

The generalized autoregressive conditional heteroscedasticity (GARCH) models, introduced by Engle [5] and Bollerslev [1], allow for time-varying volatility1 but not for time-varying skewness or time-varying kurtosis. Different GARCH models have been developed in the literature to capture dependencies in higher order moments, starting with Hansen [7] who proposed a skew-Student distribution to account for both time-varying excess kurtosis and skewness. A significant evidence of time-varying skewness found [9]. Others [11, 12] found a significant time varying in both skewness and kurtosis, while [3, 15, 16] found little evidence of either. With regard to the frequency of observation, Jondeau and Rockinger [11] found the presence of timevarying skewness and kurtosis in daily but not weekly data, while others including [2, 7, 9] found an evidence of time-varying skewness and kurtosis in weekly and even monthly data. Regarding daily data [4, 12, 18] found an evidence of time-varying skewness and kurtosis in daily data. The chapter employed GARCH(1,1) model as the performance of the model proved compared large number of volatility models; for more details, see Hansen and Lunde [8].

This paper contributes to the literature of volatility modeling in two aspects. First, we jointly estimate time-varying volatility, skewness, and kurtosis assuming Johnson SU distribution for the error term. The method is applied to two different daily returns: stock indices and exchange rates. Second, a new alternative scheme is introduced to generate the sequence of the forecasts.

The rest of the paper is organized as follows. Following this introduction, Section 2 presents the empirical results regarding the estimation of the model. Section 3 compares the models. In Section 4, the new forecasting scheme is presented, while Section 5 gives concluding remarks.
