Introduction: proved that the distribution of the financial


 Introduction:
Time variation in second and higher-order moments become very important phenomenon in an empirical econometric. Widely used volatility models such as exponentially weighted moving average (EWMA) model and General Autoregressive Conditional Heteroskedasticity (GARCH) model allow time varying volatility and assume normal distribution for the error term with constant skewness and kurtosis. However, the empirical studies have proved that the distribution of the financial series is skewed and fat tail, see for example Campbell and Siddique (1999 and 2000), Alizadeh and Gabrielsen (2011). When normal distribution assumed for the error term in financial series that leads to unreliable results in different financial issues such as risk management, option pricing, trading, hedging activities, portfolio selection and asset allocation.
Recently many researchers concern regarding higher moments to have more efficiency models. In this respect many models have been appear to allow time variation skewness and kurtosis such as Harvey (1999) who produced GARCH with skewness (GARCH-S) model, to capture time varying conditional skewness. In addition, Brooks (2005) presented a GARCH with kurtosis (GARCH-K) model to estimate time varying conditional kurtosis. These models assume a noncentral t distribution to the error terms. Although the dynamic behavior for the higher moments improves the efficiency, noncentral t distribution needs large amount of calculation.
 GARCH with skewness and Kurtosis (GARCH-SK) model proposed by Leon, et al (2005) which allows simultaneously estimating volatility, skewness and kurtosis based on assuming Gram-Charlier expansion (GCE) series for the error term.  In addition, Gabrielsen, Zagaglia, Kirchner, and Liu (2012) also applied GCE series for the EWMA model; they presented EWMA model (EWMA-SK) that jointly estimates volatility, skewness and kurtosis. Both these models have proved their efficiency to detect the dynamic behavior for empirical financial study, see for example Apergis and Gabrielsen (2012) and Gabrielsen (2015).
On the other hand, Lucas and Zhang (2015) developed a new empirical model called score-driven EWMA (SD-EWMA) model that extended the standard EWMA approach. They used the higher-moment properties of the forecasting distribution to drive the dynamics of volatilities and other time varying parameters based on the score of the forecasting distribution.
 SD-EWMA model is easy to implement, computationally simple and remains similar in spirit to the highly familiar EWMA approach. It has been used the framework presented by Harvey (2013) and Creal, Koopman, and Lucas (2011, 2013). They produce a new type of observation-driven model called generalized autoregressive score (GAS) model also named Dynamic conditional score (DCS) model which has been applied successfully to a wide range of empirical application. Its key feature is the fact that the time-varying parameter dynamics are driven by the score of the forecasting distribution. Empirical evidence of the usefulness of score-driven dynamics has provided by Creal, Schwaab, Koopman, and Lucas (2014). 
Therefore, this research aims to investigate the performance and efficiency among these competing models: GARCH-SK, EWMA-SK, and SD-EWMA model.
        Introduction: Time variation in second and higher-order moments become very important phenomenon in an empirical econometric. Widely used volatility models such as exponentially weighted moving average (EWMA) model and General Autoregressive Conditional Heteroskedasticity (GARCH) model allow time varying volatility and assume normal distribution for the error term with constant skewness and kurtosis. However, the empirical studies have proved that the distribution of the financial series is skewed and fat tail, see for example Campbell and Siddique (1999 and 2000), Alizadeh and Gabrielsen (2011). When normal distribution assumed for the error term in financial series that leads to unreliable results in different financial issues such as risk management, option pricing, trading, hedging activities, portfolio selection and asset allocation. Recently many researchers concern regarding higher moments to have more efficiency models. In this respect many models have been appear to allow time variation skewness and kurtosis such as Harvey (1999) who produced GARCH with skewness (GARCH-S) model, to capture time varying conditional skewness. In addition, Brooks (2005) presented a GARCH with kurtosis (GARCH-K) model to estimate time varying conditional kurtosis. These models assume a noncentral t distribution to the error terms. Although the dynamic behavior for the higher moments improves the efficiency, noncentral t distribution needs large amount of calculation.  GARCH with skewness and Kurtosis (GARCH-SK) model proposed by Leon, et al (2005) which allows simultaneously estimating volatility, skewness and kurtosis based on assuming Gram-Charlier expansion (GCE) series for the error term.  In addition, Gabrielsen, Zagaglia, Kirchner, and Liu (2012) also applied GCE series for the EWMA model; they presented EWMA model (EWMA-SK) that jointly estimates volatility, skewness and kurtosis. Both these models have proved their efficiency to detect the dynamic behavior for empirical financial study, see for example Apergis and Gabrielsen (2012) and Gabrielsen (2015). On the other hand, Lucas and Zhang (2015) developed a new empirical model called score-driven EWMA (SD-EWMA) model that extended the standard EWMA approach. They used the higher-moment properties of the forecasting distribution to drive the dynamics of volatilities and other time varying parameters based on the score of the forecasting distribution.  SD-EWMA model is easy to implement, computationally simple and remains similar in spirit to the highly familiar EWMA approach. It has been used the framework presented by Harvey (2013) and Creal, Koopman, and Lucas (2011, 2013). They produce a new type of observation-driven model called generalized autoregressive score (GAS) model also named Dynamic conditional score (DCS) model which has been applied successfully to a wide range of empirical application. Its key feature is the fact that the time-varying parameter dynamics are driven by the score of the forecasting distribution. Empirical evidence of the usefulness of score-driven dynamics has provided by Creal, Schwaab, Koopman, and Lucas (2014).  Therefore, this research aims to investigate the performance and efficiency among these competing models: GARCH-SK, EWMA-SK, and SD-EWMA model.        

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