INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS. VOL. NO. NOVEMBER 2023 Evaluating the Fitting Performance of AGARCH. NAGARCH. , and VGARCH. Models Didit B. Nugroho. Veny M. Ningtyas and Hanna A. Parhusip AbstractAiThis study compares the performance of the GARCH. AGARCH. NAGARCH. , and VGARCH. models fitted to real data. The observed real data are the USD exchange rate against IDR in the daily period from January 2010 to December 2017. To identify the superiority and evaluate the performance of those models in capturing the heavy-tailed and skewed character in exchange rate distribution, the return error is assumed to be the Normal. Skew Normal (SN). Skew Curved Normal (SCN), and Student-t The modelAos parameters are estimated using the GRG Non-Linear method in Excel Solver and the ARWM method in the MCMC scheme implemented in the Scilab Estimation results using ExcelAos Solver have similar values to the estimates obtained using MCMC, concluding that ExcelAos Solver has a good ability in estimating the modelAos Based on AIC values, this study concludes that the NAGARCH. model under Student-t distribution performs the best. Index TermsAiAGARCH. VGARCH. NAGARCH. INTRODUCTION In the financial market, volatility is one phenomenon that has the potential to lead the greater risks and uncertainties to the value of an investment, which causes the interest of the investors in the market to become unstable. According to . , volatility of financial markets describes the fluctuations in the value of a market asset over a certain period of time. statistics, volatility is defined to be a standard deviation of the returns . hanges in the logarithmic pric. , . It measures the difference in the value of the assetAos return movement for a given financial time series. Therefore, the existence of volatility impacts the reality of global financial markets in relation to the risk management. The existence of volatility for financial time series has raised the problem heteroscedasticity, which means that the volatility changes over different time periods. One popular class that can be used to model the time-varying volatility is the GARCH (Generalized Autoregressive Conditional Heteroscedasticit. Didit B. Nugroho and Hanna A. Parhusip are with the Data Science Study Program. Universitas Kristen Satya Wacana. Jl. Diponegoro 52Ae60 Salatiga. Jawa Tengah. Indonesia. Didit B. Nugroho is with the Study Center for Multidisciplinary Applied Research and Technology (SeMart. Universitas Kristen Satya Wacana. Jl. Diponegoro 52Ae60 Salatiga. Jawa Tengah. Indonesia e-mail: budinugroho@uksw. Veny M. Ningtyas is with the Mathematics Study Program. Universitas Kristen Satya Wacana. Jl. Diponegoro 52Ae60 Salatiga. Jawa Tengah. Indonesia. Manuscript received February 01, 2023. accepted May 25, 2023. proposed by . GARCH model has symmetrical volatility response characteristics to returns, meaning that the past positive returns . ood new. and negative returns . ad new. have the same effect on the current volatility. The relationship between return and volatility is not always symmetrical but also can be asymmetric, meaning that positive and negative returns have different effects on volatility. Therefore, the asymmetric effect is essential in modeling and forecasting volatility. Some articles have proposed several extensions and modifications of GARCH model to accommodate an asymmetric effect in volatility. This study focuses only on three asymmetric GARCH models proposed by . , namely AGARCH (Asymmetric GARCH). NAGARCH (Nonlinear Asymmetric GARCH), and VGARCH (Vector GARCH) models. They applied these models to the TOPIX (Tokyo Price Inde. data by assuming a Normal distribution for return errors. They showed that the proposed models fit data better than the GARCH. The first contribution of this study is to extend the models of . by assuming that the return errors follow four different sets of distributions: Normal. SkewNormal (SN) of . Skew-Curved Normal (SCN) of . , and Student-t. To the best of the authorsAo knowledge, there is no study investigating the performance of such distributions on the models of . In evaluating the performance of models, this study fits the models on the buying rate of USD (US Dollar. to IDR (Indonesian Rupia. in the daily period from January 2010 to December 2017. When we fit a model, it means that we estimate the model. Therefore, this studyAos second contribution is using ExcelAos SolverAos GRG (Generalized Reduced Gradien. Non-Linear and Adaptive Random Walk Metropolis (ARWM) methods to estimate the studied model. Here, we employ the ARWM method in the Markov Chain Monte Carlo (MCMC) scheme and implement this in Scilab by writing our Both methods are compared to evaluate their ability to estimate the considered GARCH. For other GARCH-type models, . Ae. showed that ExcelAos SolverAos GRG Non-Linear and ARWM methods have good ability for parameter estimation. II. STATISTICAL MODELING GARCH. Models Returns are often expressed as a normal distribution and explained in terms of mean and standard deviation . Let Rt be an asset return at time t and follows a normal distribution INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS. VOL. NO. NOVEMBER 2023 with the mean 0 and variance Et2 . The equation for the return can be expressed as follows: Rt = zt , where zt O N. Et2 ). Study in . modeled the conditional variance in Eq. as a GARCH. , . process in which p and q denote the lag length on returns and variances, respectively. The most popular GARCH-type model and often used in many empirical studies in financial time series is perhaps the GARCH. this case, the current conditional variance is calculated based on the past weighted squared return and the past weighted Mathematically, the GARCH. process is defined Et2 = O RtOe1 EtOe1 in which the weighting factors are positive, requiring O > 0 and 0 O , < 1 to ensure the positivity of variance and 0 O < 1 for stationary condition. To capture the asymmetric effect, . incorporated new information to the measure of volatility via the News Impact Curve which gives the relation between Et and RtOe1 . the news impact curve of GARCH model is symmetric and centered at RtOe1 = 0, the asymmetric GARCH model of . AGARCH, is asymmetric and centered at RtOe1 = Oe. particular, the variance process for the AGARCH. impact curve is Et2 = O (RtOe1 )2 EtOe1 zt O N 0. Et2 in which OO R represents an asymmetric parameter, and conditions for the other parameters are as in the GARCH. If = 0, the effects of positive/negative returns are if = 0, the process reduces to the GARCH. Furthermore, . modified the AGARCH. model into the NAGARCH. and VGARCH. The variance process for the NAGARCH. model assumes that Et2 = O (RtOe1 EtOe1 )2 EtOe1 whereas the VGARCH. process is given by RtOe1 EtOe1 Et2 = O EtOe1 When < 0, the distribution is skewed to the left. when > 0, the distribution is skewed to the right. So = 0 will reduce the SN distribution to a Normal distribution. Skew-Curved Normal Distribution: Arellano-Valle et al. introduced the SCN distribution to express an asymmetrical class of Normal distribution which is different from the SN distribution. The SCN probability density function for a random variable Z with skewness is given by z = 2I . 1 ( . 2 Therefore, the SCN probability density function with zeromean, variance E 2 , and skewness has an expression as Oo = exp Oe 2 y 2AE 2 1 Erf p 2E 2 . ( . 2 ) When < 0, the distribution is left-skewed. when > 0, the distribution is right-skewed. So, = 0 will reduce the SCN distribution to a Normal distribution. Student-t Distribution: Student-t distribution was introduced by William Sealy Gosset under the pseudonym AyStudentAy . The Student-t density curve is symmetrical bell-shaped like the Normal distribution but has thicker tails . ften called heavy/fat tail. than the Normal distribution. Following . , a random variable Z with zero-mean, variance E 2 , and degrees of freedom > 2, the form of the Student-t probability density function is given by Oe 1 = p E ( Oe . AE 2 ( Oe . e 2 The tail heaviness of the Student-t distribution is determined by the parameter degrees of freedom . Smaller degrees of freedom give heavier tails on both sides and increasing the degrees of freedom makes the Student-t distribution approaches to a Normal distribution for Ou 30 . ESTIMATION AND SELECTION Parameter Estimation Distributions for Return Error . Skew-Normal Distribution: Azzalini in . introduced the SN distribution to extend the Normal distribution by incorporating a parameters as parameter skewness. For a random variable Z, the general form of SN probability density function with skewness OO R is given by f . = 2I . , . where I (A) denotes the normal Probability Density Function (PDF) and (A) denotes the normal Cumulative Distribution Function (CDF). Therefore, the probability density function of the SN distribution for a random variable Z with zero-mean and variance E 2 can be expressed as follows: = Oo exp Oe 2 1 Erf Oo 2AE 2 2E 2 One standard method to estimate the parameters of GARCH-type models is the Maximum Likelihood Estimation (MLE)-based method. These methods find the parameter values which maximize the likelihood function. With the same purpose as the MLE-based method, this study first utilizes ExcelAos SolverAos GRG Non-Linear method to estimate the considered models. The GRG Non-Linear method is based on work published by . , . ExcelAos Solver is one of the add-ins available for Microsoft Excel that can be used to find an optimal value . aximum or minimu. for non-linear optimization problems. Compared with other tools which require programming knowledge. ExcelAos Solver tool is preferred by financial practitioners since numerical optimization in many situations can be done by Solver. Following steps of . , in particular, this study INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS. VOL. NO. NOVEMBER 2023 chooses the GRG Non-Linear method as an estimation method. According to . , the existing values in the worksheet cells for each decision variable are taken as an initial solution such that any small change will improve the objective value. this way, the objective value will increase if the objective is maximization, or decrease if the objective is minimized until it achieves optimal solution. Second, we employ the ARWM method introduced by . to compare the results of ExcelAos Solver. Studies in . Ae . successfully applied the method in the Bayesian MCMC The ARWM method is developed to improve the efficiency of the random walk Metropolis algorithm, a type of simplest sampler commonly used in practice. In the Bayesian framework, one makes statements about the probability of a parameter. Using Bayesian terminology, the estimated probability of a parameter after observing the data is called a Auposterior probabilityAy, and it is often stated Posterior Oy Likelihood y Prior, . where the symbol AuOyAy means Auproportional toAy. For a parameter , the posterior distribution is denoted by f ( . , the likelihood function is denoted f . ), and the prior distribution is denoted by f ( ). The ARWM method updates a parameter value of in each MCMC iteration. Given a set of values i and step size si at the i-th iteration, the next iteration of MCMC is completed as follows. Sample the proposal O = i si A qi , where qi O N. , . f ( O |A) , and . Calculate the Metropolis ratio: r. O ) = f . |A) O then defines = min. , r. , )}. If > u for u O Uniform. 0, . , then the proposal is accepted and i 1 = O . otherwise, the proposal is rejected and i 1 = i . m( O ) Oe 0. , where . Calculate: sO = max smin , si . m( O ) is the frequency of proposal acceptance O with the expected acceptance probability 0. If sO > smax , then si 1 = smax . if sO < smax , then si 1 = si . Model Selection To perform statistical model selection and comparisons, several standard statistical tests and information criteria can be Generally, information criteria such as AIC (Akaike Information Criterio. can be used to investigate the model selection among competing models . ncluding for non-nested modelsAii. , situations in which one model is not a particular case of the othe. and determine the best fit model particularly . The selection of the best model for multiple models for a given dataset is determined by an AIC score . AIC = 2K Oe 2 log(LI )), . where K is the number of estimated parameters and LI is the maximum value of the likelihood function. A lower AIC score is betterAiin other words, the model with the lowest AIC score is the best. IV. EMPIRICAL APPLICATION Data Description This study uses the daily returns of the USD currency exchange rate to IDR from January 2010 to December 2017 . onsisting of 1891 observation. The data are selected to provide evidence that the AGARCH. NGARCH. , and VGARCH. models are more suitable than the GARCH. The continuous return for the time period t Oe 1 until t is calculated in percentage as follows: Rt = 100 y . og(Pt ) Oe log(PtOe1 )) . where Pt denotes the asset price at the time t. Figure 1 displays the plot of the daily returns series of the USD/IDR exchange rate. The figure shows that the return time series data are stationary, meaning that their fluctuation is around the average. The Augmented DickeyAeFuller test . ) produces a statistic of Oe44. maller than the critical value of Oe1. with a p-value of 0. maller than 5%) which confirms that the data do not contain the unit roots anymore, which is stationary. Therefore, the USD/IDR data satisfied the modelAos underlying assumptions. 2010/1/5 2012/1/2 2014/1/2 2016/1/4 2017/12/29 Fig. Daily return of USD/IDR from January 2010 to December 2017. To examine whether the USD/IDR exchange rate exhibits conditional heteroscedasticity or the ARCH effect in the return series, we use EngleAos Lagrange multiplier test. The ARCH test confirmed that the returns have heteroscedasticity, which is indicated by the greater statistical value than the critical value of 3. 84 with a p-value of 0. Therefore, the volatility analysis needs to be done using the ARCH/GARCH model. Table I gives an overview of the statistical description for the daily return of the USD/IDR exchange rates. At the 5% significance level, the JarqueAeBera (JB) normality test has rejected the Normal distribution for the observed data. The rejection is indicated by the JB statistic . ) is greater than the critical value of 5. 99Aibased on the chi-square distribution table with 2 degrees of freedom. The departure from the non-normality of the data can also be seen from their kurtosis values greater than 3Aithe existence of heavy tailsAiand their skewness values not too close to zeroAithe distribution is not symmetrical. Therefore, the assumption of non-normal distributions is appropriate in our case. The error process is particularly allowed to follow four distribution function types: Normal. Skew-Normal. Skew-Normal Curved, and Student-t distributions. Development of Log-likelihood Suppose a vector of return series is expressed in a sequence R = {R1 . R2 , . RT }. For mathematical convenience. INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS. VOL. NO. NOVEMBER 2023 TABLE I S KEWNESS . K URTOSIS . AND JB S TATS FOR RETURNS OF USD/IDR. Min. Oe2. Mean Max. Skewness Oe0. Kurtosis JB Stat. the natural logarithm of the likelihood is usually used in computation instead of the likelihood. Based on Eqs. 7Ae10, we can derive the log-likelihood function of the GARCH. AGARCH. NGARCH. and VGARCH. The models with the SN. SCN, and Student-t distributions for return error with zero-mean and variance Et2 have a loglikelihood function at time-t given as follows, respectively: log(L ( |Rt )) = Oe log. AEt2 ) Oe t 2 log 1 Erf 2Et2 , . log(L ( |Rt )) = Oe log. AEt2 ) Oe t 2 log 1 Erf p , . 2Et2 . ( Rt )2 ) 1 log(L ( |Rt )) = log e Oe log A( Oe . Et2 Oe log e Rt2 , . ( Oe . Et2 where is the vector of estimated model parameters and the process of Et2 follows a considered GARCH-type model. Estimation Details The ExcelAos SolverAos GRG Non-Linear method is applied by following the similar steps of . Firstly, the initial values of all unknown parameters are set as follows: O = 0. 005, = 0. 2, = 0. 7, = 0. 1, = 0, = 5. In the Excel spreadsheet, for each time corresponding to the return, the variance value of Et2 and log-likelihood of log(L ( |Rt )) are calculated based on Eqs. Ae. according to the considered model. Notice that ExcelAos Solver does not have the strict inequalities (Au>Ay and Au 0 for all cases, except for the VGARCH. model with Student-t distribution. Based on the asymmetric variance process in Eq. Ae. , the positive value of implies that the past positive returns will result in a more considerable increase in current variance than negative returns of the same absolute magnitude. TABLE i HPD INTERVALS AT THE 5% SIGNIFICANCE LEVEL FOR . Dist. SCN 0853,0. 0770,0. 0565,0. (Oe0. 0623, 0. Model 1728,0. 1761,0. 0907,0. 0038,0. 0556,0. 0502,0. 0470,0. (Oe0. 0985, 0. For the skewness parameter , the 95% HPD intervals are reported in Table IV. The intervals indicate a statistical significance at the 5% level for parameter in both SN and SCN distributions in each asymmetric model since the intervals exclude 0. This result shows evidence that both skewness specifications must be considered in the distribution of the returns. TABLE IV HPD INTERVALS AT THE 5% SIGNIFICANCE LEVEL FOR . Dist. SCN 0145,0. 0113,0. Model 0140,0. 0158,0. 0101,0. 0133,0. Model Evaluation An essential task of modeling is model evaluation. This section evaluates the competing GARCH models regarding their in-sample performance and investigated using AIC. Table V presents the AIC values and ranks each distribution and model according to their AIC values. We first note that ExcelAos Solver and MCMC give similar results. The results indicate that NAGARCH. models are the best fit model, followed by AGARCH. GARCH. , and VGARCH. The only exception is for the Student-t case in which the GARCH. model outperforms the AGARCH. This result confirms the previous finding that the asymmetry parameter in both AGARCH. dan VGARCH. models is not statistically significant. Moreover. AIC selects Student-t as the best fit distribution for USD/IDR data, followed by SCN and SN distributions. This result confirms the previous finding that the skewness parameter in both SN and SCN distributions is statistically significant. Comparing all results, we can conclude that the NAGARCH. model under Student-t distribution reflects the most appropriate characteristics of the USD/IDR exchange rate time series. TABLE V AIC VALUES OF COMPETING MODELS . Mod. Dist. SCN SCN SCN SCN Using ExcelAos Solver Rank Dist. Overall AIC Using MCMC Rank Dist. Overall 1525,48 AIC CONCLUDING REMARKS This study focuses on the in-sample performance of asymmetric GARCH. models of . , including AGARCH. NAGARCH, and VGARCH, in terms of their ability to fit the volatility model for USD/IDR exchange rate return data over a period January 2010 to December 2017. The fitting performance is investigated in four different distributional assumptions for the return errors, namely: Normal. Skew Normal. Skew-Curved Normal. Student-t distributions. The GRG Non-Linear in ExcelAos Solver and MCMCAos ARWM method implemented in Scilab are employed to estimate the considered models. Even though ExcelAos Solver violates a constraint for the Student-t case. ExcelAos SolverAos GRG NonLinear method can be said to have a good ability to estimate the asymmetric GARCH models. This is indicated by their estimates similar to those from MCMCAos ARWM method. AIC values suggest that NAGARCH. model under Studentt distribution performs the best in capturing the USD/IDR The analysis confirms the result in . that showed evidence of superiority for the NAGARCH model. The result also ensures the evidence in . that Student-t distribution provides a better ability to capture heavy tails than skew normal distribution, even than skew-curved normal. ACKNOWLEDGMENT