DEVELOPMENT OF VOLATILITY MODELS WITH ASYMMETRIC ERROR INNOVATIONS

Abstract

Generalised Autoregressive Conditional Heteroskedasticity (GARCH) models have been used to model non-constant variances in financial time series models. Previous works have assumed error innovations of GARCH models of order (p,q) as: Normal, Student-t and Generalised Error Distribution (GED), but these distributions failed to captureconditional volatility ofseries adequately, leading to low forecast performance. This study is therefore aimedat developing variants of GARCH(p,q) and Asymmetric Power ARCH (APARCH(p,q)) models with asymmetric error innovations for improved forecast performance. The two error innovations considered were the Generalised Length Biased Scaled-t (GLBST) and Generalised Beta Skewed-t (GBST) distributions, obtained by remodifying Fisher Concept of Weighted Distribution and McDonald Generalised Beta Function, respectively, in the Student-t distribution. The properties of the proposed distributions were investigated. The proposed innovations were imposed on GARCH(1,1) and APARCH(1,1) models to obtain GARCH-GLBST(1,1) and APARCH-GLBST(1,1) models, respectively. Similarly, GARCH-GBST(1,1) and APARCH-GBST(1,1) models were also obtained by incorporating proposed innovations into GARCH(1,1) and APARCH(1,1) models. Data from the Nigerian Stock Index were used to illustrate the models. The proposed models were compared with GARCH and APARCH models, each for Normal, Student-t and GED error innovations. The performance of the proposed models over the existing ones were investigated using the Log-likelihood function, Root Mean Square Error (RMSE), Adjusted Mean Absolute Percentage Error (AMAPE) and Akaike Information Criterion (AIC).A Monte Carlo simulation experiment with sample sizes: 50, 100, 500, 1000 was also conducted to assess the performance of these models. Out of the 10 models under consideration, APARCH-GLBST(1,1) was the best, followed by GARCH-GLBST(1,1) and GARCH-GED(1,1) models, in terms of the AIC values (7.856,7.988 and 9.984).The forecast evaluation criteria (RMSE, AMAPE), APARCH-GLBST(1,1) model also ranked best (RMSE =0.281, AMAPE = 0.280), followed by GARCH-GLBST(1,1) model (RMSE = 0.291, AMAPE = 0.290) and GARCH-GBST(1,1) model (RMSE = 0.309, AMAPE = 0.301). The least performing in terms of forecasts was the GARCH(1,1)-Normal model.Result from the Monte Carlo simulated data were similar to those obtained from life data except for very large sample sizes (1000) where APARCH-GLBST(1,1) and GARCH-GBST models performed below others. The proposed volatility models with asymmetric innovations outperformed existing models in terms of fitness of conditional volatility and forecasts performance. The proposed models will be good alternatives for volatility modelling of symmetric and asymmetric stock returns data.