ARIMA-GARCH and ARIMA-EGARCH-t Models in Fitting and Forecasting Volatility of the Shanghai Composite Index

Abstract

Stock market volatility lies at the heart of risk management. Existing research indicates that A-share returns exhibit fat-tail characteristics, yet further refinement is required in balancing the model’s depiction of fat tails and asymmetry with predictive robustness. This paper utilises daily data from the Shanghai Composite Index spanning January 2010 to June 2025 as its sample. It compares the Autoregressive Integrated Moving Average-Generalised Autoregressive Conditional Heteroskedasticity (1,1)-Normal (ARIMA-GARCH (1,1)-Normal) and Autoregressive Integrated Moving Average-Exponential Generalised Autoregressive Conditional Heteroskedasticity (1,1)-t (ARIMA-EGARCH (1,1)-t) models in terms of volatility fitting and forecasting performance. Methodologically, the mean correlation was first filtered using ARIMA (2,0,2). Maximum likelihood estimation was employed for modelling, with performance evaluated through rolling forecasts. The research encompasses sequence stationarity testing and model construction, evaluating in-sample fit through metrics such as the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and log-likelihood values. Out-of-sample forecasting performance is measured using Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Quantile Likelihood (QLIKE) values. The results indicate that the EGARCH-t model exhibits superior in-sample fit with a higher log-likelihood value (10,606.9), successfully capturing fat tails (ν=4.7236) and the leverage effect (γ=-0.0216). However, the GARCH model demonstrated greater robustness in out-of-sample forecasting, with lower error metrics such as MSE (0.000000) and RMSE (0.000523). Research indicates that model selection requires balancing goodness-of-fit with generalizability, while EGARCH-t is well-suited for capturing historical volatility mechanisms, GARCH holds greater predictive utility.