Financial applications of ARMA models with GARCH errors

Purpose – Financial returns are often modeled as stationary time series with innovations having heteroscedastic conditional variances. This paper seeks to derive the kurtosis of stationary processes with GARCH errors. The problem of hypothesis testing for stationary ARMA(p, q) processes with GARCH errors is studied. Forecasting of ARMA(p, q) processes with GARCH errors is also discussed in some detail. Design/methodology/approach – Estimating-function methodology was the principal method used for the research. The results were also illustrated using examples and simulation studies. Volatility modeling is the subject of the paper. Findings – The kurtosis of stationary processes with GARCH errors is derived in terms of the model parameters (?), ?-weights, and the kurtosis of the innovation process. Hypothesis testing for stationary ARMA(p, q) processes with GARCH errors based on the estimating-function approach is shown to be superior to the least-squares approach. The fourth moment of the l-steps-ahead forecast error is related to the model parameters and the kurtosis of the innovation process. Originality/value – This paper will be of value to econometricians and to anyone with an interest in the statistical properties of volatility modeling.