Estimating and simulating Weibull models of risk or price durations: An application to ACD models

There is now a massive literature on both the GARCH family of risk models and the related Auto-Conditional Duration (ACD) models used for modeling the stochastic timing of trades or price changes in finance market microstructure research. Both have their origins in Engle (1982) and Bollerslev (1986). This paper uses the theory of estimating functions (EF) as a semi-parametric method for estimating the parameters of this type of model. As an example, we consider the class of ACD models with errors from the standard Weibull distribution to develop an estimation procedure. This method could equally be applied to GARCH models. Using a simulation study, it is shown that the EF approach is easier to use in practice than the maximum likelihood (ML) or quasi maximum likelihood (QML) methods. The statistical properties of the corresponding optimal estimates are investigated and it is shown that the estimates using both the EF and QML methods are comparable. However, the EF estimates are easier to evaluate than the ML and QML methods. Nevertheless, ML based estimates are superior and perform better when the true distribution is known, when this is not so EF estimates are a powerful tool.