Local scale models: state space alternative to integrated GARCH processes

State space alternative to autoregressive conditional heteroskedasticity models are proposed. The initial model, which is labelled the Gaussian local scale model, has a measurement density which is Gaussian, conditional on the unobservable precision. The precision is assumed to be a gamma variable which evolves by being scaled by a beta variable. The resulting forecast is a student's t random variable, with a scale which is approximately an exponentially weighted moving average (EWMA) of the squares of the past observations. The degrees of freedom of the student's t distribution is controlled by the size of the discount parameter of the EWMA procedure. The Gaussianity of the measurement density is potentially inadequate when the model is applied to heavy tailed finance data. Instead, this assumption can be replaced by an exponential power density which allows the observed excess kurtosis to be modelled. The choice of the exponential power means that the model still maintains conjugacy, so allowing the derivation of an exact filter and likelihood function. This model is called the generalised local scale model. It has been used to model two exchange rate series