GARCH Models, 2nd Edition

12 Parameter‐Driven Volatility Models

In this chapter, we consider volatility models in which the volatility no longer coincides with the conditional variance. In other words, volatility would remain unobservable even if the parameters of the data‐generating process were known. Such models are naturally related to GARCH and can be seen as their competitors/alternatives in financial applications.

The difference between the different volatility modellings can be perceived through the concepts of observation‐driven and parameter‐driven models introduced by Cox (1981) and recently revisited by Koopman, Lucas and Scharth (2016). Suppose that φ _t is a ‘parameter’ at time t . Let ε _t be an observed random variable at time t . In an observation‐driven model,

where Φ is a measurable function. In a parameter‐driven model,

where is an idiosyncratic innovation at time t and Φ^* is a measurable function. In the latter case, there is a latent structure in the model which, in general, cannot be directly related to the observations.

Let us a consider a GARCH model, choosing the volatility for the parameter: φ _t  = σ _t . Under stationarity and invertibility conditions, ...