Exact likelihood for inverse gamma stochastic volatility models

We obtain a novel analytic expression of the likelihood for a stationary inverse gamma stochastic volatility (SV) model. This allows us to obtain the maximum likelihood estimator for this nonlinear non-Gaussian state space model. Further, we obtain both the filtering and smoothing distributions for the inverse volatilities as mixtures of gammas, and therefore, we can provide the smoothed estimates of the volatility. We show that by integrating out the volatilities the model that we obtain has the resemblance of a GARCH in the sense that the formulas are similar, which simplifies computations significantly. The model allows for fat tails in the observed data. We provide empirical applications using exchange rates data for seven currencies and quarterly inflation data for four countries. We find that the empirical fit of our proposed model is overall better than alternative models for four countries currency data and for two countries inflation data.