A Hierarchical Approach to Covariance Function Estimation for Time Series

The covariance function in time series models is typically modelled via a parametric family. This ensures straightforward best linear prediction while maintaining positive-definiteness of the covariance function. We suggest an alternative approach, which will result in data-determined shrinkage towards this parametric model. Positive-definiteness is maintained by carrying out the shrinkage in the spectral domain. We offer both a fully Bayesian hierarchical approach and an approximate hierarchical approach that will be much simpler computationally. These are implemented on the frequently analysed Canadian lynx data and compared to other models that have been fitted to these data.