Modeling Scalar Nonstationary Covariance Time Series

In this chapter scalar nonstationary covariance time series are modeled using time varying coefficient autoregressive models, as in equation 11.1. In such a model for N observations, if the order of the AR model is m there will be N × m AR coefficient parameters and as many asNinnovations variance parameters. Fitting a time varying AR, (TVAR), model with smoothness priors constraints permits those parameters to be estimated implicitly in terms of only a small number of explicitly estimated hyperparameters. Several different mechanisms for implementing the smoothness priors constraints are possible. Our own approach to this important topic has evolved over several years, and two different methods for fitting the time varying AR coefficient model with smoothness priors constraints are shown here. (Each of the methods has implications for the modeling of multivariate nonstationary covariance data. That topic is treated in Chapter 12.) Our primary application for the scalar nonstationary covariance modeling is the evolution with time of the power spectrum. The estimated TVAR model yields what we refer to as an “instantaneous power spectral density”.