Austin Chapter of the American Statistical Association
The classical method for estimating the spectral density of a multivariate time series is to first calculate the periodogram, and then smooth it to obtain a consistent estimator. Typically, to ensure the estimate is positive definite, all the elements of the periodogram are smoothed the same way. There are, however, many situations for which different components of the spectral matrix have different degrees of smoothness, and hence require different smoothing parameters in order to obtain optimal estimates. We suggest a Bayesian approach that uses Markov chain Monte Carlo techniques to fit smoothing splines to each component, real and imaginary, of the Cholesky decomposition of the the periodogram matrix. The spectral estimate is then obtained by reconstructing the spectral estimator from the smoothed Cholesky decomposition components. Our technique allows for automatic smoothing of the different components of the spectral density matrix. We illustrate our methodology with data on the Southern Oscillation Index, as well as with a DNA sequence.