Multivariate spectral analysis using Cholesky decomposition

We propose to smooth the Cholesky decomposition of a raw estimate of a multivariate spectrum, allowing different degrees of smoothness for different elements. The final spectral estimate is reconstructed from the smoothed Cholesky elements, and is consistent and positive definite. More importantly, the Cholesky decomposition matrix of the spectrum can be used as a transfer function in generating time series whose spectrum is identical to the given spectrum at the Fourier frequencies. This not only provides us with much flexibility in simulations, but also allows us to construct bootstrap confidence intervals for the multivariate spectrum by generating bootstrap samples using the Cholesky decomposition of the spectral estimate. A numerical example and an application to electroencephalogram data are used as illustrations.