The Exponential Model for the Spectrum of a Time Series: Extensions and Applications

The exponential model for the spectrum of a time series and its fractional extensions are based on the Fourier series expansion of the logarithm of the spectral density. The coefficients of the expansion form the cepstrum of the time series. After deriving the cepstrum of important classes of time series processes, also featuring long memory, we discuss likelihood inferences based on the periodogram, for which the estimation of the cepstrum yields a generalized linear model for exponential data with logarithmic link, focusing on the issue of separating the contribution of the long memory component to the log-spectrum. We then propose two extensions. The first deals with replacing the logarithmic link with a more general Box-Cox link, which encompasses also the identity and the inverse links: this enables nesting alternative spectral estimation methods (autoregressive, exponential, etc.) under the same likelihood-based framework. Secondly, we propose a gradient boosting algorithm for the estimation of the log-spectrum and illustrate its potential for distilling the long memory component of the log-spectrum.