Bayesian Semi-Parametric Spectral Density Estimation With Applications To The Southern Oscillation Index

Standard time series modelling is dominated by parametric models like ARMA and GARCH models. Even though nonparametric Bayesian inference has been a rapidly growing area over the last decade, only very few nonparametric Bayesian approaches to time series analysis have been developed. Most notably, Carter and Kohn (1997), Gangopadhyay (1998), Choudhuri et al. (2004), and Rosen et al (2012) used Whittle’s likelihood for Bayesian modeling of the spectral density as the main nonparametric characteristic of stationary time series. On the other hand, frequentist time series analyses are often based on nonparametric techniques encompassing a multitude of bootstrap methods (Kreiss and Lahiri, 2011, Kirch and Politis, 2011).
As shown in Contreras-Cristan et al. (2006), the loss of efficiency of the nonparametric approach using Whittle’s likelihood approximation can be substantial. On the other hand, parametric methods are more powerful than nonparametric methods if the observed time series is close to the considered model class but fail if the model is misspecified. Therefore, we suggest a nonparametric correction of a parametric likelihood that takes advantage of the efficiency of parametric models while mitigating sensitivities through a nonparametric amendment. We use a nonparametric Bernstein polynomial prior on the spectral density with weights induced by a Dirichlet process. Contiguity and posterior consistency for Gaussian stationary time series have been shown in a preprint by Kirch et al (2017). Bayesian posterior computations are implemented via a MH-within-Gibbs sampler and the performance of the nonparametrically corrected likelihood is illustrated in a simulation. We use this approach to analyse the monthly time series of the Southern Oscillation Index, one of the key atmospheric indices for gauging the strength of El Nino events and their potential impacts on the Australian region.