Bayesian time series regression with nonparametric modeling of autocorrelation

Series models have several functions: comprehending the functional dependence of variable of interest on covariates, forecasting the dependent variable for future values of covariates and estimating variance disintegration, co-integration and steady-state relations. Although the regression function in a time series model has been extensively modeled both parametrically and nonparametrically, modeling of the error autocorrelation is mainly restricted to the parametric setup. A proper modeling of autocorrelation not only helps to reduce the bias in regression function estimate, but also enriches forecasting via a better forecast of the error term. In this article, we present a nonparametric modeling of autocorrelation function under a Bayesian framework. Moving into the frequency domain from the time domain, we introduce a Gaussian process prior to the log of the spectral density, which is then updated by using a Whittle approximation for the likelihood function (Whittle likelihood). The posterior computation is simplified due to the fact that Whittle likelihood is approximated by the likelihood of a normal mixture distribution with log-spectral density as a location shift parameter, where the mixture is of only five components with known means, variances, and mixture probabilities. The problem then becomes conjugate conditional on the mixture components, and a Gibbs sampler is used to initiate the unknown mixture components as latent variables. We present a simulation study for performance comparison, and apply our method to the two real data examples.