## Beyond Whittle's likelihood - new Bayesian semiparametric approaches to time series analysis

#### Project Team Members

Claudia Kirch

Renate Meyer (University of Auckland)

Alexander Meier, First Phase

Patricio M. Russel (University of Auckland), Second Phase

##### Cooperation with

Matthew Edwards (University of Edinburgh)

First Phase: 01/2016 - 12/2018 (DFG grant: KI 1443/3-1)

Second Phase: 01/2019-12/2021 (DFG grant: KI 1443/3-2)

#### Project Description

Even though nonparametric Bayesian inference has been a rapidly growing topic over the last decade, only very few nonparametric Bayesian approaches to time series analysis have been developed. The main challenge lies in the necessity to specify a likelihood function for Bayesian statistical inference.Several authors solved this problem by using Whittle's likelihood as an approximation for Bayesian modeling of the spectral density as the main nonparametric characteristic of stationary time series. Even for non-Gaussian stationary time series, which are not completely specified by their first and second-order structure, the Whittle likelihood results in asymptotically correct statistical inference in many situations but often at the cost of a loss of efficiency. Parametric models, on the other hand, are more powerful but fail if the model is misspecified. Modern nonparametric bootstrap methods for time series face similar challenges and implicitly use non- or semiparametric approximations of the true likelihood. In this project, we will take advantage of state-of-the-art developments in the bootstrap realm of time series analysis in several ways: First, we will combine Bayesian parametric time series likelihoods in the time domain with a frequency-domain correction based on nonparametric prior distributions.This yields an entirely new semiparametric approach to Bayesian time series analysis. Furthermore, multivariate extensions based on Whittle's likelihood and a Gamma process prior are investigated.

Furthermore, we will combine multivariate Bayesian parametric time series likelihoods in the time domain with a frequency-domain correction based on nonparametric prior distributions. We will then use this likelihood approximation to nonparametrically model the error term in an otherwise parametric model such as linear and non-linear regression, change point or unit-root models.

Finally, we propose and analyse a new likelihood approximation for locally stationary time series, i.e. time series with a slowly changing dependency structure so that while not being stationary globally, they can be considered approximately stationary in a certain neighbourhood of every point in time. The approximation is based on moving local Fourier coefficients which have similar statistical properties globally as the global Fourier coefficients in the stationary case.