Beyond Stationarity: Statistical Inference for Nonstationary Processes

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

The investigator develops new methods for analysing nonstationary time series and their properties. Many methods in time series are developed under the premise that the observations are stationary. This assumption simplifies both the estimation procedure and asymptotic analysis. However, in real life this assumption is often quite unrealistic. Ignoring nonstationarity in the data and treating the observations as if they were stationary, could give misleading conclusions. Therefore it is important to develop methods for dealing with data that is either temporally or spatially nonstationary. The investigator focuses on three areas where, in applications, nonstationarity can arise (i) statistical inference for time-varying ARCH-type processes (ii) nonstationary random correlated (stochastic) coefficient regression models (iii) analysis of spatially nonstationary spatio-temporal models. These are summarised below. The investigator develops methods which test or track structural changes in time-varying ARCH and GARCH processes. In order to develop sampling properties for the proposed methods, mixing of the time-varying ARCH-type processes is required, and the investigator studies the mixing properties of such processes. Random correlated coefficient regression (RCCR) models are often used to explain the nonstationarity seen in the data. Despite its advantages, until recently the statistical analysis of RCCR models has been quite limited. The investigator develops statistical sound and computationally efficient parameter estimation methods for RCCR models. Observations from spatio-temporal processes can arise frequently in several disciplines, and several factors could cause the observations to come from a spatially nonstationary process. The investigator investigates spatially nonstationary, spatio-temporal processes. In particular, the investigator considers methods which decompose estimates of the model into a global spatially stationary process, and an additional locally nonstationary term. In several disciplines, it is assumed that the main character of data observed over time (usually known as a time series), for example volatility, is not influenced by time. This time invariance property is known as stationarity and it is often the underlying assumption in many current statistical methodologies, because stationarity can often simplify the analysis. However, statistical methods which overlook the nonstationarity can lead to misleading or incorrect conclusions. There are several real data examples where there is empirical evidence to suggest that stationarity is an oversimplification. A particularly pertinent example is global temperature anomolies, where there is plenty of evidence to suggest that both the average temperature and the variation have changed over the past 150 years. In this project we develop statistical methods for nonstationary time series, in particular to identify where changes have occured and factors which have caused the changes. By developing methods that do not ignore the nonstationarity, we are better able to understand the mechanisms driving the data, which leads to better forecasts. These methods can be applied a wide range of subjects, including economics (identifying factors behind the current credit crunch) and climatology (test whether the rise in CO2 levels, has an influence on the amount of variation in the global temperatures).