Beyond Stationarity: Statistical Inference for Nonstationary Processes

研究者开发了分析非平稳时间序列及其性质的新方法。时间序列的许多方法都是在观测值是平稳的前提下发展起来的。这一假设简化了估计过程和渐近分析。然而，在现实生活中，这种假设往往是相当不现实的。忽略数据中的非平稳性，并将观察结果视为平稳的，可能会得出误导性的结论。因此，开发处理时间或空间非平稳数据的方法是很重要的。在应用中，非平稳性主要集中在三个方面：(1)时变ARCH型过程的统计推断；(2)非平稳随机相关(随机)系数回归模型；(3)空间非平稳时空模型分析。以下是这些内容的摘要。研究人员开发了测试或跟踪时变ARCH和GARCH过程中结构变化的方法。为了发展所提出方法的抽样性质，需要对时变的ARCH型过程进行混合，研究者研究了这类过程的混合性质。随机相关系数回归(RCCR)模型常被用来解释数据中的非平稳性。尽管有其优点，但直到最近，对RCCR模型的统计分析还相当有限。研究人员为RCCR模型开发了统计上可靠的和计算上有效的参数估计方法。来自时空过程的观测可能经常出现在几个学科中，几个因素可能导致观测来自空间非平稳过程。研究人员研究空间非平稳的时空过程。特别是，研究人员考虑了将模型的估计分解为全局空间平稳过程和附加的局部非平稳项的方法。在几个学科中，假设随时间观察到的数据的主要特征(通常称为时间序列)，例如波动性，不受时间的影响。这种时间不变性被称为平稳性，这通常是当前许多统计方法的基本假设，因为平稳性通常可以简化分析。然而，忽略非平稳性的统计方法可能会导致误导或错误的结论。有几个真实的数据例子表明，有经验证据表明，平稳性是过于简单化的。一个特别相关的例子是全球气温异常，有大量证据表明，平均气温和变化在过去150年里都发生了变化。在这个项目中，我们开发了非平稳时间序列的统计方法，特别是识别发生变化的位置和引起变化的因素。通过开发不忽略非平稳性的方法，我们能够更好地理解驱动数据的机制，从而产生更好的预测。这些方法可以应用于广泛的学科，包括经济学(确定当前信贷紧缩背后的因素)和气候学(测试二氧化碳水平的上升是否对全球气温的变化量产生影响)。

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).