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。许多时间序列的方法都是在观测平稳的前提下发展起来的。这个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。忽略数据的非平稳性，把观测结果当作平稳的来处理，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。研究者专注于三个领域，在应用中，非平稳性可能出现(i)时变arch型过程的统计推断（ii）非平稳随机相关（随机）系数回归模型（iii）空间非平稳时空模型的分析。这些问题总结如下。研究者开发了测试或跟踪随时间变化的ARCH和GARCH过程结构变化的方法。为了开发所提出的方法的采样特性，需要混合时变arch型过程，研究者研究了这些过程的混合特性。随机相关系数回归（RCCR）模型经常被用来解释数据中的非平稳性。尽管RCCR模型有其优点，但直到最近，其统计分析仍然相当有限。研究者为RCCR模型开发了统计健全和计算高效的参数估计方法。来自时空过程的观测可以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。研究者研究空间上不稳定的时空过程。特别是，研究者考虑将模型的估计分解为全局空间平稳过程和额外的局部非平稳项的方法。在一些学科中，假定随时间观察到的数据（通常称为时间序列）的主要特征，例如波动性，不受时间的影响。这种时不变性被称为平稳性，它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导致误导性或不正确的结论。有几个真实的数据例子，有经验证据表明，平稳性是一种过度简化。一个特别相关的例子是全球温度异常，有大量证据表明，在过去的150年里，平均温度和变化都发生了变化。在这个项目中，我们开发了非平稳时间序列的统计方法，特别是确定发生变化的位置和导致变化的因素。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，从而导致更好的预测。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温度的变化量有影响）。

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).