Residual processes of nonlinear time series models and unit root tests

本研究的第一部分是基于GARCH等非线性时间序列模型及其相关模型构建和研究残差过程。第二部分是研究基于精确极大似然估计的单位根检验。本研究的目的有三个方面。基于非线性时间序列模型构建和研究残差过程。残差过程是由特定模型的残差构造的随机过程。其中一个过程是高矩部分和过程（HMPSP）。将尝试其他非线性时间序列的新结果，如GARCH-in-mean和协整回归模型。另一个将被研究的剩余过程是一个修正的经验分布过程。新的方法是构造基于修正残差的经验分布过程。预期将出版几本关于这个问题的出版物。研究基于精确最大似然的单位根检验。精确MLE具有始终小于1的独特特征，主要用于平稳AR(1)时间序列。在具有GARCH或其他相关结构误差的集成AR(1)模型下，其极限分布有待进一步研究。本文还将研究近集成AR(1)模型的精确MLE的极限分布。单位根检验的实证和理论研究预计将产生若干出版物。使用申请人设计的R包Rmpi实现基于Mackinnon响应面回归技术的并行计算。为了进行统计推断，必须对不同的样本量和分位数进行质量模拟。一个通用的程序将在Rmpi中实现，以便用户可以快速找到P值和分位数，并获得高精度的结果。这将导致一个或两个出版物。

The first part of proposed research is to construct and study residual processes based on nonlinear time series models such as GARCH and related models. The second part is to study unit root tests based on exact maximum likelihood estimation (MLE). The objectives of this research are threefold.1. Construct and study residual processes based on nonlinear time series models. A residual process is a stochastic process constructed from residuals of a specific model. One such process is the high moment partial sum process (HMPSP). New results for other nonlinear time series such as GARCH-in-mean and cointegrated regressions models will be attempted. Another residual process that will be studied is a modified empirical distribution process. The new approach is to construct empirical distribution processes based modified residuals. Several publications on this subject are expected.2. Study unit root tests based on exact MLE. The exact MLE has a unique feature that it is always less than 1 and is mainly used in stationary AR(1) time series. Further study is needed in order to find its limiting distribution under integrated AR(1) models with GARCH or other dependence structure errors. Limit distribution of the exact MLE for nearly integrated AR(1) models will be studied as well. Empirical and theoretical studies of the unit root tests are expected to generate several publications.3. Use R package Rmpi designed by applicant to implement parallel computation based Mackinnon's response surface regression technique. In order to conduct statistical inferences, mass simulation must be computed for different sample sizes and quantiles. A general procedure will be implemented in Rmpi so that users can find out P values and quantiles quickly with highly accurate results. This will lead to one or two publications.