STATISTICAL THEORY FOR HIGHER ORDER NONSTATIONARY INTEGRATED AND COINTEGRATED PROCESSES

目前的研究已经持续了两年多。我首先描述了每年获得的结果，然后给出了进一步研究的展望。1.1993年我分析了一个时间序列回归模型，其中回归变量是高度非平稳的，遵循集成过程。特别地，研究了积分阶数大于1的过程所产生的统计量，找到并计算了它们的极限分布。在数值计算方面，很难处理这样的高阶积分过程，但我已经设计了一种方法来克服这种困难。另一方面，我发现积分阶为1的情况在分布论中是相当特殊的，可以说对应于一个奇异点。2.1994检验高度非平稳变量之间是否存在协整的一种新的检验方法。传统的检验以无协整为零，而建议的检验则以协整为零。我不仅得到了极限零点，而且还得到了局部备选序列下的极限分布。还计算了相应的百分比。根据以上和其他结果，我可以在项目期限内写一份手稿，很快就会出版。为了进一步研究，我想提出一种方法，使我们能够处理比本项目中分析的统计数据更复杂的统计数据。要做到这一点确实很困难，就像在标准情况下一样，但我会努力找到这个扩展问题的线索。

The present research has extended over two years. I first describe the results obtained each year and then give perspectives for further research.1.1993I analyzed a time series regression model, where the regressors are highly nonstationary and follow integrated processes. In particular statistics arising from processes of integration order greater than 1 are studied, and their limiting distributions are found and computed. In terms of numerical computation it is hard to deal with such higher order integrated processes, but I have devised a method for overcoming such difficulties. On the other hand I have found that the case of integration order of 1 is quite exceptional in terms of distribution theory, which corresponds, so to speak, to a singular point.2.1994A new testing procedure for testing if there exists cointegration among highly nonstationary variables. While conventional tests take no cointegration as the null, the suggested test takes cointegration as the null. I have derived, not only the limiting null, but also the limiting distributions under a sequence of local alternatives. The corresponding percent points are also computed.3. On the basis of the above and other results I could write a manuscript during the term of project, which is to be published shortly. For further research I would like to suggest a method which enables us to deal with more complicated statistics than those analyzed in the present project. It is certaily difficult to do so as in the standard case, but I will try to find a clue to this extended problem.