Testing for Multi-Step Causality in Time Series
Testing for Multi-Step Causality in Time Series
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测试时间序列中的多步因果关系
DOI：
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发表时间：
1994
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通讯作者：
Maike Müller
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作者：
H. Lütkepohl;Maike Müller
一个变量是Granger{如果前者中的信息有助于改进后者的预测，则另一个变量是因果关系。通过将变量定义为第二个变量的多阶因果性，如果前者包含用于预测未来后h个周期的有用信息，其中h可以是1以外的某个其他数字，则将该概念扩展为多阶因果性。更准确地说，如果yt中的信息有助于改进j{对某个j=1；2；xt的阶跃预测，则变量yt是另一个变量xt的阶跃因果性。。换言之，如果y t不有助于改善j{对所有j=1；2；，则y t是h{对x t是无因性的。。如果涉及两个以上的变量并且由nite阶向量自回归(VAR)过程产生，则多步无因性意味着对VAR相关矩阵的一组高度非线性的限制。对于这种类型的非线性约束，标准的Wald检验一般不具有极限分布。为此，提出了一种改进的测试方法，克服了这一问题。在模拟研究中考察了小样本容量和检验的威力，并考虑了基于我们的价格指数的说明性例子。我们感谢J·乌尔根·沃尔特斯对本文前一个版本的有益评论。这篇论文的研究是在柏林洪堡大学的Sonderforschungsbereich 373内进行的，并使用德国联邦理工学院提供的资金打印。
A variable is Granger{causal for another variable if the information in the former helps to improve the 1{step ahead predictions of the latter. This concept is extended by deening a variable to be multi{step causal for a second variable if the former contains useful information for predicting the latter h periods into the future where h can be some other number than 1. More precisely, a variable y t is h{step causal for another variable x t if the information in y t helps improving the j{step forecasts of x t for some j = 1; 2;. . .; h. In other words, y t is h{step noncausal for x t if it does not help improving the j{step forecasts for all j = 1; 2;. . .; h. If more than two variables are involved and are generated by a nite order vector autoregressive (VAR) process, multi{ step noncausality implies a set of highly nonlinear restrictions on the VAR coeecient matrices. For this type of nonlinear restrictions standard Wald tests fail to have limiting 2 {distributions in general. Therefore a modiied test is proposed which overcomes this problem. Small sample size and power of the test are investigated in a simulation study and an illustrative example based on our price indices is considered. We thank J urgen Wolters for helpful comments on a previous version of this paper. The research for this paper was carried within Sonderforschungsbereich 373 at the Humboldt University Berlin and was printed using funds made available by the Deutsche Forschungsgemeinschaft.