The Asymptotic Distribution of Unit Root Tests of Unstable Autoregressive Processes
The Asymptotic Distribution of Unit Root Tests of Unstable Autoregressive Processes
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不稳定自回归过程单位根检验的渐近分布
DOI：
10.1111/1468-0262.00184
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发表时间：
2001
期刊：
影响因子：
6.1
通讯作者：
B. Nielsen
中科院分区：
文献类型：
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作者：
B. Nielsen
自 Ž 工作以来，单位根测试已经通过大量论文得到发展。迪基和富勒 1979 年。这个想法是检验这样的假设：观察到的时间序列的差异不取决于其水平，或者换句话说，时间序列的水平具有可以通过差分消除的单位根。虽然通常可能有多个单位根，但只有恰好一个单位根的假设才被视为 Ž。这里。因此，可用的测试取决于两个假设： i 时间水平 Ž 。序列只有一个单位根，可以通过差分去除，并且 ii 时间序列的其余特征根是平稳根。在本文中，证明了似然比检验和许多其他基于似然的统计 Ž 。 Ž 。假设 ii 是多余的，而 i 是必要的。它还表明，对于一些不基于似然性的检验，确实有必要假设差异具有固定根。也许从 的含义可以最好地理解结果的后果。条件一.对于二阶或更高阶的自回归模型，在整个参数空间中不满足该条件，并且单位根的似然比检验的渐近分布取决于未知的干扰参数。在这种情况下，检验统计量并不重要；因此测试不相似，这使测试变得复杂。 Ž。对于基于非似然性的检验，条件 ii 的必要性意味着额外的相似性问题。因此，从业者面临着在具有较少相似性问题的基于可能性的测试与可能具有其他有利属性的其他测试之间的权衡。因此，该结果有两个经验意义。首先，在分析具有模数接近 1 的固定根的时间序列时， Ž 。条件 ii 几乎被违反，那么基于可能性的检验是更可取的，并且应谨慎使用其他检验。其次，如果在应用程序中发现爆炸根源，则大多数统计分析实际上是有效的，并且不一定会因为爆炸根源的存在而被忽视。第 2 节介绍了高斯自回归模型及其统计分析 Ž。结果表明条件 ii 对于基于可能性的检验是多余的。还讨论了鞅差创新的稳健性。第 2 节的结果是针对没有确定性趋势的模型给出的。在第 3 节中，这些被推广到具有确定性项的模型。数学证明 Ž.以下两个附录基于 Lai 和 Wei 1983 以及 Chan 和 Ž 的工作。魏1988.
UNIT ROOT TESTING has been developed through numerous papers since the work of Ž. Dickey and Fuller 1979 . The idea is to test the hypothesis that the differences of an observed time series do not depend on its levels, or in other words, the levels of the time series have a unit root that can be removed by differencing. While it is in general possible to have multiple unit roots, only the hypothesis of exactly one unit root is considered Ž. here. The available tests therefore hinge on two assumptions: i the levels of the time Ž. series have exactly one unit root which can be removed by differencing, and ii the remaining characteristic roots of the time series are stationary roots. In this paper it is proved that for the likelihood ratio test and a number of other likelihood based statistics Ž. Ž . the assumption ii is redundant whereas i is necessary. It is also shown that for some tests that are not likelihood based it is indeed necessary to assume that the differences have stationary roots. The consequences of the result are perhaps best understood from the implications of Ž. condition i . For autoregressive models of order two or higher, that condition is not satisfied in the entire parameter space and the asymptotic distribution of the likelihood ratio test for a unit root depends on unknown nuisance parameters. In this situation the test statistic is not pivotal; hence the test is not similar, and this complicates the testing. Ž. For non-likelihood based tests the necessity of condition ii implies an additional similarity problem. The practitioner is therefore faced with a trade off between likelihood based tests with fewer similarity problems and other tests that may have other advantageous properties. There are thus two empirical implications of the result. First, when analyzing time series with stationary roots that have modulus close to one so that Ž. condition ii is nearly violated, then the likelihood based tests are preferable and other tests should be used cautiously. Secondly, if explosive roots are found in an application, most of the statistical analysis is actually valid and should not necessarily be disregarded because of the presence of explosive roots. Section 2 presents a Gaussian autoregressive model along with its statistical analysis Ž. and the result showing that condition ii is redundant for likelihood based tests. Robustness with respect to innovations that are martingale difference is also discussed. The results of Section 2 are given for a model without deterministic trends. In Section 3 these are generalized to models with deterministic terms. The mathematical proofs Ž. following in two Appendices are based on the work of Lai and Wei 1983 and Chan and Ž. Wei 1988 .