英文文献：Bootstrap Determination of the Co-integration Rank in Heteroskedastic VAR Models-自举法确定异方差模型的协整等级
英文文献作者：Giuseppe Cavaliere,Anders Rahbek,A.M.Robert Taylor
英文文献摘要：
In a recent paper Cavaliere et al. (2012) develop bootstrap implementations of the (pseudo-) likelihood ratio [PLR] co-integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying VAR model which obtain under the reduced rank null hypothesis. They propose methods based on an i.i.d. bootstrap re-sampling scheme and establish the validity of their proposed bootstrap procedures in the context of a co-integrated VAR model with i.i.d. innovations. In this paper we investigate the properties of their bootstrap procedures, together with analogous procedures based on a wild bootstrap re-sampling scheme, when time-varying behaviour is present in either the conditional or unconditional variance of the innovations. We show that the bootstrap PLR tests are asymptotically correctly sized and, moreover, that the probability that the associated bootstrap sequential procedures select a rank smaller than the true rank converges to zero. This result is shown to hold for both the i.i.d. and wild bootstrap variants under conditional heteroskedasticity but only for the latter under unconditional heteroskedasticity. Monte Carlo evidence is reported which suggests that the bootstrap approach of Cavaliere et al. (2012) signi?cantly improves upon the ?nite sample performance of corresponding procedures based on either the asymptotic PLR test or an alternative bootstrap method (where the short run dynamics in the VAR model are estimated unrestrictedly) for a variety of conditionally and unconditionally heteroskedastic innovation processes.

在最近的一篇论文中，Cavaliere等人(2012)开发了Johansen(1996)的(伪)似然比(PLR)协整秩检验和相关顺序秩确定程序的自举实现。bootstrap样本是利用在降低秩零假设下得到的潜在VAR模型的限制性参数估计来构造的。他们提出了基于i.i.d. bootstrap再抽样方案的方法，并在与i.i.d.创新进行协整VAR模型的背景下，建立了他们提出的bootstrap程序的有效性。本文研究了它们的自举程序，以及基于无约束自举重抽样方案的类似程序在创新点的条件方差或无条件方差存在时变行为时的性质。我们证明了自举PLR测试的大小是渐近正确的，而且，相关的自举序列过程选择一个小于真实秩的秩的概率收敛到零。这个结果对条件异方差下的i - id和野自举变量都成立，但只对后者在无条件异方差下成立。据报道，蒙特卡罗证据表明，Cavaliere等人(2012)的bootstrap方法基于渐进PLR检验或另一种bootstrap方法(VAR模型中的短期动态是不受限制估计的)，对于各种有条件和无条件的异方差创新过程，cantly改进了相应程序的nite样本性能。