In this paper we re-visit a recent theoretical idea introduced by Phillips and Lee (2015). They examine an empirically relevant situation when multiple time series under consideration exhibit different degrees of non-stationarity. By bridging the asymptotic theory of the local to unity and mildly explosive processes, they construct a Wald test for the commonality of the long-run behavior of two series. Therefore, a vector autoregressive (VAR) setup is natural. However, inference is complicated by the fact that the statistic is degenerate under the null and divergent under the alternative. This is true if the parameters of the data generating process are known and re-normalizing function can be constructed. If the parameters are unknown, as is in practice, the test statistic may be divergent even under the null. We solve this problem by converting the original setting of one vector time series in a panel setting with N individual vector series. We consider asymptotics with fixed N and large T and extend the results to sequential asymptotics when T passes to infinity before N. We show that the Wald test statistic converges to nuisance parameter-free Chi-squared distribution under the null hypothesis.

中文翻译：

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关于混合到统一 VAR 的极限理论：具有弱相关误差的面板设置

在本文中，我们重新审视了 Phillips 和 Lee（2015）最近提出的理论思想。当考虑中的多个时间序列表现出不同程度的非平稳性时，他们检查了经验相关的情况。通过将局部渐近理论与统一和轻度爆炸过程的渐近理论联系起来，他们构建了一个 Wald 检验，用于检验两个系列的长期行为的共性。因此，向量自回归 (VAR) 设置是很自然的。然而，由于统计量在零下是退化的而在替代下是发散的，这一事实使推理变得复杂。如果数据生成过程的参数是已知的并且可以构造重新归一化函数，则这是正确的。如果参数未知，就像在实践中一样，即使在 null 下，测试统计量也可能发散。我们通过在具有 N 个单独向量序列的面板设置中转换一个向量时间序列的原始设置来解决这个问题。我们考虑具有固定 N 和大 T 的渐近线，并在 T 在 N 之前传递到无穷大时将结果扩展到顺序渐近线。我们表明 Wald 检验统计量在零假设下收敛到无干扰参数的卡方分布。