It is well known that the sample correlation coefficient between many financial return indices exhibits substantial variation on any reasonable sampling window. This stylised fact contradicts a unit root model for the underlying processes in levels, as the statistic converges in probability to a constant under this modeling scheme. In this paper, we establish asymptotic theory for regression in local stochastic unit root (LSTUR) variables. An empirical application reveals that the new theory explains very well the instability, in both sign and scale, of the sample correlation coefficient between gold, oil, and stock return price indices. In addition, we establish spurious regression theory for LSTUR variables, which generalises the results known hitherto, as well as a theory for balanced regression in this setting.

中文翻译：

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LSTUR回归理论与财务收益指数之间样本相关系数的不稳定性

众所周知，许多财务回报指数之间的样本相关系数在任何合理的采样窗口上都表现出很大的变化。这种风格化的事实在层次上与基础过程的单位根模型相矛盾，因为在此建模方案下统计量收敛到一个常数。在本文中，我们建立了局部随机单位根（LSTUR）变量回归的渐近理论。一项经验应用表明，新理论很好地解释了金，油和股票收益价格指数之间的样本相关系数在符号和规模上的不稳定性。此外，我们建立了LSTUR变量的虚假回归理论，该理论归纳了迄今已知的结果，以及在这种情况下的均衡回归理论。