Could the Correlation of A Stationary Series With A Non-Stationary Series Obtain Meaningful Outcomes?

While many studies report correlations between a stationary time series [Formula: see text] and a non-stationary time series [Formula: see text], it is still an open problem whether traditional correlation tests are appropriate for assessing the significance of such relationships. To address this gap, we first hypothesize that applying standard regression-based correlation tests in this context may yield spurious or non-informative results. Furthermore, we conjecture that standard correlation statistics are not suitable for evaluating such relationships, and the appropriate test statistic differs from that used for stationary or jointly random series. We first validate our conjectures through simulation studies. In our experiments, [Formula: see text] follows a stationary AR(1) process with parameter [Formula: see text], while [Formula: see text] follows a random-walk model. The empirical rejection rate exceeds the nominal 5% level, increasing from 7.86% when [Formula: see text] to around 62.3% when [Formula: see text]. Our findings support our claims about the spurious nature of the correlation and the inadequacy of standard tests in this setting. Thereafter, we develop the estimation and testing theory for the correlation between a stationary [Formula: see text] and a non-stationary [Formula: see text]. We have proved that the standard correlation statistic cannot be used in this setting and that the resulting test statistic differs from the one used to test the correlation between two random series [Formula: see text] and [Formula: see text], concluding that the traditional correlation test cannot be used to test for the correlation between a stationary time series [Formula: see text] and a non-stationary time series [Formula: see text].