Could regressing a stationary series on a non-stationary series obtain meaningful outcomes?

We have read many papers in the literature and found that some papers report results of regressing a stationary time series on a non-stationary time series (we call it the IOI1 model). However, very few studies, if there are any, examine the IOI1 model and the robustness of inference in such settings remains an open question. To bridge the gap in the literature, in this paper, we investigate whether regressing a stationary time series, Yt, on a non-stationary time series, Xt (that is, Yt = α+βXt +ut) could get any meaningful result. To do so, we first conduct a simulation and find regressing a stationary time series on a non-stationary time series could be spurious. Thereafter, we develop the estimation and testing theory for the I0I1 model and find that the statistics Tβ N for testing Hβ 0 : β = β0 versus Hβ 1 : β ̸= β0 from the traditional regression model (we call it IOI0 model) does not have any asymptote distribution with E(Tβ N) → ∞ and V ar(Tβ N) → ∞ as N → ∞, and thus, it cannot be used for the I0I1 model. We have found other interesting results as shown in our paper. Thus, our paper extends the spurious regression literature to cover a previously unexplored case, thereby contributing to a more comprehensive understanding of time series modeling and inference.