Narrow Results

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, $Y_t$, on a non-stationary time series, $X_t$ (that is, $Y_t=\alpha +\beta X_t+u_t$) 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 $T_N^{\beta }$ for testing $H_0^{\beta }:\beta ={\beta }_0$ versus $H_1^{\beta }:\beta \ne {\beta }_0$ from the traditional regression model (we call it $IOI0$ model) does not have any asymptote distribution with $E(T_N^{\beta })\to \infty$ and $\mathrm{\text{Var}}(T_N^{\beta })\to \infty$ as $N\to \infty$, 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.