Could significant regression be treated as insignificant: An anomaly in statistics?

Literature has found that regression of independent (nearly) nonstationary time series could be spurious. We incorporate this idea to examine whether significant regression could be treated as insignificant in some situations. To do so, we conjecture that significant regression could appear significant in some cases but it could become insignificant in some other cases. To check whether our conjecture could hold, we set up a model in which both dependent and independent variables Yt and Xt are the sum of two variables, say Yt=Y1,t+Y2,t and Xt=X1,t+X2,t, in which X2,t and Y2,t are independent and (nearly) nonstationary AR(1) time series such that X2,t=α1X2,t−1+εt and Y2,t=α2Y2,t−1+et. Following this model-setup, we design some situations and the algorithm for our simulation to check whether our conjecture could hold. We find that on the one hand, our conjecture could hold that significant regression could appear significant in some cases when α1 and α2 are of different signs. On the other hand, our findings show that our conjecture does not hold and significant regression cannot be treated as insignificant when α1 and α2 are of the same signs. We note that as far as we know, our article is the first article to discover that significant regression can be treated as insignificant in some situations. Thus, the main contribution of our article is that our article is the first article to discover that significant regression can be treated as insignificant in some situations and remains significant in other situations. We believe that our discovery could be an anomaly in statistics. Our findings are useful for academics and practitioners in their data analysis in the way that if they find the regression is insignificant, they should investigate further whether their analysis falls into the problem studied in our article.