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

Abstract

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 Y_t and X_t are the sum of two variables, say $Y_t=Y_{1,t}+Y_{2,t}$ and $X_t=X_{1,t}+X_{2,t}$, in which $X_{2,t}$ and $Y_{2,t}$ are independent and (nearly) nonstationary AR(1) time series such that $X_{2,t}={\alpha }_1{X}_{2,t-1}+{\epsilon }_t$ and $Y_{2,t}={\alpha }_2{Y}_{2,t-1}+e_t$. 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.

Keywords:

• Cointegration
• stationarity
• non-stationarity

AMS 2000 Subject Classifications:

• C01
• C15
• C22
• C58
• C60

Acknowledgments

The authors are grateful to the editor-in-chief, Professor Narayanaswamy Balakrishnan, and anonymous referees for substantive comments that have significantly improved this manuscript. The fourth author thank to Robert B. Miller and Howard E. Thompson for their continuous guidance and encouragement.

Additional information

Funding

This research has been supported by Xi’an Jiaotong University, University of Canberra, Asia University, China Medical University Hospital, The Hang Seng University of Hong Kong, Research Grants Council (RGC) of Hong Kong (project number 12500915), and Ministry of Science and Technology (MOST, Project Numbers 106-2410-H-468-002 and 107-2410-H-468-002-MY3), Taiwan.