This paper argues that trending time series can admit valid regression representations even when the dependent variable and the regressors are statistically independent, i.e., in situations that are presently characterized in the literature as “spurious regressions.” Our theory is directed mainly at the two classic examples of regressions of stochastic trends on time polynomials and regressions among independent random walks. But it has more general applicability and, we think, wider implications. Contrary to established wisdom, our theory justiﬁes regressions of this type as valid models for the data. The radical conclusion that emerges from this study is that there are no spurious regressions for trending time series, just alternative valid representations of the limiting dependent variable process in terms of other stochastic processes and deterministic functions of time. We ﬁnd statistical inference in such cases to be valid, not spurious, a conclusion that is in direct contrast to universal thinking about this subject since Yule (1926) ﬁrst wrote about nonsense correlations
Phillips, Peter C.B., "Spurious Regression Unmasked" (1996). Cowles Foundation Discussion Papers. 1383.