Pitfalls in the use of Time as an Explanatory Variable in Regression

Regression of a trendless random walk on time produces R-squared values around .44 regardless of sample length. The residuals from the regression exhibit only about 14 percent as much variation as the original series even though the underlying process has no functional dependence on time. The autocorrelation structure of these "detrended" random walks is pseudo-cyclical and purely artifactual. Conventional tests for trend are strongly biased towards finding a trend when none is present, and this effect is only partially mitigated by Cochrane-Orcutt correction for autocorrelation. The results are extended to show that pairs of detrended random walks exhibit spurious correlation.