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

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Abstract
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% 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 toward 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.