The expected sample autocorrelation function of residuals from regression of a random walk on time is shown to imply strongly pseudo-periodic behavior in the "detrended" series. The shape of the autocorrelation function is effectively independent of sample size and the corresponding spectral density function has a single peak at a period equal to .83 of sample size; thus the apparent dynamic properties of the residuals are artifactual. Sampling experiments are used to describe the distribution of the spectral peak and further suggest that nonzero autocorrelation in first differences of the raw data will have little effect on the spurious periodicity phenomenon.