We propose a methodology to assess the noise characteristics in time series of position estimates for permanent Global Positioning System (GPS) stations. Least squares variance component estimation (LS-VCE) is adopted to cope with any type of noise in the data. LS-VCE inherently provides the precision of (co)variance estimators. One can also apply statistical hypothesis testing in conjunction with LS-VCE. Using the w-test statistic, a combination of white noise and flicker noise turns out in general to best characterize the noise in all three position components. An interpretation for the colored noise of the series is given. Unmodelled periodic effects in the data will be captured by a set of harmonic functions for which we rely on the least squares harmonic estimation (LS-HE) method and parameter significance testing developed in the same framework as LS-VCE. Having included harmonic functions into the model, practically only white noise can be shown to remain in the data. Remaining time correlation, present only at very high frequencies (spanning a few days only), is expressed as a first-order autoregressive noise process. It can be caused by common and well-known sources of errors like atmospheric effects as well as satellite orbit errors. The autoregressive noise should be included in the stochastic model to avoid the overestimation (upward bias) of power law noise. The results confirm the presence of annual and semiannual signals in the series. We observed also significant periodic patterns with periods of 350 days and its fractions 350/n, n = 2, …, 8 that resemble the repeat time of the GPS constellation. Neglecting these harmonic signals in the functional model can seriously overestimate the rate uncertainty.