Proper analysis and subsequent interpretation of GPS position time series is an important issue in many geodetic and geophysical applications. The GPS position time series can possibly be contaminated by some abrupt changes, called offsets, which can be well compensated for in the functional model. An appropriate offset detection method requires proper specification of both functional and stochastic models of the series. Ignoring colored noise will degrade the performance of the offset detection algorithm. We first introduce the univariate analysis to identify possible offsets in a single time series. To enhance the detection ability, we then introduce the multivariate analysis, which considers the three coordinate components, north, east and up, simultaneously. To test the performance of the proposed algorithm, we use synthetic daily time series of three coordinate components emulating real GPS time series. They consist of a linear trend, seasonal periodic signals, offsets and white plus colored noise. The average detection power on individual components, either north, east or up, are 32.3 and 47.2% for the cases of white noise only and white plus flicker noise, respectively. The detection power of the multivariate analysis increases to 70.8 and 87.1% for the above two cases. This indicates that ignoring flicker noise, existing in the structure of the time series, leads to lower offset detection performance. It also indicates that the multivariate analysis is more efficient than the univariate analysis for offset detection in the sense that the three coordinate component time series are simultaneously used in the offset detection procedure.