The Global Navigation Satellite System (GNSS) is currently used in many fields, such as autonomous driving, robotics application, and Unmanned Aerial Vehicles (UAVs), where accurate position information is required. These applications require high positioning accuracy which, in turn, require precise analysis of the residual noise characteristics of the GNSS positioning solutions and their quantitative models. This paper investigates the Generalized Method of Wavelet Moments (GMWM) method for stochastic modelling of low-cost GNSS receiver signal. The paper also compares the results of GMWM to the Allan Variance (AV) which is currently the most common method to study the stochastic characteristics of different time series. Different datasets were collected using two low-cost GNSS receivers at different frequencies and were processed in Single Point Positioning (SPP) mode where position errors are expressed in the Local-Level Frame (LLF) of reference. Both techniques were used in identifying and characterizing the different latent stochastic process and their related coefficients for GNSS position residual signals where precise models of the latter have been built. The test results showed that for low-cost GNSS receivers, a white noise process alone is not sufficient for accurate position residual signals' modeling. The results also stressed out that the GNSS error signal models are complicated where the corresponding error model structures were represented as a sum of white noise and one or more 1st order Gauss-Markov (GM) processes which indicates the existence of short and relatively long correlation between consecutive observations, especially for observations collected at higher sampling rates. Moreover, the results showed that the GMWM approach in general outperforms the AV method in terms of correlated noise identification and characterization.