  The Global Navigation Satellite System (GNSS) daily position time series are
often described as the sum of stochastic processes and geophysical signals
which allow studying global and local geodynamical effects such as plate
tectonics, earthquakes, or ground water variations. In this work we propose to
extend the Generalized Method of Wavelet Moments (GMWM) to estimate the
parameters of linear models with correlated residuals. This statistical
inferential framework is applied to GNSS daily position time series data to
jointly estimate functional (geophysical) as well as stochastic noise models.
Our method is called GMWMX, with X standing for eXogeneous variable: it is
semi-parametric, computationally efficient and scalable. Unlike standard
methods such as the widely used Maximum Likelihood Estimator (MLE), our
methodology offers statistical guarantees, such as consistency and asymptotic
normality, without relying on strong parametric assumptions. At the Gaussian
model, our results show that the estimated parameters are similar to the ones
obtained with the MLE. The computational performances of our approach has
important practical implications. Indeed, the estimation of the parameters of
large networks of thousands of GNSS stations quickly becomes computationally
prohibitive. Compared to standard methods, the processing time of the GMWMX is
over $1000$ times faster and allows the estimation of large scale problems
within minutes on a standard computer. We validate the performances of our
method via Monte-Carlo simulations by generating GNSS daily position time
series with missing observations and we consider composite stochastic noise
models including processes presenting long-range dependence such as power-law
or Mat\'ern processes. The advantages of our method are also illustrated using
real time series from GNSS stations located in the Eastern part of the USA.
