Well, it is a locally weighted regression and does not account for first-order effects very well so, this is not really surprising and inherently one of the limitations of the method. Perhaps if you changed the size of the objective kernel function (bandwidth) this would be mitigated to some degree but, I would not hold my breath. I would point out that the specified distribution and bandwidth can have notable effects on the model. If you are using the ESRI implementation, don't. There is no flexibility in specifying the distributional assumption (eg., Gaussian, Poisson, binomial, multinomial) and you are stuck with the canned regression diagnostics with no access to the objects comprising the model.
Unless you have measurable nonstationarity, GWR is a dubious method and you would be better suited by just using an OLS or GLM. Please take some time to review some of the literature relating to GWR. The Wheeler and Tiefelsdorf (2005) paper demonstrates serious bias in the coefficients due to nonsystematic localized collinearity. Páez et al., (2011) showed, through simulations, that GWR did not reliably discriminate spatially varying process and sometimes exhibited spurious correlations in the local fits. I am not saying don't use GWR but, outside of exploratory analysis, it should be used with great caution and extensively tested for validity and bias.
If 1st order spatial autocorrelation is in fact, an issue affecting residuals and iid, a spatial autoregressive or mixed effects model would be in order. You can use the Lagrange diagnostics (Anselin et al., 1996) to test for spatial dependence in linear models. This will indicate if autocorrelation is influencing the residual error. If you would like to formalize a spatial model, where nonstationarity (2nd order autocorrelation) may be an issue, two methods I would recommend investigating would be principal coordinate analysis of neighbor matrices (Dray et al., 2006) or Eigenvector-based spatial filtering (Griffith 2000).
For inferential models, I particularity like spatial filtering approaches. De Jong et al., (1984) showed an empirical relationship between Eigenvectors and Moran's-I. Building on this concept one can use Eigenvector(s), in a semi-parametric approach, to partial out the effects of spatial process on regression estimates. The concept of PCNM's is to quantify multiscale spatial process, thus directly representing the autocorrelation structure, using scaled-Eigenvector matrices. The reason I cite Dray et al., (2006), for PCNM, is that his extension allows for negative autocorrelation structures. To implement a spatial filtering approach for regression models the function SpatialFiltering in the R library spdep formalizes a brute force method that make model specification quite easy. To formally explore the spatial structure of the data, the R library pcnm provides methods for specifying and visualizing principal coordinate neighbor matrices.
Anselin, L., A.K. Bera, R. Florax, & M.J. Yoon (1996) Simple diagnostic tests for spatial dependence. Regional Science and Urban Economics (26)77–104.
De Jong, P., C. Sprenger, F. van Veen, (1984) On extreme values of Moran’s I and Geary’s c. Geographic Analysis. 16:17-24.
Dray, S., P. Legendre, & P.R. Peres-Neto, (2006) Spatial modeling: a comprehensive framework for principal coordinate analysis of neighbor matrices (PCNM). Ecological Modeling, 196:483-493.
Griffith, D. A., (2000) A linear regression solution to the spatial autocorrelation problem. Journal of Geographical Systems, 2:141-156.
Páez,A., S. Farber, & D. Wheeler (2011). A simulation based study of geographically weighted regression as a method for investigating spatially varying relationships. Environment and Planning 43(12):2992–3010.
Wheeler, D. & M. Tiefelsdorf, (2005). Multicollinearity and correlation among local regression coefficients in geographically weighted regression. Journal of Geographical Systems, 7:161–187.