An adjusted parameter estimation for spatial regression with spatial confounding

Spatial regression models are often used to analyze the ecological and environmental data sets over a continuous spatial support. Issues of collinearity among covariates have been widely discussed in modeling, but only rarely in discussing the relationship between covariates and unobserved spatial random processes. Past researches have shown that ignoring this relationship (or, spatial confounding) would have significant influences on the estimation of regression parameters. To overcome this problem, an idea of restricted spatial regression is used to ensure that the unobserved spatial random process is orthogonal to covariates, but the related inferences are mainly based on Bayesian frameworks. In this paper, an adjusted generalized least squares estimation method is proposed to estimate regression coefficients, resulting in estimators that perform better than conventional methods. Under the frequentist framework, statistical inferences of the proposed methodology are justified both in theories and via simulation studies. Finally, an application of a water acidity data set in the Blue Ridge region of the eastern U.S. is presented for illustration.