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An adjusted parameter estimation for spatial regression with spatial confounding

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Abstract

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.

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Acknowledgements

We thank the Editor, an associate editor, and two anonymous referees for their insightful and constructive comments, which have greatly improved the presentation of the article. This work was supported by the Ministry of Science and Technology of Taiwan under Grants MOST 106-2118-M-018-003-MY2 and MOST 106-2811-M-018-005.

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Authors and Affiliations

  1. Department of Mathematics, National Changhua University of Education, Changhua, Taiwan

    Yung-Huei Chiou

  2. Institute of Statistics and Information Science, National Changhua University of Education, Changhua, Taiwan

    Hong-Ding Yang

  3. Graduate Institute of Statistics, National Central University, Taoyuan City, Taiwan

    Chun-Shu Chen

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  1. Yung-Huei Chiou
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  2. Hong-Ding Yang
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  3. Chun-Shu Chen
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Correspondence to Chun-Shu Chen.

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Appendix

Appendix

Proof of Theorem 1

Let \(\varvec{A}=\sigma ^2_{\varepsilon }\varvec{I}\), \(\varvec{U}=\sigma ^2_w(\varvec{I}-\varvec{P}_{\varvec{X}}), \varvec{C}=\varvec{R}_{\varvec{W}}(\phi _w,\nu _w)\), and \(\varvec{V}=(\varvec{I}-\varvec{P}_{\varvec{X}})'\), then \(\varvec{\Phi }=\varvec{A}+\varvec{U}\varvec{C}\varvec{V}\). Applying the Sherman–Morrison–Woodbury formula (see, e.g., Harville 1997), we have \(\varvec{\Phi }^{-1}=\varvec{A}^{-1}-\varvec{A}^{-1}\varvec{U}\left( \varvec{C}^{-1} +\varvec{V}\varvec{A}^{-1}\varvec{U}\right) ^{-1}\varvec{V}\varvec{A}^{-1}\). Using the fact \(\varvec{X}^{\prime }(\varvec{I}-\varvec{P}_{\varvec{X}})=\varvec{0}\) together with the Sherman–Morrison–Woodbury formula of \(\varvec{\Phi }^{-1}\), we obtain

$$\begin{aligned}&\varvec{X}^{\prime }\varvec{\Phi }^{-1} =\varvec{X}^{\prime }\left\{ \frac{1}{\sigma ^2_{\varepsilon }}\varvec{I} - \frac{\sigma ^2_{w}}{\sigma ^2_{\varepsilon }}(\varvec{I}-\varvec{P}_{\varvec{X}})\right. \\&\qquad \left. \left( \varvec{R}_{\varvec{W}}^{-1}+\frac{\sigma ^2_w}{\sigma ^2_{\varepsilon }} (\varvec{I}-\varvec{P}_{\varvec{X}})'(\varvec{I}-\varvec{P}_{\varvec{X}})\right) ^{-1}\right. \\&\qquad \left. (\varvec{I}-\varvec{P}_{\varvec{X}})' \frac{1}{\sigma ^2_{\varepsilon }} \varvec{I}\right\} \\&\quad = \frac{1}{\sigma ^2_{\varepsilon }}\varvec{X}^{\prime }. \end{aligned}$$

It implies that

$$\begin{aligned} \hat{\varvec{\beta }}_{RSR}=\left( \varvec{X}^{\prime }\varvec{\Phi }^{-1}\varvec{X}\right) ^{-1} \varvec{X}^{\prime }\varvec{\Phi }^{-1}\varvec{Y} =\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{Y}. \end{aligned}$$

This completes the proof.

Proof of Theorem 2

From Theorem 1, we have \(\hat{\varvec{\beta }}_{RSR}=(\varvec{X}^{\prime }\varvec{X})^{-1}\varvec{X}^{\prime }\varvec{Y}\). Therefore,

$$\begin{aligned}&E\left[ \hat{\varvec{\beta }}_{RSR}-\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1} \varvec{X}^{\prime }\varvec{W}\right] \\&\quad =E\left[ \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{Y}-\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{W}\right] \\&\quad =E\left[ \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\left( \varvec{X} \varvec{\beta }_{SR}+\varvec{W}+\varvec{\varepsilon }\right) -\left( \varvec{X}^{\prime } \varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{W}\right] \\&\quad =E\left[ \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{X} \varvec{\beta }_{SR}+\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime } \varvec{\varepsilon }\right] \\&\quad =\varvec{\beta }_{SR}+E\left[ \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1} \varvec{X}^{\prime }\right] E\left[ \varvec{\varepsilon }\right] \\&\quad =\varvec{\beta }_{SR}, \end{aligned}$$

where the fourth equality follows from the measurement errors \(\varvec{\varepsilon }\) are independent of \(\varvec{X}\). This completes the proof.

Proof of Theorem 3

Let \(\varvec{A}=\left( \varvec{X}^{\prime }\varvec{\Phi }^{-1}\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{\Phi }^{-1}\), we have

$$\begin{aligned} \hat{\varvec{\beta }}_{Adj}=\varvec{A}\varvec{Y}-\rho \frac{\sigma _w}{\sigma _x}\varvec{M}_{adj}(\varvec{X}^{\prime }\varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}} \varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p. \end{aligned}$$

It means that \(\hat{\varvec{\beta }}_{Adj}\) is a linear combination of \(\varvec{Y}\). Thus, the sampling distribution of \(\hat{\varvec{\beta }}_{Adj}\) given the covariates \(\varvec{X}\) is distributed as Gaussian with mean vector and covariance matrix being

$$\begin{aligned}&E\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] =\left( \varvec{X}^{\prime } \varvec{\Phi }^{-1}\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{\Phi }^{-1}E[\varvec{Y}|\varvec{X}]\\&\qquad -\rho \frac{\sigma _w}{\sigma _x}\varvec{M}_{adj}(\varvec{X}^{\prime } \varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\\&\quad =\left( \varvec{X}^{\prime }\varvec{\Phi }^{-1}\varvec{X}\right) ^{-1}\varvec{X}^{\prime } \varvec{\Phi }^{-1}\\&\qquad \left( \varvec{X}\varvec{\beta }_{RSR}+ \rho \frac{\sigma _w}{\sigma _x}(\varvec{I}-\varvec{P}_{\varvec{X}}) \varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\right) \\&\qquad -\rho \frac{\sigma _w}{\sigma _x}\varvec{M}_{adj}(\varvec{X}^{\prime } \varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\\&\quad =\varvec{\beta }_{RSR}+\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\rho \frac{\sigma _w}{\sigma _x}(\varvec{I}-\varvec{P}_{\varvec{X}}) \varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\\&\qquad -\rho \frac{\sigma _w}{\sigma _x}\varvec{M}_{adj}(\varvec{X} '\varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\\&\quad =\varvec{\beta }_{RSR}-\rho \frac{\sigma _w}{\sigma _x} \varvec{M}_{adj}(\varvec{X}^{\prime }\varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}} \varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p \end{aligned}$$

and

$$\begin{aligned}&Var\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] \\&\quad =Var\left[ \varvec{A} \varvec{Y}-\rho \frac{\sigma _w}{\sigma _x}\varvec{M}_{adj} (\varvec{X}^{\prime }\varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}} \varvec{x}_p\big |\varvec{X}\right] \\&\quad =\varvec{A}Var[\varvec{Y}|\varvec{X}]\varvec{A}'\\&\quad =\varvec{A}\left[ \sigma ^2_{w}(1-\rho ^2)(\varvec{I}-\varvec{P}_{\varvec{X}}) \varvec{R}_{\varvec{W}}(\varvec{I}-\varvec{P}_{\varvec{X}})'+\sigma ^2_{\varepsilon } \varvec{I}\right] \varvec{A}'\\&\quad =\sigma ^2_{w}(1-\rho ^2)\varvec{A}\left( \varvec{I}-\varvec{P}_{\varvec{X}}\right) \varvec{R}_{\varvec{W}}\left[ \varvec{A}\left( \varvec{I}-\varvec{P}_{\varvec{X}}\right) \right] ' +\sigma ^2_{\varepsilon }\varvec{A}\varvec{A}'\\&\quad =\sigma ^2_{\varepsilon }\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}, \end{aligned}$$

respectively. In \(E\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] \), the third equality follows from \(\varvec{X}^{\prime }\varvec{\Phi }^{-1}= \frac{1}{\sigma ^2_{\varepsilon }}\varvec{X}^{\prime }\) of Theorem 1 and the fourth equality follows from \(\varvec{X}^{\prime }(\varvec{I}-\varvec{P}_{\varvec{X}})=\varvec{0}\). In \(Var\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] \), the last equality follows from \(\varvec{A}=\left( \varvec{X}^{\prime }\varvec{\Phi }^{-1}\varvec{X}\right) ^{-1}\varvec{X}^{\prime } \varvec{\Phi }^{-1}=(\varvec{X}^{\prime }\varvec{X})^{-1}\varvec{X}^{\prime }\) and \(\varvec{A}\left( \varvec{I}-\varvec{P}_{\varvec{X}}\right) =(\varvec{X}^{\prime }\varvec{X})^{-1} \varvec{X}^{\prime }\left( \varvec{I}-\varvec{P}_{\varvec{X}}\right) =\varvec{0}\) . This completes the proof.

Proof of Theorem 4

From Theorem 3 and (12), we have

$$\begin{aligned} E\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right]& = \varvec{\beta }_{RSR} -\rho \frac{\sigma _w}{\sigma _x}\varvec{M}_{adj}(\varvec{X}^{\prime } \varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p \end{aligned}$$

and

$$\begin{aligned} \varvec{\beta }_{RSR}=\varvec{\beta }_{SR}+(\varvec{X}^{\prime }\varvec{X})^{-1}\varvec{X}^{\prime }\varvec{W}. \end{aligned}$$

Because \(\varvec{W}\) in the above equation is a random vector, the bias of \(\hat{\varvec{\beta }}_{Adj}\) is given by

$$\begin{aligned}&Bias\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] \\&\quad =E\left\{ E\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] -\varvec{\beta }_{SR}\big |\varvec{X}\right\} \\&\quad =E\left\{ \varvec{\beta }_{RSR}-\rho \frac{\sigma _w}{\sigma _x}\varvec{M}_{adj}(\varvec{X}^{\prime }\varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}} \varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\right. \\&\qquad \qquad \left. -\varvec{\beta }_{SR}\big |\varvec{X}\right\} \\&\quad =E\left\{ \varvec{\beta }_{RSR}-\varvec{\beta }_{SR}\big |\varvec{X}\right\} \\&\qquad -\rho \frac{\sigma _w}{\sigma _x}\varvec{M}_{adj}(\varvec{X}^{\prime } \varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\\&\quad =(\varvec{X}^{\prime }\varvec{X})^{-1}\varvec{X}^{\prime }E\left[ \varvec{W}|\varvec{X}\right] \\&\qquad -\rho \frac{\sigma _w}{\sigma _x}\varvec{M}_{adj}(\varvec{X}^{\prime } \varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\\&\quad =\rho \frac{\sigma _w}{\sigma _x}(\varvec{X}^{\prime }\varvec{X})^{-1} \varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\\&\qquad -\rho \frac{\sigma _w}{\sigma _x}\varvec{M}_{adj}(\varvec{X}^{\prime } \varvec{X})^{-1}\varvec{X}^{\prime }\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\\&\quad =\rho \frac{\sigma _w}{\sigma _x}\left( \varvec{I}- \varvec{M}_{adj}\right) \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime } \varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p, \end{aligned}$$

where the fifth equality follows from (13). This completes the proof.

Proof of Corollary 1

Because \(Bias\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] =\rho \frac{\sigma _w}{\sigma _x}\left( \varvec{I}-\varvec{M}_{adj} \right) \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime } \varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p,\) the desired result for \(\rho =0\) is trivial. Moreover, if \(\varvec{R}_{\varvec{x}}=\varvec{R}_{\varvec{W}},\) we have \(\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}=\varvec{I}\) and thus \(Bias\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] =\rho \frac{\sigma _w}{\sigma _x}\left( \varvec{I}-\varvec{M}_{adj}\right) \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{x}_p.\) Because \(\varvec{I}-\varvec{M}_{adj}=\hbox {diag}(1,\dots ,1,0)\) and \(\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{x}_p=(0,\dots ,0,1),'\) we obtain the desired result. This completes the proof.

Proof of Corollary 2

Since \(MSE\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] =tr\left\{ Bias \left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] Bias^{'}\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] \right\} + tr\left\{ Var\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] \right\}, \) it follows from (18) and (19) that

$$\begin{aligned}&MSE\left[ \hat{\varvec{\beta }}_{Adj}|\varvec{X}\right] =\rho ^2 \frac{\sigma ^2_w}{\sigma ^2_x}tr\\&\qquad \left\{ \left( \left( \varvec{I}-\varvec{M}_{adj}\right) \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime } \varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\right) \right. \\&\qquad \quad \left. \left( \left( \varvec{I}-\varvec{M}_{adj}\right) \left( \varvec{X}^{\prime } \varvec{X}\right) ^{-1}\varvec{X}^{\prime } \varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}}\varvec{x}_p\right) '\right\} \\&\qquad \quad +\sigma ^2_{\varepsilon }tr\left\{ \left( \varvec{X}^{\prime }\varvec{X} \right) ^{-1}\right\} \\&\quad =\rho ^2 \frac{\sigma ^2_w}{\sigma ^2_x}tr\\&\qquad \left\{ \left( \varvec{I}-\varvec{M}_{adj}\right) \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime } \varvec{B}\varvec{X}\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\left( \varvec{I} -\varvec{M}_{adj}\right) \right\} \\&\qquad \quad +\sigma ^2_{\varepsilon }tr\left\{ \left( \varvec{X}^{\prime }\varvec{X}\right) ^ {-1}\right\} \\&\quad =\rho ^2 \frac{\sigma ^2_w}{\sigma ^2_x}tr\left\{ \left( \varvec{I}-\varvec{M}_{adj}\right) \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime } \varvec{B}\varvec{X}\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\right\} \\&\qquad +\sigma ^2_{\varepsilon }tr\left\{ \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\right\} , \end{aligned}$$

where \(\varvec{B}=\varvec{R}^{1/2}_{\varvec{W}}\varvec{R}^{-1/2}_{\varvec{x}} \varvec{x}_p\varvec{x}'_p\varvec{R}^{-1/2'}_{\varvec{x}}\varvec{R}^{1/2'}_{\varvec{W}}.\) This completes the proof.

Proof of\((\varvec{X}^{\prime }\varvec{\Sigma }^{-1}_{\varvec{Y}}\varvec{X})^{-1}=\sigma ^2_{\varepsilon } \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}+\sigma ^2_w\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1} \varvec{X}^{\prime }\varvec{R}_{\varvec{W}}\varvec{X}\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}.\) Let \(\varvec{A}=\sigma ^2_{\varepsilon }\varvec{I},\)\(\varvec{U}=\varvec{V}=\varvec{I},\) and \(\varvec{C}=\sigma ^2_w\varvec{R}_{\varvec{W}},\) then we have \(\varvec{\Sigma _Y}=\varvec{A}+\varvec{U}\varvec{C}\varvec{V}.\) Applying the Sherman–Morrison–Woodbury formula (see, e.g., Harville 1997), it implies that

$$\begin{aligned} \varvec{\Sigma _Y}^{-1}& = \varvec{A}^{-1}-\varvec{A}^{-1}\varvec{U}\left( \varvec{C}^{-1} +\varvec{V}\varvec{A}^{-1}\varvec{U}\right) ^{-1}\varvec{V}\varvec{A}^{-1}\\& = \left( \sigma ^2_{\varepsilon }\varvec{I}\right) ^{-1}-\left( \sigma ^2_{\varepsilon }\varvec{I}\right) ^{-1}\left[ \left( \sigma ^2_w \varvec{R}_{\varvec{W}}\right) ^{-1}\right. \\&\quad\left. +\left( \sigma ^2_{\varepsilon }\varvec{I} \right) ^{-1}\right] ^{-1}\left( \sigma ^2_{\varepsilon }\varvec{I}\right) ^{-1}\\& = \frac{1}{\sigma ^2_{\varepsilon }}\varvec{I}- \frac{1}{\sigma ^4_{\varepsilon }}\left[ \left( \sigma ^2_w\varvec{R}_{\varvec{W}} \right) ^{-1}+\left( \sigma ^2_{\varepsilon }\varvec{I}\right) ^{-1}\right] ^{-1}. \end{aligned}$$

Similarly, let \(\varvec{A}^*= \frac{1}{\sigma ^2_{\varepsilon }}\varvec{X}^{\prime }\varvec{X},\)\(\varvec{U}^*=- \frac{1}{\sigma ^4_{\varepsilon }}\varvec{X}^{\prime },\)\(\varvec{C}^*=\left[ \left( \sigma ^2_w\varvec{R}_{\varvec{W}}\right) ^{-1} +\left( \sigma ^2_{\varepsilon }\varvec{I}\right) ^{-1}\right] ^{-1},\) and \(\varvec{V}^*=\varvec{X},\) then we have \(\varvec{X}^{\prime }\varvec{\Sigma _Y}^{-1}\varvec{X}=\varvec{A}^*+\varvec{U}^*\varvec{C}^*\varvec{V}^*.\) Applying the Sherman-Morrison-Woodbury formula again, it implies that

$$\begin{aligned}&\left( \varvec{X}^{\prime }\varvec{\Sigma _Y}^{-1}\varvec{X}\right) ^{-1}\\&\quad ={\varvec{A}^*}^{-1} -{\varvec{A}^*}^{-1}\varvec{U}^*\left( {\varvec{C}^*}^{-1}+\varvec{V}^*{\varvec{A}^*}^{-1} \varvec{U}^*\right) ^{-1}\varvec{V}^*{\varvec{A}^*}^{-1}\\&\quad =\left( \frac{1}{\sigma ^2_{\varepsilon }}\varvec{X}^{\prime }\varvec{X} \right) ^{-1}-\left( \frac{1}{\sigma ^2_{\varepsilon }} \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\left( - \frac{1}{\sigma ^4_{\varepsilon }}\varvec{X}^{\prime }\right) \\&\qquad \left[ \left( \sigma ^2_w\varvec{R}_{\varvec{W}}\right) ^{-1}+\left( \sigma ^2_{\varepsilon }\varvec{I}\right) ^{-1}+\varvec{X}\left( \frac{1}{\sigma ^2_{\varepsilon }}\varvec{X}^{\prime }\varvec{X}\right) ^{-1}\right. \\&\qquad \left. \left( - \frac{1}{\sigma ^4_{\varepsilon }}\varvec{X}^{\prime }\right) \right] ^{-1} \varvec{X}\left( \frac{1}{\sigma ^2_{\varepsilon }}\varvec{X}^{\prime } \varvec{X}\right) ^{-1}\\&\quad =\sigma ^2_{\varepsilon }\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}+\left( \varvec{X} '\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\\&\qquad \left[ \left( \sigma ^2_w\varvec{R}_{\varvec{W}}\right) ^{-1}+ \frac{1}{\sigma ^2_{\varepsilon }}\varvec{I}\right. \\&\qquad \left. - \frac{1}{\sigma ^2_{\varepsilon }}\varvec{X}\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime } \right] ^{-1}\varvec{X}\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\\&\quad =\sigma ^2_{\varepsilon }\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}+\left( \varvec{X}^{\prime } \varvec{X}\right) ^{-1}\varvec{X}^{\prime }\\&\qquad \left[ \left( \sigma ^2_w\varvec{R}_{\varvec{W}}\right) ^{-1}+ \frac{1}{\sigma ^2_{\varepsilon }}\varvec{I}\right. \\&\qquad \left. - \frac{1}{\sigma ^2_{\varepsilon }} \varvec{X}\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{X}\varvec{X}^{\prime }\left( \varvec{X} \varvec{X}^{\prime }\right) ^{-1}\right] ^{-1}\varvec{X}\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\\&\quad = \sigma ^2_{\varepsilon }\left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}+\sigma ^2_w \left( \varvec{X}^{\prime }\varvec{X}\right) ^{-1}\varvec{X}^{\prime }\varvec{R}_{\varvec{W}}\varvec{X}\left( \varvec{X} '\varvec{X}\right) ^{-1}. \end{aligned}$$

This completes the proof.

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Chiou, YH., Yang, HD. & Chen, CS. An adjusted parameter estimation for spatial regression with spatial confounding. Stoch Environ Res Risk Assess 33, 1535–1551 (2019). https://doi.org/10.1007/s00477-019-01716-9

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