Abstract
Data collected over a geographical space may exhibit some sort of dependence in the sense that closer observations are more alike than those far apart. Such behavior is modeled by including a covariance structure into the classical statistical models. In particular, spatial regression models which accommodate various types of spatial dependencies have been increasingly applied in epidemiology, geology, disease surveillance, urban planning, analysis and mapping of poverty indicators and others. An important type of spatial regression models is the Spatial Moving Average (SMA, which imposes a moving average specification on the noise term, as is the case in temporal time series regressions. In this paper we consider the SMA models and propose efficient estimators of their regression coefficients by using shrinkage and penalty approaches. We provide analytical and numerical analysis to illustrate the superiority of the proposed estimators over the classical MLE estimators. Additionally, we apply the new methodology to the Baltimore housing sale prices data.
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Appendix
Appendix
Proof of Theorem 2:
-
(i)
From Theorem 1, we have:
$$\begin{aligned} \sqrt{n} (\hat{\varvec{\beta }}-\varvec{\beta })=\sqrt{n} \left( \left( \begin{array}{cc}\hat{\varvec{\beta }}_1 \\ \hat{\varvec{\beta }}_2 \end{array}\right) - \left( \begin{array}{cc}\varvec{\beta }_1 \\ \varvec{\beta }_2 \end{array} \right) \right) \overset{D}{\longrightarrow }N_p\left( \left( \begin{array}{cc}\varvec{0}_{p_1} \\ \varvec{0}_{p_2}\end{array}\right) ,\sigma ^2\varvec{V}^{-1}\right) , \end{aligned}$$where \(\varvec{V}^{-1}\) as in (18). Therefore,
$$\begin{aligned} \varvec{T}^{(1)}_n= \sqrt{n}(\hat{\varvec{\beta }}_1-\varvec{\beta }_1)\overset{D}{\longrightarrow }\varvec{T}^{(1)}\sim N_{p_1}(\varvec{0}_{p_1},\sigma ^2\varvec{D}), \end{aligned}$$where \(\varvec{D}=\varvec{V}_{11.2}^{-1}\).
-
(ii)
Note that
$$\begin{aligned} \hat{\varvec{\beta }}^R_1= & {} \hat{\varvec{\beta }}_1+(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\hat{\varvec{\beta }}_2, \text { so}\\ \varvec{T}^{(2)}_n= & {} \sqrt{n} \Big \{\hat{\varvec{\beta }}_1+(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\hat{\varvec{\beta }}_2-\varvec{\beta }_1\Big \}\\= & {} \varvec{T}^{(1)}_n+(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\sqrt{n} (\hat{\varvec{\beta }}_2-\varvec{\beta }_2)\\+ & {} (\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\sqrt{n} \varvec{\beta }_2. \end{aligned}$$As \(n\longrightarrow \infty \), and by Slutsky’s theorem, we have: \(\varvec{T}^{(2)}_n\overset{D}{\longrightarrow }\varvec{T}^{(2)}\sim N_{p_1}(\varvec{\mu }^{(2)},\varvec{\Sigma }^{(2)})\), where
$$\begin{aligned} \varvec{\mu }^{(2)}= & {} \varvec{V}_{11}^{-1}\varvec{V}_{12}\varvec{\xi }= \varvec{\pi }\\ \varvec{\Sigma }^{(2)}= & {} Var\Big \{\varvec{T}^{(1)}+\varvec{V}_{11}^{-1}\varvec{V}_{12}\varvec{T}^{(12)}\Big \}\\= & {} Var(\varvec{T}^{(1)})+Var(\varvec{V}_{11}^{-1}\varvec{V}_{12}\varvec{T}^{(12)})+Cov(\varvec{T}^{(1)},\varvec{V}_{11}^{-1}\varvec{V}_{12}\varvec{T}^{(12)})\\+ & {} Cov(\varvec{V}_{11}^{-1}\varvec{V}_{12}\varvec{T}^{(12)},\varvec{T}^{(1)})\\= & {} \sigma ^2\Big \{\varvec{V}_{11.2}^{-1}-\varvec{V}_{11}^{-1}\varvec{V}_{12}\varvec{V}_{22}^{-1}\varvec{V}_{21}\varvec{V}_{11.2}^{-1} \Big \}\\= & {} \sigma ^2\Big \{\varvec{D}-\varvec{D}^*\Big \}. \end{aligned}$$ -
(iii)
Also note that:
$$\begin{aligned} \varvec{T}^{(3)}_n= & {} \sqrt{n}(\hat{\varvec{\beta }}_1-\hat{\varvec{\beta }}^R_1)\\= & {} \sqrt{n}\left( - (\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\hat{\varvec{\beta }}_2\right) \\= & {} -\sqrt{n}\left( (\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})(\hat{\varvec{\beta }}_2-\varvec{\beta }_2+\varvec{\beta }_2) \right) \\= & {} -(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\sqrt{n}(\hat{\varvec{\beta }}_2-\varvec{\beta }_2)\\- & {} (\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\sqrt{n}\varvec{\beta }_2. \end{aligned}$$As \(n\longrightarrow \infty \) and using Slutsky’s theorem, we have: \(\varvec{T}^{(3)}_n\overset{D}{\longrightarrow }\varvec{T}^{(3)}\sim N_{p_1}\left( \varvec{\mu }^{(3)},\varvec{\Sigma }^{(3)}\right) \), where
$$\begin{aligned} \varvec{\mu }^{(3)}= & {} \varvec{0} -\varvec{V}_{11}^{-1}\varvec{V}_{12}\varvec{\xi }= -\varvec{\pi }\\ \varvec{\Sigma }^{(3)}= & {} Var(-\varvec{V}_{11}^{-1}\varvec{V}_{12}\varvec{T}^{(12)})\\= & {} \sigma ^2\varvec{V}_{11}^{-1}\varvec{V}_{12}\varvec{V}_{22.1}^{-1}\varvec{V}_{21}\varvec{V}_{11}^{-1}\\= & {} \sigma ^2\varvec{D}^*. \end{aligned}$$ -
(iv)
Note that:
$$\begin{aligned} \varvec{T}^{(2)}_n= & {} \sqrt{n}(\hat{\varvec{\beta }}^R_1-\varvec{\beta }_1)\\= & {} \sqrt{n} \left( \hat{\varvec{\beta }}_1+(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\hat{\varvec{\beta }}_2 -\varvec{\beta }_1 \right) \\= & {} \sqrt{n}\Big ((\hat{\varvec{\beta }}_1-\varvec{\beta }_1)+(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})(\hat{\varvec{\beta }}_2-\varvec{\beta }_2\\+ & {} \varvec{\beta }_2) \Big )\\= & {} \varvec{T}^{(1)}_n+(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\sqrt{n}(\hat{\varvec{\beta }}_2-\varvec{\beta }_2)\\+ & {} (\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\sqrt{n}\varvec{\beta }_2\\= & {} \varvec{T}^{(1)}_n+\varvec{A}_n \varvec{T}^{(12)}_n+\varvec{A}_n \varvec{\xi }, \end{aligned}$$where \(\varvec{A}_n=(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\), and
$$\begin{aligned} \varvec{T}^{(3)}_n= & {} \sqrt{n}(\hat{\varvec{\beta }}_1-\hat{\varvec{\beta }}^R_1)\\= & {} \sqrt{n}\left( -(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})(\hat{\varvec{\beta }}_2-\varvec{\beta }_2+\varvec{\beta }_2) \right) \\= & {} -(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\sqrt{n}(\hat{\varvec{\beta }}_2-\varvec{\beta }_2)\\- & {} (\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{01})^{-1}(\varvec{X}_{01}^\prime {\varvec{\hat{V}}_{\varvec{n}}^{-1}} \varvec{X}_{02})\sqrt{n}\varvec{\beta }_2\\= & {} -\varvec{A}_n\varvec{T}^{(12)}_n-\varvec{A}_n\varvec{\xi }. \end{aligned}$$Therefore,
$$\begin{aligned} \left( \begin{array}{cc} \varvec{T}^{(2)}_n\\ \varvec{T}^{(3)}_n\end{array}\right)= & {} \left( \begin{array}{cc}\varvec{I}_{p_1}\\ \varvec{0}_{p_1}\end{array}\right) \varvec{T}^{(1)}_n+ \left( \begin{array}{cc}\varvec{A}_n\\ -\varvec{A}_n\end{array}\right) \varvec{T}^{(12)}_n+\left( \begin{array}{cc}\varvec{A}_n \\ -\varvec{A}_n\end{array}\right) \varvec{\xi }\\= & {} \varvec{G}_1\varvec{T}^{(1)}_n+ \varvec{G}_{2n}\varvec{T}^{(12)}_n+\varvec{G}_{2n}\varvec{\xi }, \end{aligned}$$which is a linear combination of \(\varvec{T}^{(1)}_n\) and \(\varvec{T}^{(12)}_n\), where \(\varvec{G}_1 = \left( \begin{array}{cc}\varvec{I}_{p1}\\ \varvec{0}_{p_1}\end{array}\right) \), \(\varvec{G}_{2n}=\left( \begin{array}{cc}\varvec{A}_n\\ -\varvec{A}_n\end{array}\right) \), \(\varvec{I}_{p_1}\) is a \(p_1\times p_1\) idintity matrix, and \(\varvec{0}_{p_1}\) is a \(p_1\times p_1\) matrix of zeros. Thus, as \(n\longrightarrow \infty \) and by Slutsky’s theorem, we have: \(\varvec{A}_n\overset{P}{\longrightarrow }\varvec{A} = \varvec{V}_{11}^{-1} \varvec{V}_{12}\), \(\varvec{G}_{2n} \overset{P}{\longrightarrow }\varvec{G}_2 = \left( \begin{array}{cc} \varvec{A} \\ -\varvec{A} \end{array}\right) \). So, \(\left( \begin{array}{cc}\varvec{T}^{(2)}_n\\ \varvec{T}^{(3)}_n\end{array}\right) \overset{D}{\longrightarrow }\left( \begin{array}{cc}\varvec{T}^{(2)}\\ \varvec{T}^{(3)}\end{array}\right) \sim N_{2p_1}\left( \varvec{\mu }^{(4)},\varvec{\Sigma }^{(4)}\right) \), where
$$\begin{aligned} \varvec{\mu }^{(4)}= & {} \varvec{G}_1+\varvec{G}_2 \varvec{0} +\varvec{G}_2 \varvec{\xi }\\= & {} \left( \begin{array}{cc}\varvec{V}_{11}^{-1}\varvec{V}_{12}\varvec{\xi }\\ -\varvec{V}_{11}\varvec{V}_{12}\varvec{\xi }\end{array}\right) =\left( \begin{array}{cc}\varvec{\pi }\\ -\varvec{\pi }\end{array}\right) ,\\ \varvec{\Sigma }^{(4)}= & {} Var(\varvec{G}_1 \varvec{T}^{(1)})+Var(\varvec{G}_2 \varvec{T}^{(12)})+Cov(\varvec{G}_1 \varvec{T}^{(1)}, \varvec{G}_2\varvec{T}^{(12)})\\+ & {} Cov(\varvec{G}_2 \varvec{T}^{(12)}, \varvec{G}_1 \varvec{T}^{(1)}),\\= & {} \sigma ^2\left( \begin{array}{cc} \varvec{D}-\varvec{D}^* &{} \varvec{0} \\ \varvec{0} &{} \varvec{D}^* \end{array}\right) , \end{aligned}$$using the properties of the variance and covariance functions. It is clear that \(\varvec{T}^{(2)}_n\) and \(\varvec{T}^{(3)}_n\) are asymptotically independent.
-
(v)
Using the same technique as in part (iv), we can write \(\left( \begin{array}{cc} \varvec{T}^{(1)}_n\\ \varvec{T}^{(3)}_n\end{array}\right) \) as a linear combination of \(\varvec{T}^{(1)}_n\) and \(\varvec{T}^{(12)}_n\) as below:
$$\begin{aligned} \left( \begin{array}{cc} \varvec{T}^{(1)}_n\\ \varvec{T}^{(3)}_n\end{array}\right)= & {} \left( \begin{array}{cc} \varvec{I}_{p_1} \\ \varvec{0}_{p_1}\end{array}\right) \varvec{T}^{(1)}_n+ \left( \begin{array}{cc} \varvec{0}_{p_1\times p_2} \\ -\varvec{A}_n\end{array}\right) \varvec{T}^{(12)}_n+\left( \begin{array}{cc} \varvec{0}_{p_1\times p_2} \\ -\varvec{A}_n\end{array}\right) \varvec{\xi }\\= & {} \varvec{G}_1\varvec{T}^{(1)}_n+\varvec{F}_n\varvec{T}^{(12)}_n+\varvec{F}_n\varvec{\xi }, \end{aligned}$$where \(\varvec{F}_n =\left( \begin{array}{cc} \varvec{0}_{p_1\times p_2} \\ -\varvec{A}_n \end{array}\right) \), and \(\varvec{0}_{p_1\times p_2}\) is a \(p_1\times p_2\) matrix of zeros. Therefore, as \(n\longrightarrow \infty \) and by Slutsky’s theorem, we have: \(\varvec{F}_n\overset{P}{\longrightarrow }\varvec{F}=\left( \begin{array}{cc} \varvec{0} \\ -\varvec{A}\end{array}\right) \), and hence \(\left( \begin{array}{cc} \varvec{T}^{(1)}_n\\ \varvec{T}^{(3)}_n\end{array}\right) \overset{D}{\longrightarrow }\left( \begin{array}{cc} \varvec{T}^{(1)}\\ \varvec{T}^{(3)}\end{array}\right) \sim N_{2p_1} \left( \varvec{\mu }^{(5)},\varvec{\Sigma }^{(5)}\right) \), where
$$\begin{aligned} \varvec{\mu }^{(5)}= & {} \varvec{G}_1\varvec{0} +\varvec{F}\varvec{0} +\left( \begin{array}{cc}\varvec{0} \\ -\varvec{A} \end{array}\right) \varvec{\xi }= \left( \begin{array}{cc}\varvec{0} \\ -\varvec{A} \varvec{\xi }\end{array}\right) =\left( \begin{array}{cc}\varvec{0} \\ -\varvec{\pi }\end{array}\right) ,\\ \varvec{\Sigma }^{(5)}= & {} Var(G_1\varvec{T}^{(1)})+Var(\varvec{F} \varvec{T}^{(12)})+Cov(\varvec{G}_1\varvec{T}^{(1)},\varvec{F}\varvec{T}^{(12)})\\+ & {} Cov(\varvec{F} \varvec{T}^{(12)},\varvec{G}_1\varvec{T}^{(1)})\\= & {} \sigma ^2\left( \begin{array}{cc} \varvec{D}&{} \varvec{D}^*\\ \varvec{D}^* &{} \varvec{D}^* \end{array}\right) , \end{aligned}$$using the same procedure as in part (iv).
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Al-Momani, M., Ejaz Ahmed, S., Hussein, A.A. (2020). Efficient Estimation Strategies for Spatial Moving Average Model. In: Xu, J., Ahmed, S., Cooke, F., Duca, G. (eds) Proceedings of the Thirteenth International Conference on Management Science and Engineering Management. ICMSEM 2019. Advances in Intelligent Systems and Computing, vol 1001. Springer, Cham. https://doi.org/10.1007/978-3-030-21248-3_38
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