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Random effects specifications in eigenvector spatial filtering: a simulation study

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Abstract

Eigenvector spatial filtering (ESF) is becoming a popular way to address spatial dependence. Recently, a random effects specification of ESF (RE-ESF) is receiving considerable attention because of its usefulness for spatial dependence analysis considering spatial confounding. The objective of this study was to analyze theoretical properties of RE-ESF and extend it to overcome some of its disadvantages. We first compare the properties of RE-ESF and ESF with geostatistical and spatial econometric models. There, we suggest two major disadvantages of RE-ESF: it is specific to its selected spatial connectivity structure, and while the current form of RE-ESF eliminates the spatial dependence component confounding with explanatory variables to stabilize the parameter estimation, the elimination can yield biased estimates. RE-ESF is extended to cope with these two problems. A computationally efficient residual maximum likelihood estimation is developed for the extended model. Effectiveness of the extended RE-ESF is examined by a comparative Monte Carlo simulation. The main findings of this simulation are as follows: Our extension successfully reduces errors in parameter estimates; in many cases, parameter estimates of our RE-ESF are more accurate than other ESF models; the elimination of the spatial component confounding with explanatory variables results in biased parameter estimates; efficiency of an accuracy maximization-based conventional ESF is comparable to RE-ESF in many cases.

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Notes

  1. The expectation of MC, E[MC], equals \( - \frac{1}{N - 1} \) when M = I–11′/N, whereas it equals \( - \frac{N}{{{\mathbf{1^{\prime}C1}}}}\frac{{tr[({\mathbf{X^{\prime}X}})^{ - 1} {\mathbf{X^{\prime}CX}}]}}{N - K - 1} \) when M = I–X(X′X) -1 X′. E[MC] < MC, MC < E[MC], and MC = E[MC] imply positive, negative, and no spatial dependence, respectively.

  2. \( {\hat{\mathbf{\beta }}} \) in Eq. (5) is identical to the generalized least squared estimator (Henderson 1975).

  3. Models with known mean structure are widely applied in geostatistics (e.g., Cressie and Wikle 2011).

  4. Although the covariance matrix of ω usually is defined by σ 2NS I + σ 2 γ (I + C), where σ 2NS is a variance parameter, because it can be expanded as follows: σ 2NS I + σ 2 γ (I + C) = (σ 2NS +σ 2γ)I + σ 2 γ C = σ 2 I + σ 2 γ C, the assumption of zero diagonal elements is consistent with the usual assumption.

  5. When M = I–11 ′/N, RE-ESF approximates the geostatistical model with known constant mean.

  6. The asymmetric matrix W has real eigenvalues and eigenvectors (see Griffith 2000).

  7. Expanding the variance of Eγ in Eq. (22) as \( {\text{Var}}[{\mathbf{E\gamma }}] = \frac{{{\mathbf{\gamma^{\prime}E^{\prime}E\gamma }}}}{N} = \frac{{{\mathbf{\gamma^{\prime}\gamma }}}}{N} = \frac{1}{N}\sum\nolimits_{l} {{\text{diag}}[\sigma_{\gamma }^{2} k{\varvec{\Lambda}}(\alpha )]_{l} } = k\frac{{\sigma_{\gamma }^{2} }}{N}\sum\nolimits_{l} {\lambda_{l}^{\alpha } } \), where \( {\text{diag}}[ \cdot ]_{l} \) returns the lth diagonal of the matrix \( \cdot \). This equation shows that Var[Eγ] equals σ 2 γ when \( k = \frac{N}{{\sum\nolimits_{l} {\lambda_{l}^{\alpha } } }} \).

  8. Equation (5) is expanded using X ′ E = 0, which M = I–X(X ′ X) -1 X ′ implies, as \( \left[ {\begin{array}{*{20}c} {{\hat{\mathbf{\beta }}}} \\ {{\hat{\mathbf{\gamma }}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{X^{\prime}X}}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{I}} + \frac{{\sigma_{{}}^{2} }}{{\sigma_{\gamma }^{2} }}{\varvec{\Lambda}}^{ - 1} } \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {{\mathbf{X^{\prime}y}}} \\ {{\mathbf{E^{\prime}y}}} \\ \end{array} } \right] \) \( = \left[ {\begin{array}{*{20}c} {({\mathbf{X^{\prime}X}})^{ - 1} {\mathbf{X^{\prime}y}}} \\ {\left( {{\mathbf{I}} + \frac{{\sigma_{{}}^{2} }}{{\sigma_{\gamma }^{2} }}{\varvec{\Lambda}}^{ - 1} } \right)^{ - 1} {\mathbf{E^{\prime}y}}} \\ \end{array} } \right] \). Thus, \( {\hat{\mathbf{\beta }}} \) equals the estimate of the OLS estimate.

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Acknowledgments

We received helpful comments from Professor Arthur Getis at the 59th Annual North American Meetings of the Regional Science Association International. Also, we received generous support from Professor Morito Tsutsumi, University of Tsukuba, and Professor Yoshiki Yamagata, National Institute for Environmental Studies. We would particularly like to thank them. In addition, we are grateful to anonymous referees for their many suggestive comments. This work was supported by a grant-in-aid from the Japan Society for the Promotion of Science Fellows. In addition, this is an achievement of the Global Climate Risk Management Strategies (S10) project.

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  1. Center for Global Environmental Research, National Institute for Environmental Studies, 16-2 Onogawa, Tsukuba-shi, Ibaraki, 305-8506, Japan

    Daisuke Murakami

  2. School of Economic, Political and Policy Sciences, The University of Texas at Dallas, Richardson, Dallas, TX, 75083, USA

    Daniel A. Griffith

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  1. Daisuke Murakami
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Correspondence to Daisuke Murakami.

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Murakami, D., Griffith, D.A. Random effects specifications in eigenvector spatial filtering: a simulation study. J Geogr Syst 17, 311–331 (2015). https://doi.org/10.1007/s10109-015-0213-7

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