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Linear Spatial Dependence Models for Cross-Section Data

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Spatial Econometrics

Part of the book series: SpringerBriefs in Regional Science ((BRIEFSREGION))

Abstract

This chapter gives an overview of all linear spatial econometric models with different combinations of interaction effects that can be considered, as well as the relationships between them. It also provides a detailed overview of the direct and indirect effects estimates that can be derived from these models. In addition, it critically discusses the stationarity conditions that need to be imposed on the spatial interaction parameters and the spatial weights matrix, as well as the row-normalization procedure of the spatial weights matrix. The well-known cross-sectional dataset of Anselin (1988), explaining the crime rate by household income and housing values in 49 Columbus, Ohio neighborhoods, is used for illustration purposes.

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Notes

  1. 1.

    In this book, we use the acronyms most commonly used in the spatial econometrics literature to refer to the model specifications (see e.g., LeSage and Pace 2009).

  2. 2.

    For an explanation of this terminology see Hendry (1995).

  3. 3.

    The superscript T indicates the transpose of a vector or matrix.

  4. 4.

    LeSage and Pace (2009) neither name nor assign an equation number to model (2.5a, b), which reflects the fact that this model is typically not used in applied research.

  5. 5.

    We use the symbol r rather that ω to denote that both symmetric and asymmetric spatial weights matrices are covered here.

  6. 6.

    http://www.regroningen.nl/elhorst and http://community.wvu.edu/~djl041/.

  7. 7.

    Lee (2003) introduces the optimal instrument 2SLS estimator, but Kelejian et al. (2004) show that the 2SLS estimator based on this set of instruments has quite similar small sample properties.

  8. 8.

    This also holds if the spatial autoregressive parameter is negative. The first term that produces feedback effects is δ 2 W 2. This term will always be positive. The second term is δ 3 W 3. Since δ is restricted to the interval (1/r min , 1) and the non-negative elements of W after row-normalisation are smaller than or equal to 1, the diagonal elements of δ 3 W 3 are smaller in absolute value than those of δ 2 W 2. Since the series δ 2 W 2 + δ 3 W 3 + δ 4 W 4 + … alternates in sign if δ is negative, the sum of the diagonal elements of the matrix represented by this series will always be positive.

  9. 9.

    The default value is 1,000, but for models with large N this number might be decreased.

  10. 10.

    One of the constants in the log-likelihood function of the routines of James LeSage is ln(π), while this should be ln(2π). This error is probably based on Anselin’s (1988) textbook, where the same mistake is made. See, e.g., Eqs. (6.15), (8.4), and p. 181. This innocent error has been removed from the SARp, SEMp and SACp routines.

  11. 11.

    The latter tests are called robust because the existence of one type of spatial dependence does not bias the test for the other type of spatial dependence.

  12. 12.

    The coefficient of the spatially autocorrelated error term in the SAC model amounts to 0.166. The corresponding t-value is so low that this coefficient plus two times its standard error also covers the coefficient estimate of the spatially autocorrelated error term in the SEM model of 0.562. The fact that the latter is significant does not change this conclusion.

  13. 13.

    Broader in the sense that it is based on theory, statistics, flexibility and possibilities to parameterize W.

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Authors and Affiliations

  1. Faculty of Economics and Business, University of Groningen, Nettelbosje 2, 9747 AE, Groningen, The Netherlands

    J. Paul Elhorst

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Elhorst, J.P. (2014). Linear Spatial Dependence Models for Cross-Section Data. In: Spatial Econometrics. SpringerBriefs in Regional Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40340-8_2

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  • DOI: https://doi.org/10.1007/978-3-642-40340-8_2

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