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Critical issues in spatial models: error term specifications, additional endogenous variables, pre-testing, and Bayesian analysis

Abstract

This paper is a polished and mildly extended version of the Getis-Ord Lecture I gave at the WRSA conference in Tucson Arizona, February 2015. The force of that lecture was to critically evaluate the literature, and in doing so, suggest alternative directions in our research. In some cases, these suggestions could greatly widen the scope of discussions on a number of important social issues. In sticking close to that lecture, this paper contains only a few references in the text so that it will read as the lecture that was presented. Relevant background references are included in the bibliography.

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Notes

  1. 1.

    Consistency is a large sample property. The formal statement is that consistency relates to \(\overline{VC}\) where

    $$\begin{aligned} \overline{VC}=Lim_{N\rightarrow \infty }(N^{-1}X^{\prime }X)^{-1}\,\, N^{-1}X^{\prime }DX\,\,(N^{-1}X^{\prime }X)^{-1} \end{aligned}$$
  2. 2.

    Formal results would be based on the following. Under further reasonable conditions

    $$\begin{aligned}&N^{1/2}(\hat{\beta }-\beta )\overset{D}{\rightarrow }N(0,V_{XDX}) \\ V_{XDX}= & {} Lim_{N\rightarrow \infty }N(X^{\prime }X)^{-1}N^{-1}(X^{\prime }DX)\, N(X^{\prime }X)^{-1} \end{aligned}$$

    Small sample inferences would be based on the approximation

    $$\begin{aligned} \hat{\beta }\simeq N(\beta ,N^{-1}\hat{V}_{XDX}) \end{aligned}$$

    where

    $$\begin{aligned} \hat{V}_{XDX}=N(X^{\prime }X)^{-1}X^{\prime }\hat{D}X(X^{\prime }X)^{-1} \end{aligned}$$

    and

    $$\begin{aligned} p\lim _{N\rightarrow \infty }\hat{V}_{XDX}=V_{XDX} \end{aligned}$$
  3. 3.

    A HAC procedure is a non-parametric procedure for consistently estimating a VC matrix. The term HAC stands for “Autocorrelated and Heteroskedastic Consistent”. In general terms, this is described below.

  4. 4.

    As an illustration, using evident notation, consider the model

    $$\begin{aligned} y=Xa+\varepsilon \end{aligned}$$

    where \(X\) is an \(N\times 1\) vector of observations on an exogenous variable and \(\varepsilon \) has mean and \(VC\) matrix \((0,\sigma ^{2}I_{N})\). The variance of the least squares estimator \(\hat{a}\), say, \(\sigma _{\hat{a} }^{2}\) is \(\sigma ^{2}/X^{\prime }X\rightarrow 0\) if, as would typically be assumed \(X^{\prime }X\rightarrow \infty \). Thus, \((\sigma _{\hat{a} }^{2})^{-1}\rightarrow \infty .\)

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Correspondence to Harry H. Kelejian.

Additional information

I would like to thank Henk Folmer for numerous suggestions which helped me put a proper focus on my lecture. I would also like to thank him and Raymond Florax for helpful comments when I presented that lecture in Tucson. Finally, I would like to thank Arthur Getis and Rachel Franklin for inviting me to give that lecture. Of course, I am solely responsible for any shortcomings in this paper.

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Kelejian, H.H. Critical issues in spatial models: error term specifications, additional endogenous variables, pre-testing, and Bayesian analysis. Lett Spat Resour Sci 9, 113–136 (2016). https://doi.org/10.1007/s12076-015-0146-2

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Keywords

  • Specifications of spatial models
  • Additional endogenous variables
  • Pre-testing
  • Bayesian analysis

JEL Classification

  • C01
  • C12
  • C13