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Specifying a joint space- and time-lag using a bivariate Poisson distribution

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Abstract

Separate space- or time-lags have been considered regularly in data analyses; as space–time models are more recently being studied extensively in data analytic fashion, joint estimation of both lags has to be considered explicitly. This paper addresses this issue, taking into special consideration parametric parsimony together with specification richness; use of the bivariate Poisson frequency distribution is advocated and applied to an empirical case. The relation of this approach to random effects specifications is investigated. Data for Belgian regional products constitute the empirical case study.

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Notes

  1. This decomposition is similar to that for principal components analysis (PCA) in multivariate statistics. A key difference is that PCA eigenvectors are used to construct synthetic variates that are linear combinations of attribute variables, whereas here the eigenvectors themselves are used as synthetic variates. De Jong et al. (1984), and Tiefelsdorf and Boots (1995), show that the eigenvalues associated with these eigenvectors cover the complete range of spatial autocorrelation possibilities for a given connectivity matrix. As such, the eigenvectors identify global, regional, or more local components of spatial autocorrelation.

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Correspondence to Daniel A. Griffith.

Appendix A

Appendix A

1.1 Significance of least absolute difference (LAD) estimators

LAD estimators are derived from a square sub-matrix of the usual X-matrix in the linear model (see Taylor 1974, pp. 169–190):

$$ {\mathbf{y}} = {\mathbf{X}{\varvec \upbeta}} + {\varvec\upvarepsilon }. $$
(7)

Call that sub-matrix X *; then (the capped \( {\hat{\varvec{\upbeta }}}\) being the vector of estimators):

$$ {\hat{\varvec{\upbeta }}} = {\mathbf{X}}^{*{ - 1}} {\mathbf{y}}^*, $$
(8)

where y * is a sub-vector of y corresponding to X *; if \( \varvec{\upbeta} \) is k × 1, then X * is k × k and y * is k × 1.

From Eqs. 7 and 8,

$$ {\hat{\varvec{\upbeta }}} = {\varvec{\upbeta}} + {\mathbf{X}}^{*{ - 1}} {\varvec{\upvarepsilon }}, $$
(9)

and if ε is symmetrically distributed around zero, even if X * is a random matrix, then \( \varvec{\upbeta} \) is unbiased (Taylor, Theorem 5); under the usual assumptions, its variance is given by

$$ E\left( {\hat{\varvec\upbeta } - {\varvec{\upbeta}}} \right)\left( {{{\hat{\varvec\upbeta }}} - {\varvec{\upbeta}}} \right)^{\prime } = \sigma^{2} {\mathbf{X}}^{*{ - 1}} \left( {{\mathbf{X}}^{*{ - 1}} } \right)^{\prime } . $$
(10)

To derive X *, one computes, for all observations,

$$ {\mathbf{y}} - {\mathbf{X\hat{\varvec\upbeta }}}\mathop = \limits^{\Updelta } {\mathbf{e}}, $$
(11)

where \( \mathop = \limits^{\Updelta } \) denotes “per definition,” and for the relevant range of y, e* = 0 (from the properties of the LAD-estimators; as in practice, here a reduced gradient method—Fylstrom a.o., 1998—is used instead of linear programming; e* should correspond to the k smallest errors).

Then \( \hat{\sigma }^{2} \) can be computed from RSS/(n – k − 1), as the average of e is not necessarily equal to zero. The square root of the diagonal of the expression in Eq. 10 (with estimated σ2) gives the desired standard deviations, which then can be used to conduct classical t tests.

From RSS and the total sum of squares, \( {\text{TSS}} = \left[ {{\mathbf{y}} - {\varvec{\upmu }}({\mathbf{y}})} \right]^{\prime } \left[ {{\mathbf{y}} - {\varvec{\upmu }}({\mathbf{y}})} \right], \) and the pseudo- R 2 can be computed as

$$ \left( {1 - {\text{RSS}}} \right)/{\text{TSS}} $$
(12)

and submitted for a classical F test.

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Griffith, D.A., Paelinck, J.H.P. Specifying a joint space- and time-lag using a bivariate Poisson distribution. J Geogr Syst 11, 23–36 (2009). https://doi.org/10.1007/s10109-008-0075-3

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