Academic journal article Geographical Analysis

Estimation Bias in Spatial Models with Strongly Connected Weight Matrices

Academic journal article Geographical Analysis

Estimation Bias in Spatial Models with Strongly Connected Weight Matrices

Article excerpt

This article shows that, for both spatial lag and spatial error models with strongly connected weight matrices, maximum likelihood estimates of the spatial dependence parameter are necessarily biased downward. In addition, this bias is shown to be present in general Moran tests of spatial dependency. Thus, positive dependencies may often fail to be detected when weight matrices are strongly connected. The analysis begins with a detailed examination of downward bias for the extreme case of maximally connected weight matrices. Results for this case are then extended by continuity to a broader range of (appropriately defined) strongly connected matrices. Finally, a simulated numerical example is presented to illustrate some of the practical consequences of these biases.

Introduction

In a recent simulation study, Mizruchi and Neuman (2008) showed that, for spatial lag (SL) models with strongly connected (high-density) weight matrices, a severe downward bias is often present in maximum likelihood estimates of the spatial dependency parameter. (1) A similar finding is reported by Farber, Paez, and Volz (2009) in their simulation analysis of the influence of network topology on tests of spatial dependencies. Hence, the central purpose of this article is to clarify the nature of this bias from an analytical perspective. In addition, the same type of bias is present in both spatial error (SE) models and in the more general Moran test of spatial dependency. In all cases, this bias implies that significantly positive spatial dependencies may not be detected when weight matrices are strongly connected.

To establish these results, the analytical strategy employed considers the extreme case of maximally connected weight matrices and obtains exact results for this case. The rest follows from simple continuity considerations. To avoid repetition, the analytical development of spatial regression models here focuses on SL models. Parallel results for SE models are simply sketched.

We begin with the following a standard SL model for n spatial units:

y = [rho]Wy + X[beta] + [epsilon], [epsilon] ~ N(0, [[sigma].sup.2] [l.sub.n]) (1)

where y [member of] [R.sup.n] is some variable of interest, and X = [[1.sub.n], [x.sub.1],..., [x.sub.k]] [member of] [R.sup.[nx(k+1)]] represents a relevant set of k explanatory variables, with [1.sub.n] = = (1,..., 1)' denoting the unit n-vector (corresponding to the intercept term in this linear model). (Throughout the following analysis X is always X is always assumed to have full column rank, k+1, so that [(X' X).sup.-1] exists.) The unknown parameters of the model include the vector of beta coefficients [beta] = ([[beta].sub.0], [[beta].sub.1],..., [[beta].sub.k])', the variance [[sigma].sup.2] of each residual in [[epsilon], and the spatial dependence parameter [rho], which is of primary interest in the present analysis.

Also of major interest is the structure of the spatial weight matrix W. For our analysis, it is convenient to begin by characterizing these matrices in the following way. First, we choose a fixed positive scalar, b, to serve as an upper bound on weight values. With respect to this bound, an n-square matrix, W = ([W.sub.ij]: i, j = 1,..., n), is designated as a weight matrix if and only if (i) [w.sub.ij] = 0 and (ii) 0 [less than or equal to] [W.sub.ij] [less than or equal to] b for all i,j = 1,..., n. As usual, condition (i) specifies that dependencies are defined only between distinct spatial units. Condition (ii) can be thought of as a normalization condition that allows each weight, [w.sub.ij], to be interpreted as the "degree of connectivity" between i and j, where [w.sub.ij] = b implies a maximal degree of connectivity. This is particularly appropriate for applications of model (1) to social networks among n agents. For the present, the bound b only serves as a convenient conceptual device and can be set equal to one without loss of generality. …

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