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Testing for spatial error dependence in probit models


In this note, we compare three test statistics that have been suggested to assess the presence of spatial error autocorrelation in probit models. We highlight the differences between the tests proposed by Pinkse and Slade (J Econom 85(1):125–254, 1998), Pinkse (Asymptotics of the Moran test and a test for spatial correlation in Probit models, 1999; Advances in Spatial Econometrics, 2004) and Kelejian and Prucha (J Econom 104(2):219–257, 2001), and compare their properties in a extensive set of Monte Carlo simulation experiments both under the null and under the alternative. We also assess the conjecture by Pinkse (Asymptotics of the Moran test and a test for spatial correlation in Probit models, 1999) that the usefulness of these test statistics is limited when the explanatory variables are spatially correlated. The Kelejian and Prucha (J Econom 104(2):219–257, 2001) generalized Moran’s I statistic turns out to perform best, even in medium sized samples of several hundreds of observations. The other two tests are acceptable in very large samples.

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  1. For example, using the Columbus data from Anselin (1988) (N \(=\) 49), truncated in the same fashion as in McMillen (1992) and others, with y \(=\) 1 for crime \(>40\), yields values of respectively 2.48, 1.22 and 2.89 for the statistics in a probit estimation of crime on income and housing values. Whereas the first two values do not reject the null, the last statistic weakly rejects the null.

  2. Recall that the LM test statistic for spatial error autocorrelation in the standard linear regression takes the form \(LM=[(\mathbf e^{\prime } \mathbf W \mathbf e ) / \hat{\sigma }^2]^2 / tr(\mathbf{WW }+\mathbf{W^{\prime }W })\), where \(\mathbf e \) is a vector of OLS residuals, \(tr\) is a trace operator and \(\hat{\sigma }^2\) is the usual estimate of error variance ((Anselin 1988, p. 104)). Using the notation from Eq. 5, \(v=\hat{\sigma }^4 tr(\mathbf{WW }+\mathbf{W^{\prime }W })\).

  3. For all computations, we used PySAL, a Python library for spatial analysis Rey and Anselin (2007).

  4. All the experiments were also performed for irregular layouts and unbalanced samples. The results were similar to those for a balanced sample with regular lattice and were omitted from this paper.

  5. The spatial transformation induces a change in the mean and variance of the \(x\) variables. In order to ensure that the sample remains balanced and that the approximate \(R^2\) in the samples is comparable to the other samples, we carry out a transformation of the \(\mathbf x \) vector and adjust the variance of the error term \(\varepsilon \) such that the \(R^2 \approx 0.67\) in all settings. The transformation of the \(\mathbf x \) vector required to maintain a balanced sample is \(\mathbf x \sim -2(1-\gamma )^{-\theta }+U(-5,5)\).

  6. We take the 95th percentile of the distribution under the null obtained from our simulations as the “correct” critical value, rather than the value of 3.84 for a \(\chi ^2(1)\). This correction will become negligible for the larger sample sizes, since all three tests achieve their asymptotic distribution. The correction is most pronounced for \(LM_{PS}\) in the smaller samples.


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This project was supported by Award No. 2009-SQ-B9-K101 by the National Institute of Justice, Office of Justice Programs, U.S. Department of Justice. The opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect those of the Department of Justice.

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Correspondence to Pedro V. Amaral.

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Amaral, P.V., Anselin, L. & Arribas-Bel, D. Testing for spatial error dependence in probit models. Lett Spat Resour Sci 6, 91–101 (2013).

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  • Spatial econometrics
  • Spatial probit
  • Moran’s I

JEL Classification

  • C21
  • C25