Spatial groupwise heteroskedasticity and the SCAN approach

Transcription

1 Spatial groupwise heteroskedasticity and the SCAN approach Coro Chasco, Julie Le Gallo, Fernando A. López and Jesús Mur (*) Universidad Autónoma de Madrid. (**) Université Bourgogne Franche-Comté, UMR1041 CESAER, INRA, AgroSup Dijon. ( ) Universidad Politécnica de Cartagena; fernando.lopez@upct.es ( )Universidad de Zaragoza; jmur@unizar.es Abstract: We evaluate the usefulness of the Scan approach in order to test for the presence of spatial groupwise heteroskedasticity in cross-sectional data. This approach has been used in different fields before, including spatial econometric models, but to detect instability in the mean values. In this paper, we extend its use to second order moments, searching for spatial clusters in the variance. Using large Monte Carlo simulations, we check the reliability of the Scan procedure to detect instabilities in the variance, the size and power of the test and its accuracy to find spatial clusters of observations with similar variances. JEL classification: C21, C52, C63, R15 Key words: Spatial scan procedure, spatial group-wise heteroskedasticity, spatial variance clusters, Monte Carlo simulation. 1

2 1 Introduction Spatial econometrics is a subfield of econometrics dealing with two main specificities of spatial data: spatial autocorrelation and spatial heterogeneity (Anselin, 1988). Spatial autocorrelation, or the coincidence between value similarity and locational (Anselin and Bera, 1998), is now well documented with multiple possible specifications for cross-sectional, spatio-temporal and panel data and extensive testing procedures (see Arbia 2014; Darmofal 2015; Dubé and Legros 2014; Elhorst 2014; LeSage and Pace 2009 for recent textbooks on the topic). On the other hand, spatial heterogeneity means that the spatial process is not uniform over space. Frequent causes of heterogeneity are instability (i) in the mean, (ii) in the variance or (iii) in both. Mean instability implies local clustering of the values of a spatial variable. For instance, in the case of parameter instability in a regression, regression coefficients may follow a number of distinct spatial regimes such as North-South or Center-Periphery patterns, or they can evolve continuously over space (Brunsdon et al. 1999; Leung et al. 2000; Páez et al. 2002). The result is, usually, a form of clustering of the values of the variable. Hence, as is well known, spatial autocorrelation and spatial heterogeneity in the mean values are observationally equivalent in cross-sections, necessitating specification and test procedures able to disentangle these effects. In other cases, the variance, which evolves over space, is the source of instability in the model. This phenomenon is known as spatial heteroskedasticity. The variance can vary continuously over space or it can take different values between separate parts of the area. This is the case of spatial Group-Wise Heteroskedasticity (SGWH from now one). Mur and Angulo (2009) show that most patterns of SGWH are indistinguishable from the cases of spatial dependence or heterogeneous mean values. The consequence of omitted spatial effects are well known. Omitted spatial autocorrelation in the mean equation, or the wrong assumption of a common vector of parameters for the whole sample, leads to biased and inconsistent inference (LeSage and Pace, 2009, chapter 2), as is the case for patterns of omitted heteroskedasticity in spatially autocorrelated models (see Zhang and Lin, 2016, for the case of the bias of the Moran s I under heteroskedasticity). We may suspect that both sources of heterogeneity happen simultaneously. This is why joint tests for spatial error autocorrelation and heteroscedasticity have been proposed by Anselin (1988) and Kelejian and Robinson (1998). Moreover, conditional tests of heteroscedasticity or instability in the 2

3 regression coefficients, under the presence of spatial autocorrelation, can be found in Anselin (1988, 1990) and Páez et al. (2001), who introduce a Chow test for spatially switching regressions in a spatial lag model (see also López et al., 2009). In spite of that, heteroskedasticity has attracted less attention in spatial econometrics. This omission can be partially attributed to the generalization of the HAC methods, which allow performing robust inference in the mean equation for different departures of the iid clause. For instance, Kelejian and Prucha (2007) suggest a non-parametric heteroscedasticity and autocorrelation consistent (HAC) estimator of the variance-covariance matrix within a spatial context. This route has been followed by Kim and Sun (2015), who generalize the spatial HAC estimator for nonlinear spatial models, and Dorn and Egger (2015), who analyze the small sample performance of the spatial HAC estimators. Other spatial HAC estimators based on a continuous-index random field have been suggested by Conley (1999) and Bester et al. (2014). Hetereroskedasticity in spatial models, that do not follow a spatial pattern, is not of interest in this paper as it can be treated as usual. In fact, the White test (1980), for example, should have power against these patterns of unkown heteroskedasticity. In the case of SGHW we may think in the class of the so-called constructive test (Spanos, 1999) whose alternative hypothesis focuses on particular skedastic functions, such as the classical Breusch-Pagan test (Breusch and Pagan, 1980), the Koenker and Basset test (Koenker and Basset, 1982) or the Harvey test (1976). The test of Kelejian and Robinson (1998) can also be included in this group of targeted test which, in general, offers higher power. Note however that all these tests necessitate prior information on the variables at the source of heteroskedasticity and that their results depend totally on the correct specification of the alternative hypothesis (Wooldridge, 2010). Our point of departure differs from the previous papers and considers the perspective initiated by Ord and Getis (2012). They consider the problem of local instability in the variance introducing the so-called statistic H i, whose aim is to identify the limits of the area where the variance changes. The authors draw the attention to the lack of papers directed at examining the spatial structure of the variance (p. 530): Spatial statistics cluster identification is now common to many fields. ( however) these studies have focused attention upon local means, to the extent that variability is considered at all it is typically assumed that the process has a constant variance (i.e., that it is 3

4 homoscedastic). A moment s thought indicates that such an assumption could overlook important information. This paper aims to fill such gap. Indeed, we introduce a formal test for SGWH with a null hypothesis of constant variance. Then, in the case of rejecting the null, it is natural to ask for the location of the clusters of observations which share a similar variance. This information is of vital importance in order to treat adequately the SGWH structure with the aim of improving inference. It may also serve as a guide to specification in empirical analysis. We follow the approach originally suggested by Openshaw et al. (1987) in the so-called Geographical Analysis Machine, GAM, and later improved by Kulldorff et al. (1995, 2009) in the nowadays popular Scan algorithms. The paper is organized as follows. Section 2 introduces some basic results from the Scan methodology, including our proposal to detect SGWH. The design of a Monte Carlo experiment is presented in Section 3, together with the main results in relation to estimated size and power. Section 4 focuses on the so-called accuracy of the technique understood as the ability to identity exactly the location and composition of clusters in the variance. Section 5 illustrates the use of this methodology for the case of house prices in the city of Madrid. Main conclusions appear in Section 6. 2 References References [] Abrams, A.M., Kleinman, K., Kulldorff, M. (2010): Gumbel based p-value approximations for spatial scan statistics. International Journal of Health Geographics, 9(1): 61. [] Anselin, L. (1988): Spatial Econometrics: Methods and Models. Kluwer: Dordrecht [] Anselin, L. (1990): Spatial dependence and spatial structural instability in applied regression analysis. Journal of Regional Science, 30: [] Anselin, L. (2010): Thirty years of spatial econometrics. Papers in Regional Science, 89(1), [] Anselin, L., Griffith, D. (1988): Do spatial effects really matter in regression analysis? Papers of the Regional Science Association. 65, [] Anselin, L., Bera, A. (1998): Spatial dependence in linear regression models with an introduction to spatial econometrics. In 4

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