Testing for Spatial Group Wise Testing for SGWH. Chasco, Le Gallo, López and Mur, Heteroskedasticity.

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1 Testing for Spatial Group Wise Heteroskedasticity. A specification Scan test procedure. Coro Chasco (Universidad Autónoma de Madrid) Julie Le Gallo (Université de Franche Comté) Fernando A. López (Universidad Politécnica de Cartagena) Jesús Mur (Universidad de Zaragoza) 6JP 6th Jean Paelinck - Spatial Econometric Madrid October 2013

2 Motivation What is Spatial Group Wise Hereroskedasticity (SGWH)? Variability of the spatial data is systematically higher in some areas than in others, e.g. the variance present a form of spatially clustered or spatial trend What happens if in OLS residual has Spatial Group Wise Hereroskedasticity? The typical problem with heteroskedasticity and It is possible to be confused with spatial dependence!!! In Anselin 1999 pag 4 Anselin L A working paper on the use of spatial interactions and spatial structure in regression analysis. Center for Spatially Integrated Social Science, a spatial cluster ( ) of extreme residuals may be interpreted as due to spatial heterogeneity (e.g., groupwise heteroskedasticity) or as due to spatial autocorrelation (e.g., a spatial stochastic process yielding clustered values). This requires that both aspects of the problem be structured very carefully to obtain identifiability of the model parameters, There is a huge literature on the topic of spatial dependence but, unfortunately, the detection SGWH is much less developed.

3 Motivation In Anselin 1999 pag 4, in a single cross-section, spatial autocorrelation and spatial heterogeneity may be observationally equivalent Figure show two maps of the OLS residual, one with spatial dependence and another one with SGHW. SGHW σ=0.9/σ=0.1 Who's who? Map 1 Map 2 SAR (ρ=0.3)

4 Background and Objetive There is a huge literature on the topic of spatial dependence but, unfortunately, the detection SGWH is less developed. Don t there are, far as we know, specific tests to identify SGWH There are several heteroskedasticity tests (BP, White, GQ are most popular) that we can to adapt to test SGWH butitisnecessarygiveaprioriinformationabout the spatial structure present in the data that the researcher must supply. Kelejian and Robinson (1998) suggested tests for spatial heteroskedasticity. This method seems unsatisfactory, as it requires a specification of the causes of the changing variance. Ord and Getis (2012) consider the problem of local instability in the variance introducing a new statistic, called H i. The authors draw the attention to the lack of papers directed at examining the spatial structure of the variance. However, the SGWH is a frequent phenomenon when working with real data, which involves serious inference problems. López, Chasco and Le Gallo (2013) Exploring Scan methods to test spatial structure with an application to housing prices in Madrid. Papers in Regional Science The objective of this paper is developing a flexible and powerful statistic test based on the Scan methodology (usual meth. is spatial epidemiology) to detect SGWH. The test DO NOT need a priori information about the spatial structure, and As secondary output, the test identify the spatial cluster of localizations with different variance.

5 The Scan Tests: Scan σ and Scan μ,σ

6 The Scan Test: Scan σ Suppose {x i } to be a spatial process with i=1,..,r a set of spatial coordinates H : x i.i.d. N( ; ) H 0 A i There are a set Z xi i.i.d.n( ; Z) for i Z : xi i.i.d.n( ; Z) for i Z Z Z l I ( H Scan H H I =max l ( A) l( 0) Z A l( H 0 ) R 2 ln 2 2 ˆ H H 2 H A ) R 2 ln21 R Z 2 ln ˆ ( Z ) R R Z lnˆ ( Z ) A ˆ I 0 R Z H ˆ ( ) Z HA l ( HA) l( H0) 2R ln ln 2 ( Z) R 2 ˆ ˆ ( Z) H A HA The Scan σ test SCAN the surface looking for the window Z where the difference in log likelihood is maximum Using permutational bootstrapping we can assigns a p-value. And, as secondary output, the test define a spatial cluster Z*, the most likely cluster when the difference in variance inside/outside is maximum.

7 An example using Scan σ test Map 1 Most Likely Cluster SGHW Map 2 σ=0.9/σ=0.1 Number of Windows Explored: Scanning Windows Scan (p value with 999 boots) (Z) 0.90 (Z) 0.52 SAR (ρ=0.3) Scan 3.07 (p value with 999 boots) (Z) 1.82 (Z) 0.98

8 The Scan Test: Scan μ, σ Scan μ, σ is a Test for SGWH and/or Spatial Structure in mean H : x i.i.d. N( ; ) H A 0 i There are a set Z xi i.i.d. N( Z; Z) for i Z : xi i.i.d. N( Z; Z) for i Z Z Z ; Z Z Scan H H I, =max l ( A) l( 0) Z The Scan μ,σ test SCAN the surface looking for the window Z where the difference in log likelihood is maximum

9 Monte Carlo: Size, Power, Precision and Sensitivity

10 Monte Carlo: The size of Scan tests We evaluate the performance of the Scan test when it is applied on residuals of a linear regression model y i =3+2x i +e i in a square regular lattice. Normal residuals ei~n(0,1) Non-Normality in residuals Several alternatives Residuals with random Heteroskedasticity Residual with spatial dependence (SAR, SMA with ρ=0.5)

11 Monte Carlo: The power of Scans tests Power of the Scan test when there are different pattern of SGWH σ=0.9/σ=0.1 σ=0.9/σ=0.1 σ=0.9/σ=0.1 σ=0.9/σ=0.1

12 Monte Carlo: Local Sensibility and Spatial Precision

13 Monte Carlo: Local Sensibility TheScantestgiveanimportantinformation:TheMostLikelyCluster with different variance. Several measures can give me information about the ability to identify the true set LS(i)= Number of times that localization i is assign to the MLC Number of times that the test rejects the null hypothesis It is clear that Scan tests offer valuable information about the spatial structure of the variance.

14 Monte Carlo: Spatial Precision Global Sensitivity (Sens) = % cells correctly classified. Inverse of Sensitivity (Isens) = % cells wrongly assigned. Sensibility does not depend on the sample size. For small sample size we get hig values of Sens and Low values of Isens

15 Scan Test: Conclusion The SGWH is an important topic in spatial econometric unfortunately this topic is not explored in the literature. Main results: The Scan test is a powerful and simple test to check SGWH and do not need a priori information. Moreover, Scan test identify the pattern of instability in the variance (MLC) and can help us to make a correct specification of the regression model. Working in process: The Scan test is sensitive to the presence of Spatial dependence. We are working in a new tests to identify correctly the source of instability. The Scan test is computing intensive. Newsdevelopmentbasedin Gumbel distribution could be solve this problem. Thanks for your patience!!