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TestingforSpatialGroup‐WiseHeteroskedasticity.

AspecificationScantestprocedure.

CoroChasco(UniversidadAutónomadeMadrid)JulieLeGallo(UniversitédeFranche‐Comté)FernandoA.López(UniversidadPolitécnicadeCartagena)JesúsMur(UniversidaddeZaragoza)

6JP Madrid18-19 October 20136th Jean Paelinck - Spatial Econometric

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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 spatialtrend

…, a spatial cluster (…) of extreme residuals may be interpreted as due to spatialheterogeneity (e.g., groupwise heteroskedasticity) or as due to spatial autocorrelation (e.g.,a spatial stochastic process yielding clustered values). This requires that both aspects ofthe problem be structured very carefully to obtain identifiability of the model parameters,…

• What happens if in OLS residual has Spatial Group‐WiseHereroskedasticity?– The typical problemwith heteroskedasticity and…– It is possible to be confused with spatial dependence !!!

• There is a huge literature on the topic of spatial dependence but,unfortunately, the detection SGWH is much less developed.

• InAnselin1999pag4Anselin L 1999. A working paper on the use of spatial interactions and spatial structure in regression analysis. Center for Spatially Integrated Social Science

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Motivation

In Anselin 1999 pag 4…, in a single cross-section, spatial autocorrelation and spatial heterogeneity may beobservationally equivalent

Figure show two maps of the OLS residual, one with spatial dependence and another one with SGHW.

SGHW

SAR (ρ=0.3)

Who's who?

Map 1 Map 2

σ=0.9/σ=0.1

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BackgroundandObjetive

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 but it is necessary give a priori information about the spatial structure present in the data that theresearcher must supply.

◦ Kelejian and Robinson (1998) suggested tests for spatial heteroskedasticity. This method seemsunsatisfactory, 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 Hi. The authors draw the attention to the lack of papers directed at examining the spatial structureof the variance.

However, the SGWH is a frequent phenomenon when working with real data,which involves serious inference problems.

The objective of this paper is developing a flexible and powerful statistic testbased on the Scan methodology (usual meth. is spatial epidemiology) todetect 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.

López, Chasco and Le Gallo (2013) Exploring Scan methods to test spatialstructure with an application to housing prices in Madrid. Papers in RegionalScience

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TheScanTests:Scanσ andScanμ,σ

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TheScanTest:Scanσ

i0H : i.i.d . N( ; )x

• Suppose{xi}tobeaspatialprocesswithi=1,..,Rasetofspatialcoordinates

i ZA

i Z

Z Z

for i Zfor i Z

There are a set Zi.i.d .N( ; )x

:H i.i.d .N( ; )x

l( 0H ) R2

ln 2 0H

2̂ 1

l I( AH ) R2

ln2 1 RZ2

lnAH

2̂ ( Z )R RZ

2ln

AH2̂ ( Z )

IA 0

ZScan H H= max l ( ) l( )

The Scanσ test SCAN the surface looking for the window Z where thedifference 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.

0 A

A A

2 2I Z

A 0 2 2

ZRH H2RH H RZ ZH H

( )ˆ ˆl ( ) l( ) ln ln

( ) ( )ˆ ˆ

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AnexampleusingScanσ test

Scan 85.54 (p value 0.001 with 999 boots)

(Z) 0.90 (Z) 0.52

MostLikelyCluster

Scan 3.07 (p value 0.693 with 999 boots)

(Z) 1.82 (Z) 0.98

NumberofWindowsExplored:13940

ScanningWindows

SGHW σ=0.9/σ=0.1

SAR (ρ=0.3)

Map 1

Map 2

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TheScanTest:Scanμ ,σ

i0H : i.i.d . N( ; )x

Scanμ ,σ isaTestforSGWHand/orSpatialStructureinmean

i Z ZA

i Z Z

Z ZZ Z

There are a set Zi.i.d.N( ; )x

:H i.i.d.N( ; )x

for i Zfor i Z

;

IA 0

ZScan H H, = max l ( ) l( )

The Scan μ,σ test SCAN the surface looking for the windowZwhere the difference in log‐likelihood is maximum

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MonteCarlo:Size,Power,PrecisionandSensitivity

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MonteCarlo:ThesizeofScantests

Residual with spatial dependence

(SAR, SMA with ρ=0.5)

WeevaluatetheperformanceoftheScantestwhenitisappliedonresidualsofalinearregressionmodelyi=3+2xi+ei inasquareregularlattice.

Residuals withrandom

Heteroskedasticity

Non-Normality in residuals

Several alternatives

Normal residualsei~N(0,1)

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MonteCarlo:ThepowerofScanstests

PoweroftheScantestwhentherearedifferentpatternofSGWH

σ=0.9/σ=0.1 σ=0.9/σ=0.1 σ=0.9/σ=0.1 σ=0.9/σ=0.1

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MonteCarlo:LocalSensibilityandSpatialPrecision

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MonteCarlo:LocalSensibility

The Scan test give an important information: The Most Likely Clusterwith different variance. Several measures can give me information aboutthe ability to identify the true set

LS(i)= Number of times that localization i is assign to the MLCNumber of times that the test rejects the null hypothesis

It is clear that Scan tests offer valuable information about the spatialstructure of the variance.

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MonteCarlo:SpatialPrecision

Global Sensitivity (Sens) = % cells correctly classified.

Inverse of Sensitivity (Isens) = % cells wrongly assigned.

• Sensibilitydoesnotdependonthesamplesize. ForsmallsamplesizewegethigvaluesofSensandLowvaluesofIsens

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ScanTest:Conclusion

• The SGWH is an important topic in spatial econometricunfortunately this topic is not explored in the literature.

Main results:• The Scan test is a powerful and simple test to check SGWH and donot 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 theregressionmodel.

Working in process:• The Scan test is sensitive to the presence of Spatial dependence. Weare working in a new tests to identify correctly the source ofinstability.

• The Scan test is computing intensive. News development based inGumbel distribution could be solve this problem.

Thanksforyourpatience!!