Measures of Spatial Dependence Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Junel 30th, 2016 C. Hurtado (UIUC - Economics) Spatial Econometrics
On the Agenda 1 Moran s I 2 Geary s Index 3 4 Moran Scatter Plot C. Hurtado (UIUC - Economics) Spatial Econometrics
Moran s I On the Agenda 1 Moran s I 2 Geary s Index 3 4 Moran Scatter Plot C. Hurtado (UIUC - Economics) Spatial Econometrics
Moran s I Moran s I Using the concept of spatial lag Let us deffine Moran s I as where Y = X X I = i j w ij (X i X)(X j X) i (Xi X) 2 = However, Moran s I is not ranging ( 1, 1) but Var(WX) I Var(X) Y WY Y Y Burridge (1980) proved that Moran s I test corresponds to a LM test when an explicit alternative hypothesis of uncorrelation is formulated. The presentation of alternative test statistics for the hypothesis of residual correlation cannot be treated in more detail until we present some explicit formulations for the alternative hypothesis. C. Hurtado (UIUC - Economics) Spatial Econometrics 1 / 7
Moran s I Moran s I Using the concept of spatial lag Let us deffine Moran s I as where Y = X X I = i j w ij (X i X)(X j X) i (Xi X) 2 = However, Moran s I is not ranging ( 1, 1) but Var(WX) I Var(X) Y WY Y Y Burridge (1980) proved that Moran s I test corresponds to a LM test when an explicit alternative hypothesis of uncorrelation is formulated. The presentation of alternative test statistics for the hypothesis of residual correlation cannot be treated in more detail until we present some explicit formulations for the alternative hypothesis. C. Hurtado (UIUC - Economics) Spatial Econometrics 1 / 7
Moran s I Moran s I Using the concept of spatial lag Let us deffine Moran s I as where Y = X X I = i j w ij (X i X)(X j X) i (Xi X) 2 = However, Moran s I is not ranging ( 1, 1) but Var(WX) I Var(X) Y WY Y Y Burridge (1980) proved that Moran s I test corresponds to a LM test when an explicit alternative hypothesis of uncorrelation is formulated. The presentation of alternative test statistics for the hypothesis of residual correlation cannot be treated in more detail until we present some explicit formulations for the alternative hypothesis. C. Hurtado (UIUC - Economics) Spatial Econometrics 1 / 7
Moran s I Moran s I Using the concept of spatial lag Let us deffine Moran s I as where Y = X X I = i j w ij (X i X)(X j X) i (Xi X) 2 = However, Moran s I is not ranging ( 1, 1) but Var(WX) I Var(X) Y WY Y Y Burridge (1980) proved that Moran s I test corresponds to a LM test when an explicit alternative hypothesis of uncorrelation is formulated. The presentation of alternative test statistics for the hypothesis of residual correlation cannot be treated in more detail until we present some explicit formulations for the alternative hypothesis. C. Hurtado (UIUC - Economics) Spatial Econometrics 1 / 7
Geary s Index On the Agenda 1 Moran s I 2 Geary s Index 3 4 Moran Scatter Plot C. Hurtado (UIUC - Economics) Spatial Econometrics
Geary s Index Geary s Index The index is j C = (n 1) wij(xi Xj)2 i 2W i (Xi X) 2 Ranges between 0 and 2 c > 1 indicates NEGATIVE spatial correlation c < 1 indicates POSITIVE spatial correlation C. Hurtado (UIUC - Economics) Spatial Econometrics 2 / 7
Geary s Index Geary s Index The index is j C = (n 1) wij(xi Xj)2 i 2W i (Xi X) 2 Ranges between 0 and 2 c > 1 indicates NEGATIVE spatial correlation c < 1 indicates POSITIVE spatial correlation C. Hurtado (UIUC - Economics) Spatial Econometrics 2 / 7
On the Agenda 1 Moran s I 2 Geary s Index 3 4 Moran Scatter Plot C. Hurtado (UIUC - Economics) Spatial Econometrics
Global measures allow to test for spatial patterning over the study area as a whole. It could happen that there is significant spatial correlation in smaller sections and they remain unombserved in a global averaging. A way of measuring this effect is to pass a moving window accross the data and examine spatial correlation within each window. This is a general procedure to derive Local Indicatiors of Spatial Association (LISA, Anselin 1995) C. Hurtado (UIUC - Economics) Spatial Econometrics 3 / 7
Global measures allow to test for spatial patterning over the study area as a whole. It could happen that there is significant spatial correlation in smaller sections and they remain unombserved in a global averaging. A way of measuring this effect is to pass a moving window accross the data and examine spatial correlation within each window. This is a general procedure to derive Local Indicatiors of Spatial Association (LISA, Anselin 1995) C. Hurtado (UIUC - Economics) Spatial Econometrics 3 / 7
Global measures allow to test for spatial patterning over the study area as a whole. It could happen that there is significant spatial correlation in smaller sections and they remain unombserved in a global averaging. A way of measuring this effect is to pass a moving window accross the data and examine spatial correlation within each window. This is a general procedure to derive Local Indicatiors of Spatial Association (LISA, Anselin 1995) C. Hurtado (UIUC - Economics) Spatial Econometrics 3 / 7
Getis-Ord Statistics The Getis-Ord satistics is given by G i = j wijxi j Xj It is a measure of local clustering based on the concentration of values in the neighborhood of a unit. Getis and Ord (1992) derived the expected value and variance of G i when W are elementes of a binary weight matrix. C. Hurtado (UIUC - Economics) Spatial Econometrics 4 / 7
Getis-Ord Statistics The Getis-Ord satistics is given by G i = j wijxi j Xj It is a measure of local clustering based on the concentration of values in the neighborhood of a unit. Getis and Ord (1992) derived the expected value and variance of G i when W are elementes of a binary weight matrix. C. Hurtado (UIUC - Economics) Spatial Econometrics 4 / 7
Local Moran s I it can be written as I i = j wij(xi X)(X j X) n 1 j (Xj X) 2 It represents a decomposition of the global Moran s I (Anselin, 1995). A local version of Geary s C can also be derived. Significance can be assessed using the expected value and the varince (and the asymptotic normality). Positive values indicate clustering of high or low values. HH or LL Negative values indicate spatial outliers. HL or LH C. Hurtado (UIUC - Economics) Spatial Econometrics 5 / 7
Local Moran s I it can be written as I i = j wij(xi X)(X j X) n 1 j (Xj X) 2 It represents a decomposition of the global Moran s I (Anselin, 1995). A local version of Geary s C can also be derived. Significance can be assessed using the expected value and the varince (and the asymptotic normality). Positive values indicate clustering of high or low values. HH or LL Negative values indicate spatial outliers. HL or LH C. Hurtado (UIUC - Economics) Spatial Econometrics 5 / 7
Local Moran s I it can be written as I i = j wij(xi X)(X j X) n 1 j (Xj X) 2 It represents a decomposition of the global Moran s I (Anselin, 1995). A local version of Geary s C can also be derived. Significance can be assessed using the expected value and the varince (and the asymptotic normality). Positive values indicate clustering of high or low values. HH or LL Negative values indicate spatial outliers. HL or LH C. Hurtado (UIUC - Economics) Spatial Econometrics 5 / 7
Local Moran s I it can be written as I i = j wij(xi X)(X j X) n 1 j (Xj X) 2 It represents a decomposition of the global Moran s I (Anselin, 1995). A local version of Geary s C can also be derived. Significance can be assessed using the expected value and the varince (and the asymptotic normality). Positive values indicate clustering of high or low values. HH or LL Negative values indicate spatial outliers. HL or LH C. Hurtado (UIUC - Economics) Spatial Econometrics 5 / 7
Local Moran s I it can be written as I i = j wij(xi X)(X j X) n 1 j (Xj X) 2 It represents a decomposition of the global Moran s I (Anselin, 1995). A local version of Geary s C can also be derived. Significance can be assessed using the expected value and the varince (and the asymptotic normality). Positive values indicate clustering of high or low values. HH or LL Negative values indicate spatial outliers. HL or LH C. Hurtado (UIUC - Economics) Spatial Econometrics 5 / 7
Moran Scatter Plot On the Agenda 1 Moran s I 2 Geary s Index 3 4 Moran Scatter Plot C. Hurtado (UIUC - Economics) Spatial Econometrics
Moran Scatter Plot Moran Scatter Plot Introduced by Anselin in 1995. An exploratory tool for assessing local patterns. It places the valu of X on the horizontal axes and the corresponding spatially lagged value on the vertical axes. C. Hurtado (UIUC - Economics) Spatial Econometrics 6 / 7
Moran Scatter Plot Moran Scatter Plot Introduced by Anselin in 1995. An exploratory tool for assessing local patterns. It places the valu of X on the horizontal axes and the corresponding spatially lagged value on the vertical axes. C. Hurtado (UIUC - Economics) Spatial Econometrics 6 / 7
Moran Scatter Plot Moran Scatter Plot Introduced by Anselin in 1995. An exploratory tool for assessing local patterns. It places the valu of X on the horizontal axes and the corresponding spatially lagged value on the vertical axes. C. Hurtado (UIUC - Economics) Spatial Econometrics 6 / 7
Moran Scatter Plot Moran Scatter Plot C. Hurtado (UIUC - Economics) Spatial Econometrics 7 / 7