Framework: Examples Using Data on Urban Crime, Regional Inequality, and Government

Expenditures

Arthur Getis

If autocorrelation is found, we suggest that it be corrected by appropriately transforming the model so that in the transformed model there is no autocorrelation (Gujarati, 1992, p. 373).

14.1 Introduction

In a recent paper Getis (1990), I develop a rationale for filtering spatially depen-dent variables into spatially independepen-dent variables and demonstrate a technique for changing one to the other. In that paper, the transformation is a multi-step procedure based on Ripley’s second order statistic (1981). In this chapter, I will briefly review the argument for the filtering procedure and propose a simplified method based on a spatial statistic developed by Getis and Ord (1992). The chapter is divided into four parts: (1) a short discussion of the rationale for filtering spatially dependent variables into spatially independent variables, (2) a review of a Getis–Ord statis-tic, (3) an outline of the filtering procedure, and (4) three examples taken from the literature on urban crime, regional inequality, and government expenditures.

14.2 Rationale for a Spatial Filter

One of the most difficult problems facing those who develop regression models of spatial series is finding ways to estimate parameters of stochastically autocorrelated variables. A typical stochastically autocorrelated spatial variable is a modified or spatially lagged autocorrelated variable. It is made up of the original autocorrelated variable (y) multiplied by a spatial weight matrix (W) and a spatial autocorrela-tion coefficient (ρ).ρW_ydoes not fulfill the required fixed-effects linear regression assumption that correlated variables are not to be stochastic. In this case, since A. Getis

Department of Geography, San Diego State University, San Diego, CA, USA e-mail:arthur.getis@sdsu.edu

L. Anselin and S.J. Rey (eds.), Perspectives on Spatial Data Analysis, Advances in Spatial Science, DOI 10.1007/978-3-642-01976-0 14, c Springer-Verlag Berlin Heidelberg 2010

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ordinary least squares yields biased parameter estimates andR-squared values, other estimation techniques must be considered, such as maximum likelihood estimation.

In addition, any remaining spatial dependence in the regression equation, as may be evident in error terms, must be accounted for. In multiple variable cases, it may be necessary to develop a series of W matrices, thus further complicating the meaning of the various tests on the significance of the parameters.

Because of the complexity of the typical spatial regression formulation, I pro-pose that the spatial dependence within each dependent and independent variable be filtered out before the estimation procedure is adopted. This proposal has the fol-lowing ramification: one must somehow reintroduce into the regression equation the removed spatial dependence in order to avoid misspecification. No spatial depen-dence should be evident in the error term since supposedly it has been removed from all of the possible sources. Since each variable on the right hand side is no longer stochastically correlated nor spatially autocorrelated, and there is only the usual spherical error, ordinary least squares can be used for estimation.

This argument for filtering from spatially autocorrelated variables the spatial dependence effects is sound only insofar as there is a way to accomplish the task.

Before demonstrating the variable filtering procedure, let me briefly describe a statistic that will act as its foundation.

14.3 The G

Statistic

This statistic measures the degree of association that results from the concentration of weighted points (or areas represented by weighted points) and all other weighted points included within a radius of distancedfrom the original weighted point.¹We are given an area subdivided intonregions,i= 1,2, . . . , n, where each region is identified with a point whose Cartesian coordinates are known. Eachihas associated with it a valuex(a weight) taken from a variableX. The variable has a natural origin and is positive.²The statistic is written as

G_i(d) = _n

j=1w_ij(d)x_{j n}

j=1x_j , j=i, (14.1)

wherew_ijis a one/zero spatial weight matrix with ones for all links defined as being within distancedof a giveni; all other links are zero. The numerator is, therefore, the sum of allxwithindofiexcept wheniequalsj. The denominator is the sum of allx_jexcept wheniequalsj. The mean is

E(G_i) = W_i

(n−1), (14.2)

whereW_i=

jw_ij(d).

1For a full discussion seeGetis and Ord (1992).

2A more recent version of this statistic in Ord and Getis (1993) avoids these restrictions.

If we set

jx_j/(n−1) =Y_i1and

jx²_j/(n−1)−Y_i1² =Y_i2then:

V ar(G_i) = W_i(n−1−W_i) (n−1)²(n−2)(Y_i2

Y_i²₁). (14.3)

G_i(d) measures the concentration or lack of concentration of the sum of values associated with variableX in the region under study.G_i(d)is to be differentiated from a statisticG^∗_i(d)that takes into account the value ofxati, that is, j equal toi.G_i(d)is a proportion of the sum of allx_j values that are withind ofi. If, for example, high-valuedx_js are near to the pointi, and dincludes these high values, so that a large proportion of the sum of allx_jsis withindofi, thenG_i(d) is high. Whether theG_i(d)value is statistically significant depends on the statistic’s expectation.

For our purposes here, the most important characteristic of the statistic is that it gives the proportion of the summed variable within a specified distance from a particular pointi as a part of the entire summed variable. When this value is compared to the statistic’s expectation, the difference tells us the degree of clustering of the sum of thexvariable in the vicinity ofithat is greater or less than chance would have it.

14.4 The Filtering Procedure

The rationale for transforming a spatially dependent variable into a spatially inde-pendent variable is that the spatial dependence can be removed from the spatially dependent variable and replaced as a separate independent variable. An easy solu-tion to this problem, but useless, would be to set all values of the spatially dependent variable to the mean. This “variable” would not be spatially dependent and it would not correlate with any other variable. The solution I outline below attempts to adjust the spatial dependent variables only to the point where spatial dependence is no longer embodied in them. That which is filtered from the original variable becomes a new spatial variable. It may be that the autocorrelation filtered from one variable is highly correlated with that which is filtered from another. In a regression equation, in order to avoid multicollinearity, in the final equation it may be necessary to use only one spatial variable rather than all spatial variables extracted from the original variables.

Suppose that within distance dof a point,x₁, there are two other points with values,x₂, andx₃, which when summed are a greater proportion of allx(minus x₁) than what one should expect in a similar spatial configuration with the samed value when allxvalues are randomly distributed. This means thatG₁(d)is greater than the random expectation,W₁/(n−1). SupposeG₁(d)is the proportion 0.40 andE[G₁(d)] = 0.30. Then 40% of the sum of all observedx(not countingx₁ itself) is contained withindofx₁, while the expectation is only 30%. We then call 30/40 ofx₁ the filtered value ofx₁. The difference between the original value and

the filtered value is that which is filtered out due to the spatial clustering ofxvalues in the vicinity ofx₁. In this example, the ratio 30/40 represents the degree to which x₁ is similar to its expectation. The degree of dissimilarity, 10/40, represents that which is due to positive spatial association. Negative spatial association is found in like manner. Thus,

x^∗_i =x_i(_n−^Wi₁)

G_i(d) (14.4)

which when solved for allx_i, represents the filtered variableX^∗. The difference betweenXandX^∗is a new variable,L, that represents the spatial effects embedded inX.

For realistic filtering it is essential to find an appropriated value. The value should represent the distance within which spatial dependence is maximized. In Getis (1990), d corresponds to the maximum total sum of squared differences between the observed and the expectedG_i values. On reflection, however, a dif-ferentd value seems more appropriate. This is the value that corresponds to the maximum absolute sum of the normal standard variate of the statistic G_i(d) for alliobservations of the variableX. This single value is chosen since it represents overall the distance beyond which no further association or nonassociation effects increase the probability that the observed value is different than the expected value.

One might argue that beyond thisdvalue there is an overall cessation of spatial effects for the variable in question. For each variable, we use this value in the exam-ples given below. A more detailed approach, but less general, would be to identify a criticaldvalue for each individual pointi. Clearly, more research is needed on this subject.

Our approach to spatial filtering can be given credence by showing that the following four conditions hold:

1. There is no spatial autocorrelation embodied inX^∗.

2. IfX is a variable with spatial dependence embedded in it, then the difference betweenX andX^∗is a spatially autocorrelated variable (L).

3. In any regression model where all variables have been filtered using an appropri-ate distanced, residuals are not spatially associated.

4. In a regression equation, appropriate variables should be statistically significant after spatial dependence has been removed from them. Of course, appropriate-ness in this case requires theoretical justification. In this chapter, we will be satisfied if intuitively correct variables are statistically significant after spatial dependence has been removed.

If these conditions are met, one might conclude that a reasonable estimate of spa-tial autocorrelation has been found and that ordinary least squares is appropriate for regression modeling where this type of filtering has been used. In the next section, these conditions are demonstrated by way of three examples.

14.5 Filtering Variables: Three Examples 14.5.1 Example 1: Urban Crime

Anselin (1988) provides data on three variables (crime, income, and housing) by neighborhoods (given as points withx,ycoordinates) for Columbus, Ohio, for 1980 (n= 49). The autocorrelated variable, crime (CR), is constructed from the number of burglaries and thefts per thousand households, and income (IN) and housing (HO) are given by household in thousands of dollars. Whend= 4km spatial association is at a maximum for crime,d= 3for housing and income. In the tests that follow, these values fordwill be used. The trial OLS model is

CR= 68.62 −1.597IN −0.274HO

(t) (14.49) (−4.78) (−2.65) (14.5) The adjustedR²= 0.533 and the standardized Moran’sIon the residuals is 2.765.

The criterion for spatial dependence in this and all subsequent tests is the 0.05 level of the normal curve calculated from the / statistic (randomization, distance effect =1/d²) of Moran (Cliff and Ord, 1973). This statistic will not be reviewed here. In this case then, the residuals are spatially statistically significant. In addi-tion, it is important to note that two of the variables reveal strong spatial auto-correlation:

Variable Z(I)

CR 7.345

IN 4.168

HO 1.903

Test 1: There is no spatial autocorrelation embodied inX^∗.

This test requires that the transformed variables,CR^∗,IN^∗, andHO^∗are not spatially autocorrelated. The results are as expected:

Variable Z(I)

CR^∗ −0.152 IN^∗ −0.280 HO^∗ −0.454

Test 2: IfXis a variable with spatial dependence embedded in it, then the difference betweenX andX^∗is a spatially autocorrelated variable (L).

The results of the test (see below) show that all threeLvariables are spatially autocorrelated. Although it is not required for the housing variable to have a high Z(I)value, it, too, has spatial dependence embedded within it.