Regional Science and Urban Economics 39 (2009) 360–364

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Regional Science and Urban Economics

j ourna l homepage: www.e lsev ie r.com/ locate / regec

Dartboard tests for the location quotient☆

Paulo Guimarães a,b, Octávio Figueiredo b,c, Douglas Woodward a,⁎a University of South Carolina, United Statesb CEMPRE, Portugalc Universidade do Porto, Portugal

☆ The authors acknowledge the support of FCT, thScience and Technology. An earlier version of this papSouthern Regional Science Association Annual Meetin2008.⁎ Corresponding author. Division of Research and Dep

School of Business, University of South Carolina, ColumFax: +1 803 777 9344.

E-mail address: woodward@moore.sc.edu (D. Wood1 The location calculator is available at http://www.bl

0166-0462/$ – see front matter © 2009 Elsevier B.V. Aldoi:10.1016/j.regsciurbeco.2008.12.003

a b s t r a c t

a r t i c l e i n f o

Article history:

In this paper we reinterpre Received 4 April 2008Received in revised form 23 December 2008Accepted 27 December 2008Available online 13 January 2009

JEL classification:R10R12C12

Keywords:Location quotientDartboard modelIndustrial locationAgglomeration

t the location quotient, the commonly employed measure of regional industrialagglomeration, as an estimator derived from Ellison and Glaeser [Ellison, G., and Glaeser, E., 1997. Geographicconcentration in U.S. manufacturing industries: a dartboard approach. Journal of Political Economy 105 (5),889–927.] dartboard framework. This approach provides a theoretical foundation on which to build statisticaltests for the measure. With a simple application, we show that these tests provide valuable informationabout the accuracy of the location quotient. The tests are relatively easy to implement using regionalemployment and establishment data.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

First introduced by Florence (1939), the location quotient is oftenemployed to quantify industrial concentration in regions. In fact, itspopularity is so widespread that the U.S. Bureau of Labor Statistics hasadded a tool for calculating location quotients to their web site.1

Moreover, the location quotient has long been applied to estimate thestrength of regional economic impacts and export (economic-base)activities (Isserman, 1977, 1980). Typically constructed with employ-ment data, the measure is the ratio of two shares: the employmentshare of a particular industry in a region and the employment share ofthat industry in a wider area, such as a country. Researchers usuallyassume that if the quotient is above one, then the industry isconcentrated in the region. Yet they do not have any suitable statisticaltests to determine whether location quotients provide evidence of anexcessive degree of geographic concentration of an industry. Thus, themeasure remains awidely applied empirical techniquewithout a solidtheoretical and statistical foundation.

e Portuguese Foundation forer was presented at the 47thg, Washington DC, March 28,

artment of Economics, Moorebia, SC 29208, United States.

ward).s.gov/cew/home.htm.

l rights reserved.

Location quotients purport to reveal distinct specializations inregional activities based on natural assets like coastal locations, alongwith other comparative and competitive advantages, including thepositive agglomeration economies of existing local industry clusters.Obviously, they must reflect firm location decisions. Yet, as stressed byDuranton and Overman (2005), industry location is in part a randomphenomenon. Firms may exhibit some level of spatial concentrationby chance. This idea can be illustrated using the well-knowndartboard example of Ellison and Glaeser (1997). If 10 firms choosetheir locations by throwing darts at a map with 10 equally sizedregions the probability of ending upwith a single firm in each region isvery small. Most likely, in some regions by chance pockets ofconcentration will occur and that is perfectly compatible with theidea that firms' decisions were independent and random. The locationquotient is unable to account for this problem. Because of the discretenature of the phenomena it is possible to observe spuriousconcentration; that is, concentration that occurs by chance alone.

Hence, industry concentration (or localization) indices should offeran indication of the statistical significance of the results. To ourknowledge, the only attempts to provide statistical properties for thelocation quotient are O'Donoghue and Gleave (2004) and Moineddinet al. (2003). In the former, the approach consists of a set of rulesdesigned to elicit the empirical regularities of the data, without a clearrationale for the proposed methodology. Likewise, the latter paper'smethods are ad hoc, with standard deviations for location quotientscalculated, without any reference to an underlying theory (in this case,applied to health care utilization, not industry location). The essential

3 Note, however, that, as shown in Guimarães et al. (2007), this assumption is

361P. Guimarães et al. / Regional Science and Urban Economics 39 (2009) 360–364

problem persists in the literature: The location quotient lacks atheoretically based statistical foundation. Without one, it cannotaccount for the inherent randomness of the underlying locationdecisions.

In this paper we reinterpret the location quotient as an estimatorderived from Ellison and Glaeser's (1997) dartboard location model.This approach has the distinct advantage of providing a theoreticalbasis onwhich to build statistical tests for themeasure. Importantly, inthis framework, the location quotient is derived from a probabilisticmodel. It is then possible to account for sampling uncertainty, animportant concept in statistical inference.

The rest of the paper is structured as follows. In the next sectionweshow how to derive the location quotient as an estimator in thecontext of the Ellison and Glaeser's (1997) dartboard location model.In Section 3, we develop statistical tests for the measure. Section 4,shows, using a simple application, how to implement the tests, andSection 5 concludes.

2. The location quotient as an estimator derived from theEllison–Glaeser framework

This section takes as its starting point the approach in Figueiredoet al. (2007). Consider an economywith J distinct regions. In accordancewith the natural advantage model of Ellison and Glaeser's (1997)2 weassume that the spatial distribution of firms in a given industry, sayindustry k, reflects their profit maximizing location choices. Profits forfirm i in industry k, if it locates in region j, are given by:

logπijk = logπ j + ηjk + eijk: ð1Þ

The term π−j reflects the expected profitability of locating in region jcommon to all industries, while ηjk is a variable that captures thestrength of external economies and (or) natural advantages specific tothat region that attract firms from that particular industry. One way tointerpret ηjk is to assume as in Ellison and Glaeser (1997) that at thestart of the process nature assigns resource endowments to eachregion and these affect industry profits differently. Yet anotherinterpretation discussed in Ellison et al. (2007) is to assume that theηjk are picking endogenous factor advantages of the types described ineach of Marshall's theories, as for example, the availability of laborerswith particular skills or the existence of industry-specific knowledgein the region. Ultimately what matters is that when firms make theirlocation decisions they take this industry/region specific advantage asgiven and know that their individual decision will not affect the ηjk.Thus, from a modeling perspective we may assume that the ηjk aregenerated by some random mechanism and that firms observe arealization of that distribution. As we will see below, the locationquotient may then be interpreted as an estimator of these realizations[or, more precisely, of a realization of exp (ηjk)]. Finally, the term εijk inEq. (1) is a random effect that picks all other idiosyncratic factors thataffect the profit of individual firms. As commonly done we assumethat the εijk are i.i.d. with an Extreme Value Type I distribution.Conditional on a realization of ηk=(η1k,η2k,…,ηJk), we can useMcFadden's (1974) well-known result to derive an expression forthe probability that a firm from industry k will locate in region j:

pjkjηk =exp logπ j + ηjk

� �PJ

j = 1 exp logπ j + ηjk� � : ð2Þ

2 The fact that we base the discussion on the model of natural advantages of Ellisonand Glaeser (1997) is not crucial. Observationally this model is equivalent to thespillover version also developed by the authors.

This means that firms' location probabilities are affected byconditions specific to the region and the sector. Borrowing fromEllison and Glaeser (1997), we assume that the spatial distribution offirms will, on average, replicate the distribution of overall economicactivity which is measured by total manufacturing employment. As inEllison and Glaeser (1997) this means that the expected value of pjk|ηk

(calculated over the multivariate distribution of ηk) equals:

E pjk� �

=xjPJj = 1 xj

=xjx; ð3Þ

where xj is total manufacturing employment in region j and x is totalmanufacturing employment in the economy.We also admit as valid anadditional assumption introduced in Figueiredo et al. (2007) thatrequires that on average the effect of the ηjk's cancel out in such a waythat:

E pjk� �

=π jkPJj = 1 π jk

: ð4Þ

While this assumption imposes some restriction on the admissibledata generating processes for the ηjk it allows us to rewrite thelocation probabilities in terms of the known xj,3

pjkjηk =exp logxj + ηjk

� �PJ

j = 1 exp logxj + ηjk� � =

xjexp ηjk� �

PJj = 1 xjexp ηjk

� � : ð5Þ

Because we have observed multiple location decisions in the sameregion we can treat the realizations of ηjk as constants that need to beestimated. To see how this is implemented suppose that there are nkplants in industry k. We assume that these correspond to a total of nkindependent location decisions. This means that we can construct thelikelihood of observing a particular spatial distribution of investmentsas the product of all location probabilities, each decisionweighted by afactor of wjk. Hence, the likelihood function for a given sector is:

Lk =YJj = 1

pwjk

jjηk =YJj = 1

xjexp ηjk� �

PJj = 1 xjexp ηjk

� �0@

1A

wjk

: ð6Þ

Notice that by introducing weights we are allowing for the possibilitythat two location decisions in the same region offer differentcontributions to the likelihood function. For example, it may beargued that the location decision of larger plants should be givenmoreimportance. To account for this, we can make the weights propor-tional to employment or any other measure of firm size. In any case, ifweights are used, then their sum across regions, wk, should benormalized to equal the number of location decisions observed in thesector.4

Maximization of the likelihood function in Eq. (6) is straightfor-ward. The individual elements of the vector s containing the first orderconditions for maximization with respect to ηjk are generically givenby:

sjkuAlogLkAηjk

=wjk − wk

xjexp ηjk� �

PJj = 1 xjexp ηjk

� � ð7Þ

compatible with the framework of Ellison and Glaeser (1997). In Guimarães et al.(2007), the authors proposed a specific distribution for ηjk, the gamma distribution,that satisfies Eq.(4) as well as all the parametric assumptions laid out by Ellison andGlaeser (1997).

4 For example, if the weights are proportional to employment then wjk=(xjk /xk)×nk.

6 Note that the relevant alternative hypothesis, the existence of localization, is Ha:ηjkN0. Hence, in subsequent applications we will use one-sided tests.

7 For example, to test the hypothesis of equality of the degree of localization ofindustry k in two regions, ηjk=ηlk, the Wald statistic becomes:

362 P. Guimarães et al. / Regional Science and Urban Economics 39 (2009) 360–364

In turn, the generic elements of the Hessianmatrix,H, are given by,

hjlkuA2logL

AηjkAηlk=

−wk

xjexp ηjk� �

PJj = 1 xjexp ηjk

� � 1−xjexp ηjk

� �PJ

j = 1 xjexp ηjk� �

0@

1A; j = l

wk

xjexp ηjk� �

PJj = 1 xjexp ηjk

� � xlexp ηk� �

PJj = 1 xjexp ηjk

� � ; j≠l

:

8>>>>>>><>>>>>>>:

ð8Þ

The first order conditions lead to an indeterminate solution, becausethe ηjk's are not identifiable. To solve the problem we need tointroduce a restriction on the parameters. Hence, for identificationpurposes, we add the restriction that:

XJ

j = 1

xjexp ηjk� �

= x: ð9Þ

With this restriction in place we now have a metric for the ηjk's. Thisrestriction also makes intuitive sense. From Eq. (5) we see that if theparameter ηjk for region j equals 0 that means that the actual locationprobability equals its expected value and thus industry k is notlocalized in region j. Solving the first order conditions we obtain thefollowing maximum likelihood estimator:

bη jk = logLjk; ð10Þ

where Ljk=(wjk /wk) / (xj /x) is a location quotient for industry k inregion j. The structure of weights given to the location probabilitiesdetermines the different types of location quotients that areconstructed. If we assume that the contribution of each firm to thelikelihood function should be weighted by its size (say employment)then we obtain the traditional employment location quotient. On theother hand, if we assume that all firms have identical contributionsthen we get a location quotient that is based on plant counts.5

This derivation of the location quotient–now seen as an estimatorobtained from a probabilistic model–provides a convenient frame-work to construct hypothesis tests. However, we should bear in mindthat hypotheses should be formulated in terms of the ηjk's; that is, interms of the unknown variables that capture the locational advantagesof the specific regions.

3. Tests based on the location quotient

3.1. Testing the hypothesis of non-localization of one industry in a region

Consider the general expression for the Wald test of a linearcombination(s) of the ηjk's given by:

W = Rbη − q� �0 RVR0

h i− 1Rbη − q� �

; ð11Þ

where R is a matrix containing the coefficients of the linear combination(s),q is avectorof constants,η are themaximumlikelihoodestimatesofηandV is the estimatedvariance-covariancematrix associatedwithη . Thistest follows a chi-squared distribution with degrees of freedom equal tothe number of restrictions being tested. To calculateVwe use the inverseof the negative of the Hessian evaluated at the maximum likelihoodestimates. Taking advantage of the first order conditions, we can rewriteEq. (8), the expression for the generic elements of the Hessian, as:

hjlk =−wk

wjk

wk1−

wjk

wk

� �; k = l

wkwjk

wk

wlk

wk

� ; k≠l

:

8>><>>: ð12Þ

5 Figueiredo et al. (2007) argued that if the objective is to measure the intensity oflocalization economies, then this should be the preferred approach.

Or, in matrix form, as:

H = − wk diag ~wð Þ− ~w ~w0� �; ð13Þ

where w̃=(w1k /wk, w2k /wk,…,wJk /xk) is a vector containing theregional shares of the variable used to weight the location probabil-ities. The expression for the negative of the Hessian, −H, is the same asthe variance–covariance matrix of the multinomial distribution. Thisallows us to use the result of Tanabe and Sagae (1992) to obtain thegeneralized inverse of −H which we denote by V̂. Thus,

bV =w− 1k I −

1Jii0

� �diag ~wð Þ½ �− 1 I −

1Jii0

� �; ð14Þ

where I is the identitymatrix and i is a J×1 vectorwith elements (1,1,…,1).Using V̂ in place of V we can now test for any linear combinations of theηjk's. Of particular interest is the test of the hypothesis that ηjk=0, whichcan be interpreted as a test of non-localization of an industry in a region.6

Without loss of generalityassume that j=1. Thismeans thatR=[1,0,0,…,0]andq=[0]. Replacing these vectors into Eq. (11) and simplifyingweobtainthe statistic:

Wjk =J log Ljk

� � �2J − 2ð Þw− 1

jk +w− 1k

; ð15Þ

which is asymptotically distributed as chi-square with one degree offreedom. The term w− 1

k is the average of all wjk−1. In a similar fashion we

could implement other tests using the general expression in Eq. (11).7

3.2. Testing the hypothesis of non-localization of one industry across aset of regions

Another test that can be implemented based on the locationquotient is a score test of the hypothesis that the ηjk's are identical forall the regions. If that is the case then the location probabilities for theindustry will coincide with the share of each region in overallmanufacturing employment. This can be seen as a test of whether ornot one industry is localized across a set of regions and should lead tosimilar conclusions as other tests for localization of an industry, suchas Ellison and Glaeser (1997), Maurel and Sedillot (1999), Mori et al.(2005) and Guimarães et al. (2007). To implement the test, let ηr

denote the value of η under the null hypothesis. Then, Rao's score testis obtained by the statistic:

T = − s0 ηr� �

H ηr� � �− 1s ηr

� �; ð16Þ

which is known tobeasymptoticallydistributedas chi-squarewithdegreesof freedom equal to the number of restrictions being tested. In the aboveformula, s(ηr) is the score vector evaluated under the null hypothesis andH(ηr) is defined similarly. Thus, the generic element of s(ηr) equals:

sj ηr� �

=wjk − wkxjx; ð17Þ

and the Hessian matrix is,

H ηr� �

= − wk diag ~xð Þ− ~x~x0ð Þ; ð18Þwhere x̃=(x1/x, x2/x,…,xJ/x) is a vector containing the regional shares oftotalmanufacturing employment. Replacing the elements in Eq. (16) by theexpressions derived above, and using again the result of Tanabe and Sagae

W =log Ljk

� �− log Llkð Þ �2

w− 1jk +w− 1

lk

;

which is also asymptotically distributed as chi-square with one degree of freedom.

Table 1Compared with the U. S. average is any manufacturing sector localized in the state of Connecticut?

NAICS Employment Establishments Ljk J wjk−1 w− 1

k Wjk

U.S. Connecticut U.S.

311- Food 1,451,633 7978 17,927 0.382 51 – – –

312- Beverage and tobacco 168,651 449 2225 0.185 51 – – –

313- Textile mills 332,036 2910 3179 0.609 50 – – –

314- Textile product mills 213,512 1645 4380 0.535 51 – – –

315- Apparel manufacturing 507,236 1256 6856 0.172 51 – – –

316- Leather and allied products 68,111 197 1469 0.201 50 – – –

321- Wood products 592,443 1871 13,247 0.219 50 – – –

322- Paper 550,212 5718 4590 0.722 51 – – –

323- Printing and related support activities 805,271 12,354 14,706 1.066 51 0.004 0.016 0.884324- Petroleum and coal products 106,604 676 1783 0.440 50 – – –

325- Chemicals 860,534 15,178 10,374 1.225 51 0.005 0.062 6.368326- Plastics and rubber products 1,041,322 9625 10,221 0.642 50 – – –

327- Nonmetallic mineral products 518,138 3144 12,393 0.421 51 – – –

331- Primary metal manufacturing 600,995 5289 5423 0.611 50 – – –

332- Fabricated metal products 1,777,533 40,848 29,744 1.596 51 0.001 0.031 108.639333- Machinery 1,360,732 22,341 20,421 1.140 51 0.003 0.246 2.243334- Computer and electronic products 1,559,624 21,011 11,460 0.936 51 – – –

335- Electrical equipment and components 586,219 13,976 5538 1.656 51 0.008 1.112 8.747336- Transportation equipment 1,875,332 52,009 9841 1.926 51 0.004 0.180 60.999337- Furniture and related products 628,944 3176 10,998 0.351 51 – – –

339- Miscellaneous manufacturing 727,169 13,475 15,472 1.287 51 0.003 0.021 16.978Total manufacturing 16,332,251 235,126 212,247 – – – – –

Source: U. S. Census Bureau, County Business Patterns, 2000 (as in Thomas J. Holmes web page).

363P. Guimarães et al. / Regional Science and Urban Economics 39 (2009) 360–364

(1992) to calculate the inverseof theHessian,wearrive at the following teststatistic for an industry localization across a set of regions expressed interms of location quotients:

Tk =XJ

j = 1

wjkLjk − 1� �2

Ljk: ð19Þ

The above statistic follows (asymptotically) a chi-square distributionwith J−1 degrees of freedom. This statisticmakes intuitive sense. If mostlocation quotients are around one then the industry does not exhibit anydegree of localization, but the further apart the location quotients arefrom 1 the more likely the evidence is in favor of localization.

4. An illustration

We now illustrate the implementation of the tests developedabove using a simple study of industry localization in the state ofConnecticut. Connecticut is suitable for our purpose becausewith only8 counties it makes for a simple example with almost all required datafitting in a few tables. We use employment based location quotientsbecause this is common practice among applied researchers.

The questions that we want to answer are:

(1) Compared with the U. S. average, is any manufacturing sectorlocalized in the state of Connecticut?

(2) If so, is this sector localized within Connecticut?(3) And, in which counties is this sector localized within

Connecticut?

To do this we needed data on employment and number ofestablishments by manufacturing sector for the 50 U.S. states and thedistrict of Columbia, as well as for the 8 counties in Connecticut.8

8 We obtained the data from the web page of Thomas J. Holmes at the University ofMinnesota. The data are for the year of 2000 and is taken from the U.S. Census Bureau,County Business Patterns (CBP). The county data in the CBP have a large number ofdisclosure problems. The advantage of the Holmes data is that the author used animputation procedure to fill all the empty cells. The data can be found at www.econ.umn.edu/~holmes/data/cbp.

In Table 1, we present location quotients calculated at the statelevel for the 21 industries of the 3-digit North American IndustrialClassification System (NAICS). We find that location quotients areabove one for 7 of the 21 industries in Connecticut. Next, we use thestatistic in Eq. (15) to check whether these location quotients provideevidence of geographic concentration in excess of what would beexpected to arise randomly.9 This statistic is asymptotically distrib-uted as chi-square with one degree of freedom and thus the criticalvalue for the one-sided test is 2.71 for a level of significance of 95%. Aninspection of Table 1 shows that only 5 of the 7 industries withlocation quotient above the unity pass the test. Hence, with referenceto the United States, we can conclude that Chemicals, FabricatedMetalProducts, Electrical Equipment & Components, Transportation Equip-ment, and Miscellaneous Manufacturing are localized in Connecticut.

Next, in Table 2, we examine whether Transportation Equipment(the industry with the highest statistically significant locationquotient in Table 1) is localized within Connecticut. To answer thisquestion, we calculate location quotients for this industry at thecounty level and use the statistic in (19). Since this statistic follows(asymptotically) a chi-square distribution with J−1 degrees offreedom, the critical value is 14.07 for a level of significance of 95%.Given that we obtained 42.74 for this statistic, we conclude thatTransportation Equipment is localized within Connecticut.

Finally, in Table 3, we address our last question: Within the set ofConnecticut counties, where is Transportation Equipment localized?To find out, we again used the statistic in Eq. (15). The results in Table 3indicate that Transportation Equipment is localized within Connecti-cut in the county of New London (using a level of significance of 95%).It should be noted that this industry (NAICS code 336) includes Shipand Boat Building (NAICS code 33661). At this level, it is not surprisingto find that a strong localization exists for Transportation Equipmentwithin Connecticut. The town of Groton in New London county is the

9 Note that with the exception of column 8, Table 1, all calculations can be replicatedusing the employment and establishment data shown in the tables. It is also worthy tonote in Table 1, column 6, that some sectors are not present in all the regions. This is aproblem likely to show up in applied work. In these instances, when computing thetests, we can overcome the problem by subtracting from J the number of regions withmissing sectors.

Table 2Is NAICS 336 localized within Connecticut?

County Employment Establishments Ljk wjk Tk

NAICS336

Totalmanufacturing

NAICS 336

Fairfield 12,085 58,045 6 0.941 31.369Hartford 21,203 68,049 11 1.409 55.037Litchfield 1638 15,387 37 0.481 4.252Middlesex 2506 11,390 7 0.995 6.505New Haven 4171 51,359 7 0.367 10.827New London 9621 18,867 11 2.305 24.973Tolland 180 3930 31 0.207 0.467Windham 605 8099 25 0.338 1.570Total Connecticut 52,009 235,126 135 – – 42.744

Source: U. S. Census Bureau, County Business Patterns, 2000 (as in Thomas J. Holmesweb page).

Table 3In which counties is NAICS 336 localized within Connecticut?

County Ljk wjk−1 w− 1

k Wjk

Fairfield 0.941 0.032 – –

Hartford 1.409 0.018 0.419 1.780Litchfield 0.481 0.235 – –

Middlesex 0.995 0.154 – –

New Haven 0.367 0.092 – –

New London 2.305 0.040 0.419 8.471Tolland 0.207 2.140 – –

Windham 0.338 0.637 – –

Source: U. S. Census Bureau, County Business Patterns, 2000 (as in Thomas J. Holmesweb page).

364 P. Guimarães et al. / Regional Science and Urban Economics 39 (2009) 360–364

“submarine capital” of the world–which includes a highly concen-trated manufacturing hub based on the area's legacy of ship-buildingand sustained by large government contracts.10

5. Conclusion

The location quotient professes to assess industry localization in aparticular region. Researchers usually assume that if the calculatedlocation quotient is above one, then the industry is concentrated inthat particular region. They do not, however, have any suitablestatistical technique to determine whether the measure providesevidence of geographic concentration in excess of what would beexpected to arise randomly. Despite its practical popularity, then, thelocation quotient has lacked a theoretical basis. In particular, it cannotaccount for the inherent randomness of underlying location decisions.Even if plants are randomly distributed across locations, by chancepockets of concentration are likely to emerge.

In this paper we construe the traditional employment locationquotient to be an estimator derived from Ellison and Glaeser's (1997)

10 Clearly, in any calculation of location quotients, it helps to have disaggregated data.In this illustration, however, we restricted the analysis to 3-digit NAICS codes, becausewe wanted to make available to the readers in a few tables the data required for thetests.

dartboard location model. This approach has the advantage ofproviding a framework on which to build statistical tests. It is thenpossible to account for sampling uncertainty, an important concept instatistical inference. Estimates for the parameter without an asso-ciated level of significance (as usually reported in location quotientanalysis) are of little value. In reporting point estimates for thelocation quotient, statistical tests should be given in future work.

It is also worth noting that because of the modifiable areal unitproblem application of the location quotient may provide unreliableresults. Thus, it would be important to develop other methodologiesthat could overcome the problem. Duranton and Overman (2005,2008) used a particularly promising approach that relies on the theoryof spatial point processes. However, their approach requires detailedgeographic information data which are often unavailable. Futureresearch should explore alternatives that account for spatial depen-dence making use of commonly available aggregated data.

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