Grid and shake: spatial aggregation and the robustness of regionally estimated elasticities


This paper proposes a simple and transparent method for measuring spatial robustness of regionally estimated coefficients and considers the role of the administrative districts and of the size of regions. The procedure offers a new solution for a practical empirical issue: comparing the variables of interest across spatially aggregated units. It improves upon existing methods, especially when spatial units are heterogeneous. To illustrate the method, we use Hungarian data and compare estimates of agglomeration externalities at various levels of aggregation. Using the procedure, we find that the method of spatial aggregation seems to be of equal importance to the specification of the econometric model.

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  1. 1.

    See, for instance, Ciccone and Hall (1996) on the relationship between labor productivity and agglomeration, Ács and Armington (2004) on entrepreneurship. Examples on local aspect of activity include Iyer and Seetharaman (2008) on retail prices, Hortaçsu and Syverson (2007) on local competition and vertical integration, Gobillon and Milcent (2013) on hospital performance.

  2. 2.

    Note that our questions are closely related to that of Amrhein (1995) who argued that aggregation may be indeed an important methodological choice.

  3. 3.

    There is a growing amount of related literature on a different type of setup, when the data are geocoded. For instance, Dubé and Brunelle (2014) argued that distance-based concentration measures may offer a good solution.

  4. 4.

    We use the 2007 euro per capita measure of GDP and the 2006–2008 average population density data from the Eurostat regional databases. The countries that constitute our sample: the UK, Germany, Belgium, France, Spain, Portugal, the Netherlands and Italy.

  5. 5.

    For instance, trade proximity matters for localization for about 50 km as suggested by Duranton and Overman (2008) for manufacturing industries in Britain. Rosenthal and Strange (2008) estimated that in the USA wage increasing effect of being close to educated people falls to just 25% as the distance rises from 5 to 15 miles in the USA. Lychagin et al. (2010) found that the crucial range for proximity to laboratories is limited to 100–200 km. Andersson et al. (2016) document and attenuation of wage density elasticity even within cities.

  6. 6.

    Note that, shape of units may be of interest. Briant et al. (2010) showed it does not matter. However, even though shape is not of our immediate concern here, the method presented caters for different shapes as well.

  7. 7.

    When regressing the (log) number of municipalities within a district on (log) wages, we find a negative correlation. This is primarily due to the fact that there are plenty of small municipalities close to Vienna with easy market access and higher wages. Using a cross section, we have: \(log(Num)=11.8+0.8\times log(\textit{Wage})\), significant at 5%; \(log(No)=1.41+ 0.05\times log(Wage)+0.45\times log(Distance_{Budapest}) -0.24\times log(Distance_{Vienna})\), where both distance variables are significant at 1%.

  8. 8.

    In some countries, like Hungary, municipality also contains the agricultural land. In Hungary, Eastern municipalities have larger plots of land and hence, Western municipalities will be calculated as denser.

  9. 9.

    See Dewhurst and McCann (2007) looking at a relationship between size and specialization in different types of administrative levels in the UK, or Devereux et al. (2007) studying UK “Regional Selective Assistance” grants given to regional political units.

  10. 10.

    Mercator WGS84 EPSG:41001 projection.

  11. 11.

    Polygon matching can be done in a GIS mapping software; we used Mapinfo. In Fig. 9 in appendix, they are illustrated by using a small part of southwest Hungary close to the Croatian border.

  12. 12.

    This approach would in some way, mimic the Swedish example, where 290 municipalities have been defined with a minimum population requirement, yielding similarly sized areas—except for the large Nordic regions.

  13. 13.

    This is illustrated in the left panel of Fig. 10 in “Appendix” section, where the random grid joins only three municipalities at the border, including Őrtilos, Zákány, Surd and Mezőkeresztúr. The right panel of Fig. 10 shows the adjusted spatial units. In this case, the four municipalities are added to the larger region to the east, hence increasing its size.

  14. 14.

    Unfortunately, we only know about headquarters. However, this is not a major problem for manufacturing firms in Hungary: in their appendix, Békés and Harasztosi (2013) showed that only 7% of firms have multiple sites and that, on average, these firms have 1.15 plants. This may give rise to a relatively small potential bias only.

  15. 15.

    Note that the variables are recalculated on different spatial aggregation levels and artificial spatial units by recalculation and using frequency weights. For example, if the average year of schooling (ys) in municipality i is \(ys_i\) calculated over \(n_i\) individuals, then the average year of schooling for spatial unit that includes municipalities i and j will be \((ys_i\times n_i+ys_j\times n_j)/(n_1+n_j)\).

  16. 16.

    This selection issue might affect the comparison between municipalities and 15 km grids because the latter will include the areas of zero employment areas in the denominator of the density measure, but it does not affect grids of larger sizes, like the 26 and 36 km grids. The sign of the bias caused by the selection is ambiguous and is an issue in the western part of Hungary, owing to the highly fragmented municipality structure.

  17. 17.

    Comparing them with the results presented by Briant et al. (2010) using a different aggregation methodology, we find that the order of elasticities is different. In this paper, we find coefficients rising with size, but a significant difference can only be established between small- and medium-sized units. Briant et al. (2010) found no linear relations, and the difference was only between their medium- and large-sized units.

  18. 18.

    KS results are available on request.

  19. 19.

    Also, the DtB method generates a variety of “shapes,” but importantly, they show that this feature does not matter.

  20. 20.

    The method is simple: once the map is digitized, it is fast and may be modified easily. One additional difference is the speed of calculating robustness measures; our code is substantially faster. In lack of available codes, we coded the Briant et al. (2010) method ourselves. On a laptop, our code run was about 10–20 times faster.


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Corresponding author

Correspondence to Gábor Békés.

Additional information

Péter Harasztosi: The opinions expressed are those of the author(s) only and should not be considered as representative of the European Commission’s official position.



See Table 3. See Figs. 91011121314 and 15.

Table 3 Summary of Hungarian administrative spatial units and grids
Fig. 9

Markers serve as basic elements to be shaken

Fig. 10

Adjustment for small areas near the border: \(26\times 26\) km. a Before adjustment. b After adjustment

Fig. 11

Population density in European regions

Fig. 12

Additional SD. This figure shows the distribution of estimated coefficients on population in Model 1—agglomeration elasticity of wage—(blue line) as well as an extended distribution (red dashed line). The extended distribution takes into account the uncertainty of each elasticity estimate:\(\beta ^\mathrm{km26}_i\) where \(i=\{1\ldots 300\}\). For each, we randomly draw 50 coefficients from normal distribution \(N(\beta ^\mathrm{km26}_i\),\(\sigma ^\mathrm{km26}_i\)). All together red dashed line is a density over \(50\times 300\) elasticity value

Fig. 13

Additional SD2. This figure shows the distribution boxplots for population coefficient estimates in Model 1 (agglomeration elasticity of wage) produced by various approaches. These are from top to bottom: (1) elasticity estimate using municipality-level data, confidence intervals are calculated using the standard deviation estimates of the elasticity. (2) Estimates on grids using km15 sizes, confidence interval is created using the deviation from the 300 estimates. (3) Estimates on grids using km15 sizes, confidence interval takes into account estimation uncertainty of each of the 300 estimates by adding 50 random normal draws: \(N(\beta ^\mathrm{km15}_i\), \(\sigma ^\mathrm{km15}_i\)). (4) Elasticity estimate using micro-region-level data, confidence intervals are calculated using the standard deviation estimates of the elasticity. (5) Estimates on grids using km26 sizes, confidence interval is created using the deviation from the 300 estimates. (6) Estimates on grids using km26 sizes, confidence interval takes into account estimation uncertainty of each of the 300 estimates by adding 50 random normal draws: \(N(\beta ^\mathrm{km26}_i\), \(\sigma ^\mathrm{km26}_i\))

Fig. 14

A realization of \(26\times 26\) km units (above) and 150 seeded DtB (below)

Fig. 15

Comparison to dots-to-boxes results. This figure shows the exact replication of Fig. 4, and it additionally displays DtB results too. It shows the coefficient estimate on population in Model 1 (agglomeration elasticity of wage). The thick blue (solid) line gives the point estimate for the municipality estimate. The thin blue (solid) line gives the distribution of point estimates over 300 point estimates on the \(15\times 15\) grid. The thick red (dashed) line gives the point estimate for the micro-region estimate. The thin red (dashed) line gives the distribution of point estimates over 300 point estimates on the \(26\times 26\) grid. The very thin dashed lines represent the 5% confidentiality thresholds. Additionally, the very thick mint-green (short dash) line shows the distribution of point estimates over 300 realizations of Dots-to-boxes spatial arrangements using 150 seeds

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Békés, G., Harasztosi, P. Grid and shake: spatial aggregation and the robustness of regionally estimated elasticities. Ann Reg Sci 60, 143–170 (2018).

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JEL Classification

  • R12
  • R30
  • C15