In Search of ‘W’

Abstract

The paper discusses the standard approaches in constructing the spatial weights matrix, W, and the implications of using such approaches in terms of the potential mis-specification of W. We then look at more recent attempts to measure W in the literature, including: Bayesian (searching for ‘best fit’); non-parametric techniques; the use of spatial correlation to estimate W; other iteration techniques; and alternative approaches. Lastly, an illustration is provided based on estimating spatial lag models determining establishment level R&D spending in the UK, finding that differently constructed W matrices produce different estimates of spatial spillovers.

A la recherche du « W »

Résumé La présente communication se penche sur les méthodes standards de la structure de la matrice de poids spatiale «W», et les implications de l'emploi de ces méthodes sur le plan d'une erreur de spécification potentielle de W. Elle se penche ensuite sur des tentatives plus récentes de mesure de W dans certains ouvrages, y compris des inférences bayésiennes (recherche de meilleure probabilité); des techniques non paramétriques; l'emploi d'une corrélation spatiale pour l’évaluation de W; des techniques d'itération diverses; et d'autres méthodes en alternative. Enfin, elle contient une illustration basée sur l'estimation de modèles à décalage spatial permettant de déterminer le niveau d’établissement de dépenses en R&D au Royaume-Uni, qui conclut que des matrices W à structure différente produisent différentes évaluations de débordement spatial.

En busca de ‘ W ’

ExtractoEste trabajo trata los planteamientos típicos al construir la matriz de ponderaciones espaciales, W, y las implicaciones de utilizar dichos planteamientos en términos de la posible especificación errónea de W. Seguidamente, examinamos intentos más recientes de ponderar W en la bibliografía, incluyendo: bayesiano (búsqueda de lo que ‘mejor encaja’); técnicas no paramétricas; el uso de correlación espacial para estimar W; otras técnicas de iteración; y planteamientos alternativos. Finalmente, se ofrece una ilustración basada en estimar modelos de retardo (lag) espacial que determinan el gasto en I+D a nivel de establecimiento en el Reino Unido; se descubre que matrices W construidas de forma diferente producen estimaciones diferentes de excedentes (spillovers) espaciales.

Keywords:

• Spatial weights
• spatial dependence
• spatial models

JEL CLASSIFICATION:

• C31
• O18
• R11

Notes

1. For evidence and more discussion on spillovers being spatially bounded, see Niebuhr (2000), Thornton & Flynne (2003), Bottazzi & Peri (2003), Vernon Henderson (2007), Baldwin et al. (2008) and Puga (2010). Recently, Peri (2005) used patent data for a panel of 113 European and North American regions over 22 years, finding that the externally accessible stock of R&D had a positive impact on firm innovation but that only about 20% of average knowledge is learned outside the region of origin and only 10% outside the country of origin. In contrast, Lehto (2007) used R&D data for Finnish firms and found that only when other firms’ R&D is located in the same sub-region is there any positive spillover effect. On this evidence, R&D spillovers appear to be (very) localized. However, there are also studies that find stronger support for international knowledge spillovers, rather than localized spillovers. These emphasize transmission through international trade, FDI, international technology transfer, and other forms of internationalization (see e.g. Gong & Keller (2003), Niosi & Zhegu (2005), and a recent review by Harris & Li (2006)).

2. A third approach to agglomeration is the work of Porter (1998) which emphasizes inter-firm local competition within his ‘diamond’ model.

3. Note, a more generalized model is to also weight x, giving: —the spatial Durbin model (see LeSage & Fischer (2008) for a discussion).

4. Where ρ≠0 and λ≠0, Anselin & Rey (1991) argue that whichever statistic is larger probably indicates which model is to be preferred.

5. For an example where this does occur, see Andersson & Gråsjö (2009) who state: ‘… a well-formed model should most likely not produce spatial autocorrelation at all. From this perspective spatial autocorrelation is not (pure) statistical nuisance but a sign … that a model lacks representation of an important economic phenomenon’. McMillen (2010) states that the spatial lag model ‘… actually makes more sense than the spatial error model because it tries to specify how the dependent variable responds to its neighboring values’.

6. Contrary to such views, there would seem to be an (implicit) expectation in the spatial econometrics literature that spatial models are not particularly sensitive to the W used in such models; our evidence presented in Section 5 suggests that estimates of spatial spillovers do differ significantly depending on how W is constructed.

7. As McMillen (2010) states, functional form mis-specification of a non-spatial model can cause the error term in the model to be spatially correlated; more fundamentally, omission of a spatially correlated variable could mean that ‘… a misspecified spatial econometric approach may be accepted in place of the true model’ (p. 120).

8. Of course there are many forms of physical distance functions, from inverse distances raised to some power (often 2) to various forms of bandwidth approaches, as well as variations on n nearest neighbours, etc.

9. See Maggioni & Uberti (2009) who consider these various ‘distance’ concepts when modelling knowledge networks across Europe.

10. Differences in economic intensity are measured by the square-root of a region i’s GDP relative to the GDP of region j. Essentially this is a proxy for absorptive capacity, as regions with a large technology gap are assumed to be less-able to ‘absorb’ potential spillovers from another area.

11. Van Stel & Nieuwenhuijsen (2004) used a similar approach, estimating a model using NUTS3 data, but they also tested for the statistical significance of NUTS1 and NUTS2 dummies on the premise that inter-regional spillovers across NUTS3 areas (if present) can be captured by higher-level regional dummies. They recognized the limitations of their approach (‘… distant regions may interact more than neighbours because they contain important cities and are well connected by communications networks’ (van Stel & Nieuwenhuijsen, 2004, p. 400)), but did not go further and include dummies in their model for regions located outside the higher-level region, although in principle this would seem an obvious extension of their approach.

12. If the model includes only cross-section data, then by definition β=θ=0. Otherwise Equation (7) is a more general (panel data) model incorporating spatial and temporal effects.

13. There is also a separate literature that looks specifically at exporting and technology diffusion, based on whether firms that export learn about foreign technology through their experience of exporting (i.e. through a ‘learning-by-exporting’ effect). See Greenaway & Kneller (2005), Table 1) and Greenaway & Kneller (2007), Table 3) for a summary of the evidence.

14. Hillberry & Hummels (2005) find using very detailed data for US manufacturing that shipments had a median radius of just four miles.

15. In 2007, London and the South East accounted for over 34% of UK gross-value-added.

16. See Oikarinen (2006) for an example on the diffusion of housing price movements.

17. See note 1.

18. The list of cooperation partners comprises: supplier/customer/competitor companies; consultants, commercial labs, or private R&D institutes; universities or other higher education institutions; and government or public research institutes.

19. Any cut-off point to define the ‘core’ R&D regions is somewhat arbitrary, but we found our results were robust to including between 12 and 16 regions (i.e. the top 10–13%). Included in our list of 16 are 11 NUTS3 regions in London, the South East and Eastern England (such as Berkshire, Buckinghamshire, Cambridgeshire, Hampshire, Hertfordshire, Oxfordshire and Surrey), and as well as Bristol, Cheshire, Lancashire, Manchester South and Warwickshire.

20. We used the PC-cluster at Glasgow which limited us to 33Gbytes of physical memory (and at times another 45Gbytes of virtual memory). We therefore omitted some 47.9% of the total observations available, by omitting those industries (such as wholesale and retail trade; hotels and restaurants; construction; transport, storage; real estate and renting) which contained firms with mostly very small values of R&D. Overall, the 2,373 firms we included accounted for 82.6% of total R&D undertaken in the CIS dataset.

21. The ‘standardize’ option in ‘spatwmat’ was used, which requests that a row-standardized weights matrix be generated; i.e. all non-zero spatial weights are rescaled so that—within each row—their sum equals 1.

22. Outside these boundaries w _ij =0. Note, we experimented with different lower and upper bounds to obtain the largest significant value of the spatial spillover effect (ρ).

23. That is, the impact of including different weight matrices did change the vector in the model, but usually not significantly. With such a large dataset and with a large number of (highly significant) regressors in the model, with relatively large partial R ² values attached to them, this is to be expected. For example, dropping any significant variable (such as one of the ‘size’ dummies) can be shown to have only a small impact on other variables (in the case of a size dummy, the main affect is on the remaining size dummies in the model). The significant change in the intercept term in Table 3 for the model when ‘core’ R&D regions are also given a weighting of w _ij =1, is mainly due to these ‘core’ regions having a large number of firms in the dataset undertaking relatively high levels of R&D spending.

24. An alternative and more informative approach is that set out in Folmer & Oud (2008) who model both the overall strength of spatial spillovers (as represented by ρ) and the linkages between W and the underlying indicators used to make up W, thus allowing them to statistically test whether different types of contiguity are statistically significant. Overall their approach allows the incorporation and testing of more information on spatial dependence and thus offers more flexibility than the standard representation in terms of, say, a spatial lag model with a pre-specified and thus untested W.

25. Table A1 in the Appendix also reports results based on the spatial error model. These are overall very similar to the results obtained from the spatial lag model, although overall, the diagnostic tests for the spatial error model suggest that it performs less well compared to the spatial lag model (although the difference is rather small). Set against this though is that λ>ρ (see note 4). A better approach to deciding which model is preferable has been suggested by Born & Breitung (2009), who test spatial lag and error models that are nested within a ‘hybrid model’ that encompasses both specifications. Unfortunately, this cannot be currently run in STATA and anyway goes beyond the scope of this study (which is to show that different ‘W’ matrices produce different results).