1262 Spatial Analysis of Economic Convergence
the underlying exogenous variables in the data-generating process are uncorrelated.
This collinearity can degrade coefficient precision in GWR and lead to counter-
intuitive signs for some regression coefficients. Another methodological problem
has been pointed out by Paezet al. (2002) and Pace and LeSage (2004). Indeed, one
of the motivations of this approach is that, if spatial autocorrelation only arises
due to inadequately modeled spatial heterogeneity, GWR can potentially elimi-
nate this problem. However, this is not necessarily the case as substantive spatial
interactions may coexist with parameter heterogeneity, as we will show in the next
section. Therefore, Pace and LeSage (2004) have generalized GWR to allow simulta-
neously for spatial parameter heterogeneity and spatial autocorrelation: the spatial
autoregressive local estimation (SALE). Formally, estimates are produced usingn-
models, wherenrepresents the number of cross-sectional sample observations,
using the locally linear spatial autoregressive model:
U(i)y=ρiU(i)Wy+U(i)Xγi+U(i)ε, (27.9)whereU(i)represents an(N×N)diagonal matrix containing distance-based weights
for observationithat assign the weight of one to themnearest neighbors to obser-
vationiand weights of zero to all other observations. The productU(i)ythen
represents an(m× 1 )sub-sample of observed per capita income rates associated
with themobservations nearest in location to observationi. The other products
are interpreted in a similar fashion. Asm→N,U(i)→In, and the local estimates
approach the global estimates from (27.4) as the sub-sample size increases. This
model is estimated by recursive ML forεi→N(0,σi^2 U(i)IN). This approach has
been implemented by Erturet al. (2007) for a sample of 138 European regions for
the period 1980–95. Moreover, they also control for non-constant variances with
a Bayesian spatial autoregressive local estimation. They show that, while the mean
of the estimates forρis near zero, there are still a number of regressions for which
spatial dependence estimates take on large and significant values. Country-level
differences are also obvious for the different estimates of the convergence parame-
terβ, with negative and significant values across EU regions in Spain, Portugal and
some French regions.
27.2.4 Theoretical foundations of spatial effects
As pointed out by Islam (2003), the specifications of growth regressions used in
initial studies ofβ-convergence were not derived from theoretical growth models.
Only at a subsequent stage were the regression specifications formally derived from
the neoclassical growth models by Barro and Sala-i-Martin (1992) and Mankiwet al.
(1992). The literature focusing on the relationships between space and growth has
evolved quite similarly. All the studies surveyed above included spatial effects in
anad hocway, allowing for spatial autocorrelation and/or spatial heterogeneity in
conditionalβ-convergence models or Verdoorn models in order to obtain a better
fit and consistent estimates. Spatial autocorrelation in this context may reflect
spatial spillovers arising between economies but can also be the result of some
model misspecification or omitted variables. Recently, some authors have tried to