1256 Spatial Analysis of Economic Convergence
econometrics: the autoregressive and the moving average specifications. It is inter-
esting to note that only the former has been extensively used for the modeling
of spatial error dependence in convergence or Verdoorn’s law models. A spatial
autoregressive specification for the(N× 1 )error vector can be expressed as:
ε=λWε+u, (27.5)whereλis the spatial autoregressive parameter andu∼i.i.d.(0,σu^2 IN). Note that
this model can be rewritten in another form, called the spatial Durbin model: if
(27.3) is premultiplied by(I−λW),weget(I−λW)y=(I−λW)Xγ+u. Hence:
y=λWy+Xγ+δWX+u, (27.6)withδ=−λγ. These restrictions can be tested by the common factor test (Burridge,
1981). If it cannot be rejected, then model (27.6) reduces to model (27.5). In this
model, the average growth rate of a regioniis influenced by the average growth
rate of neighboring regions, by the initial per capita income of neighboring regions
and the spatial lags of the other explanatory variables.
Conversely, the spatial moving average specification can be expressed as:
ε= ̃λWu+u, (27.7)where ̃λis the spatial autoregressive parameter andu∼i.i.d.(0,σu^2 IN). Both models
can be estimated by ML, under the normality assumption, or generalized methods
of moments (GMM) (see Kelejian and Prucha, 1999, for the AR case and Fingleton,
2008, for the moving average (MA) case). These two specifications differ in the
terms of the range of spatial dependence in the variance-covariance matrix and
of the diffusion process they imply. More precisely, in the first case, the variance-
covariance matrix forεis=σu^2 (A′A)−^1 withA=IN−λW. WhileWmay be
sparse, the inverse term(A′A)−^1 is not. As a consequence, a random shock at one
locationiis transmitted to all other locations of the sample: the spillovers are
global. Rey and Montouri (1999) and Le Galloet al. (2005) illustrate this property
in the context of aβ-convergence model and show how a random shock in one US
state or in one European region impacts upon the per capita income growth rates of
all the regions in the sample. Conversely, in the MA case, the variance-covariance
matrix does not involve a matrix inverse: ̃=σu^2 ̃A′ ̃A. Therefore, the spillovers
remain local: a shock at locationiwill only affect the directly interacting locations
as given by the non-zero elements inW. Finally, higher-order spatial models have
been investigated by Kosfeldet al. (2006).
This basic framework has been extensively used to analyze the convergence pat-
terns among several sets of countries and regions: convergence among US states
(Rey and Montouri, 1999; Lee, 2004; Garrettet al., 2007), European regions (Fingle-
ton, 1999; Maurseth, 2001; Carrington, 2003; Le Galloet al., 2003; Arbia, 2006; Le
Gallo and Dall’erba, 2006), Brazilian states (Lall and Shalizi, 2003; Magalhãeset al.,
2005; Azzoni and Silveira-Neto, 2006), Spanish regions (Villaverde, 2005, 2006)
and Turkish regions (Gezici and Hewings, 2004; Yildirim and Öcal, 2006). Similarly,