Sergio J. Rey and Julie Le Gallo 1255function model. While exchangeability problems have not been discussed in this
specific context, it is also relevant given the type of assumption made on the errors
terms.
27.2.2 The cross-sectional approach to growth and convergence
We focus in this section on another possible violation of the exchangeability
assumption: spatial dependence in the error terms (Ertur and Koch, 2007). Indeed,
in the cross-sectional context, the observational units are spatially organized and
the i.i.d. assumption may therefore be overly restrictive. Various alternative spec-
ifications are appropriate to deal with different forms of spatial dependence (Rey
and Montouri, 1999). They are best described if we rewrite equations (27.1) and
(27.2) in matrix form:
y=Xγ+ε, (27.3)whereyis the(N× 1 )vector containing the observations on the dependent variable,
Xis the matrix containing the observations on all explanatory variables including
the constant term,εis the(N× 1 )vector of error terms, the properties of which we
describe below, andαandγare the unknown parameters to be estimated. In the
convergence case,ycontains the vector of average growth rates of per capita income
between datet 0 andt 0 +TandXcontains the initial log per capita income and all
the other control variables. In the Verdoorn case,ycontains the labor productivity
growth rate. Several spatial econometric specifications have been used to control
for spatial dependence in growth econometrics models: the spatial lag model, the
spatial error model and the spatial Durbin model.^1 In the spatial lag model, or
mixed regressive spatial autoregressive (AR) model, a spatially lagged dependent
variableWyis added to the right-hand side of the regression specification:
y=ρWy+Xγ+ε, (27.4)whereWis an(N×N)spatial weights matrix, usually row-standardized, andρis
the spatial autoregressive parameter. In a convergence context, for instance, this
specification allows measuring how the growth rate in a region may relate to the
ones in its surrounding regions (as defined inW) after conditioning on the starting
levels of per capita income and the other variables. Unlike the time series case, the
spatial lag term is endogenous, since it is always correlated withε(Anselin, 1988).
Therefore, this specification must be estimated using instrumental variables (IVs)
or, assuming thatεfollows a multivariate normal distribution with zero mean and a
constant scalar diagonal variance-covariance matrixσ^2 IN, by maximum likelihood
(ML). In the former case, Kelejian and Prucha (1999) show that the low-order spatial
lags of the exogenous variables can be used as instruments forWy. Of course, if
additional endogenous variables are present in the specification, this approach can
easily be extended by adding additional instruments.^2
The spatial error model is a special case of a non-spherical error covariance matrix
in which the spatial error process is based on a parametric relation between a loca-
tion and its neighbors. Two specifications have commonly been used in spatial