Sergio J. Rey and Julie Le Gallo 1257these spatial econometric models have been fitted to Verdoorn’s law specifications
by Bernat (1996), Fingleton and McCombie (1998), Pons-Novell and Viladecans-
Marsal (1999) and Dall’erbaet al. (2008).^3 Several methodological issues should be
kept in mind when dealing with these models.
First, in the absence of sound theoretical foundations for the specific form taken
by spatial autocorrelation in these models, most of these papers apply a classical
“specific to general” specification search approach outlined in Anselin and Rey
(1991) and Anselin and Florax (1995) to discriminate between the two forms of
spatial dependence – spatial error autocorrelation or spatial lag. Several Lagrange
multiplier (LM) tests are used for that purpose. Floraxet al. (2003) show by means
of Monte Carlo simulation that this classical approach outperforms Hendry’s “gen-
eral to specific” approach. Note that this classical approach has several drawbacks,
including the problem of multiple comparisons highlighted by Savin (1984): the
significance levels of the sequence of tests conducted in this section are unknown.
Second, particular attention should be given to the interpretation of the co-
efficients in the spatial lag model, as compared to those of the spatial error model
or the simple model estimated by ordinary least squares (OLS). Indeed, in these
latter models, the marginal effect of one explanatory variablexkcorresponds to
the associated parameterβk. Conversely, the spatial lag model (27.4) can be rewrit-
ten asy=(I−ρW)−^1 (ρWy+Xγ+ε). Since (in most cases) the elements of the
row-standardized weights matrixWare less than one, a Leontief expansion of
the matrix inverse follows as:(I−ρW)−^1 =I+ρW+ρ^2 W^2 +.... Consequently,
the growth rate of per capita income (in the convergence context) or of labor pro-
ductivity (in the Verdoorn context) of one regioniis not only affected by a marginal
change of the explanatory variables of regionibut is also affected by marginal
changes of the explanatory variables in the other regions, more importantly so for
closer regions. As a consequence, the estimated coefficients in a spatial lag model
include only the direct marginal effect of an increase in the explanatory variables,
excluding all indirect induced effects, while in the standard model estimated by
OLS, they represent the total marginal effect. It is therefore not relevant to compare
OLS and ML or two-stage least squares (2SLS) estimates for a spatial lag. This aspect
has so far been overlooked in the spatial econometrics literature in general (Pace
and LeSage, 2007) and in the spatial growth empirics literature in particular (Abreu
et al., 2005) and should be kept in mind when drawing inference on determinants
of the economic growth process using, for instance, the computationally feasible
means of summarizing the output into direct and indirect impacts of each variable
of interest suggested by Pace and LeSage (2007).
Third, endogeneity in cross-country regression models is a problem that is com-
monly encountered as output growth, investment rates, and so on, in a particular
period are likely to be jointly determined. Caselliet al. (1996) note that “At a
more abstract level, we wonder whether the very notion of exogenous variables is
at all useful in a growth framework (the only exception is perhaps the morpho-
logical structure of a country’s geography).” In a non-spatial context, they deal
with this issue using the Arellano and Bond (1991) GMM procedure. Similarly,
while Verdoorn’s law is usually treated as a single equation and estimated via OLS,