Seemingly Unrelated Regressions with Spatial Effects. An Application to the Case of the European Regional Employment
The seemingly unrelated regressions (SUR) equations are a traditional multivariate econometric formulation employed in very different fields including, obviously, spatial analysis. The basis of the approach is very well known due to the initial works of Zellner (1962), Theil (1971), Malinvaud (1970), Schmidt (1976) and Dwivedi and Srivastava (1978). In this paper, we address the case of a SUR model that involves spatial effects, under the configuration of a given number of equations, G, a finite number of crosssections, T, and a large number of spatial units, R. The problem that we pose is testing for the presence of spatial effects, as in Mur and LÃ³pez (2008), and to select the most adequate spatial model for the data, as in Mur et al (2010). Following these papers, we also assume a maximum-likelihood framework that facilitates the obtaining of simple Lagrange Multipliers, with good behaviour in small-sized samples. Then, we focus on the assumption of constancy (among equations, between cross-sections) of the parameters of spatial dependence. In a standard framework, these coefficients are allowed to vary between equations but not in time. In general terms, this is an unnecessary assumption to start with the econometric modelling. For this reason, we extend the discussion to the problem of the instability of the coefficients of crosssectional dependence, both in a spatial dimension and among equations. We present the results of a small Monte Carlo experiment to study the behaviour of the Lagrange Multipliers developed in order to analyze the assumption of parameter stability. Finally, an application of these techniques to the case of the European regional employment, at NUTS II level and disaggregated by sectors of activity, in the period 1980 to 2008, is also included. We specify a SUR model where each equation corresponds to a sector of activity. Provisional results indicate that there exist strong symptoms of instability in the spatial structure of the equations, although this structure appears to be stable in time.
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