A straightforward combination of the causal model and the autoregressive process gives the
causal-autoregressive specification
y = 'Wy + Xβ + μ , μ ~ N(0,)2I)
One complication about this specification is the suggested independence between the causal
model and the autoregressive process. Actually, as the sociodemographic conditions used to
specify the workforce are of a rather ad-hoc or proxy nature they may partly capture features
which are due to the underlying spatial processes. Formally, one may expect the spatial Durbin
process to be more adeqate, specified as
y = λWy + Xβ + λWxβ + μ , μ~N(0,σ2I).
In this specification, the autoregressive process is specified simultaneously outside the causal
model by the term λWy and inside the causal model by the spatial spill-over term λWXβ.
Following Anselin (1988a) and the common practice in spatial econometric literature, the spatial
Durbin process may be rewritten, using simple algebraic manipulations, as
y = Xβ + e
e = λWe + μ , μ ~N(0,σ2I),
i.e. the spatial autocorrelated error model. Finally, the spatial Durbin specification may be
combined with the autoregressive process, reading as
y = 'Wy + Xβ + e
e = λWe + μ , μ ~N(0,σ2I),
whereby the spatial dependence is specified as a model specific part (the spatial Durbin process)
and a model independent part (the autoregressive process).
The spatial autoregressive, the spatial Durbin and the combined models may be estimated using
asymptotically justified maximum likelihood estimation as described in Anselin (1988a).
However, these estimations is not without problems. First, it is quite computer-demanding and
the methods are not implemented in commonly used packages. Second, methodological problems
arise pertaining to the asymptotic justification of the estimated models, which especially renders
traditionally used model selection criteria invalid. Consequently, in order to evaluate the relative
impacts of different spatial specifications and to support model selection, a Lagrange Multiplier
pre-test strategy is applied. Specifically, this strategy consists of an estimation of the simple
linear causal model without any spatial processes, followed by a comparison with different
alternative specifications using a Lagrange Multiplier test for each (see Anselin 1988a, 1988b for
a detailed outline of the tests).
3. Empirical results.
Based on the availability of census data for the 275 Danish municipalities (see the map in Figure
1) three analysis variables were selected to describe commuting behaviour: