well known linear regression formulation, i.e.
y = Xβ + μ , μ ~ N(0,σ2I)
where y is an n-vector of observations for one of the considered commuting variable (n being the
number of municipalities, i.e. i = 275), X an n by k matrix of n observations for k
sociodemographic variables, β a k-vector of regression coefficients, μ an n-vector of random
errors, assumed to be independent of each other and identically normally distributed with mean
value 0 and common variances σ2, as I is an n by n identity matrix.
To this point, the causal relationship between sociodemographic characteristica - describing the
workforce - and commuting patterns has been formulated much in accordance with known
empirical findings. What has not been formulated is the spatial patterning of industries and the
implications of this patterning for commuting behaviour. Actually, only a few relevant statements
about industrial patterning has been provided: Some industries with preferences for low wages
locate in low-wage areas, i.e. rural areas and small- and medium sized cities. Other industries
with emphasize on infrastructure, knowledge and highly skilled employees prefer urban areas.
From this, it is obvious that a spatial clustering of some industries should emerge and that
commuting occurs due to the contradiction between this clustering and the spatial pattern in
choice of livingplaces, as people necessarily have to follow the supplied jobs.
Due to the aforementioned lack of data for industrial localization on a municipality base, it is
impossible to estimate a causal relationship between this determinant and commuting. Another
route to recognize this relationship will be followed in this investigation, as illustrated by the
following informal argumentation: Suppose that the in-commuting to a specific municipality is
high. Then two conclusions may be drawn: First, there is a high concentration of industries
attracting commuters in this municipality. Second, there may be a high concentration of these
industries in neighbourhood municipalities, giving rise to a high in-commuting to these
municipalities. Conversely, a high out-commuting from a specific municipality is caused by a
lack of firms in certain industries, a lack which may also be found in neighbourhood
municipalities whereby a high out-commuting is expected from these too.
These ideas may be formalized using a spatial autoregressive specification as suggested by
Anselin (1988a). Define an n by n contiguity matrix W as
wij = 1 if municipalities i and j are neighbours,
wij = 0 otherwise, and
wii = 0
and a spatial autoregressive process as
y = 'Wy + μ , μ~ N(0,σ2I)
where ` is an autoregression parameter between -1 and 1. For the present case, ` is restricted to
the interval between 0 and 1 in order to be meaningfully interpreted. Further, W is - according
to common practice - rowstandardized, i.e. each element in the matrix is divided by the number
of elements in the row to which it belongs. By this, the i’th elements of the n-vector Wy is simply
the average of the y variable in the neighbours to the i’th municipality.