Thresholds for Employment and Unemployment - a Spatial Analysis of German Regional Labour Markets 1992-2000

cussion. The W matrix stores the information on the geographic map patterns, and is
derived from a binary contiguity matrix. The elements of the latter are equal to 1 for
neighbourhood regions and 0 otherwise. Moran's I can be expressed as a weighted sum
of the eigenvalues of the matrix

(3.2) C=(IN -11'/N)W(IN -11'/N),

where IN is the N-dimensional identity matrix and 1 a vector of ones, see Tiefelsdorf
and Boots (1995) and Griffith (1996). The eigenvectors of the C-matrix are utilized to
separate spatial from non-spatial components. Generally, spatial dependencies are rep-
resented by the system of eigenvectors, which identify distinct geographic map patterns.
The non-spatial part of a variable is given by the OLS residuals of a regression of that
variable on the significant eigenvectors, see Griffith (1996, 2000). Since the eigenvec-
tors are both near-orthogonal and near-uncorrelated,¹ forward stepwise regression can
easily applied for selection. Based on this approach, the model

(3.3) ^yt =^β0 ^+β1 ■ ^xt ⁺∑^γ j ∙<^ω j ^{+ v}t^,    t=^1,2^,^^,T,

S

can be estimated via OLS. Here, x* refers to the non-spatial component of the regressor;
the set S is formed by the relevant eigenvectors ωj of C-matrix. Eigenvectors must rep-
resent substantial spatial autocorrelations in order to be considered as relevant. Griffith
(2003, p. 107) suggests to assess substantial spatial autocorrelation on the basis of the
ratio MI/MImax, where MImax denotes the largest Moran coefficient of any eigenvector of
the C-matrix. According to his qualitative classification we use a threshold value of 0.25
for the selection of candidate eigenvectors. As the linear combination of eigenvectors
accounts for spatial dependencies, the errors are whitened.

Note that the decomposition is required for each point t in time, as the spatial patterns
may vary. Thus, (3.3) should be interpreted as a cross-section regression. However, de-
pendencies also exist over time. For example, shocks arising in the regional labour mar-

¹ Although the standardisation of the weight matrix generally comes along with a loss of orthogonality
and uncorrelatedness of the eigenvectors (Griffith, 1996), we make uniformly use of it because of its
more natural interpretation, see Ord (1975). In our application it turns out, that nearly all correlations
between the spatial components lie in a narrow range about zero.

<<   5   6   7   8   9   10   11   >>

More intriguing information