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The existence of differential spatial regimes has been revealed on different occasions (see for instance Arbia and Paelinck, 2003a, b; also see Chap. 14). Hence the necessity exists for developing workable specifications to compute possible frontiers or thresholds between those regimes.

The next section describes one possible method. Sects. 12.2 and 12.3 then apply it to two spatial models using Dutch data: an income generating model, and an activity complex model.

1 Method

Assume the following model

$${\textrm{y}}={\textrm{ax}},$$
((12.1))
$${\textrm{a = }}{{\textrm{a}}_{\textrm{1}}}{\textrm{ }}|{\textrm{ x }} \le {\textrm{x*,}}$$
((12.2))
$${\textrm{a = }}{{\textrm{a}}_{\textrm{2}}}|{\textrm{ x}} \ge {\textrm{x*}}{\textrm{.}}$$
((12.3))

The model can be respecified as follows, for each observation i

$${{\textrm{y}}_{\textrm{i}}}{\textrm{ = }}{{\textrm{a}}_{\textrm{1}}}{\textrm{ }}{{\textrm{u}}_{\textrm{i}}}{\textrm{ }}{{\textrm{x}}_{\textrm{i}}}{\textrm{ + }}{{\textrm{a}}_{\textrm{2}}}{\textrm{ (1 }} - {\textrm{ }}{{\textrm{u}}_{\textrm{i}}}{\textrm{) }}{{\textrm{x}}_{\textrm{i}}}{\textrm{ + }}{{\upvarepsilon }_{\textrm{i}}},$$
((12.4))
$${\textrm{(}}{{\textrm{u}}_{\textrm{i}}}-{ \upeta )(}{{\textrm{x}}_{\textrm{i}}} - {\textrm{x*) }} \le { 0, 0 <\upeta< 1,}$$
((12.5))
$${{\textrm{u}}_{\textrm{i}}}{\textrm{ = }}{{\textrm{u}}_{\textrm{i}}}^{\textrm{2}}{\textrm{,}}$$
((12.6))

the estimation being, e.g., performed by minimizing ɛ'ɛ. This model was first tested on the following data, appearing in Table 12.1.

Table 12.1 Test data for the model defined by Eqs. (12.4)–(12.6)
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The values for a 1 and a 2 are respectively 2 and 1, the first four observations being governed by a 1 . The value of x *, which is not unique, should lie between 8 and 11. Estimation reproduced the values of a 1 and a 2 , x * = 8,1333, with the u i s correctly partitioning the observations. The objective function adopted the value zero, and all restrictions and optimality conditions were satisfied.

If one wants to replace Eq. (12.2) or (12.3) by a strict inequality, the following specification could be used

$${\textrm{(}}{{\textrm{u}}_{\textrm{i}}} - {\upeta )(}{{\textrm{x}}_{\textrm{i}}}{\textrm{ }} - {\textrm{ x* }} - { \uptheta )} \ge {\textrm{0}}$$
((12.7))

where η is defined as in Eq. (12.5), and θ is an appropriately chosen small positive number. If x i – x * > θ, u i = 1; if x i – x * = θ, u i could be equal to either 0 or 1, whichever value would give the best estimation result; if x i –x * < θ, u i = 0. In both Eqs. (12.5) and (12.6) the specification (parameter η) prevents u i from being zero if the second factor of Eq. (12.7) is non-zero (positive or negative, depending on the case).

2 Spatial Income Generating Model

This model was initially developed in Paelinck and Klaassen (1979, pp. 21–23). Its aim is to measure the spatial interdependence between regional incomes or products.

Let y be the column vector of regional incomes; then the model in its simplest form is specified as

$$\textbf{y = Ay + b,}$$
((12.8))

where matrix A integrates some spatial interaction operator. In the present case, a first-order contiguity matrix, C 1 , has been selected for matrix A to compute total incomes over neighboring regions.

As can be seen from Fig. 12.1, in the Netherlands all regions, except for the provinces of Flevoland and Utrecht, are peripheral, with three purely maritime (Friesland, Noord- and Zuid-Holland), so five “pseudo-border” correcting (additive) parameters have been introduced into the model (Paelinck, 1996b, pp. 4–8). Moreover, the “reaction” parameters a and b have been split, according to Eq. (12.4) together with constraints (see Appendix).

Fig. 12.1
figure 12_1_214082_1_En

Map of the Dutch provinces

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Table 12.2 presents the data for regional (provincial) products (1987, Dfl 106; source: van Gastel and Paelinck, 1995, p. 152). Table 12.3 presents the spatial contiguity structure (degrees of contiguity) for the Netherlands (same source).

Table 12.2 Provincial products in the Netherlands, 1987
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Table 12.3 Contiguity structure of the Netherlands
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A typical equation for a region r now may be written as follows

$${{\textrm{y}}_{\textrm{r}}}{\textrm{ = }}{{\textrm{a}}_{\textrm{1}}}{{\textrm{u}}_{\textrm{r}}}{{\textrm{x}}_{\textrm{r}}}{\textrm{ + }}{{\textrm{a}}_{\textrm{2}}}{\textrm{(1}} - {{\textrm{u}}_{\textrm{r}}}{\textrm{)}}{{\textrm{x}}_{\textrm{r}}}{\textrm{ + }}{{\textrm{b}}_{\textrm{1}}}{{\textrm{z}}_{\textrm{r}}}{\textrm{ + }}{{\textrm{b}}_{\textrm{2}}}{\textrm{(1}} - {{\textrm{z}}_{\textrm{r}}}{\textrm{) + }}{{\textrm{c}}_{\textrm{r}}}{\textrm{ + }}{{\upvarepsilon}_{\textrm{r}}}{\textrm{,}}$$
((12.9))

where y r denotes a regional product and x r the sum of products in neighboring regions (here divided by 10, for reasons of magnitude similarity), u r and z r are binary variables, and c r denotes the “pseudo-border” coefficients previously mentioned.

Figure 12.2 hereafter has served to specify the condition referred to in Eq. (12.5). It suggests replacing x i of Eq. (12.5) by the ratio x r / y r .

Fig. 12.2
figure 12_2_214082_1_En

Specifying condition (12.5): the xr – yr relation

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Table 12.4 presents the results of the estimation procedure (derived by means of a constrained gradient method; see Fylstrom et al., 1998). Model (12.8) including interdependent endogenous variables, an SDLS estimation procedure (see Chap. 11, Sect. 11.1.3) was used, whose optimization principle, as said there, is the minimization of the sum of squared differences between the observed and the endogenously computed y r variables.

Table 12.4 Results of the estimation procedure of model (12.8)
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Rather than indicating two regimes, all four combinations of the reaction parameters are present: a 1 –b 1 (four times), a 2 –b 1 (five times), a 1 –b 2 (twice) and a 2 –b 2 (once, the case of Flevoland, a recent small new province). The value of the threshold is 0.3486. Corrective constants c r are positive for Noord- and Zuid-Holland (two heavily exporting provinces, with important harbors), and negative for Flevoland (see the remark above), the other corrections being negligible.

Figure 12.3 compares the observed and computed values of the endogenous variables; Theil’s U (Theil, 1961, p. 32) has value 0.0380, showing the close connection between observed and computed values.

Fig. 12.3
figure 12_3_214082_1_En

Observed and computed values of yr

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3 A Spatial Activity Complex Model

To subject the model to a more terse test, a so-called “attraction model” (first developed by Klaassen; see Paelinck and Klaassen, 1979, pp. 23–30) was evaluated.

Let y ir represent the production level (or value added) of activity sector i in region r; the model here is specified as

$${{\textrm{y}}_{{\textrm{ir}}}}{\textrm{ = }}{\sum _{\textrm{j}}}{\textrm{ }}{{\textrm{a}}_{{\textrm{ijl}}}}{\textrm{ y}}{{\textrm{*}}_{{\textrm{jr}}}}{\textrm{ + }}{{\textrm{b}}_{{\textrm{il}}}}{\textrm{,}}$$
((12.10))

where the y * jr represent spatially discounted (from Table 12.3) aggregations (potentials) of activity sector production levels (index j). The index l denotes the relevant regime to which the parameters belong.

Nine endogenous sectors were employed (Table 12.5) for the Netherlands case study.

Table 12.5 Endogenous activity sectors
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Appendix 12.5 reports their production (value added) levels. The following sectors have been considered as exogenous: agriculture, forestry and fishery (10), crude oil and gas plants (11), public sector (12).

Because only 12 observations per activity sector are available, aggregated explanatory variables were constructed, whereby contiguous regions have been exogenously discounted at 50%, the aggregation being the following ones: 10, 11; 1, 2; 3 through 9, 12.

Figures 12.4 through 12.7 portray the y ir / y * jr ratios (inverses of the ratios used in Sect. 12.2) and the y ir / ∑ r y ir one for sector 1; they suggest (at least) three regimes.

Fig. 12.4
figure 12_4_214082_1_En

Distribution of xt/yt ratios fot the four Dutch regions

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Forty-eight binary variables are to be used for only 2 regimes, generating a heavy 0–1 mathematical program; no solution could be reached within 40 hours, so a different strategy was developed.

For n observations and k relevant parameters, combining the relative frontiers between the values of Figs. 12.4, 12.5, 12.6, and 12.7, generates [n(n–1)/2] k simple quadratic programs. In the present exercise the cut-off points have been selected visually, so sub-optimal outcomes might be expected.

Coming back to parameters a ijl, and b il of Eq. (12.10), some outliers might produce reverse results compared to a general trend as revealed, e.g., in Fig. 12.2; therefore an extra condition was introduced, to wit the sign equality of parameters a ijl , , ∀l. A way to impose that constraint would be

$${\textrm{abs}} \left({\sum _{\textrm{l}}}{\textrm{ }}{{\textrm{a}}_{{\textrm{ijl}}}}\right) ={\sum _{\textrm{l}}}{\textrm{ abs (}}{{\textrm{a}}_{{\textrm{ijl}}}}{\textrm{),}}$$
(12.11)

or, alternatively, binary conditions such as

$${{\textrm{a}}_{{\textrm{ijl}}}}{\textrm{*}}{{\textrm{a}}_{{\textrm{ijm}}}} \ge {\textrm{0,}}$$
((12.12))

which would lead to a quadratic program with non-linear constraints. In practice, and in order to obtain classical results, necessary conditions [like those exposed through Eqs. (12.11) or (12.12)] have been replaced by inspecting all possible sign combinations (i.e., 23=8, as the level parameters, b il , were left free).

Finally, to allow for comparability of results, logarithms are used, yielding non-dimensional elasticities as parameters. Table 12.6 presents the estimation results.

Table 12.6 Estimation results of model (12.10)
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One finding is that only two sectors (1 and 9) satisfy without constraint the equal sign condition; they also show great similarity inside each parameter group, and, as should be expected, the lowest U-values. Sectors with one active constraint number four, sectors with two number two, and only one sector needed all three constraints. The sectors needing two constraints also show the highest U-values (the only ones exceeding 0.05), even higher than the sector needing all three constraints.

At this stage, no SDLS-computations were performed, despite the interdependent specification of the model. Seven out of nine U-values are sufficiently low that no further corrections were deemed necessary (the two high U-values are probably related, to regional accountancy for sector 5, and the requirement of another specification, in terms of explanatory variables, for sector 8).

No significance magnitudes have been shown. However, they could be computed deleting the series of variables corresponding to the zero values.

Moreover no detailed study was made of the different combinations of a ijl and b jl parameters as presented in Table 12.6 (nine tables would be necessary for such a comparison). From that table, however, a remarkable finding can be derived, namely that parameters a 2 l are all non-negative in the solution. As for the other a il parameters, five out of nine are non-positive, but in this exploratory study no detailed analysis has been made of the various cases.

4 Conclusion

A workable method to flexibly select parameter regimes has been presented. It is a member of a class of non-standard estimators, many of which will be used in further spatial econometric work. They are indispensable companions of non-standard specifications that will also be required in the field of spatial econometrics.

Recent experience has indeed shown that systems of regions often reveal two regimes. In Coutrot et al. (2009), the introduction of a second regime lifted the R 2 from 0.5156 to 0.9990. Moreover, the regions were behaviorally very different, which separated the main regional activity poles from the minor centers. Another example will be provided in Chap. 14.

In the light of this, still limited, experience, it seems that an appropriate specification-cum-estimation strategy is to systematically test for the presence of multiple regimes.