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1998
La science régionale : Journées d'études en l'honneur de Jean Paelinck
46

Provincial-level spatial statistical modelling of the change in per capita disposable Family Income in Spain, 1975-1983

Modélisation spatiale du changement de revenu disponible par famille au niveau des provinces en Espagne, de 1975 à 1983
Daniel A. Griffith

Résumé

Computational simplifications for a space-time autoregressive response model specification are explored for the change in Spain's per capita disposable family income between 1975 and 1983. The geographic resolution for this analysis is the provincial partitioning of part of the Iberian peninsula into Spain's 47 coterminous provinces coupled with its 3 island clusters provinces. In keeping with the Paelinckian tradition of spatial econometrics, exploration focuses on both new spatial econometric estimators and model specifications that emphasize the capturing of spatial dependency effects in the mean response term. One goal of this analysis is to differentiate between spatial, temporal, and space-time interaction information contained in the per capita disposable family income data. A second objective of the application is to illustrate the utility of extending computational simplifications from the spatial to the space-time domain. And a third purpose is to gain some substantive insights into the economic development of one country in a changing Europe. A serendipitous outcome of this investigation is a detailed analysis of locational information latent in Spain's regionally disaggregated per capita disposable family income.

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Introduction

1Attempts to model space-time data series are hampered by a lack of theoretical understanding of parameter estimators (econometric estimation remails a difficult exercise), structural complexity (attributable to multiple sources of latent correlations), and data availability (information tagged to both location and time). Paelinck (1990) has addressed this first issue in an enlightening fashion, proposing five potentially useful additions to the assortment of existing estimators. He outlines a rationale for, relevant properties of, empirical applications of, and possible extensions to each of his proposed estimators. One of his conclusions is that "enormous possibilities ... still remain for contributing to this" body of knowledge. A principal objective of this paper is to make such a contribution. Paelinck also addresses the second aforementioned difficulty, but in a less formal manner, when he publicly promotes his preference to represent spatial autocorrelation in first rather than second moment model specifications. Another goal of this paper is to advance a means of achieving this particular end. Synthetic mean response surrogates of spatial autocorrelation are proposed and analyzed. A third, and final, major purpose of this paper is substantive in nature, and seeks to provide insights into the economic development of one country in a changing Europe, namely Spain. This substantive theme is in keeping with the abundance of empirical work Paelinck has undertaken concerning the regional economics of Europe.

Purely Spatial Models

2Popular spatial autoregressive models involve specifications that cast spatial autocorrelation as a covariance parameter. Frequently the detected spatial autocorrelation is moderate in degree, and positive. A principal estimation difficulty in this case involves the normalizing factor (the Jacobian of the transformation from an autocorrelated to an unautocorrelated mathematical attribute space). Griffith and Sone (1995) discuss one computational simplification for this estimation process. But following Paelinck's suggestion may well eliminate the need for explicitly dealing with this normalizing factor altogether.

3Each of the five individual years of Spanish data may be analyzed with purely spatial models. In order to do so, the Jacobian approximation discussed by Griffith and Sone yields the following results for a row-standardized version of the geographic connectivity matrix (see Figure 1 for the associated surface partitioning) for the 50 Spanish provinces:

λ1 = 1 λ50 = -0.58293

α1 = 5.93090 α2 = 19.06230

δ1 = 1.03947 δ2 = 1.05423

10-3

4The computational simplification based upon this approximation yields estimates of the autoregressive response (AR) model spatial autocorrelation parameter, ρ, reported in Table 1.

5Results appearing in Table 1 suggest a number of informative implications based upon the log-income data. First, weak-to-moderate positive spatial autocorrelation prevails across the five years of these log-income data. Second, on average this spatial autocorrelation accounts for roughly 50% of the variation displayed by these log-income data across Spain. These log income data tend to exhibit homogeneity of variance across provinces. And, each year of these log-income data seems to conform to a normal distribution.

6These preceding findings can be contrasted with spatially adjusting ones. The log-income for each year was regressed on the eigenvectors of

7(I - 11t/n)C(I - 11t/n), (1)

8which represent orthogonal and independent components of spatial autocorrelation latent in these data (see Griffith, 1984) having means of zero [the matrix (I - 11t/n) appearing in equation (1) reprojects the eigenvectors of matrix C in order to achieve this result]. Three eigenvectors consistently were found to be significant correlates of the log-income data over the five years, and are denoted here by E1, E2, and E21. The percentages of variance accounted for in the log-income data by, as well as the corresponding Moran Coefficients (MC) and Geary Ratios (GR) for, these three eigenvectors are as follows:

9On average these three eigenvectors account for approximately 50% of the variation exhibited by log-income across the provinces of Spain. The first two of these eigenvectors represent very strong positive spatial autocorrelation patterns, whereas this third eigenvector represents a weak-to-moderate positive spatial autocorrelation pattern. As depicted in Table 1, the OLS residuals after regressing log-income on these three eigenvectors (i.e., the spatial adjustment) exhibit dramatically less spatial autocorrelation. The single province revealing somewhat aberrant behavior across these time periods is Baleares (comprised of the islands of Mijorca, Minorca, Iviza, and Formentera; see Figure 1). Oddly enough, behavior of the other two island provinces (Las Palmas and Santa Cruz de Tenerife) seems to conform to that of the rest of the country. Meanwhile, the following nine provinces appear to be influential observations in this regression: Ciudad Real, Córdoba, Jaén, León, Navarra, Teruel, Valladolid, Zamora, and Zaragoza. Interestingly, this set of provinces forms three clusters of units that essentially triangulate Madrid. Finally, each of the residual sums of squares are roughly 86% of the PRESS statistic, suggesting a respectable degree of agreement amongst the 50 jackknife-type sub-regressions.

10Results for the mean response spatial adjustment also are reported in Table 1. For every one of the five years the measures of spatial autocorrelation have plummeted while conformance to a normal distribution has markedly improved, whereas displayed variance homogeneity has tended to noticeably but not bothersomely decrease. The percent of variance accounted for in log-income by the three eigenvectors is very similar to that accounted for by the spatial lag variable, WY, appearing in the AR model. Since these three eigenvalues account for only a portion of the variance displayed in the spatial lag variable, WY, the spatially adjusted estimation of ρ must be done sequentially (i.e., on the residuals from the OLS regression of Y on E1, E2, and E3), which places it into a diagnostic mode. The original AR estimates of ρ roughly average 0.72, which intuitively seems significant. The spatially adjusted AR estimates of ρ average roughly 0.26, whose significance is not quite so intuitive. One of the findings of Griffith and Can (1996) is that the value of sρ   computed by SAS may well be a poor estimator of its corresponding standard error. Further exploring this contention within the context of the Spanish data being analyzed in this research has yielded the following estimates (see Upton and Fingleton, 1985):

1975 is at the 10% level, and  ρ 1983 is at the 5% level, using df = 45). Following a suggestion put forth in Griffith and Can (1996), a "quick and dirty" sρ estimator is given by , which in this case equals 0.09578. This particular approximation appears to underestimate σρ though. Perhaps a much better estimator is the average of this "quick and dirty" estimator and the SAS estimator (a Stein-type shrinkage estimator); the foregoing results for this estimator show that it holds some promise.

11Because five years of data are being analyzed simultaneously, then a Bonferroni adjustment should be made to the Type I error rate in order to maintain it at a prespecified "family-wide" level, ideally converting it from, say, 5% to 1%. This adjustment appears to reverse the statistical decisions regarding the 1975 and 1983 estimates of ρ. But a Bonferroni adjustment assumes independence, while these geographic distributions of log-income residuals have the following correlation structure:

12Because some of these correlations exceed 0.9, the actual Bonferroni levels will be discernibly greater than 0.01.

13In conclusion, the purely spatial models estimated for each of the five time periods suggest that Paelinck's recommendation to incorporate spatial autocorrelation into the mean response can be done successfully by utilizing eigenvectors of the geographic weights matrix. Eigenvectors still need to be computed, which is a computationally intensive and computer resources demanding task, but pursuing this specification allows spatial statistical analysis to remain in the realm of traditional regression theory. This approach also alludes to a nice simplification for space-time modelling, since the necessary geographic weights matrix eigenvectors remain constant across time periods.

Space-time Modelling

14The Jacobian approximation discussed by Griffith and Sone has been used in the preceding section to simplify estimation of the purely spatial AR model. This simplification has been extended by Griffith (1996) to selected space-time models. The neighborhood forecasting model specification to be explored here is of the form

15where Y is the log-income variable, ρs and ρT respectively are the spatial and temporal autocorrelation parameters, and ni is the number of neighboring provinces of a given province i. The Jacobin of the transformation for this space-time model asymptotically has the following eigenvalues:

λj,s are the n eigenvalues for the purely spatial geographic weights matrix C, and ωk are the k eigenvalues for the time series structure at each location i. Here the total number of observations equals nT = 50x5 = 250, and ωk = 2COS(kπ/6) which is the set {, 1 0, -1, }. Behavior of this Jacobian term across its feasible parameter space is portrayed at a rather coarse level of resolution in Figure 2. The three-dimensional Jacobian term surface has a trough at point (0,0,0), and ascends in all directions from this low point. All four of the visual perspectives appearing in Figure 2 indicate that this Jacobian term surface acts very much like its pure spatial autoregressive counterpart; there is a relatively smooth non-linear, concave-upwards curve spanning the extremes of the feasible parameter spaces for ρs [upper right-hand perspective in Figure 2] and ρT [lower left-hand perspective in Figure 2], and an asymmetry exists in the spatial domain [lower right-hand perspective in Figure 2].

Figure 2

16Equation (3) highlights an important feature of space-time autocorrelation, which accents part of the asymmetry depicted in Figure 2, namely that in order to ensure interpretability the two autoregressive parameters ρs and ρT need to be constrained; their respective separate feasible parameter spaces are (-0.36479, 0.18619) and (-1, 1). Equation (3) is based upon a matrix premultiplied by its transpose, which necessarily means the equation always is positive. The original matrix is asymmetric, because of the uni-direction of time dependencies, which complicates an analytical evaluation of equation (3). Nevertheless, one attractive feature of equation (2) is that the geographic surface partitioning remains constant over time, meaning the mean response accommodation of spatial autocorrelation outlined in the preceding section has the potential of becoming easier to implement.

17An initial step in the identification of a model describing some space-time data set is to check for non-stationarities. Variance heterogeneity already has been addressed in the preceding section by subjecting income data to a logarithmic transformation. Another common concern has to do with the presence of any systematic change in the level of the log-income data over time. Such a complication is managed in conventional time series analysis by including differencing. Invoking the notion of space-time exchangeability discussed by Arbia (1989), this issue can be explored by concatenating the five sequential geographic distributions of log-income (e.g., creating a 250-by-1 vector Yt by stacking Y1983 on top of Y1981, and then stacking this new vector on top of Y1979, and then this newer vector on top of Y1977, and finally this newest vector on top of Y1977), and the corresponding five sequential (spatially adjusted) geographic distributions of residuals from the OLS regressions computed as described in the preceding section. Time series structure in this aggregate vector can be analyzed by pretending that only seasonalities exist. Thus, since there are 50 Spanish provinces, using a seasonal ARIMA model of order 50 will exploit the temporal correlation between values for the same provinces in successive years. This specific formulation of the data analysis problem enables 50 time series, each consisting of five points in time, to be pooled together (the exchangeability concept) for estimation purposes. Estimation results from this specification for the Spanish data are as follows:

18Because the residuals of an OLS regression have a mean of zero, by construction, the intercept term for this spatially adjusted ARIMA model being essentially zero is not surprising. In both cases the differencing was required in order to achieve stationarity; the estimate of ρT for the undifferenced log-income model tried to exceed unity, and hence convergence was not achieved, whereas the estimate of ρT for the undifferenced spatially adjusted log-income model was 0.951 but the back-forecasts failed to rapidly dampen. Therefore, these exploratory findings suggest that the yi,t values in equation (2) first need to be temporal differenced.

19The second step in the identification of a model describing some space-time data set is estimation of the parameters of equation (2), which for the log-income data yields

20These results reveal two salient features of this particular space-time data set. First, the presence of spatial autocorrelation suppresses the serial correlation parameter estimate. Second, the spatially adjusted data (residuals from regressions of the log-income variable on the three aforementioned eigenvectors, E1, E2, and E21) render estimates consistent with those reported for the corresponding ARIMA modelling efforts, mainly that a first order temporal differencing is called for here. Of note, again, at this point is that this spatially adjusted estimation is sequential in nature, which is somewhat counter to prescriptions described in Griffith (1981); space-time correlation structures tend not to be decomposable into independent space and time components. As mentioned earlier, a sequential procedure is necessary because only three of the 50 eigenvectors summarizing spatial structure are being employed. Furthermore, removing different means for the individual geographic distributions (one for each time period) is consistent with the preceding ARIMA finding of a non-zero constant for the differenced series. The implication here, then, is that equation (2) needs to be rewritten as

21where Δt is the first order temporal difference operator, which transforms the log-income data by subtracting a given province's values of log-income for immediately adjacent time periods.

22The third step in this modelling task is estimation of the parameters of equation (4), which for the log-income data yields

23Of note is the similarity between these estimation results for ρT, and those previously reported ones obtained with the purely time-series ARIMA model. The estimate of µ has changed signs principally because (1 - ρsni)µ is the intercept term now. Meanwhile, the relative spatial autocorrelation parameter estimate decreases from 0.10090/0.18619 0.54192 to 0.03253/0.18619 0.17471 through use of the sequential eigenvector mean response spatial adjustment, a marked decline in the spatial autocorrelation present in the adjusted data, and a magnitude of decline that is in keeping with findings reported earlier in this paper for the purely spatial models.

24One striking difference between the spatial adjustments utilized for equations (2) and (4) is that equation (2) employs eigenvectors E1, E2, and E21 for all time periods, whereas equation (4) employs a varying assortment of eigenvectors. The percentages of variance accounted for in the differenced log-income data by, as well as the corresponding Moran Coefficients (MC) and Geary Ratios (GR) for, these sundry eigenvectors are as follows:

25These four subsets of eigenvectors were delimited by including only ones with marked percentages of variance accounted for in each response variable, by including only ones that essentially have a statistically significant MC (E42 is marginal), and by eliminating ones portraying negative spatial autocorrelation. On average, the various uncovered subsets of eigenvectors account for roughly 43% of the geographical variance displayed by a given differenced log-income. The loss of a constant set of statistically important eigenvectors across the time periods again suggests the presence of non-stationarities. Nevertheless, relatively few synthetic variates are able to capture much of the positive spatial autocorrelation displayed by each temporally differenced geographic distribution of log-income.

26In conclusion, the space-time models estimated using the five time periods suggest that Paelinck's recommendation to incorporate spatial autocorrelation into the mean response can be done successfully by utilizing eigenvectors of the geographic weights matrix for each time period. Pursuing this specification allows spatial statistical analysis to better align with traditional Box-Jenkins methodology.

Implications

27Contents of this paper constitute another contribution to the spatial econometric estimation literature in a three-fold way. First, concatenating geographic distributions of a certain variable for sequential points in time, and then estimating a seasonal ARIMA model with the resulting nT-by-1 data vector is a imaginative and enlightening procedure for pooling space and time series for exploratory purposes. This approach is especially useful when the temporal dimension of space-time data is the dominant one, which often is the case. Second, using equation (3) to computationally simplify the space-time Jacobian term aids in implementing equations (2) and (4). The possibility of further simplification, like that derived for purely spatial models, is suggested by Figure 1 and is outlined in Griffith (1996). Third, the extraction of eigenvectors from expression (1) enables spatial autocorrelation to be successfully incorporated into a mean response, as preferred by Paelinck, in order to create spatially adjusted synthetic variates. The success of this technique is demonstrated here in the purely spatial domain as well as in the space-time domain.

28Uncovering non-stationarities in this empirical case is not surprising, since regional economic theory describes the prevalence of leading and lagging regions constituting a space economy. This notion is highlighted in the space-time data analysis presented in Griffith (1992), too. But the emphasis throughout this paper is on the computation of parameter estimators and the contrasting of model specifications incorporating spatial structure, not on this underlying theory. Even though very little is known about the spatial autocorrelation adjustment mechanism proposed in this paper, the author's experience to date is that the results it yields, taken as a whole, can lead to valid statistical outcomes. Findings summarized in this paper hopefully will encourage practitioners to use this simple mean response spatial autocorrelation specification as a routine part of their exploratory spatial data analyses.

29Finally, the data analysis example employed in this paper also furnishes insights into the economic development trajectory of Spain. Not surprisingly, income by province follows a skewed distribution, one that becomes quite symmetric through the use of a logarithmic transformation. The time series of income is non-stationary, as well, requiring first order differencing in order to regain stationarity. The average log-income is time dependent. There is an interaction between the time and spatial series manifesting itself in such a way that: the temporal autoregressive parameter is positive in the presence of spatial autocorrelation, and negative in the filtered absence of spatial autocorrelation. As is frequently the case, the latent level of spatial autocorrelation is moderate and positive. Income for the island provinces tends to behave the same way it does for the coterminous provinces. Clusters of influential regression analysis provinces circumscribe Madrid. And, the neighborhood model provides a good description of the space-time distribution of disposable family income across Spain. Therefore, this model should be able to render useful forecasts of the geographic distribution of disposable family income, aiding economic development decision making in Spain.

APPENDIX I

SAS CODE FOR COMPUTING THE MORAN COEFFICIENT AND THE GEARY RATIO

FILENAME ATTRIBUT 'C:\SPAIN\SPAIN.DAT'; 1

FILENAME CONN 'C:\SPAIN\SPAIN.CON'; 2

OPTIONS LINESIZE=72; 3

TITLE 'MC AND GR FOR SPAINISH INCOME DATA'; 4

DATA STEP1; 5

INFILE ATTRIBUT; 6

INPUT PROVNUM1 DUM1 DUM2 INCOME1-INCOME5 NAME$; 7

Y = LOG(INCOME1); 8

RUN; 9

PROC STANDARD MEAN=0 STD=1 OUT=STEP1(REPLACE=YES); VAR Y; RUN; 10

 

DATA STEP2 (REPLACE=YES); 12

SET STEP1; 13

INFILE CONN; 14

INPUT PROVNUM2 C1-C50; 15

ARRAY CONY{50} CY1-CY50; 16

ARRAY CON{50} C1-C50; 17

CSUM = 0; 18

DO I=1 TO 50; 19

CSUM = CSUM + CON{I}; 20

CONY{I} = Y*CON{I}; 21

END; 22

Y2 = CSUM*Y**2; 23

RUN; 24

PROC MEANS NOPRINT; 25

VAR CY1-CY50; 26

OUTPUT OUT=CYOUT1 SUM=CY1-CY50; 27

RUN; 28

PROC TRANSPOSE DATA=CYOUT1 PREFIX=WY OUT=CYOUT2;29

VAR CY1-CY50; 30

RUN; 31

DATA STEP2 (REPLACE=YES); 32

SET STEP2; 33

SET CYOUT2; 34

CY = WY1; 35

INT=1; INT2=1; 36

RUN; 37

***********************

* *

* COMPUTING MC AND GR *

* *

***********************;

 

PROC REG DATA=STEP2 OUTEST=OUTNUM; MODEL CY = Y/NOINT; RUN; 45

PROC REG DATA=STEP2 OUTEST=OUTDEN; MODEL CSUM = INT/NOINT; RUN; 46

PROC REG DATA=STEP2 OUTEST=OUTGR; MODEL Y2 = INT2/NOINT; RUN; 47

 

DATA STEP3; 49

SET OUTNUM (KEEP=Y); 50

MCNUM=Y; 51

SET OUTDEN (KEEP=INT); 52

MC = MCNUM/INT; 53

SET OUTGR (KEEP=INT2); 54

 

N = 50; 56

CSUM = N*INT; 57

 

GR = INT2/INT - ((N-1)/N)*MC; 59

 

ZAPP = (MC+1/(N-1))/SQRT(2/CSUM); 61

PROBMC = 1 - PROBNORM(ZAPP); 62

RUN; 63

 

PROC PRINT; VAR N CSUM MC ZAPP PROBMC GR; RUN; 65

Lines 1-2 define the paths to the digital files housing the attribute data and connectivity matrices.

Lines 5-9 retrieve the attribute data and compute the logarithm of the first income variable.

Line 10 converts the log-income values to z-scores.

Lines 12-22 retrieve the connectivity matrix, compute the row sums of this matrix, and begin computing the spatially lagged log-income z-scores.

Line 23 computes the weighted squared deviations needed for calculating GR.

Lines 25-31 complete the calculation of the spatially lagged values.

Lines 45-47 execute the regression analyses needed to compute MC and GR.

Line 53 computes MC, and Line 59 computes GR.

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Bibliographie

Anselin, L. (1995) "Local indicators of spatial association LISA," Geographical Analysis, 27: 93-115.

Arbia, G. (1989) "Estimating economic relationships with temporal and spatial data," paper presented to the NARSA, Santa Barbara, CA, November 10-12. Revised version published in Italian as "L'use e l'interpretazione di regressioni basate su dati territoriali nella modellistica economica," in Sistemi Spaziali Approcci e Methodologie, edited by M. Bielli and A. Reggiani. Milan: FrancoAngeli, 1991, pp. 57-72.

Griffith, D. (1981) "Interdependence in space and time: numerical and interpretative considerations," Dynamic Spatial Models, edited by D. Griffith and R. MacKinnon. Alphen aan den Rijn: Sijthoff and Noordhoff, pp. 258-287.

Griffith, D. (1984) "Measuring the arrangement property of a system of areal units generated by partitioning a planar surface," in Recent Developments in Spatial Analysis: Methodology, Measurement, Models, edited by in G. Bahrenberg, M. Fischer and P. Nijkamp. Aldershot, UK: Gower, pp. 191-200.

Griffith, D. (1992) "Model identification for estimating missing values in space-time data series: monthly inflation in the U.S. Urban System, 1977-1990," Economic Evolution and Demographic Change: Formal Models in Social Sciences, edited by G. Haag, U. Mueller, and K. Troitzsch. Berlin: Springer-Verlag, pp. 192-218.

Griffith, D. (1996) "Spatial statistical analysis and GIS: exploring computational simplifications for estimating the neighborhood spatial forecasting model," Spatial Analysis: Modelling in a GIS Environment, edited by P. Longley and M. Batty. London: Longman GeoInformation, pp. 255-268.

Griffith, D. and A. Can. (1996) "Spatial statistical/econometric versions of simple urban population density models," in Handbook of Spatial Statistics, edited by A. Arlinghaus. Boca Raton, FL: CRC Press, pp. 231-249.

Griffith, D., and A. Sone. (1995) "Trade-offs associated with normalizing constant computational simplifications for estimating spatial statistical models," Journal of Statistical Computation and Simulation, 51: 165-183.

Haining, R. (1990) "The use of added variable plots in regression modelling with spatial data," The Professional Geographer, 42: 336-345.

Paelinck, J. (1990) "Some new estimators in spatial econometrics," in Spatial Statistics: Past, Present, and Future, edited by D. Griffith. Ann Arbor, MI: Institute of Mathematical Geography, pp. 165-178.

Upton, G., and B. Fingleton. (1985) Spatial Data Analysis by Example, vol. 1. NY: Wiley.

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Daniel A. Griffith, « Provincial-level spatial statistical modelling of the change in per capita disposable Family Income in Spain, 1975-1983 », Cybergeo : European Journal of Geography [En ligne], Dossiers, document 46, mis en ligne le 20 février 1998, consulté le 28 novembre 2020. URL : http://journals.openedition.org/cybergeo/385 ; DOI : https://doi.org/10.4000/cybergeo.385

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Auteur

Daniel A. Griffith

Department of Geography, Syracuse University, United States of AmericaGRIFFITH@MAXWELL.SYR.EDU

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