





































Microsoft Word - 01_Szulc_E.docx


DYNAMIC ECONOMETRIC MODELS 
Vol. 11 – Nicolaus Copernicus University – Toruń – 2011 

Elżbieta Szulc 
Nicolaus Copernicus University in Toruń 

Identification of the Structures of Spatial and Spatio-
Temporal Processes and a Problem of Data Aggregation 

A b s t r a c t. The paper concerns the measurement of the dependence between economic spatial 
and also spatio-temporal processes at various levels of data aggregation. The considerations refer 
to the investigations confirming efficiency of the so-called quasi-congruent spatial model as a tool 
of the measurement of the dependence between economic processes. The main problem of the 
paper is to discuss such a description of the spatial and spatio-temporal connections, which would 
be adequate to express the autodependence of the investigated processes. The application of the 
economic distance, which characterizes the similarity of the regions on the ground of the values 
of the analyzed processes, is proposed. 

K e y w o r d s: quasi-congruent spatio-temporal model, autodependence, connectivity matrix, 
economic distance.  

Introduction 

 Spatial models are estimated for the given spatial units, such as: countries, 
regions, provinces (voivodeships), census tracts and so on. Data, usually 
spatially correlated, collected for these units create the established 
configurations which determine quantification of the spatial autodependence. 
Configurations of the aggregated data simplify the connections of the smaller 
spatial units and then they may lead to the simplified description of some 
properties of the economic spatial processes and the dependence between them1. 
This is also the case of the spatio-temporal processes. 

 In the paper attention is paid to the spatial aspect of the data aggregation. 
The question of the time aggregation is not taken into account. Since the paper 
concerns the investigation of the spatial processes in time, some dynamic 
elements were introduced to the analysis. 

                                                 
1 See e.g. Arbia (1988) for an early discussion of some of the issues. 



Elżbieta Szulc 6 

 The presented considerations refer to the previous investigations carried out 
on the basis of generated data (Szulc et al., 2011) and empirical data for one 
year (Szulc, 2011). At present the time range of the analysis has been extended 
to six years. 

 In Szulc et al. (2011) there was formulated and verified the thesis that 
finding out the value of the parameter which measures the actual influence of 
one spatial process on another spatial process needs to identify the properties 
and componential structures of the processes and to take them explicitly into 
account in the spatial regression model. 

 Formulation of the hypothesis mentioned above means the recommendation 
of construction of the quasi-congruent spatial model2 in which, thanks to taking 
into account the components describing the internal structure of the processes, 
the filtration of the processes takes place and the estimation of the dependence 
is more precise, as not disturbed by the so-called indirect influences. Finally, on 
condition of the precise filtration, the regression coefficients in the quasi-
congruent spatial models estimated at the individual level should be the same 
(identical) as the ones estimated at the aggregated level. The hypothesis and its 
consequences may be extended to the case of the analysis of the spatio-temporal 
processes. 

 In the quoted work there was shown, that the quasi-congruent spatial model 
was an effective tool of the measurement of the dependence between spatial 
processes (actual value of the regression coefficient), if the model description of 
the structures of the processes was correct and sufficient. The differences 
between the values of the regression coefficient obtained for the quasi-
congruent models before and after the aggregation of the data were dependent 
on the structures of the separate processes. For the processes only with the trend 
structure, regardless of the degrees of the trend and of the values of their 
parameters, the regression coefficients estimated at the various levels of data 
aggregation did not differ from one another significantly and oscillated around 
the actual value. On the contrary, in terms of the spatial autocorrelation, the 
value of the regression coefficient after the aggregation was overestimated, 
when y > x , and underestimated, when y < x

3.  

 The regression coefficients in the spatial quasi-congruent models for the 
primary data and for the aggregated data with the spatial autocorrelation will  
differ in value first of all because the connectivity matrix in the aggregated 
model does not reflect the connections of the data before the aggregation. Thus, 
the autodependence existing at the individual level cannot be precisely 
described at the aggregated level. Moreover, the quantification of the spatial 

                                                 
2 The procedure of constructing such a model was described in the quoted work. It was also 

explained what the quasi-congruency of the spatial model meant. 
3 y and x denote respectively the autoregression coefficients of the explaining and 

explanatory process. 



Identification of the Structures of Spatial and Spatio-Temporal Processes… 7

dependence by means of the standard matrixes of the connections, within the 
scheme of the structure of the spatial connections existing at the given level of 
data aggregation as well, may be inadequate. This is the situation, when the 
strength of the connection between the spatial units depends not only on the 
physical distance and the fact of being contiguous, but also on e.g. the economic 
similarity of the units. 

 Looking for the more precise measurement of the spatial autodependence 
with the help of other than the standard matrixes of connections, in Szulc (2011) 
there was investigated the question of how the application of the certain matrix 
of the economic distance would influence the estimation of the dependence 
between the spatial processes. In the quoted investigation the difference 
between the parameters of the models for the data before aggregation and for 
the aggregated data slightly decreased, but it should not be treated as regularity. 
All the more it should not be expected that the difference will disappear because 
the proposed matrix is still based on the common border criterion. 

 The present investigation is the continuation of the works quoted above. 
Analogical analyses are carried out for the successive years in the period 2004–
2009. This approach allows to verify whether the observed regularities for 2007 
(see Szulc, 2011) are the same for other years. Moreover, the spatio-temporal 
models are considered. In the investigation both the standard connectivity 
matrixes and the matrixes of the economic distance are used. The economic 
distance matrixes in the spatio-temporal models have the dynamic features in 
the sense that for each of the years another connectivity matrix is applied. 

1. Quantification of the Spatial Connections and the Measurement 
of the Spatial Autodependence 

 The basis of measurement of the spatial (auto)dependence is establishing  
the connections among spatial units. The spatial neighbours can be defined in 
a number of ways. 

 The spatial connections are represented in the form of connectivity matrix 
W. Assuming that there are N regions (spatial units), the matrix has as many 
rows and columns as there are the regions, i.e. N by N matrix W is considered. 
Each row of the matrix contains non-zero elements in columns which 
correspond to the connected objects (the so-called neighbours), according to the 
received criterion. Furthermore, the given object cannot be connected to itself, 
i.e. it cannot be a neighbour of itself, so wij = 0 for all i = j. Thus, the diagonal 
elements of W are zeros. 

 Starting point in establishing the spatial connections is the binary matrix S, 
with elements: 

 
 








,if,0

if,1

iNj

iNj
sij  (1) 



Elżbieta Szulc 8 

where N(i) denotes the set of  neighbours of spatial units i. 

 The neighbours usually are established according to the common border 
criterion. Then the rows in S are normalized, so that the row sums equal 1, as 
a result of dividing each entry on a row by the sum of the row values (the so-

called row standardization to one). So, if  iddiagD , where 




 N

j
ij

i

s
d

1

1
, i = 1, 

2, …, N, then W = DS and 1
1




N

j
ijw . 

 The weights wij which are established in this way signify that each j-th 
neighbour of the i-th spatial units is treated identically, and the greater the 
strength of its interactions with the neighbours is, the less neighbours it has. 

 Another is the case when weights wij are the functions of some properties of 
cells of the lattice, e.g. of the length of the common border, of the distance 
between the centres of the regions, or of other measures of similarity between 
the regions, e.g. of the so-called economic distance between them as well. 
Various types of weights wij may be pointed out according to the established 
criteria (see Haining, 2005, p. 83–84 ). As above, for all units i and j wij  0, 
when i  j, and wij = 0, when i = j. Such weights create the generalized matrix of 
neighbourhood, which is row standardized to one by transforming its elements 

according to the formula 




 
N

j
ij

ij
ij

w

w
w

1

. 

 In the paper the connections between the regions will be defined by using 
two approaches. The first one is traditional. In it the standard matrix of 
neighbourhood with the equal weights is used. The second one consists in that 
in the matrix of connections the economic distance (the essence of which is to 
establish similarity of the regions on the basis of the value of the analyzed 
economic process and the economic processes which determine its 
changeability) is taken into consideration. 

 In the general form the economic distance between regions i and j is 
expressed as follows (see Pietrzak, 2010, p. 75): 






























 














,for,0

for,||
1

1
...

||
1

1
||

1

1
||

0
,,

0
,2,2

20
,1,1

1

1

2
3

1
21

ji

jixx
k

xx
k

xx
k

yy

d
n

n

k

z
ztnjztni

n

k

z
ztjzti

k

z
ztjztiji

ij




(2) 



Identification of the Structures of Spatial and Spatio-Temporal Processes… 9

where: yi, yj – values of the investigated spatial economic process at spatial 
location i and j respectively; xli, xlj (l = 1, 2, …, n) – values of explanatory 
processes, which determine changeability of the explained process at spatial 
locations as above; k1, k2, …, kn – constants denoting time lags for the 
considered processes; 1, 2, …, n+1 – normalizing constants. 

 In this approach, the elements wij of the matrix of neighbourhood will be as 
follows: 












.

,
1

ji

ji
dw ijij

if0,

if
 (3) 

 Finally, as a result of row standardization to one the matrix of 
neighbourhood based on the economic distance is obtained. 

 The more exact, because based on the economic criterion, measurement of 
the spatial autodependence should effect more precise filtration of the 
investigated processes (explained and explanatory), and as a consequence, lead 
to the appropriate estimation of the regression dependence. In other words, the 
more precise description of the spatial autodependence the more efficient 
filtration of the processes and the measurement of the dependence between the 
processes takes place at the level of the data which are more and more cleaned 
from the autodependence. The pure dependence between the processes would 
be the same, apart from the level of the data aggregation. 

 In the approach proposed the effect of spatial aggregation of the data as the 
change of the regression parameter in the spatial model will be not eliminated, 
because the aggregated matrix of neighbourhood has still referred to another 
one than before aggregation of the data configuration. 

2. Empirical Example 

 The dependence between the unemployment rate and investment outlays of 
enterprises (in PLN) in Poland in years 2004–2009 across poviats (districts) and 
sub-regions was investigated. The data are from the internet source: 
http://www.stat.gov.pl. The conception of the spatial and spatio-temporal quasi-
congruent model was used. The trend-autoregressive structures of the individual 
processes at the both levels of the data aggregation were investigated by 
constructing the appropriate fundamental models4. For describing the trend 
structure the models of the spatial and spatio-temporal trend were used. The 
autoregressive structure was defined in two versions, with the use of (1) the 

                                                 
4 The fundamental model denotes here the model describing the componential structure of the 

spatial and spatio-temporal process. The considerations are accompanied by the foundation, that 
in economic processes at least two components are potentially present: the trend and 
autoregressive connections. 



Elżbieta Szulc 10

standard matrix with the equal weights – variant I and (2) the matrix of the 
economic distance (different weights) – variant II. Then the quasi-congruent 
models describing the dependence between the investigated processes were 
built and the obtained results were compared. 

 For the investigated processes the following symbols were taken:  
 ptY  – spatial process of unemployment (unemployment rate in the region at 

spatial location p = [p1, p2] in established year t),  ptX  – spatial process of 
investments (investment outlays per capita in the region at spatial location and 
time – as above),  tY ,p  – spatio-temporal process of unemployment,  tX ,p  – 
spatio-temporal process of investments. 

Investigation of the dependence in years – variant I 

 Investigation of the dependence between the unemployment rate and the 
investment outlays per capita across poviats in the successive years allows to 
ascertain the following: 

1. The models of unemployment and investments contained spatial trends of 
the first degree in all the years and spatial autodependence of the first order, 
except for 2004 and 2005, where the spatial autodependence in investments 
appeared statistically insignificant. 

2. The quasi-congruent models in their full version took into consideration the 
trend structure and the autoregressive structure of the separate processes. 
Thus, they were the models of the following form: 

         ppWppWp ttttt XXYppY   20111000 .(4) 

3. After reduction of the statistically insignificant components, for all the 
years the models with analogous structures were obtained, i.e. the models 
of the form: 

       pppWp tttt XYppY   20111000 , (5) 

except for  2007, where the spatial trend was also reduced. The theoretical 
notation of the model for 2007 is following: 

       pppWp 444004   XYY . (6) 

 In the investigation of the dependence between the unemployment and the 
investment outlays per capita across sub-regions in the successive years the 
following was established: 

1. The fundamental models of unemployment and investments did not contain 
any significant spatial trend, but they had autoregressive components. Only 
in 2004 and 2009 the spatial autoregression in investments appeared 
statistically insignificant. 



Identification of the Structures of Spatial and Spatio-Temporal Processes… 11

2. In the face of settlement 1. the full quasi-congruent models for 2005–2008 
took the form: 

         ppWppWp ttttt XXYY   00 , (7) 

and for 2004 and 2009: 

       pppWp tttt XYY   00 . (8) 

3. The reduction of the statistically insignificant components led to the models 
of the form (8), for all the years. 

 Table 1 presents specification of the values of the chosen parameters of the 
reduced quasi-congruent models for poviats and sub-regions. The comparison 
of the values leads to the statement, that in the successive years the regression 
coefficients estimated at the level of poviats are, as regards the absolute value, 
smaller than the ones obtained for sub-regions. In turn, the coefficients of 
autoregression for poviats are bigger than for sub-regions. 

Table 1. Coefficients of regression and autoregression in the models estimated at the 
level of poviats and sub-regions – variant I 

Level of 
aggregation 

Years 
2004 2005 2006 2007 2008 2009 

Coefficients of regression  
Poviats -1.6371 -1.6489 -1.3799 -0.8802 -0.6379 -0.5991 

Sub-regions -2.5394 -2.5007 -1.9284 -1.4969 -1.2207 -1.3018 
Coefficients of autoregression  

Poviats 0.6833 0.6747 0.6724 0.6851 0.6632 0.6381 
Sub-regions 0.5112 0.5139 0.4353 0.3370 0.3544 0.4321 

Note: All the calculations for the investigation presented in the paper were done with the use of R-CRAN. 

Investigation of the dependence in years – variant II 

 The influence of investment outlays on unemployment rate was also 
investigated by using the matrix of economic distance to define spatial 
autodependence in the considered processes. The matrix with differentiated 
weights, taking into account similarity of poviats/sub-regions established on the 
ground of the values of unemployment and investment outlays in the connected 
regions (i.e. in poviats, and then in sub-regions) was used. Thus, the economic 
distance was defined as follows: 









,for0,

for,|||| 21

ji

jixxyy
d jiji

ij



 (9) 

where 1 = 2 = 0.95. 

 In this case, for poviats, it was established, that: 

                                                 
5 The value has been established on the ground of the separated investigations.  



Elżbieta Szulc 12

1. The fundamental models of unemployment and investments identified at 
the level of poviats, for all the years, contained spatial trends of the 1st 
degree and the autodependence of the 1st order. It is the analogous result as 
that one obtained in variant I. 

2. Also, likewise in variant I of the analysis the full quasi-congruent models 
contained spatial trends, spatially lagged unemployment, current 
investments and spatially lagged investments (see formula (4)). 

3. After reduction of the statistically insignificant components, for 2004–2007 
the spatial autoregressive regression models were obtained (see, formula 
(8)). For 2008 from the full model only the spatial trend was eliminated, yet 
for 2009 the full model did not need any component to be reduced. 

 The analysis of the data at the level of sub-regions led to the following 
settlements: 

1. The unemployment and investments in all the years were described with 
the help of the pure autoregressive spatial models (without trend). 

2. It led to the quite simple structure of the full quasi-congruent models, i.e.: 

         ppWppWp ttttt XXYY   
00 . (10) 

3. For all the years there were obtained the reduced models of the form: 

       pppWp tttt XYY   
00 . (11) 

Table 2. Coefficients of regression and autoregression in the models estimated at the 
level of poviats and sub-regions – variant II 

Level of 
aggregation 

Years 
2004 2005 2006 2007 2008 2009 

Coefficients of regression  
Poviats -1.2893 -1.3225 -1.1025 -0.7173 -0.4401 -0.4075 

Sub-regions -2.3358 -2.2648 -1.7279 -1.3171 -1.0979 -1.1736 
Coefficients of autoregression  

Poviats 0.8158 0.8026 0.7954 0.7727 0.7506 0.7110 
Sub-regions 0.6170 0.5938 0.5506 0.5046 0.4829 0.5472 

 Table 2 presents specification of the values of the chosen parameters of the 
reduced quasi-congruent models for poviats and sub-regions. Comparing the 
results of the analysis for poviats and sub-regions, which are presented in Table 
2, it should be stated that in all the years the regression coefficients estimated at 
the level of poviats, are as regards the absolute value smaller than their 
estimates at the level of sub-regions. However, the coefficients of 
autoregression in the considered years are always bigger for poviats than for 
sub-regions. 

 



Identification of the Structures of Spatial and Spatio-Temporal Processes… 13

Spatio-temporal models – variant I 

 Constructing the spatio-temporal models the analogous structure of 
components as in the case of the spatial models was established. In the analysis 
the deterministic spatio-temporal trends were taken into consideration. 
Identification of the spatio-temporal autodependence was done with the use of 
the matrix of the form: 





















6

2

1

W00

0W0

00W

W







, (12) 

where 621 ... WWW  – standard matrixes of the spatial connections, the 
same for all the considered years. 

 With specification of the fundamental models it was confirmed, that both 
the unemployment rate and the investment outlays showed the spatio-temporal 
trends of the 1st degree and spatial autodependence of the 1st order. The 
components were observed as well at the level of poviats as at the level of sub-
regions and they were taken into account in the quasi-congruent spatio-temporal 
models constructed afterwards. Since the full models contained insignificant 
variables they were eliminated and in this way the reduced models were 
obtained. Table 3 contains the characteristics of the quasi-congruent models 
obtained in the investigation at the level of poviats, yet Table 4 presents the 
analogous specification for sub-regions. 

 The reduced models do not contain the spatially lagged investments. This 
means that the unemployment in the given poviat does not depend on the 
investment outlays in the neighbouring poviats. The current unemployment in 
the given poviat is influenced by the current investment outlays in the same 
poviat (each thousand PLN spent causes the unemployment rate to decrease on 
the average by 0.93 percentage point) and by the level of unemployment in the 
neighbouring poviats (the change of the unemployment rate in the given poviat 
by about 0.69 percentage point is connected with the one percentage change of 
the unemployment rate in the neighbouring poviats). 

 Likewise, the unemployment in the given sub-region does not depend on the 
investment outlays in the neighbouring sub-regions. The current unemployment 
rate in the given sub-region is influenced by the current investment outlays in 
this sub-region (each thousand PLN spent causes the unemployment rate to 
decrease by about 1.74 percentage point) and by unemployment rate in  
the neighbouring sub-regions (the increase of the unemployment rate in the 
neighbouring sub-regions by one percentage point is connected with the 
increase of the unemployment rate in a given sub-region by about 0.39 
percentage point). 



Elżbieta Szulc 14

 The coefficient which measures the influence of the investment outlays on 
the unemployment rate in a sub-region differs from the analogous coefficient 
which is estimated at the level of poviats. The main reason of the difference 
observed is the spatial autocorrelation in unemployment and also in 
investments. The value of coefficient  for sub-regions is bigger than for 
poviats. In turn, autoregression coefficient , which measures the connections 
among the unemployment rates in the neighbouring areas at the level of sub-
regions is visibly smaller than the analogous parameter estimated for poviats. 
The results are analogous to the ones obtained previously for the spatial models 
constructed for the successive years. They are also convergent with the results 
of the previous analyses, pointed out in the introduction, which were obtained 
on the basis of the generated data (especially, when y > x, the coefficient of 
regression, estimated on the basis of the aggregated data, is overestimated). 

Table 3. Characteristics of quasi-congruent models for poviats – variant I 

Full model:  
         ttXtXtYtpptY ,,,,, 00120101100000 ppWppWp    

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
100 
010 
001 
 
 

7.6305 
-0.1892 
0.3472 
-0.3433 
-0.9315 
-0.0332 

0.6915 
0.0652 
0.0664 
0.0665 
0.0504 
0.1085 

11.0347 
-2.9025 
5.2255 
-5.1612 
-18.4788 
-0.3060 

0.0000 
0.0037 
0.0000 
0.0000 
0.0000 
0.7596 

 = 0.6872  
Test LR: 967.48;  p-value: 0.0000 

Wald statistic: 1409.3; p-value: 0.0000 
AIC: 13592 (AIC for lm: 14557) 

Residual autocorrelation 
Test LM: 4.2911; p-value: 0.0383;  

Moran statistic: -0.8026; p-value: 0.2111 
Reduced model:  

       ttXtYtpptY ,,,, 00120101100000 pppWp    

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
100 
010 
001 
 

7.5202 
-0.1816 
0.3464 
-0.3472 
-0.9334 

0.6092 
0.0613 
0.0664 
0.0647 
0.0499 

12.3439 
-2.9644 
5.2171 
-5.3681 
-18.6968 

0.0000 
0.0030 
0.0000 
0.0000 
0.0000 

 = 0.6890  
Test LR: 1090.4; p-value: 0.0000 

Wald statistic: 1538; p-value: 0.0000 
AIC: 13590  (AIC for lm: 14679) 

Residual autocorrelation  
Test LM: 3.7569; p-value: 0.0526 

Test Moran statistic: -0.8867; p-value: 0.1876 



Identification of the Structures of Spatial and Spatio-Temporal Processes… 15

Table 4. Characteristics of quasi-congruent models for sub-regions – variant I 

Full model:  
         ttXtXtYtpptY ,,,,, 00120101100000 ppWppWp    

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
100 
010 
001 
 
 

17.2312 
-0.0459 
0.0482 
-0.5223 
-1.6817 
-0.6643 

2.0420 
0.0156 
0.0151 
0.1628 
0.1431 
0.3645 

8.4384 
-2.9479 
3.2023 
-3.2082 
-11.7497 
-1.8225 

0.0000 
0.0032 
0.0014 
0.0013 
0.0000 
0.0684 

 = 0.3541 
Test LR: 30.415;  p-value: 0.0000 

Wald statistic: 35.928; p-value: 0.0000 
AIC: 2279.5  (AIC for lm: 2308) 

Residual autocorrelation  
Test LM: 6.2037; p-value: 0.0127 

Moran statistic: 0.6251; p-value: 0.2660 
Reduced model:  

       ttXtYtpptY ,,,, 00120101100000 pppWp    

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
100 
010 
001 
 

14.8309 
-0.0321 
0.0522 
-0.6181 
-1.7460 

1.5208 
0.0139 
0.0149 
0.1565 
0.1365 

9.7522 
-2.3164 
3.4839 
-3.9504 
-12.7951 

0.0000 
0.0205 
0.0005 
0.0000 
0.0000 

 =  0.3883 
Test LR: 40.419; p-value: 0.0000 

Wald statistic: 50.843; p-value: 0.0000 
AIC: 2281 (AIC for lm: 2319.5) 

Residual autocorrelation 
Test LM: 0.2836; p-value: 0.5943 

Moran statistic: -0.1386; p-value: 0.4449 

Spatio-temporal models – variant II 

 In the discussed variant for the purpose of quantification of the spatio-
temporal connections there was used the matrix of the form: 































6

2

1

W00

0

0W0

00W

W







, (13) 

where: 621 ...   WWW – matrixes of spatial connections, taking into 
account the economic distance between regions, different for the successive 
years. 



Elżbieta Szulc 16

 Various specifications of the fundamental models of unemployment and 
investments and the models of the dependence between the investigated 
processes, which resulted from them, were used. Tables 5–6 present the results 
of the investigation, in which the time lags of the spatial dependence are not 
taken into account. Tables 7–8, on the contrary, refer to the investigation, in 
which such the lags were considered. 

 The structure of the full models results from the structures of the individual 
processes described by the fundamental models. In the first specification the 
fundamental models with the spatio-temporal trend and with the spatial 
autoregression were established. In the full model identified at the level of 
poviats all the components were statistically significant. Other than in variant I, 
the measurement of spatial (auto)dependence caused the spatially lagged 
investments had not been removed from the model. It denotes some connection 
of the current unemployment rate in a given poviat with the investment outlays 
in the neighbouring poviats. In turn, in the analysis at the level of sub-regions 
the reduced model without the spatially lagged investments was obtained.  

 Like in variant I of the investigation, the difference in the estimation of the 
coefficient measuring the influence of investments on unemployment at the 
level of poviats and sub-regions, was observed. For poviats its value amounted 
to circa -0.69, while for sub-regions to circa -1.52. Also the coefficient of 
autoregression, as in variant I, is bigger for poviats (about 0.76) than for sub-
regions (about 0.53).  

Table 5. Characteristics of quasi-congruent models for poviats – variant II a) 

Full model:  

         ttXtXtYtpptY ,,,,, 00120101100000 ppWppWp     

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
100 
010 
001 
 
 

5.9703 
-0.1403 
0.2307 
-0.1419 
-0.6937 
-0.4661 

0.5611 
0.0548 
0.0548 
0.0537 
0.0453 
0.1219 

10.6413 
-2.5583 
4.2073 
-2.6408 
-15.3140 
-3.8231 

0.0000 
0.0105 
0.0000 
0.0083 
0.0000 
0.0001 

 =  0.7595 
Test LR: 1467.6;  p-value: 0.0000 

Wald statistic: 2912.2; p-value: 0.0000 
AIC: 12943 (AIC for lm: 1.4408) 

Residual autocorrelation 
Test LM: 68.725; p-value: 0.0000  

Moran statistic: -2.9692; p-value: 0.0010 

 In the more extended specification of the structures of the investigated 
processes the time lags of the spatial autodependence were additionally taken 
into account. The characteristics of the full quasi-congruent model and of the 



Identification of the Structures of Spatial and Spatio-Temporal Processes… 17

reduced model for poviats are presented in Table 7, whereas for sub-regions – 
in Table 8. 

Table 6. Characteristics of quasi-congruent models for sub-regions – variant II a) 

Full model:  

         ttXtXtYtpptY ,,,,, 00120101100000 ppWppWp     

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
100 
010 
001 
 
 

12.3451 
-0.0240 
0.0315 
-0.3442 
-1.4485 
-0.4057 

1.7497 
0.0143 
0.0138 
0.1466 
0.1495 
0.3925 

7.0555 
-1.6530 
2.2901 
-2.3488 
-9.6917 
-1.0337 

0.0000 
0.0983 
0.0220 
0.0188 
0.0000 
0.3013 

 =  0.5178 
Test LR: 77.168;  p-value: 0.0000 

Wald statistic: 117.56; p-value: 0.0000 
AIC: 2235.4 (AIC for lm: 2310.5) 

Residual autocorrelation 
Test LM: 37.529; p-value: 0.0000  

Moran statistic: -0.7447; p-value: 0.2282 
Reduced model:  

       ttXtYtpptY ,,,, 00120101100000 pppWp     

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
100 
010 
001 
 

11.1540 
-0.0161 
0.0343 
-0.4002 
-1.5245 

1.3366 
0.0128 
0.0135 
0.1362 
0.1266 

8.3450 
-1.2578 
2.5450 
-2.9374 

-12.0426 

0.0000 
0.2085 
0.0109 
0.0033 
0.0000 

 =  0.5315 
Test LR: 86.996; p-value: 0.0000 

Wald statistic: 137.67; p-value: 0.0000 
AIC:  2234.5 (AIC for lm: 2319.5) 

Residual autocorrelation 
Test LM: 14.549; p-value: 0.0001 

Moran statistic: -1.1656; p-value: 0.1219 

 In the full quasi-congruent model of unemployment for poviats the 
following were taken into consideration: spatio-temporal trend, current 
unemployment rate in the neighbouring poviats, unemployment rate from the 
previous period in the neighbouring poviats, current investment outlays in the 
given poviat and in the neighbouring poviats and also investment outlays in the 
neighbouring poviats from the previous period. Most of the mentioned 
components were statistically significant. Only the spatial trend appeared 
insignificant. After elimination of it the reduced model was obtained. 

 Starting from the analogous structure of the full model for sub-regions, as 
a result of eliminating the statistically insignificant components the reduced 



Elżbieta Szulc 18

model was obtained. In the model there are the following components: current 
unemployment in the neighbouring sub-regions, current investment outlays in 
a given sub-region, current and time lagged investment outlays in the 
neighbouring sub-regions. 

 Coefficient of regression () estimated at the level of poviats amounted to 
circa  -0.63, and at the level of sub-regions – to circa -1.21. The coefficients of 
autoregression amounted to circa 0.45 and 0.33 respectively. 

Table 7. Characteristics of quasi-congruent models for poviats – variant II b) 

Full model:  

     
       ttXtXtX

tYtYtpptY

,1,,,

1,,, 00120101100000

ppWpWp

pWpWp












 

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
100 
010 
001 
 

 
 
 

-1.6307 
0.1506 
-0.0589 
0.4975 
0.4882 
-0.6319 
-0.7915 
0.9709 

0.8154 
0.0632 
0.0621 
0.0794 
0.0253 
0.0461 
0.1780 
0.1823 

-1.9998 
2.3803 
-0.9496 
6.2677 

19.2694 
-13.7052 
-4.4470 
5.3247 

0.0455 
0.0173 
0.3423 
0.0000 
0.0000 
0.0000 
0.0000 
0.0000 

 =  0.4466 
Test LR: 189.85;  p-value: 0.0000 

Wald statistic: 351.93; p-value: 0.0000 
AIC: 10456 (AIC for lm: 10643) 

Residual autocorrelation 
Test LM: 3.4071; p-value: 0.0649 

 Moran statistic: 0.7442; p-value: 0.2284 
Reduced model:  

     
       ttXtXtX

tYtYttY

,1,,,

1,,, 001000

ppWpWp

pWpWp












 

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
001 
 

 
 
 

-0.2661 
0.4718 
0.4622 
-0.6332 
-0.8660 
0.8658 

0.5758 
0.0775 
0.0233 
0.0462 
0.1757 
0.1784 

-0.4622 
6.0847 

19.8427 
-13.7195 
-4.9275 
4.8538 

0.6439 
0.0000 
0.0000 
0.0000 
0.0000 
0.0000 

 = 0.4483 
Test LR: 191.29; p-value: 0.0000 

Wald statistic: 359.94; p-value: 0.0000 
AIC: 10458 (AIC for lm: 10648) 

Residual autocorrelation  
Test LM: 7.136; p-value: 0.0076 

 Moran statistic: 1.0981; p-value: 0.1361 



Identification of the Structures of Spatial and Spatio-Temporal Processes… 19

Table 8. Characteristics of quasi-congruent models for sub-regions – variant II b) 

Full model: 

     
       ttXtXtX

tYtYtpptY

,1,,,

1,,, 00120101100000

ppWpWp

pWpWp












 

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
100 
010 
001 
 

 
 
 

4.5273 
0.0047 
-0.0006 
0.2194 
0.4392 
-1.1887 
-1.6994 
1.7360 

2.8085 
0.0165 
0.0165 
0.2483 
0.0837 
0.1569 
0.5483 
0.5679 

1.6120 
0.2861 
-0.0371 
0.8838 
5.2446 
-7.5757 
-3.0992 
3.0568 

0.1070 
0.7748 
0.9704 
0.3768 
0.0000 
0.0000 
0.0019 
0.0022 

 =  0.3148 
Test LR: 14.607;  p-value: 0.0001 

Wald statistic: 26.774; p-value: 0.0000 
AIC: 1820.7 (AIC for lm: 1833.3) 

Residual autocorrelation 
Test LM: 0.2322; p-value: 0.6299 

Moran statistic: 0.2175; p-value: 0.4139 
Reduced model: 

       
     ttXtX

tXtYtYtY

,1,,

,1,,, 000

ppWpW

ppWpWp












 

Parameters 
Estimates 

of parameters 
Standard 
deviations 

Statistics z Pr(>|z|) 

000 
 

 
 
 

6.3379 
0.3855 
-1.2131 
-1.7387 
1.8079 

1.4305 
0.0566 
0.1491 
0.5353 
0.5220 

4.4306 
6.8075 
-8.1375 
-3.2479 
3.4631 

0.0000 
0.0000 
0.0000 
0.0012 
0.0005 

 = 0.3256 
Test LR: 16.022; p-value: 0.0000 

Wald statistic: 29.457; p-value: 0.0000 
AIC: 1815.6 (AIC for lm: 1829.6) 

Residual autocorrelation 
Test LM: 1.1578; p-value: 0.2819 

Moran statistic: 0.4223; p-value: 0.3364 

Conclusions 

 In the modelling of the dependence of spatial processes and also spatio-
temporal processes the principle of congruency of the structures of the separate 
processes should be applied. Limited efficiency of the spatial and spatio-
temporal quasi-congruent models as the tools of discovering the real 
dependence between the processes (in particular the differences in estimating 
the dependence at different levels of data aggregation observed) result, first of 



Elżbieta Szulc 20

all, from imperfection of the measurement of autodependence in the 
investigated processes. 

References 

Arbia, G. (1988), Spatial Data Configuration in Statistical Analysis of Regional Economic and 
Related Problems, Kluwer Academic Press, Dordrecht. 

Haining, R. (2005), Spatial Data Analysis. Theory and Practice, Cambridge University Press, 3th 
ed., Cambridge. 

Pietrzak, M. B. (2010), Dwuetapowa procedura budowy przestrzennej macierzy wag 
z uwzględnieniem odległości ekonomicznej (Two-stage Procedure of Building a Spatial 
Weight Matrix with the Consideration of Economic Distance), Oeconomia Copernicana, 
1, Wydawnictwo UMK, Toruń, 65–78. 

Szulc, E., Müller-Frączek, I., Pietrzak, M. B. (2011), Modelowanie zależności między 
ekonomicznymi procesami przestrzennymi a poziom agregacji danych (Modeling of 
Dependence between Economic Spatial Processes and the Level of Data Aggregation), in 
Suchecka J. (ed.), Ekonometria przestrzenna i regionalne analizy ekonomiczne (Spatial 
Econometrics and Regional Economic Analyses), Folia Oeconomica, Wydawnictwo 
Uniwersytetu Łódzkiego, 327–344. 

Szulc, E. (2011), Ekonometryczna identyfikacja struktur procesów przestrzennych wobec 
problemu agregacji danych (Econometric Identification of the Structures of Spatial 
Processes and a Problem of Data Aggregation), Acta Universitatis Nicolai Copernici, 
Ekonomia XLII – Nauki Humanistyczno-Społeczne, z. 392, Toruń, in press. 

Identyfikacja struktur procesów przestrzennych i przestrzenno-
czasowych wobec problemu agregacji danych 

Z a r y s  t r e ś c i. Artykuł dotyczy odkrywania zależności między ekonomicznymi procesami 
przestrzennymi a także przestrzenno-czasowymi, gdy są one mierzone na różnych poziomach 
agregacji danych. Rozważania nawiązują do badań, potwierdzających efektywność tzw. quasi-
zgodnego modelu przestrzennego jako narzędzia pomiaru rzeczywistych zależności między 
procesami. Wykorzystuje się koncepcję modelu quasi-zgodnego także w odniesieniu do procesów 
przestrzenno-czasowych. Poszukuje się opisu powiązań przestrzennych i przestrzenno-czasowych 
adekwatnego do wyrażenia autozależności w badanych procesach. Proponuje się wykorzystanie 
odległości ekonomicznej, mierzącej podobieństwo regionów na podstawie wartości 
analizowanych procesów. 

S ł o w a  k l u c z o w e: przestrzenny i przestrzenno-czasowy model quasi-zgodny, 
autozależności, macierz sąsiedztwa, odległość ekonomiczna.  


