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

Iwona Müller-Frączek, Michał Bernard Pietrzak 
Nicolaus Copernicus University in Toruń 

Space-Time Modelling of the Unemployment Rate  
in Polish Poviats 

A b s t r a c t. The purpose of the article is to model the unemployment rate in Poland in 
its spatial and time dimensions. The spatial lag models with the neighbourhood matrix 
based on the common border of poviats were used in the study. The analysis of the 
changes over time of the parameters in the space models gave a foundation to apply the 
space-time model with the parameters linear dependent on time. Such an approach ena-
bled us to work out models with good statistical properties and with a possibility of 
providing a correct economic interpretation of the parameters. 

K e y w o r d s: spatial econometrics, spatial model, space-time model.  

Introduction 

 The authors show interest in modelling economic phenomena that are char-
acterized by spatial dependencies. The spatial approach requires the use of spe-
cific methods and models taken from spatial econometrics. The term of spatial 
econometrics was introduced by J. H. P. Paelinck in 1974. Since that year the 
problems of spatial econometrics have been discussed in abundant works, such 
as, for instance, Clif and Ord (1981), Anselin (1988), Zeliaś (1991), Anselin, 
Florax and Rey (2004), LeSage and Pace (2004), Haining (2005), Arbia (2006), 
Szulc (2007), Bivand, Pebesma and Gómez-Rubio (2008), LeSage and Pace 
(2009), Suchecki (2010). In the present paper the authors will conduct a spatial 
analysis of the unemployment in Poland with a view to researching its proper-
ties and existing dependencies within the selected determinants. A correct de-
scription of this phenomenon, regarded as one of the major social and economic 
problems, constitutes a significant task of spatial econometrics. 

 The article is an attempt to present two research approaches used in model-
ling of economic processes, i.e. the spatial and space-time modelling the pro-
cesses. The models falling into the former category were applied to provide 



Iwona Müller-Frączek, Michał Bernard Pietrzak 204 

a spatial description of the character of the unemployment rate, though only in 
its static aspect. The models consider the spatial dependencies showing the in-
fluence of the unemployment observed in one poviat on the unemployment 
level in other poviats. In addition, the global spatial tendency of the phenome-
non was characterized by means of the spatial trend. The dependencies between 
the unemployment rate and the spatial processes defined as ‘determinants’ were 
adequate interpreted. 

 The purely spatial analysis of the unemployment rate carried out in the years 
2004–2009 constituted a starting point for conducting a dynamic spatial analy-
sis. It was observed that the models in specific years possessed some common 
features such as the same spatial trend and a similar parameter of the spatial 
autocorrelation. These settlements formed a basis for the space-time approach 
also referred to as ‘dynamic’. 

 The next step was to estimate the space-time model for the unemployment 
rate in Polish poviats in the years 2004–2009, with time-varying parameters of 
the space-time trend and the structural parameters. The analysis of the parame-
ters showed linear dependence with regard to time. Consequently, the final ver-
sion of the hypothetical space-time model with parameters that change linearly 
relative to time was selected. This choice allowed the number of the parameters 
of the model to be significantly limited. The space-time approach is infrequent-
ly used in spatial econometrics, though it allows the enriching of spatial anal-
yses with a time dimension. The methodology relating to space-time modelling 
and few of applications can be found e.g. in the works of Anselin (1988), Szulc 
(2007), LeSage and Pace (2009). 

1. Methodology 

 The models used for the purposes of the paper are to reflect the spatial and 
space-time dependencies relative to the unemployment rate. The models that 
were applied included static ones as well as two types of dynamic spatial mod-
els. Throughout the present paper n denotes the number of the ordered units of 
space. The letters X and Y denote the processes observed in space only  
(the static version) or in space and time (the dynamic version) and they possess 
either a single or double index dependent on their use.  

 One of the methods of taking into consideration the spatial relations ob-
served in economic processes is to introduce into the formal description the  
so-called the spatial weights matrix (denoted as )( ijwW , nji ,...,1,  ). This 

matrix determines the intensity of connections between separated areas of 
space. Usually the form of the matrix can reflect only the geographical structure 
of the researched area; however, it may consider some other characteristics 
(properties) of the space, such as economic ones (see Anselin, 1988). For the 
dynamic models our assumption is that the spatial matrix is constant over time.  



Space-Time Modelling of the Unemployment Rate in Polish Poviats 205

 With the specified W matrix, we can determine a lag operator for the spatial 
process Y, which determines the influence on the observation of the process in 
the given location 1,...,i n  of the observations in other locations, that is: 

  


nj jjii YwY
,...,1 ,)(W  – in the spatial version, 

  


nj jtjiit YwY
,...,1 ,)(W , where niitt YY ,...,1)(   – in the space-time version. 

Moreover, it is assumed that for a given area it is possible to determine the 
global force of the relations occurring in the process between various locations. 
This is the so-called spatial autocorrelation of the process, incorporated into the 
model by adding the spatial autoregression component with the parameter ρ. 

 When defining models the so-called spatial or space-time white noise ε, i.e. 
collections of the uncorrelated random variables with the following properties: 

),,0(~ 2 Ni  ,,...,1 ni   (1) 

),,0(~ 2 Nit  ni ,...,1  and ,,...,1 Tt   (2) 

are used. 

 In the static part of the research the spatial lag model was applied and it had 
the following form (see Anselin,1988): 

,)( iiii XYY   W  (3) 

where: 

ni ,...,1  stands for the number of the location, 

)( ijwW , nji ,...,1,   is the weight matrix, 

i  is the spatial white noise. 

 For the dynamic part of the research two types of models were applied. The 
first model is autoregressive with time-varying structural parameters and with 
constant spatial autocorrelation and it is defined as follows: 

,)( itittitit XYY   W  (4) 

where: 

ni ,...,1  is the number of the location, 

Tt ,...,1  is the time index, 

)( ijwW , nji ,...,1,   is the weight matrix, 

it  is the white space-time noise. 

 Due to its slight research value the above model constitutes rather a starting 
point for further research and not a final form of the model that would be ready 
for adaptation.  



Iwona Müller-Frączek, Michał Bernard Pietrzak 206 

 The other model under consideration is a peculiar case of the model (4), in 
which changes of coefficients for explanatory variables are linear relative to 
time, that is: 

,)()( 10 itititit XtYY   W  (5) 

where: 

ni ,...,1  is the number of the location, 

Tt ,...,1  is the time index, 

)( ijwW , nji ,...,1,   is the weight matrix, 

it  is the white space-time noise. 

 In the empirical part of the research all the spatial and space-time models 
were extended by the introduction of the second explanatory variable and by the 
spatial time-varying trend.1  

2. Empirical Results 

  The empirical research was conducted based on the statistical data obtained 
from the official Central Statistical Office website and they concerned the regis-
tered unemployment rate in Polish poviats ( 379n ) at the end of the years 
2004–2009 ( 6T ). The research conducted earlier by the authors (see Müller-
Frączek and Pietrzak (2011a, 2011b)) showed that the unemployment rate is 
characterized by strong spatial interdependencies, which justified the use of the 
models presented in this section for the purpose of the description of the phe-
nomenon. Also, it was observed that the unemployment rate is subject to 
a global spatial trend; therefore, the models used in the empirical analyses were 
extended by the addition of the spatial trend. 

 For the purpose of this research the normalized weights matrix (W) was 
used and this matrix reflects the geographical neighbourhood of poviats in the 
sense of the common border line. Moreover, centres of gravity were determined 
for the poviats whose geographic coordinates were taken for the estimation of 
spatial trends.  

 The analysis of statistical data showed that among potential unemployment 
determinants only two were statistically significant. Table 1 presents economic 
processes and their designations used in the research. 

Table 1. Economic spatial processes used in modelling the unemployment rate 

Process Designation 
Unemployment rate  Y 

Investment outlays in thousands of PLN per capita X1 
One hundred of entities of the national economy per 10000 inhabitants X2 

                                                 
1 The calculations were performed in R-CRAN. 



Space-Time Modelling of the Unemployment Rate in Polish Poviats 207

2.1. Spatial Models 

 In the static approach the spatial model of the unemployment rate was esti-
mated separately for each considered year. In all the cases of the estimation of 
the static models contained in this section, extended by the spatial trend, only 
the coefficients of a first degree trend turned out to be statistically significant. 
Therefore, the static models used for successive years took the following hypo-
thetical form expressed by the following equation: 

,21   XXcybxaYY W  (6) 

where: 

x  and y  are geographic coordinates for the centres of gravity of poviats, 

)( ijwW , nji ,...,1,   is the weight matrix, 

  stands for the white noise. 

 Table 2 shows the estimated parameters of the models determined by the 
equation (6) for individual years, where the designations used were consistent 
with the equation. 

Table 2. The results of the estimation of the models determined by the equation (6) 

Parameter 
Static models 

2004 2005 2006 2007 2008 2009 

a 
Estimates 16,51 15,69 13,90 11,42 9,86 12,30 
p-value ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 

b 
Estimates 0,61 0,58 0,48 0,42 0,41 0,52 
p-value ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 

c 
Estimates -0,84 -0,73 -0,62 -0,43 -0,30 -0,45 
p-value ≈0,00 ≈0,00 ≈0,00 ≈0,00 0,02 ≈0,00 

α 
Estimates -1,20 -1,27 -1,07 -0,68 -0,49 -0,42 
p-value ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 

β 
Estimates -0,73 -0,67 -0,59 -0,55 -0,53 -0,66 
p-value ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 

ρ 
Estimates 0,66 0,65 0,65 0,64 0,64 0,60 
p-value ≈0,00 ≈0,00 ≈0,00 ≈0,000 ≈0,00 ≈0,00 

 The statistical quality of the models was examined by calculating the re-
spective characteristics for the residuals. These were the following: the value of 
the determination coefficient, the value of the Global Moran’s statistics, and 
significance of the parameters. The values of the above-mentioned characteris-
tics are shown in Table 3. All of them prove a proper fit each model to the sta-
tistical data. The estimates of the parameters of the static models changes over 
time. However, prior to the year 2008 the changes had been subject to some 
regularity with the exclusion of the 2009 model. Its being different can be spot-
ted especially in the parameters of the trend that reflects the main spatial ten-
dency of the examined phenomenon. The years 2004–2008 in Poland were the 
period of economic prosperity. The unemployment rate was decreasing every 



Iwona Müller-Frączek, Michał Bernard Pietrzak 208 

subsequent year, investment and salaries were increasing. At that time Poland 
was perceived as a quickly developing country and was attracting foreign inves-
tors. The 2008 crisis slowed down all world economies significantly. Obvious-
ly, this affected the Polish economy. Because of that two models for the years 
2004–2008 and 2004–2009 will be considered in the dynamic approach.  

Table 3. The selected measures of quality of the models determined by the equation (6) 

Characteristics 
Static models 

2004 2005 2006 2007 2008 2009 
R2 0,640 0,630 0,600 0,570 0,550 0,536 
I 0,032 0,028 0,036 0,013 -0,003 0,003 

(I-E(I))/S(I) -0,003 -0,003 -0,003 -0,003 -0,003 -0,003 
p-value 0,157 0,185 0,128 0,320 0,509 0,439 

 When analyzing the models presented in Table 2, it can be stated that up to 
the year 2008 the intercept was decreasing. That means that the unemployment 
rate was, globally, becoming lower and lower in the period 2004–2008. Moreo-
ver, the slope of the surface of the trend, determined by the b and c values, be-
comes less steep, so the disproportions in the unemployment rate between indi-
vidual poviats decrease. These positive effects, as seen from an economic point 
of view, started to disappear in the year 2009. 

 Although the signs of the coefficients do not change with the spatial trend, 
their absolute values increase in relation to the previous year. The regression 
parameters α, β for all the examined years had negative values. This reflects the 
beneficial impact of the growth of investments made and the number of eco-
nomic subjects on the decrease of the unemployment rate in poviats. However, 
it can be seen that in course of time the influence of the 1X  process is constant-

ly decreasing and the slightly decreasing impact of the 2X  process changed its 
character in the year 2009. 

 In all the six models there can be noticed strong spatial autodependence, 
which gives the average value of the autoregression parameter ρ at the level of 
0.63. Moreover, this dependency appeared to be quite stable over time, which 
resulted in the constant spatial autocorrelation being accepted in the dynamic 
models. A slight decrease in the autoregression parameter over time indicates 
the weakening of spatial dependencies of the unemployment rate along with the 
improvement in the level of social and economic development. This should be 
read as a positive phenomenon since the existing spatial dependencies create 
a negative mechanism countering changes in the spatial system of the unem-
ployment rate. 

2.2. Space-Time Models 

 Due to the change in the developmental trend of the major phenomena in 
Poland both space-time models used for the purposes of the present paper were 



Space-Time Modelling of the Unemployment Rate in Polish Poviats 209

estimated twice – with and without the inclusion of the observations from the 
year 2009 ( 6T  or 5T ). 

 First the spatial lag model with time-varying structural parameters, the trend 
parameters and the stable over time spatial autocorrelation was considered. 
What turned out to be statistically significant during the estimation was the pa-
rameters for the trend model of the first degree; hence the final hypothetical 
form of the model was as follows: 

,21 tttttttttt XXycxbaYY   W  (7) 

where: 

,,...,1 Tt   
)( ijwW , nji ,...,1,   is the weight matrix, 

x  and y are the geographic coordinates of the gravity centres of the poviats, 

t  stands for the space-time noise.  

 Tables 4 and 5 include the results of the estimation of the parameters of the 
models designated by the equation (7) with and without the inclusion of the 
observations from the year 2009. The analysis of the results presented allows 
a statement to be formulated that the autoregression parameter and the parame-
ters for the determinants model of the unemployment rate are statistically valid. 
Also, in the case of the spatial trend all the parameters proved to be statistically 
significant. It needs to be emphasized that in order to identify a major spatial 
trend it is enough to have a significant parameter at least for one of the spatial 
coordinates.  

Table 4. The results of the estimation of the models determined by the equation (7) in 
the years 2004–2009 

Parameter 
Dynamic model  

2004 2005 2006 2007 2008 2009 

at 
Estimates 16.87 13.71 12.45 11.60 11.30 11.69 
p-value ≈0.00 0.590 0.170 0.010 0.001 0.015 

bt 
Estimates 0.68 0.63 0.52 0.44 0.43 0.50 
p-value ≈0.00 ≈0.00 ≈0.00 ≈0.00 0.039 0.003 

ct 
Estimates -0.86 -0.74 -0.63 -0.44 -0.31 -0.45 
p-value ≈0.00 ≈0.00 ≈0.00 0.003 0.003 ≈0.00 

αt 
Estimates -1.20 -1.27 -1.07 -0.68 -0.48 -0.41 
p-value ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 

βt 
Estimates -0.73 -0.67 -0.59 -0.56 -0.53 -0.65 
p-value ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 

ρ 
Estimates 0.65 
p-value ≈0.00 

 The statistical quality of both dynamic models was evaluated applying the 
same measures that were used for static models. The aggregate results are 



Iwona Müller-Frączek, Michał Bernard Pietrzak 210 

shown in Table 7 and they show that the models were well fitted to the empiri-
cal data and that the autocorrelation in the residuals did not occur.  

Table 5. The results of the estimation of the models determined by the equation (7) in 
the years 2004–2008 

Parameter 
Dynamic model 

2004 2005 2006 2007 2008 

at 
Estimates 16.68 16.14 15.74 15.33 15.28 
p-value ≈0.00 0.600 0.180 0.011 0.001 

bt 
Estimates 0.67 0.62 0.51 0.43 0.42 
p-value ≈0.00 ≈0.00 ≈0.00 0.005 0.041 

ct 
Estimates -0.88 -0.74 -0.63 -0.43 -0.31 
p-value ≈0.00 ≈0.00 0.001 0.003 0.004 

αt 
Estimates -1.20 -1.27 -1.07 -0.67 -0.48 
p-value ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 

βt 
Estimates -0.73 -0.66 -0.59 -0.55 -0.52 
p-value ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 

ρ 
Estimates 0.68 
p-value ≈0.00 

 The comparison of the results presented in Table 4 and Table 5 with the 
results contained in Table 2 shows that the values of the parameters of the two 
dynamic models are similar to the corresponding parameters of the static mod-
els. Consequently, their economic interpretation will be similar.  

 The analysis of the changes in the time of the parameters of the dynamic 
models indicates that for the model in which the last observations made  
(i.e., from the year 2009) were omitted, the changes are linear relative to time. 
The linear property concerns both the trend and the structural parameters of the 
model. The consequence of the assumption of the linear changes of the parame-
ters over time was the acceptance of the theoretical model reflecting this  
property: 

0 1 0 1 0 1

0 1 1 0 1 2

( ) ( )

( ) ( ) ,
t t

t t t

Y Y a a t b bt x c ct y

t X t X


    

        
    

W
  (8) 

where: 

,,...,1 Tt   
)( ijwW , nji ,...,1,   is the weight matrix, 

x  and y are the geographic coordinates of the gravity centres of the poviats, 

t  stands for the space-time noise. 

 Despite the fact that the analysis of the results presented in Table 4 and Ta-
ble 5 showed that in 2009 the linear character of the changes of the parameters 
relative to time was altered, the estimation of the models designated by the 
equation (8) was made both with and without the inclusion of the observations 



Space-Time Modelling of the Unemployment Rate in Polish Poviats 211

from that year. The estimation results are shown in Table 6 and the residuals of 
both models are contained in Table 7.  

 For both models the autoregression parameter and the parameter for the per 
capita investment are statistically significant. In the case of the number of eco-
nomic subjects the linear variation over time is insignificant for the model esti-
mated in the years 2004-2009. However, for the years 2004–2008 it is just on 
the edge of significance. Attention must be paid to the fact that the analysis 
covered five or six years. Increasing the number of periods would certainly 
enhance the significance of the parameters in the estimated models. The esti-
mate of the autoregression parameter indicates strong spatial dependencies of 
the unemployment rate and the estimates obtained for the parameters of the 
determinants show negative dependencies between the unemployment rate, 
investment, and the number of economic subjects per capita. 

Table 6. The results of the estimation of the models determined by the equation (8) 

Parameter 
Model for the years 2004-2009 Model for the years 2004-2008 

Estimates p-value Estimates p-value 
a0 17.44 ≈0.00 18.81 ≈0.00 
a1 -1.28 ≈0.00 -1.81 ≈0.00 
b0 -0.91 ≈0.00 -1.01 ≈0.00 
b1 0.10 0.005 0.14 0.004 
c0 0.65 ≈0.00 0.74 ≈0.00 
c1 -0.04 0.210 -0.07 0.140 
α0 -1.57 ≈0.00 -1.58 ≈0.00 
α1 0.20 ≈0.00 0.21 ≈0.00 
β0 -0.70 ≈0.00 -0.76 ≈0.00 
β1 0.02 0.300 0.05 0.110 
ρ 0.66 ≈0.00 0.65 ≈0.00 

Table 7. The selected measures of quality of the models determined by the equation (7) 
and (8) 

Property 
Models described by the equation (7) Models described by the equation (8) 

2004-2009 2004-2008 2004-2009 2004-2008 
R2 0.69 0.70 0.69 0.69 
I 0.019 0.023 0.012 0.022 

(I-E(I))/S(I) 1.436 1.546 0.915 1.483 
p-value 0.075 0.061 0.180 0.069 

 When analyzing the results presented in Table 6 and Table 7 it can be in-
ferred that the observations from the year 2009 did not affect considerably the 
obtained models either in the statistical or the interpretational aspect. The  
acceptance of the observation for the years following 2009, where the economic 
slow down would be continued, would undoubtedly affect the evaluation of the 
parameters and the properties of the model. Both models are characterized by 
a high degree of fitting to the empirical data and the differences in the estimates 
of the parameters remain slight. If the change of the character of the unemploy-
ment rate observed in the year 2009 were to be of a lasting nature, then the 



Iwona Müller-Frączek, Michał Bernard Pietrzak 212 

model created on the basis of all the observations would certainly not be correct 
from a forecasting point of view, either. 

Conclusions 

 The article discussed a spatial analysis of the registered unemployment rate 
in Polish poviats at the end years 2004–2009. Within the first approach, a static 
one, spatial models of the unemployment rate in subsequent years were estimat-
ed. Next, the analysis was enhanced by a time dimension and within the dynam-
ic approach estimations were made of appropriate space-time models. A spatial 
linear trend as well as two determinants of the unemployment rate, investment 
made and the number of economic subjects per capita were assumed for all the 
models.  

 The analysis conducted allows the identification of a linear spatial trend and 
negative dependencies occurring between the unemployment rate, investment 
made and the number of economic subjects per capita.  

 For the trend parameters and assumed processes of determinants a linear 
character of the time-varying changes was determined. All of the estimated 
models were characterized by good statistical properties and economic inter-
pretability of the parameters. 

 The question of the dissimilarity of the observations from the year 2009 
remains unsettled. If the economic cycle in Poland and the related unemploy-
ment, level of investment and general condition of enterprises changed perma-
nently, then the model with a linear dependency of time-varying parameters 
could result in an erroneous simplification of changes of the parameters. 

References  

Anselin, L. (1988), Spatial Econometrics: Method and Models, Kluwer Academic Publishers, 
Netherlands. 

Anselin, L., Florax, R. J. G. M., Rey, S. J. (2004), Advances in Spatial Econometrics. Methodolo-
gy, Tools and Applications, Springer-Verlag, Berlin. 

Arbia, G. (2006), Spatial Econometrics, Springer-Verlag GmbH. 
Bivand, R. S., Pebesma, E. J., Gómez-Rubio, V. (2008), Applied Spatial Data Analyses with R, 

Springer, New York. 
Clif, A., Ord, J. (1981), Spatial Processes, Models and Applications, Pion, London. 
Müller-Frączek, I., Pietrzak, M. B. (2011a), Analiza stopy bezrobocia w Polsce z wykorzystaniem 

przestrzennego modelu MESS (Model MESS of the Unemployment Rate in Poland), Folia 
Oeconomica 253, Wydawnictwo Uniwersytetu Łódzkiego, Łódź, 215–223. 

Müller-Frączek, I., Pietrzak, M. B. (2011b), Przestrzenna analiza stopy bezrobocia w Polsce 
w latach 2004–2008 (Spatial Analysis of the Unemployment Rate in Poland in the Years 
2004–2008), Economic Development and Management of Regions, Part II, Hradec 
Kralove, 205–209. 

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

LeSage, J. P., Pace, R. K. (2004), Advances in Econometrics: Spatial and Spatiotemporal Econo-
metrics, Elsevier, Amsterdam. 



Space-Time Modelling of the Unemployment Rate in Polish Poviats 213

LeSage, J. P, Pace, R. K. (2009), Introduction to Spatial Econometrics, CRC Press. 
Suchecki, B. (2010), Ekonometria Przestrzenna (Spatial Econometrics), Wydawnictwo 

C.H.Beck, Warsaw. 
Szulc, E. (2007), Ekonometryczna analiza wielowymiarowych procesów gospodarczych (Econo-

metric Analysis of Multidimensional Economic Processes), Wydawnictwo UMK, Toruń. 
Zeliaś, A. (1991), Ekonometria Przestrzenna (Spatial Econometrics), PWE, Warsaw.  

Przestrzenno-czasowe modelowanie stopy bezrobocia w Polsce  

Z a r y s  t r e ś c i. Celem artykułu jest modelowanie stopy bezrobocia w ujęciu przestrzennym 
i przestrzenno-czasowym. Dane wykorzystane w badaniach empirycznych dotyczyły powiatów 
Polski w latach 2004-2009. Wykorzystano modele o charakterze regresyjno-autoregresyjnym 
z macierzą sąsiedztwa opartą na zasadzie wspólnej granicy powiatów. Analiza zmian w czasie 
parametrów modeli przestrzennych dała podstawy do zastosowania modeli czasowo-
przestrzennych z parametrami liniowo zależnymi od czasu. Podejście takie pozwoliło na otrzy-
manie modeli o dobrych własnościach statystycznych oraz charakteryzujących się poprawną 
ekonomiczną interpretowalnością parametrów. 

S ł o w a  k l u c z o w e: ekonometria przestrzenna, model przestrzenny, model przestrzenno-
czasowy.  

 

 

 

 

 

 



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


