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Spatial regression model of households: a case of the Czech Republic

Modèle de régression spatiale des ménages: un cas de la République tchèque
Jaroslav Kraus

Résumés

Au cours des dernières décennies, les schémas des ménages et des familles ont considérablement changé. D’une part, on peut remarquer un plus petit nombre d’enfants vivant dans des familles, ainsi qu’une réduction du nombre de familles en général en raison du faible taux de fécondité et du report de la maternité en Europe. En revanche, la proportion de ménages d’une personne et de familles sans enfants parmi les personnes plus âgées a augmenté de manière significative en raison du vieillissement de la population.
La question de recherche est de savoir dans quelle mesure ces changements dans la structure des ménages sont spatialement homogènes et si la proportion de ménages d’un type donné sur un territoire donné peut être expliquée par d’autres variables démographiques telles que l’âge, l’éducation, l’état civil ou l’activité économique.
Les principes susmentionnés peuvent être démontrés par l’exemple des données sur les ménages, dont la détection et l’analyse ultérieure font partie intégrante des recensements de la population. Pour résoudre ce problème, il est possible d’utiliser des méthodes d’analyse de données spatiales, qui peuvent être définies comme une analyse de données quantitatives, dans lesquelles l’explication dépend de variables spatiales explicites lors de la prédiction du phénomène étudié sur la base de l’autocorrélation spatiale.
L’hypothèse de régression spatiale est l’existence d’une autocorrélation. Les résultats du I de Moran et du C de Geary montrent que l’autocorrélation pour les deux types de ménages s’est avérée statistiquement significative et augmente à mesure que la distance entre les éléments adjacents (c’est-à-dire les municipalités) diminue.
L’âge est un facteur important affectant la structure des ménages. Les résultats pour les deux types de ménages montrent que les tranches d’âge les plus influentes sur la création de ménages d’une personne ou de ménages unifamiliaux peuvent également être utilisées pour créer un modèle spatial. Une affirmation similaire s’applique à l’âge moyen. L’éducation a montré que la part des personnes ayant une éducation primaire n’a pas d’influence sur la régression spatiale, contrairement à la part des personnes ayant une éducation secondaire ou universitaire. Dans le cas de l’état matrimonial, il y a une régression spatiale statistiquement significative pour les ménages unipersonnels, mais pas clairement pour les ménages unifamiliaux. L’activité économique ou l’emploi est statistiquement significatif pour une simple régression, même dans un petit territoire comme la République tchèque.
Pour la solution, il est possible d’utiliser plusieurs types de modèles proposés par la théorie (économétrique). Dans le cas des ménages, le modèle spatial de Durbin (SDM) est relativement largement utilisé, du fait de la prise en compte à la fois des effets d’interaction endogènes et exogènes, selon les critères choisis (Log Likelihood, AIC, SBC). Cependant, les résultats des autres modèles (SAR, SEM, SDEM) ne sont pas significativement différents.

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Notes de l’auteur

This paper is supported by grant GAČR, 2018–2020 No. 18-12166S.

Texte intégral

1. Introduction – sources and references

1Household trends are among the phenomena of modern times and are a long-term interest of demographers. The changing structure of households is one of the fundamental attributes of the second demographic transition. The classic contribution of demographic literature states that changes in the propensity to marry, divorce, separate, remarry, or cohabitant changes in fertility behavior, and in the ages at which children leave home along with mortality trends and differentials, have had a marked impact on household patterns in Europe. There are several causes for this phenomenon. The basis is the already mentioned demographic transition, where changes can be characterized as a shift from uniform to pluralistic family and household [van Kaa, 1987] structures.

2In recent decades, household and family patterns have changed significantly [van Imhoff et al., 1995]. On one hand, one can notice a smaller number of children living in families, as well as a reduction of family numbers in general as a result of the low fertility rate and the postponement of childbearing in Europe. On the other hand, there has been a significant increase in the proportion of one-person households and families without children among people at an older age because of population ageing [Habartová, 2018].

3The situation in the Czech Republic is similar to that of other European countries. Although the basic trend in the development of private households has been in a steady decline since the 1980s, in the absolute number of private households made up of one-completed family, almost two thirds of people lived in this type of household in 2011. The number of one-completed family households peaked in 1980 and has been showing a slow long-term decline since. From the early 1990s an increased intensity in the decline was observed. Similarly, the average size of a household comprising one complete family is decreasing. While there were 3.35 persons in this type of household in 1970, 40 years later the average size decreased by 0.3 persons. To a large extent, the less frequent coexistence of other persons with a complete family, as well as a significant reduction in the number of dependent children in the family, have had a significant share in this situation. As regards the household of individuals, it is one of the most dynamically growing types of households in addition to single-parent families. Since 1970, the number one-person households more than doubled to 1 422 thousand in 2011 and their share increased to one third of all households [Habartová, 2013].

4As a data source, it is possible to use the results of population censuses, the advantage of which are the time and space-comparable results provided in large territorial detail, e.g. for municipalities. Such an approach is described in [Howell, et al., 2016], where the local nature of demographic data is applied to find universal principles for spatial distribution through spatial analysis. It can be seen retrospectively that the fertility decline in Europe is a classic example of spatial autocorrelation. In general, demographic transition – including household structures - has an impact on all aspects of society.

5The research question is to what extent these changes in the structure of households are spatially homogeneous and whether the proportion of households of a given type in a given territory can be explained by other demographic variables such as age, education, marital status or economic activity. The Czech Republic is one of the smaller European countries and is characterized by relatively homogeneous demographic behaviour, which has its historical causes [Rychtaříková, 2008]. During the socialist era and especially since the 1960s, a historically earlier model of family behaviour dominated in Czechoslovakia. Marriage was contracted early and became frequent. Children were born within marriage to younger parents and at short intervals. Procreative behaviour in each of the 1930–1960 birth cohorts was at the level of simple reproduction. The Czech Republic ranked among countries with a fairly high divorce rate. Until the early1960s, infant mortality risk decreased rapidly in both countries due to particularly active maternity and welfare policies [Rychtaříková, 2018].

6In terms of territorial differences, they were found both in the size of households and in their structure. From the west to the east of the Czech Republic, the share of family households, especially complete families, is increasing and the share of individual households is decreasing. The exceptions are large cities, which have their own specifics. The shares of single-family households with dependent children within family households were balanced on a national average. Overall, households without dependent children slightly predominated and almost all regions did not differ significantly in the representation of households without dependent children. The size of the municipality also affects the average number of members or the number of dependent children in the household. With the growing size of the municipality, both the average number of people living together in one household and the share of families with a higher number of dependent children decreases [CZSO, 2013].

7The share of one-person households in larger municipalities in 2011 ranged between 23 and 40%, depending on the geographical location and the number of inhabitants in the municipalities. In 15% of larger municipalities, the representation of individual households was one third or more. The highest share of individual households within municipalities with more than 10,000 inhabitants was found in Mariánské Lázně, Karlovy Vary (West Bohemia Region), Litvínov (North Bohemia Region) and Prague (Capital) - more than 39%. These are generally cities in the west or north of the Czech Republic. In contrast, less than 9% of one-person households are placed in the towns of Jablunkov, Kravaře (Moravia-Silesian Region), Valašské Klobouky (Zlín Region) and Veliké Meziříčí (Vysočina Region) [CZSO, 2014].

8The lowest average number of dependent children in complete families (e.g. not only one-family households) with dependent children was found in the municipalities with the highest population and in the municipalities of the western and north-western part of the Czech Republic, especially in the pelvic areas of the Ore Mountains (Krušné hory). The absolute minimum was found in the town of Most (Ústí nad Labem Region), only 1.50 dependent children. On average, the most dependent children in the whole family traditionally lived in municipalities in the Vysočina, Zlín and Pardubice Regions, specifically in municipalities of the so-called inner periphery on the border of Bohemia and Moravia (poorer transport accessibility and civic amenities, economically weaker regions), where an average of 1.72 children lived in a complete family. These are the cities of Nové Město na Moravě, Telč (Vysočina Region) and Valašské Klobouky (Zlín Region). In contrast, in suburban zones around the capital city of Prague with a relatively high proportion of complete families, the average number of dependent children in complete families was only around 1.58–1.64 children [CZSO, 2014].

9Therefore, it makes sense to ask the question whether local changes in the age structure (there are regions with an older population as in the case of large cities) or education (similar to the previous example) can affect the spatial structure of households.

10If this proves to be the case, it would be possible to model the development of households on the basis of annual data provided by demographic (intercensal) statistics. Data on the number of households are traditionally the result of population censuses and are thus available in a ten-year period, which is not sufficient for many users. Estimates of the number of households based on demographic statistics could thus be used virtually on an ongoing basis.

Map 1: Share of one-person households in all households by municipality in the CR

Image 100000000000056C000003D20C6DB09C5E455D1D.jpg

Map 2: Share of one-family households in all households by municipality in the CR

Image 1000000000000568000003D6381A7E1268A9A5AB.jpg

Source: author’s own calculation based on data from the 2011 Population and Housing Census

11To solve this problem, it is possible to use spatial data analysis methods, which can be defined as a quantitative data analysis, in which the explanation is dependent on explicit spatial variables when predicting the investigated phenomenon based on spatial autocorrelation. Spatial distribution can be seen as either a spatial dependence or a spatial heterogeneity of a given phenomenon. Spatial dependence is associated with Tobler’s law of geography – everything is related to everything else, but near things are more related than distant things [Tobler, 1970].

12The above mentioned principles can be demonstrated by the example of data for households, whose detection and subsequent analysis is an integral part of population censuses. Spatial analysis of household distribution is a relatively new topic, but several interesting contributions with focus on the Czech Republic have been published on this subject, see [Bleha, et al. 2019] or [Blažek, et al., 2012]. The development and structure of all types of households are becoming a key element of demographic developments in many countries; their methodological definition is well standardized and coordinated.

13The modern concept of defining households is based on the principle of common living and is thus identified in population censuses. The simplest type of households is the one-person household, which consists of a person who lives alone in a separate housing unit or who occupies, as a lodger, a separate room (or rooms) of a housing unit, but does not join with any of the other occupants of the housing unit to form part of a multi-person household. One-family households are a subcategory of family households. A one-family household consists of a single family (e.g. a couple with or without children). The family concept is defined as the limits of relationships between children and adults to direct (first degree) relationships that exists between parents and children [UNECE, 2011].

2. Data

14The results of the 2011 Population and Housing Census in the Czech Republic are described in a number of places - for example in [CZSO, 2013]. In the Czech Republic, in 2011, one-person households and one-family households made up almost 94 percent of the total number of households. Therefore, this paper focuses on the spatial structure and modeling of one-person and one-family households. Variable ratios, and for analysis purposes, transformed values of one-person households and one-family households are used to ensure data normality.

Table 1 Number of households in census years 1970–2011

1970

1981

1991

2001

2011

index 2011/1970

index 2011/2001

Total number of households

3 365 407

379 097

3 983 858

4 216 085

4 375 122

130

104

One-family households

2 526 778

2 760 247

2 856 608

2 803 340

2 667 867

106

95

One-person households

668 859

897 447

1 047 221

1 276 176

1 422 147

213

111

Source: [CZSO, 2013]

15The basic characteristics of the development of households include the relative decline in the share of family households at the expense of uncompleted family households, as well as the absolute and relative growth of one-person households. While in 1970 private households consisting of one complete family made up two thirds of all private households, four decades later they made up hardly half. A decrease in their proportion was caused mainly by a marked absolute increase in one-person households. In 2011, one-person households comprised already a third of all households. The average size of a household has also been constantly decreasing for a long period of time. In 1970, on average 2.89 persons lived in a private household, whereas in 2011 it was only 2.34 persons. This, together with a change in the structure of private households, is a result of a long-term demographic trend, especially due to declines in the fertility rate, long-term high levels of divorce rates and an increasing availability of independent living (i.e. by frequent and simpler decomposition of complete families to singles and incomplete families) [CZSO, 2013].

16The spatial regression model is based on the fact that the number and proportion of households of a given type are related to the spatial distribution of basic demographic variables such as age, education, marital status or economic activity. For example, the highest proportion of women living in one-person households in younger age groups was recorded at the age of 27 in the 2011 Census. At this age, 13% of women lived in this way. With increasing age, the proportion of women in the household of an individual continued to decline at the expense of the increasing proportion of women in family households. In the male population, the distribution is more even, with a peak at 29 years and a sharp increase in the oldest age groups. The considerable predominance of women over men in the elderly population is due to the over-mortality of men. This causes a higher probability of disintegration of the complete family through the loss of the male partner and the consequent emergence of an individual widowed woman.

17Similarly, in the case of one-family households, the underlying trend is a steady decline, but still almost two-thirds of people occupied this type of household in 2011. The average size of households made up of one-family is decreasing. A separate sub-group of one-family households to which attention is often drawn with regards to aging populations, are households of seniors. This type of household predominates mainly in older seniors (over 76 years). Older people live more often in households resulting from the break-up of a complete family, i.e. in individual households, single-parent families or as another person in a complete family. The share of two-parent families is 14-25% of the total number of two-parent families [Habartová, 2018].

Table 2 Descriptive statistics of variables by municipalities

Variable

Number of
Nonmissing
Observations

Minimum

Mean

Maximum

Standard
Deviation

50th
Percentile
(Median)

Mode

Test
Statistic
for
Normality

Age20_24Share

6122

0,000

6,018

18,200

1,757

6,000

6,400

0,063

Age25_29Share

6122

0,000

6,078

19,200

1,751

6,000

6,300

0,069

Age65+Share

6122

1,300

15,895

53,300

4,240

15,500

15,400

0,085

AvgAge

6122

30,500

41,074

63,100

2,734

40,800

40,800

0,081

ShareSH100

6122

14,634

27,829

47,541

5,599

27,426

25,000

0,031

Share1FH100

6122

45,946

65,987

80,597

5,916

66,444

66,667

0,037

TShareSH100

6122

-1,461

-1,072

-0,666

0,137

-1,072

-1,134

0,008

TShare1FH100

6122

-0,343

-0,256

-0,167

0,031

-0,256

-0,255

0,011

Age20_24Share - Share of persons aged 20-24 of total population, Age25_29Share - Share of persons aged 25-29 of total population, Age65+Share - share of persons aged 65 and over of the total population, AvgAge - average age, ShareOnePersonHH100 - Share one-person households from the total number of households times 100, ShareOneFamilyHH100 - Share of one-family households from the total number of households times 100, TShareOnePersonHH - transformed ShareOneFamilyHH100 variable, TShareOneFamilyHH - transformed ShareOneFamilyHH100 variable

18All these facts lead to an awareness of the importance of age for individual types of households. In terms of some simplification, the model uses the proportions of persons aged 20-24, 25-29 (modeling the age of early adulthood), age 65+ (representing the highest age groups) and average age (representing the entire age range). This covers the entire age range. The model uses the transformed form of the variables one-person households and one-family households, the transformation method is described in detail [Kraus, 2020]. A similar procedure was used in [Griffith, et al., 2017].

3. Methods and methodology

3.1. Spatial autocorrelation

19Spatial autocorrelation may be the result of unobserved or difficult to quantify processes combined at different locations and consequently causing spatial structuring of a given phenomenon. In the context of spatial regression (i.e. searching for explanatory variables), measurement of spatial autocorrelation can be considered as a diagnostic tool. In the case of spatial autocorrelation, it is determined whether the variable value for a given (geolocalized) observation is associated with values ​​of the same variable for adjacent observations. Spatial autocorrelation may be positive, negative, or the data may not be spatially autocorrelated [INSEE, 2018]. However, if we want to investigate spatial autocorrelation, it is first of all necessary to evaluate whether the examined data - in this case the distribution of households of a given type - meet the normality requirement [Krivoruchko, 2011].

20The basic tools of spatial autocorrelation include Moran’s I, along with others such as Geary C or Getis-Ord G [Smith et al., 2020]. Moran’s I statistic is defined as:

Image 10000000000000BF0000002424F4CB57A8D8DC5E.png

where Image 10000000000001400000001564CB90CD1F94C442.png, wijis a matrix of spatial weights with zeroes on the diagonal and Image 100000000000006400000013D8D0CF7B4D7A42A2.png.

21Another measure of spatial autocorrelation is Geary’s C statistic, defined as

Image 10000000000000E400000044CD44B23BB27F1FDD.png

22These expressions indicate that Moran’s coefficient makes use of the centered variable, whereas the Geary’s expression uses the noncentered values in the summation. Other relationships are based on the normality assumption, which relate to moment measures - mean and variance.

23For Moran’s I coefficient, I > E[I] indicates positive autocorrelation. Positive autocorrelation suggests that neighboring values si and sj tend to have similar feature values zi and zj, respectively. When I < E[I], this is a sign of negative autocorrelation, or dissimilar values at neighboring locations. A measure of strength of the autocorrelation is the size of the absolute difference |I - E[I]|. Geary’s C coefficient interpretation is analogous to that of Moran’s I. The only difference is that C > E[C] indicates negative autocorrelation and dissimilarity, whereas < E[C] signifies positive autocorrelation and similarity of values [Anselin, 2016].

24Another useful tool for exploring spatial dependence is the Moran scatter plot. It is a visual tool for exploratory analysis, because it enables you to assess how similar an observed value is to its neighboring observations. Its horizontal axis is based on the values of the observations and is also known as the response axis. The vertical Y axis is based on the weighted average or spatial lag of the corresponding observation on the horizontal X axis. The Moran scatter plot provides a visual representation of spatial associations in the neighborhood around each observation. The observations are represented by their standardized values; therefore only nonmissing observations are shown in the plot.

25Moran scatter plot spatial autocorrelation plot begins to establish a link between global and local spatial autocorrelation and hence spatial regression investigation.

3.2. Spatial regression

26There are a number of ways to study spatial data, including the use of spatial regression (econometrics) models. Its aim is to spatially model and analyze data using spatial information - see e.g. [Cressie, 1993] or [Fischer, 2010]. One vital component of spatial modeling is appropriately accounting for spatial dependence in the data. In general, spatial dependence can arise from three sources: endogenous interaction, exogenous interaction, and correlated error terms. Endogenous interaction effects are typically considered as the formal specification for the equilibrium outcome of a spatial or social interaction process, in which the value of the dependent variable for one agent is jointly determined with that of neighboring agents. The second are exogenous interaction effects, where the dependent variable of a particular unit depends on independent explanatory variables of other units. Interaction effects among the error terms do not require a theoretical model for a spatial or social interaction process, but instead, are consistent with a situation where determinants of the dependent variable omitted from the model are spatially autocorrelated, or with a situation where unobserved shocks follow a spatial pattern [Elhorst, 2014].

27Particularly important for spatial modeling is how to choose a model that can describe data. Although the choice of model is often a specific problem, there are general guidelines that can be followed. Spatial autoregressive models (SAR) are used to account for the effects of endogenous interactions. Spatial Durbin models (SDM) models are used to address both the effects of endogenous and exogenous interactions. You use spatial error models (SEM) or spatial moving average model (SMA) to track spatial dependence on random members. SDMA and SDEM models can be used when taking into account exogenous interaction effects and spatial dependence in random members. Spatial autoregressive moving average models (SARMA) and spatial autoregressive confused models (SAC) can be used to address the common effects of endogenous interactions and the spatial dependence of random members. In the case of endogenous interaction, when effects, exogenous interactions and spatial dependence in random members coexist, SDARMA and SDAC models can be used. Spatial lag of X model (SLX) models are used to describe the effects of exogenous interactions and linear models in pure linear regression [Elhorst, 2014].

Table 3. Overview of selected types of spatial models

Image 1000000000000913000001F6345C47B5761E4AB5.png

Where y = (y1, y2, .. yn) and yi is the value of the continuous dependent variable that corresponds to region i for i = 1, 2, .., n, where n is the number of observations in the data. X and Z are  n x p  and n x q matrices of regressors, respectively, and Image 1000000000000081000000117AD5F17DE807A202.png, where Image 1000000000000073000000129530E2A9133C202C.png for i = 1, 2, .. n. W is a nonnegative n x n matrix describing the spatial configuration or arrangement of the units in the sample. In addition, 𝛽 and 𝜃 are a p x 1 parameter vector and a q x 1 parameter vector, 𝜆 is the spatial autocorrelation coefficient [[Wu et al., 2016] in according to [Elhorst, 2014]].

3.3. Spatial weights matrix

28The spatial weights matrix plays an important role in geostatistical data analysis and thus in spatial modeling. Spatial weights matrices commonly used in applied research are: (i) p-order binary contiguity matrices (if p = 1 only first-order neighbors are included, if p = 2 first and second order neighbors are considered, and so on); (ii) inverse distance matrices (with or without a cut-off point); (iii) q-nearest neighbor matrices (where q is a positive integer); (iv) block diagonal matrices where each block represents a group of spatial units that interact with each other but not with observations in other groups [Elhorst, 2014].

29In this paper, two basic approaches for determining the neighborhood matrix were chosen: (i) binary contiguity edges corners (Queen’s Case) matrix and (ii) distance metric matrix. For contiguity edges corners, polygons that share an edge or a corner will be included in computations for the target polygon. If any portion of two polygons overlap, they are considered neighbors and will be included in each other’s computations. Use one of these contiguity conceptualizations with polygon features in cases where you are modeling some type of contagious process or are dealing with continuous data represented as polygons [ESRI, 2020].

30The fixed distance band method is a good option for polygon data when there is a large variation in polygon size and need to ensure a consistent scale of analysis. Another reason is knowledge about the geographic extent of the spatial processes promoting clustering for the phenomena you are studying. This is based on the Euclidean distance between centroids of the two spatial units. Let (loni, lati) and (lonj, latj) be the centroids of units i and j, where 1 ≤ i, j ≤ n, and lon and lat denote the longitude and latitude, respectively. Under the Euclidean distance metric, the distance dij between units i and j is

Image 10000000000000E90000002AB8A560FFFB2B1397.png

31Following this distance metrix is defined for (i, j) as

Image 10000000000000AE0000002343E3FFB7C0BA7B31.png

where dcutoff is a prespecified threshold distance [Shekhar et al., 2008].

3.4. Method of establishing neighborhood

32A key element for calculating indices of spatial autocorrelation is to determine the neighborhood, which means selecting spatial entities that are neighbors by definition. The defining of the neighborhood is a rather complex issue, which should always be based on the knowledge of the examined issue (i.e. determination of a working hypothesis on why given spatial elements are selected to be neighbors) and that has a major influence on the result of calculation of a spatial autocorrelation.

33At the beginning of the selection of a particular solution there are always theoretical bases, which are built on the factual knowledge of the issue. In this paper, two previously mentioned methods were used to conceptualize the spatial relationship of a spatially weighted matrix: (i) contiguity edges and corners method and (ii) method of fixed distance. The starting point for the first case was neighborhoods as a central institution in the organization of residential space in cities. They function simultaneously as institutional, sociological, economic, political, and geographic entities at multiple levels. At the scale of the individual and the family, neighborhoods exert a major influence over property values and the webs of social relations that tie people to those who live in close proximity to them. In general, property values tend to be more similar within neighborhoods than among them [Barney, 2006]. The starting point for the second method are theories that work with the influence of locally important communities on their surroundings. As an example, influential central place theory can be mentioned, which is a model of city systems that posits them as retail centers (central places) that distribute goods and services to their surrounding hinterlands. A hierarchy of goods and services leads to a hierarchy of central places, with lower-order ones nested within the hinterlands of higher-order ones [Barney, 2006], but also many others (gravity models, spatial diffusion). In this paper, three distances have been gradually set, their justification and the result are given in the next part of the paper.

34It is also necessary to check the distribution of households in terms of surface trends and paired distribution. In order to determine the spatial correlation, a sufficient number of distance classes should be established to capture the extent of this correlation. Each class must contain a minimum number of data pairs. There are different recommendations leading to recommendations, the rule being that there should be at least 30 pairs per lag class.

35The lag size determination has a large influence on the calculation of the empirical semivariogram and thus the intensity of the spatial autocorrelation. If the lag size is large, then short-range autocorrelation can be hidden. If, on the other hand, it were set too large, the representation in the individual bin would be low.

36Lag class is determined by

Image 10000000000000C000000036DDF824E692FB0AD6.png

where the directed line segment P1P2 is superimposed on the coordinate system that shows the distance or lag classes and lag distance is valued by Δ. The result is a lag distance of 27.2 km. This distance can be used to calculate autocorrelation statistics. Further distance determination is based on the methods discussed above: (i) the contiguity edges and the corners method and (ii) the method of fixed distance. The first case is based on the fact that the Czech Republic is relatively evenly populated and the average distance of centroids of neighboring municipalities is 4 km. In the second case, the average distance of municipalities over one thousand persons is usually 8 km, and virtually every municipality in this case has a neighboring village with more than one thousand usual residents.

4. Results

4.1. Global measures of spatial autocorrelation

37It is now possible to proceed with the calculation of Moran’s I and Geary’s statistics, as well as the graphical representation using the Moran scatter plot. These statistics were calculated assuming normalization using binary weights. Basic information can be found in [Shekhar et al., 2008].

Table 4 Autocorrelation statistics – share of one-person households in all households by municipality in the Czech Republic

One-Person Households

Distance 27 km

Assumption

Coefficient

Observed

Expected

Std Dev

Z

Pr > |Z|

Normality

Moran’s I

0.111

0.000

0.001

79.900

<0.000

Normality

Geary’s c

0.886

1.000

0.002

-66.600

<0.000

Distance 8 km

Normality

Moran’s I

0.184

0.000

0.005

38.600

<0.000

Normality

Geary’s c

0.815

1.000

0.005

-37.400

<0.000

Distance 4 km

Normality

Moran’s I

0.201

0.000

0.010

20.000

<0.000

Normality

Geary’s c

0.809

1.000

0.011

-18.100

<0.000

One-Family Households

Distance 27 km

Normality

Moran’s I

0.114

0.000

0.001

82.200

<0.000

Normality

Geary’s c

0.882

1.000

0.002

-68.500

<0.000

Distance 8 km

Normality

Moran’s I

0.184

0.000

0.005

38.600

<0.000

Normality

Geary’s c

0.815

1.000

0.005

-37.400

<0.000

Distance 4 km

Normality

Moran’s I

0.182

0.000

0.010

18.100

<0.000

Normality

Geary’s c

0.826

1.000

0.011

-16.500

<0.000

38Based on Table 2, the value of Moran’s I, and Geary’s C decreases for both one-person households and one-family households as the distance shifts. The highest value was found in the case of one-person households for distances of 4 km (0.201) and in the case of one-family households for distances of 8 km (0.184), but the difference compared to 4 km (0.182) is negligible. Although spatial autocorrelation values are not very high, they are statistically significant in all cases. Since the Z score for Moran is positive (ZI> 0) and the Z score for Geary is negative (ZC<0), positive autocorrelation is indicated. The standard deviation is minimal in all cases.

39The results show that the chosen distance of 27.2 km does not lead to a result that would be worthy of consideration and it makes no sense to further work with this distance. From the results of both Moran I and Geary C it is evident that with decreasing distance the size of the spatial autocorrelation increases based on the neighborhood method. For further work on the regression model, two neighboring matrices will be used: (i) contiguity edges corners (which is approximately 4 km distance when determining 4 neighbors) and (ii) distance band (8 km).

Graph 1 Moran scatter plot for the share of households of the given type in the total number of households by municipalities in the Czech Republic (distance 4 km)

Image 10000000000007D00000035DC205ACAA001ECA6C.png

One Family HH – one-family households, One Person HH – one-person households

40The results of both graphs show similarity of distribution for both types of households. Moran scatterplot is an example of the relationship between the values of the share of households of a given type in each locality and the average value of these shares in neighboring locations. In the case shown (Graph 1), there are cases where higher than average household shares (points on the right side of the graph) tend to be associated with high local average values (points slightly upwards). Similarly, in the lower-left quadrant there are cases where both the value and local average value of the attribute are lower than the overall average value. Both types of households are an example of a positive autocorrelation, where the local value of the weighted average increases with the growing share of households (e.g. standardized variable).

4.2. Spatial regression model

41When choosing a spatial regression model, it is necessary to start from the factual knowledge of the issue, interpretation of the results and theoretical options of the choice of the individual classes of models. Another important element in model selection is the determination of the distance matrix. In accordance with the previous results, a distance matrix based on contiguity edges corners (approx 4 km) and distance band (8 km) will be used.

Table 5 Model selection using AIC criterion for share of one-person households on all households

Model

SAR

SDM

SEM

SDEM

SMA

SDMA

Distance 8 km

Number of Observations

6115

6115

6115

6115

6115

6115

Log Likelihood

4185

4244

4192

4217

4127

4174

AIC

-8334

-8447

-8348

-8392

-8219

-8306

SBC

-8213

-8306

-8227

-8251

-8098

-8165

Min Distance (4 km)

Number of Observations

6122

6122

6122

6122

6122

6122

Log Likelihood

4121

4173

4112

4146

4075

4119

AIC

-8206

-8304

-8188

-8250

-8113

-8196

SBC

-8085

-8163

-8067

-8109

-7993

-8055

Table 6 Model selection using AIC criterion for share of one-family households on all households

Model

SAR

SDM

SEM

SDEM

SMA

SDMA

Distance 8 km

Number of Observations

6115

6115

6115

6115

6115

6115

Log Likelihood

13166

13228

13175

13200

13111

13159

AIC

-26295

-26415

-26314

-26357

-26185

-26276

SBC

-26174

-26274

-26193

-26216

-26064

-26135

Min Distance (4km)

Number of Observations

6122

6122

6122

6122

6122

6122

Log Likelihood

13071

13125

13064

13096

13024

13067

AIC

-26107

-26208

-26092

-26150

-26011

-26091

SBC

-25986

-26067

-25971

-26009

-25891

-25950

Note The different number of observations results from the requirement that the given municipality should have at least three neighboring municipalities at distance 8km. The lowest AIC and SBC criteria are highlighted in yellow

42When evaluating individual models, the question is which model describes the data best. One of the criteria that may be used for this purpose is the likelihood ratio (LR) test based on log-likelihood function values of the different models. The LR test is based on minus two times the difference between the value of the log-likelihood function in the restricted model and the value of the log-likelihood function of the unrestricted model. Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (SBC) can also be used for model selection.

43For a set of candidate models, the model with the smallest AIC or SBC is often preferred. As shown in tables 3 and 4, the results mentioned here are not fundamentally different and all models can work, with a preference for SDM and SDEM models for both distances. SDM models has been used to address both endogenous and exogenous interaction effects. This model is referred to below as ModelA.

Table 7 Parameter estimates for SDM model and contiguity edges cornes (4km) matrix (ModelA) - age variables

Image 10000000000004070000034102498A25F2751C28.png

TShareOne-familyHH - share of one-family households on all households - transformed variable, TShareOne-personHH - share of one-person households on all households - transformed variable, AvgAge - average age, Age20_24Share - share of persons aged 20 -24 years of the total number of persons, Age25_29Share - share of persons aged 25 - 29 of the total number of persons, Avg65 + Share - share of persons aged 65 and more of total number of persons, W_Age20_24Share - spatial effect of variable Age20_24Age, W_Age25_29Share - spatial effect of variable Age25_29Share, W_Age65 + Share - The spatial effect of the variable Age65 + Share, "_rho" is name of the autoregressive coefficient, the t statistic given for "_rho" is the test of autoregressive coefficient. _sigma2 is the name of the variance parameter. Statistically significant results are marked in yellow.

Table 8 Parameter estimates for SDM model and 8km distance matrix, age variables

Image 100000000000040700000342BE76E5EC1BD1CE95.png

The explanations of the variables are the same as for Tab. 7

44Tables 4 and 5 show parameter estimation results for both distances and variables. The model was constructed as a regression with multiple classification variables. This means that individual age variables were included in the model either separately, or in combination with each other - see tables 4 and 5. The SDM model also includes spatial effects of each variable. In order to avoid potential collinearity with the intercept term in the model, the spatial effects exclude the intercept term. All age variables were included in the calculation of spatial effects to construct spatial lag of covariate effects – again see table 4 and 5. In the SDM model, estimates are also displayed for the autoregressive coefficient and the variance of the error term σ2.

45The results show different meanings of variables in individual models. The average age variable (AvgAge) is always significant; for one-family households the estimate is positive, while for one-person households it is negative, which is difficult to interpret. The proportion of persons aged 20-24 (Age20_24Share) is not statistically significant in any of the models. This means that this proportion is homogeneous in terms of spatial distribution and does not affect territorial differentiation. The situation is different for the share of persons aged 25-29 (Age25_29Share), but only for one-family households. Territorial differentiation plays a role in constituting this type of household (due to age, it can be assumed that it is a family with a small child or children). The situation is similar for the share of persons aged 65 and over - in the case of one-family households, although it is a different type of households than in the previous case.

46However, when considering the commutative effects of several of the above variables, the situation is different. In the case of one-family households, the variable average age plays an important role in the iteration with the share of persons in individual age groups, resp. with a share in ages 25-59. In the case of one-person households, it is slightly different they are particularly important for shares of persons aged 25-29, resp. 65 or more years. The spatial effect allowing the specification of covariates was added to the model and is statistically significant in all cases. This, moreover, corresponds to the previous finding that the commutative involvement of individual variables in the model increases its predicative ability.

47The value of the autoregressive coefficient is relatively high in all four models and shows good predicative ability of variable ages in terms of capturing the territorial structure of selected types of households.

4.3. Alternative models

48In the previous case, only the age - selected age groups and average age - were taken as an explanatory variable of both types of spatial structure of households. Age (population aging in general) is one of the key parameters in analyzing the structure and distribution of households, but it is not, of course, the only one. Therefore, the impact of education (ModelB), marital status (ModelC) or economic activity (ModelD) is investigated in the next section of this paper. The effect of homogeneous variables is always considered in individual models. Since the average age as a synthesizing indicator proved statistically significant in the previous case (see ModelA), it is also included in the following three considered models.

49Furthermore, taking the previous findings (see ModelA) into account, only a space matrix of relations at a minimum distance of 4 km is considered and it is assumed that the results for distances of 8 km are similar.

Table 9 Parameter estimates for SDM model and contiguity edges cornes (4km) matrix (ModelB) - education variables

 

TSShare One-family HH

TSShare One-person HH

Parameter

Estimate

Standard
Error

t Value

Approx
Pr > |t|

Estimate

Standard
Error

t Value

Approx
Pr > |t|

Intercept

-0.106

0.012

-8.79

<0.000

-1.043

0.057

-18.21

<0.000

AvgAge

-0.003

0.000

-17.02

<0.000

0.014

0.001

21.48

<0.000

ShareBasicEd15+

0.000

0.000

-0.48

0.633

0.000

0.001

-0.11

0.911

ShareSecondSt15+

0.001

0.000

6.39

<0.000

-0.002

0.001

-3.71

0.000

ShareUniversity15+

0.000

0.000

-2.03

0.042

0.003

0.001

4.68

<0.000

W_AvgAge

0.001

0.000

1.82

0.070

-0.002

0.001

-1.81

0.070

W_ShareBasicEd15+

0.000

0.000

1.59

0.112

-0.002

0.001

-2.25

0.025

W_ShareSecondSt15+

0.000

0.000

1.04

0.300

-0.001

0.001

-1.36

0.174

W_ShareUniversity15+

0.001

0.000

3.74

0.000

-0.007

0.001

-6.45

<0.000

_rho

0.358

0.017

21.30

<0.000

0.370

0.017

22.03

<0.000

_sigma2

0.001

0.000

54.85

<0.000

0.015

0.000

54.81

<0.000

ShareBasicEd15 + - share of persons with basic education of the total number of persons aged 15+, ShareSecond15 + - share of persons with secondary education of the total number of persons aged 15+, ShareUniversity15 + - share of persons with university education of the total number of persons aged 15+ , W_ShareBasicEd15 + - spatial effect of variable ShareBasicEd15 +, W_ShareSecondEd15 + - spatial effect of variable ShareSecondEd15 +, W_ShareUniversityEd15 + - spatial effect of variable ShareUniversityEd15 +, the meaning of other variables is the same as in Tab. 7

50Table 9, which provides the results of educational variables (ModelB), shows that secondary education variables, as well as university education are statistically significant for both types of households (e.g. one-family households and one-person households). The spatial effect of university education is also significant. This can be explained by the fact that while basic education is spatially similar, the level of secondary and higher education differs spatially. Thus, there are localities characterized by the concentration of the population with higher education (Prague and Central Bohemia), as well as localities where the opposite is true. The mean age is also statistically significant in combination with education.

Table 10 Parameter estimates for SDM model and contiguity edges cornes (4km) matrix (ModelC) - marital status variables

 

TSShare One-family HH

TSShare One-person HH

Parameter

Estimate

Standard
Error

t Value

Approx
Pr > |t|

Estimate

Standard
Error

t Value

Approx
Pr > |t|

Intercept

-1,430

0,235

-6,090

<0.000

1,368

1,018

1,340

0.179

AvgAge

-0,002

0,000

-10,970

<0.000

0,011

0,001

12,330

<0.000

ShareSingle

0,006

0,001

5,900

<0.000

-0,015

0,004

-3,760

0.000

ShareMarried

0,007

0,001

7,620

<0.000

-0,021

0,004

-5,080

<0.000

ShareDivorced

0,003

0,001

3,050

0.002

-0,002

0,004

-0,600

0.551

ShareWidowed

0,004

0,001

4,080

<0.000

-0,006

0,004

-1,520

0.129

W_AvgAge

-0,002

0,000

-3,750

0.000

0,006

0,002

3,430

0.001

W_ShareSingle

0,008

0,002

3,530

0.000

-0,013

0,010

-1,330

0.183

W_ShareMarried

0,008

0,002

3,610

0,000

-0,013

0,010

-1,350

0.176

W_ShareDivorced

0,007

0,002

3,310

0.001

-0,012

0,010

-1,220

0.222

W_ShareWidowed

0,009

0,002

3,930

<0.000

-0,017

0,010

-1,730

0.084

_rho

0,185

0,020

9,330

<0.000

0,231

0,019

11,920

<0.000

_sigma2

0,001

.

.

.

0,01 2

0,000

55,100

<0.000

ShareSingle - share of single persons of the total number of persons, ShareMarried - share of married persons of the total number of persons, ShareDivorced - share of divorced persons of the total number of persons, ShareWidowed - share of widowed persons of the total number of persons, W_ShareSingle - spatial effect of variable ShareSingle, W_ShareMarried - spatial effect of variable ShareMarried, W_ShareDivorced - spatial effect of variable ShareDivorced, W_ShareWidowed - spatial effect of variable ShareWidowed, the meaning of other variables is the same as in tab. 7.

51The results of the spatial model for variables of marital status are interesting; while for one-family households, the results are statistically significant without exception, for one-person households the statistical significance is confirmed only for single and married persons. This is logically justified by the fact that divorce and widowhood are spatially neutral events, for whose explanation it is necessary to look for an explanation other than spatial localization. Even with Model C, the mean age is statistically significant in all cases.

Table 11 Parameter estimates for SDM model and contiguity edges cornes (4km) matrix (ModelD) – variables of economic activity and employment

 

TSShare One-family HH

TSShare One-person HH

Parameter

Estimate

Standard
Error

t Value

Approx
Pr > |t|

Estimate

Standard
Error

t Value

Approx
Pr > |t|

Intercept

-0.128

0.018

-7.130

<0.000

-0.910

0.081

-11.300

<0.000

AvgAge

-0.003

0.000

-17.130

<0.000

0.014

0.001

22.020

<0.000

ShareEconActive

-0.002

0.000

-7.070

<0.000

0.005

0.001

5.700

<0.000

ShareEmployed

0.002

0.000

10.790

<0.000

-0.008

0.001

-8.420

<0.000

W_ShareEconActive

0.001

0.000

2.810

0.005

-0.004

0.001

-3.220

0.001

W_ShareEconActive

0.001

0.000

3.400

0.001

-0.006

0.002

-3.800

0.000

W_ShareEmployed

-0.001

0.000

-3.440

0.001

0.004

0.001

3.130

0.002

_rho

0.365

0.017

21.790

<0.000

0.373

0.017

22.210

<0.000

_sigma2

0.001

0.000

54.830

<0.000

0.015

0.000

54.800

<0.000

ShareEconActive – share of economically active persons of the total number of persons, Share_Employed - share of employed persons of the total number persons, W_ShareEconActive – spatial effect of variable ShareEconActive, W_Share_Employed – spatial effect of variable Share_Employed, the meaning of other variables is the same as in tab. 7.

52The results for the two economic variables (share of economically active and share of employed persons) are also interesting. For both types of households (one-family households, one-person households) the results are statistically significant, even in combination with the spatial effect of variables or with average age. Since employment levels vary in the Czech Republic (although these differences are not essential), these variables can be used to create spatial regression models.

Table 12 Model selection based on AIC and SBC criteria

Model Fit Summary

One-family Households

One-person Households

ModelA

ModelB

ModelC

ModelD

ModelA

ModelB

ModelC

ModelD

Number of Observations

6122

6122

6122

6122

6122

6122

6122

6122

Log Likelihood

13125

13058

13844

13098

4173

4103

4820

4123

AIC

-26208

-26095

-27661

-26178

-8304

-8185

-9614

-8228

SBC

-26067

-26021

-27574

-26117

-8163

-8111

-9527

-8167

ModelA – age variables, ModelB – variables of education, ModelC – variables of marital status, ModelD – variables of economic activity

53Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (SBC). AIC or SBC can be used for model selection. For a set of candidate models, the model with the smallest AIC or SBC is often preferred. The results of individual models show that ModelC achieves the lowest values for both the AIC and SBC criteria. It appears that the marital status of the Czech Republic is the most suitable parameter in the spatial modeling of household structures in the Czech Republic. Alternatively, age indicators (i.e. selected age groups and average age) could be used as a second option to determine the spatial structure of one-family households or one-person households.

5. Conclusion - discussion

54It follows from the previous sections of this paper that it is reasonable to assume that there is a spatial regression relationship between households (in this case one-person households and one-family households) and explanatory variables such as age, education, marital status or economic activity.

55The assumption of spatial regression is the existence of autocorrelation. The results of both Moran’s I and Geary’s C show that the autocorrelation for both types of households was found to be statistically significant and increases as the distance between adjacent elements (i.e. municipalities) decreases.

56Age is an important factor affecting the structure of households. The results for both types of households show that the age groups with the greatest influence on the creation of one-person households or one-family households can also be used to create a spatial model. A similar claim applies to the average age.

57Education showed that the share of persons with primary education has no influence on spatial regression, unlike the share of persons with secondary or university education. In the case of marital status, there is a statistically significant spatial regression for one-person households, but not clearly for one-family households. Economic activity or employment is statistically significant for simple regression even in a small territory such as the Czech Republic.

58The individual models were designed to maintain their internal homogeneity and spatial regression models for age, marital status, education, or economic activity - including the average age in all models were created separately as a synthesizing element. The advantage of this approach is easier interpretation. In the case of the Czech Republic, the age composition for municipalities is an annual output of demographic (intercensal) statistics and the household model could be upgraded annually.

59Religion deserves separate consideration as a factor that generally has an undoubted influence on the structure of households, even in terms of territorial differences. The main problem why it was not possible to use this variable in the creation of alternative models in the case of this analysis is the very low response rate to this traditional question of population censuses. Data on religious beliefs from post-war censuses document a steady decline in religiosity in the Czech Republic.

60After a society-wide change after 1989, this resulted in the number of unanswered answers in the Czech Republic reaching almost five million in Population and Housing Census 2011 - 44.7% of the total population, compared to less than 9% in Population and Housing Census 2001. The number of unanswered answers to beliefs was more than twice as high as the total number of believers. The extent to which even potential believers who did not answer the question intentionally are included in these numbers can only be speculated. However, it is a fact that they were free to profess their faith and were called upon by a number of churches before the census. Even more detailed data according to individual classification aspects do not specify a person without the given answer. For the vast majority of individual characteristics, such as gender, age, size of the municipality of residence, territorial dislocation, etc., there was agreement or significant similarity with the national average. The only distinguishing feature is education, which is directly proportional to the willingness to answer. Thus, the higher the education, the greater the number of answers about religious faith (CZSO, 2013).

61From the number of people who have professed a religion, it is clear that believing women predominate over men. In 2011, a total of 1,209,000 women and 906,000 men professed a religious belief, representing a ratio of 55.7% to 44.3%. The decrease in the total number of believers, which occurred mainly in higher age categories (from the age of 50), was mainly reflected in a decrease in the share of people with only basic education but also in the share of people with secondary education. The opposite is true of full secondary and tertiary education. The proportions of these people professing a faith have increased significantly compared to the census ten years ago (2001) and are also higher than in the general population. It is most striking among believers with a university degree, whose share was 16% compared to only 7.4% in 2001 (in the entire population in 2011 it was 12.5%) (CZSO, 2013).

62The increase in religiosity from west to east, which is a basic feature of the territorial division of the faithful in our republic, is evident since the foundation of Czechoslovakia (1918) and is confirmed by the 2011 census. The order of regions (NUTS2) according to religiosity has practically not changed since 1991, only the rate of religiosity has gradually decreased, both the highest and the lowest. In 2011, the Ústí nad Labem, Karlovy Vary and Liberec regions continue to be the areas with the lowest religiosity - the shares of believers in the total population of the region only slightly exceed ten percent. On the contrary, the territories with the highest degree of religiosity remain the Moravian regions - Zlínský, Jihomoravský and Vysočina region, where the shares of believers ranged between 37% and 29% of the total population. All the above facts are not affected by the size class of the place of usual residence (CZSO, 2013).

63All of these are, in short, the reasons that led the author of this paper to resign implementing religion to the issue of spatial models, although he has no problem describing these facts. Probably the most interesting is the old statistical sentence, which says that more interesting than the answered questions (eg census), are the unanswered, respectively reasons that led to non-response, which also applies to the detection of religion in the Czech Republic.

64For the solution, it is possible to use several types of models offered by (econometrics) theory. In the case of households, the spatial Durbin model (SDM) is relatively widely used, because of the inclusion of both endogenous and exogenous interaction effects, based on the criteria chosen (Log Likelihood, AIC, SBC). However, the results for other models (SAR, SEM, SDEM) are not significantly different.

65A frequently discussed, comparable example can be found in [Elhorst, 2014], which is already cited in [Anselin, 1988] with a cross-sectional dataset of 49 Columbus, Ohio neighbourhoods to explain the crime rate as a function of household income and housing values. The spatial weights matrix W is specified as a row-normalized binary contiguity matrix, with elements wij = 1 if two spatial neighbourhoods share a common border, and zero otherwise. The input variables of both examples are therefore similar and the spatial weights matrixes are not fundamentally different. To compare multiple models and the parameter estimates are obtained by applying the maximum likelihood function, and this procedure was also chosen by the author of this paper.

66The main question asked by all authors is which model best describes the data. In both mentioned cases, the likelihood ratio (LR) test based on the log-likelihood function values of the different models was used. The test statistic has a Chi squared distribution with degrees of freedom equal to the number of restrictions imposed.

67The interpretation of the results must then be based on the nature of the researched problem. In the case of census data (i.e. this paper), then mainly from the input assumptions of the population situation and development in the Czech Republic.

68At the very end, it is possible to refer to the classic words, which state: "All models are wrong but some are useful" or even better to ask: "Is the model illuminating and useful?" (Box, 1976). The author of this paper considers that the answer to this question is in the affirmative, because it is possible to explain - with some degree of caution - the spatial changes in households that occur in modern society on the basis of classical demographic indicators.

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Annexe

To better understand the settlement of the territory of the Czech Republic, a map with a territorial-administrative structure was created: the NUTS2 border and municipalities with more than 20,000 usually living. The municipalities mentioned in the text should be described in this map.

Image 10000201000009C4000006E9877F411F41BAADBD.png

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Référence électronique

Jaroslav Kraus, « Spatial regression model of households: a case of the Czech Republic »Espace populations sociétés [En ligne], 2022/1 | 2022, mis en ligne le 01 mars 2022, consulté le 20 mai 2022. URL : http://journals.openedition.org/eps/12307 ; DOI : https://doi.org/10.4000/eps.12307

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Auteur

Jaroslav Kraus

Faculty of Science, Charles University, Department of Demography and Geodemography, Albertov 6, Čzech Republic, kraus[at]natur.cuni.cz

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Espace Populations Sociétés est mis à disposition selon les termes de la licence Creative Commons Attribution - Pas d'Utilisation Commerciale - Pas de Modification 4.0 International.

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