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Miranda, Rubens Augusto de; Domingues, Edson Paulo Commuting to work and residential choices in the metropolitan area of Belo Horizonte, Brazil Urban Public Economics Review, núm. 12, 2010, pp. 41-71 Universidad de Santiago de Compostela España Disponible en: http://redalyc.uaemex.mx/src/inicio/ArtPdfRed.jsp?iCve=50414006002

Urban Public Economics Review ISSN (Versión impresa): 1697-6223 [email protected] Universidad de Santiago de Compostela España

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Commuting to work and residential choices in the metropolitan area of Belo Horizonte, Brazil Rubens Augusto de Miranda*, Edson Paulo Domingues** This paper aims at presenting some of the main residential location theories and supporting empirical evidence using data for the metropolitan area of Belo Horizonte, Brazil, from the research “OriginDestination 2001” of Fundação João Pinheiro. Concerning the empirical evidence, our understanding is that the most appropriate methodology is the multilevel approach, given that the data has a hierarchized and nested structure. Thus, the time spent in commuting to work was modeled as a function of some individual, household and regional characteristics. The results corroborate some of the theories about the household structure in the process of residential choice and about gender differences in commuting to work. Keywords: residential choices, commuting to work, multilevel models Este trabajo tiene como objetivo presentar algunas de las principales teorías de localización residencial y evidencia empírica de apoyo utilizando datos del survey "Origen-Destino 2001" de la Fundación João Pinheiro, para la región metropolitana de Belo Horizonte, Brasil. En cuanto a la evidencia empírica, para nuestro entendimiento la metodología más adecuada es el enfoque de niveles múltiples, dado que los datos tienen una estructura jerarquizada y anidada. Así, el tiempo empleado en ir al trabajo fue modelado como una función de algunas características individuales, de hogares y regionales. Los resultados corroboran algunas de las teorías sobre la estructura de los hogares en el proceso de elección de vivienda y sobre las diferencias de género en los desplazamientos al trabajo. Palabras clave: opciones residenciales, ir al trabajo, modelos multinivel JEL classification: C49, R20, R21, R23, R41

* Researcher from Centro de Pós Graduação e Pesquisas em Administração of the Federal University of Minas Gerais (CEPEAD-UFMG) ** Researcher and Professor from Centro de Desenvolvimento e Planejamento Regional of the Federal University of Minas Gerais (CEDEPLAR-UFMG)

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1. Introduction In market economies land is allocated among alternative uses, mostly by means of private markets, with greater or lesser public regulations. In these societies, the current spatial structure of a city is, therefore, the result of millions of individual actions taken in the past. One could think that the result of such action would be very close to chaos. However, history suggests the opposite: the larger is the number of individual actors in a system, the stronger are the regularities shown. Taking these “evidences” into account, the orthodoxy of urban economics, substantiated in the New Urban Economics (NUE), built “an image of the residential structure based on the principle of market coordination of autonomous and decentralized family location decisions” (ABRAMO, p. 126, 2001, free translation). That is, from the chaos of individual location decisions, the market would be able to deliver an urban order, leading to the construction of the “urban invisible hand” hypothesis. When a household moves to a given city, and has to choose a residence, it faces a complex set of decisions. In this situation, according to the New Urban Economics, households face a trade-off between issues of accessibility, space and amenities. Accessibility includes pecuniary costs as well as time costs associated with journeys to work, leisure, shopping, and other activities. Concerning space, we have the physical aspects themselves. Finally, environmental amenities include natural characteristics (landscape, for instance) as well as neighborhood features related to the quality of schools, security and even the racial composition of the population. Thus, in the formulation of residential choice, a household must consider all these three factors accordingly and, in addition, it has to face budget and time constraints. The option of studying decision-making regarding residential choices, aiming at a better understanding of urban issues, has the significant advantage of focusing on the actions of individuals who make decisions. The understanding of the location process of intra-urban residential mobility is of great importance for the elaboration of urban policies, such as the formulation of land use plans and of projects of urban systems. From theory we can understand the reason for some unexpected results of public investment on some intra-urban localities. An example would be investment in infrastructure, or even in landscaping, in a low-income neighborhood; a likely result is the rise in prices of the land due to such investments, which would lead to the expulsion of poorer families. Therefore, it is important to develop disaggregated investigation at

individual or household level regarding the main motivations that explain individuals’ options for a given residence, including the influence of environmental aspects. Given these aspects, and in light of residential location theories, this paper intends to investigate the importance of accessibility (to work) in the choice of residence location in the metropolitan area of Belo Horizonte (RMBH). Through the use of statistical modeling, we evaluate the role of the spatial interaction residence-workplace as decisive in this choice, given the various aspects influencing location, related to household and environmental characteristics, and the contribution of these different factors is assessed. 2. The New Urban Economics and residential choices The so-called New Urban Economics (NUE) was founded in the 1960s and early 1970s as representative of neoclassical synthesis in urban economics1. Starting with the seminal work of Alonso (1964), this School tried to formulate a theory of residential location based on the conversion of von Thünen’s insights for an urban environment 2. According to the NUE, households make their locational choices aiming to maximize utility, under conditions of perfect competition and freedom of choice. Therefore, commuting costs, housing costs and income are considered altogether in order to predict the location of households and of different income groups in a city. If a household’s demand for space remains constant whereas leisure time becomes more valuable, the family could choose a location closer to downtown, reducing the time spent in commuting to work, which would increase leisure time and, therefore, the household’s welfare. Another situation would be the one in which valuation of time remains constant, but demand for space increases. In this case, the household would move to peripheral regions in the city in order to find cheaper housing. In general, the rate of substitution between commuting costs 1

It started mainly from contributions of Alonso (1964), Mills (1967, 1972), Beckmann (1969), Muth (1969), Solow & Vicrey (1971), Solow (1972). Simultaneously, but not

2

less importantly, we have the contributions of Wingo (1961a, 1961b).

Von Thünen (1826) develops his classical model of the determination of land use and land rent for agriculture in hinterland around an isolated town. He aims to respond how land is alocated if there is a non-planned competition between farmers and landlords, in which each individual acts in his/her own interest.

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and living costs is given by the household’s valuation of time and by its preferences for living in a specific living density. These results are derived in the next section using NUE’s extended-time model of residential choice. 2.1 The extended-time residential choice model A representative household aims to choose a home which maximizes its welfare (utility), subject to some restrictions. Let’s assume this household’s utility function is given by U(z,s,tl), where z represents the amount of all consumption goods except land, which we’ll call composite good, s is the consumption of land, given by the size of the house, or size of the lot, and tl represents leisure time. The composite consumption good is usually chosen as numeráire, i.e. its price is equal to one (FUJITA, 1989). The total disposable time of the household or individual, given by t , is divided between leisure time tl, working time tw, and commuting time br, where b is a constant which represents commuting time by distance. Therefore, the household’s temporal restriction is given by: tl + tw + br = t It is assumed that the household’s total income can be seen as the sum of a non-wage component YN, and a wage component Wtw, where W represents the wage rate. As before, the family will spend its income in the consumption of a composite good, in land costs R(r)s, and in transportation costs ar, where the constant a represents the pecuniary cost of commuting by distance r. Now the budget constraint can be given by: z + R(r)s + ar = YN + Wtw Assuming the households are free to choose the amount of time spent with leisure and work, we can define the new household’s problem of residential choice, given by:

max U (z , s, t l )

r , z , s ,t l

s.t z + R(r)s + ar = YN + Wtw tl + tw + br = t

[2.1]

This is the so-called extended-time residential choice model. The household’s residential choice can be obtained by solving the maximization problem [2.1]. However, Alonso (1964) brings from von Thünen a very rich conceptual device for the analysis of locational choices. He reformulates the agricultural rent supply curves for a urban environment, which are usually called bid rent curves. The bid rent function essentially describes the household’s ability to pay for housing under given levels of utility. Fujita (1989) states the following definition for the bid rent function: Definition 1: The bid rent function ψ (r,u) is the maximum rent per unit of land which the household can afford to pay in order to live at a distance r while keeping a given level of utility, u. The steps for obtaining the bid rent functions in the basic model can € we can simplify the constraints by reoralso be taken in this model. First, ganizing the temporal constraint, tw = t - tl - br, and replacing it in the budget constraint. By doing so, we obtain a problem with only one restriction, as follows:

max U (z , s, t l )

r , z , s ,t l

s.t. z + R(r)s + Wtw = I(r)

[2.2]

Where I(r)= YN + Iw(r) - ar and Iw(r) = W( t - br). This problem can be interpreted in the following way: the household sells its available time at a wage rate W and it also buys leisure time at the same rate, due to opportunity costs. Fujita (1989) calls I(r) and Iw(r), respectively, potential net income and potential wage income at distance r. Given these definitions, we can also determine the total commuting costs by distance r. Thus, we define commuting costs as: T(r) = ar + Wbr

[2.3]

Given these modifications, we can define the bid rent function as:

()

⎧⎪ I r − z −Wt ⎫⎪ l ψ r ,u = max ⎨ |U z ,s ,tl = u ⎬ z ,s ,t l ⎪ s ⎪⎭ ⎩

( )



(

)

[2.4]

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Solving the utility constraint with respect to z, we obtain the indifference curve z = Z(s,tl,u). Substituting in equation [2.4], we get an expression for the unrestricted residential choice problem:





( )

ψ r ,u = max

()

(

)

I r − Z s ,tl ,u −Wtl s

s ,t l



[2.5]

In order to continue with the analysis in this model, it is important to obtain the Marshallian demand for land, sˆ(R, Pl , t l ), where Pl is the unit price of leisure time. It is obtained from the solution of the following maximization problem:

max U (z , s, t l )

s.t.

z , s ,t l

z + Rs + Pltl = I

[2.6]

The Marshallian demand under land rent R and leisure price W is precisely the bid-max demand for land under utility u, that is, S (r,u ) ≡ sˆ (ψ (r,u ),W , I (r )) . Given these considerations, one can analyze the effects of non-wage income and of wage income on the residential location. The non-wage income effect is quite similar to the basic model discussed in the previous section, i.e., given that ψr = −T' (r ) sˆ (ψ (r,u ),W , I (r )) , we have:



[

=−

T' (r ) ∂sˆ ∂I (r ) sˆ 2 ∂I ∂YN

=−

a + Wb ∂sˆ sˆ 2 ∂I





]

∂ T' (r ) sˆ (ψ (r,u ),W , I (r )) ∂ψr € |dψ=0 = |ψ (r ,u )=const ∂YN ∂YN





Assuming normality3, the income effect is negative. As a result, the bid rent function becomes steeper as YN increases. Thus, we can conclude € with the following proposition: 3

The income effect on the Marshallian demand for land is positive.

proposition 1. Households with high non-wage income are located farther from CBD than households with low non-wage income, ceteris paribus. Concerning the wage income effect, the wage rate has effects both on the transportation costs function and on the land demand function. Therefore, the wage income effect on the steepness of rent supply is given by:



⎛ 1 ∂T' (r ) T' (r ) ∂S (r,u ) ⎞ ∂ψr ⎟ |dψ=0 = ⎜⎜ − 2 ∂W S (r,u ) ∂W ⎟⎠ ⎝ S (r,u ) ∂W

dψ=0





T' (r ) ⎛ ∂T' (r ) W ∂S (r,u ) W ⎞ = − ⎜ ⎟ S (r,u )W ⎝ ∂W T' (r ) ∂W S (r,u ) ⎠dψ=0



[2.7] 46 47

Therefore, the wage income effect will be positive if the wage elasticity of the marginal transportation cost,

€ ∂T' (r ) W ∂W T' (r )

is greater than the wage elasticity of the size of the lot,







∂S (r,u ) W ∂W S (r,u ) and it will be negative otherwise. The wage income effect thus depends on the elasticities. In this sense, given that T'(r) = a + Wb, we can reformulate the wage elasticity of the marginal transportation cost as follows: −1 ⎛ ∂T' (r ) W ∂T' (r ) W a ⎞ |dψ =0 = = ⎜1+ ⎟ ∂W T' ∂W T' (r ) ⎝ bW ⎠

[2.8]

∂S (r,u ) W I (r ) |dψ =0 = h w +e ∂W S (r,u ) I (r )

[2.9]

where

€ h=



∂sˆ I (r ) , e = ∂sˆ Pl ∂Pl sˆ ∂I sˆ



[2.10]

Here, h and e represent, respectively, the net potential income elasticity of the lot size and the cross elasticity of the lot size on leisure time. Therefore, equation [2.7] can be rewritten as:

Urban Public Economics Review | Revista de Economía Pública Urbana





T' (r ) ∂ψr |dψ=0 = ∂W S (r,u )W

−1 ⎡⎛ ⎛ I (r ) ⎞⎤ a ⎞ ⎢⎜1+ − + e⎟⎥ ⎜h w ⎟ ⎢⎣⎝ bW ⎠ ⎝ I (r ) ⎠⎥⎦

[2.11]

Since the difference in elasticities is a function of r and W, we will call it f(r, W). In order to understand the behavior of households, we can consider two representative situations, one in which the families earn only wage income (YN = 0) and the pecuniary costs are negligible in relation to temporal costs (a=0), and another situation in which again the households only earn wage income but in which pecuniary costs are significant in relation to temporal costs. In the first case, the difference of the elasticities is simplified to: f(r, W) = 1 – (h + e) Therefore, we can define a proposition for this first situation: proposition 2. Given that households consist on pure wage earners and that pecuniary costs are null, then: (i) If h + e > 1, the household’s location equilibrium moves outside the CBD as wage rates increase; (ii) If h + e < 1, the household’s location equilibrium moves toward CBD as wage rates increase; (iii) If h + e = 1, the wage rates do not affect location. In the situation in which pecuniary costs are significant, f(r, W) remains unchanged. We can see that when r = 0, f(r, W) ≤ 0 if h + e ≥ 1. However, the ratio Iw(r) / I(r) increases as r increases, thus f(r, W) < 0 for all r, if h + e ≥ 1. We can conclude that: if h + e ≥ 1, then −

∂ψr |dψ =0 < 0 ∂W

This result suggests that households earning high wages live farther from the CBD than households earning low wages. The wage € effect on bid rent has a very specific pattern when 0 < h + e < 1.

Given that the wage elasticity of the marginal transportation cost is increasing from 0 to 1 and that the wage elasticity of the lot size is constant at h + e, we see that, as the wage changes, f(W) is negative when ˆ , where W ˆ and positive when W > W ˆ is the wage rate that equates W 0), then: (i) If h + e ≥ 1, the household’s location equilibrium moves farther from the CBD as wage rates increase; (ii) If 0 < h + e < 1, the increase in the wage rate initially moves the locational equilibrium away from the CBD; however, after a certain ˆ , wage increases lead to a location closer to the CBD. rate W In this scenario, the rich households would be allocated in two clusters, one close and the other far from downtown. In such distribution, the rich € individuals located close to downtown place more value to time, whereas the rich individuals in the suburbs put more value on space. This variation in preferences can be attributed to high income households in different stages of the life cycle; households in earlier stages of the life cycle tend to value time while households in later stages tend to prefer more space. 2.2 The literature on commuting to work and residential choices According to Cadwallader (1992), urban residential mobility can be inter-

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preted as a phenomenon of the residential market, with households changing their stock of homes as they experience changes in their familiar status and in their socio-economic status. Therefore, the decision of moving is a function of the expected level of satisfaction of living in other places, given that households’ preferences have been “changed”. The different types of people, in terms of socio-economic status, race and other aspects, living in a household’s neighborhood, give an idea of the relative importance of the social environment. The way the household sees itself in relation to that environment can be very important in taking the decision regarding residential moves. As suggested by Simmons (1968), residential mobility at times becomes a spatial expression of social mobility. The inflow of a different socio-economic group in a neighborhood can be decisive in inducing old time residents to move. Lee et al (1994) and Bolan (1997) criticize various researches on location and residential mobility for focusing on the specific features of homes in each stage of the live cycle, and on “types” of people, and for ignoring contextual aspects of the location of the houses. Among these contextual aspects, one could mention: the physical qualities, the socio-economic status of the environment, diverse amenities and affection to the neighborhood. For instance, when a household enters the stage in life which corresponds to the raise of children, the features of the neighborhood and of the house are normally judged according to new standards. Some studies reject the location of the workplace as an important variable in the decision to move (Simmons, 1968), but some evidence suggest that the duration of the commuting trip to work represents a significant factor on the location of families, as observed in the works of Clark & Burt (1980), Madden (1981), Bem-Akiva & Bowman (1998), Magalhães (2002). Ben-Akiva & Bowman (1998) argue that a family normally considers the accessibility of each of its members by following a certain hierarchy of priorities, when choosing its living location. For example, accessibility to the head of the household can have greater weight than the one for other members of the family. Arguing about commuting distances in standard urban models, Ng (2008, p. 116) affirmed that ‘In nearly all of the underlying theoretical models, household preferences for amenities are ignored’. In order to solve this problem, the author developed a model, with two possibilities of job locations: the CDB and a sub-center, and that considers the fact of ‘households are willing to accept a longer commute to work if proximity to

certain amenities is important to them’. The model shows that the preferences for amenities increase the average of commuting distances in the cities. Barbonne, Sheamur and Coffey (2008) used the 1998 and 2003 OD Survey of Montreal’s Metropolitan Transport Agency to support their study about the evolution of commuting distances and mode choice associated with employment distribution in metropolitan area of Montreal. The authors argue that depending on the classification of job centers, these being aggregated or disaggregated, it leads to the results of a short or a long time of commuting. While a less detailed classification of a polynucleation could result in less “sustainable” commuting, in the other hand a more detailed classification of job centers, which makes distinction between the core and frame of job centers, presents as a result a more “sustainable” commuting. Given the difficulties in finding a good proxy for the utility of households, the empirical works on residential location, theoretically related to NUE, focused on the analysis of distance or commuting time. Using this approach, multilevel models are used by Magalhães (2002), who tries to examine the relative importance of socio-economic and demographic features of households, as well as environmental and neighborhood aspects, on the processes of mobility and residential location in urban areas. For that, he used data from “Pesquisa Domiciliar de Origem Destino da Região Metropolitana de Belo Horizonte”4, developed in 1992 by NUCLETRANS, in partnership with TRANSMETRO and PLAMBEL. In his study, Magalhães uses a multilevel logistic regression model to investigate how household and contextual factors affect the probability of living close to the workplace (commuting time under 20 minutes). Among other results, it was suggested that for married workers, especially women, the probability of living close to work increases 60%. The proximity to the workplace is also a characteristic of higher income groups, independently of gender. This relates to the ideas behind the extended-time model, in which an increase in the individual’s purchasing power makes time more valuable. The same approach is also used by Bottai et al (2006) for modeling distance and the number of trips per day, focusing on the effects of gender and age of the residents of Pisa, in Italy. They have used a three-level hierarchical model to capture the influence of intra-household and intra4

Household Survey on Origin and Destination in Belo Horizonte metropolitan area.

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regional correlation on individual mobility. The study did not specify the purpose of the trip, thus the response variable is the distance covered in trips for any reason. Among the main results, we have that household characteristics explain 37% of the variance of commuting time and regional characteristics explain only 6% of variance. Another important result refers to gender differences, since distances covered by women are on average 46% smaller. The difference between sexes regarding daily commuting, as reported by Bottai et al (2006), corroborate in part arguments made by Madden (1981). According to this author, women look for jobs close to home for two reasons: 1) lower wage rates and lower working hours decrease the commuting return rate and; 2) household responsibilities increase the cost of long trips. However, the gender gap in commuting time decreases as wage rates and working hours increase. In other words, with smaller differences in wages and working hours, women tend to have similar trips to the ones observed for men. 3. Data and methodology The idea that individuals interact with the social contexts in which they are inserted means that they are influenced by the social groups where they belong. The features of such groups, on the other hand, depend on the individuals who are part of the group. Individuals and groups may be seen as units belonging to some hierarchical structure. In such systems we could observe different hierarchical levels, in which the individual and group variables would be defined in different levels (individuals in a lower hierarchical level and groups in higher levels). Facing data which present a given hierarchical structure, it can be noticed that the observations from the same group are in general more similar than those from other groups, which violates the hypothesis of independence among observations. Heteroscedasticity is another common problem of nested data, since the error term is usually correlated with the explanatory variables, whereas common multiple regression models assume homoscedasticity. When this hypothesis is violated, ordinary least squares (OLS) regression may be inadequate. Therefore, given these features of hierarchical data, it is necessary to use an alternative approach. The so-called multilevel, or hierarchical, or random coefficients approach is a methodology that tries to capture the relations between individuallevel variables and variables related to some groups. The name random coefficients is due to the fact that the intercept and slope coefficients

vary between families and regions, which are second- and third-level groups, respectively. The investigation on the determinants of location and urban residential mobility in light of the main theoretical approaches clearly involves some hierarchical levels of analysis. In principle, at least three levels can be defined: the first one relates to individual characteristics such as: age, gender, condition in the household, and means of transportation; the second one refers to household characteristics, involving, among other aspects: income, household composition; and the third, relating to the environmental features of the urban region where the household is located, such as: neighborhood characteristics, accessibility to services, stores, etc. Regarding the database, this study used the Survey Origin Destination (OD) of 2001, from Fundação João Pinheiro. In order to understand the process of intra-urban mobility within the Belo Horizonte metropolitan area5, 121,296 permanent urban residents were interviewed, which corresponds to 2.68% of the households in the metropolitan area. The OD survey built a database in which census sectors using IBGE’s 6 classification were aggregated into spatial units characterized by an “urbanistic similarity of the settlements (households, institutions, stores, industries), of occupation density, of the environment, declivity, topographic convergence area, physical barriers and road nets of internal and external articulation”7 which present some degree of homogeneity. Therefore, such spatial units where named homogeneous areas (HA). According to this criterion, 1003 homogeneous areas were defined. However, since not all 5

The survey also aimed to understand the structures of land use, urban land market, external migration currents and the metropolitan demographic growth: To delineate the socio-economic picture of the population from Belo Horizonte metropolitan area, given the close relation between these variables and the others (income / occupation / moving; income / occupation / daily commuting): To provide information on the population’s patterns of commuting in the metropolitan area, in order to forecast future transportation demands: To provide background information for research on demographic information (population growth) using different aggregation levels (IBGE’s census sectors), districts and municipalities with political-administrative units and OD zones, developed by PLAMBEL (Planning Agency of the Belo Horizonte metro

6 7

area, in three previous surveys – 1972, 1981/1982 e 1991/1992 (FJP, 2004a , p. 1). Brazilian Institute of Geography and Statistics Fundação João Pinheiro, 2004a, p.9

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the HAs can be taken as urban agglomerations, or residential spaces, the areas which had less than 100 households were excluded. Thus, the OD survey was made in only 795 HAs. In comparison with census sectors, regarding the number of households, we have that: The census sectors of demographic censuses are administrative units for information collection, comprising from 300 to 400 households on average. Therefore, a typical homogeneous area comprises between 900 and 1600 households. (FJP, 2004a, p. 4). The research comprised four large steps. The first step consisted on a household survey, in which individuals in 31,416 households were interviewed, as pointed out in table 3.1. Four questionnaires were used in the interviews: one focused on characteristics of the household and the family; the second one approached socio-economic aspects of the individuals; another one inquired about all the trips and commuting made by the residents on the day before; and, lastly, one related to the residents’ evaluation of the means of transportation, accessibility to the bus system, and the number of vehicles in the household. In the second stage, the interviews were made in the so-called “Contour Line”, which refers to the limits of the Belo Horizonte metropolitan area (RMBH), by interviewing drivers and passengers and by counting the vehicles in 14 survey stations located in the main federal and state roads in the region. The goal in this stage was to quantify the inbound commuting. The third step of the research took place in the main bus station in Belo Horizonte8, and aimed to complement the information obtained in the “Contour Line” survey, including all the bus lines external to the RMBH. Thus, we interviewed a sample of passengers from each line according to arrival or departure time. The last step consisted on the survey of the “Passage Line”, which is a railway that divides the RMBH. In this stage, counting of vehicles and assessment of passengers per vehicle was made in survey stations located on the railway crossings. 3.1. The multilevel approach As discussed before, in the multilevel approach it is assumed that the data structure presents some hierarchy; thus, some variables are defined in each 8

Terminal Rodoviário Governador Israel Pinheiro – TERGIP, in downtown Belo Horizonte, where all intercity and interstate bus lines depart and arrive.

hierarchical level. However, variables can be moved between different levels by aggregation or disaggregation. In aggregation, variables from a given level are moved to a higher level, whereas in disaggregation the opposite occurs, variables from a higher level are moved to a lower level. Therefore, when choosing the multilevel approach, one of the goals of the researcher is to investigate hypothesis between hierarchical levels, called multilevel problems. According to Hox (1995, p. 5), “a multilevel problem is a problem that concerns the relationships between variables that are measured at a number of different hierarchical levels”. A question that well illustrates this problem is simply how individual- and group- explanatory variables influence the dependent, individual, variable. In this sense, the goal is to determine if the explanatory variables at group level work as moderators of the relations at individual level. In other words, we aim to understand how individual-level variables are standardized among the different groups. The equations below describe a three-level model with intercept and random coefficients. It assumes the existence of three nested levels: individual, household and region. Subscripts i, j, k describe the position of each individual in the hierarchical structure of the data; nj is the number of individuals in the jth household and mk is the number of households in the kth region. Thus we have i = 1,2,...,njk individuals in family j of region k; j = 1,2,...,mk households in region k and; k = 1,2,...,K regions. According to Raudenbush & Bryk (2002), a generalized multilevel hierarchical linear model (GMHL) consists of three parts: a sample model, a link function and a structural model. The multilevel hierarchical linear model (MHL) which will be used in this paper is a special case of the GMHL, in which the sample model is normal, the link function is the link identity and the structural model is linear. Before developing the three-level model, it is necessary to analyze these three parts in a single model. 3.1.1. The non-conditional model Before developing a complete model, it is important to analyze the simplest three-level model: the non-conditional model9. In this model, the explan9

Also known as null model or intercept-only model. We can see that it is equivalent to the ANOVA model with random effects

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atory variables are not included in each level of analysis. The basic idea of the model is to describe how changes in the explanatory variable affect the different levels. In this paper, it allows us to estimate the variability associated with the three levels of analysis: individuals, families and regions. In the individual-level model, commuting time is modeled for each individual as a function of his family’s and region’s average and an error term. Yijk = b0jk + eijk

[3.1]

where, Yijk is the commuting time of individual i, in household j, in region k; b0jk is the average commuting time of household j in region k and; eijk is an individual random effect, i.e. the deviations in commuting time of the individual ijk in relation to his household average. It is assumed that these effects are normally distributed, with zero mean and variance s2e . In the household-level model, b0 jk varies randomly around the mean of a given region: b0jk = g00k + u0jk

[3.2]

where, g00k is the average commuting time in region k; u0jk is the household random effect, which is the deviation from the mean of commuting time of household jk in relation to the region’s average. Again, we assume these are normally distributed, with zero mean and variance s2b. Within each one of the k regions, variability among households is assumed to be the same. Finally, there is the regional-level model. This three-level model presents variability among regions. Here it is assumed that the regional mean, g00k, varies randomly around a general mean: g00k = f000 + w00k

[3.3]

where, g000 is the general mean; w00k is the regional random effect, which is the deviation from the

mean of commuting time in region k in relation to the general mean. As before, we consider these effects to be normally distributed, with zero mean and variance s2g. A single-equation model can be obtained by replacing equations [3.3] and [3.2] in equation [3.1]: Yijk = g000 + w00k + u0jk + eijk

[3.4]

This simple three-level model partitions the variance of the response variable Yijk in three independent components: s2e , individual-level variance; s2b , household-level variance; and s2g , regional-level variance. The residuals of the three levels are considered to be mutually independent, thus there are no covariance elements in this partition. Therefore, the variance of the dependent variable can be described as: Var (Yijk) = Var (w00k + u0jk + eijk) = s2g + s2b + s2e

[3.5]

which corresponds to the sum of the individual-level variable, the household-level variance, and the regional-level variance. Covariance between two individuals in the same household (called i1and i2) is given by:

cov(u0 jk + ei1 jk , uojk + ei 2 jk ) = cov(uojk , uojk ) = s 2b



In turn, covariance between two households in the same region (named j1and j2) is given by:

cov(w00k + u0j 1 k + eij 1 k ,w00k + u0 j 2 k + eij 2 k ) = cov(w00k ,w00k ) = s g2



These results allow us to estimate the proportion of variance within households, among households, within regions, and among regions. Thus we have:

r2 = s 2b r3 = s g2 € €

(s (s

) +s )

2 e

+ s 2b + s g2

[3.6]

2 e

+ s 2b

[3.7]

2 g

The parameter r2 indicates the proportion of the dependent variable variance within the household in relation to the total variance and r3

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€ €

represents the proportion of the dependent variable variance between regions, in relation to the total variance. The parameters r2 and r3 are known as intra-class correlation coefficients10, and they can be interpreted as a measure of the degree of dependence in the observations belonging to the same group. The existence of intra-class correlations different from zero, resulting from the presence of more than one residual term in the model, indicates that estimation procedures such as Ordinary Least Squares (OLS) are not applicable, since the covariance terms are not zero. The use of a non-conditional model, i.e. a completely random model, aims to obtain those intra-class correlation coefficients. However, we expect to explain part of this variance by introducing explanatory variables at three levels in the model. The following conditional model does exactly this, considering the existence of explanatory variables in three levels. 3.1.2. The conditional model The non-conditional model presented in the previous section allows us to estimate the variance associated with the three levels of analysis. However, part of the variance in each level can be explained, or quantified, by the inclusion of variables in each level of analysis. In other words, individual, household and regional characteristics can be used as predictors. Within each household, commuting time is modeled as a function of individual-level predictors and an error term:

Yijk = b0 jk + b pjk X pijk + eijk



[3.8]

When we include the second level of analysis, we need to assume that the intercept and/or slopes of the coefficients vary by household, i.e. the estimated parameters become random variables, called household effect. Thus, for each household effect, b0jk and bpjk, we have:

b0 jk = g 00k + g 0 qk Z qjk + u0jk

[3.9]

b pjk = g p0k + g pqk Z qjk + u pjk

[3.10]

where, 10 According to Raudenbush & Bryk (2002) these coefficients can be called cluster effect.

Zqjk are the q = 1,...,Q household characteristics used as predictors of the household effect. The covariance between the residuals u0jk and upjk is usually different from zero, thus cov(u0 jk , u pjk ) = s b 0 p . We can see that the household-level model has P + 1 equations, i.e. one for each individual-level coefficient. It is assumed that the random effects in these equations are correlated. Therefore, the terms upjk are multivariate normally distributed terms with € zero mean and variance s2bpp , and some covariance among the elements upjk and up' jk of s2bpp' . The dimension of the variance-covariance matrix Tb depends on the number of individual-level coefficients specified as random. In order to illustrate this, one can take a case in which the household effect bpjk is considered to be fixed. In this case, no predictor at the household level will be included in equation [3.11] so that the effect of the correspondent upjk will be zero. Substituting equations [3.9] and [3.10] into equation [3.8], we get: Ypjk = g 00k + g 0 qk Z qjk + g p0 k X pijk + g pqk Z qjk X pijk + u pjk X pijk + u0 jk + eijk



The segment g 00k + g 0 qk Z qjk + g p0 k X pijk + g pqk Z qjk X pijk is the deterministic part, since it contains all the fixed coefficients. The stochastic part contains all the error terms u pjk X pijk + u0jk + eijk , where the components upjk and u0jk€refer to individuals in the same household. From the stochastic part, it can be seen how OLS estimation in inadequate to deal with nested data, since one can explicitly observe a heteroscedasticity problem, € given that the term u pjk X pijk + u0jk varies according to household units and individual attributes (due to the term Xpijk). The interaction term ZqjkXpijk appears in the model as a consequence of modeling the variations in the individual-level coefficients as a function of household-level vari€ ables. According to Hox (1995, p. 13), “the moderator effect of Z on the relationship between the dependent variable Y and X is expressed as a crosslevel interaction”. Another modeling process, similar to the one used to include variations in individual-level coefficients, can be used when the intercept and the household-level coefficients vary by region. In this case, the householdlevel estimated parameters also become random variables. Thus, we have the following region effects:

g 00k = f000 + f00lWlk + w00k g 0qk = f0 q0 + f0 qlWlk + w0qk € €

[3.11]

[3.12] [3.13]

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g p0 k = f p00 + f p0lWlk + w p0 k g pqk = f pq0 + f pqlWlk + w pqk

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€ €







[3.14] [3.15]

where, Wjkare the l = 1,...,L regional characteristics used as predictors of the region effect. As before, we substitute equations [3.12] to [3.15] and obtain the complex equation below: Yijk = f 000 + f 00 l Wlk + f 0 q 0 Z qik + f 0 ql Wlk Z qjk + f p 00 X pijk + f p 0 l Wlk X pijk +



f pq 0 Z qjk X pijk + f pql Wlk Z qjk X pijk + w 0 qk Z qjk + w p 0 k X pijk + w pqk Z qjk X pijk [3.16] +u pij X pijk + w 00 k + u 0 jk + e ijk

The deterministic part is now given by the segment

f000 + f00lWlk + f0 q0 Z qik + f0 qlWlk Z qjk + f p00 X pijk + +f p0lWlk X pijk + f pq0 Z qjk X pijk + f pqlWlk Z qjk X pijk and the stochastic part, besides incorporating household components, also incorporates regional components such as w00k, w0qk, wp0k and wpqk, forming the segment

w0qk Z qjk + w p0k X pijk + w pqk Z qjk X pijk + u pij X pijk + w00k + u0jk + eijk . The number of interaction terms also increased significantly, and these are now also present in the stochastic part. 3.2. Analysis procedures for hierarchical models When there are not available theories to formulate a model to be estimated, it is common to use an exploratory procedure. Thus, in the multilevel analysis, one usually begins with a non-conditional model, and the inclusion of new variables is done step-by-step, observing their significance. The models estimated in this paper are not theory-free, however they are not explicitly derived. The specifications of the econometric models estimated here are based on arguments following the insights of theories described in section 2. The first important point in hierarchical analysis is that the p-values produced by different software used in multilevel analysis may differ. Most of this software produces parameter estimates and asymptotic standard

errors for these estimates using the maximum likelihood estimation procedure. In this type of procedure, the usual significance test is the Wald test11. However, Bryk & Raudenbush (2002, p.58) argue that in a fixedeffect analysis, for instance in a model with two levels of analysis, a t-test would be more appropriate to test the effects of second-level explanatory variables. The software used in this paper, HLM512, produces p-values based on t-tests, instead of the usual Wald tests. The difference between these two procedures is greater when the number of groups is small. The authors also argument that the Z-test is not adequate for variances, due to the fact that the sample distribution of the variances in asymmetrical. Therefore, they propose a Chi-square test for the residuals (ibidem, p. 55 and p. 64). The HLM, due to the use of the maximum likelihood estimation procedure, produces an estimate that indicates the fit of the model to the data, called deviance. In other words, deviance is an index of bad model fitting. Usually, models with smaller deviance estimates present a better fit, but this is not always valid. Given that, when we compare two models, the difference in deviances has a Chi-square distribution in which the degrees of freedom are given by the difference in the number of estimated parameters in the models. An appropriate method to test if a given model which uses additional variables has a better fit is to divide the difference in deviances by the number of degrees of freedom. If the result is smaller than -2 or greater than 2, according to the rule of thumb proposed by Snijers & Bosker (1999), then the expanded model has a better fit. That is, one tests if a more general model adjusts significantly better than a impler model. Besides, this test can also be used to compare random-effect models and fixed-effect models. As mentioned before, one of the greatest advantages of multilevel modeling is the variance partition that is produced and that can be used to generate intra-class correlation indexes, which indicate how much of the total variance is explained by each level. However, as we consider the randomness among certain groups of coefficients, the models become more and more complex and the concept of explained variance may 11 A Z-test in the form Z=(estimate)/(standard error of the estimate), where Z refers to the standard normal distribution. 12 HLM (Raudenbush, Bryk & Congdon, 2000) is a set of software for the analysis of two- and three-level hierarchical data, called HLM2 and HLM3, respectively, and a special software for meta-analysis called VKHLM.

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include other definitions. In order to solve this problem, it is usually proposed to examine the proportion of explained variance through the analysis of the variance of errors in sequential models (models in which variables are included progressively). The idea is to interpret the differences in explained variance with the inclusion of new variables. 3.3 The residential choice model Using the hierarchical approach described in the previous sections, this paper proposes a multilevel model. As a response variable (dependent) we will use the natural log of the commuting time from home to work of all individuals. The option for the variable commuting time, instead of the traditionally used distance to CBD, presents the advantage of not being stuck to the idea of a monocentric urban structure in which all the jobs are located in downtown. In the first hierarchical level the following explanatory variables (covariables) are included: a gender dummy is included in order to control for gender differences in individuals’ commuting; regarding the age of individuals, dummies for age levels are included in order to capture the life cycles of individuals; since it is important to consider the means of transportation used for commuting, given the specific operational characteristics of each transportation mode, dummies are also included in order to capture such effects; the weight of the individual’s household situation in the household is captured by dummies of household situation. In the model, we also aim to evaluate how the household income of the workers affects a greater proximity of the home in relation to the workplace. This income will be treated in the model as a co-variable in the second hierarchical level (household). Thus, we included the natural log of the household per capita income and also the log of the square of income. The goal is to capture a non-linear relation between relative income and commuting time. Besides income, and according to economic theories, the space of the household also needs to be considered. However, since this information is not available, we will use the presence of dependent children (under 18 years of age) as a proxy, given that families with children may prefer types and locations for living which are different from families without children. The presence or absence of children may have greater influence in the location of the home than simply the size of the family does. Therefore, the presence of children may also indicate a different lifestyle of the families. Thus, the presence of

children will be treated in the model as a dummy variable in the second hierarchical model. In the third hierarchical model (regional), three co-variables were included: accessibility variables of the region, represented by the log of individual’s average time of commuting for access to services, shopping, and leisure. The explanation for including these variables is that when a region has many services, shopping and leisure options, the family would be willing to live farther from the workplace. Table 1 presents a description of the co-variables used in the model. The dummy for private motorized commuting (motorpri) refers to commuting as car drivers, car passengers and motorcycle drivers; the dummy for public motorized commuting (motorpub) refers to commuting by bus and by train/subway. Regarding the household situation of the individual, the dummy for relatives/others refers to individuals of a group cohabiting with members of the family, to the father or mother of the household head, to relatives of the household head and, finally, to guests. The age dummies will be named, respectively, age2, age3. Table 1: Description and codification of the co-variables used in the residential location model Level Co-variable Type of description variable Category Code/Value 1 - individual gender dichotomic male base female 1 = yes; 0 = no age dichotomic 1-29 years-old base 30-59 years-old 1 = yes; 0 = no > 60 years-old 1 = yes; 0 = no mean of dichotomic others base transportation private motorized 1 = yes; 0 = no public motorized 1 = yes; 0 = no individual’s dichotomic head base family situation spouse 1 = yes; 0 = no child 1 = yes; 0 = no relative/others 1 = yes; 0 = no

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2 - household family income children under 18 3 - Region average income accessibility to services accessibility to shopping accessibility to leisure

quantitative dichotomic

R $ has

continuous 1 = yes; 0 = no

quantitative

R $

continuous

quantitative

minutes

continuous

quantitative

minutes

continuous

quantitative

minutes

continuous

Source: authors elaboration

4. Econometric results The general model presented in the previous section uses the log of commuting time to work, in minutes, as dependent variable. However, for estimation purposes, we considered only individuals who made at least one trip to work, and whose commuting time was no longer than two hours, since longer trips may be considered outliers. Table 2 presents the number of observations after these two restrictions were considered. We can see that the number of individuals decreased to 65950, divided in 21570 households. From 795 original HAs, we now consider 784. Table 2: Number of observations in the databases by level of analysis

Level of analysis Individual Household Regional

Total number of observations 65950 21570 784

Source: Authors elaboration based on data from the OD Survey 2001

The initial step was to estimate the models (models 1 to 3) with only the intercept varying randomly (intercepts vary among groups). The results can be seen in table 3. In a second step, we estimated models in which the slope of some variables would randomly vary among groups (models 4 and 5). Table 4 presents the results of these estimations. Thus, the first estimated model (model 1) was the non-conditional one, which has only the intercept as linear predictor. As mentioned before, the goal of the estimation in this model is to calculate the intra-class correla-

tion coefficients, which measure how much of the total variation derives from variation among households, r2, and regions, r3. The results inform that 44.9% of variations in commuting time are explained by household differences, and only 6.8% are explained by regional differences. Bottai et al (2006) found similar results, 37% and 6%, respectively. In model 2, explanatory variables at individual level were included in order to explain part of the variance. The result was that the inclusion of these variables explained 12.59% of the variance at individual level, 13.62% of the variance at household level, and 53.14% of the variance at regional level. According to the variable Age3, the fact that an individual is older than 60 is negatively related to commuting time. We can assume that the individuals who walk or ride a bike to work usually live relatively close to the workplace. Therefore, when we include control variables to the individuals who use motorized means of transportation, commuting time tends to increase, since these individuals should cover larger distances to work. The relation between the gender dummy and time is negative, i.e. women live on average closer to the workplace than men do. In model 3, explanatory variables at household and regional levels were included. The variable Age3 becomes not significant in this case. Both income variables at household level are significant and have opposite signs. This result is compatible with other results in the literature and it is in line with the conclusions of the extended-time model, in which two different agglomerations of wealthy families, one close by and one far away from the workplace. Another important result is that families with children under 18 usually live closer to work. The inclusion of these variables helped to explain 28.5% of the variance at regional level, although the explained variance among households is practically null. As it is known, it is difficult to find good information to explain such variance, related to the families’ preferences. When the random effects are extended to the slope of the regressions, besides the intercept, we aim to capture the idea that some characteristics have different impacts in various groups. The age variables and mode of transportation variables are highly correlated with the variables Motorpri, Motorpub, and Regionincome. The multicolinearity resulting from the inclusion of these variables as random effects would lead to estimated correlations very close to 1, which prevents us from estimating covariance components and from computing initial values in the maximum likelihood process. There are two possibilities for solving this problem: either to exclude the problematic variables from the model, or to treat them as having constant slopes. In this paper, we chose the second option.

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Table 3: Results of commuting time equations for fixed effects models variables Model 1 Model 2 Intercept 3,0850 * 2,8129 * Individual Age2 0,0059 ns Age3 - 0,0513 ** Motorpri 0,0986 * Motorpub 0,6224 * Spouse - 0,0663 * Children - 0,0167 ns Relatives -0,0033 ns Gender - 0,0642 * Household Income Income2 Childund18 Region Accesserv Accesshop Accessleis Regionincome VariancePartition Individual 0,2486 * 0,2173 * Household 0,2744 * 0,2370 * Region 0,0382 * 0,0179 * Intraclass Corr Coef r 2 0,4489 r 3 0,0680 Deviance 126360,50 116987,18 # estim. paramet. 4 12

Model 3 2,7994 * 0,0136 ns - 0,0242 ns 0,1343 * 0,6147 * - 0,0627 * - 0,0014 ns - 0,0182 ns - 0,0554 * - 0,1336 * 0,0104 * - 0,0905 * 0,2305 * 0,0462 * 0,0257 * - 0,0602 * 0,2173 * 0,2364 * 0,0128 *

116821,38 19

Source: Authors’ elaboration from the models estimations Note: * significant at 1%; ** significant at 5%; *** significant at 10%; ns not significant.

Models 4 and 5 were estimated following this approach. We included explanatory variables at the three levels and assumed that the variables of household condition, gender, and children under 18 vary randomly among households. Besides, these models present interaction terms. Since

Table 4: Results of commuting time equations for random effects models variables Model 4 Intercept 2,7994 * Individual Age2 0,0136 ns Age3 - 0,0242 ns Motorpri 0,1343 * Motorpub 0,6147 * Spouse - 0,0627 * Children - 0,0014 ns Relatives - 0,0182 ns Gender - 0,0554 * Household Income - 0,1336 * Income2 0,0104 * Childund18 - 0,0903 * Region Accesserv 0,2305 * Accesshop 0,0462 * Accessleis 0,0257 * Regionincome - 0,0602 * Interaction terms Gender*Childund18 - 0,0004 ns Childund18*Accessleis Deviance 103555,41 # estim. paramet. 39

Model 5 2,7994 * 0,0136 ns - 0,0242 ns 0,1343 * 0,6147 * - 0,0627 * - 0,0014 ns - 0,0182 ns - 0,0554 * - 0,1336 * 0,0104 * - 0,0905 * 0,2305 * 0,0462 * 0,0257 * - 0,0602 *

0,0009 ns 103555,09 36

Source: Authors’ elaboration from the models estimations Note: * significant at 1%; ** significant at 5%; *** significant at 10%; ns not significant.

the interpretation of such terms is often not intuitive, we decided to include interaction terms only in cases we had good theoretical arguments for it. In model 4, we have the interaction term Gender*Childund18, and the justification for including it is that women in households with underage children would try to work closer to home. This argument is developed by Madden (1981), according to which the existence of children under 18 would increase the women’s work load at home, and induce them to look for jobs closer to home, in order to reduce commuting time. However, the

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results do not confirm this argument, given that the interaction term is not significant, even though the sign is as expected. The reason to include the interaction term Childund18*Accessleis, in model 5, comes from the assumption that regions with greater access to leisure are farther away from regions characterized by a large number of jobs. Thus, families with small children would tend to live in regions with better leisure options and, as a consequence, would have larger commuting distances. In other words, we aimed to capture some effect from the life stage of the family. Again, despite the arguments for including this variable, it is not significant although the sign is as expected. Using the deviances we made some chi-square tests in order to find out which models present a better fit to the data. The tests suggested that as more variables were included, such as random effects, the models’ fit was gradually increasing. Thus, among the fixed effect models, model 3 has the best fit. In turn, model 4 presents the best fit among models with random coefficients. Considering all the models, model 4 was also the one presenting the best fit. Lastly, some of the main conclusions of the estimated models can be summarized as follows: The differential between men’s and women’s commuting to work, as found in the literature, was confirmed. Women usually work closer to their homes; Among individuals who use motorized means of transportation, the additional commuting time is significantly higher for users of public transportation; The household income is significant in the determination of commuting time; The age variables and the existence of young children were able to partially capture the idea of life cycle; Household differences were very important to explain the different commuting patterns. 5. Conclusions This paper aimed to offer a theoretical revision and an empirical application of the main residential choice theories. The sample design required special attention to the choice of the statistical method for analysis. Given the understanding that the data have a hierarchized structure, we chose the multilevel approach in order to capture some correlation within different levels. One of our main results is that household differences

are very important to explain the different commuting patterns, i.e., the kind of household structure reveals family’s preferences. Therefore, by applying such methodology for the data of the OD survey at Belo Horizonte metropolitan region, we could conclude that commuting behavior is more similar among individuals from the same family than among individuals from different families. Also, there are gender differences in commuting time. Income and gender variables had the expected result. Thus, the gender differential observed in the literature also appears in the RMBH. The income variable behaves in line with the most recent residential location theories, which consider families with different preferences, since it shows a non-linear relation between income and commuting time. The importance of the household structure was also examined, and results suggest that households with children under 18 tend to live closer to the workplace. This paper provides elements to better understand the dynamics of residential choice and daily commuting to work at Belo Horizonte metropolitan region. However, the complexity of a topic such as intra-urban structure requires continuing investigation in the area. A suggestion for future research is to apply this methodology as well as the theories to other cities or metropolitan regions in Brazil, in order to elucidate the peculiarities of each urban center, regarding the characteristics of its intra-urban structure.

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