Catherine Baumont
email: catherine.baumont@u-bourgogne.frcem.ertur@u-bourgogne.friulie.le-gallo@u-bourgogne.fr

Cem Ertur

Julie Le Gallo Janvier

Catherine Bau

Julie Le Gallo
email: ilegallo@uiuc.edu

Cem E rtu C Atherine Baumont

Intra-Urban Spatial Distributions of Population and Employment: the Case of the Agglomeration of Dijon, 1999

Keywords: exploratory spatial data analysis, monocentric and polycentric configurations, population density, spatial autocorrelation, spatial heterogeneity, employment subcenters JEL Classification: C21, R14

The aim o f this paper is to analyze the intra-urban spatial distributions ofpopulation and employment in the agglomeration o f Dijon (regional capital of Burgundy, France). We study whether this agglomeration has followed the general tendency o f job decentralization observed in most urban areas or whether it is still characterized by a monocentric pattern. In that purpose, we use a sample o f 136 observations at the communal and at the IRIS (infra-urban statistical area) levels with 1999 census data and the employment database SIRENE (1NSEE). First; we study the spatial pattern o f total employment and employment density using exploratory spatial data analysis. Apart from the CBD, few IRIS are found to be statistically significant, a result contrasting with those found using standard methods o f subcenter identification with employment cut-offs. Next, in order to examine the spatial distribution o f residential population density, we estimate and compare different specifications: exponential negative, spline-exponential and multicentric density functions. Moreover, spatial autocorrelation is controlledfor by using the appropriate spatial econometric techniques. The splineexponential is found to perform the best, highlighting the monocentric character o f the agglomeration of Dijon.

Résumé

L fobjectif de ce papier est d'analyser la distribution spatiale de l'emploi et de la population à une échelle intra-urbaine pour la COMADI en 1999. Nous étudions en particulier si l'agglomération dijonnaise a suivi la tendance générale à la décentralisation des emplois ou si elle reste caractérisée par une organisation monocentrique de l'emploi et de la densité de population. Cette étude est menée sur un échantillon de 136 unités spatiales à l'échelle communale ou intra-urbaine (IRIS) à partir de données issues du RGP et du fichier SIRENE pour l'emploi salarié du secteur privé. Dans un premier temps nous réalisons une étude d'identification des centres d'emplois en utilisant les techniques de

VAnalyse Exploratoire des Données Spatiales (ESDA). En dehors du CBD, seuls quelques autres IRIS présentent un profil significatif de centres secondaires. Ce résultat montre que l'organisation spatiale de l'emploi dans la COMADI est plutôt de type monocentrique et contraste avec celui obtenu avec les méthodes standards d'identification des centres avec seuils. Ensuite, nous étudions si la distribution de la densité de population suit un schéma monocentrique ou multicentrique. Partant de la fonction de densité exponentielle négative, nous estimons différentes spécifications intégrant la distance au CBD et à d'autres centres secondaires et tenons également compte dline forme d'hétérogénéité spatiale de type spline exponentielle négative. La présence d'autocorrélation spatiale est systématiquement contrôlée. Il apparaît que la COMADI est de nouveau caractérisée par une organisation monocentrique de la population marquée cependant par une variation du gradient de densité résidentielle entre Dijon et les communes périphériques.

In t r o d u c t io n

Over the last decades, there has been considerable interest in the analysis of urban spatial structure. Indeed, urban growth has exhibited complex spatial patterns including both population spread and employment decentralization from the central city towards the suburbs.

The validity of the monocentric model [START_REF] Alonso | Location and Land Use: Toward a General Theory o f Land Rent[END_REF]Muth, 1969) to explain urban patterns has therefore been questioned since employment decentralization has recently taken a polycentric form, with a number of employment subcenters influencing the spatial distribution of employment and population. The polycentric urban phenomenon has been extensively documented for many years. Most studies have been carried out on North American urban agglomerations: Chicago (MacMillen and MacDonald, 1998a, b), Dallas-Fort Worth (Wadell and Shukla, 1993), Los Angeles [START_REF] Gordon | The distribution of population and employment in a polycentric city: the case of Los Angeles[END_REF][START_REF] Heikkila | What happened to the CBD-distance gradient? Land values in a policentric city[END_REF]Guiliano and Small, 1991;Small and Song, 1994;Sivitanidou, 1996), San Francisco (Cervero andWu, 1997, 1998), Montréal [START_REF] Coffey | Examining the thesis of Central Business District decline: evidence from the Montreal metropolitan area[END_REF]... Moreover, this trend toward employment decentralization is not limited to North American areas (see, for example, Alperovitch and Deutsch, 1996 for Jerusalem; Chen, 1997 for Taipei;Wu, 1998 for Guangzhou;[START_REF] Gaschet | La structure d'un espace urbain polycentrique: la métropole bordelaise[END_REF] for Bordeaux, Boiteux-Orain and Guillain, 2002 for Ile-de-France).

Few studies have been carried out on middle-sized urban areas. Therefore, it is interesting to investigate whether these particular areas have experienced a similar trend toward employment decentralization or whether the monocentric model is still valid to explain employment and population spatial distributions. In this paper, we are interested in this empirical challenge applied to the agglomeration of Dijon, which is the capital of Burgundy (France). Following Baumont andLe Gallo (1999, 2000), an economic center is then defined as a cluster of economic activities that influences the spatial organization of population, employment and land values. From an empirical point of view, studying polycentric rather than monocentric urban configurations raises a set of challenges [START_REF] Anas | Urban spatial structure[END_REF]Baumont andLe Gallo, 1999, 2000) which can be summarized as follows. How many economic subcenters can be identified apart from the traditional Central Business District (CBD)? What are their sizes and their boundaries? Does the emergence of urban subcenters result in the CBD decline? Finally, how do these multiple economic centers influence land values, population and employment distributions?

Two types of identification methods can be used in empirical studies. On the one hand, exogenous methods are based on a set of a priori characteristics on an economic center.

Exploratory statistical and cartographic methods applied to total employment or employment densities allow the identification of a potential set of economic employment centers. On the other hand, endogenous methods rely on the analysis of influence: an economic employment center has a significant influence on the spatial organization of the city. Econometric estimations of population density functions, ànd values gradient functions or employment density functions specified with the distances from each potential economic center allow concluding if a particular economic center has such an influence.

In this paper, we apply both exogenous and endogenous identification methods combined with spatial statistic and spatial econometric techniques to study the characteristics of the agglomeration of Dijon in 1999. The study covers the territory of the Communauté de l ' Agglomération Dijonnaise (COMADI), which is made up of 16 towns. The identification of subcenters in this area using the methods suggested by MacDonald and Guiliano and Small (McDonald, 1987;Guiliano and Small, 1991) has already been carried out in [START_REF] Baumont | Centres secondaires et recomposition économique des espaces urbain, le cas de la Communauté de l'Agglomération Dijonnaise[END_REF]. In this paper, we study the usefulness of an alternative identification methodology, namely exploratory spatial data analysis [START_REF] Anselin | Small sample properties of tests for spatial dependence in regression models[END_REF][START_REF] Anselin | The Moran scatterplot as an ESDA tool to assess local instability in spatial association[END_REF][START_REF] Getis | The analysis of spatial association by use of distance statistics[END_REF]Ord and Getis, 1995) in the spirit of Scott and Lloyd W.J. (1997), which used these methods in the case of Los Angeles. Indeed, these methods allow detecting both spatial autocorrelation, in the form of spatial clusters of population and employment, and spatial heterogeneity. Furthermore, they constitute an improvement over existing methodologies that necessitate toe definition of arbitrary cut-offs. Next, we analyze whether the employment clusters detected in the previous step have a significant influence on the distribution of population using both monocentric and multicentric population density functions. Given the presence of spatial autocorrelation and spatial heterogeneity in the population density distribution, this issue is investigated using the appropriate spatial econometric methods [START_REF] Anselin | Spatial Econometrics: Methods and Models[END_REF][START_REF] Anselin | Spatial econometrics[END_REF]. Indeed, the presence of spatial autocorrelation yields inconsistent and inefficient OLS estimators in population density functions. Only a few studies have used spatial econometric techniques in this framework [START_REF] Griffith | Modelling urban population density in a multi-centered city[END_REF]Anselin and Can, 1986;Stem, 1993;Griffith and Can, 1996;McMillen, 2002).

The paper is organized as follows. In the following section, we describe the data and the spatial weight matrices used in this study. In the third section, we study the spatial pattern of total employment and employment density using exploratory spatial data analysis. In the fourth section, we provide a spatial econometric analysis of monocentric and multicentric population density functions. Different specifications are estimated and compared: exponential negative, spline-exponential and multicentric density functions. The paper concludes with a summary of key findings.

. D a t a a n d s p a t ia l w e ig h t m a t r ix

Our study focuses on the COMADI (Communauté de l'Agglomération Dijonnaise), which is an urban area in the French region of Burgundy currently composed of 16 adjacent towns: the central city of Dijon (the capital of Burgundy) and 15 suburban towns around Dijon: Ahuy, Chenôve, Chevigny-Saint-Sauveur, Daix, Fontaine-lès-Dijon, Longvic, Marsannay-la-Côte, Neuilly-lès-Dijon, Ouges, Perrigny-lès-Dijon, Plombières-lès-Dijon, Quétigny, Saint-Apollinaire, Sennecey-lès-Dijon and Talant. As this urban area has its own administrative entity (< Communauté d 'Agglomération) it may be described as a conurbation.

In terms of the zoning defined by the INSEE (i.e. the National Institute fo r Statistics and Economic Surveys), the COMADI is more than the Dijon urban pole1 structuring the Dijon urban area (composed of 214 towns in 1999). Furthermore, we consider 22 additional suburban towns immediately surrounding the COMADI area. These towns, that we will label "urban fringe", are introduced in order to reduce edge effects present in spatial data analysis.

The spatial configuration of the COMADI area and its urban fringe is displayed in Maps 1 and 22 and the main geographic and demographic characteristics of this area are shown in Table 1. From Table 1, it can be seen that the COMADI is a small area compared with the urban areas usually analyzed in urban studies (North American cities, major cities in Europe, Asia or in developing countries).

[Maps 1 and 2 about here]

[Table 1 about here]

We use data on employment and total population without double counting in the 1999 census. They are drawn from the RGP census and the SIRENE3 data provided by Direction Régionale Bourgogne de l 'INSEE for all communes in the region of Burgundy. These data are collected at the communal level and at a finer scale (IRIS-2000®) for places of more than 5000 inhabitants (cf. Appendix 1 for a description of IRIS-2000® level). Of the 16 towns that make up the COMADI, IRIS data are collected for nine of them (cf. Table 2) and none of the which different demographic, economic and geographic data are available4.

[Table 2 about here] INSEE's procedures for breaking down employment or population data in the IRIS system are relatively recent and they require relevant information to be available on the localization of individuals and firms. Sometimes these localizations cannot be identified or assigned to an IRIS zone. There is therefore some discrepancy between the data provided for the whole town and the data for the town computed from the IRIS data. Such discrepancies are relatively minor, however, both for population and for employment data in our sample.

The employment data from the SIRENE files correspond to salaried employment.

They relate primarily to private-sector employment and are far from complete when it comes to public-sector employment and agricultural employment as well as employment in some major financial organizations. There are a number of reasons for this: agricultural employment figures come from the farming census, public-sector employment and employment in large financial organizations cannot be broken down in the IRIS system because it is assigned for the regional head office and not for the actual place of business.

We now turn to the choice of the spatial weight matrix, which is the fundamental tool used to model the spatial interdependence between spatial units. Each unit is connected to a set of neighboring units by means of a spatial pattern introduced exogenously in this spatial weight matrix W. The elements wu on the diagonal are set to zero whereas the elements wtj indicate the way the unit i is spatially connected to the unit j . These elements are non stochastic, non-negative and finite. In order to normalize the outside influence upon each unit, the weight matrix is standardized such that the elements of a row sum up to one. Various matrices can be considered: a simple binary contiguity matrix, a binary spatial weight matrix with a distance-based critical cut-off, above which spatial interactions are assumed negligible, more sophisticated generalized distance-based spatial weight matrices with or without a critical cut-off.

Given the specific geographical configuration of the spatial units, we choose not to consider inverse distance matrices. Indeed, the analysis of the distance distribution between all pair of units reveals that the minimal allowable cut-off5 is above the first quartile. This particular feature is a consequence of the important size heterogeneity of the spatial units since two spatial scales are used in this study: the IRIS and the communal level. Furthermore, residential IRIS are generally much smaller than business IRIS. Therefore, the IRIS situated in the centrally urbanized areas of some subdivided communes are very small (for example in Dijon, Chenove or Saint-Apollinaire) whereas the communes that are not subdivided are much larger spatial units and are generally located in the periphery of the Dijon. Finally, this minimal allowable cut-off is very large (almost 5 km) compared to the size of the urban area (18 km by 16 km). Therefore, if we consider such a distance cut-off, central IRIS will be connected to almost a quarter of the whole urban area, which is probably too much.

For these reasons, distance-based matrices with fixed cut-offs are not very relevant since small units in the central areas would have a lot more neighbors than the large units in the periphery. Instead, we consider two other types of weight matrices. First, we use a simple binary contiguity matrix, where an element wy is one if the units i and j share a common border, and 0 otherwise. Second, we use a ^-nearest neighbors matrix computed from the distances between the units' centroids. Considering a nearest neighbors weight matrix implies that each spatial unit is connected to the same number of neighbors, wherever it is localized, and implies a variable distance cut-off for each IRIS. Therefore, we think that such a weight matrix matches the irregular spatial configuration of our urban area. The general form of the ^-nearest neighbors weight matrix W(k) is defined as following:

w (k) = 0 if i= j , Vft v <V (*)=1 if d9<dt(k) and %(£) = h £ (£ )/£ w '(£) (1) w* (k) = 0 if dtj > di (k)
where w >* (&) is an element of the unstandardized weight matrix; w0(k) is an element of the standardized weight matrix; dt{k) is a critical cut-off distance defined for each unit i. More precisely, d,(k) is the k'1' order smallest distance between unit i and all the other units such that each unit i has exactly k neighbors. Since the average number of neighbors is 5.76, we use one nearest neighbor matrix with k = 66.

The identification of subcenters is often carried out using [START_REF] Giuliano | Subcenters in the Los Angeles region[END_REF] methodology (see, for example, [START_REF] Giuliano | Subcenters in the Los Angeles region[END_REF]Small and Song, 1994;Song, 1994;Sivitanidou, 1996;McMillen and McDonald, 1998a, b). These authors define a center as a zone or a set of contiguous zones for which a measure of employment concentration is higher than for all adjacent zones and for which the employment and employment density are above some predetermined cut-offs. The critical values chosen for these levels depend on the metropolitan area and may even vary over the metropolitan area if one observes strong variations in the employment or density employment distributions. Therefore, this identification method depends heavily on the choice of arbitrary cut-offs and may generate conflicting results. In the case of Los Angeles, for example, the various authors recognize from 6 to 54 different centers [START_REF] Giuliano | Subcenters in the Los Angeles region[END_REF] with only a small portion of Los Angeles' total employment actually contained within the identified centers.

Different methods have been suggested to overcome these problems and to avoid the determination of arbitrary cut-offs. For example, [START_REF] Craig | Using quantile smoothing splines to identify employment subcenters in a multicentric urban area[END_REF] and McMillen (2001) use nonparametric techniques. In this paper, we suggest to define a potential employment subcenter as an area having significantly higher employment and employment density than neighboring sites. To identify such areas, we use exploratory spatial data analysis (ESDA).

ESDA is a set of techniques aimed at describing and visualizing spatial distributions, at identifying atypical localizations or spatial outliers, at detecting patterns of spatial association, clusters or hot spots, and at suggesting spatial regimes or other forms of spatial heterogeneity [START_REF] Haining | Spatial Data Analysis in the Social and Environmental Sciences[END_REF][START_REF] Bailey | Interactive Spatial Data Analysis[END_REF]Anselin, 1998a, b). These methods provide measures of global and local spatial autocorrelation.

We apply ESDA both on total employment (Emp99) and employment density (Demp99) distributions in 1999 for the sample "COMADI + Urban fringe" of 136 observations at IRIS and communal levels. All computations are carried out for two spatial weight matrices: contiguity and 6 nearest neighbors weight matrices.

Global spatial autocorrelation

First, we consider global spatial autocorrelation, which is usually based on Moran's / statistic.

This statistic is written as follows for a row-standardized weight matrix:

z'z where z is the vector of the n observations in deviation from the mean; n = 136; W is the spatial weight matrix. Values of I larger (resp. smaller) than the expected value

E(I) = -l/( « -l) indicate positive (resp. negative) spatial autocorrelation.
Inference is based on the permutation approach with 9999 permutations. It appears that total employment and employment density are positively spatially autocorrelated since Moran's / statistics are positive and significant with at least 5% significance level for both variables and both weight matrices (Table 3)7.

[Table 3 about here]

Local spatial autocorrelation and Moran scatterplots

For positive global spatial autocorrelation, Moran's / global statistic doesn't allow discriminating between a spatial clustering of high values and a spatial clustering of low values. Moreover, given the definition of a potential employment subcenter used in this paper, we need to compare each zone's total employment or employment density to that of its neighbors. In that purpose, we use Moran scatterplots [START_REF] Anselin | The Moran scatterplot as an ESDA tool to assess local instability in spatial association[END_REF] (8.1% in quadrant HL and 24.2% in quadrant LH). Moran's scatterplot also allows visualizing the presence of outliers, i.e. the observations further than two units away from the origin.

There are 7 such outliers with total employment more than two standard deviations above the mean: 2 in central COMADI (D4 and D65), 2 in South of COMADI (Co8 and Lo4), 2 in the North (D62 and D63) and one eastern IRIS (Qu5). For employment density, positive association associations are even stronger (66.2% in quadrant LL and 14.6% in quadrant HH) than those of negative spatial associations (11% in quadrant LH and 8.8% in quadrant HL).

Moreover, 4 outliers with employment density more than two standard deviations above the mean are detected, all being located in central COMADI.

[ [START_REF] Anselin | Small sample properties of tests for spatial dependence in regression models[END_REF]. The local version of Moran's / statistic for each observation i is written as:

I , = -■ -X wv (xi ~ **) with mo = X ( ^) 2/« (3) mo j i
where x, is the observation in unit i; » = 136; ¡1 is the mean of the observations across spatial units and where the summation over j is such that only neighboring values of j are included. A positive value for 7; indicates spatial clustering of similar values (high or low) whereas a negative value indicates spatial clustering of dissimilar values between a zone and its neighbors.

Due to the presence of global spatial autocorrelation, inference must be based on the conditional permutation approach with 9999 permutations. The pvalues obtained for the local Moran's statistics are then pseudo-significance levels. Note that inference is this case is further complicated by the problem of multiple comparisons since the neighborhood sets of two spatial units contain common elements [START_REF] Anselin | Small sample properties of tests for spatial dependence in regression models[END_REF]Ord and Getis, 1995;[START_REF] Gallo | Exploratory Spatial Data Analysis of the distribution of regional per capita GDP in Europe[END_REF]. Therefore, the overall significance of 5% is not jestricted enough and we also consider significance levels at 1% and 0.1%.

Detailed results for total employment and employment density and both weight matrices are displayed in Table 4. Maps 3 and 4 display the Moran significance maps for total employmeit and employment density. These maps combine the information in a Moran scatterplot and the significance of LISA by showing the IRIS with significant LISA and indicating by a color code the quadrants in the Moran scatterplot to which these IRIS belong.

For total employment, at the 5% pseudo-significance level, 7 IRIS are significantly HH, 3 being located in the center of the COMADI and the other 3 in the south. One eastern IRIS, that is significantly HL, can be interpreted as an isolated employment pole, tbwever, only 3 It is worthwhile to compare these results to those obtained by [START_REF] Baumont | Centres secondaires et recomposition économique des espaces urbain, le cas de la Communauté de l'Agglomération Dijonnaise[END_REF] where standard subcenter identification methods have been used. Following [START_REF] Giuliano | Subcenters in the Los Angeles region[END_REF], Baumont and Bourdon define a center as a zone or a set of contiguous zones where total employment is greater than a given level E and greater than total employment in surrounding zones and where employment density is greater than a given level D and greater then the density in surrounding zones. For the urban area of the COMADI composed of 114 contiguous zones, the authors consider the employment level that allows taking into account a sufficient number of IRIS to include more than 50% of the total employment of the COMADI. This level is fixed to E -I 400. The density employment level is D = 10 jobs per acre. Eleven IRIS containing more than 1400 jobs are identified but among them, only 5 IRIS have the sufficient level for employment density. Moreover, six other IRIS characterized by an employment density level greater than 10 jobs per acre, have more than 1 000 jobs but less than 1 400 jobs. [START_REF] Baumont | Centres secondaires et recomposition économique des espaces urbain, le cas de la Communauté de l'Agglomération Dijonnaise[END_REF] named "Employment poles" the zone or the set of contiguous zones that have more than 1 400 jobs and named "Potential center" a zone that have more than 1000 jobs whatever the level of employment density. According to these definitions, an IRIS belonging to an employment pole is a potential center. The different characteristics of these IRIS are displayed in Table 5.

IRIS
[Table 5 about here]

Five employment poles have been identified. Some are composed of central IRIS, like the CBD, or of IRIS from several contiguous different towns, like the "Multi-pole" South and North. Others are single IRIS like the "Isolated Poles" Quetigny and Chevigny. Seven other IRIS are defined as potential centers and they are either centrally located or located in peripheral areas (Map 5). We note that many peripheral IRIS have many jobs although they have low employment densities. On the contrary, central IRIS have high employment densities. This is a traditional feature due to the heterogeneity of the spatial scale that we have mentioned in the preceding section. Therefore, [START_REF] Baumont | Centres secondaires et recomposition économique des espaces urbain, le cas de la Communauté de l'Agglomération Dijonnaise[END_REF] prefer not considering the employment density as a relevant indicator to define economic center.

According to these different results, these authors conclude that the COMADI exhibits a multicentric economic pattern for employment: the traditional CBD and four economic subcenters (South, North, Quetigny and Chevigny). This result contrasts with that found with ESDA.

[Map 5 about here]

Let us now analyze the spatial association characteristics of these economic centers with the exploratory spatial analysis that we have carried out above. Although a different sample has been used, we consider that we can compare the two studies for the following reasons. First, our sample is composed of the IRIS belonging to the COMADI and of its urban fringe. If we analyze the employment composition of the towns belonging to the urban fringe (Table 1), only 3.54% of jobs are added to the analysis. None of these towns have more than 1 000 jobs and more than 10 jobs per acre. Second, the main statistical characteristics (mean and quartiles) of the total employment and employment density distributions are quite similar in the two samples. Local spatial association indicators associated to the employment poles and potential economic centers are displayed in Table 5. We can easily note that the IRIS belonging to the CBD have significant Lisa statistics. On the contrary, the peripheral employment poles are not significant according to the ESDA except the isolated pole "Chevigny", which has to be considered with caution. Therefore, the multicentric pattern of employment highlighted by standard methods of employment subcenter identification is not fully confirmed by ESDA.

The present study and the results obtained in [START_REF] Baumont | Centres secondaires et recomposition économique des espaces urbain, le cas de la Communauté de l'Agglomération Dijonnaise[END_REF] highlight some contradictory results concerning the employment pattern of the COMADI. Furthermore, two arguments make it difficult to determine the correct analysis. On the one hand, in [START_REF] Baumont | Centres secondaires et recomposition économique des espaces urbain, le cas de la Communauté de l'Agglomération Dijonnaise[END_REF], the multicentric pattern is quite robust to a large variation of the employment level cut-off: at least one central IRIS and one peripheral IRIS belonging to each "Employment Pole" South and North are identified when the critical employment level varies from 1 000 jobs to 4 000 jobs. However, no statistical test assessing the significance of this result has been done. Moreover, only the restrictive definition of contiguity given by [START_REF] Giuliano | Subcenters in the Los Angeles region[END_REF] is adopted, like all the papers using this definition. On the other hand, by using ESDA, the present study allows taking into account less restrictive definitions of neighbors. Furthermore, the computation of LISA statistics uses the average total employment. In the sample, this average is 539 jobs, which is largely below the cut-off of Therefore, our study is a further contribution to the ongoing debate about exogenous identification methods. However, if we consider that an economic center influences residential location choices, we must also estimate the effects of all subcenters on population density. In that purpose, different population density functions, with different functional forms and including one or more potential economic centers, are considered in the following section.

. S p a t ia l e c o n o m e t r ic a n a l y s is o f p o p u l a t io n d e n s it y f u n c t io n s

The analysis of urban structures is usually conducted using population residential density functions including the distance from the CBD as an explanatory factor. The negative exponential density function defined by [START_REF] Clark | Urban population densities[END_REF] is the most used theoretical specification. It has been largely improved in order to better capture the irregularities of the population density distribution in real urban areas10. For example, [START_REF] Anderson | Cubic spline urban density functions[END_REF][START_REF] Anderson | The changing structure of a city: temporal changes in cubic spline urban density patterns[END_REF] suggests the use of cubic spline specifications when population densities do not homogeneously decrease as the distance from the CBD increases. [START_REF] Brueckner | A switching regression analysis of urban population densities[END_REF] estimates distance-oriented density functions, with an unknown number of possible regimes, using switching regressions. [START_REF] Alperovich | An application of a switching regimes regression to the study of urban structure[END_REF] find evidence of two distinct regimes in the urban area of Tel-Aviv. In fact, all these studies take into account in different ways spatial heterogeneity: the estimated coefficients are different depending on their distance from the CBD or on the spatial regime they belong. Moreover, other economic centers than the CBD may influence the spatial population distribution on the urban area. In this case, the distance from several potential economic subcenters are added as explanatory variables. If more than one of the associated coefficients is statistically significant, then the urban pattern is considered as multicentric and it is considered as monocentric otherwise.

In this section, we estimate population residential density functions for the COMADI using different specifications including the distance from the CBD and from the potential economic subcenters detected in the previous section and by [START_REF] Baumont | Centres secondaires et recomposition économique des espaces urbain, le cas de la Communauté de l'Agglomération Dijonnaise[END_REF].

Furthermore, as for total employment and employment density, we first carry out an ESDA on population density in order to identify possible patterns of spatial heterogeneity and/or spatial autocorrelation. The latter effect should be systematically detected. Indeed, if it is ignored, the OLS estimators are inefficient and statistical inference is biased. Wrong conclusions can therefore be drawn out of the results. Except in some isolated studies [START_REF] Griffith | Modelling urban population density in a multi-centered city[END_REF][START_REF] Can | Model comparison and model validation issues in empirical work on urban density functions[END_REF]Stem, 1993;Griffith and Can, 1996;McMillen, 2002), spatial autocorrelation is not taken into account in the estimation of population density functions.

ESDA on population density

The population density is named Dpop99 and is defined as the population per acre. It is measured for each observation of the sample of 136 observations of the COMADI and its urban fringe. We use the same spatial weight matrices as in the ESDA on employment: the contiguity weight matrix and the 6 nearest neighbors weight matrix.

Concerning the detection of global spatial autocorrelation, the value of the Moran's I statistic is positive and significant with p = 0.001 (Table 6). This result suggests that the IRIS with relatively high values (resp. low) of population density are surrounded by IRIS with relatively high values (resp. low) of population density.

[Table 6 about here]

If we look at the spatial distribution of LISA statistics (Table 7 andMap With the 6 nearest neighbors matrix, an additional cluster of high population density is found in "Le Belvédère", a neighborhood in Talant (see appendix 2). In conclusion, this spatial autocorrelation pattern is not in contradiction with the standard theoretical distribution of residential density associated with the monocentric assumption, except for the three local peaks located at the western part of the city of Dijon that may reflect a form a spatial heterogeneity. We investigate these issues further with population density functions.

[Map 6 about here]

[Table 7 about here] suggested by Alperovitch (1995). This specification is an extension of the negative exponential function and is adapted when the population density does not decrease monotonously with distance from the CBD. One or more "knots" are specified defining distance intervals. The function is then exponential between knots and the gradient of the function is allowed to vary along different distance intervals. Note that cubic-spline functions are often used in empirical analyses. However, we prefer the use of a spline-exponential since it avoids the use of high-order distance terms and therefore limits the amount of multicollinearity (Alperovitch, 1995). We define one knot located at 4 km of the CBD because it is approximately the distance at which the three local population density peaks are situated. The spline-exponential function can then be written as:

In £)(«) = In Z)(0)-7iw +/]«" +£ (5)
where D(u), D{0), y and u are defined as before; yx is the parameter describing the change of the gradient of the function occurring within the distance interval defined by the 4 km knot.

Finally, ua is defined by:

[0 if u < a u" = < with a = 4 km (6) [m -a if u > a
The results of the estimation by OLS of this model are given in the third column of Table 8.

While the gradient is still significant and negative, the coefficient yx is not significant.

Therefore it seems that the gradient does not change after 4 km of the CBD. The model fit is approximately the same as in the simple exponential density function and the splineexponential does not perform better in terms of information criteria. However, this model is strongly misspecified since the spatial autocorrelation tests indicate the omission of a spatial error and the inference based on it is not reliable.

Both models must therefore be modified to integrate spatial autocorrelation explicitly in the form of a spatial error model in order to achieve reliable inference. In equation ( 4) and

(5), the following error structure is added:

e = lW e + u u~ N(0,<y2 uI) (7)
where W is the spatial weight matrix and X is the coefficient indicating the extent of spatial autocorrelation between the error terms.

The estimation results of the exponential negative density function by maximum likelihood (ML) are displayed in the second column of Table 8. The coefficients are all strongly significant with an estimated gradient slightly lower than in the model estimated by OLS. It is as well important to note that a significant positive spatial autocorrelation of the errors is found (A = 0.439 ). Furthermore, the LMLAG* test does not reject the null hypothesis of the absence of an additional autoregressive lag variable in the spatial error model.

According to information criteria this model performs better than the preceding one [START_REF] Akaike | A new look at the statistical model identification[END_REF]Schwarz, 1978).

The estimation results of the spline-exponential density function by ML are presented in the forth column of table 8. Compared to the model estimated by OLS, i appears that the estimated gradient is higher and that y, is positive and now significant at 5%, highlighting a change of the gradient after 4 km. Others things being equal, the value of the constant term and of the absolute value of the density gradient are lower for units located farther than 4 km from the CBD ( D(0) = 3.146 and y= -0.388) than for areas located closer than 4 km from the CBD (I>(0) = 5.062 and y --0.867). A significant positive spatial autocorrelation of the errors is also found (A = 0.509 ) while the LMLAG* test does not reject the null hypothesis of the absence of an additional autoregressive lag variable in the spatial error model. According to information criteria, this model performs better than all the preceding ones.

[Table 8 about here]

All these results indicate that the monocentric model explains well the spatial pattern of population density, provided that some heterogeneity in the distance decay is allowed and that spatial autocorrelation is tested. Indeed, if spatial autocorrelation had not been tested for, we would have concluded that the spline-exponential density function was not appropriate.

However, in the preceding section, a potential set of subcenters has been detected. Therefore, it is interesting to study how multicentric density functions perform compared to the monocentric model.

Estimation results fo r multicentric population density functions

We consider the following model:

m \aD(u) = \nD{Qi)-yu+^[j yiui x +e ( 8
)
where D(u) , D(0), y are defined as before; ut is the distance from subcenter i; y; are the parameters associated to the subcenters to be estimated; m is the number of subcenters. All distances from subcenters are expressed in inverse form because this specification allows the effect of distance from the subcenter i to decline rapidly with distance. Furthermore, it limits the amount of multicollinearity (McMillen and McDonald, 1998a, b). Indeed, the maximum condition number found in the following regressions is 7.388. In this specification, a positive coefficient yt indicates that population density falls with distance from subcenter i.

We consider three sets of regressions. The first one contains the distance from the 2 IRIS that were significant in ESDA on total employment: Ch5 ( y ) and Ma5 ( y2). The second one considers the subcenters that were detected in [START_REF] Baumont | Centres secondaires et recomposition économique des espaces urbain, le cas de la Communauté de l'Agglomération Dijonnaise[END_REF]: Ma512 (y2), Qu5 (y3) and D63° ( y4). The last one contains all four potential subcenters. The estimation results of these specifications by OLS are displayed in Table 9. It appears that the CBD gradient is always strongly significant and negative, highlighting the importance of the CBD, even when other subcenters are included. However, the distances from the various subcenters included are either not significant (Ch5, Qu5), or are significant but do not have the expected sign (Ma5, D63). This latter result is in contradiction with the standard conclusions of residential choice models since it indicates that population densities increase with the distance from an employment pole. Therefore, such results are not further analyzed here even if they question the nature of the attractiveness of these employment poles for residential choice. The model fits are comparable to monocentric models estimated by OLS (AE2-adjusted = 61% ) and these specifications do not perform better in terms of information criteria. However, all three specifications are misspecified due to the omission of spatial autocorrelation, as indicated by the five spatial autocorrelation tests.

[Table 9 about here]

The inclusion of distances to various subcenters did not remove spatial autocorrelation. It is not clear, however, which spatial specification is the most appropriate.

Indeed, the degrees of significance of LMERR and LMLAG are very similar in all three cases, while the robust tests are never significant. Therefore, in each case, we estimate both a spatial error and a spatial lag model.

The latter is defined as following:

m In D(u) = plV*ln D(u) + In D(0) -yu+ % yi +e (8) i= l
where IV is the spatial weight matrix; fV*ln D(u) is the endogenous spatially lagged variable and p is the spatial autoregressive coefficient indicating the amount of correlation between population densities of neighboring observations. The estimation results of these specifications by ML are presented in Table 10. Whatever the specification estimated, the CBD gradient is always strongly significant and negative. However, as in the preceding specifications estimated by OLS, the distances from the various subcenters are not significant or do not have the expected sign. In each case, the spatial coefficient (spatial error or spatial autoregressive) is positive and strongly significant. However, none of these models perform better in terms of information criteria than the monocentric models with spatial error autocorrelation estimated by ML.

Finally, these results show few evidence of a multicentric urban pattern in the COMADI and are in conformity with those found in the ESDA on total employment and employment density. Rather, the monocentric model allowing for some spatial heterogeneity is the most appropriate specification in the form of the spline-exponential model indicating that the density gradient is different before and after a distance of 4 km from the CBD. This distance corresponds more or less to the boundaries of Dijon where several high density housing projects were realized in the seventies. Until this distance, population densities are measured at the IRIS scale. On the contrary, beyond this distance are located peripheral residential towns often characterized by low population density levels and measured at the communal level. The spline-exponential model is then an appropriate model to capture the spatial heterogeneity in middle size agglomerations where the downtown still dominates.

[Table 10 about here]

. C o n c l u s io n

In this paper, we have analyzed the intra-urban spatial distributions of population and employment in the agglomeration of Dijon (regional capital of Burgundy, France). Our aim was to study whether this agglomeration has followed the general tendency of job decentralization observed in most urban areas or whether it is still characterized by a monocentric pattern. In that purpose, a sample of 136 observations at the communal and at the IRIS (infra-urban statistical area defined by INSEE) levels with 1999 census data and the employment database S IRENE (INSEE) was used.

First, the spatial pattern of total employment and employment density using Exploratory Spatial Data Analysis (ESDA) has been studied. Contrary to standard methods of employment subcenter identification, ESDA does not require the determination a priori of arbitrary cut-offs and it allows assessing the significance of clusters of high employment. The application of these procedures to total employment and employment density shows that, apart from the CBD, few IRIS are found to be statistically significant. These results contrast with those found using standard methods, where potential employment subcenters were detected in the North, South and East of the COMADI.

Second, in order to examine the spatial distribution of residential population density, different specifications have been estimated and compared. On the one hand, an exponential negative and a spline-exponential density function have been considered. The latter has been estimated due to the presence of local clusters of high population density located in the western part of the COMADI. Both functions have been found to perform quite well. On the other hand, multicentric density functions including various subcenters yield rather unsatisfactory results with distances from the subcenters that are not significant or that do not have the expected sign. In each case, spatial autocorrelation is controlled for by using spatial econometric techniques.

Finally, all the results highlight the monocentric character of the agglomeration of Dijon. Although some job decentralization, following urban policies, has taken place in the last years, there are no clusters of employment having a significant impact on the distribution of population density. These findings could be extended by considering the distribution of land and housing values and/or by studying a larger area surrounding the COMADI.

Tables, maps and figures

Ch5

Zone -27- -28- -30- -31- [START_REF] Cliff | Spatial Processes: Models and Applications[END_REF]. LMERR is the Lagrange multiplier test for residual spatial autocorrelation and R-LMERR is its robust version. LMLAG is the Lagrange multiplier test for spatially lagged endogenous variable and R-LMLAG is its robust version [START_REF] Anselin | Small sample properties of tests for spatial dependence in regression models[END_REF]Anselin et al.9 1996). LMLAG is the Lagrange multiplier test for an additional spatially lagged endogenous variable in the spatial error model [START_REF] Anselin | Spatial Econometrics: Methods and Models[END_REF]. SARMA is the joint test of residual spatial autocorrelation and spatially lagged endogenous variable. 1978). MORAN is the Moran's / test adapted to OLS residuals [START_REF] Cliff | Spatial Processes: Models and Applications[END_REF]. LMERR is the Lagrange multiplier test for residual spatial autocorrelation and R-LMERR is its robust version. LMLAG is the Lagrange multiplier test for spatially lagged endogenous variable and R-LMLAG is its robust version [START_REF] Anselin | Small sample properties of tests for spatial dependence in regression models[END_REF]Anselin et al.9 1996). SARMA is the joint test of residual spatial autocorrelation and spatially lagged endogenous variable.

OLS-white OLS-white OLS-wfaite

ML-err ML-lag ML-err ML4ag ML-err ML-lag

In 0(0) 1978). LMERR* is the Lagrange multiplier test for an additional spatial error in the spatial lag model [START_REF] Anselin | Spatial Econometrics: Methods and Models[END_REF]. LMLACr is the Lagrange multiplier test for an additional spatially lagged endogenous variable in the spatial error model [START_REF] Anselin | Spatial Econometrics: Methods and Models[END_REF].

Map 1: The COMADI and its urban fringe Map 2: The 114 IRIS of the COMADI

1

  The towns of the COMADI correspond to the towns within the urban pole of Dijon plus the town of Ahuy. 2 Maps are created using Arc-View©3.2 software on the basis of maps provided by Direction Régionale Bourgogne de I 'INSEE. 3 SIRENE: Système Informatique pour le Répertoire des Entreprises et des Etablissements.

  Figures 1 and 2 display the Moran scatterplots for total employment and employment density using the contiguity weight matrix9. For total employment, it appears that most of the observations are characterized by spatial positive association (51.5% in quadrant LL and 16.2% in quadrant HH) while the other IRIS are characterized by negative spatial association

1 000 .

 000 Therefore, what we identify as HH type IRIS are in fact IRIS having a total employment above 539 jobs and that are surrounded by other IRIS having in average total employment above 539 jobs. 22 IRIS match this definition. Following the same line of reasoning, 11 IRIS are HL type, i.e. they are surrounded by other IRIS having in average total employment below 539 jobs. However, only a few of these spatial associations are significant, with all weight matrices used.

  6), two additional features appear. First, more than 82% of the spatial observations exhibit a positive spatial autocorrelation pattern (HH and LL types). In the case of the contiguity matrix, 51 IRIS are of type HH and 61 IRIS are of type LL. Only 8 IRIS are of type HL and 16 IRIS of type LH. Second, using a pseudo-significance level of 5%, we detect clusters of high values in the central part of the COMADI and cluster of low values in the peripheral towns of the agglomeration. Note also that two clusters of high population density \alues are detected in the district of "Fontaine d'Ouche" (a neighborhood in Dijon) and in the town of "Chenôve".

  pseudo-significance level; **1% pseudo-significance level; ***0.1% pseudo-significance level; inference based on 9999 permutations.

  5% pseudo-significance level; **1% pseudo-significance level; ***0.1%pseudo-significance level; inference based on 9999 permutations.

  are significant at the 1% pseudo-significance level: central IRIS Monge (Dl) in HH, southern IRIS ZAC de Marsannay (Ma3) in HH and Zone Economique (Ch5) in HL. These results may indicate the existence of potential subcenters in the COMADI. However, a quite

	considered, only the central IRIS are found to be statistically significant. Finally, it appears
	that ESDA mainly detects the central business district of the COMADI and highlights the
	monocentric character of the agglomeration of Dijon.

different picture is obtained when considering employment density. Indeed, in this case, only 11 central IRIS are significantly HH, possibly pointing out the monocentric character of the agglomeration This overall picture is quite similar for 6 nearest neighbors matrix (see appendix 2).

[Maps 3 and 4 about here]

[Table

4

about here] Concerning the identification of employment subcenters in the COMADI, these results can be interpreted as following. First, both total employment and employment density distributions are characterized by significant local positive spatial autocorrelation. The local clusters of high employment are primarily located in the inner center of the agglomeration. If total employment distribution is considered, it appears that several southern IRIS located in the South of the agglomeration can be considered as employment subcenters (i.e. the economic district of Marsannay-la-Cote and its neighborhood). Moreover, one IRIS located in the East of the agglomeration has significantly more employments than its neighbors and may be considered as an isolated economic center. However, this latter result should be considered with caution since this IRIS is surrounded by a lot of open areas. When employment density is

Table 1 : Main geographic and demographic characteristics of study area

 1

	Characteristics COMADI	COMADI +	Urban fringe/	Côte d'Oi***	Burgundy
			Urban Fringe	COMÀDI	(% COMADI/	Region.
				(%)	Côte d'Or)
	Total area	172.4 km2	412.1 km2		8 763.2 km2	31 581.96 W
		(42 600 acres)	(101 833		(2%)	(0.5%)
			acres)
	Width	16km
	Length	18 km
	Population	238 309	257 844	8.08%	504 950	1 608 262
	1999*				(47%)	(14.8%)
	Working pop.	111 028	120 587	8.6%	232 152	708 174
	1999*				(47.8%)	(15.7%)
	Employment	70 770	73 277	3.54%	125 781	387 183
					(56.3%)	(18.3%)

Sources: * Recensement Général de la Population 1999 (RGP Census) ** SIRENE (INSEE) *** Côte d'Or is the French department in Burgundy where the COMADI is located

Table 2 : Towns with and without IRIS data Towns with IRIS data Number Of m i s Towns with no IRIS data

 2

	Chenôve	9	Ahuy
	Chevigny	5	Daix
	Dijon	66	Neuilly
	Fontaine	5	Ouges
	Longvic	4	Perrigny
	Marsannay	3	Plombières
	Quétigny	5	Sennecey
	St-Apollinaire	4	Urban fringe (22)
	Talant	6
	Total spatial units	107	29

Table 3 : Moran's / statistics for total employment and employment density in 1999

 3

	Variable	I I	Contiguity weight matrix Moran's I St. dev. St. value	6 nearest neighbors weight matrix Moran's / St. dev. St. value
	Emp99'		0.117	0.049	2.255	0.136	0.042	3.365
	Demp99*		0.378	0.045	8.467	0.436	0.039	11.249
	* The expected value for Moran's / statistic is -0.007 for Emp99 and Demp99. All statistics
	are significant at 5% level.

Table 6 : Moran's I statistics for population density in 1999

 6

	I	Contiguity weight matrix	6 nearest neighbors weight matrix
	Variable 1 I Moran's I	St. dev.	St. value	Moran's /	St. dev.	St. value
	Dpop99* 1 0.474	0.0494	9.734	0.367	0.043	8.771
	* The expected value for Moran's / statistic is -0.008 for Dpop99. All statistics are significant
	at 5% level.

Table 4 : LISA statistics for total employment and employment density in 1999 Total employment 1999 Contiguity 6 near, neighbors Employment density 1999 Contiguity 6 near, neighbors D20 Dijon (continuai)

 4

		Petites Roches	LL	LL	LL	LH
	D21	Mansart	LL	LL	LL	LL
	D22	Abattoirs	LH	LH	LL	LL
	D23	Castel	LH	LL	LH*	LL
	D24	Stearlnerie	LL	LL	LL	LL
	D25	Carrousel	LL	LL	LL	LL
	D26	Greuze	LH*	LL	LL	LL
	D27	Arsenal	HH*	HL	LL	LL
	D2Q	Bel Air	LL	LL**	LL	LL
	D29	Larrey	LL	LL	LL	LH
	D30	Bourraches Ouest	LL	LL	LL	LL
	D31	Bourraches Est	LL	LL	LH	LL
	D32	Trois Forgerons	HL	HL	HL	HL
	D33	Les Valendons	LL	LL*	LL	LL
	D34	La Montagne	LL	LL	LL*	LL
	D35	Tire Pesseau	LL	LL	LL	LL
	D36	Le Lac	LL	LL	HL	HL
	D37	E. Belin	LL	LL	LL	LL
	D38	Champ Perdrix	LL	LL	HL	HL
	D39	Chartreuse	HH	HH	HH*	HH**
	D40	Arquebuse	HH	HH**	HH**	HH***
	D41	Tanneries	LH	LH	LH*	LH*
	D42	Providence	LH	LL	LH	LL
	D43	Carrières Basquin	HH	HL	HH**	HH
	D44	F. Pompom	LL	LL	LL	LL
	D45	Hauts Montchapet	LL	LL	LL	LL
	D46	E. Spuller	LL	LL	LH	LH
	D47	La Charmette	LL	LL	LL	LL
	D48	Fauconnet	LL	LL	LL	LL
	D49	Jouvence Ouest	LL	LL	LH	LH
	D50	Jouvence Est	LL	LL	HH	HH
	D51	Balzac	LH	LL	LL	LL
	D52	Stalingrad	LH**	LL	LH	LL
	D53	Casernes	LL	LL	LH	LH
	D54	Sacré Coeur	LH	LL	LH	LH
	D55	York	LH	LL	LH	LL
	D56	Lochères	LH	LL	LH	LL

Total employment 1999 Contiguity 6 near, neighbors Employment density 1999 Contiguity 6 near, neighbors Ahuy

		Ahuy	LH	LH	LL	LL
		Chenôve
	Co1	Piscine-Valendons	LL	LL	LL	LL
	Co2	Petignys -Chaufferies	LL	LL	LL	LL
	Co3	Chapitre-Bibliothèque	LL	LL	LL	LL
	Co4	Saint-Exupery	LL	LL	HL	HL
	Co5	Vieux Bourg-Grands Crus	LH	LL	LL	LH
	Co6	Atliers SNCF	HH	HH	HL	HL
	Co7	Mairie	LH	LH	LH	LH
	Cod	Zone industrielle	HH*	HL	HH	HH
	Co9	STRD	HH	HH	HL	HL
		Chevlgny-Salnt' Sauveur
	Ch1	Breuil	LL	LL	LL*	LL
	Ch2	Corcelles	LL	LL	LL**	LL*
	Ch3	Centre-Ville	LL	LL	LL*	LL
	Ch4	Château	LH	LH	LL	LL

Table 4 (continued): LISA statistics for total employment and employment density in 1999 Total employment 1999 Contiguity 6 near, neighbor* Employment density 1999 Contiguity 6 near, neighbors Sapol Salnt-Apolllnalre

 4

		Nord Vlllage	LH	LH	LL	LL
	Sapo2	Sud Vlllage	LH	LH	LL	LL
	Sapo3	Nord-Est	HH	HH	LL	LL
	Sapo4	Sud-Est	HH	HH	LL	LL
	Senec	Sennecey-lés-DIJon	LL	LL	LL**	LL
		Talant
	Ta1	Vleux Talant-Cllnique	LL	LL*	LL	LL
	Ta2	Maronniers-Neruda	LL	LL	LL	LL
	Ta3	Mall-Canzio-Jouvet	LL	LL	LL	LL
	Ta4	Prévert-plein ciel	LL	LL	LL	LL
	Ta5	Borte Vlan-Triolet	LL	LL	LL	LL
	Ta6	Montoillots-ZA	HL	HL	LL*	LL
		Urban fringa of COMAL 01
	Asn	Asnléres-léa-DIJon	LH	LH	LL	LL
	Belt	Bellefond	LH	LH*	LL	LL
	Bres	Bressoy-sur-Tilla	LL	LL	LL	LL*
	Brat	Bretenlére	LL	LL*	LL**	LL***
	Core	Corcelles-les-Monts	LL	LL	LL***	LL***
	Couch	Couchey	LL	LL	LL**	LL**
	Cout	Couternon	LL	LL	LL**	LL**
	Crim	Crlmolols	LL	LL	LL**	LL**
	Danois	Darois	LL	LL	LL**	LL***
	Fauv	Fauvemey	LL	LL*	LL*	LL**
	Fenay	Fénay	LH*	LH*	LL	LL
	Fixin	Flxln	LL	LL	LL*	LL*
	Flav	Flavlgnerot	LL	LL	LL***	LL***
	Hautv	Hautevllle-Iéa-Dljon	LL	LL	LL**	LL*
	Lant	Lantenay	LL	LL*	LL**	LL***
	Magny	Magny-aur' Tllle	LL	LL	LL*	LL**
	Mess	Messlgny et-Vantoux	LL	LL	LL*	LL***
	Pran	Preñóla	LL	LL	LL**	LL***
	Row	Rouvres-en-Plalne	LL	LL*	LL***	LL***
	Ruff	Ruffey-lés-Echlrey	LH*	LH*	LL	LL
	Varols	Varols-et' Chalgnot	LL	LL	LL*	LL*
	Velars	Voiars-sur-Ouche	LL	LL	LL***	LL***

Total employment 1999 Contiguity 6 near, neighbors Employment density 1999 Contiguity 0 near,: neighbors D57 Dijon (continued)

		Grésilles Centre	LH*	LL	LL	LL
	D58	Castelnau	LL	LL	LH	LL
	D59	Charles de Gaulle	LH	LL	LL	LL
	D60	Concorde	HH	HL	HL	HL
	D61	Clos de Pouilly	LH	LH	HL	HL
	D62	La Toison d'Or	HH	HL	LL	LH
	D63	ZI Nord Est	HH	HL	HL	HL
	D64	La Gare	HH	HH*	HH***	HH***
	D65	Le Bocage	LH	LL	LL	LL
	D66	Combe à la Serpent	LL*	LL**	LL	LL
		Fontalne-lés-DiJon
	Fo1	Vieux Vlllage	LL	LL	LL	LL
	Fo2	Saverney	LL	LL	LL	LL
	Fo3	Saint Martin	LL	LL	LL	LL
	Fo4	Majnoni	LL	LL	LL	LL
	Fo5	Activités économiques	HL	HL	HL	HL
		Longvlc
	Lo1	Bief du Moulin	LH*	LH	LL	LL
	Lo2	Bourg	LH	LH	LL	LL
	Lo3	Parc Poussot	LH	LL	LL	LL
	Lo4	Zone indust. Colomblères	HH	HH	LL	LL
		Marsannay-la-Côte
	Ma1	Le Bourg	LH*	LH	LL	LH
	Ma2	Champagne Haute	LL	LL	LL*	LH
	Ma3	ZAC	HH**	HH**	LL	LH
	Nauil	Neullly-lès-Dijon	LL	LL	LL***	LL**
	Ouges	Ouges	LH	LH	LL*	LL
	Perry	Perrlgny-lès-DIJon	HL	HH**	LL*	LH
	Plomb	Plombières-lè s-Dijon	LL	LL	LL***	LL
		Quétlgny
	Qui	Atrias-Vieux Village	LH	LH	LL	LL
	Qu3	La Fontaine aux Jardins	LH	LH	LL	LL
	Qu3	Les Huches	LL	LH	LL	LL
	Qu4	Place Centrale	LH	LH	LL	LL
	Qu5	Zone Activités Cap Vert	HL	HL	HL	HL

Notes: * 5% pseudo-significance level; **1% pseudo-significance level; ***0.1% pseudo-significance level; inference based on 9999 permutations.

Table 5 : Employment poles and potential centers in 1999 detected in Baumont and Bourdon (2002) compared to LISA statistics

 5

		■ ... m nt 1999	Emplfltpfiept density 1999
	Contiguity	" Sneer. neighbor*	contiguity	6 neer. neighbors

Table 7 : LISA statistics for population density in 1999

 7

	Petites Roches	HH	HH
	021 Mansart	HL	HH
	022 Abattoirs	LL	LL
	D23 Caste)	LH	LH
	D24 Stearlnerie	HL	HH
	D25 Carrousel	LH	LH
	026 Greuze	LL	LH
	027 Arsenal	LH	LH
	D28 Bel Air	LH	LH
	D29 Larrey	HL	HH
	D30 Bourroches Ouest	HL	HH
	D31 Bourraches Est	HH	HH
	D32 Trois Forgerons	HH	HH
	D33 Les Valendons	HL	HH**
	D34 La Montagne	LL	LH**
	D35 Tire Pesseau	HH*	X X
	D36 Le Lac	H H*	HH**
	D37 E. Belln	HH*	HH
	D38 Champ Perdrix	HH**	HH**
	D39 Chartreuse	LL	LH
	D40 Arquebuse	HL	HH
	041 Tanneries	HH	HH
	042 Providence

' Continuity --......... D20 Dijon (contlnuod)

LL LH 043 Carrières Basquin HH HH 044 F. Pompom HL HH 045 Hauts Montchapet HH HH D46 E. Spuller HH* HH*

		C ontlw dtv
	Ahuy Ahuy	LL**	LL
	Chenôva
	Co1 Piscine-Valendons	HH	HH*
	Co2 Petlgnys -Chaufferies	HH**	HH*
	Co3 Chapitre-Bibliothèque	HH*	HH*
	Co4 Saint-Exupéry	HH	HH
	Co5 Vieux Bourg-Grands Crus	LH	LH*
	Co6 Atllers SNCF	LH*	LH*
	Co7 Mairie	HH	HH
	Co8 Zone industrielle	LL*	LH
			047 La Charmette	HH	HH
			D48 Fauconnet	HH	HH
			049 Jouvence Ouest	H H*	HH*
			D50 Jouvence Est	HH	HH*
			051 Balzac	HL	HH
			052 Stalingrad	HL	HH
			D53 Casernes	HH	HH*
			054 Sacré Coeur	LH	LH*
			D55 York	HH	HH
			D56 Lochères	HH	HH

Table 7 (continued): LISA statistics for population density in 1999 Contiguity ........... . , i, tneatneiQ hb ors , ' D57 Dijon ( continuaci)

 7

		Grésilles Centre	LL	LH
	D58	Castelnau	HH	HH
	D59	Charles de Gaulle	LL	LL
	D60	Concorde	HL	HL
	D61	Clos de Pouilly	LL	LL
	D62	La Toison d'or	LL*	LL
	D63	ZI Nord Est	LL	LL
	D64	La Gare	LH	LH*
	D65	Le Bocage	LL	LH
	D66	Combe à la Serpent	LH	LH***
		Fontalna-lès-DIJon
	Fo1	Vieux Village	LL	LL
	Fo2	Saverney	LH	LH
	Fo3	Saint Martin	LH	LH
	Fo4	Majnoni	HL	HL
	Fo5	Activités économiques	LL	LL
		Longvlc
	Lo1	Bief du Moulin	HL	HL*
	Lo2	Bourg	LL*	LL
	Lo3	Parc Poussot	LL	LL
	Lo4	Zone indust. Colombières	LL*	LL

Ma1 Marsan nay-la-Cô te Le Bourg LL* LH Ma2

							Salnt-Apolllnalre
						Sapol	Nord Village	LL	LH
						Sapo2	Sud Village	LL	LH
						Sapo3	Nord-Est	LL*	LL
						Sapo4	Sud-Est	LL	LL
						Sanac	Sennecey-lés~DIJon	LL*	LL
							Talant
						Tal	Vieux Talant-Clinique	LL	LH
						Ta2	Maronniers-Neruda	LH	LH**
						Ta3	Mail-Canzlo-Jouvet	HH	HH**
						Ta4	Prévert-plein ciel	HH	HH*
						Ta5	Boris Vian-Triolet	HH	HH**
						Ta6	Montoillots-ZA	LL	LH
							Urban frlnge of COMADI
						Asn	Asnléraa-lés-DIJon	LL**	LL*
						Belf	Bellefond	LL*	LL*
						Bras	Bressay-sur-TII le	LL*	LL*
						Bret	Bretenlére	LL**	LL***
						Core	Corcelles-le&Monts	LL***	LL**
						Couch	Couchey	LL***	LL**
						Cout	Coutemon	LL**	LL*
						Crim	Crlmolols	LL**	LL
						Darois	Darols	LL**	LL***
						Fauv	Fauvemey	LL**	LL**
		Champagne Haute	LL**	LH
	Ma3	ZAC			LL**	LL*
	Neuil	|	i	!	LL**	LL*
	Ouges	Ouges			LL**	LL*
	Perry	Parrlgny-lès-DIJon	LL***	LL**
	Plomb	Plomblères-lèa-DIJon	LL***	LH
		Quétlgny
	Qui	Atrias-Vieux Village	LL	LL
	Qu3	La Fontaine aux Jardins	LL	LL
	Qu3	Les Huches	HL	HL
	Qu4	Place Centrale	HL	HL
	Qu5	Zone Activités Cap Vert	LL	LL

Fenay Fénay LL*** LL* Flxln Flxln LL* LL** Flav Flavlgnerot LL** LL*** Hautv Hautevllle-Iéa-Dljon LL** LL Lant

		Lantenay	LL*	LL***
	Magny	Magny'sur-Tllle	LL*	LL**
	Mess	Messlgny at-Vantou)	LL*	LL***
	Pran	Pranols	LL**	LL***
	Rouv	Rouvres-en-Plalne	LL**	LL***
	Ruff	Ruffey-lés-Echlray	LL***	LL**
	Varois	Vamls-et-Chalgnot	LL**	LL**
	Velars	Velars-sur-Ouche	LL**	LL***

Notes: * 5% pseudo-significance level; **1% pseudo-significance level; ***0.1% pseudo-significance level; inference based on 9999 permutations.

Table 8 : Estimation results for the monocentric density functions

 8 : /^-values are in parentheses. OLS-White indicates the use of the White (1980) heteroskedasticity consistent covariance matrix estimator for statistical inference in the OLS estimation. MLrerror indicates maximum likelihood estimation of the spatial error model. Sq. Corr. is the squared correlation between predicted values and actual values. LIK is the value of the maximum likelihood function. AIC is the[START_REF] Akaike | A new look at the statistical model identification[END_REF] information criterion. BIC is the Schwarz information criterion(1978). MORAN is the Moran's / test adapted to OLS residuals

		Negative exponential	Spline-exponential
		OLS-White	ML-error	OLS-White	ML-error
	In 2>(0)	4.038	4.178	4.368	5.062
		(0.000)	(0.000)	(0.000)	(0.000)
	y \d(CBD)\	-0.535	-0.519	-0.678	-0.867
		(0.000)	(0.000)	(0.000)	(0.000)
	7, ld(>4km)]	-	-	0.208	0.479
				(0.238)	(0.028)
	X	-	0.439	-	0.509
			(0.000)		(0.00Ó)
	R1	0.5997	-	0.6050	-
	R*-adj	0.5967	-	0.5991	-
	Sq.corr	-	0.600	-	0.600
	LIK	-232.159	-225.490	-231.247	-219.656
	AIC	468.318	454.981	468.494	445.313
	BIC	474.143	460.806	477.232	454.051
	a*	1.806	1.552	1.795	1.481
	Condition number	3.081	-	10.388	-
	MORAN	4.372	-	4.702	-
		(0.000)		(0.000)
	LMERR	15.734	-	17.161	-
		(0.000)		(0.000)
	R-LMERR	1.295	-	4.850	-
		(0.255)		(0.028)
	LMLAG	12.945	-	13.110	-
		(0.000)		(0.000)
	R-LMLAG	0.129	-	0.798	-
		(0.720)		(0.372)
	LMLAG*	-	1.636	-	1.877
			(0.201)		(0.349)
	SARMA	15.862	-	17.960	-
		(0.000)		(0.000>

Notes

  Notes: / 7-values are in parentheses. OLS-White ndicates the use of the White (1980) heteroskedasticity consistent covariance matrix estimator for statistical inference in the OLS estimation. LIK is the value of the maximum likelihood function. AIC is the[START_REF] Akaike | A new look at the statistical model identification[END_REF] information criterion. BIC is the Schwarz information criterion (

	lnD (0)	4.282	4.819	4.789
		(0.000)	(0.000)	(0.000)
	y WCBD)]	-0.554	-0.577	-0.584
		(0.000)	(0.000)	(0.000)
	K W C hS)}	0.522	-	0.441
		(0.748)		(0.793)
	r2[d(Mu3)]	-1.245	-1.614	-1.581
		(0.016)	(0.012)	(0.000)
	r3 w o » m	-	0.292	0.225
			(0.745)	(0.810)
	rAW B 4 %	-	-1.403	-L390
	/ ' ' -• ,		(0.031)	(0.033)
	R*	0.615	0.630	0.632
		0.606	0.619	0.617
	LIK	-229.563	-226.818	-226.511
	AIC	467.126	463.637	465.022
	i l i l B B H 1 I	478.777	478.200	482.498
		1.764	1.708	1.713
	Condition number	4.487	6.956	7.388
	MORAN	3.975	4.091	3.857
		(0.000)	(0.000)	(0.000)
	LMERR	11.475	11.069	9.071
		(0.000)	(0.000)	(0 .002)
	R-LMERR	0.346	1.237	0.325
		(0.556)	(0.266)	(0.568)
	LMLAG	11.795	10.018	9.509
		(0.000)	(0.001)	(0.002)
	R-LMLAG	0.666	0.187	0.762
		(0.414)	(0.666)	(0.382)
	SARMA	12.140	11.255	9.834
		(0.002)	(0.002)	(0.007)

  -values are in parentheses. MLrerr indicates maximum likelihood estimation of the spatial error model. MLrlag indicates maximum likelihood estimation of the spatial lag model. Sq. Corr. is the squared correlation between predicted values and actual values. LIK is the value of the maximum likelihood function. AIC is the[START_REF] Akaike | A new look at the statistical model identification[END_REF] information criterion. BIC is the Schwarz information criterion (

		4.518	2.917	5.053	3.448	5.115	3.452
		(0 .000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	r WCBD)]	-0.528	-0.373	-0.565	-0.407	-0.561	-0.413
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	YjdfChS)]	-0.363	0.286	-	-	-0.360	0.246
		(0.557)	(0.595)			(0.555)	(0.648)
	y2 ld(Mß3)]	-1.236	-0.965	-1.418	-1,282	-1.411	-1.269
	' '	(0.067)	(0.103)	(0.003)	(0.032)	(0.035)	(0.035)
	y,ld(Qu5)i	-	-	-0.212	0.186	-0.225	0.150
				(0.709)	(0.663)	(0.697)	(0.727)
	rJd(D63)]	-	-	-1.445	-1.148	-1.457	-1.145
				(0.032)	(0.035)	(0.032)	(0.035)
	' * .	0.446	-	0.425	-	0.451	-
		(0 .000)		(0.000)		(0.000)
	P	_	0.377	-	0.353	-	0.347
			(0.000)		(0.000)		(0.001)
	Sq. corr	0.608	0.653	0.626	0.663	0.621	0.663
	LIK	-223.698	-224.263	-221.497	-222.150	-221.347	-222.046
	AÍC	455.396	458.525	452.995	456.299	454.693	458.093
	BIG	467.047	473.088	467.558	473,775	472.169	478.481
	G 2	1.510	1.541	1.468	1.500	1.457	1.498
	LMERR*	-	2.318	-	2.953	-	2.099
			(0.127)		(0.086)		(0.147)
	LMLAG*	0.619	-	0.696	-	0.295	-
		(0.431)		(0.404)		(0.587)
	Notes:/7

A complete list of the observations and their codes is displayed inTable 4.

i.e. the cut-off for which each unit has at least one neighbor.

Complete results with k = 5 are available upon request from the authors.

See Mills and Tan (1980) and McDonald (1989) for surveys of results and methodology.

All results are presented using the contiguity weight matrix. Complete results with the 6 nearest neighboB matrix are available upon request from the authors.

Its centroid is taken as the centroid of the southern employment pole.

Its centroid is taken as the centroid of the northern employment pole.

Urban Economics, 26, 361-385. McMillen D. (2001) Nonparametric employment subcenter identification, Journal o f Urban Economics, 50, 448-473. McMillen D. (2002) Employment densities in large metropolitan areas: spatial autocorrelation versus functional form misspecification, paper presented at the 49th North American Congress o f the Regional Science Association International, Porto Rico (USA), November 14-16, 2002. McMillen D.P., McDonald J.F. (1998a) Population density in Chicago: a bid rent approach, Urban Studies, 1, 1119-1130. McMillen D.P., McDonald J.F. (1998b) Suburban subcenters and employment density in metropolitan Chicago, Journal o f Urban Economics, 43, 157-180. Mills E.S., Tan J.P. (1980) A comparison of urban population density functions developed and developing countries, Urban Studies, 17, 313-321. Muth R.F. (1969) Cities and Housing: The Spatial Pattern o f Urban Residential Land-Use, The University of Chicago Press, Chicago and London. Ord J.K., Getis A. (1995) Local spatial autocorrelation statistics: distributional issues and an application, Geographical Analysis, 27,286-305. Schwarz G (978) Estimating the dimension of a model, The Annals o f Statistics, 6,461-464. Scott, L., Lloyd W.J. (1997) Spatial Analysis in a GIS environment: employment patterns in greater Los Angeles, 1980 -1990, Proceedings o f the University Consortium fo r Geographic Information Science, Orono, Maine. Sivitanidou R (1996) Do office-commercial firms value access to service employment centers? A hedonic value analysis within polycentric Los Angeles, Journal o f Urban Economics, 40, 125-149. Small K., Song S. (1994) Population and employment densities: structure and change, Journal o f Urban Economics, 36, 292-313. Song S. (1994) Modelling worker residence distribution in the Los Angeles Region, Urban Studies, 31, 1533-1544. Stem D.I. (1993) Historical path-dependence of the urban population density gradient, Annals o f Regional Science, 27, 259-283. Waddell P., Shukla V. (1993) Employment dynamics, spatial restructuring and the business cycle, Geographical Analysis, 25, 35-52. White H. (1980) A heteroskedastic-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica, 48, 817-838. Wu F. (1998) Polycentric urban development and land-use change in a transitional economy: the case of Guangzhou, Environment and Planning A, 30, 1077-1100. McDonald J.F. (1989) Econometric studies of urban population density: a survey, Journal o f

Estimation results fo r monocentric population density functions

Let us take as a starting point the following negative exponential function:

where D(u) is the population density at distance u from the CBD; D(0) is the population density at the CBD; y is the density gradient and measures the proportional rate at which population density falls with distance; £ is the error term with the usual properties. Note that this particular form can be derived from the monocentric model with several restrictive assumptions, i.e. constant returns Cobb-Douglas production function for housing, consumer with identical tastes and incomes and unit price elasticity of demand of housing. The function is then estimated using OLS by taking logs on both sides:

All distances are measures in straight-line km from the centroid of the IRIS Monge (Dl). The results of the estimation by OLS of this model are given in the first column of Table 8. The density gradient is significant at 5% and negative, y = -0.535, which confirms the decay of population density from the center. Note also that the model fit is quite good: i?2-adjusted -60%. Given the presence of spatial autocorrelation detected in the ESDA, we also carry out five spatial autocorrelation tests: Moran's I test adapted to regression residuals [START_REF] Cliff | Spatial Processes: Models and Applications[END_REF] indicates the presence of spatial autocorrelation. To discriminate between the two forms of spatial autocorrelation -spatial autocorrelation of errors or endogenous spatial lag -we perform the Lagrange Multiplier tests: respectively LMERR and LMLAG and their robust versions [START_REF] Anselin | Spatial Econometrics: Methods and Models[END_REF][START_REF] Anselin | Simple diagnostic tests for spatial dependence[END_REF]11. It appears that both LMERR and LMLAG are significant whereas their robust versions are not.

Furthermore LMERR is more significant than LMLAG. These tests therefore seem to indicate the presence of spatial error autocorrelation rather than a spatial lag variable. A direct implication of these results is that the OLS estimator is inefficient and that all the statistical inference based on it is unreliable.

Before turning to the spatial error exponential density function, we investigate the presence of some form of spatial heterogeneity given the presence of local population density peaks detected in the ESDA. In that purpose, we use a spline-exponential function as -residential IRIS: IRIS-2000® with populations of 1800 to 5000 inhabitants. They are homogeneous in respect of types of housing.

-business IRIS: IRIS-2000® clustering more than 1000 employees and with twice as many salaried jobs as resident inhabitants.

-miscellaneous IRIS: IRIS-2000® covering large areas and for special purposes (woods, parkland, docklands, etc.).

In data bases covering several communes, IRIS data correspond either to l'IRIS-2000® for subdivided communes or to the entire commune for small non-subdivided communes (34 800 communes).

B ID H IE